FFLO correlation and free fluids in the one-dimensional attractive Hubbard model
Song Cheng Yi-Cong Yu Murray T Batchelor Xi-Wen Guan

























    GState Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics,
        Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, ChinaUniversity of Chinese Academy of Sciences, Beijing 100049, China.Department of Theoretical Physics, Research School of Physics and Engineering,
        Australian National University, Canberra ACT 0200, AustraliaState Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics,
        Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, ChinaUniversity of Chinese Academy of Sciences, Beijing 100049, China.Centre for Modern Physics, Chongqing University, Chongqing 400044, ChinaDepartment of Theoretical Physics,
        Research School of Physics and Engineering,
        Australian National University, Canberra ACT 0200, AustraliaMathematical Sciences Institute, Australian
        National University, Canberra ACT 0200, Australia[]xiwen.guan@anu.edu.auState Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics,
        Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, ChinaDepartment of Theoretical Physics,
        Research School of Physics and Engineering,
        Australian National University, Canberra ACT 0200, AustraliaCenter for Cold Atom Physics, Chinese Academy of Sciences, Wuhan 430071, China71.10.Fd, 75.40.Cx,02.30.Ik
         In this Rapid Communication  we show  that low energy macroscopic properties of the one-dimensional (1D)
         attractive Hubbard model exhibit two fluids of bound pairs and of unpaired fermions.
         
         Using the thermodynamic  Bethe ansatz equations  of the model, we  first  determine the low temperature phase diagram  and analytically calculate
         the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) pairing correlation function for  the partially-polarized phase.
         
        We  then  show  that for such a FFLO-like  state  in the low density regime the  effective chemical potentials of bound pairs and unpaired fermions
        behave like  two free fluids.
        
         Consequently, the susceptibility, compressibility and specific heat  obey  simple  additivity rules, indicating
         the `free' particle  nature of interacting fermions on a 1D lattice.
         
         In contrast to the  continuum Fermi gases,   the correlation critical exponents and  thermodynamics of the
         attractive Hubbard  model  essentially depend on two lattice  interacting parameters.
         
        Finally, we  study scaling functions, the Wilson ratio  and susceptibility  which  provide universal  macroscopic
        properties and dimensionless constants   of interacting fermions at low energy.
        
        
        
         FFLO correlation  and  free  fluids  in the one-dimensional  attractive Hubbard model
        
            Xi-Wen Guan
            December 30, 2023
        =======================================================================================
        
        
        The notion of Landau  quasiparticles  gives rise to the  Fermi liquid  theory  successfully used for describing
        properties of a large variety of systems, such as Fermi liquid ^3He and electrons in metals <cit.>.
        In contrast, it is generally accepted that Fermi liquid theory is not applicable in 1D,
        where  the description of the low-energy physics of strongly correlated electrons, spins, bosonic and fermionic atoms relies
        on the  Tomonaga-Luttinger liquid (TLL) theory <cit.>.
        Such an understanding of the TLL  in 1D is based on collective
        excitations which are significantly different from  Landau  quasiparticles in higher dimensions.
        However, concerning macroscopic properties, there are many universal properties/quantities
        which are common for both 2D/3D and 1D systems <cit.>.
        
        The  1D repulsive Fermi-Hubbard model describing interacting fermions on a lattice provides a paradigm for understanding many-body physics,
        including  spin-charge separation, fractional excitations, quantum dynamics of spinons,  a Mott insulating  phase and magnetism <cit.>.
        Very recently, ultracold atoms trapped in   optical lattices <cit.>
        offer promising opportunities  to test such fundamental concepts <cit.>.
        In contrast, the 1D attractive Fermi-Hubbard model <cit.>
        is a notoriously  difficult problem due to the complicated  bound states of multi-particles and multi-spins  on lattices.
        Despite there being a mapping by Shiba transformations between the repulsive and attractive regions of the Hubbard model <cit.>,
         such a mapping cannot be used for a study of  the low energy themodynamics  of the attractive Hubbard model
         due to the different  cut-off  processes in terms of  such multi-spin and multi-charge bound states.
        Of central importance to  this attractive Hubbard model  is the understanding  of quantum  correlations of charge  bound states,  for example,
        the  Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) like  pairing <cit.>  on a 1D lattice <cit.>.
        
