title: Generic transport formula for a system driven by Markovian reservoirs
authors: Tony Jin Michele Filippone Thierry Giamarchi
text: 
  
  
  
  
  
  Department of Quantum Matter Physics, Ecole de Physique University
  of Geneva, Quai Ernest-Ansermet 24, CH-1211 Geneva 4, Switzerland
  
  
  We present a generic, compact formula for the current flowing in interacting and non-interacting systems which are driven out-of-equilibrium by biased reservoirs described by Lindblad jump operators. We show that, in the limit of high temperature and chemical potential, our formula is equivalent to the well-known Meir-Wingreen formula, which describes the current flowing through a system connected to fermionic baths, therefore bridging the gap between the two formalisms.
  Our formulation gives a systematic way to address the transport properties of  correlated systems strongly driven out of equilibrium. As an illustration, we provide explicit calculations of the current in three cases : i) a single-site impurity ii) a free fermionic chain iii) a fermionic chain with loss/gain terms along the chain. In this last case, we find that  the current across the system has the same behavior for loss or gain terms and depends on the loss/gain rate in a non-monotonic way.
  
  Generic transport formula for a system driven by Markovian reservoirs
      Thierry Giamarchi
      December 30, 2023
  =====================================================================
  
  
  
  
  
  § INTRODUCTION
  
  
  The formulation by Landauer and Büttiker <cit.> of the current through mesoscopic regions underpins our understanding of  electron transport in quantum-coherent systems. It makes explicit the connection between the current and the local properties of the finite region (its transmission coefficients) and the distribution functions of connected reservoirs, and has been extremely successful to deal with transport in non-interacting systems, such as disordered systems or Fermi liquids. Moving from the transport in non-interacting systems to the understanding of strongly correlated systems  remains one of the most challenging and not yet fully achieved tasks in quantum Physics. Beyond  the well-established and practical interest in the context of transport measurements in bulk solid-state systems <cit.> and nanoscopic devices <cit.>, the recent realization of novel experimental platforms, probing stationary quantum transport in synthetic quantum matter systems  relying on circuit-QED <cit.>, quantum dot arrays <cit.> and  atomtronics <cit.>, paves the way to accessing novel and unexplored transport regimes also far away from equilibrium.
  
  One step towards the understanding of such transport properties in interacting systems, was provided by a remarkable generalization, by Meir and Wingreen <cit.> (MW), of the Landauer-Büttiker formula to the case of an interacting system.
  This generalization, expressing the current in terms of local Green's functions of the system in presence of the reservoirs, provided a unified framework in which understanding the transport was akin to finding approximate (or exact) ways of computing such
  Green's function of the system in presence of two fermionic reservoirs (see Fig. <ref>). Indeed in such class of systems a stationary current is usually generated by letting the system exchange particles at different rates with two (or multiple) reservoirs.
  
  
  An alternative way to view the coupling of a quantum system to the external world, used in particular routinely in the context of quantum optics <cit.>, is to describe the evolution of the system by
  Lindblad-type generators <cit.>. Such description consists of Markovian processes by which the system has a non-unitary evolution due to some coupling to the external world. The Lindblad description where the operators would either
  inject or absorb particles could thus replace the coupling to external fermionic reservoirs in order to generate a steady state current through a quantum system, as depicted in Fig. <ref>. The Lindblad evolution, fully Markovian, is a priori
  simpler, although of course not equivalent to the fermionic reservoirs, and as such has been widely used coupled with a Liouvillian formalism <cit.>,
  to tackle out of equilibrium issues. Beyond experimental interest, for which the Lindblad coupling is the proper microscopic description, on the theory side, this approach has
  allowed to unveil  non-trivial properties of highly excited and correlated systems: integrable structures, traditionally
  restrained to closed systems in the quantum realm <cit.>; the existence of ballistics spin-transport <cit.> and anomalous diffusion <cit.> in the integrable XXZ model, thus allowing for the discovery of Kardar-Parisi-Zhang correlations <cit.> in the quantum realm <cit.>. Additionally, it has allowed to characterize the anomalous transport properties of disordered  <cit.> and quasi-periodic <cit.> interacting systems, the persistence of ballistic transport in the presence of level repulsion induced by single impurities <cit.> and ballistic-to-diffusive transition induced by integrability-breaking in finite-sized systems <cit.>.
  
  However, the study of transport with Lindblad boundary conditions is mostly done on a case by case basis, and it remains unclear which properties of the interacting region determine the current in systems driven by Lindblad  reservoirs.
  In a similar way, although some connection between the fermionic reservoir description and the Lindblad one can be found in the literature <cit.>, generic consequences for the transport properties have not been carried out.
  In particular,  an equivalent of the Meir and Wingreen's formulation <cit.> for Lindblad boundary conditions was not worked out yet, and it is yet unclear whether a systematic evaluation of transport properties in arbitrary Markovian settings is possible.
  
  We address this question in the present paper and develop such formalism. We show, by using a Keldysh description <cit.> of an arbitrary system in presence of Lindblad boundary conditions injecting and extracting particles,
  that one can derive a generic transport formula in the spirit of the one of Meir-Wingreen.
  Quite remarkably, this formula relates the transport properties of an arbitrary system uniquely  to its  Keldysh Green's function and the injection/extraction rates of the Lindblad boundaries.
  We also generalize the transport formula to the case when the system itself can have losses and gains of particles by coupling to other Lindblad reservoirs. We then illustrate the usefulness of our generic formula by deriving the current in various settings, summarized in Fig. <ref>, and in particular a one dimensional tight-binding chain in presence of losses/gains in the bulk.
  
  The paper is structured as follows. In Sec. <ref>, we introduce the general setup and its description in the Hamiltonian and Lindbladian formulation. In Sec. <ref>, we introduce the Keldysh formalism and compute
  the effective contribution of reservoirs onto the system in both cases. We show how
  they map onto each other in the limit of high chemical potential and temperature for the fermionic reservoirs.
  In Sec. <ref>, we apply our mapping to derive the generic transport formula for Lindblad type boundaries, and a generic system, potentially containing both interactions and dissipation.
  In Sec. <ref>, we show applications of our formula to three examples. The two first ones, a single level and a one-dimensional chain, were already studied in the literature by other methods
  and serve to show how or formula allows to recover easily the previous results. The last example, a one-dimensional chain with dissipation, is to the best of our knowledge new and exhibits unusual properties of
  the current in presence of the dissipative terms. Sec. <ref> is the conclusion and perspectives. Finally, several technical points have been put in appendices.
  
  
  
  
  
  
  
  § MODELS
  
  
  In this section, we detail the two sets of models that we examine in the present paper, as sketched in Fig. <ref>a.
  In the fist class of models, generically denoted by a subscript ℒ, Lindblad  jump operators inject and extract particles at the left (L) and right (R) edge of a generic  system at different rates.
  In the second class, generically denoted by a subscript ℱ, the system exchanges particles with two fermionic baths, at the temperature T, with different chemical potential μ_L,R.
  
