Rectification in Nonequilibrium Steady States of Open Many-Body Systems

We study how translationally invariant couplings of many-body systems and nonequilibrium baths can be used to rectify particle currents. We propose novel setups to realize bath-induced currents in nonequilibrium steady states of one-dimensional open fermionic systems. We first analyze dissipative dynamics associated with a nonreciprocal Lindblad operator and identify a class of Lindblad operators that are sufficient to acquire a nonreciprocal current. Remarkably, we show that rectification can in general occur even when a Lindblad operator is reciprocal provided that the inversion symmetry and the time-reversal symmetry of the microscopic Hamiltonian are broken. We demonstrate this new mechanism on the basis of both analytical and numerical approaches including the Rashba spin-orbit coupling and the Zeeman magnetic field. Our findings will play fundamental roles for exploring rectification phenomena in homogeneous open many-body systems.

In this Letter, we reveal new mechanisms to obtain a nonreciprocal current in nonequilibrium steady states (NESS) of one-dimensional open fermionic systems, where a nonequilib-rium bath is uniformly coupled to the system in addition to an equilibrium heat bath (see Fig. 1). We first consider a nonreciprocal Lindblad operator, which is translationally invariant and conserves the particle number of the system, and elucidate a general condition to acquire a nonreciprocal current in NESS. We numerically calculate the current by considering a specific dissipator that can be realized in ultracold atoms [1,2]. Remarkably, we demonstrate that even a reciprocal Lindblad operator can rectify the current in NESS provided that the inversion symmetry and the time-reversal symmetry of the Hamiltonian are broken. We consider spin-dependent dephasing as a reciprocal Lindblad operator and evaluate the current in NESS by analytical and numerical methods in the presence of the Rashba spin-orbit coupling and the Zeeman magnetic field [77]. Our results demonstrate that open manybody systems that are uniformly coupled to nonequilibrium baths provide ideal platforms to explore novel mechanisms of realizing current rectification.
Model.-We consider a one-dimensional lattice model coupled to both an equilibrium heat bath and a nonequilibrium Markovian bath. Such a situation is described by the FIG. 1. Schematic illustration of our setup. Fermionic atoms are trapped in a one-dimensional lattice and uniformly coupled to a nonequilibrium bath. An equilibrium heat bath with the inverse temperature β is also coupled to the system to ensure that the system goes to the Gibbs state in the absence of nonequilibrium driving. The coupling strength to each bath is given by γeq and γneq = 1 − γeq, respectively. Nonreciprocal current I can arise in NESS only when the system is driven out of equilibrium. Lindblad master equation with dissipators where H 0 is a noninteracting Hamiltonian governing the internal dynamics, L m is a so-called Lindblad operator, γ eq and γ neq = 1 − γ eq denote the relative coupling strengths between two baths, γ eq ∈ [0, 1]. We assume that the baths are weakly coupled to the system with a small dimensionless parameter . Here and henceforth, we set = 1. We remark that the present model is an intrinsically interacting many-body problem because the dissipator cannot in general be expressed in terms of quadratic annihilation/creation operators as detailed below.
When the integrability of the translationally invariant many-particle system is weakly broken, the time evolution of the system can be described by a time-dependent generalized Gibbs ensemble (tGGE) [78][79][80], which is justified for times t of the order of 1/ and larger, where I q is an approximately conserved quantity as a consequence of weak driving. Previous studies [78][79][80] have shown that, by applying a perturbation theory to the Lindblad equation, one can obtain a simple differential equation that determines the dynamics of Lagrange parameters up to the order of where · · · GGE = tr[· · · ρ GGE (t)]. In the following, we evaluate the current in NESS by using tGGE approach with Eqs. (4)- (6).
