Generation of spin currents by a temperature gradient in a two-terminal device
R. E. Barfknecht, A. Foerster, N. T. Zinner, A. G. Volosniev
GGeneration of spin currents by a temperature gradient in a two-terminal device
R. E. Barfknecht,
1, 2, ∗ A. Foerster, † N. T. Zinner,
4, 5, ‡ and A. G. Volosniev § INO-CNR Istituto Nazionale di Ottica del CNR, Sezione di Sesto Fiorentino, 50019 Sesto Fiorentino, Italy LENS, European Laboratory for Non-Linear Spectroscopy, 50019 Sesto Fiorentino, Italy Instituto de F´ısica da UFRGS, Av. Bento Gon¸calves 9500, Porto Alegre, RS, Brazil Department of Physics and Astronomy, Aarhus University, Ny Munkegade 120, Denmark Aarhus Institute of Advanced Studies, Aarhus University, DK-8000 Aarhus C, Denmark Institute of Science and Technology Austria, Am Campus 1, 3400 Klosterneuburg, Austria (Dated: January 7, 2021)Theoretical and experimental investigations of the interaction between spins and temperature gradi-ents are vital for the development of spin caloritronics, and can dictate the design of future spintron-ics devices. In this work, we propose a two-terminal cold-atom simulator to study that interaction.The proposed quantum simulator consists of strongly interacting atoms that occupy two reservoirsconnected by a one-dimensional link. The reservoirs are kept at different temperatures. We showthe existence of a spin current in this system by studying the dynamics that follows a spin-flip of anatom in the link. We argue that the dynamics in the link can be described using an inhomogeneousHeisenberg chain whose couplings are defined by the local temperature. A temperature gradientaccelerates the impurity in one direction more than in the other, leading to an overall spin current.Therefore, our study offers a way to simulate certain features of the spin Seebeck effect with coldatoms.
INTRODUCTION
The coupling between charge and heat currents – thethermoelectric effect – was discovered more than twocenturies ago. Today, this effect is a standard topic inphysics textbooks [1], and is at the heart of thermoelec-tric generators and thermocouples. The spin thermoelec-trical effect – the interaction between heat and spin cur-rents – has a much shorter history [2–4], but it alreadydemonstrates the potential to complement the success ofits older sibling. Spin thermoelectrics encompasses spinSeebeck [5], spin-dependent Seebeck [6], spin-dependentPeltier [7] and related physical phenomena, which maylead to conceptually new devices based on the spin de-gree of freedom. While solid-state setups have providedcrucial insight into the problem of spin transport, theirlimited degree of tunability does not allow one to go be-yond the parameter regime given by the material at hand.Therefore, a logical next step is to explore spin thermo-electric effects using quantum simulators, in particular,cold-atom simulators, which provide a highly controllableenvironment for studying transport phenomena [8, 9].Features of cold-atom systems such as the possibility torealize low-dimensional geometries and to control inter-actions are particularly favorable for the study of spintransport.It has been proposed to simulate certain featuresof spin caloritronics using three-dimensional cold gaseswhere spin-up particles are separated from spin-downparticles by applying a spin-dependent temperature gra- ∗ [email protected]fi.it † [email protected] ‡ [email protected] § [email protected] dient [10, 11]. However, those proposals are experi-mentally challenging, especially in a strongly interactingregime. In this work, to study spin thermoelectrics forstrongly interacting systems, we propose a two-terminaldevice. Two-terminal cold-atom systems are a state-of-the-art interpretable platform for studying transportphenomena [12–15], which can allow for testing our the-oretical framework, and developing involved quantumtechnologies.We study the dynamics of a strongly interacting two-component atomic system in a small one-dimensional linkbetween two reservoirs at different temperatures. Thesystem has a large population imbalance, and we focuson the dynamics of the minority component. To describethe system, we employ a basic theoretical model, whichdoes not manifest any heat and charge transfer. Instead,it captures certain features of the magnon-driven spinSeebeck effect [16]. In particular, it contains microscopicphysics of a spin current in a ferromagnetic insulator.In our study, we rely on the bijection between astrongly interacting one-dimensional system and a spinchain. For zero temperature, this correspondence wasconfirmed using different approaches [17–21]. It hasbeen extensively developed and applied in subsequentworks [22–26]. Experimentally, the validity of the spin-chain description was verified in a few-body setup [27].In the present work, we employ the bijection to studythe dynamics of a finite-temperature system. In mostcases, we will focus on the dynamics that follows a singlespin flip in the link, namely the “impurity” model, whichconnects our findings to the physics of spin excitationsin cold-atom simulators [28]. Our observations can beeasily generalized to the spin current, which occurs whenspin impurities are placed in the reservoirs.For convenience of the reader, let us summarize ourmain findings: a r X i v : . [ c ond - m a t . qu a n t - g a s ] J a n
1. We introduce an effective Hamiltonian for studyingthe time dynamics of a strongly interacting one-dimensional system at finite temperatures. Thisallows us to circumvent the need to explicitly in-clude excited states in the analysis.2. Using the effective Hamiltonian from 1, we presenta microscopic theory in which a spin current is in-duced by a temperature gradient.Although, in this paper, we study a cold-atom system,our results are general and can be applied to other phys-ical systems described by our fundamental Hamiltonian(see the main text for details). For example, they can beused to describe quantum wires at low electron densitysuch GaAs [29].
