title: Imbalanced Fermi systems and exotic superconducting phases
authors: Madhuparna Karmakar
text: 
  Self-Supervised Human Activity Recognition by Augmenting Generative Adversarial Networks
      Fillia Makedon
      
  ========================================================================================
  
  
  Quantum many body phases, in competition or coexistence has always been a 
  fascinating area of research in condensed matter physics. One of the most 
  widely investigated scenario of phase competition is that between superconductivity 
  and magnetism, which in spite of several decades of research continues to bring up 
  new surprises till date. While superconductivity and magnetism are naively considered 
  to be antagonists, there are several circumstances where these many body phases can be 
  found to be cooperating, coexisting and even promoting each other. The results of such 
  cooperative behavior are found to show up as exotic quantum phases.
  
  In this chapter we discuss about such an exotic superconducting phase, which comes to life 
  when a particular class of superconducting material viz. “Pauli limited superconductors” 
  are subjected to an imbalance between the population of the fermionic species undergoing pairing. 
  The consequence of such an imbalance is the emergence of a non-trivial spatially modulated superconducting 
  phase, popularly known as the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase. Characterized by a 
  finite momentum pairing between the fermionic species the FFLO phase hosts a non zero magnetization, 
  leading to an unconventional coexistent phase of superconductivity and magnetism.
  
  Starting with an introduction to the Pauli limited superconducting systems, in this article we 
  will review the solid state and ultra cold atomic gas setups which hosts the FFLO superconducting 
  phase, along with the relevant experimental diagnostics and their reported observations. This would 
  be followed up by the theoretical attempts made to understand the physics of these novel superconducting 
  systems, and the progress made therein. Finally, a non perturbative numerical technique would be discussed, 
  which is suitable to capture the behavior of such many body systems, and edges over the existing ones in 
  several aspects. Being a vibrant area of strong correlation physics, the Pauli limited superconductors 
  hold several exciting promises for future investigations. We will touch upon them towards the end of 
  the chapter.
  
  
  
  
  § INTRODUCTION
  
  
  Superconductivity is one of the most fascinating phenomena of quantum many body systems. 
  Right from its discovery in mercury (Hg) by Kamerlingh Onnes in 1911 <cit.>, superconductivity 
  has continued to intrigue the condensed matter physics community over the past more than a century. 
  The basic idea that underlies any superconducting material is the formation of a macroscopic 
  condensate with long range order, which manifests itself via properties like perfect diamagnetism, 
  absence of electrical resistivity etc. below a transition temperature (T_c). It was established by Bardeen, 
  Cooper and Schrieffer (BCS) in their pioneering work that the microscopic mechanism that gives rise to 
  the superconducting behavior in a system is the formation of “Cooper pairs”,  wherein, two 
  fermions with equal and opposite momenta (K_↑, - K_↓) pairs 
  up by overcoming the repulsion between them <cit.>. 
  
  The BCS theory was originally devised to capture the physics of “conventional” superconductors, 
  with low transition temperature such as, lead, mercury etc., and it was shown that in these 
  materials the lattice phonons served as the attractive “glue” to pair up the fermions 
  <cit.>. While the BCS theory is still the most celebrated microscopic theory 
  in the history of superconductivity, the discovery of high temperature superconducting cuprates 
  brought forth the limitations of the BCS 
  theory <cit.>. The transition temperatures reported for the cuprates and subsequently for 
  several other categories of superconducting materials were well above what could possibly be  
  achieved by a phonon mediated coupling between the fermions <cit.>. 
  In other words, there was some alternate mechanism which served as the attractive “glue” 
  between the fermions in these “unconventional” superconductors with high transition temperature. 
  
  Inspite of the change in the mechanism of superconductivity, some of the aspects however remained 
  unaltered between the conventional 
  and unconventional superconductors. For example the superconductivity was still dictated by the pairing 
  between the fermions having equal and opposite momenta, i. e. the net momentum of the Cooper pair was zero. 
  Keeping it consistent with the existing literature, we would refer to such superconducting pairing as 
  the “zero-momentum” superconductivity, so as to distinguish them from the “finite-momentum” 
  superconductivity, which would be introduced in the later sections of this chapter. 
  
  It is obvious that once this pairing between the fermions is broken by some perturbation, superconductivity 
  is destroyed. One 
  of the most prominent superconducting pair breaking perturbation is the magnetic field. It is now well 
  established that an orbital magnetic field when applied to a type-II superconductor leads to the formation of 
  superconducting vortices beyond a lower critical field, H≥H_c1^orb <cit.>.
   As the 
  magnetic field is increased the number of vortices penetrating the superconductor progressively increases 
  and gives rise to the Abrikosov vortex lattice <cit.>. As large number of vortices crowd the 
  superconducting system the vortex cores begin to overlap each other at an upper critical field, 
  H_c2^orb=Φ_0/2πξ^2, where 
  Φ_0=πħ c/| e| is the flux quantum, and ξ is the superconducting coherence length. 
  The upper critical field (H_c2^orb) thus sets the strength of the orbital magnetic field required to 
  destroy the superconductivity of a material <cit.>. 
  
  Superconductivity can also be destroyed by magnetic field via an alternate mechanism known as the 
  Pauli paramagnetic pair breaking effect, originating from the Zeeman splitting of the single electron
  energy levels. When a Zeeman field is applied to a normal metal the electrons undergo polarization because 
  the Zeeman effect splits the Fermi surfaces corresponding to the up (↑) and down (↓) 
  fermion species. This is known as Pauli paramagnetism.
  On the other hand, a superconductor comprising of Cooper pairs is not readily polarized and one needs to 
  break the pair in order to polarize the system. This pair breaking takes place when the Pauli paramagnetic 
  energy, E_p=(1/2)χ_nH^2 equals the superconducting condensation energy E_c=(1/2)N(0)Δ^2. 
  Here, χ_n=(1/2)(gμ_B)^2N(0) is the spin susceptibility of the normal state, g is the spectroscopic 
  splitting factor of an electron, μ_B is the Bohr magneton, Δ is the superconducting energy 
  gap and N(0) is the density of states at the Fermi level. The upper critical field at which the Pauli 
  paramagnetic pair breaking effect takes place in a superconductor is estimated to be 
  H_c2^P = √(2)Δ/gμ_B and is known as the Chandrasekhar-Clogston 
  limit <cit.>.      
  
  In general, the upper critical magnetic field of a superconductor is determined based on both the orbital and 
  Pauli paramagnetic pair breaking effects. The relative importance of these effects is determined 
  in terms of the Maki parameter α, defined as <cit.>, 
  
  
      α    =    √(2)H_c2^orb/H_c2^P∼Δ/ϵ_F
  
  which is the ratio of the two upper critical magnetic fields at zero temperature (T=0). Here, ϵ_F 
  is the Fermi energy. 
  For most of the superconductors α < 1, and thus the system loses superconductivity via the 
  overlap of superconducting vortices much before the applied magnetic field reaches H_c2^P. 
  There are however situations when α > 1 and thus H_c2^P < H_c2^orb, viz,  
  (i) the Fermi energy is strongly suppressed, 
  (ii) the superconducting system is two dimensional and is subjected to an in-plane magnetic field, 
  such that the orbital effects can be neglected, and  
  (iii) the system is charge neutral. 
  The class of superconductors with α>1 and which loses its superconductivity by Pauli paramagnetic 
  pair breaking effect are known as the Pauli limited superconductors. Prominent examples include 
  heavy fermion superconductor CeCoIn_5 <cit.>, layered two-dimensional organic superconductor 
  κ-(BEDT-TTF)_2Cu(NCS)_2 <cit.>, iron superconductor KFe_2As_2 <cit.> 
  and ultracold Fermi gas with imbalance in the populations of fermionic species <cit.>.   
  
  
  
   §.§ Pauli limited superconductivity
  
  
  
  
  
  
  A Pauli limited superconductor in the absence of any orbital effect is known to demonstrate 
  non trivial behavior in the in-plane magnetic field-temperature (H-T) phase diagram 
  <cit.>,  as shown by the schematic in Figure 1(a). 
  In the vicinity of T_c,  dH_c2/dT ∝ 
  √(T_c/(T_c-T)) and the initial slope of H_c2^P is infinite, unlike the orbital 
  limiting case for which the slope is finite. In the absence of Pauli limiting effect the loss 
  of superconductivity at H_c2^orb is via a second order phase transition. For a Pauli limited 
  superconductor, however, the order of thermal phase transition changes below a tricritical 
  temperature, T^*, say. While the transition is of second order for T>T^*, a first order 
  transition is realized for T<T^*. In Figure 1(a) we represent the second and first order 
  phase transitions by solid and dashed curves, respectively. The second order transition line 
  (dotted) at T<T^* correspond to a metastable state. The low temperature high magnetic field regime of
  this phase diagram thus correspond to a first order thermal transition regime, and the associated 
  field at T=0 is H_c2^P.
  
  This Chandrasekhar-Clogston limit H_c2^P was considered to be the upper critical field of Pauli 
  limited superconductivity until Fulde-Ferrell (FF) <cit.> and Larkin-Ovchinnikov (LO) <cit.> 
  pointed out that this critical field can be enhanced further, by taking into account the possibility of 
  an inhomogeneous superconducting state in the high magnetic field regime. The Pauli paramagnetic pair 
  breaking effect is reduced by a new 
  pairing state (K_↑, - K+ Q_↓) between the Zeeman split Fermi surfaces,  
  where Q≠ 0 is the centre of mass momentum of the Cooper pair. Thus, rather than a zero momentum 
  superconducting state, the system hosts a finite momentum superconducting state in the regime of high 
  magnetic field and low temperatures. We represent this possibility of finite momentum pairing by the 
  schematic shown in Figure 1(b). Below the tricritical point (T^*) the transition is always first order, as 
  was demonstrated later for continuum two and three-dimensional systems <cit.>. 
  This exotic finite momentum superconducting phase is known as the Fulde-Ferrell-
  Larkin-Ovchinnikov (FFLO) state. The schematic presented in Figure 2 shows the finite momentum pairing between 
  the fermions in the FFLO state in comparison with the zero momentum pairing in the BCS state. As is 
  evident from the figure an FFLO state comprises of a mismatch in the size of the Fermi surfaces 
  corresponding to the up and down fermion species. Such a Fermi surface mismatch gives rise to unequal 
  populations of the fermionic species i. e. a population imbalance.
  
  
  
  
  The symmetry breaking arising due to the finite momentum Q gives rise to spatial modulations in the 
  superconducting order parameter. There had been several proposals regarding the form of this spatially modulated 
  order parameter with the first one suggested by Fulde and Ferrell being <cit.>, 
  
  
      Δ( r)     =    |Δ| e^i Q. r
  
  Here, the superconducting amplitude is homogeneous but the phase undergoes periodic modulations in the real space. 
  In a similar spirit Larkin and Ovchinnikov suggested that a linear combination of two plane wave solutions 
  corresponding to degenerate superconducting states as <cit.>, 
  
      Δ( r)     =    |Δ|(e^i Q. r+e^-i Q. r) = 2|Δ|cos( Q. r)
  
  can give rise to a spatially modulated superconducting state, such that the amplitude rather than the phase 
  of the superconducting order parameter is modulated. In general, depending upon the symmetry of the system 
  more than two plane waves can undergo linear combination to give rise to a spatially modulated 
  superconducting order parameter <cit.>. It has been found that for a two-dimensional thin film the linear combination
  of two and three plane waves give rise to square and hexagonal states, respectively <cit.>. The spatial modulations give rise to nodes in the superconducting order parameter, which serve as hosts 
  to the unpaired fermions arising out of the Pauli paramagnetic pair breaking. The FFLO phase can thus 
  be envisaged as a coexistent phase wherein finite momentum superconductivity and finite magnetization 
  (arising out of the unpaired fermions) compliment each other in real space. 
  
  For a long time, the FFLO superconducting state though theoretically proposed early on, failed to garner the 
  interest it deserved. One of the principal reasons for the same was its inaccessibility to the existing 
  experimental probes. The FFLO state in two-dimensional materials is highly susceptible to impurities 
  <cit.>, consequently 
  the sample needs to be very clean in order to observe the FFLO state. Even for a clean sample the 
  applied magnetic field should be applied precisely parallel to the plane because  
  any small perpendicular component of the magnetic field would give rise to 
  Abrikosov vortices and destroy the FFLO state <cit.>. Secondly, the transition 
  temperature of the FFLO state 
  is strongly suppressed and the state gets rapidly destroyed by thermal fluctuations even in three 
  dimensions, which makes its experimental detection difficult <cit.>. 
  It was however suggested that crystal anisotropy of real materials can 
  stabilize the FFLO phase against thermal fluctuations <cit.>. 
  