  
  
   §.§ Lindblad boundary conditions
  
  
  For the Lindblad boundary condition, the contact with the environment is described
  by the action of a Lindblad operator
  L <cit.>. The non-unitary evolution of the  density matrix of the system ρ_ S obeys the Lindblad master equation
  
      d ρ_ S/dt=-i[H_ S,ρ_ S]+ L(ρ_ S) ,
  
  where H_ S is the Hamiltonian of the system. In general,  ℒ(ρ)=∑_n,mλ_m,n(Λ_nρΛ_m^†-1/2{Λ_m^†Λ_n,ρ})
  where {} denotes anticommutation of generic jump  operators Λ_n, acting with rates λ_m,n on the system.
  We focus here for simplicity on spinless fermions on a lattice. The extension including additional degrees of freedom is straightforward. The experimentally relevant situation, sketched in Fig. <ref>, involves an external  environment which injects particles at site r=L/R at rates α_r and extracts them at rates β_r. Injecting and extracting particles at different rates at each end of the system allows to drive currents through it.
  
  This situation is described by the Lindblad operator <cit.>
  
      L(ρ_ S)  =∑_r=L,R   {α_r[2c_r^†ρ_ S c_r-{c_rc_r^†,ρ_ S}]
         +β_r[2c_rρ_ S c_r^†-{c_r^†c_r,ρ_ S}]} ,
  
  in which (c_r,c_r^†) are  fermionic annihilation and creation operators acting on the system on site r.
  
  The description (<ref>) of the reservoirs relies on the  Markovian approximation, which discards  all memory effects and correlations between the system
  and the environment. This is visible in the fact that the action of L only depends on the current
  state ρ_ S of the system.
  
  
  
  
   §.§ Fermionic reservoirs
  
  
  The other canonical description, in particular heavily used in solid-state mesoscopic systems <cit.>, consists of coupling the system to two free fermionic  baths, see Fig.<ref>b.
  These baths mimic large metallic contacts exchanging particles with the edges r=L/R of the system. This setting is described by a Hamiltonian of the form
  
      H=H_ S+H_ F+H_τ ,
  
  in which H_ S is the Hamiltonian of the system and H_ F the Hamiltonian describing the L- and R-reservoirs:
  
      H_ F =∑_k,r∈{L,R }(ϵ_k-μ_r)a_k,r^†a_k,r ,
  
  where the a_k,r denote the usual fermionic annihilation operators. We assume that these fermions have a continuous spectrum ϵ_k and chemical potential μ_r. The exchange of fermions between system and reservoirs is described by the tunnelling Hamiltonian
  
      H_τ = - τ∑_k,r∈{L,R }(a_k,r^†c_r+ h.c.) .
  
  
  The systems described by (<ref>–<ref>) and (<ref>–<ref>) correspond in general to different physical situations. In general they thus lead to different transport properties. For the case of the Lindblad boundaries the current is controlled by the asymmetry of the injection and extraction rates between the L and R sides. For the fermionic case the current is controlled by the chemical potential difference between the two reservoirs. It is thus interesting to connect these two situations, this would allow to gain  physical insight by transferring well-established results  obtained in each of the two formulations of the problem. 
  
  
  
  
  
  
  
  
  § KELDYSH FORMULATION
  
  
  In this section we give a Keldysh description <cit.> of the two situations of the previous section. Using the Keldysh formalism allows us to trace out in both cases the reservoirs and boundary conditions and provides a natural path for connecting the two
  approaches. Given the fact that the reservoirs are coupled locally at each end of the system, it is enough to consider the case where reservoirs are coupled to a single site to see how its action gets modified.
  In this case, the system is described by the simple Hamiltonian
  
      H_ S=ϵ_0c^†c ,
  
  in which ϵ_0 describes the local chemical potential for the site occupation.
  
  We operate in the Keldysh path-integral formalism  <cit.>, that we summarize here mainly to fix notations. The  object of interest is the Keldysh action 𝒮, appearing in the partition function 𝒵 of the system:
  
      𝒵=∫ D[ψ,ψ̅]e^i𝒮[ψ,ψ̅] .
  
  We assume implicitly the standard Keldysh matrix structure in which ψ=(ψ^+,ψ^-) are vectors of fermionic Grassmann variables defined on the upper and lower Keldysh branches 𝒞_±. We also follow the Larkin and Ovchinnikov convention <cit.> to perform the Keldysh rotation.
  
  In this basis, the Keldysh action is expressed in terms of the retarded, advanced and Keldysh green functions
  G^ℛ, G^𝒜 and G^𝒦:
  
      𝒮=∫dϵ/2π(ψ̅^1,ψ̅^2)_(ϵ)[ [G^ℛ]^-1 [G^-1]^𝒦;        0 [G^𝒜]^-1 ]_(ϵ)[ ψ^1; ψ^2 ]_(ϵ) .
  
  For the single-site Hamiltonian (<ref>), initially at thermodynamic equilibrium of temperature T_ S and chemical potential μ_ S, the Green functions read
  
      [G_ S^ℛ(𝒜)]^-1    =ϵ-ϵ_0± iη ,
      
      G_ S^𝒦   =-2π i  tanh(ϵ-μ_ S/2T_ S)δ(ϵ-ϵ_0) ,
  
  in which η is an infinitesimally small quantity. We remind that [G^-1]^ K=2iηtanh[(ε-μ)/2T_ S] is also infinitesimal and formally keeps memory about the initial state of the system <cit.>. As we are going to illustrate below, by comparing the effect of adding Lindblad and fermionic reservoirs on the single level,  this infinitesimal term can be neglected as soon as the system is coupled to external baths.
  
  
  
  
  
  
  
  
   §.§ Lindblad reservoirs
  
  
  In the case of reservoirs described by Lindblad operators as in Eq. (<ref>), the description within the Keldysh formalism is given in Appendix <ref>, following the method outlined in <cit.>. A single reservoir, injecting and extracting particles with rates α and β, leads to an additional contribution to the action (<ref>), which reads
  
      S_ L    =i∫dϵ/2π(ψ̅^1,ψ̅^2)_(ϵ)[     α+β -2(α-β);       0  -(α+β) ][ ψ^1; ψ^2 ]_(ϵ).
  