I. SYSTEM
We consider a quasi-one-dimensional quantum wirewhose ends are connected to two infinite reservoirs, seeFig. 1. The system is spin-polarized at t <
0, i.e., it con-tains only ‘spin-up’ fermionic particles. The reservoirsare at two different temperatures, T and T , which arekept constant at all times. Moreover, the system is in asteady state at t (cid:39)
0, i.e., there are no mass currents. At t = 0, there is a spin flip of a particle in the link, whichcan be implemented in cold-atom experiments using mi-crowave or radio frequency pulses. The goal of this paperis to understand the quench dynamics at t > t <
0. Indeed, if one simply pre-pares a cold-atom system with a temperature mismatch,then particles will first flow from the hot reservoir intothe cold reservoir due the difference in chemical poten-tials, in agreement with the Landauer picture [38]. Later,the particle current might be reversed due to the differ-ence in the particle number between two reservoirs, and,eventually, the system will come to a thermal equilib-rium. We note that the timescale of particle transfer canbe tuned in these experiments [14, 39]. In particular, thistimescale can be made comparable or smaller than thetimescale associated with spin transfer in small stronglyinteracting one-dimensional systems, which is typically
FIG. 1. Sketch of the system studied in this work: a smalllink connects a hot (red, right) reservoir at temperatures T to a cold (blue, left) reservoir at T ( T > T ). Initially,the system is completely polarized (black ‘spin-up’ particles),and by assumption there is a time-independent temperaturegradient across the link. a) At t = 0 a particle at the centerof the system has its internal state changed by a spin-flippulse. b) Due to the presence of the temperature gradient,one observes a spin current across the system, which is causedby the motion of the impurity from the low- to the high-temperature region. . − µ s [20]. This makes the steady-state assumptionadequate.We consider the link between the reservoirs as a one-dimensional strongly interacting gas whose size and num-ber of particles is fixed. To simulate the effect of thereservoirs whose properties are not affected by the emis-sion and absorption of particles from the link, we em-ploy a Lindblad master equation (see Section IV), withspin-flip operators acting at the edges of the system [40].These play the role of the impurity being exchanged witha particle from a reservoir. We further include a ‘pump’term, which realizes a spin-flip at the center of the chain(an opposite effect to that of the operators acting at theedges). In this regime, we find that a steady-state currentarises, and we study its dependence on the temperaturegradient.After this brief description of the system, the rest ofthe paper is organized as follows: in Section II, we de-scribe the model of the link, assuming that the link is dis-connected from the reservoirs. We review the mappingof a strongly interacting confined system onto a spin-chain Hamiltonian. In Section III, we describe spin-flipdynamics in the link, assuming that there is no temper-ature difference, i.e., when T = T . We show that thesedynamics at small temperatures can be described usingan effective spin-chain Hamiltonian whose exchange coef-ficients depend on temperature. Such a Hamiltonian im-plies a homogeneous diffusion of the spin-impurity in thesystem, a feature that has been verified in experimentswith a bosonic Heisenberg chain [28]. In Section IV, weuse the effective Hamiltonian, along with the local den-sity approximation, to study the dynamics of the systemwhen T > T . We start with a single flipped spin in aclosed system, where the temperature gradient is mod-elled as a simple step function. Then, we take into con-sideration the effect of the spin reservoirs at the edgesof the system, as well as the possibility of having a spinpump acting at the center of the system. We find thatthe presence of the temperature gradient leads to a di-rectional motion of the spin-impurity, i.e., to an overallspin current. II. DESCRIPTION OF THE LINK