  Though the FFLO phase continues to be the most sought after superconducting phase in systems with 
  Fermi surface mismatch, there are other candidate phases which can be stabilized in these 
  systems, viz. Breached Pair (BP) <cit.> and deformed Fermi surface superfluid 
  phases <cit.>. Unlike FFLO, both the BP and deformed Fermi surface phases 
  comprise of zero momentum pairing between the fermions. The 
  breached pair phase is characterized by the coexistence of finite magnetization and “gapless” 
  zero momentum superconductivity. The issue of BP phase was first raised by Sarma in his seminal 
  work where he discussed the possibility of self consistent mean field solution with gapless mode 
  in s-wave superconductors, in presence of an applied magnetic field <cit.>. He however found that such a
  phase even if it exists would be energetically unfavorable as compared to the BCS state. Later Liu 
  and Wilczek demonstrated that while a contact interaction between the species in a population imbalanced 
  system does not allow 
  the BP phase to be a stable ground state, the same can indeed be achieved under non trivial circumstances 
  such as, imbalance in fermion effective masses with contact interaction between them, momentum dependent 
  interaction, same species repulsion etc <cit.>. 
  Another possible mechanism to achieve stable ground state in an imbalanced Fermi system is 
  through deformed Fermi surfaces which spontaneously breaks the rotational symmetry <cit.>. 
  
  
  
  § EXPERIMENTAL SIGNATURES
  
  
  The renewed interest on imbalanced superconducting systems is attributed to 
  two key experimental breakthroughs viz. (i) the advent of ultracold atomic gases and (ii) improvisation 
  of experiments on solid state materials. Before going into the details of the experimental observations 
  we highlight the candidate systems amenable to the realization of imbalanced superconductivity 
  and superfluidity. In general, the term superconductivity is used in the context of solid state 
  systems containing charged electrons, while for the charge neutral ultracold atomic gases, superfluidity 
  is the correct choice of nomenclature to describe the state with long range macroscopic phase coherence. 
  In the remaining sections of this chapter we will follow this protocol to refer to the solid 
  state and neutral cold atomic gases.   
  
  We mentioned in the earlier sections that in order to realize FFLO phase the superconducting material should 
  have large Maki parameter (α >1). This requirement is stringent and is achievable only under 
  specific conditions.
  
  
    * In a three dimensional superconductor if H_c2^orb is significantly larger than 
  H_c2^P then it would lead to a large Maki parameter. In heavy fermion superconductors 
  such as CeCoIn_5 the large mass of the electrons give rise to low electron velocity 
  and thus a high H_c2^orb. The resulting high Maki parameter α∼ 5 makes it 
  a suitable candidate to realize FFLO superconductivity.
  
    * In layered two-dimensional materials the orbital effects are strongly suppressed when the 
  magnetic field is applied, which makes a large Maki parameter plausible in such systems. 
  A candidate material is quasi two-dimensional organic superconductor κ-(BEDT-TTF)_2Cu(NCS)_2, 
  with Maki parameter of α∼ 8.
  
    * If the system is a neutral Fermi gas, i. e. comprising of neutral atoms then the applied magnetic 
  field can couple only to the spin degrees of freedom. This makes the ultracold atomic gases a promising 
  candidate to realize FFLO phase. In these systems 
  the superfluid pairing takes place between the fermions present in two hyperfine energy 
  states of the neutral atoms or it can take place between the fermions in the energy states 
  of the atoms belonging to two different atomic species or to two different isotopes of the same 
  atomic species.
  
  
  We now discuss the experimental observations made on these candidate systems, which suggests 
  that these systems host exotic phases such as, FFLO and BP, in presence of applied magnetic field 
  or population imbalance of fermionic species.   
  
  
  
   §.§ Heavy fermion superconductor:- CeCoIn_5
  
  
  One of the earliest experiments which suggested the possibility of FFLO phase in CeCoIn_5 
  (T_c∼2.3K) were the specific heat measurements reported by Bianchi et al. 
  <cit.>. They demonstrated that below a critical temperature T_0∼ 0.3T_c and in presence 
  of high magnetic field (H > 10T) applied either in-plane or out of plane of the material, the thermal 
  transition of superconducting to normal state changes from second to first order. They identified a 
  specific heat anomaly in this regime of first order 
  transition and attributed it to an underlying spatially modulated superconducting state. 
  Similar observations were earlier made by Tayama et al. <cit.> in their dc magnetization measurements 
  on CeCoIn_5, which showed a first order phase transition, indicating strong Pauli paramagnetic 
  suppression of the superconducting pairing in the regime of high magnetic field and low temperatures 
  (T_0∼ 0.3T_c). The possibility of FFLO phase in CeCoIn_5 was further established 
  by thermal conductivity <cit.> as well as by resistivity and penetration depth 
  measurements <cit.>. In presence of an in-plane applied magnetic field the thermal 
  conductivity data showed a discontinuous jump indicating a change from second to first order phase 
  transition below a critical temperature of T_0∼1K. Similar signatures were detected 
  by Martin et al. in their resistivity measurements <cit.>.
  
  Based on the nuclear magnetic resonance (NMR) measurements carried out on single crystal of CeCoIn_5,  
  Koutroulakis et al. <cit.> mapped out the (in-plane) magnetic field-temperature 
  phase diagram of this system. Based on the results of NMR shift they identified three different regimes in 
  the thermal phase diagram. In the regime of low magnetic field (H≲ 9.2T) a long range ordered 
  (LRO) homogeneous superconducting phase was identified. In the high magnetic field regime (H≥ 10.2T), 
  in-commensurate spin density wave (IC-SDW) order was found to coexist with FFLO at low temperatures. The 
  intermediate magnetic field regime (9.2T ≤ H < 10.2T) was inferred to comprise of FFLO phase surviving 
  with antiferromagnetic (AFM) correlations. This phase was found to undergo first order thermal transition. 
  Similar inferences were
  made earlier by Kumagai et al. <cit.> via NMR Knight shift measurements on CeCoIn_5. They 
  observed that the FFLO superconductivity is realizable for both in-plane and out of plane magnetic 
  fields and the corresponding phase diagrams are qualitatively different from each other. 
  
  By carrying out high magnetic field neutron scattering experiments on CeCoIn_5 Gerber et al.
  <cit.> identified a “Q-phase” in the high magnetic field low temperature regime, corresponding 
  to a finite 
  momentum pairing. They further confirmed that the finite momentum pairing coexists with an SDW order 
  and undergoes first order thermal phase transition at H∼ 9.8T. In a relatively recent work Kim et al.
  <cit.> suggested that though there is indeed a Q-phase in the high field low temperature regime of 
  the CeCoIn_5 phase diagram in presence of an in-plane applied magnetic field, it does not correspond to a FFLO 
  phase. It was inferred based on their thermal conductivity measurements that the Q-phase comprises 
  of intertwinned SDW, d-wave superconducting and inhomogeneous p-wave pair density wave (PDW) 
  orders. Evidently, the true nature of the Q-phase seems to be unsettled. A very recent 
  thermal conductivity measurement by the same group suggests that the characteristic of the Q-phase 
  is intimately connected to the direction of the applied magnetic field <cit.>. For a magnetic field 
  applied along the ab-plane of the crystal the Q-phase is likely to host a SDW order. On the other hand 
  for a field applied perpendicular to the ab-plane a FFLO order is realizable in the high field 
  low temperature Q-phase of CeCoIn_5 <cit.>.   
  
  
  
   §.§ Layered organic superconductor:- 
   κ-(BEDT-TTF)_2Cu(NCS)_2
  
   
  In two-dimensional layered organic superconductor κ-BEDT (T_c∼ 9.5K), the FFLO 
  phase was inferred from specific heat measurements 
  in presence of applied in-plane magnetic field <cit.>. Unlike CeCoIn_5 it was observed that in κ-BEDT the thermal transitions from both the zero momentum and FFLO superconducting phases are of 
  first order. A Zeeman field driven phase transition was detected in κ-BEDT via NMR measurements
  by Wright et al. <cit.>. They suggested that this transition and the associated discontinuity in the 
  derivative of magnetization, at H∼213±3kOe amounts to a possible indication of spatially 
  inhomogeneous superconducting phase. Similar inference was made by Bergk et al. <cit.> 
  and Agosta et al. <cit.> based on magnetic torque and penetration depth measurements, 
  respectively. The high magnetic field (H ≥ 21T) regime was suggested to host FFLO superconducting 
  phase.  The most clear evidence of FFLO phase in 
  κ-BEDT was given by Mayaffre et al. through NMR measurements <cit.>. They observed 
  that the NMR spin-lattice relaxation rate gets significantly enhanced in the high magnetic field 
  regime, and attributed this observation to the Andreev bound states arising out of the unpaired 
  quasiparticles residing in the nodes of the spatially modulated FFLO state.   
  
  In addition, the presence of FFLO superconducting phase has been recently proposed in 
  iron superconductors such as KFe_2As_2, LiFeAs etc. <cit.>, 
  which are suggested to be two-band superconductors. 
  
  
  
   §.§ Cold atomic gases and Fermi superfluids
  
  
  Over the past decade ultracold Fermi gases have proved to be the testbed of research on imbalanced 
  superfluid systems, owing to the flexibility of tuning the interaction parameters (via Feshbach resonance), 
  optical lattice and trap geometries, that these systems provide. Unlike the solid state superconducting 
  systems where a magnetic 
  field is applied, population imbalance in ultracold atomic gases is created by loading unequal populations 
  of fermionic species in the optical lattice. Alternatively, equal populations of different atomic 
  species with unequal effective masses 
  can be loaded in the optical lattice to create mass imbalanced Fermi gases. Clear evidence of 
  superfluid pair correlations have been found in case of both balanced <cit.> as well 
  as imbalanced <cit.> Fermi gases. Most of these experiments are carried out at or close 
  to unitarity. 
  
  The concept of unitarity is intimately related to the s-wave scattering 
  length which can be controlled via the Feshbach resonance. In three-dimensional (3D) continuum Fermi gas the 
  scattering length a_3D→∞ as g → g_c. Here, g_c is finite in 3D 
  and defines the interaction strength at which the first two-body bound state formation takes place in 
  vacuum. The dimensionless coupling 1/k_Fa_3D=0 (where, k_F is the Fermi momentum) at g_c 
  and also corresponds to the regime of maximum T_c of the 3D Fermi gas. For two-dimensional 
  (2D) Fermi gases unitarity corresponds to the coupling strength at which 
  ln(k_Fa_2D) → 0 (a_2D is s-wave scattering 
  length in 2D), which once again corresponds to the interaction for maximum T_c. For a lattice the 
  notion of k_F is not very meaningful particularly if one is away from the regime of small filling. 
  In the context of lattice fermion models, the crossover regime or unitarity corresponds to the interaction 
  strength at which T_c is maximum, where neither the fermionic nor bosonic description of the 
  system holds true completely <cit.>. We will revisit the issue of 
  unitarity in the 
  later sections of this chapter, when we discuss the BCS-BEC crossover picture.     
  We now discuss some of the important experiments on imbalanced Fermi gas in ultracold  
  cold atomic gas setups, and their principal observations.
  
  Wolfgang Ketterle and his group mapped out the thermal phase diagram of strongly interacting imbalanced spin 
  mixture created between two hyperfine states of ^6Li atoms <cit.>. 
  Based on radio frequency (rf) spectroscopy 
  measurements they identified a gap in the single particle excitation spectra which indicated that 
  the minority species is paired up. It was shown that in presence of 
  large imbalance, at high temperature the spectra shows only the atomic peak which was considered to be 
  the evidence of unpaired fermions in the system. As the temperature is lowered, in addition to the 
  atomic a pairing peak also appeared in the spectra indicating a coexistent phase with superfluid 
  pairs and unpaired fermions. 
  Further decreasing the temperature left behind only the pairing peak. The work showed high temperature 
  coexistence of paired and unpaired fermions but did not find any indication of superfluidity at low temperatures 
  in the regime of large imbalance. Note that at large imbalance there would not be enough minority 
  spins to pair up with the majority species. Consequently, signatures of pair correlations (if any) 
  would be strongly suppressed.   
  
  Shin et al. <cit.> reported the experimental observation of phase separation between 
  superfluid and normal regions in a strongly interacting Fermi gas of ^6Li with population 
  imbalance. They used a special phase contrast imaging technique and 3D image reconstruction to 
  determine the density difference between 
  the two components of the gas in an optical trap. A shell structure was observed,  in which the superfluid 
  region containing equal densities of the components is surrounded by a normal region of unequal densities. 
  Furthermore, using phase contrast imaging they characterized the normal to superfluid phase transition. 
  
  A quantum phase transition was detected in the unitary imbalanced Fermi gas of ^6Li at a critical polarization 
  (magnetization) of ∼ 36 % by Shin et al. <cit.>. Further, they detected the change 
  in the order of thermal  phase transition as a function of polarization. In the regime of low temperatures 
  and increasing polarization 
  a first order phase transition was detected via a jump in the polarization. On the other hand the transition 
  at high temperatures and weak polarization was found to be of second order. These two orders of phase transition 
  are connected at a tricritical point. While they carried out the experiment in presence of trapping potential, 
  using tomographic reconstruction of local Fermi temperatures and spin polarization they determined the phase 
  diagram of the homogeneous system free from any inhomogeneity arising due to the trapping potential. The resulting 
  phase diagram comprised of a superfluid phase in the regime of weak polarization characterized by  
  spatial coexistence of superfluid order and non zero polarization at finite temperatures, akin to the 
  BP phase. This regime undergoes second order thermal 
  phase transition to a normal phase. The large polarization regime is a phase separated (unstable) regime 
  which undergoes a first order thermal phase transition. At still larger polarizations the system is in the normal 
  phase. Note that signatures of FFLO phase was not detected in this experiment, however Shin et al. did not 
  rule out the possibility of such a phase at large polarizations <cit.>. 
  