Rectification by nonreciprocal dissipator.-We first consider the one-dimensional tight-binding model where J is the hopping amplitude and k = −2J cos(k) is the eigenspectrum. The system is subject to periodic boundary conditions and periodic dissipation of length L. In this case, I q in Eq. (4) is given by the local number operator in the momentum space I q = c † q c q . The Lindblad operators corresponding to the equilibrium heat bath satisfy [L m , H 0 ] = ζ m L m with ζ m = k − l , m = (k, l) ∈ {−π, −π + 2π/N, ..., π − 2π/N } to ensure the detailed balance condition L † kl = L lk e −β( k − l )/2 [5,81] in such a way that, without nonequilibrium driving, the system goes to the Gibbs state ρ can = e −βH0 /tr(e −βH0 ) irrespective of the initial state [see Fig.2(a) and Supplemental Materials]. For the sake of simplicity, we here employ the following Lindblad operator corresponding to the equilibrium heat bath To realize current rectification in NESS, we consider a nonreciprocal Lindblad operator corresponding to the nonequilibrium bath and assume that it is translationally invariant and conserves the particle number of the system. In this case, the Lindblad operator can in general be labeled by a wave number with coefficients ∆ kq as Using Eq. (5) and Lindblad operators (8) and (9), we obtain the rate equation that governs the dynamics of the system [82] where We numerically solve the rate equation (10) to obtain the dynamics of Lagrange parameters [82] and their steady state values [see Fig. 2(a)]. We see that Lagrange parameters depart from the Gibbs state when the system is driven out of equilibrium as the nonequilibrium dissipation rate γ neq is increased.
We now derive a general condition to realize a nonzero nonreciprocal current in NESS. The current I generally consists of two terms including Hamiltonian current and dissipative current of order . For such a small that justifies  (8) and (15). The parameters are set to β = 2/J, δ = 1 + i, and δ = 1 + 0.5i. the tGGE approach, the dissipative current can be ignored, which is consistent with a general description of the current in open quantum systems [83,84]. We obtain the current from the continuity equation for the density matrix as In the present model, the current is then given by Thus, to obtain a nonreciprocal current, the Lagrange parameter λ q must not be an even function of q. More specifically, as inferred from Eq. (12), this condition requires a set of (k, q) ∈ [−π, π) to satisfy (at least) one of the following conditions: We note that an even function λ q prohibits the rectification of the current even when the dissipative current of order is included because it leads to a parity-even distribution of particles in real space and thus the current (that is parity-odd) cannot exist. Let us apply the condition (14) for obtaining the nonreciprocal current to a specific example, which is proposed in ultracold atoms [1,2], where the subscript j denotes the lattice site. After the Fourier transformation, it is rewritten as where we set the lattice constant a = 1. From Eq. (14), the Lindblad operator (16) should give rise to a nonreciprocal current when either δ or δ has the imaginary part. This is demonstrated in Fig. 2(b), where the current in NESS is plotted as a function of γ neq for δ = 1 + i, δ = 1 + 0.5i. We see that the current rectifies as it is driven out of equilibrium [82] though it exactly vanishes in equilibrium (γ neq = 0). Rectification by reciprocal dissipator.-We now consider a more intriguing mechanism to realize a nonzero nonreciprocal current in NESS, namely, rectification by a reciprocal Lindblad operator at the expense of the broken inversion and time-reversal symmetries of the internal Hamiltonian. To be concrete, we include the Rashba spin-orbit coupling and the Zeeman magnetic field into the one-dimensional tight-binding model [77] where h denotes the Zeeman splitting, σ y,z are the Pauli matrices, α y,z denote the Rashba hopping with spin flips, σ =↑↓ and ν = ± label spin and band indices, respectively, and the system is subject to periodic boundary conditions and periodic dissipation of length L. The Rashba spin-orbit coupling and the Zeeman magnetic field break the inversion symmetry and the time-reversal symmetry of the Hamiltonian, respectively (see Fig. 3). The Hamiltonian is diagonalized with eigenvalues k± = −2J cos(k) ± (2α y sin(k) + h) 2 + 4α 2 z sin 2 (k) and quasiparticle operators η k± , which are given by a unitary transformation as c kσ = ν u σν (k)η kν [82] and obey the anticommutation relation {η kµ , η † k ν } = δ kk δ µν . In this case, local conservation laws of few-body observables are given by the number operators of quasiparticles To identify the Lindblad operators L eq m that satisfy the detailed balance condition, we consider µ(ν) dependence for the energy bands of quasiparticles in addition to Eq. (8): As the reciprocal Lindblad operator of the nonequilibrium bath, we consider the spin-dependent dephasing given by where j labels the lattice site and the dissipation rates of up and down spins satisfy γ ↑ + γ ↓ = 1. We calculate the rate  (5) with Lindblad operators (18) and (19), which is given bẏ with the force We see from Eq. (22) that the system goes to the infinite temperature state, i.e., λ q = 0 for all q, without equilibrium heat bath. Nevertheless, the current can rectify if the system couples to both equilibrium and nonequilibrium baths. When the dynamics is determined from the rate equation (20), the current in NESS can be obtained from the continuity equation for the density matrix as where the