        In the expansion dynamics of the FFLO state   in 1D <cit.>,  a nature of two fluids of bound pairs and  free fermions was indicated. 
        
        In this Rapid Communication, building on the thermodynamic Bethe ansatz (TBA) equations  of the attractive Hubbard model,
        we  analytically obtain the  FFLO pairing correlation and  the universal two free quantum fluids of the FFLO-like state,
        where the lattice effects are seen to drive the system differently to the continuous Fermi gas <cit.>, see Fig. <ref>.
        More detailed  studies of  this  model will be  presented  elsewhere <cit.>.
        
         The  Bethe ansatz  solution. The 1D single  band Hubbard model  is described by the Hamiltonian <cit.>
            H=   - ∑^L_j=1, a=↑,↓( c_j,a^† c_j+1,a +  h.c.)  
               + u ∑_j=1^L (2n_j,↑-1) (2 n_j,↓-1),
        
        where c_j,a^† and c_j,a are the creation and annihilation operators of electrons (fermionic atoms) with
        spin a (internal degrees of freedom)  (a=↑ or a=↓) at site j on  a 1D  lattice with length L.
        
        They satisfy the anticommutation relations { c_j, a, c_k,b}= { c_j, a^† , c_k,b^†}=0 and
        { c_j, a, c_k,b^†}=δ_jkδ_ab.
        
        Meanwhile n_j,a=c_j,a^† c_j,a is the density operator, n_e=1/L∑_j=1^L∑_a n_j,a is the total fermion number per lattice  site
        and u is the dimensionless interaction strength between particles (u>0 for  repulsion and u<0 for  attraction).
        
        In 1968  Lieb and Wu  <cit.> derived  the Bethe ansatz (BA) equations for  the 1D Hubbard  model by means  of Bethe's hypothesis <cit.>.
        
        
         
         
        
        
        Takahashi <cit.> discovered the solutions of the BA equations which in general are
        classified as real quasimomenta k, k-Λ strings and complex spin rapidities of  Λ strings, see <cit.>.
        These  roots respectively count for the quasimomenta of the single fermions, bound states of different lengths of fermions and
        bound states of magnons with different lengths. At  high energy or momentum, such bound states can  coexist.
        Building on Takahashi's string hypothesis, we obtain the TBA equations for the 1D attractive Hubbard model <cit.>
            ε(k)   =    g_0(k)-∑_n=1^∞ a_n ∗( F[ε^'_n]-F[ε_n])(k),
            ε_n(Λ)   =   2nB-a_n^t ∗ F[ε](Λ)
             -∑_m=1^∞ A_n m∗ F[ε_m](Λ), 
            ε_n^'(Λ)   =   g_n(Λ) -a_n^t ∗ F[ε](Λ)  -∑_m=1^∞ A_n m∗ F[ε^'_m](Λ)
        with  the notation F[x](y)=-Tln[1+exp(-x(y)/T)] and n=1,…,∞.
        The kernel function a_n(x)=1/2π2n|u|/(n|u|)^2+x^2.
        The driving terms  are g_0(y)= -2cos y-μ-2u-B and g_n(y) =-4√(1-(y+i  n   |u|)^2)-n(2μ+4u).
        In the above equations we denoted the convolutions
        a_n ∗ F[x] (k) = ∫_-∞^∞y   a_n( k-y) F[x(y)] and a_n^t ∗ F[x](Λ)  =∫_-π^π  y  cos y  a_n(sin y-Λ) F[x(y)].
        The functions ε,  ε_m^' and ε_n  stand for the dressed energies  for unpaired fermions,
        bound states of 2m fermions (the k-Λ strings) and   length-n spin strings of magnons, respectively.
        The function A_nm(x) is given in  <cit.>.
        