  In a different experimental set up,  containing arrays of one-dimensional tubes constructed using a two- 
  dimensional optical lattice, Liao et al. <cit.> investigated the phases of an imbalanced Fermi gas of 
  ^6Li. Due to nesting of Fermi surfaces the FFLO phase is expected to have a larger regime of stability 
  in one dimension as compared to the higher dimensions. Liao et al. <cit.> mapped out the phase 
  diagram of this system in polarization-temperature plane and demonstrated that, at low imbalance a 
  partially polarized phase is realized at the centre of the trap, which radially spreads out with increasing 
  imbalance. This partially polarized region extends to the edge of the cloud at a critical imbalance. 
  Further increase in imbalance leads the edges of the cloud to become completely polarized. Liao et al. 
  found their experimental phase diagram to be in agreement with theoretical predictions and suggested that even 
  though signatures of FFLO phases was not visible in their experiment it was certainly a possibility <cit.>. 
  
  Experimental investigations of mass imbalanced Fermi-Fermi mixtures are relatively fewer. 
  A mass imbalanced Fermi-Fermi mixture is achievable for example in a ^6Li-^40K mixture. While 
  superfluidity in such a system is yet to be attained in experiments, the Fermi degenerate 
  regime <cit.> as well as the Feshbach resonance between ^6Li 
  and ^40K atoms <cit.> and formation 
  of ^6Li-^40K heteromolecules <cit.> are already a reality. Furthermore, experimental 
  realization of mixtures of other fermion species (such as,  ^161Dy, ^163Dy, ^167Er) 
  are expected in future <cit.>. It has been reported that for a double-degenerate 
  ^6Li-^40K mixture the Fermi temperatures are T_Li=390 nK and T_K=135 nK, 
  for Li and K species, respectively <cit.>. In comparison, 
  for a balanced Fermi gas of ^6Li, the Fermi temperature is known to be T_F=1.0μK 
  <cit.> with the corresponding T_c scale being T_c∼ 0.15T_F <cit.>.
  In other words, imbalance in mass strongly suppresses the superfluid T_c of the Fermi-Fermi mixtures, 
  making their experimental detection non trivial. However, short range superfluid pair correlations are 
  likely to survive at temperatures significantly higher than the superfluid T_c and accessible to 
  experimental probes. 
   
  
  
  § THEORETICAL FORMALISM
  
  
  The experimental observations across different classes of solid state materials and ultracold atomic gases 
  lead to a flurry of activities in the theoretical front. The theoretical studies can broadly 
  be classified in two categories as, (i) the lattice fermion models and (ii) the continuum models. 
  In this chapter we will mainly focus on (i). For a detail discussion on continuum 
  models with imbalance the readers are referred to <cit.>. 
  Before getting into the details of the results obtained from different lattice fermion based calculations, 
  let us first set up the theoretical framework which can be used to investigate imbalanced Fermi superfluids and 
  superconductors. For this we resort to the simplest tool of mean field theory. We will use the 
  framework of BCS theory and Bogoliubov-Valatin transformation 
  to set up the mean field formalism. The generic Hamiltonian of the two component Fermi gas 
  with an on-site interaction between the fermion species is used to model the system. The BCS 
  mean field approximation is used such that the order parameters are striped of all 
  fluctuations. We take into account the imbalance between the population of the fermion species 
  and allow for the possibilities of exotic superfluid phases such as, FFLO and BP, in addition to 
  the homogeneous BCS phase. Bogoliubov-Valatin transformations <cit.> are used to 
  diagonalize the Hamiltonian and the resulting eigenvalues and eigenfunctions are used to compute different observables 
  relevant from the point of view of experiments. Once the basic formalism is in place we 
  will discuss the modifications required to address different families of imbalanced
  superconductors and superfluids. 
  
  
  
   §.§ Mean field theory
  
  
  We consider the Hamiltonian of a two component Fermi gas comprising of up (↑) 
  and down (↓) fermion species, with an attractive interaction between them. 
  For the sake of simplicity we do not take into account any external potential acting 
  on the fermions. The Hamiltonian corresponding to such a system reads as, 
  
  
      Ĥ    =    ∫ d r∑_σψ̂_σ^†( r) (K̂_σ( r)-μ_σ)
      ψ̂_σ( r) 
             + ∫∫ d rd r^' V_↑↓( r, r^') 
      ψ̂_↑^†( r)ψ̂_↓^†( r^')ψ̂_↓( r^')ψ̂_↑( r)
     
  where, σ = ↑, ↓ correspond to the two fermion species. V_↑↓( r, r^') is the interaction between the species, with r and r^' being the position coordinates, corresponding to a two dimensional system. The kinetic energy contribution to the Hamiltonian is given as, 
  K̂_σ( r) = -ħ^2/2m_σ∂^2/∂ r^2 and 
  μ_σ is the species dependent chemical potential. Note that the effective mass (m_σ) 
  in the kinetic energy contribution is dependent on the species. For a system with only imbalance in fermionic 
  populations m_↑=m_↓=m since the mass of the fermions of both the species are the same. 
  On the other hand, for a system comprising of two different isotopes of the same atom or of two different 
  atomic species, m_↑≠ m_↓. Such systems can be realized in ultracold atomic gases 
  and are called “mass imbalanced” Fermi-Fermi mixtures. For the present discussion we consider equal effective 
  masses for both the species and set m_↑ = m_↓ = m.
  
  We simplify the model by approximating the interaction to be a contact interaction as, 
  
      V_↑↓( r, r^')     =     V_0δ( r- r^')
  
  This approximation is particularly good for ultracold atomic gases in which the interaction is short ranged 
  <cit.>. With this approximation the Hamiltonian takes the form, 
  
  
      Ĥ    =    ∫ d r∑_σψ̂_σ^†( r) (K̂_σ( r)-μ_σ)
      ψ̂_σ( r) 
             + V_0∫ d rψ̂_↑^†( r)ψ̂_↓^†( r)ψ̂_↓( r)ψ̂_↑( r)
  
  
  We consider an underlying lattice potential and for that expand the continuum Fermi operators in terms of Wannier 
  functions <cit.> as, 
  
      ψ̂_σ^†( r)     =    ∑_ n, iĉ_σ,  n, i^†w_ n^*( r- r_i)
  
  where, ĉ_σ, n,i^† creates a σ particle in the lattice site i in the energy band 
  n, and w_ n^*( r- r_i) are the Wannier functions. The Wannier functions can be 
  approximated by harmonic oscillator states in the tight-binding limit. At low temperatures and weak interactions only 
  the lowest energy band n=0 would be occupied by the fermions, we can thus omit the band index henceforth. 
  Using the above expansion the continuum Hamiltonian is re-written as, 
  
  
      Ĥ    =     -∑_σ∑_⟨ ij⟩t_σ, ijĉ_σ,i^†ĉ_σ,j + ∑_σ,i(ϵ_σ-μ_σ)ĉ_σ,i^†ĉ_σ,i
             + U∑_iĉ_↑,i^†ĉ_↓,i^†ĉ_↓,iĉ_↑,i
  
  
  Here, ⟨ ij⟩ corresponds to the summation over nearest neighbors of a lattice and t_σ,ij is the hopping amplitude between the nearest neighbors; ϵ_σ is the onsite energy and U is the on-site interaction, and are expressed as <cit.>, 
  
  
      t_σ,ij    =     -∫ d rw^*( r- r_i)K̂_σ( r)w( r- r_j) 
      ϵ_σ    =    ∫ d rw^*( r- r_i)K̂_σ( r)w( r- r_i) 
       
      U     =     V_0∫ d r| w( r- r_i)|^4
  
  
  Equation (8) corresponds to the standard single band Hubbard model for a two component Fermi gas. In order to 
  take into account the imbalance in the population of fermionic species, we set μ_↑≠μ_↓.
  We consider the system to be mass balanced and the hopping amplitude to be a 
  constant, such that t_↑,ij = t_↓,ij=t_ij=t. We also consider ϵ_↑=ϵ_↓=ϵ = 0, i. e. the on-site energy is taken to be a constant and set to zero. The resulting Hamiltonian 
  thus takes the form, 
  
      Ĥ    =     -t∑_σ∑_⟨ ij⟩ĉ_σ,i^†ĉ_σ,j-∑_σ,iμ_σĉ_σ,i^†ĉ_σ,i+U∑_iĉ_↑,i^†ĉ_↓,i^†ĉ_↓,iĉ_↑,i
   
  Here, U<0 corresponding to the attractive interaction between the fermions. 
  
  
  
    §.§.§ BCS theory of uniform superconductors
  
  
  Equation (12) comprises of a four-fermion interaction term which is not readily solvable. In order to make the 
  Hamiltonian tractable we use the BCS mean field approximation. Within this approximation we introduce a pairing field
  (Δ̂_i) to describe the Cooper pairs. We do not go into the details of the BCS theory here and guide the 
  interested readers to <cit.>. The pairing field thus introduced serves 
  as the superconducting “order parameter” and is related to the expectation value of ψ̂_↓ψ̂_↑ as, 
  
  
      ⟨Δ̂_i⟩≡Δ_i    =     V_0⟨ψ̂_↓( r) ψ̂_↑( r)⟩ = U⟨ĉ_i↓ĉ_i↑⟩
   
  
  The standard BCS theory describes the transition from a normal Fermi gas to a zero momentum uniform superconducting 
  state, and the pairing field is considered to be a constant, Δ_i=Δ. In order to capture the 
  inhomogeneous superconducting state such as FFLO or BP, one needs to retain the spatial dependence 
  of the pairing field. 
  
  Note that the pairing field defined so far is on-site and does not take into account any directional dependence. 
  In the language of superconducting pairing state symmetry, such an on-site pairing correspond to a s-wave 
  superconducting state. For the sake of simplicity we will discuss the theoretical formalism in detail for a 
  s-wave superconducting state and will outline the modifications of this formalism for a non s-wave pairing 
  state, at the end of this section.
  
  The BCS approximation in addition to the pairing field introduces Hartree field corresponding to the 
  fermionic number densities, which is defined as, n_σ=⟨ĉ_σ^†ĉ_σ⟩.
  Including the Hartree field in the formalism renormalizes the chemical potential as, 
  
  
      (-μ_σ+Un_-σ)ĉ_σ^†ĉ_σ   ≡    -μ̃_σĉ_σ^†ĉ_σ
  
  
  Here, μ̃_σ is the renormalized chemical potential and the densities are considered to be uniform. 
  In terms of the average pairing and the Hartree fields the mean field Hamiltonian describing a  
  s-wave superconducting pairing of the fermions in a two component Fermi gas, reads as, 
  
  
      Ĥ    =     -t∑_σ∑_⟨ ij⟩ĉ_σ,i^†ĉ_σ,j -∑_σ∑_iμ̃_σĉ_σ,i^†ĉ_σ,i
             + ∑_i(ĉ_↑,i^†ĉ_↓,i^†Δ_i + Δ_i^*ĉ_↓,iĉ_↑,i - |Δ_i|^2/U)
  
  
  In the Hamiltonian given by Equation (15) the pairing field is replaced by its approximated average value 
  (see Equation 13), but we do not assume any specific form of this average. 
  
  Note that the approximation of uniform density should be used carefully. The approximation breaks down in case 
  of superconductors with spatial inhomogeneities, such as those arising in a FFLO of BP phase or in disordered 
  superconductors. In such cases the correct tool of choice is Bogoliubov-de-Gennes (BdG) formalism which takes into 
  account the spatial inhomogeneities of the pairing field amplitude (|Δ_i|) as well as of 
  the number densities <cit.>.  
  
  
  
    §.§.§ Mean field theory of FFLO superconductivity
  
  
  The FFLO phase comprises of spatially modulated superconducting state 
  coexistent with non zero magnetization. The corresponding Cooper pair centre of mass momentum 
  Q is determined by minimizing the energy of the system. While the FF state corresponds to 
  a superconducting state with modulation in the pairing field phase 
  (Δ( r) = Δ e^i Q. r), the LO (Δ( r) = Δcos( Q. r)) 
  state on the other hand contains modulation in the pairing field amplitude. 
  
  There is now a consensus that at the ground state of two component population imbalanced Fermi gas the LO 
  phase is energetically favorable as compared to the FF phase, both 
  in case of lattice as well as continuum systems <cit.>. 
  An exception however is the population imbalanced system with spin-orbit 
  coupling, where the FF phase is found to have lower energy as compared to that of the LO phase. 
  Strictly speaking, the regime of 
  stability of the FF and LO phases should depend upon the system under consideration, however for many systems 
  they are found to overlap. 
  