  Notice that the upper right element of the action is now finite and the contribution from the initial Keldysh component (<ref>) can be neglected. As an important consequence, the energy of the single level ϵ_0 fully disappears from the action, by making the shift ϵ→ϵ+ ϵ_0 in the integral. This has direct consequences for a certain number of physical properties of the system which will become fully controlled by the bath.
  For instance, for the case of a single site, the level occupation in the stationary state  is given, in terms of  Green functions, by
  
      ⟨ n⟩_∞ = i/2∫dϵ/2π[G^ℛ(ϵ)-G^𝒜(ϵ)-G^𝒦(ϵ)] ,
  
  where the Green functions are derived by inverting the full action obtained by adding (<ref>) and (<ref>). One obtains, in the case of Lindblad boundaries <cit.>:
  
      ⟨ n⟩_∞^ L=α/α+β,
  
  which exclusively depends on the injection and extraction rates α and β, irrespective of the local chemical potential ϵ_0. The Lindblad case will thus “erase” certain characteristics of the system.
  
  It is also interesting to note that in (<ref>) the retarded and advanced parts of the action depend only on the sum α+β and thus are insensitive on whether the boundary condition is injecting or extracting particles. The difference
  between extraction and injection only appears in the Keldysh component. As we will see below, this has remarkable consequences on some transport properties of systems with losses.
  
  
  
  
  
  
  
   §.§ Fermionic reservoirs
  
  In the case of a fermionic bath, it is useful to obtain simple analytical expressions by making certain approximations on the properties of the reservoirs which correctly describe typical metallic contacts, without  loss of generality.
  In particular, we assume in what follows that the reservoir has a constant density of states.
  We also make the approximation that the tunnelling takes place on a single site of the system so that the momentum is not conserved during the tunnelling.
  
  These two assumptions allow us to analytically
  integrate over the reservoirs in (<ref>) and obtain the effective boundary terms for the system.
  The details of this derivation are given in Appendix <ref>. The single level action thus becomes (we adopt the convention e=ħ=k_B=1):
  
      S_ℱ=i∫dϵ/2π(ψ̅^1,ψ̅^2)_(ϵ)Δ[             1 2tanh(ϵ-μ/2T);             0            -1 ][ ψ^1; ψ^2 ]_(ϵ) ,
  
  where we introduced the hybridization constant Δ=τ^2/v_F, in which v_F is the Fermi velocity of the reservoirs and thus a direct measure of the density of states.
  
  
  
   §.§ Mapping between the two boundary conditions
  
  
  The  comparison between (<ref>) and (<ref>) shows that the main difference lies in the ϵ–dependence of the Keldysh component of the
  action for S_ℱ. This energy dependence has for consequence that the fermionic boundary term is non-local in time and thus encodes the memory
  effects of the fermionic bath. Contrarily, the absence of such energy dependence in the action for the Lindblad boundaries S_ℒ is directly encoding the Markovian aspect.
  
  It is possible to get rid of the ϵ dependence of the fermionic reservoirs
  by taking the limit μ→∞, T→∞, while keeping the ratio μ/T
  fixed. One thus obtains the mapping S_ℱ→ S_ L  by making the identification
  
      α    =1/2Δ[1+tanh(μ/2T)] ,
      β    =1/2Δ[1-tanh(μ/2T)] .
  
  Such limiting mapping between the fermionic reservoirs and the Lindblad boundaries was already noted in general terms in the literature <cit.>. This precise mapping between the two formalisms allows us to derive the transport properties, as will be done in the next section.
  
  In connection with this limit, it is instructive to consider the single level occupation for the fermionic reservoirs. It is readily derived relying on (<ref>):
  
      ⟨ n⟩_∞^ℱ    =∫dϵ/2π[1-tanh(ϵ-μ/2T)]Δ/(ϵ-ϵ_0)^2+Δ^2 .
  
  
  First we note that in the limit Δ→ 0 the coupling to the reservoirs becomes extremely small. The density then becomes:
  
      lim_Δ→0⟨ n⟩_∞^ℱ=1/1+e^ϵ_0-μ/T .
  
  In that case we recover the Fermi-Dirac distribution for a single site with energy ϵ_0
  and at a temperature of the reservoirs, showing that in this limit the only effect of the reservoirs is to thermalize the single site.
  Note that in our formalism we always implicitly take first the limit of infinitely long time and assume that we have reached a stationary state before taking other limits.
  
  On the other hand, if we take the infinite chemical potential and temperature limit (<ref>), we get back the Lindblad result (<ref>), as can be expected.
  It is however important to stress that the spectrum of the system has to be bounded to allow to take such limit.
  In the specific case of the single site case considered in (<ref>), the condition  μ,T ≫Δ,ϵ_0 is sufficient to enforce the correspondence with (<ref>).
  
  
  
  
  
  
  
  
  § GENERIC TRANSPORT FORMULA FOR LINDBLAD BOUNDARIES
  
  
  We are now in a position to tackle the main question of the paper, namely a generic transport formula for Lindblad (ℒ) type boundaries.
  
  
  
   §.§ Generic formula
  
  
  Let us thus consider a generic quantum system driven out of equilibrium by a a L- and R- fermionic reservoir (see Fig. <ref>).
  Since we want to be able to address the more general case in which the system itself can potentially lose or gain particles, we define two currents
  
      d n_L/dt = -J_L,   d n_R/dt = J_R ,
  
  in which n_L/R is the occupation of the sites L/R  attached to the reservoirs L/R. As a consequence, J_L is the current leaving the left reservoir, while J_R is the one entering the right reservoir. As a result two generic currents can be defined: J = (J_L + J_R)/2 is the current going through the system, while J_D = (J_L-J_R) is the current representing the loss (or gain) of particles in the bulk. In the absence of such extraction or injection of charges in the system J_D=0 and J is the usual conserved current. Let G^a with a∈{ℛ,𝒜,𝒦}, be the matrices whose elements are the different Green's function in a given basis. A generic derivation of the current for Lindblad boundary conditions (ℒ) is presented in Appendix <ref> and leads to
  
  
      J_ L   = 1/2{(α_L-β_L-α_R+β_R) +i/2∫dϵ/2π Tr[((α_L+β_L)γ_L-(α_R+β_R)γ_R)G^𝒦]} ,
      
      J_D, L   =i/2∫dϵ/2π Tr[((α_L+β_L)γ_L+(α_R+β_R)γ_R)G^𝒦].
  
  
  with γ_r,ij=2δ_i,rδ_i,j in the position basis.
  This is the main result of the paper. 
  
  While Appendix <ref> presents the full derivation of (<ref>–<ref>), we show in the main text how one can also derive (<ref>) by using the mapping (<ref>) and the MW formula for fermionic reservoirs (ℱ).
  We consider the Hamiltonian of the system in the form H_ S= H_0+V, in which H_0=∑_i,jc^†_ih_i,jc_j is the kinetic part of the Hamiltonian written in position basis and h_i,j describes hopping between sites i and j. The operator V describes  many-body interactions. For such a system the MW formula for the current reads <cit.>
  
  
      J_ℱ    =i/2∫dϵ/2π Tr{[(f_L(ϵ)-1/2)Γ_L-(f_R(ϵ)-1/2)Γ_R](G^ℛ-G^𝒜)+1/2(Γ_L-Γ_R)G^𝒦}
  
  
  where the matrices Γ_r,ij=Δ_rγ_r,ij in position basis describe the coupling between the system and the baths at the edge site r, and f_r(ϵ) is the Fermi-Dirac distribution associated to the r-bath.
  