        It is particularly important to observe that the  longer k-Λ strings are  involved in the thermodynamics  as  temperature increases <cit.>.
        The free energy per site is thus given by
        
            f=u+∫_-π^πk/2π F[ε](k)
            +∑_n=1^∞∫_-∞^∞Λ/2πξ_n(Λ) F[ε_n^'](Λ)
        
        with ξ_n(Λ)=∫_-π^π k
          a_n(Λ-sin k). We also observe that in the dilute limit, u→ 0, n_e→ 0 with n_e/|u| constant <cit.>,
        the TBA equations (<ref>)-(<ref>) reduce to those of  the Gaudin-Yang model <cit.>.
        
        We note that the Shiba transformation between the repulsive and attractive regions of the Hubbard model does not help to obtain universal low energy physics from the TBA equations.
        
        This is mainly because the cut-off processes regarding the above spin and  charge bound states are quite different <cit.>, unlike the case of  the ground state <cit.>.
        
        As we shall see, in the attractive regime, the low energy physics of the model
         is no longer described by the spin-charge separated theory, rather it is described by the FFLO-like quantum liquids of pairs and single fermions.
        
        
         Quantum phase diagram and Wilson ratio.
         In contrast to the repulsive case, the ground state  of the attractive Hubbard model has  charge bound states, i.e.,
         length-1k-Λ strings,  forming  a lattice version of the FFLO state.
         The quantum phases  and phase diagram at T=0  can be directly determined   from the  TBA equations   (<ref>)-(<ref>) in the limit T→ 0, which are  called the dressed energy equations <cit.>.
        
        
        
        
        
        
        The dressed energy equations  determine  five quantum phases in the μ-B plane: vacuum I, fully-polarized phase II,
        half-filled phase III, FFLO-like state IV and fully-paired state V, see Fig. <ref>.
        
        
        
        The zero temperature  phase boundaries can also be  determined by  the Shiba transformation <cit.>.
        
        Here we show that the Wilson ratio, namely, the dimensionless ratio of the  compressibility κ and the specific heat divided by the temperature T,
        
            R_W^κ=π^2 k_B^2/3κ/C_v/T,
        
        provides a convenient way for revealing the full phase diagram at low temperatures, see Fig. <ref>.
        In the above k_B is Boltzmann's constant.
        This  ratio can be directly calculated from the finite temperature TBA equations   (<ref>)-(<ref>) with a suitable spin and charge bound state cut-off  process,  see <cit.>.
        
        
        We find that the ratio R_W^κ  is capable of distinguishing all phases of quantum states,
        including the FFLO-like state in the phase diagram  Fig. <ref>.
        We observe that an enhancement of this ratio  occurs  near a phase transition.
        It gives a finite value  at the critical point  unlike the divergent values of compressibility and  susceptibility for T→ 0.
        
         Indeed, the phase boundaries determined  by the Wilson ratio  (<ref>) coincide with the ones determined by  the
         dressed energy equations  at T=0.
        
        
         The phases IV  and V  in  Fig. <ref>  reveal  significant features, namely  the quasi-long range order  and free-fermion quantum criticality.
         
         A constant Wilson ratio implies that
        the two types of fluctuations are on an equal footing, regardless of the
        microscopic details of the underlying many-body systems.
         
         Regarding the sudden change of the Wilson ratio near a phase transition,  we observe that  the  particle number and energy
         fluctuations become temperature dependent, see Fig. <ref>(a).
          At the critical point,  the vanishing of the Fermi points, i.e., ε_1(0)=0 and ε_1^' (0)=0,
          in the Fermi sea of pairs and of unpaired fermions leads to a universality class of quantum criticality.
          In the critical regime, the scaling functions of thermodynamic  properties can be cast into  universal forms.
          From the TBA equations   (<ref>)-(<ref>),  we obtain the scaling functions of compressibility and susceptibility
        
            κ(μ,B,T)   =   κ_0(μ,B)+T^d/z+1-2/ν zλ_κℱ( μ-μ_c/T^1/ν z) , 
            χ(μ,B,T)   =   χ_0(μ,B)+T^d/z+1-2/ν zλ_χ𝒦( μ-μ_c/T^1/ν z)  .
        