  In this section we consider the simpler FF ansatz to discuss the mean field theory of the spatially modulated 
  superconducting phase. The theory can however be easily generalized to describe the LO superconducting phase. 
  Using the ansatz for the FF state (Δ_i = Δ_i e^i Q. r = Δ_i e^i(Q_xx+Q_yy)) 
  the superconducting Hamiltonian for the population imbalanced two component Fermi gas in two-dimensions is given as, 
  
  
      Ĥ    =     -t∑_σ∑_⟨ ij⟩ĉ_σ,i^†ĉ_σ,j-∑_σ∑_iμ̃_σĉ_σ,i^†ĉ_σ,i
             + ∑_i(Δ_i e^i(Q_x.x+Q_y.y)ĉ_↑,i^†ĉ_↓,i^†+Δ_i e^-i(Q_x.x+Q_y.y)ĉ_↓,iĉ_↑,i - Δ_i^2/U)
    
  
  Here, Q_x and Q_y correspond to the components of the Cooper pair centre of mass momentum 
  Q,  in two dimensions.
  In order to determine the superconducting gap and number equations this Hamiltonian needs to be diagonalized. 
  Since the Hamiltonian is not diagonal in the electronic basis we carry out a basis transformation using 
  Bogoliubov-Valatin transformations defined as <cit.>,   
  
  
      ĉ_↑,i    =    ∑_n(u_n,↑^iγ_n,↑-v_n,↑^*iγ_n,↓^†) 
      ĉ_↓,i    =    ∑_n(u_n,↓^iγ_n,↓+v_n,↓^*iγ_n,↑^†)
  
  
  Here, n is a complete set of states, u_n and v_n are the eigenfunctions of the Hamiltonian Ĥ, 
  which satisfies the condition u_n^i2+v_n^i2=1 <cit.>. The new fermionic operators 
  γ_n are quasiparticle operators which satisfies the anti commutation relation 
  γ_n^†γ_n+γ_nγ_n^† = 1 and are chosen in a way such that,
  
  
      Ĥ    =     E_0 + ∑_n,σϵ_nγ_n,σ^†γ_n,σ
    
  Thus, we have, 
  
      [Ĥ,γ_n,σ]     =     -ϵ_nγ_n,σ 
      [Ĥ,γ_n,σ^†]     =    ϵ_nγ_n,σ^†
   
  where, ϵ_n corresponds to the eigenvalues of the Hamiltonian Ĥ. Using the 
  above expressions one can determine the coefficients u_n^i and v_n^i by solving the matrix eigenvalue 
  equation. For this we define the matrices A and Ψ, such that, 
  
      A     =    [ -t-μ_↑   Δ̂_i; Δ̂_i^*  t+μ_↓ ]
   
  and 
  
      Ψ    =    [ {u_n}; {v_n} ]
  
  The resulting eigenvalue equation is then given as, 
  
      AΨ    =    ϵ_nΨ
  
   
  In order to solve the eigenvalue equations, one first needs to determine the superconducting pairing field 
  (Δ_i) as well as the chemical potentials μ_↑ and μ_↓. For this the set 
  of coupled equations corresponding to Δ_i and fermionic number densities n_iσ is to be solved 
  self consistently. In terms of the BdG eigenfunctions and eigenvalues the self consistent mean field equations are 
  given as, 
  
  
      Δ_i    =     U∑_n⟨ĉ_i,↓ĉ_i,↑⟩
          =     U∑_n{v_n,↓^*iu_n,↑^if(ϵ_n)+u_n,↓^*iv_n,↑^if(ϵ_-n)}
  
  
      n_i↑    =    ∑_n⟨ĉ_i,↑^†ĉ_i,↑⟩
          =    ∑_n{| u_n,↑^i|^2f(ϵ_n)+| v_n,↑^i|^2f(ϵ_-n)}
  
  
      n_i↓    =    ∑_n⟨ĉ_i,↓^†ĉ_i,↓⟩
         =   ∑_n{| u_n,↓^i|^2[1-f(ϵ_n)] + | v_n,↓^i|^2
      [1-f(ϵ_-n)]}
  
  
  where, f(ϵ_n) and f(ϵ_-n) are the Fermi functions corresponding to the positive 
  and negative eigenvalues, respectively. 
  Starting from an initial guess these equations are iterated to obtain self consistent values of the pairing field and 
  the number densities. 
  
  For a lattice of dimension L × L the dimension of the matrix is 2N × 2N, where, N=L^2. 
  The net chemical potential of a population imbalanced Fermi gas is defined as, 
  μ=μ_↑+μ_↓/2. The difference between the chemical 
  potentials act as a Zeeman field defined as, h=μ_↑-μ_↓/2. In terms 
  of μ and h we can thus write the spin resolved chemical potentials as, μ_↑=μ+h and 
  μ_↓=μ-h. While working in grand canonical ensemble we work at a fixed chemical potential 
  μ and tune the Zeeman field h such that, the system undergoes phase transition between an uniform 
  superconducting phase and inhomogeneous FFLO or BP phases, at a fixed interaction U. 
  
  By examining the real space self consistent mean field solutions of the pairing field and fermionic number 
  densities one can analyze the underlying state. A non zero Δ_i corresponds to a superconducting 
  state, while a state with Δ_i=0 is non superconducting. In the same spirit, 
  m_i=(n_i↑-n_i↓)≠ 0 corresponds to a state with finite magnetization. Before analyzing 
  the self consistent solution it is however essential to determine whether the particular solution is 
  stable, in other words whether it corresponds to the minimum energy state for the given choice of parameters. 
  
  We thus next calculate the free energy of the system and for that we go back to Equation 19, which gives the 
  Hamiltonian in the quasiparticle basis. Here E_0 correspond 
  to the ground state energy and in order to determine E_0 we note that, 
  
  
      ⟨Ψ_G|Ĥ|Ψ_G⟩    =     E_0
  
  where, |Ψ_G⟩ correspond to the ground state. The free energy of the system reads as,
   
  
      F     =     E_0 + ∑_ϵ_n(-ϵ_-n-1/β[ln(1+e^-βϵ_n)+ln(1+e^-βϵ_-n)] 
             -Δ_i^2/U)
  
  
  The global minima of this free energy functional gives the stable ground state of the system. Based on 
  the pairing field amplitude and average magnetization (⟨ m_i⟩) the ground state of 
  two component imbalanced Fermi gas can be broadly classified as, (i) uniform (BCS) superconducting 
  phase (Δ_i( Q=0) ≠ 0, ⟨ m_i⟩ = 0), (ii) (partially) polarized Fermi 
  liquid (PPFL) (Δ_i=0, ⟨ m_i⟩≠ 0), (iii) FFLO phase (Δ_i( Q≠ 0)≠ 0, 
  ⟨ m_i⟩≠ 0), (iv) BP phase (Δ_i( Q=0)≠ 0, ⟨ m_i⟩≠ 0). 
  Note that even though both BCS and BP phases are characterized by Q=0 pairing state, the BP phase 
  comprises of gapless excitations unlike the BCS superconducting state <cit.>. These gapless excitations are 
  essential to accommodate the unpaired fermions which in turn gives rise to finite magnetization.    
  
  
  
   §.§ Beyond the mean field approximation
  
  
  So far we have taken into account only the inhomogeneity in the amplitude of the pairing field 
  (BdG formalism). 
  The complex superconducting pairing field defined as, Δ_i=|Δ_i| e^iθ_i 
  however comprises of amplitude and phase, both of which in principle can fluctuate and become inhomogeneous. 
  The fluctuations in the pairing field amplitude and phase becomes progressively important with increasing 
  temperature where thermal 
  fluctuations can serve as pair breaking agent. In order to understand the importance of fluctuations 
  on the superconductivity of a system let us take a detour to briefly summarise the BCS-BEC crossover in 
  Fermi gases <cit.>. 
  
  
  
    §.§.§ BCS-BEC crossover
  
  
  Figure 3 shows the schematic phase diagram of the BCS-BEC crossover of the two component Fermi gas 
  comprising of equal populations of up and down fermion species, in the interaction-temperature (U-T) 
  plane. The phase diagram comprises of four distinct regimes viz. superconductor, (bosonic) insulator, 
  Fermi liquid, non Fermi liquid and two thermal scales as T_c and T_pg. While in a solid state 
  system tuning the interaction strength is non trivial in ultracold atomic gases, Feshbach resonance is 
  used to control the s-wave scattering length and in turn the interspecies interaction 
  <cit.>. 
  The ground state of the BCS-BEC crossover can be captured by the BCS theory with suitably modified coefficients, 
  as shown by Leggett <cit.>.
  The two thermal scales that we show in this phase diagram can naively 
  be understood to be the one determined through the experiments on ultracold atomic gases (T_c) and 
  the one determined through BCS-meanfield approximation (T_pg). Evidently, they do not agree with 
  each other except for the weak coupling regime (U≪t) i. e. deep inside the BCS regime. 
  
  
  
  
  According to the BCS mean-field theory, superconductivity of a system vanishes due to the 
  suppression of the superconducting pairing field amplitude.  This obeys the thermal scale 
  T_BCS∼ e^-1/U and the corresponding superconducting 
  pairing field amplitude varies as Δ∼ T_BCS. Thus, T_BCS increases linearly with increasing 
  Δ and in turn with U which governs the magnitude of Δ <cit.>. 
  As shown in Figure 3,  T_pg bears 
  out this linear relation with respect to U. The agreement of T_pg with the experimentally 
  determined T_c in the weak coupling regime suggests that the BCS theory is well suited to capture 
  the finite temperature behavior of the system in the weak coupling regime. The high temperature regime 
  which can be captured by the BCS theory is marked as the Fermi liquid. 
  
  Unlike the BCS prediction the 
  experimental T_c however behaves non monotonically and in the strong coupling regime T_c∼ 1/U, 
  as shown in Figure 3. 
  It is obvious that the BCS theory ceases to be applicable in the strong coupling regime where the pair 
  of fermions tightly bound together and undergoes Bose-Einstein condensation, unlike the weak coupling 
  regime where the Cooper pairs are characterized by large coherence length <cit.>. An efficient way to understand the BEC regime of this crossover picture is by 
  mapping the fermionic attractive Hubbard model on to an equivalent spin model. It is now established 
  that the attractive Hubbard model yields an effective field theoretic description in terms of an 
  SO(3) non linear sigma model. In the strong coupling regime this model maps on to the Heisenberg 
  model in an uniform field. Except for the regime of weak magnetic field (i. e. low filling of the attractive 
  Hubbard model) the Heisenberg model reduces to the quantum XY model <cit.>.
  In the regime of strong coupling, the system undergoes transition from the superfluid state to a bosonic 
  insulator with increasing temperature. The bosonic insulator is characterized by phase uncorrelated large 
  pairing field amplitudes and a hard single particle spectral gap at the Fermi level.        
  
  The regime of intermediate coupling is the most complex part of this phase diagram and none of the 
  perturbative schemes are suitable to capture its behavior. In this regime marked as the “non Fermi liquid” 
  in Figure 3,  neither a completely fermionic nor a bosonic description holds true because the phase comprises of 
  coexisting unpaired fermions and preformed pairs (bosons). The intuitive basis of understanding the BCS-BEC crossover was 
  provided by the earlier works of Leggett <cit.> and Nozeires and Schmitt-Rink <cit.>. 
  Later powerful semi analytic <cit.>
  and sophisticated numerical schemes such as determinant quantum Monte Carlo (DQMC) <cit.>, dynamical mean field 
  theory (DMFT) <cit.> 
  and static path approximation (SPA) <cit.> were utilized with success to investigate the physics 
  of the intermediate coupling regime of the BCS-BEC crossover picture. This non Fermi liquid regime of 
  the BCS-BEC crossover is now known as the pseudogap phase and leaves distinct signatures in 
  the quasiparticle spectrum of the system. The quasiparticle spectra of the pseudogap regime (T_c <T<T_pg) 
  is characterized by renormalized phase coherence peaks arising out of the existing short range pair 
  correlations in this region. Further, at the Fermi level the quasiparticle spectra is no longer 
  hard gapped (unlike T < T_c) and contains finite spectral weight due to unpaired fermions arising out 
  of thermal pair breaking. 
  
  The two thermal scales are thus defined as the T_c, at which the superconducting state loses its long 
  range macroscopic phase coherence and T_pg, at which short range pair correlations are lost. The 
  pseudogap regime with short range pair correlations but no long range order, is characterized by spatial 
  fragmentation of the state into isolated islands akin to uncorrelated Josephson junctions. 
  The finite temperature behavior 
  of the BCS-BEC crossover shows that in the regime of intermediate (and strong) coupling it is the thermally 
  induced phase fluctuations which leads to the loss of long range phase coherence and of the global 
  superconductivity,  unlike the weak coupling regime where superconductivity is destroyed via the suppression 
  of pairing field amplitude. One thus needs to take into account the effect of phase fluctuations 
  in order to correctly capture the behavior of superconductors or superfluids with interactions away from 
  the weak coupling regime, at finite temperatures. 
  
  
  
    §.§.§ BCS-BEC crossover and FFLO systems
  
   
  Having established the importance of fluctuations in superconducting 
  systems we now once again take a look at the candidate systems where FFLO or BP phases are reported. The key 
  candidates as discussed earlier are, three dimensional heavy fermion superconductor CeCoIn_5, 
  two-dimensional layered organic superconductor κ-BEDT, both with large Maki parameter of 
  α>1 and neutral ultracold atomic gases. 
  