  Using the correspondence (<ref>) in (<ref>) one notices that, after performing the transformation, the term between square brackets becomes constant as a function of energy.
  This allows to use the additional relation
  
      i∫dϵ/2π(G^ℛ-G^𝒜)=𝕀 ,
  
  which follows from the fermionic anticommutation relations. Using that [Γ^r]=2Δ_r, we obtain the formula (<ref>) giving the current for a generic system driven by Lindblad boundary conditions.
  Note that this result is also valid in presence of dissipation in the system (see Appendix <ref>).
  
  One of the remarkable properties of the Lindblad boundary condition is the fact that the current is fully determined by the Keldysh component of the local Green's function G^𝒦.
  Note that the first term in (<ref>) depends on the difference between injection and extraction rates α-β for each of the reservoirs. We could naively expect that this difference plays a similar role to the voltage or chemical potential difference for fermionic reservoirs and thus control the current flow. Nevertheless, this term is not sensitive to the properties of the system, which are encoded in the Keldysh Green's function appearing in the second term. The second term is also sensitive to the sum of the injection and extraction α+β rates of each of the reservoirs. 
   These considerations  apply also for the current (<ref>), which quantifies dissipative gains and losses in the system. The Keldysh Green function G^𝒦 is  thus the central object to understand  transport in dissipative systems driven by Lindblad boundaries,  as we will examine in the examples of the next section.
  
  
  
   §.§ Non-interacting systems
  
  
  As for the case of fermionic reservoirs, a non-interacting and non-dissipative system allows for further simplifications of the transport formula. In that case, we can rely on two additional relations <cit.>:
  
      1/2(   G^𝒜-G^ℛ+G^𝒦)  =
         if_L(ϵ)G^ℛΓ_LG^𝒜+if_R(ϵ)G ^ℛΓ_RG^𝒜 ,
      G^ℛ-   G^𝒜 =iG^ℛ(Γ_L+Γ_R)G^𝒜 ,
  
  leading to the following expression for the current in the fermionic and Lindblad setting
  
      J_ℱ    =∫dϵ/2π[f_L(ϵ)-f_R(ϵ)] Tr[Γ_ RG^ℛΓ_ LG^𝒜],
      
       J_ L    =∫dϵ/2π[α_Lβ_R-β_Lα_R] Tr[γ_RG^ℛγ_LG^𝒜].
  
  In position basis these expressions become:
  
      J_ℱ    =∫dϵ/2π[f_L(ϵ)-f_R(ϵ)]4Δ_RΔ_L|G_L,R^ℛ|^2 ,
      
      J_ L    =∫dϵ/2π[α_Lβ_R-β_Lα_R]4|G_L,R^ℛ|^2 ,
  
  where we used the fact that G^ℛ*=G^𝒜 and G^ℛ is
  symmetric. 
  
  
  For the fermionic reservoirs, Eq. (<ref>) reproduces Landauer-Büttiker formula, where the current is directly related to the probability of transmission through the system at a given energy ϵ. The transmission probability is given by the Green's function G^ℛ_L,R connecting the two reservoirs. The expressions of the currents for the Lindblad and the fermionic bath boundaries both depend on this transmission probability. These probabilities coincide for these two cases, as can be seen from Eqs. (<ref>) and (<ref>), by making the identification Δ=α+β. It is thus possible to draw a connection between the two driving protocols in terms of transport coefficients. 
  
  Indeed, the corresponding formula for Lindblad boundaries allows further simplifications. For non-interacting and non-dissipative systems, the retarded Green function  G^ℛ only depends on the hybridization coefficients Δ_r =α_r + β_r.
  This has the remarkable consequence in (<ref>) that the current J_ L is always linear in the bias α_Lβ_R-β_Lα_R. This allows us to exactly connect the linear response for fermionic systems to the
  Lindblad driving. For the fermionic case,  we consider the linear response limit of  (<ref>–<ref>), in which δμ=μ_L-μ_R→0 and T_R=T_L=T. In this limit, f_L(ϵ)-f_R(ϵ)∼ - ∂_ε f(ε)δμ/T, which, in the limit of large temperatures, scales as δμ/4T. We stress again that such limit makes sense if the transmission amplitude Tr[Γ_ RG^ℛΓ_ LG^𝒜] is non-zero only on a finite energy window. In such a high temperature limit the conductance becomes
  
      g(T→∞)   =lim_δμ→ 0J_ℱ/δμ=1/4T∫dϵ/2π Tr[Γ_ RG^ℛΓ_ LG^𝒜] ,
  
  which vanishes as the inverse temperature T, as expected. Comparing (<ref>) with (<ref>), one gets
  
      J_ L=c Tg(T→∞) ,
  
  where c=(α_Lβ_R-β_Lα_R)/4Δ_RΔ_L is a constant which depends on the choice of the Lindblad driving. There is thus a perfect connection between the large temperature conductance of a fermionic system and the transport measured with
  Lindblad driving. Note that the condition of large chemical potential  necessary  to derive the mapping (<ref>) is not required here, the limit of large temperature is sufficient. Whether such an exact connection applies in the presence of interactions  remains an open question, which is left for further investigations.
  
  
  
  
  
  
  
  § APPLICATIONS
  
  
  We provide some applications of the general formula (<ref>) for the systems sketched in Fig. <ref>. We will compute the current for : i) a single site of energy ϵ_0 as described by the Hamiltonian (<ref>); ii) a free fermionic chain of N sites;  iii) the same free-fermionic chain but with loss or gain terms modelled by injecting or extracting Lindblad terms acting throughout the chain.
  
  
  
  
  
   §.§ Single site connected to biased reservoirs
   
  
  A single site connected to biased reservoirs leads to standard Breit-Wigner resonances <cit.>. In this  case, Γ_L/R=2Δ_L/R and the Green function reads, in the case of fermionic bath boundaries, G^ℛ(𝒜)=(ϵ-ϵ_0± i(Δ_R+Δ_L))^-1. Equation (<ref>) leads to the well known result for the current <cit.>:
  
      J_ℱ=∫dϵ/2π4Δ_RΔ_L[f_L(ϵ)-f_R(ϵ)]/(ϵ-ϵ_0)^2+(Δ_L+Δ_R)^2 .
  