        Here  the scaling functions ℱ(x)=𝒦(x)= Li_-1/2(x) indicates a free-fermion criticality classified by the
        dynamical critical exponents z=2 and correlation critical exponent
        ν=1/2, see <cit.>.
        The  terms κ_0 and χ_0 are the regular part and the factors λ _κ,χ are phase dependent constants.
        Fig. <ref>(b) and  Fig. <ref>(c) show such universal scaling behaviour of the susceptibility and compressibility across the phase boundaries (V,IV).
        Similar scaling invariant behaviour occurs whenever the model parameters are driven across the phase boundaries in Fig. <ref>.
        
        
        
          FFLO correlation.
        For the fully paired state V,  the pairing correlation length is larger than the average interparticle  spacing.
         In this phase, the single particle Green's function decays exponentially, whereas  the singlet pair correlation function decays as a power of
        distance <cit.>.
        However,  once the external field exceeds the critical line between phases IV and V, the Cooper pairs start to break apart. Thus both of these
        correlation functions decay as a power of distance, indicating a quasi-long range correlation.  In the phase IV, Cooper pairs  and excess fermions form a
        1D analogue of the  FFLO pairing-like state <cit.>.
        However,  analytical result for  the FFLO pairing correlations for the Hubbard model  is still lacking.
         For obtaining a universal  form of   the FFLO-like correlation function, we  first  focus on the  case of  low density n_e≪ 1 and low energy.
        In the FFLO-like phase IV, the spin wave bound states ferromagnetically couple to  the Fermi sea of  the unpaired fermions.
        Thus   the spin wave fluctuations can be ignored at low temperatures due to this ferromagnetic nature.
        Then we simplify the TBA equations  (<ref>)-(<ref>)  as  <cit.>
            ε(k)   ≈   k^2-μ_1
            -a_1 ∗ F[ε^'_1](k),
            ε_1^'(Λ)   ≈   α_1 ( Λ^2- μ_2)-a_1 ∗ F[ε](Λ)
            - a_2 ∗ F[ε^'_1](Λ).
        
        The free energy (<ref>) reduces to
        f≈ u+∫_-π^πk/2π F[ε](k)
        +∫_-∞^∞Λ/2πβ_1 F[ε_1^'](Λ)<cit.>.
        
        In this new set of TBA equations (<ref>) and (<ref>)  we have introduced two effective chemical potentials
        
            μ_1   =   μ-2|u|+B+2,
            μ_2   =   1/α_1 [2μ+
            4(√(u^2+1)-|u|)],
        
        for understanding the FFLO correlation and  free-fermion nature of  the attractive Hubbard model.
        In the above equations the  parameters α_n  and β_n reflect the interacting effect of the length-nk-Λ bound states on a lattice. They are given by
        
            α_n    =   ∫_-π^π k  cos^2 k a_n(sin k)    2|u| cos^2 k (n^2u^2-3 sin^2 k)/(n^2u^2+sin^2 k)^3,
            β_n   =   ∫_-π^π k   a_n(sin k).
        
        At low energy physics only length-1k-Λ strings  are involved.
        In this region, the lattice parameters α_1 and β_1  approach 2 when u tends to zero.
        However, for large |u|,  the band of  pairs becomes flat <cit.>.  The TBA equations (<ref>) and (<ref>)
        are reminiscent  of the `feedback interaction' equation in the  Landau Fermi liquid theory <cit.>.
        The driving term in  (<ref>)  can be expressed  as
        ħ^2/2mα_1(k^2-μ_2)=p^2_0/2α_1 m-ħ^2 /2mα_1 μ_2 with 2m=ħ=1,
        which  is the first-order coefficient describing the excitation energy of a single bound pair.
        The lattice parameter  α_n characterizes  the effective mass  of length-nk-Λ strings (bound state of 2n atoms on a lattice).
        