  Apart from the large Maki parameter there is another property which serves as a common link between 
  these apparently very distinct systems. An important characteristic of any superconductor is 
  the ratio between the superconducting gap at zero temperature and T_c. The BCS estimate of 
  the same is 2Δ_0/k_BT_c∼ 3.5 <cit.>, which corresponds 
  to the weak coupling regime of Figure 3. The BCS estimate sets the scale,  based on which the interaction 
  regime of other superconducting systems are determined. Based on the point contact <cit.> 
  and scanning tunnelling spectroscopy <cit.> used to determine the magnitude of the gap 
  of any superconducting material,  it has been observed that for CeCoIn_5 (α∼ 5) 
  2Δ_0/k_BT_c∼ 7.73 <cit.>, 
  while for κ-BEDT (α∼ 8) it is 2Δ_0/k_BT_c∼ 4.4 <cit.>. 
  The iron superconductor is conjectured to be a two gap superconductor with the ratio for the smaller gap 
  being 2Δ_0/k_BT_c∼ 1.4 and that of the larger gap being 2Δ_0/k_BT_c∼ 4.6 
  <cit.>. 
  These experimental observations indicate that all the candidate Pauli limited solid state 
  superconductors are situated away from the weak coupling regime. In case of the ultracold 
  atomic gases the interaction can be tuned externally and most of the experiments are carried out 
  at unitarity, which roughly corresponds to the regime of maximum T_c in the 
  BCS-BEC crossover, shown in Figure 3. 
  
  The experimental observations thus bring forth a common 
  link between the various Pauli limited superconducting systems, viz. all these systems belong 
  to the regime of intermediate coupling. While at one hand this sets up the stage for complex 
  physical properties to play out, on the other hand it calls for a non perturbative approach to 
  address these systems. It is thus imperative to take into account the effect of fluctuations 
  while addressing the finite temperature behavior of the imbalanced Fermi superconductors and 
  superfluids. Consequently,  a mean field approach to the problem will not suffice. 
  
  We discussed above that apart from the semi analytic and T-matrix approaches 
  <cit.> there are powerful numerical techniques 
  which takes into account the effect of fluctuations in quantum many body systems 
  <cit.>.  
  We now discuss a path integral formalism based on which the different techniques and their 
  regimes of validity can be understood. 
  
  
  
    §.§.§ Path integral formalism for imbalanced Fermi systems
  
  
  We begin by rewriting the two-dimensional attractive Hubbard model with population 
  imbalance, introduced in Equation (12), which can be used to model the solid state as well as 
  the ultracold atomic gas systems. We will discuss about the system specific modifications in the later 
  sections and for the sake of simplicity discuss here the theoretical formalism in detail for a square 
  lattice with on-site attractive interaction | U | which gives rise to s-wave superconductivity. 
  The corresponding Hamiltonian reads as, 
  
      Ĥ    =    ∑_⟨ ij⟩,σ(t_ij-μδ_ij)ĉ_iσ^†ĉ_jσ-h∑_iσ_iz-| U|∑_in̂_i↑n̂_i↓
    
  where, t_ij = -t only for the nearest neighbor hopping and is zero otherwise; σ_iz is the Pauli matrix. 
  The hopping amplitude t is set to unity and is chosen to be the 
  reference energy scale for the problem. μ is 
  the net chemical potential and the Zeeman field h is applied along the ẑ direction; | U| > 0. 
  
  The Hubbard partition function is a functional integral over the Grassman fields ψ_iσ(τ) 
  and ψ̅_iσ(τ), 
  
      Z  =  ∫ Dψ Dψ̅e^-∫^β_0 dτ L(τ) L(τ) =  L_0(τ) +  L_U(τ)  L_0(τ)=∑_⟨ ij⟩, σ{ψ̅_iσ((∂_τ-μ_σ)
      δ_ij+t_ij)ψ_jσ} L_U(τ) = -| U |∑_iσσ^'ψ̅_iσψ_iσψ̅_iσ^'ψ_iσ^'
  
  where, β is the inverse temperature, μ_σ takes into account the field h. 
  Our strategy is to decompose the interaction in terms of the 
  bosonic auxiliary s-wave pairing field Δ_i(τ) = |Δ_i(τ)| e^iθ_i(τ),  
  using Hubbard-Stratonovich transformation <cit.>. Here |Δ_i(τ)| and 
  θ_i(τ) corresponds 
  to the amplitude and phase of the pairing field, respectively, and encodes spatial (i) and imaginary 
  time (τ) dependence. This leads to,   
  
      Z = ∫ Dψ Dψ̅ DΔ DΔ^*e^-∫^β_0 
      dτ L(τ) L(τ) =  L_0(τ) +  L_U(τ) +  L_cl(τ)  L_0(τ)=∑_⟨ ij⟩, σ{ψ̅_iσ(τ)((∂_τ-μ_σ)δ_ij+
      t_ij)ψ_jσ(τ)} L_U(τ) = ∑_i [Δ_i(τ)ψ̅_i↑(τ) ψ̅_i↓(τ) + h. c.]  L_cl(τ) = ∑_i|Δ_i(τ)|^2/| U|
  
  
  The ψ integral is now quadratic, but an additional integration over the field Δ_i(τ) has been 
  introduced. The weight factor for the Δ configurations can be determined by integrating out the ψ,
  ψ̅ and using these weighted configurations, one goes back and compute the fermionic properties. Formally, 
  
      Z = ∫ DΔ DΔ^* e^-S_eff{Δ,Δ^*}
      S_eff = ln Det [ G^-1{Δ,Δ^*}] + ∫^β_0dτ L_cl(τ)
  
  where, G is the electronic Green's function in the {Δ} background.
  The weight factor for an arbitrary space-time configuration Δ_i(τ) involves computation 
  of the fermionic determinant in that background. If we write the auxiliary field Δ_i(τ) in 
  terms of its Matsubara modes as, Δ_i(Ω_n) then the different options are readily 
  recognized, 
  
  
    * Quantum Monte Carlo retains the full “i,Ω_n” dependence of Δ computing 
  ln Det[ G^-1{Δ}] iteratively for importance sampling. The approach is valid 
  at all T, but does not readily yield real frequency spectra.
  
    * Mean field theory (MFT) is time independent, neglects the phase fluctuations completely but 
  can handle spatial inhomogeneity in amplitude of the pairing field. Thus, 
  Δ_i(iΩ_n) →|Δ_i|. When the mean field order parameter 
  vanishes at high temperature the theory trivializes.
  
    * Dynamical mean field theory (DMFT) retains the full dynamics
  but keeps Δ at  effectively one site, i.e,
  Δ_i(Ω_n) →Δ(Ω_n).
  This is exact when dimensionality D →∞.
  
    * Static path approximation (SPA) <cit.> 
  approach retains the full spatial dependence
  in Δ but keeps only the Ω_n=0 mode,
  i.e, Δ_i(Ω_n) →Δ_i.
  It thus includes classical fluctuations of arbitrary magnitude but no quantum (Ω_n ≠ 0)
  fluctuations.  One may consider different
  temperature regimes. (1) T = 0: Since classical fluctuations die off at T = 0, SPA reduces to
  standard Bogoliubov-de-Gennes (BdG) MFT.
  (2) At T≠ 0 we consider not just the saddle-point configuration but all configurations 
  following the weight e^-S_eff above.
  These involve the classical amplitude and phase fluctuations of the order parameter, and the BdG equations
  are solved in all these configurations to compute the thermally averaged properties.
  This approach suppresses the order much quicker than in MFT. (3) High T : Since the Ω_n=0
  mode dominates the exact partition function, the SPA approach becomes exact as T→∞. 
  It is thus akin to the MFT only at T=0 but captures the thermal behavior 
  of the system accurately. Within this approach the system can be envisaged as free fermions moving in a 
  random background of Δ_i.
   
  
  
  
    §.§.§ Static path approximation (SPA)
  
  
  Evidently, all the numerical approaches have different set of merits and caveats, and consequently 
  different regimes of applicability. We now discuss the numerical implementation of the SPA technique in 
  some detail <cit.>. The merits of this technique 
  rests on its applicability, in principle to systems of arbitrary 
  dimensions and lattice sizes, with moderate computation cost and its access to real frequency 
  properties of the system. The technique takes into account spatial fluctuations and agreement
  with the results of numerically exact DQMC suggests that it can capture the thermal scales and high 
  temperature phases accurately <cit.>. Using the Hubbard Stratonovich decomposition to 
  decompose the interaction term in Equation (29) and then dropping the imaginary time dependence 
  of the bosonic auxiliary field (Δ_i(τ)) the effective Hamiltonian reads as, 
  
      H_eff    =     -t∑_⟨ ij⟩, σĉ_iσ^†ĉ_jσ
      + ∑_iΔ_iĉ_i↑^†ĉ_i↓^† + h. c. -μ∑_i, σn̂_iσ
             - h∑_iσ_i^z + 
      ∑_i|Δ_i|^2/| U |
  
  where, the last term H_cl=|Δ_i|^2/| U| is the stiffness 
  cost associated with the now “classical” auxiliary field. In the standard mean field approach the
  superconducting pairing field Δ_i is assumed to be a real number, but here we retain the degrees of freedom
  associated with the pairing field phase and amplitude. 
  
  The background field Δ_i obeys the Boltzmann distribution,
  
      P{Δ_i}∝ Tr_cc^†e^-β H_eff
  
  
  which is connected to the free energy of the system. For large and random Δ_i the 
  trace is taken numerically. We generate the random background of {Δ_i} by using Monte Carlo, 
  diagonalizing H_eff for each attempted update of Δ_i. The relevant fermionic correlators are computed
  on the optimized configurations at different temperatures. 
  
  Evidently, the technique is numerically expensive 
  and involves large computation cost in diagonalizing the matrix to compute the free energy for each 
  attempted update of the auxiliary field. For an N × N matrix the computation cost scales as  
  ∼ O(N^3) per update and ∼ O(N^4) per lattice sweep. This imposes constraint 
  on the lattice size that can be accessed within reasonable computation time. In order to access larger 
  system sizes a cluster based update scheme could be utilized wherein instead of diagonalizing for the entire 
  lattice at each attempted update, a smaller cluster of size N_c=L_c× L_c surrounding the 
  update site is diagonalized <cit.>. This reduces the cost per update 
  to ∼ O(N_c^3) and to 
  ∼ N× O(N_c^3) per lattice sweep, which scales linearly with the system size. 
  This allows one to access system sizes of upto ∼ 40× 40, which is beyond the reach of DQMC. 
  Access to larger system sizes aid in to explore lower interaction regimes where the coherence length 
  of the Cooper pairs are large. Moreover, in order to faithfully capture spatially inhomogeneous 
  orders, access to lager lattice sizes is essential. 
  
  We have now set up a theoretical framework to investigate the behavior of spatially inhomogeneous 
  superconducting and superfluid phases. Different numerical tools to access these systems 
  both at the level of ground state as well as at finite temperature through mean field and non perturbative 
  techniques are discussed. In order to understand the theoretical formalism we considered that the 
  interaction between the fermion species is onsite and the resulting pairing state symmetry is 
  s-wave. This is a valid approximation as far as ultracold atomic gas set ups are concerned 
  and most of the experiments carried out on imbalanced Fermi systems in ultracold atomic gases 
  are for s-wave superfluids. Furthermore, here a single channel decomposition of the interaction term 
  was carried out. In principle, along with the pairing channel the interaction can be decomposed 
  in density and spin channels as well. This however would add to the computation cost and more 
  importantly would not lead to any qualitative difference in the behavior of the present 
  system. 
  
  The behavior of the imbalanced Fermi superfluid (or superconductor) is analyzed based on 
  different thermodynamic and quasiparticle indicators, some of which are,
  
  
    * Magnetization:- m_i=n_i↑-n_i↓.
  
    * Pairing field structure factor:- S_Δ( q) = 1/N^2∑_i,j⟨Δ_iΔ_j^*⟩ e^i q.( r_i- r_j)
  
    * Spin resolved single particle density of states (DOS):- N_↑(ω)=
  1/N⟨∑_i| u_n↑^i|^2δ(ω-ϵ_n)⟩, 
  N_↓(ω) = 1/N⟨∑_i| v_n↓^i|^2δ(ω-ϵ_n)⟩
  
    * Spectral functions and lineshapes:- A_σ( k,ω)=(-1/π)Im G_σ( k,ω).
  
    * Low energy spectral weight distribution:- A( k,0).
   
  Here, G( k,ω) = lim_δ→ 0G( k,iω_n)|_iω_n→ω+iδ, with G( k,ω) being the imaginary frequency transform of ⟨ c_ k(τ)
  c_ k^†(0)⟩.
  
  Note that the static path approximation approach, as mentioned above does not take into account the 
  quantum fluctuations at the ground state. While in continuum this could be a poor approximation, on 
  a lattice it is reasonable. The low energy fluctuations in a FFLO like state arises from, (i) the phase 
  symmetry of the U(1) order parameter, (ii) the translational symmetry breaking and (iii) rotational 
  symmetry breaking. As a result, in continuum a long range order can not survive even at T=0 in two 
  dimensions. The mean field theory, which predicts such order is therefore invalid. 
  