  
  For Lindblad type boundaries we have instead
  
      G^ ℛ(𝒜)=(ϵ-ϵ_0± i(α_R+β_R+α_L+β_L))^-1 .
  
  Using the relation <ref> for the current leads to
  
      J_ L=2e(α_Lβ_R-β_Lα_R)/α_L+β_L+α_R+β_R .
  
  Comparing (<ref>) and (<ref>) one
  sees that, as can be expected, all dependence  of the current on ϵ_0 is lost for the Lindblad
  case. Thus, the current is fixed entirely by the boundary conditions, in analogy to the occupation (<ref>) of the impurity in the presence of  a single Lindblad reservoir.  On the other hand, it is clear that for the fermionic baths case, the conductivity
  depends on the relative values of the chemical potentials of the bath
  and the Fermi energy of the system.
  
  
  
  
   §.§ 1D free fermionic chain
  
  The current flowing through a non-interacting fermionic chain attached to Lindblad reservoirs has been recently derived relying on variational <cit.> and third-quantization methods <cit.>. Our formulation allows to derive the above results in a systematic way relying on standard techniques.
  Let N≥2 be the size of the system.  The
  Hamiltonian is H_ S, chain=J∑_j=1^N-1[c_j^†c_j+1+].
  According to (<ref>) and (<ref>),
  we need to compute for fermionic baths boundaries
  
      Tr(Γ_R G^ℛΓ_LG^𝒜)=4Δ_RΔ_L|G_L,R^ℛ|^2
  
  where the indices L,R refer to the sites where the left and right reservoir are connected, i.e 1 and N. For Lindblad
  boundaries we have similarly
  
      Tr(γ_RG^ℛγ_LG^𝒜)=4|G_L,R^ℛ|^2 .
  
  In the position basis, for fermionic bath boundaries these functions
  are given by tridiagonal matrices of the form
  
      [G^ℛ(𝒜)]^-1=[ ϵ+iΔ_L      J      ⋯      0      0;      J      ϵ      J      ⋯      0;      ⋮      J      ⋱      J      ⋮;      0      ⋯      J      ϵ      J;      0      0      ⋯      J ϵ+iΔ_R ] ,
  
  in which the presence of boundaries affects  only the first and last
  diagonal term. The Green functions for Lindblad type boundaries
  are simply obtained by making the substitution Δ_L→α_L+β_L,
  Δ_R→α_R+β_R. To simplify notations, we will suppose
  in what follows that Δ_L=Δ_R=Δ and |J|=1.
  The inverse of such tridiagonal matrices has been derived in Ref. <cit.>.
  For the element (i,j) with i<j, the Green function reads:
  
      G_i,j^ℛ(𝒜)=(-J)^i+jB_i-1^ℛ(𝒜)B_N-j^ℛ(𝒜)/(ϵ± iΔ)B_N-1^ℛ(𝒜)-B_N-2^ℛ(𝒜) ,
  
  where B_i^ℛ(𝒜)=(r_+± iΔ)r_+^i-(r_-± iΔ)r_-^i/(r_+± iΔ)-(r_-± iΔ) (one has to take the + sign for ℛ and the - sign for 𝒜),
  and r^±_ϵ=(ϵ±√(ϵ^2-4J^2))/2.
  Equations (<ref>-<ref>) lead to
  
  
      |G^ℛ_L,R(ε,Δ)|^2=4|(ϵ^2-4)|/|(ϵ-Δ^2ϵ+4iΔ)[(r^+_ϵ)^N-(r^-_ϵ)^N]+(Δ^2+1)√(ϵ^2-4)[(r^+_ϵ)^N+(r^-_ϵ)^N]|^2 ,
  
  
  which,  inserted in (<ref>-<ref>), gives the current flowing through the system. Equation (<ref>) is plotted in Fig. <ref> as a function of energy ϵ. It features N peaks whose width is controlled by the hybridization constant Δ. These peaks  correspond to the N single-particle resonances of a chain of N sites, appearing within an energy band of width 4J.
  
  
  
  
  
  
      J_ B=∫dϵ/2π(f_L(ϵ)-f_R(ϵ))16Δ^2|(ϵ^2-4)|/|((ϵ-Δ^2ϵ+4iΔ)((r^+)^N-(r^-)^N)+(Δ^2+1)√(ϵ^2-4)((r^+)^N+(r^-)^N))|^2 ,
  
  
      J_ L=∫dϵ/2π16(α_Lβ_R-β_Lα_R)|(ϵ^2-4)|/|((ϵ-Δ^2ϵ+4iΔ)((r^+)^N-(r^-)^N)+(Δ^2+1)√(ϵ^2-4)((r^+)^N+(r^-)^N))|^2 .
  
  
  
  
  For the fermionic baths, there is no simple way of carrying the integral
  in general but, for large values of N, there is a simple way of bounding
  J_ℱ up to corrections of order e^-N:
  
      J_3<J_ℱ<J_1 ,
  
  with
  
      J_α    =∫_-π/2^π/2dθ/π[f_L(2sinθ)-f_R(2sinθ))cos^α(θ)]
  
  where we took Δ=1 to simplify and made the trigonometric change of variable ϵ→ 2 sinθ.
  
  For Lindblad boundaries, a numerical evaluation of the expression (<ref>) shows that the current is independent of the system size N, namely
  
      J_ L=(α_Lβ_R-β_Lα_R)/Δ(1+Δ^2) ,
  
  in agreement with the result of Ref. <cit.>. As it may be expected for ballistic systems, this result does not depend on the size of the system N. Nevertheless, this should be contrasted with the case in which the system is coherently driven by fermionic reservoirs. In this case, the current depends in general on the system size N, as it is shown in Fig. <ref>. This is particularly clear when considering the  conductance as defined in Eq. (<ref>), but at zero temperature (T→0). In this limit,  the conductance corresponds to the transmission probability |G^ℛ_L,R(μ,Δ)|^2, taken at the  chemical potential μ. When plotted as a function of the system size N, it displays pronounced even/odd oscillations corresponding to the appearance/disappearance of  Fabry-Perot resonances at the energy μ=0 (see also Fig. <ref>). As expected, and accordingly to the Lindblad limit exemplified by Eq. (<ref>), the size dependence disappears in the T→∞ limit, in which the temperature acts as if effectively broadening the  single-particle peaks, and making them indistinguishable.
  