        
        
        
        
        In light  of the conformal field theory approach  <cit.>
        and using the TBA equations  (<ref>) and (<ref>),
        we calculate  the  asymptotic form  of  the FFLO  correlation function of the attractive Hubbard model in  the low density region <cit.>
            G_p(x,t)   =   ⟨Ψ_↑^†(x,t) Ψ_↓^†(x,t) Ψ_↑(0,0) Ψ_↓(0,0) ⟩
               ≈    A_p,1cos( π(n_↑-n_↓)x )/|x+ i  v_u   t|^2θ_1 |x+i  v_b   t|^2θ_2
                  + A_p,2cos( π(n_↑-3n_↓)x )/|x+i  v_u   t|^2θ_3 |x+i  v_b   t|^2θ_4,
        
        with the exponents θ_1 ≈ 1/2,   θ_2 ≈ 1/2+n_2/|u|β_1,  θ_3 ≈1/2-4  n_2/|u|β_1 and
        θ_4 ≈5/2-4 n_1/|u|-3  n_2/|u|β_1. Here n_2,1=N_2,1/L are  the dimensionless  densities  of pairs and  unpaired fermions, respectively.
        The sound velocities are given by v_b=√(α_1 )/β_1π n_2(1+1/|u|β_1 (2n_1+n_2)) and v_u=√(2)π n_1 (1+4/|u|n_2 ).
        
        In the above equation the coefficients A_p,1 and A_p,2 are  constant factors. In this phase IV
        the spatial oscillation in  the pairing correlation is a characteristic  of the FFLO state, where the
        imbalance n_↑-n_↓ in the densities of spin-up and spin-down fermions gives rise to a mismatch in
        Fermi surfaces between both species of fermions.
        In 1D,  the spatial oscillation signature in pair correlation is  a consequence of  the  backscattering for bound pairs and unpaired fermions, see also the results for the
        Gaudin-Yang model <cit.>.
        Here we observe that the critical exponent θ_2 depends essentially on the lattice  parameter β_1.
        So do the critical exponents in other types of correlation functions <cit.>.
        The Fourier transform of  G_P(x,0^+) gives
        G̃_p(k) ∼[ (k-π(n_↑-n_↓) ) ]^2s_p|k-π(n_↑-n_↓)|^ν_p
        with 2s_P≈ 0 and ν_p ≈ n_2/(|u|β_1).
        
        
         Two free fluids  and spin gapped phase.
        At low temperatures, we find  a significant  nature of two fluids in phase IV.
        For the ground state, the energy can be regarded as two  TLLs  of unpaired fermions and of pairs due to the quasi-long range correlation.
        Without losing generality,  we consider a physical regime  of  low density (n_e small),  low temperature  and finite strong magnetic field.
        This region is reachable in cold atoms <cit.>.
        In this regime, the chemical potentials for the unpaired fermions and pairs are given explicitly by
        
            μ_1   =   π n_1^2A_1^2+4π^2α_1/3β_1^3|u|n_2^3A_2^3, 
            μ_2   =   π^2n_2^2/β_1^2A_2^2+4π^2/3α_1|u|n_1^3 A_1^3 + 2π^2/3β_1^3|u|n_2^3 A_2^3 ,
        
        where    A_1=1+2n_2/|u|+(2n_2/|u|)^2 and A_2=1+2n_1+n_2/β_1 |u|+(2n_1+n_2/β_1 |u|)^2  indicate  interacting effects among pairs and unpaired fermions like that of the Fermi gas  <cit.>.
        
        
        
        The effective chemical potential μ_2 in  the 2D interacting Fermi gases  shows
        a  crossover from a Bose-Einstein condensate to a Bardeen-Cooper-Schrieffer superconductor in ultracold fermions <cit.>.
        
        Moreover, from the relations (<ref>) we   demonstrate    the free-particle nature of two fluids  through  the  additivity rules in compressibility and susceptibility:
        
            κ=κ_1+2/α_1 κ_2,     1/χ=1/χ_1 +α_1 /21/χ_2,
        
        where κ_r=( ∂  r  n_r/∂μ_r)|_B and χ_r=( ∂ r  n_r/∂μ_r )|_n with r=1,2 for unpaired fermions and pairs, respectively.
        