  On a lattice, the spatial symmetry is already broken and thus the translational and rotational modes are 
  gapped out. The only low energy excitations thus left are the XY type, in presence of which the two-dimensional 
  system can have long range order at T=0 and undergoes Berezinskii-Kosterlitz-Thousless (BKT) transition at the 
  finite temperatures. The issue of fluctuations thus reduces to how well the U(1) phase fluctuations 
  and the corresponding T_c are captured by this approach.
   
  
  
    §.§.§ Non s-wave pairing
  
   
  The superconducting pairing state symmetry in solid state materials depend on 
  the material parameters and the coupling constants involved. As discussed earlier, the all the
  known Pauli limited superconductors belong to the regime of intermediate coupling. This in 
  turn naturally categorises them to be superconductors with short coherence length, whose 
  pairing state symmetry can no longer be s-wave. Indeed the pairing state symmetry of 
  CeCoIn_5 and κ-BEDT are established to be d-wave <cit.>. Unlike  
  s-wave, the d-wave pairing state requires an intersite interaction between the fermions. 
  
  The Hubbard Hamiltonian can be suitably modified to take into account the nearest 
  neighbor interaction which gives rise to d-wave pairing state symmetry as, 
  
      Ĥ    =    ∑_⟨ ij⟩, σt_ijĉ^†_i,σĉ_j,σ-U∑_⟨ ij⟩n̂_in̂_j -μ∑_i,σn̂_i,σ - h∑_iσ_i^z
  
  where, the attractive interaction | U | > 0 is now between the nearest neighbors, 
  on a square lattice. After carrying out the Hubbard-Stratonovich transformation to decompose the 
  four-fermion interaction, the corresponding partition function reads as, 
  
      Z = ∫ Dψ Dψ̅ DΔ DΔ^*e^-∫^β_0 dτ L(τ)
       L(τ) =  L_0(τ) +  L_U(τ) +  L_cl(τ)  L_0(τ)=∑_⟨ ij⟩, σ{ψ̅_iσ((∂_τ-μ_σ)
      δ_ij+t_ij)ψ_jσ} L_U(τ) = -∑_i ≠ jΔ_ij(ψ̅_i↑ψ̅_j↓+
      ψ̅_j↑ψ̅_i↓) + h. c.  L_cl(τ) = 4∑_i ≠ j|Δ_ij|^2/| U|
  
  The bosonic auxiliary field Δ_ij encodes the direction dependence of the d-wave 
  superconducting order parameter.
  
  Apart from the superconducting pairing state symmetry there is another aspect which needs to be taken 
  into account while constructing a model Hamiltonian for the solid state Pauli limited superconductors, 
  viz the lattice structure. The heavy fermion CeCoIn_5 comprises of a three dimensional cubic lattice 
  structure <cit.>, while on the other hand the quasi two-dimensional κ-BEDT is a 
  material with triangular lattice structure <cit.>. 
  KFe_2As_2 contains an added level of complexity which needs to be taken into 
  account while setting up a theoretical framework, this material is conjectured to be a multiband 
  superconductor with two superconducting gaps (and thus the pairing fields) of different magnitudes 
  <cit.>. 
  
  We conclude this section by stating that a realistic theoretical model of solid state Pauli limited 
  superconductors would be significantly complicated owing to the various interaction parameters and complex 
  band structure that needs to be taken into account. However, a relatively simplistic model which takes into 
  account the dominant coupling scales and energy bands is amenable. Such a theoretical framework should 
  be able to capture the experimental observations on these materials with qualitative and in some cases 
  with quantitative accuracy. It should however be kept in mind that solving even a minimal quantum many body 
  Hamiltonian is a demanding and computationally extensive task. Consequently, there still remains several 
  open areas to be explored in the playground of Pauli limited superconductors as well as of Fermi superfluids
  in ultracold atomic gas setups.   
  
  
  
  § MEAN FIELD AND BEYOND MEAN FIELD STUDIES
  
  
  This section discusses some of the studies carried out on lattice fermion models to understand population 
  imbalanced Fermi gases, Pauli limited superconductors and mass imbalanced Fermi-Fermi mixtures. Most of 
  the T=0 investigations are carried out within the framework of mean field theory and there is a large 
  volume of literature discussing the same. We do not attempt to be comprehensive and would discuss only 
  some of the important results and the inferences drawn based on them. 
  
  Regarding the beyond mean field attempts, broadly the observations made by three different numerical 
  approaches are covered in this section, viz. (i) dynamical mean field theory (DMFT) 
  <cit.>, (ii) determinant quantum Monte Carlo (DQMC) 
  <cit.> and (iii) static path approximation (SPA) <cit.>.    
  
  The thermal transitions pertaining to two-dimensional systems, discussed in this section 
  are Berezinskii-Kosterlitz-Thouless (BKT) transitions corresponding to algebraic decay 
  of quasi long range order in two dimensions <cit.>. 
  
  
  
   §.§ Population imbalanced s-wave superfluids
   
  
  Within the framework of lattice mean field theory the two-component Fermi mixture with population 
  imbalance in an optical lattice was studied by Koponen et al. <cit.>. 
  The authors found that both the FF and BP
  phases correspond to stable ground state of the system, with the FF state corresponding to 
  the global minimum in a fixed chemical potential calculation, and the BP phase corresponding 
  to the minimum energy state for a fixed fermionic number density calculation. The same authors further 
  investigated the finite temperature behavior of this system, wherein they accessed the finite 
  temperature physics by using the mean field theory. The resulting temperature-magnetization 
  phase diagram showed that at fixed number density the FF phase constitutes a large part of the 
  phase diagram along with the BP and phase separated regimes <cit.>.   
  
  Using an elaborate variational mean field scheme based on BdG method with Hartree corrections, 
  Loh and Trivedi <cit.> investigated the LO phase in a cubic lattice. Their mean field scheme comprised 
  of six variational parameters corresponding to the complex valued pairing field, chemical potential 
  and three Zeeman fields. The study found that for a cubic lattice the LO phase is a stable ground state 
  over a significant parameter regime. The authors further discussed the quasiparticle behavior of the 
  LO state and showed that the state is characterized by additional van-Hove singularities in the 
  single particle density of states (DOS). Based on the momentum 
  resolved fermionic occupation number (n_σ( k) = 
  ⟨ĉ_ k,σ^†ĉ_ k,σ⟩) they inferred that the LO phase breaks 
  the four fold symmetry but preserves the inversion symmetry, unlike the FF phase.  
  
  
  
  
  
  An extensive numerical study of the ground state was carried out by Chiesa and Zhang as well as by Rosenberg 
  et al on a two-dimensional square lattice, using Hartree-Fock-Bogoliubov-de-Gennes 
  (HFBDG) theory <cit.>. 
  It was observed that an uniaxial LO phase covered a large part of the phase diagram 
  in the magnetization-number density (m-n) plane, at an intermediate interaction strength. 
  In the regime of small magnetization and large 
  number density, diagonal stripe orders were found for intermediate interaction, corresponding to 
  a “supersolid” phase, in which charge and pairing orders coexisted. 
  Apart from these,  mean field based comparative study with LO and FF ansatz showed that for both 
  two and three dimensional systems the LO state correspond to lower energy phase as compared to 
  the FF <cit.>. The observation was in agreement with the preceding attempts along 
  this direction <cit.>. 
  
  Mean field investigations are carried out in one dimensional systems as well, for example by 
  Sun and Bolech, who studied the system of coupled one dimensional tubes <cit.>. It was found that the 
  stability of the FFLO state was dependent on the magnitude of pair tunnelling between the tubes. 
  In addition to the FFLO phase, an exotic η-phase was realized in the mean field study of this 
  one dimensional system. In the η-phase the FFLO wave vector was found to occur at the edge 
  of the first Brillouin zone.
  
  The behavior of the LO state across dimensional crossover between one to three dimensions was 
  investigated using DMFT by Törmä and co-authors <cit.>. 
  It was shown that at the ground state the LO phase is stable over large regime of parameters, 
  extending well into the 3D-like regime, unlike the predictions of mean field <cit.> 
  and effective field theory <cit.>. 
  When extended to finite temperatures by using QMC solver in single site DMFT impurity problem, the 
  same system showed that the transition temperature of the LO phases (T_FFLO) is 
  suppressed to upto one third of that of uniform superfluid at zero imbalance <cit.>. 
  Effect of non local fluctuations were taken into 
  account in this framework by using cluster-DMFT and it was demonstrated that CDMFT correctly 
  captured the behavior of the system in terms of decreasing robustness of long range phase 
  coherence with reducing dimensionality of the system <cit.>. 
  
  In their DQMC study of imbalanced s-wave superfluid Wolak et al. <cit.> 
  observed signatures of finite momentum pairing, suggesting instability towards the FFLO phase. 
  At high temperature indications of a magnetized paired phase was observed. The work however failed to explore 
  the low temperature regime (T < 0.1t, with t being the hopping scale), owing to the fermionic sign 
  problem of DQMC. Note that the access to low temperature regime is important in case of FFLO phase, for which 
  the transition temperature is expected to be strongly suppressed, due to the population imbalance. This 
  work therefore failed to give an estimate of the T_c scale associated with the FFLO phase. 
  The authors predicted the zero imbalance T_c to be ∼ 0.1t but suggested that FFLO 
  correlations survive up to a scale T >> T_c. 
  
  
  
  
  
  Gukelberger et al. <cit.> used diagrammatic QMC to map out the thermal phase 
  diagram of imbalanced s-wave superfluid in the magnetization-temperature plane at intermediate lattice filling, 
  based on their observation of the divergence of the pair susceptibility. 
  Interestingly, they found that the weak magnetization low temperature regime of the phase diagram 
  comprises of a zero momentum pairing state with finite magnetization. As discussed in the earlier 
  sections, such a coexistence of superfluidity and magnetization corresponds to the BP phase. The 
  intermediate magnetization regime was found to be a FFLO phase, followed by the high magnetization 
  regime corresponding to a Fermi liquid. The lowest temperature that was probed in this study 
  was T∼ 0.02t which in principle can be higher than the T_FFLO. Thus, it remains unclear 
  whether the BP phase shown in the phase diagram by Gukelberker et al. <cit.> 
  corresponds to a true ground state or a finite temperature phase. It must however be noted that earlier mean field 
  studies of similar system, carried out at fixed lattice filling suggested the BP phase to be a 
  possible stable ground state in the regime of weak magnetization <cit.>. 
  
  A comprehensive analysis of this system in grand canonical ensemble (fixed chemical potential) was 
  carried out using SPA, close to the half filling and at an intermediate interaction <cit.>. 
  Apart from mapping out the ground state and finite 
  temperature phase diagrams and capturing the thermal scales accurately, the work discussed real space and 
  quasiparticle signatures of the LO phase as well as their thermal evolution. The ground state of the 
  system was found to comprise of uniform superfluid (USF), LO and partially polarized Fermi 
  liquid (PPFL) phases, as the function of increasing magnetic field.  
  It was demonstrated, as in Figure 4, that inclusion of thermal fluctuations significantly suppresses 
  the T_c corresponding to both uniform and LO superfluid phases. Comparison with the results of mean 
  field calculations showed that the mean field theory overestimates the T_c by a factor of ∼ 4. 
  In the LO phase the T_c was about 20% of its mean field estimate and in the absolute term only 2% of 
  the hopping scale. The T_c corresponding to the two component balanced Fermi gas in two dimensions 
  is estimated to be T_c∼ 0.15t, based on DQMC <cit.> as well as SPA <cit.> 
  calculations. The maximum T_c of the LO phase is thus ∼0.03t, which is far lower than the lowest 
  temperature accessed by Wolak et al. <cit.> and very close to that accessed by Gukelberger 
  et al. <cit.>.  
  
  
  
  
  
  The work showed that in the regime of weak magnetic field the system loses uniform superfluid order 
  to PPFL via a second order thermal transition, while for the strong magnetic field LO phase, the thermal 
  transition is of first order. Moreover, a first order thermal transition regime was identified near the 
  quantum critical point between the uniform superfluid and 
  FFLO phases. In this regime, the uniform (Q=0) superfluid phase undergoes first order thermal 
  transition to the PPFL phase. Over the entire regime of magnetic field (magnetization) signatures of superfluid 
  pairing were observed at the high temperatures as short range pairing correlations, which survives upto 
  temperatures T ∼ 1.5T_c and gives rise to the pseudogap phase. Akin to the 
  T_c the pseudogap scale is strongly suppressed by the magnetic field.   
  
  Based on the real space maps the low magnetic field high temperature phase was identified to be a BP 
  phase with spatial coexistence of zero momentum superfluid correlations and non zero 
  magnetization, as shown in Figure 5 <cit.>. In Figure 5, along with the pairing field 
  amplitude (|Δ_i|), phase coherence (cos(θ_0-θ_i)) and magnetization 
  (m_i), the momentum resolved pairing field (S_Δ( q)) and magnetic (S_m( q)) 
  structure factors, as well as spatially resolved fermionic number density (n_i) are shown.
  The lowest temperature state corresponds to a phase cohered uniform superfluid with vanishingly 
  small magnetization. Increase in temperature leads to suppression of long range phase coherence 
  and fragmentation of the superfluid state. Concomitantly, magnetization begins to develop 
  spatially in the system, in the form of isolated islands, which are roughly complementary to 
  the suppression of the pairing field. The real space observation is further attested by the momentum 
  resolved structure factor corresponding to the pairing field and magnetization.  
  