  
  
  
   §.§ 1D fermionic chain with loss/gain terms
  
  We restrict in this case to Linblad type boundary conditions and add on top of the free fermionic Hamiltonian H_S, chain the following Lindblad terms in the dynamics ν∑_j2c_jρ_tc_j^†-{c_j^†c_j,ρ_t},
  which describes the loss of particles at rate ν on each site. The gain of particles is instead described by a Lindblad term of the form ν∑_j2c^†_jρ_tc_j-{c_jc_j^†,ρ_t}. Remarkably, the advanced and retarded Green's
  functions in the position basis do not discriminate between losses and gains and read, in both cases <cit.>:
  
      [G^ℛ(𝒜)]^-1=
         [ ϵ± iΔ_L± iν           J           ⋯           0           0;           J       ϵ± iν           J           ⋯           0;           ⋮           J           ⋱           J           ⋮;           0           ⋯           J       ϵ± iν           J;           0           0           ⋯           J ϵ± iΔ_R± iν ]
  
  The equivalence of gains and losses for the advanced and retarded Green functions is also apparent in Eq. (<ref>), where the injection/extraction rates α/β appears with the same sign on the diagonal.
  
  Importantly, this is not the case for the Keldysh Green function, which, in the case of losses, reads
  
      [G^𝒦]^-1=
         [ -2i((α_L-β_L)-ν)                0                ⋯                0;                0              2iν                                 ⋮;                ⋮                0                ⋱                0;                0                ⋯                0 -2i((α_R-β_R)-ν) ]
  
  while the case with gains is obtained by making the substition ν→ -ν. Differently from the retarded and advanced components, the Keldysh Green's function discriminates between losses and gains in the bulk, with important consequences on transport. For instance,  the presence of loss/gain terms invalidates the relations (<ref>,<ref>). Thus, standard identities for non-interacting fermion do not apply  anymore, but the complexity remains manageable because the loss/gain terms are quadratic in fermionic operators. We thus compute
  the current from (<ref>). More specifically, we have to compute G_1,1^𝒦
  and G_N,N^𝒦 (recall that the left site index L is 1 and the right-site index is N here). As before, we will consider the symmetric
  case. In addition, to lighten the final formulae we choose α_R/L+β_R/L=Δ and take |J|=1. We also suppose
  that α_L-β_L=-(α_R-β_R)=δμ̃, to be distinguished from the bias in chemical potential δμ in Eq. (<ref>). We also recall the
  relation G^𝒦=-G^ℛ[G^𝒦]^-1G^𝒜 <cit.>.
  Since the loss term simply adds a term on the diagonal, we get the
  inverse of [G^ℛ(𝒜)]^-1
  by making the substitution ϵ→ϵ± iν in the
  expression of the inverse for the free case in Section <ref>.
  
  As it is shown in App. <ref>, it is also in this case possible to reexpress the current as a function of G^ℛ, but differently from the Landauer-Büttiker like expressions (<ref>) and (<ref>), namely:
  
      J_ L=δμ̃[1+2Δ∫dϵ/2π(-|G_1,1^ℛ|^2+|G_1,N^ℛ|^2)] .
  
  G_i,j^ℛ is given by the expression (<ref>)
  where one made the substitution ϵ→ϵ+ iν. The behavior of the current with respect to ν is shown in Fig. <ref> for different system sizes. In this case the current does depend on the system size and,
  for N>2, we observe first a decrease of the current and then an
  increase with respect to the loss of particles ν. For large values of ν
  all the curves collapse towards the same value J_ℒ=α_R-α_L=δμ̃. The reason for this counter-intuitive non-monotonous behavior is due to the fact that, in the large ν limit, the bulk dynamic doesn't matter anymore: a particle injected from the left never reaches the right reservoir. As a consequence, the current  J_L injected from the left is equal to the rate of injection of the left reservoir 2α_L and, inversely, J_R to minus the rate of injection at the right reservoir 2α_R.
  
  Another counterintuitive feature of this model, which is not apparent in Fig. <ref>, is that the behavior of the current J_ℒ does not discriminate between loss and gain terms. Indeed, J_L is entirely expressed in terms of elements of G^ℛ (see App. <ref>) which, as previously discussed, has the same expression whether we inject particles at rate ν along the chain or we extract them. This interesting dependence of the current show that the dissipative model deserves further scrutiny, which will be done in future studies.
  
  
  Additionally, we can use the formula (<ref>) to compute the current of
  particles going from the system to the environment. Again, we provide the proof in  App. <ref>. It leads to
  
      J_D, L=±4∫dϵ/2π∑_j=1^Nν |G_1,j^ℛ|^2 ,
  
  where the +/- sign corresponds to the situation in which we consider a loss/gain term weighted by ν in the Lindbladian. Differently from the direct current J_ℒ, the current J_D, L discriminates between injection
  or extraction of particles, as expected. For loss terms, we show the
  behavior of J_D, L for different system sizes on Fig. <ref>. For
  a gain term, we would just have gotten the symmetric of this curve
  with respect to the horizontal axis. Notice that J_D,ℒ does not depend on the bias δμ̃ and that it tends  towards 2 in the ν→∞ limit.
  
  
  
  
  § CONCLUSION AND PERSPECTIVES
  
  
  In this work, we derived a generic expression for the stationary current flowing in a general interacting system, potentially with loss or gain of particles, driven by Lindblad jump operators. This expression is the equivalent for Markovian reservoirs of the Meir-Wingreen formula for the fermionic reservoirs. These two situations are related by a limit of high temperature and chemical potential for the fermionic reservoirs. In the non-interacting regime, an additional number of simplifications are possible in this generic transport formula.
  Using our general transport formula, we showed that, for Lindblad boundaries, the current is always in a linear regime no matter the values of the injecting and extracting rates. We illustrated how our approach can be systematically  applied on three different examples concerning non-interacting fermions. For each application, we  witnessed drastic differences in the behavior of the current between fermionic bath boundaries and Lindblad ones. Of particular interest is the example of a fermionic tight-binding chain with loss of gain of particles. This system shows a non monotonic 
  dependence of the current in the strength of the dissipation. In addition it has the very counterintuitive feature that the behavior of the current is independent on whether one injects or extracts particles in the bulk.
  
  Our work thus provides a path to tackle transport properties of open systems driven by Lindblad boundaries, with methodology similar to the one used in for mesoscopic systems in contact 
  with fermionic reservoirs and should help bridge the gap between the phenomena in these two situations. 
  
  There are of course several directions in which it would be interesting to extend our study.  
  
  Concerning the reservoirs, the study of the present paper has been done for Lindbladians that described injection and extraction of particles. It would be interesting to understand whether similar mappings exist for other types of Lindblad action and whether it can be done in a systematic way. For instance, it is possible to generate diffusive behaviors in lattice models of spins or fermions by putting local, independent dephasing noise at each site whose mean actions are described by quartic Lindblad terms <cit.>. In this case, the noise is introduced by hand and is supposed to model interaction with an external environment. It would be interesting to make this description more physically grounded by seeing it as emerging from the interaction with an actual environment taken within a certain limit.
  