To model the link, we adopt the following ‘fundamen-tal’ Hamiltonian H = N ↑ (cid:88) i =1 h ( x i ) + N ↓ (cid:88) j =1 h ( y j ) + g (cid:88) i,j δ ( x i − y j ) , (1)where h ( x ) = − ¯ h m ∂ ∂x + V ( x ). Below, the trapping po-tential V ( x ) is a box potential, although our findings canbe extended to inhomogeneous potentials that changeweakly on the length scale given by the density of thegas. Particles in the system have identical masses m ,but can be differentiated by some internal degree of free-dom, which we label as ↑ , ↓ and refer to as the ‘spin’.The total number of particles, N = N ↑ + N ↓ , is time-independent, but the value of N ↑ ( N ↓ ) can be changed,e.g., by a spin-flip protocol (see below). In cold-atomexperiments, spin can be simulated using hyperfine [41]or nuclear spin states [42]. The interaction in Eq. (1) ismodelled by a delta-function potential. Its strength g isrelated to the three-dimensional scattering length and todetails of the trapping geometry [43]. For simplicity, weuse ¯ h = m = 1 in what follows.We assume that particles in the link are strongly inter-acting, in the sense that the energy scale associated withinteractions is much larger than any other energy scaleof the problem. This limit is often denoted as g → ∞ .It can be simulated with cold atoms at a few- and many-body levels [27, 44–46]. For 1 /g = 0 the Hamiltonian (1)can be related to a problem of N ↑ + N ↓ spin-polarizedfermions [47]. Let us assume that the spectrum of spin-polarized fermions is { E , E , ... } ( E ≤ E ≤ E ≤ ... );the set of the corresponding wave functions is given by { Ψ , Ψ , ... } . The function Ψ A yields ( N ↑ N ↓ )! /N ↑ ! N ↓ !wave functions of H each with the energy E A . Thesefunctions can be written as φ A,i ( x , ..., y N ↓ ) = ( N ↑ N ↓ )! N ↑ ! N ↓ ! (cid:88) P =1 a P Ψ A ( x , ..., y N ↓ ) G P , (2)where G P is an indicator function, G P determines a spe-cific ordering of particles, e.g., G = x < x < x < ... The parameter A determines the energymanifold (note that (cid:15) A,i (cid:39) E A for all values of i ), and i determines the position of the state within this manifold.We introduce the following notation to express this fact H (cid:39) (cid:88) A H A , (5)which means that any eigenstate in H (for g → ∞ ) canbe obtained using a set of eigenstates of H A from Eq. (2).To date, the focus of theoretical works was on the zero-temperature properties of the system, in particular, onthe lowest energy manifold described by H . To studythis manifold one uses Ψ – the Slater determinant madeof the N lowest states. The corresponding exchange co-efficients in Eq. (4) have been calculated with great pre-cision in systems as large as N ≈ 30 [49, 50], allowingone to study in depth static and dynamic properties ofthe system at T = 0 [22–25, 35, 51–54]. Here, we extendthe discussion of spin-flip dynamics incorporating excitedmanifolds to account for the role of temperature. III. SPIN-FLIP DYNAMICS AT FINITETEMPERATURES In this section, we study the quench dynamics thatfollow a spin-flip in the link [as in the previous section,the link is disconnected from the reservoirs]. We considerthe following spin-flip protocol at T (cid:54) = 0: At t < 0, thereare N spin-polarized fermions at temperature T , whichare described by the density matrix ρ = (cid:88) P A ( T ) | Ψ A (cid:105)(cid:104) Ψ A | , (6)where | Ψ A (cid:105) is a state whose spatial representation is Ψ A . P A ( T ) = e − βE A / (cid:80) A e − βE A , where β = 1 / ( k B T ), and k B is the Boltzmann constant. At t = 0, a single spin isflipped somewhere in the system, it can be a single-siteflip or a quantum superposition involving a few sites.Our goal is to understand the time dynamics at t > O ( t )can be calculated by considering different manifolds sep-arately, i.e., O ( t ) = (cid:88) P A ( T ) O A ( t ) , where O A ( t ) describes the time dynamics of the observ-able in a