spin-resolved current I σ is given by [82] We have confirmed that, by numerical calculations using Eq. (23), the current I is nonzero only when both the Zeeman magnetic field and the Rashba spin-orbit coupling exist. This can be understood as follows. Since dissipation by an equilibrium bath does not rectify the current, one must resort to a nonequilibrium bath for obtaining a nonzero nonreciprocal current. From Eq. (22), we see that nonreciprocity of the distribution of Lagrange parameters is determined from the property of the unitary transformation of quasiparticles, namely, the symmetry of the internal Hamiltonian H 0 . In fact, due to the structure of the matrix component u σν (k) (see Supplemental materials), dephasing by the nonequilibrium bath in Eq. (22) contributes to the Lagrange parameters as an even function with respect to q if either one of the Zeeman magnetic field or the Rashba spin-orbit coupling is broken. As inversion-symmetric Lagrange parameters give the parity-even distribution of particles in real space, the current does not rectify even if dissipative correction of the order of is included. Figures 4(a) and (b) show the currents in NESS in the presence of the Rashba spin-orbit coupling and the Zeeman magnetic field. As shown in Fig. 4(a), the dephasing applied to up spins leads to the nonreciprocal current I in NESS and it becomes larger as the system is driven out of equilibrium. We recall that the system goes to the Gibbs state for γ neq = 0 and the infinite temperature state for γ neq = 1, both of which do not rectify the total current I. When the dephasing is applied to both up and down spins with equal rates [see Fig. 4(b)], the total current I vanishes irrespective of the dissipation rate γ neq , as up spins and down spins contribute to the currents I in the opposite directions and cancel out [82]. Here, we note that the sharp peak of the current in Fig. 4(a) comes from the sudden heating up to the infinite temperature due to the nonequilibrium bath and the peak position can be controlled by the system parameters, e.g., the Zeeman magnetic field h.
Physically, rectification of the current in NESS can be understood from the change of spin distribution near the Fermi surface. As shown in the left panel of Fig. 4(c), the spin distribution forms an effective Fermi surface in the steady state [see also the right panel in Fig. 3], reflecting the half-filled initial state. When dephasing is applied to up spins [see the right panel in Fig. 4(c)], they heat up and those near the Fermi surface are most likely to move to the other eigenstates of the Hamiltonian. As a result, the number of particles near the Fermi surface where up spins exist decreases, thereby contributing to the current in the positive direction [see also Eqs. (S12) and (S13) in Supplemental Materials].
Discussions.-We demonstrate a new type of rectification mechanism unique to homogeneous open quantum systems, which arises from the interplay between nonequilibrium dis- 4. (a,b) NESS current and its spin dependence as a function of γneq in the presence of the Zeeman magnetic field and the Rashba spin-orbit coupling. Dephasing is applied to up spins in (a), and to both up and down spins with equal rates in (b). (c) Distribution of the upper band (blue) and the lower band (red) in NESS for the equilibrium Gibbs state (left) and the nonequilibrium state where dephasing is applied to up spins (right). Population changes are enhanced near the Fermi surface due to dephasing (marked by grey dotted circles). The parameters are set to β = 2/J, αy = 1.1J, αz = 0.9J, and h = J. The initial state is at infinite temperature. sipator and internal Hamiltonian dynamics. Our finding is distinct from most of the previous studies that focused on inhomogeneous setups, where a system is coupled to different baths at its boundaries, thus relying on temperature biases or boundary driving. In particular, our open-system formulation provides a new versatile platform for studying current rectification, which gives a completely different framework from the conventional approach to nonreciprocal phenomena, such as magnetochiral effect, i.e., unidirectional electrical resistance due to DC electric fields [38,86,87], or transmissions of an electron current in the presence of a potential barrier [77,88]. From an experimental perspective, our results can be tested in ultracold atoms; the use of Raman-type spin-orbit coupling can be another possible candidate to break the inversion symmetry. One can also consider semiconductor quantum dots in GaAs as possible experimental candidates, where the spin relaxation time is very long [89][90][91][92]; spin-resolved dephasing should be realized by using the Zeeman shift.
To summarize, we have proposed a novel mechanism to realize a nonreciprocal current in open many-body systems. In contrast to conventional approaches, our finding provides a unique avenue for rectification, namely, the current is neither generated by temperature gradients nor boundary driving, but via the translationally invariant couplings to nonequilibrium baths. We have demonstrated that a nonreciprocal Lindblad operator in general rectifies the current in NESS. Importantly, we have revealed that even a reciprocal Lindblad operator can be used to rectify the current when the inversion symmetry and the time-reversal symmetry of the internal Hamiltonian are broken. The present analysis opens up various avenues of possible future research such as current rectification in higher dimensions or changes on transport properties by strong integrability breaking.