  
  
  
  
  At the ground state the LO phase was characterized by uniaxial stripes with 
  the exact pairing momentum being dictated by the magnitude of the Zeeman field for a fixed 
  interaction and chemical potential. The thermal disordering of the LO phase is demonstrated via 
  the real space maps in Figure 6 <cit.>. The maps show that the thermal scales are strongly suppressed 
  in the LO phase and the long range order is lost even at T∼0.005t, even though short range 
  correlations survive upto high temperatures. The superfluid order and magnetization are spatially 
  complementary to each other and the minimum of one corresponds to the maximum of the other.
  
  Distribution of spin and momentum resolved fermionic occupation number which maps out the Fermi 
  surface shows that in the LO phase the Fermi surface is two-fold 
  symmetric, rather than the four-fold symmetry of the uniform superconducting state. Signatures of 
  LO phase was observed upto T ∼ 2T_c, at still higher temperatures the signatures of pairing 
  correlations are lost and the distribution was found to be akin to the PPFL phase, as shown in 
  Figure 7.  
  
  
  
  
  
  The quasiparticle properties such as single particle DOS, spectral functions and distribution 
  of low energy spectral weight carried intriguing imprints of the underlying modulated superfluid 
  phase <cit.>. The single particle DOS contained additional van-Hove singularities,  
  in agreement with the mean field predictions <cit.>. This was the first and so far the only 
  work on population imbalanced s-wave superfluids which mapped out the spectral function (A( k,ω)) 
  of the LO phase, within a lattice fermion model. 
  Owing to the multiple scatterings that the fermion undergoes,  the dispersion spectra of the LO phase 
  was found to contain six branches, in contrast to the two branches of the BCS (uniform) superconductors. 
  Each fermion species gives rise to three dispersive branches, separated by soft gaps, located at (ω=0)
  and away (ω = ± h), from the Fermi level. In agreement with such a non trivial 
  dispersion spectra the corresponding low energy spectral weight distribution (A( k, 0)) 
  was found to be highly intriguing. It was observed that in spite of a s-wave pairing state symmetry 
  a nodal superfluid gap structure emerges out of the LO correlations. Figure 8 shows the thermal 
  evolution of the spectral function (for one of the fermionic species) and low energy spectral weight 
  distribution of the LO phase. Thermal fluctuations lead to the smearing out of the nodal architecture 
  of the low energy spectral weight distribution and at high temperatures the four-fold symmetry of the 
  s-wave superfluid gap is restored. 
  
  
  
  
  
  This work drew comparison between the numerically computed <cit.> thermal phase diagram with that obtained 
  experimentally in ultracold atomic gas set up <cit.>. As shown in Figure 9, the agreement 
  between the two, both in terms of the order of phase transitions as well as thermal scales, is fairly good. 
  Numerical calculations suggest that the large magnetization regime hosts the LO phase. Though the signature of 
  spatially modulated LO phase was not observed in the experiment, its possibility has not been ruled out either.
   In a related work the same authors determined an anomalous 
  pseudogap phase with non trivial spectral signatures, near the finite temperature BP-LO boundary 
  at low magnetizations and a re-entrant pseudogap phase at large magnetizations <cit.>.   
  
  
  
   §.§ Pauli limited d-wave superconductors
  
  
  Studies carried out on the FFLO phase of non s-wave superconductors are relatively fewer, 
  inspite of all the solid state Pauli limited superconductors being d-wave. Vorontsov et al. 
  used self consistent theory to study the effect of in-plane Zeeman field on a two-dimensional 
  superconducting system with d_x^2-y^2 pairing state symmetry <cit.>. 
  They solved Eilenberger equations in presence of magnetic field for the superconducting order parameter 
  with spatial dependence, to map out the phase diagram in the temperature-magnetic field (T-H) plane. Their 
  study showed the LO phase to be more stable as compared to the FF, over a large part of the parameter space. 
  The single particle DOS as determined from their calculations showed van-Hove singularities akin 
  to Andreev bound states, which are characteristic to FFLO state. 
  
  
  
  
  
  Within the framework of Ginzburg-Landau theory  Beaird et al. studied Pauli limited superconductors 
  with d-wave pairing state symmetry <cit.>. They included thermally induced classical 
  magnetic fluctuations 
  in their theoretical framework to explain the experimentally observed phase diagram of CeCoIn_5. 
  Their work showed that the inclusion of magnetic fluctuations affect the thermodynamic phase 
  diagram significantly such that, a regime of first order thermal transition from the normal state 
  emerges near the quantum critical point where the system undergoes transition between the uniform 
  d-wave superconducting state and a spatially modulated LO phase. Beaird et al. <cit.> 
  showed that the regime of first order thermal transition straddles both the uniform and LO superconducting 
  phases in agreement with the experimental observations of the heavy fermion Pauli limited superconductor 
  CeCoIn_5 <cit.>.
  
  The effect of non magnetic impurities on Pauli limited d-wave superconductor was studied by 
  Vorontsov et al. using quasiclassical Eilenberger equations in which the effect of impurities 
  was included via self consistent T-matrix approximation <cit.>. It was inferred that 
  contrary to the naive expectations, the FFLO phase is robust against impurities in the limit of 
  strong disorder. 
  
  
  
  
  
  Using the BCS weak coupling model Yang and Sondhi made certain intriguing predictions about d-wave 
  superconductor subjected to an in-plane Zeeman field <cit.>. They showed that owing to the nodal architecture 
  of the d-wave pairing state the Zeeman field would destroy superconductivity in parts of the Fermi 
  surface and give rise to pockets of normal electrons. The authors suggested that this phenomenon should 
  be observable in the quasiparticle signatures of the system.
  
  Using the BdG mean field theory on a two-dimensional lattice model Zhou and Ting mapped out the thermal 
  phase diagram of Pauli limited d-wave superconductor as a function of in-plane Zeeman field <cit.>. 
  Apart from the uniform superconducting phase in the low field regime they observed one and two-dimensionally 
  modulated FFLO phases in the regime of high field and low temperatures. Since the approach does not take 
  into account the effects of thermal fluctuations the stability of the modulated phases remain unknown. 
  The single particle DOS at the ground state as obtained by the BdG mean field theory showed signatures 
  of LO state in terms of Andreev bound states. 
  
  The beyond mean field studies of d-wave Pauli limited superconductors are much lesser than 
  that of s-wave. In a series of papers Ikeda and co-authors studied the problem of FFLO phase 
  in d-wave superconductor in presence of antiferromagnetic (AFM) fluctuations, relevant for materials 
  such as CeCoIn_5 <cit.>. Within the 
  framework of repulsive Hubbard model which gives rise to d-wave superconductivity as well as AFM order, 
  the effect of paramagnetic pair breaking was taken into account through an in-plane Zeeman field. Using fluctuation 
  exchange approximation (FLEX) the authors showed that the high magnetic field low temperature regime 
  comprises of coexisting AFM and d-wave superconducting orders, as well as a FFLO phase. The authors claimed 
  their results to be in agreement with the experimental observations on CeCoIn_5. 
  A Ginzburg Landau theory based study was carried out by Adachi and Ikeda on two-band Pauli limited 
  superconductors, which is expected to provide useful insights to the behavior of Pauli limited iron superconductors
  such as KFe_2As_2 <cit.>. 
  
  
  
  
  
  The static path approximation (SPA) technique was used at a fixed chemical potential and intermediate 
  interactions to address the behavior of two-dimensional population imbalanced d-wave 
  superconductor, in detail <cit.>. The corresponding phase diagram as shown in Figure 10 
  was mapped out in the magnetization-temperature (m-T) as well as in the field-temperature 
  (h-T) plane. It was demonstrated 
  that the low field (magnetization) regime comprises of a “quantum breached pair” (QBP) phase, 
  characterized by spatial coexistence of “gapless” d-wave superconductivity and non zero 
  magnetization. This was the first numerical realization of the coexistent phase predicted by 
  Yang and Sondhi, in a lattice fermion model <cit.>. The QBP phase can be understood as the T=0 counterpart 
  of the finite temperature BP phase realized in the population imbalanced s-wave superconductors/superfluids. 
  It was observed that increase in field (magnetization) leads to a Lifshitz transition from the uniform 
  d-wave superconductor to the QBP phase, quantified by the average magnetization of the system. 
  Both the uniform d-wave superconductor and QBP phases 
  undergo second order thermal phase transition to the PPFL phase. The high magnetic field (magnetization) 
  regime hosts FFLO state at low temperatures, which undergoes first order thermal phase transition to PPFL. 
  Across the quantum critical point corresponding to the transition between the QBP and FFLO phases the 
  thermal phase transition was found to be of first order, in agreement with the earlier predictions <cit.>. 
  Inclusion of fluctuations showed significant suppression of the superconducting transition temperature, both for 
  the uniform superconducting as well as FFLO regimes. 
  
  Based on the real space signatures shown in Figure 11 and Figure 12 it,  was inferred that both field and temperature 
  leads to spatial inhomogeneity in the superconducting state, which shows up in the quasiparticle signatures 
  as the “pseudogap” behavior. The phase diagram thus comprises of an additional thermal 
  scale T_pg as shown in Figure 10, where short range superconducting pair correlations die out, long 
  after the loss of long range superconducting phase coherence at T_c <cit.>.  
  
  
  
   §.§ Mass imbalanced s-wave superfluids
  
  
  In ultracold atomic gases pairing between equal populations of two atomic species with unequal atomic 
  masses (Fermi-Fermi mixture) can give rise to mass imbalanced Fermi superfluid. Experimentally a mass 
  imbalanced Fermi-Fermi mixture can be achieved in ^6Li-^40K mixtures. The theoretical studies 
  are primarily restricted to continuum models, using both mean field and beyond mean field techniques
  <cit.>. 
  The cumulative outcome of these works brought forth two important observations, (i) imbalance 
  in mass promotes the instability of the system towards a supersolid phase accompanied by a Lifshitz 
  transition, (ii) the mass imbalanced Fermi-Fermi mixture contains more than one pseudogap scales at 
  high temperature. 
  
  Works on lattice fermion models of mass imbalanced Fermi-Fermi mixtures are very limited. 
  Wang et al. <cit.> studied this problem in one dimension using time-evolving block 
  decimation technique and found that increasing mass imbalance ratio shrinks the FFLO regime of the 
  phase diagram and at large imbalance the FFLO phase disappears. 
  
  
  
  
  
  Using density matrix renormalization group (DMRG) and quantum Monte Carlo (QMC) techniques Dalmonte et al. 
  <cit.> showed that mass imbalance favors paired phases as compared to population imbalance,  and 
  concluded that in mass imbalanced systems it might be possible to detect FFLO in experimentally realizable 
  parameter regimes. A mean field study on two-dimensional square lattice by Pahl and Koinov revealed 
  that the ground state phase diagram of this system comprises of normal, FF and BP regimes and that a majority of 
  heavy species promotes the FF phase <cit.>. A non perturbative lattice Monte Carlo study was carried 
  out in two-dimensions, which revealed that a mean field approach to the problem grossly overestimates 
  the ground state energy and therefore makes the stability of the phases questionable <cit.>.  
  
  The only beyond mean field study of mass imbalanced Fermi-Fermi mixture in two-dimensions, at 
  and above the ground state was carried out using the SPA <cit.>. Within the 
  framework of attractive Hubbard model the imbalance in mass was taken into account through imbalance 
  in the hopping amplitude of the two species as, t_H/t_L=m_L/m_H=η. Here, H and L 
  corresponds to the heavy and light fermion species, respectively and η serves as the tuning parameter to 
  control the ratio of imbalance. The work demonstrated that imbalance in mass significantly alters 
  the dispersion spectra of the fermionic species and gives rise to subgap and supergap 
  features in the species resolved single particle DOS, as shown in Figure 13. 
  
  
  
  
  
  At a fixed imbalance of population, the phase diagram comprises of LO, BP and unstable (phase separated) 
  regimes in the imbalance-temperature plane and a larger imbalance in mass reduces the regime of stability 
  of these phases. The thermal phase diagram in the imbalance-temperature 
  (η-T)-plane presented in Figure 14 shows the intriguing feature of two pseudogap phases as PG-I and 
  PG-II, in agreement 
  with the observation of continuum models <cit.>. Owing to the mismatch in mass and 
  thus of the hopping amplitudes the two species are subjected different “scaled” temperatures. Consequently, 
  while in the pseudogap regime PG-I both the fermion species are pseudogapped, in the pseudogap regime PG-II 
  it is only the lighter 
  species which is pseudogapped while the heavier species is a polarized Fermi liquid. At η=1 corresponding 
  to zero imbalance in mass,  both the pseudogap scales collapses to one. The phase diagram shows that,  while 
  akin to the population imbalanced system the T_c is strongly suppressed by imbalance in mass, the short 
  range pair correlations survive over a large temperature regime of T ∼ 2T_c <cit.>.
  