given manifold. The unitary map that deter-mines the corresponding time dynamics reads as e − i H A t .We assume that V ( x ) is a box potential, i.e., the potentialis zero if − L/ < x < L/ 2, and infinite otherwise. In thiscase the exchange coefficients are position-independentfor the ground state manifold [55], i.e., α i = α j . Wehave checked that α A ; i = α A ; j also for excited mani-folds (within numerical accuracy) [56], which allows usto simplify the notation: α A ≡ α A ; i . The correspondingHeisenberg Hamiltonians are homogeneous: H A = E A − α A g N − (cid:88) l =1 (1 − σ l · σ l +1 ) . (7)The coefficients α A can be calculated analytically: α A =2 E A /L . To calculate them, we notice that α A = (cid:80) N − i =1 α A ; i / ( N − α A allows usto integrate over the whole space ( x i ∈ [ − L/ , L/ α A isstraightforward.The quantity α A /g is the only parameter in Eq. (7)that can define a non-trivial time scale for the spin dy-namics. Time evolution of any observable can be writtenas O A ( t ) = f ( α A t ), where f is some function that de-pends on O A ( t ). The observable O is then given by O ( t ) = (cid:88) A P A ( T ) f ( α A t ) . (8)Our focus is on low temperatures, for which only lowenergy states are populated, and, hence, α A = α (1+ δ A ),where δ A (cid:28) 1. Using that (cid:80) P A = 1, we rewrite O as O ( t ) = f (cid:32) t (cid:88) A P A ( T ) J A (cid:33) + (cid:88) A O ( δ A ) . (9)This expression allows us to map the spin dynamics (upto the terms O ( δ A )) governed by H to the time dynamicsgoverned by the effective Hamiltonian h = α ( T ) g N − (cid:88) l =1 σ l · σ l +1 , (10)where α ( T ) = 2 (cid:15) ( T ) /L , and (cid:15) ( T ) = (cid:80) A P A ( T ) E A isthe average energy of a system of N spinless fermions attemperature T . Our derivation shows that the exchangecoefficient α ( T ) is a natural extrapolation of the zero-temperature coefficient, α (0) = 2 (cid:15) (0) /L [54], to finite T .Equation (10) is the main result of this section. We propose to use the Hamiltonian (10) for the analy-sis of the time dynamics in strongly interacting systemsat finite temperatures. The advantage of this Hamilto-nian over Eq. (5) is that it can be analyzed using well-developed approaches to the spin-chain Hamiltonians,see, e.g., [57]. In particular, the spin-flip dynamics canbe studied using the spectrum of magnons, which for ourXXX system reads as E M ( p, T ) = α ( T ) g (1 − cos( p )) , (11)here p is a quasi-momentum, p = 2 πn/N with an integer n . In the limit of N (cid:29) n (infrared limit), we can write E M ( p, T ) (cid:39) α ( T ) g p , which allows us to study the dynam-ics using a free-particle picture with the temperature-dependent effective mass: m eff = g/α ( T ). This renor-malization of the effective mass of a magnon can be ob-served in cold-atom experiments. In particular, if weassume that at t = 0 the wave packet of a spin-impurityhas a Gaussian profile, then the probability density atlater times is | ψ ( x, t ) | = e − x t /m (cid:115) π (1 + 4 t /m ) . (12)This formula shows how the effective mass changes thetime dynamics, which can be observed in situ [28]. Byexploring a single-particle picture further, we concludethat the experiment should observe a diffusive behaviorof an initial spin flip with the diffusion coefficient thatdepends on the temperature as: D = [2 m eff ( T )] − .Let us discuss the dependence of the coefficients α ( T )on temperature in the thermodynamic limit. For smalltemperatures, the expansion of the energy (cid:15) ( T ) allows usto write α ( T ) ≈ α (cid:34) (cid:18) k B T L N π (cid:19) π + 12 (cid:35) . (13)where α = α ( T = 0) (that is, the exchange coeffi-cient calculated using solely the ground state manifold).The expression shows a T -dependence of the coefficients α ( T ) in this limit. For large temperatures, the equipar-tition theorem requires (cid:15) ( T ) be proportional to T . Notethat the mapping onto the effective Hamiltonian (10) failsin this limit where the values of