        We see that  the effective binding energy e_b=-(2u+2)n_1-4(u+√(u^2+1))n_2 of a bound pair is  absorbed into the effective chemical potentials.
        
        The compressibility and susceptibility can be explicitly calculated from
        the chemical potentials (<ref>) and  (<ref>) via the  relations
        
            1/κ_1   =   J/(  ∂μ_1/∂ n_2
            -α_1/2∂μ_2/∂ n_2 ),    1/κ_2=-1/α_1J/( ∂μ_1/∂ n_1
            -α_1/2∂μ_2/∂ n_1),
            χ_1   =   1/(∂μ_1/∂ n_1
            -1/2∂μ_1/∂ n_2) ,    χ_2=-1/(∂μ_2/∂ n_1
            -1/2∂μ_2/∂ n_2) ,
        
        where the Jacobi determinant
        J=-α_1/2
        (∂μ_1/∂ n_1∂μ_2/∂ n_2
        -∂μ_2/∂ n_1∂μ_1/∂ n_2).
        The explicit forms are given in  <cit.>.
        The additivity rules in the thermodynamic properties reveal a significant free-particle feature in the phase of multiple quantum liquids on a 1D lattice.
        
        Furthermore, using the TBA equations (<ref>)-(<ref>) and the BA equations with the  length-1k-Λ strings,
        we show that the specific heat, i.e., a measure  of the energy fluctuations,
        is given by C_v=π T/3( 1/v_u + 1/v_b).
        Here the sound velocities v_b,u are as given above.
        
        A second-order  phase transition occurs when the system is driven across the phase boundary in the μ-B plane, see Fig. <ref>.
        Fig. <ref>(a) and Fig. <ref>(b)  show  the compressibility and susceptibility  vs  magnetic  field at different temperatures.
         They  are temperature independent in  phase IV, whereas the specific heat depends linearly on the temperature, having thus a common feature of the Fermi liquid in higher dimensions.
        We observe that in phase IV the compressibility and susceptibility curves at different temperatures collapse into the zero temperature ones   obeying  the additivity rules (<ref>).
         Fig. <ref>(c) shows the susceptibility vs  temperature for different magnetic  fields.
        For  B>B_c, the susceptibility  displays a flat region in the χ-T plane, the small region  to the left of  the green dashed line, indicating the two free fluids.
        For B<B_c, the susceptibility illustrates the exponential decay as temperature decreases (blue lines).
        In this case, the susceptibility is given by χ_s=T^-1/2/4√(π)e^-Δ/T with the energy gap
         
            Δ=-R^2+4 (2π-R^3/3)/3 |u| π^2( 1+ 2|u|πμ/2π-R^3/3)^3/2,
        
         indicating the behaviour of dilute magnons.
        Here we have denoted  R=Re√(μ+2u+B+2).
        
        
        In summary,  for the attractive Hubbard model, we have analytically calculated  the  FFLO pair correlation and critical exponents,  along with
        scaling functions of thermal and  magnetic properties for  which the lattice effect becomes prominent.
        We have obtained   the effective chemical potentials  of the bound pairs and of the unpaired fermions  and demonstrated  the
        additivity rules of  the susceptibility and the compressibility in the FFLO-like state.
        While we  have found  that  the susceptibility and the compressibility are temperature independent, the specific heat depends linearly on the temperature in this phase.
        These  results provide strong evidence for the existence of two free fluids of bound pairs and of unpaired fermions,
        which were predicted in expansion dynamics of the FFLO state in 1D <cit.>.
        
        
         Acknowledgments.  The authors SC and YCY contributed equally to the calculations in this paper.
        The authors  thank M. Takahashi and R. Hulet  for helpful discussions. This work is supported by
        Key NNSFC grant number 11534014, MOST grant number 2017YFA0304500,
        NNSFC grant numbers 11374331, 11174375 and ARC Discovery Projects DP130102839, DP170104934.
        
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