Supplemental Material for "Rectification in Nonequilibrium Steady States in Open Many-Body Systems" Detailed calculations of rate equations for Lagrange parameters
We here explain the detailed calculations to obtain the rate equations for Lagrange parameters. For the first model discussed in the main text, the local conservation laws of few-body observables are given by I q = c † q c q . Thus, χ qp in Eq. (6) in the main text is nonzero only for the diagonal components, given by where we have omitted the subscript · · · GGE and the same applies hereafter. Then, we calculate İ q = tr [I q L 1 ρ GGE ] on the right hand side of the rate equation (5) for the Lindblad operator (8) as where we used Wick's theorem. Here, we note that the terms that do not include q in kl becomes zero because such terms correspond to flows k → l or l → k (k, l = q) and do not contribute to the dynamics of I q . In the same way, we calculate İ q on the right hand side of the rate equation (5) for the Lindblad operator (9) as .
By using Eqs. (S1)-(S3), we obtain the rate equation (10) for the first model in the main text. For the second model (with the reciprocal dissipator), we can calculate the rate equation almost in the same way as discussed above. As the local conservation law is given by I qν = η † qν η qν (ν = ±), χ qν,pµ in Eq. (6) in the main text is zero for the off-diagonal components and the diagonal component is calculated as We see from Eq. (S4) that the degrees of freedom in momentum space are doubled by upper and lower energy bands compared to Eq. (S1). Then, İ qν on the right hand side of the rate equation (5) The contribution from the nonequilibrium bath, denoted as İ qν neq , can also be simplified by using the expression of the Lindblad operator (19): To use Wick's theorem, we substitute the Bogoliubov transformation c kσ = u σν (k)η kν in Eq. (S6) (for the detailed form of u σν (k), see the section "Detailed derivation of the quasiparticle operators" in Suppplemental Materials below). We note that, though we have to calculate 2 4 times as many terms as Eq. (S6) as a result of the substitution, many of which become zero since tGGE ensemble is defined by local conservation quantities. Then, we obtain .

Dynamics of Lagrange parameters without nonequilibrium driving
We numerically verify that Lagrange parameters go to the Gibbs state in NESS if there is no nonequilibrium driving. Figure S1 shows such relaxation dynamics of Lagrange parameters for the first model in the main text, which obeys Eq. (10) with γ eq = 1 satisfying the detailed balance condition Eq. (8). We see that the system goes to the Gibbs state (grey dashed lines) after sufficiently long time evolution. The behaviors are qualitatively the same for the second model except the fact that the degrees of freedom are doubled.  (S11) We see from Eq. (22) in the main text and Eq. (S11) that the contribution to Lagrange parameters from the nonequilibrium bath is inversion symmetric with respect to q if either one of the Zeeman magnetic field or the Rashba spin-orbit coupling is absent, which does not rectify the current. As a result, we need to break both the inversion symmetry and the time-reversal symmetry of the Hamiltonian to obtain the nonreciprocal current in NESS. We can also calculate the current (24) in the main text by using these quasiparticle operators as L q cos(q) sin(q) η † k+ η k+ − η † k− η k− (2α y sin k + h) 2 + 4α 2 z sin 2 k (S12) I ↓ = 2J L q sin(q) c † q↓ c q↓ − 2α y L q cos(q) c † q↓ c q↓ − 2iα z L q cos(q) c † q↓ c q↑ = 1 L q (J sin(q) − α y cos(q))   −2α y sin(q) − h (2α y sin k + h) 2 + 4α 2 z sin 2 k + 1   η † k+ η k+ +   2α y sin(q) + h (2α y sin k + h) 2 + 4α 2 z sin 2 k (S13) Results of the nonreciprocal current in NESS with down-spin dephasing We here give the numerical results of the current in NESS when dephasing is applied to down spins in the second model discussed in the main text. From Fig. S2(a), we see that the current rectifies in the opposite direction and the total current I has the reversed value of that in Fig. 4(a) in the main text. As shown in Fig. S2(b), the change of population near the Fermi surface where dephasing is applied becomes large (grey dotted circles) compared to the Gibbs state, which contributes to the current in the negative direction [see Eqs. (S12) and (S13)].