  
  
  
  
  The work further discussed the BCS-BEC crossover picture for mass imbalanced Fermi-Fermi mixture at a fixed 
  population imbalance and established the presence of two pseudogap regimes across the crossover. The 
  BCS-BEC crossover revealed a crucial observation. It showed that a critical interaction strength (U_c) 
  is required to realize superfluid order in mass imbalanced Fermi-Fermi mixture in presence of a population imbalance. 
  This observation is in striking contrast with those of two component balanced Fermi gases where superfluid 
  phase is realized for any arbitrary strength of attractive interaction. Figure 15 presents the BCS-BEC crossover 
  of the Fermi-Fermi mixture, which shows that the superfluid T_c collapses to zero at U∼ 3t_L for 
  the parameter regime mentioned in the figure caption. 
  
  A similar study was carried using SPA by Karmakar <cit.>, for Fermi-Fermi 
  mixtures with non local d-wave interaction between the fermion species. In the context of solid state 
  systems, materials with unequal masses of the fermions can be envisaged as different fermion species 
  belonging to different electronic bands. The thermal phase diagram of this system was found to be similar 
  to the one obtained for a mass imbalanced s-wave Fermi gas, with two finite temperature pseudogap regimes 
  even for a population balanced system.
  The ground state calculations showed that over a large regime of mass imbalance the QBP phase is 
  stablized with coexisting gapless d-wave superfluidity and non zero magnetization, which undergoes 
  Lifshitz transition to an uniform d-wave superfluid phase, with decreasing imbalance in mass. 
  
  
  
  § IMBALANCE AND SPIN-ORBIT COUPLING IN FERMI SYSTEMS
  
  
  Over the past few years there has been immense progress both in the experimental and theoretical 
  front to understand the spin-orbit coupled ultracold atomic gases. Since the ultracold atomic gases are 
  one of the prospective systems to realize exotic superfluid phases such as,  FFLO and BP, there has 
  been obvious interest to investigate the effect of spin-orbit coupling on imbalanced Fermi gases. 
  In this section we will summarize some of the major developments in this area and the observations 
  therein. We will also touch upon some of the open problems in this area towards the end of this section. 
  For further details on spin orbit coupled Fermi gases the readers are encouraged to consult the excellent 
  reviews by Galitski and Spielman <cit.> and Dalibard et al. <cit.>.
  The subject is also discussed in a recent review by Kinnunen et al. <cit.>.  
  
  Spin orbit coupling has remained a subject of interest since the early days of quantum mechanics and 
  can be understood as follows. If an electron is allowed to move in an electric field and we 
  move the electron's rest frame, this will lead to the generation of a magnetic field by the associated 
  Lorentz transformation. This magnetic field will couple itself to 
  the electron spin leading to spin-orbit coupling. The exact form of the spin-orbit coupling depends upon 
  the direction of the electric field and the electron velocity; one of the commonly used form of the 
  spin-orbit coupling is the Rashba spin-orbit coupling (RSOC), given as,
  
      H_RSOC    =    α(k_yσ̂_x - k_xσ̂_y)
   
  where, α is the strength of the spin-orbit coupling interaction, σ̂_x and σ̂_y 
  are the Pauli matrices. In the absence of inversion symmetry the Rashba spin-orbit coupling takes the form 
  of Dresselhaus spin-orbit coupling and is given as,
  
      H_DSOC    =    α(k_yσ̂_x + k_xσ̂_y)
  
  In real space the Rashba spin-orbit coupling is given as, 
  
      H_RSOC    =     -∑_⟨ ij⟩, σt_ij,αĉ_i,σ^†{σ·α_ij}_σ,σ^'ĉ_j,σ^'
  
  where, σ is the vector of Pauli matrices and α_ij = αẑ× r_ij
  is the spin-orbit vector on each bond of a two-dimensional square lattice. t_ij,α correspond 
  to the strength of the spin-orbit coupling.
  
  In ultracold atomic gas experiments spin-orbit coupling is generated using synthetic electric field <cit.>
  and pseudospins of the atomic species. The basic principle that is followed in such experiments is to couple 
  different internal energy states of atoms through a two-photon Raman transition. The atom first absorbs 
  the photon from one laser beam then emits the photon to a second laser beam, by stimulated emission. 
  The process conserves momentum by imparting some of the momentum to the atom. The Doppler shift between 
  the laser beams and the atom gives rise to the momentum dependent spin-orbit coupling <cit.>. 
  Spin-orbit coupling in ultracold atomic gases was experimentally realised for bosons <cit.> 
  as well as for fermions <cit.>. A detail discussion on the experimental realization 
  of spin-orbit coupling in ultracold atomic gases can be found in 
  <cit.>. 
  
  Theoretically, the effect of spin-orbit coupling on the FFLO superfluid state has largely been investigated 
  using mean field theory. This has been carried out either by determining the ground state energy using 
  the FF ansatz in presence of spin-orbit coupling <cit.>, or by solving the self consistent BdG 
  equations for arbitrary 
  choice of spatially inhomogeneous superfluid order parameter, so as to determine the global minima of the 
  energy landscape <cit.>. Obviously, the second approach is computationally 
  more demanding than the first one but allows for a larger family of possible ground state solutions. 
  
  Based on BdG mean field theory in two-dimensions, Xu et al. showed that in a continuum system in 
  presence of spin-orbit coupling the FF state is energetically favoured over 
  the LO state, particularly at strong spin-orbit couplings <cit.>. This observation was reconfirmed 
  by several other works.  A continuum system in three-dimensions with spin-orbit coupling and in presence 
  of in-plane Zeeman field was studied by Zheng et al. <cit.>. They showed that the FF state 
  is stable over a large parameter regime.      
  In a similar spirit, Hu and Liu pointed out that at sufficiently large spin-orbit coupling the superfluid 
  state is always an FF state <cit.>. They worked out the finite temperature phase diagram at 
  a broad Feshbach resonance and predicted a superfluid transition temperature of T_c∼ 0.2T_F, 
  where, T_F is the Fermi temperature. Similar estimates were made by these authors when they extended 
  their model by including out of plane Zeeman field <cit.>. 
  
  Beyond mean field corrections were incorporated for the continuum model of three dimensional spin-orbit 
  coupled Fermi gas by Liu <cit.>. Once again the superfluid transition temperature was found to be  
  upto ∼ 0.2T_F. Similar conclusions were drawn by Dong et al. <cit.>
  and Zhou et al. <cit.> for spin-orbit coupled three-dimensional Fermi gases.
  
  Within the framework of lattice fermion models, Iskin investigated the spin-orbit coupled Fermi gas in 
  presence of an optical trap in two dimensions, using BdG mean field theory <cit.>. 
  He showed that in presence of spin-orbit coupling and in-plane Zeeman field the FF state is stabilized,  
  in agreement with the observations of continuum model. By investigating the finite temperature behavior 
  Iskin suggested that since the modulations of the superfluid order parameter are restricted near the edge 
  of the trap, it would be difficult to detect them at high temperatures <cit.>. 
  
  The topological aspect of spin-orbit coupled Fermi gases was investigated by Tewari et al. <cit.>, 
  who showed that spin-orbit coupling along with Zeeman field can be used to drive a phase transition between a 
  topologically trivial and a topologically non-trivial superfluid. Similar observations were reported 
  for imbalanced Fermi gases in presence of both spin-orbit coupling and Zeeman field <cit.>. 
  In their studies carried out on two dimensional continuum system Zhang et al. <cit.> and 
  Qu et al. <cit.> identified a topological FF state in addition to gapped FF and gapless 
  nodal FF states. They further confirmed the existence of chiral edge states of the topological FF state, 
  on a two-dimensional square lattice <cit.>. 
  Observation of gapless topological FF states was also reported by Cao et al. <cit.>. 
  They further confirmed the robustness of the Majorna modes by 
  adding disorder potential to the system. Presence of topological 
  FF phase has also been confirmed in one-dimensional lattices <cit.>. 
  
  Mean field theory has been pushed to investigate the finite temperature behavior of spin-orbit coupled 
  imbalanced Fermi gases, as well <cit.>. It was inferred that together with in-plane and 
  out of plane Zeeman field, spin orbit coupling gives rise to BKT transition in two dimensions, both for 
  gapped as well as gapless FF phases <cit.>.
  
  In a recent work, Wang et al. investigated the spin-orbit coupled Fermi gas within the framework 
  of continuum model in two and three dimensions <cit.>. They took into account the effect of pairing field 
  fluctuations and studied the system using two schemes, viz. (i) T-matrix approximation and 
  (ii) fluctuation exchange approximation (FLEX). Based on their observations from both these approaches 
  Wang et al. inferred that a finite temperature realization of FFLO phase is not possible in this system. 
  They remarked that the FFLO phase is unstable against pair fluctuations in continuum in both two and three 
  dimensions <cit.>. Within the purview of lattice fermion models, an auxiliary field QMC study was 
  carried out for balanced two-component Fermi gas with spin-orbit coupling at the ground state, by 
  Rosenberg et al <cit.>. It was shown that such a system comprises of exotic phases 
  such as, supersolid phase containing both singlet and triplet pairings. Rosenberg et al. showed 
  the presence of edge currents arising out of the spin-orbit coupling. So far no such investigation has 
  been carried out in the context of spin-orbit coupled imbalanced Fermi gases.   
  
  As is evident from the above discussions, the theoretical works on spin-orbit coupled imbalanced Fermi 
  gases are largely the ones based on mean field theory. This leaves a void in our understanding as far 
  as the effects of fluctuations are concerned. For example, the exact thermal scales of the spin-orbit 
  coupled imbalanced Fermi gas in an optical lattice is so far unknown. The presence of lattice breaks 
  the translational symmetry and can potentially stabilize phases which are unstable in continuum. It 
  is worth investigating whether a modulated superfluid phase can be stabilized in a lattice with spin 
  orbit coupling in presence of pair fluctuations. Moreover, spin-orbit coupling allows for the possibility of 
  realizing triplet superfluid pairing. Most of the works we discussed here do not take this possibility 
  into account. Even if a triplet pairing state does not lead to a long range order at the 
  ground state, it is certainly possible that triplet pair correlations are present at high temperatures. 
  In the same spirit, with the energy of the FF and LO phases being comparable,  even though the ground 
  state hosts FF superfluid phase in presence of spin-orbit coupling, competing LO correlations 
  can be present at finite temperatures. Signatures of the presence of such competing correlations would be borne out 
  in the quasiparticle properties of the system. Investigations of such effects require one to include 
  fluctuations beyond the mean field theory, and are certainly worth pursuing.  
  
  
  
  § CONCLUSIONS AND OUTLOOK
  
  
  The last few years have witnessed a great deal of activity in the theoretical as well as experimental 
  front to unveil the mysteries of the exotic superconducting phases like FFLO and BP. Though 
  the possibility of superconducting phases with finite centre of mass momentum of the Cooper pairs was 
  predicted decades ago, it was the advent of ultracold atomic gases, and significant advancement of 
  material science experiments, that has paved the way to access and understand such novel quantum many 
  body phases. 
  
  Amongst the solid state materials, so far three candidate materials have stood out as possible hosts of 
  FFLO phase viz., heavy fermion superconductor CeCoIn_5, organic superconductor 
  κ-(BEDT-TTF)_2Cu(NCS)_2 and iron superconductor KFe_2As_2. With the continuous 
  development that is going on in material science, the hope is that other materials would join the class 
  of Pauli limited superconductors in not so distant future. The ultracold atomic gas systems with the unprecedented 
  control of the tuning parameters, that it provides, have taken the centre stage to engineer and analyze 
  exotic quantum phases, such as FFLO. While a clear spatial signature of the modulated superfluid phase 
  is yet to be realized in these systems, coexistence of superfluid pair correlations and unpaired fermions 
  are indeed observed, suggesting a possible BP phase. 
  
  The theoretical attempts are not far behind either, and the problem of imbalanced Fermi systems have  
  been addressed by mean field as well as non mean field approaches. We now have a broad understanding 
  of not just the ground state but also of how the effects of fluctuations play out and dictate the thermal 
  scales associated with these systems. 
  
  The world of quantum many body physics is however a fast evolving one and newer experimental breakthroughs 
  are not infrequent. For example, experimental realization of spin-orbit coupling in fermionic and bosonic 
  systems in ultracold atomic gas setups have been reported recently. In the theoretical front however, we 
  have not moved beyond the realms of mean field theory so far, except occasionally. It is obvious that 
  even for the population balanced scenario, these spin-orbit coupled bosonic and fermionic systems are theoretically 
  not well understood at this moment and a concerted effort is required to achieve the same.    
  In the same spirit, the effect of disorder adds another interesting dimension to these systems. While 
  disorder is largely expected to be detrimental for the FFLO phase, it can possibly aid in to stabilize 
  the BP phase, at the ground state of a s-wave superfluid. 
  
  This chapter has covered some of the important theoretical and experimental advances and 
  observations pertaining to imbalanced Fermi systems, and is no way comprehensive. The subject is not yet 
  well understood and promises a plethora of avenues and areas which needs to be explored inquisitively. 
  
  
  unsrt
  
  
  