  Concerning the systems themselves, the models for which we provided explicit evaluations of the transport formulas (<ref>) and (<ref>) here were only non-interacting models. It would be natural
  to apply our formula to interacting models and give a general interpretation in terms of Keldysh Green functions of the emergence of ballistic and anomalous diffusion in integrable systems <cit.>. In this spirit, future and interesting research directions could be concerned with single impurity problems <cit.>. Given the simplifications that we observed in the studies of this paper for the case of Markovian reservoirs, one can have 
  the hope that tackling such systems will be simpler than for their fermionic counterparts. Finally, extending our  
  approach to describe transport of either matter or energy in interacting systems driven by Lindbladians <cit.> beyond the strictly one-dimensional case could be tackled by addressing the transport properties in quantum ladder systems.
  
  
  The authors thank Enrico Arrigoni, João Ferreira, Pierre Le Doussal and Marko Žnidarič for useful discussions. This work has been
  supported by the Swiss National Science Foundation under Division II. M. F. acknowledges support from the FNS/SNF AmbizioneGrant PZ00P2_174038.
  
  
  
  
  
  
  § DERIVATION OF THE ACTION ASSOCIATED TO LINDBLAD TYPE BOUNDARIES
  
  
  Here, for completeness, we briefly show how to derive (<ref>)
  from the main text. This derivation was carried out in <cit.>
  and we simply transcribe it here.
  
  Let 𝔏 be the Liouvillian generating the total evolution,
  i.e 𝔏(ρ)=-i[H,ρ]+ L(ρ). where we choose
  for L the injecting and extracting terms introduced in the
  main text : L(∙)=α(2c^†∙ c-{cc^†,∙}_+)+β(2c∙ c^†-{c^†c,∙}_+.
  
  By definition
  
      ρ_t=lim_N→∞(𝕀+δ t𝔏)^Nρ_0
  
  with Nδ t=t. To get the Keldysh action, we have to insert
  2N identity resolutions in the above equation, N in the forward
  direction of time and N in the backward direction of time. The
  contribution of L to the action for an elementary time step
  is :
  
      ⟨ψ_n+1^+| L(|ψ_n^+⟩⟨ -ψ_n^-|)|-ψ_n+1^-⟩
          =-⟨ψ_n+1^+|ψ_n^+⟩⟨ψ_n^-|ψ_n+1^-⟩
         (α(2ψ̅_n+1^+ψ_n+1^-+(2-ψ̅_n+1^+ψ_n^+-ψ̅_n^-ψ_n+1^-))
         +β(2ψ_n^+ψ̅_n^-+(ψ̅_n+1^+ψ_n^++ψ̅_n^-ψ_n+1^-)))
  
  where ψ_n^±ψ̅_n^± are the usual Grassman
  fields. Summing over all elementary steps and performing the Keldysh
  rotation one gets the contribution to the action :
  
      S_L    =∫dϵ/2π(ψ̅_1,ψ̅_2)_(ϵ)[    α+β 2(β-α);      0 -(α+β) ][ ψ_1; ψ_2 ]_(ϵ)
  
  
  
  
  
  § DERIVATION OF THE EFFECTIVE ACTION FOR FERMIONIC BATH BOUNDARY CONDITION
  
  
  The starting point is the Hamiltonian (<ref>) from the main
  text :
  
      H     =H_ S+H_ F-μ N_ F+H_τ,
      
      H_ F    =∑_kϵ_kc̃_ F,k^†c̃_ F,k,
      
      H_τ    =-τ(c_ F,0^†c_ F,j+ h.c).
  
  As explained in the main text, we take the continuous limit in k
  and the dispersion relation is linearized around the Fermi points
  
      ϵ_k=
      v_F(k-k_F)     -Λ≤ k≤Λ
      
      -v_F(k+k_F)     -Λ≤ k≤Λ
  
  The total Keldysh action of the full system is composed of three
  parts,
  
      S_ tot=S_ S+S_ F+S_τ
  
  The goal is to get an effective action for the system alone by integrating
  over the degrees of freedom of the bath contained in S_ F
  and S_τ.
  
  The Keldysh action for the reservoir is <cit.>
  
      S_ F=
         ∫_-Λ^Λdk/2π∫dϵ/2π(ψ̅̃̅_ F^1,ψ̅̃̅_ F^2)_(ϵ,k)[ [G_ F^ℛ]^-1 [G_ F^-1]^𝒦;           0 [G_ F^𝒜]^-1 ]_(ϵ,k)[ ψ̃_ F^1; ψ̃_ F^2 ]_(ϵ,k)
  
  with
  
      G_ F^ℛ(𝒜)(ϵ,k)     =1/ϵ-(ϵ_k-μ)± iδ,
      
      G_ F^𝒦(ϵ,k)     =-2π itanh(ϵ-μ/2T)δ(ϵ-(ϵ_k-μ)).
  
  where δ=0^+.
  
  The Keldysh action associated to H_τ is
  
      S_τ=
         τ∫_-Λ^Λdk/2π∫dϵ/2π((ψ̅̃̅_ F^1,ψ̅̃̅_ F^2)_(ϵ,k)[ ψ_j^1; ψ_j^2 ]_(ϵ)
          +(ψ̅_j^1,ψ̅_j^2)_(ϵ,k)[ ψ̃_ F^1; ψ̃_ F^2 ]_(ϵ))
  
  Since the theory is quadratic, we can use usual Gaussian integrals
  formula to integrate over ψ̃_ F,ψ̅̃̅_ F
  
      ∫∏ dψ̅_jdψ_je^-∑_ijψ̅_iM_ijψ_j+∑_jψ̅_jχ_j+χ̅_jψ_j
          =(M)e^∑χ̅_iM_ij^-1χ_j
  
  identifying ψ→ i[ ψ^1; ψ^2 ], M→ i[ [G_ F^ℛ]^-1 [G_ F^-1]^𝒦;           0 [G_ F^𝒜]^-1 ]_(ϵ,k), χ→ -iτ[ ψ^1; ψ^2 ]_(ϵ)in the above formula and using (M)=1 yields the effective action
  of the bath on the system :
  
      S_ℱ=-∫_-Λ^Λdk/2π∫dϵ/2πτ^2(ψ̅_j^1,ψ̅_j^2)_(ϵ)[ G_ F^ℛ G_ F^𝒦;      0 G_ F^𝒜 ]_(ϵ,k)[ ψ_j^1; ψ_j^2 ]_(ϵ)
  
  So we need to compute ∫_-Λ^Λdk/2πG_ F^ℛ(𝒜)(ϵ,k)
  and ∫_-Λ^Λdk/2πG_ F^𝒦(ϵ,k):
  
      ∫_-Λ^Λdk/2πG_ F^ℛ(𝒜)(ϵ,k)
          =∫_-Λ^Λdk/2π1/ϵ-(ϵ_k-μ)± iδ
          =∫_-Λ v_F^Λ v_Fdϵ_k/π v_F1/ϵ-(ϵ_k-μ)± iδ
          =∫_-Λ v_F^Λ v_Fdϵ_k/π v_F( vp(1/ϵ-(ϵ_k-μ))∓ iπδ(ϵ-(ϵ_k-μ)))
          =1/π v_F(lnϵ-(-Λ v_F-μ)/ϵ-(Λ v_F-μ)∓ iπ)+O(1/Λ)
          =_limΛ→∞∓i/v_F,L.
  
  
      ∫_-Λ^Λdk/2πG_ F^𝒦(ϵ,k)
          =-∫_-Λ^Λdk/2π2π itanh(ϵ-μ/2T)δ(ϵ-(ϵ_k-μ))
          =-∫_-Λ v_F^Λ v_Fdϵ_k/π v_F2π itanh(ϵ-μ/2T)δ(ϵ-(ϵ_k-μ))
          =_limΛ→∞-2i/v_Ftanh(ϵ-μ/2T),
  
  which proves (<ref>)
  
      S_ℱ=i∫dϵ/2π(ψ̅_j^1,ψ̅_j^2)_(ϵ)Δ[             1 2tanh(ϵ-μ/2T);             0            -1 ][ ψ_j^1; ψ_j^2 ]_(ϵ)
  
  
  
  
  § DERIVATION OF THE TRANSPORT FORMULA FOR LINDBLAD BOUDARIES
  
  
  In this appendix we give the full derivation of the equation for the current (<ref>) for a generic system, potentially with interactions and loss/gain of particles.
  The dynamical evolution of the density n_L= tr(ρ_tc_L^†c_L)
  of the site connected to the left reservoir is given by
  
      dn_L/dt=2α_L(1-n_L)-β_Ln_L+𝔏^*(n_L)
  
  𝔏^* is the dual of the Liouvillian, generating the time evolution of operators in the Heisenberg picture.
  The first term corresponds to the current from the reservoir to the site, the second term represents interaction with the rest of the system and/or other external degrees of freedom. In the stationary state, these two are equal. We have a similar equation for the right site:
  
      dn_R/dt=2α_R(1-n_R)-β_Rn_R+𝔏^*(n_R)
  
  Hence the stationary current is given by
  
      J_ L=1/2(J_L+J_R)
  
  with J_L=2α_L(1-n_L)-β_Ln_L, J_R=-(2α_R(1-n_R)-β_Rn_R).
  
  The expression of the density at site j in terms of Green's functions is given by n_j=i/2∫dϵ/2π(G_j,j^ℛ-G_j,j^𝒜+G_j,j^𝒦). We then have
  
      n_L    =i/4dϵ/2π tr(γ^L(G^ℛ-G^𝒜+G^𝒦))
      
      n_R    =i/4dϵ/2π tr(γ^R(G^ℛ-G^𝒜+G^𝒦))
  
  Inserting these two equations in (<ref>) and using
  that i∫dϵ/2π(G^ℛ-G^𝒜)=𝕀,
  we arrive at (<ref>).
  
  Similarly, one can get the current from the chain to the external environment J_D, L=(J_L-J_R). When there is no loss in the system, this current is just 0 by
  conservation of the number of particle. If J_D, L>0 it means, that
  there are loss of particles in the system while if J_D, L<0,
  it means that there is a gain of particles in the system. Using again (<ref>,<ref>) and i∫dϵ/2π(G^ℛ-G^𝒜)=𝕀, we get after some elementary computations that
  for Lindblad type boundary conditions:
  
      J_D, L=i/2∫dϵ/2π tr(((α_L+β_L)γ_L+(α_R+β_R)γ_R)G^𝒦).
  
  which is Eq.(<ref>) of the main text.
  
  
  § TRANSPORT THROUGH DISSIPATIVE FERMIONIC CHAINS
  
  
  In this appendix, we provide more details on the calculations presented
  in Sec. <ref>.
  
  As explained in the main text, we compute the current from (<ref>). For
  that, we need to calculate the elements G_1,1^𝒦 and
  G_N,N^𝒦. The starting point is the relation G^𝒦=-G^ R[G^𝒦]^-1G^𝒜:
  
      G_k,l^𝒦=-∑_ijG_k,i^ℛ[G^𝒦]_i,j^-1G_j,l^𝒜
          =∑_ijG_k,i^ℛ2iδ_i,j((α_L-β_L)δ_i,1
          +(α_R-β_R)δ_i,L-ν)G_j,l^𝒜
  
  Thus,
  
      G_1,1^𝒦=     2i(|G_1,1^ℛ|^2(α_L-β_L)+|G_1,N^ℛ|^2(α_R-β_R)
          -∑_jν|G_1,j^ℛ|^2)
      
      G_N,N^𝒦=     2i(|G_1,N^ℛ|^2(α_L-β_L)+|G_N,N^ℛ|^2(α_R-β_R)
          -∑_jν|G_j,N^ℛ|^2)
  
  Now let us make the simplifying approximations α_L+β_L=α_R+β_R=Δ,
  α_L-β_L=-(α_R-β_R)=δμ̃. The
  expression for the current then simplifies into :
  
      J_ L    =δμ̃+i/2Δ∫dϵ/2π(G_1,1^𝒦-G_N,N^𝒦) ,
      
      J_D, L    =iΔ∫dϵ/2π(G_1,1^𝒦+G_N,N^𝒦)
  
  Substituting the found values for G_1,1^𝒦 and G_N,N^𝒦
  in these last expressions, we arrive at
  
      J_ L    =δμ̃-Δ∫dϵ/2π[2δμ̃(|G_1,1^ℛ|^2-|G_1,N^ℛ|^2)
         -∑_jν(|G_1,j^ℛ|^2-|G_j,N^ℛ|^2)] ,
      
      J_D, L    =2Δ∫dϵ/2π∑_jν(|G_1,j^ℛ|^2+|G_j,N^ℛ|^2)
  
  Now from (<ref>) one can remark that G_1,N-j+1^ℛ=G_j,N^ℛ.
  By re indexing the terms in the sum we have that ∑_j|G_1,j^ℛ|^2=∑_j|G_j,N^ℛ|^2
  so that :
  
      J_ L    =δμ̃(1-2Δ∫dϵ/2π(|G_1,1^ℛ|^2-|G_1,N^ℛ|^2)) ,
      
      J_D, L    =4Δ∫dϵ/2π∑_jν|G_1,j^ℛ|^2
  
  For gain terms instead of loss terms, the proof follows exactly the
  same lines. We end up with the same expression for J_ L
  while
  
      J_D, L=-4Δ∫dϵ/2π∑_jν|G_1,j^ℛ|^2
  
  
  
  apsrev4-2
  
  
  