α ( T ) are large. There-fore, large temperatures are outside the scope of this pa-per.Finally, we discuss α ( T ) for a finite number of par-ticles, since we are mainly interested in experimentallyrelevant small links between the reservoirs. In Fig. 2,we show the behavior of α ( T ) for N = 5 , k B T by the energy per particlein the thermodynamic limit (with density ρ = 1), thatis (cid:15) = π / 6. All in all, the results for finite N agreewell with the predictions of Eq. (13) for k B T /(cid:15) (cid:28) FIG. 2. The temperature dependence of the exchange co-efficients α ( T ) for different values of N . α denotes thevalue of these coefficients at T = 0. The black dashed curveshows the prediction of Eq. (13). The inset zooms in the re-gion k B T /(cid:15) (cid:28) 1. All presented quantities are dimensionless. which implies that finite-size effects are not important inthis limit. To ensure the convergence of the curves inFig. 2, we use A = 10 . Small values of the number ofmanifolds ( A (cid:39) ) do not lead to accurate results for α ( T ) for the considered parameters. For the time dy-namics, this implies that a simultaneous consideration of A = 10 manifolds is needed. The effective Hamiltonian h provides a convenient way to incorporate these manymanifolds. It is worthwhile noting that α ( T ) /α (cid:39) k B T /(cid:15) (cid:46) 2; therefore, a necessary condition for the va-lidity of the mapping of H onto h can be satisfied in anexperimentally accessible window of temperatures.In the following, we will use Eq. (10) along with the lo-cal density approximation to study dynamical propertiesof an impurity in the presence of a temperature gradient.The Hamiltonian h can be used to describe also spin dy-namics that follow more than a single spin flip (arbitrarynumber of magnons). However, it is important to no-tice that this Hamiltonian cannot be used to study staticproperties, since we explicitly rely on time dynamics inits derivation. IV. DYNAMICS IN THE PRESENCE OF ATEMPERATURE GRADIENT Once we have established the mapping (10), we can useit to investigate the quench dynamics. Quench dynamicsare often considered to understand transport propertiesof the many-body Heisenberg model and related Hamilto-nians, especially the transition from ballistic to diffusiveregimes, see, e.g., [58–61]. Here we study quench dynam-ics to investigate the effect of a temperature gradient onthe motion of an impurity. We stress that, within ourformalism, the simplest case of a constant temperaturefield across the system leads only to faster dynamics incomparison to the zero-temperature limit, and does notintroduce any additional effects. The temperature gradi-ent is essential for the findings discussed in this section. We first consider an infinite link ( L → ∞ ) where thegradient is defined by a step function, i.e., T = T for x < T = T for x > 0. We apply the local densityapproximation to write α ( T ) [ α ( T )] for the exchangecoefficients at x < x > x (cid:39) − m eff ( x ) ∂ ∂x f = Ef, (14)which describes the infrared dynamics of the impurity inthe thermodynamic limit; m eff = g/α ( T ) for x < m eff = g/α ( T ) for x > 0. The solution we are afterreads as f = (cid:40) e − ik x , x < e ik x , x > , where k α ( T ) = k α ( T ) = 2 gE . These expressionsconstitute a phenomenological description of a source ofparticles with a given energy E at x (cid:39) 0. The flux thatcorresponds to these solutions is given by j P ( x < 0) = (cid:112) Eα ( T ) /g and j P ( x > 0) = (cid:112) Eα ( T ) /g . We seethat the particle is more likely to move into the regionwith high temperature. The ratio of the probability cur-rents reads as j P ( x > j P ( x < 0) = (cid:115) α ( T ) α ( T ) (cid:39) T − T ) (cid:18) k B L N π (cid:19) π + 14 . (15)Notice the quadratic dependence of the currents on tem-perature for T → 0. This dependence is typical for thelow-temperature spin currents in our model.The derivations above rely on the thermodynamiclimit, and we need to investigate finite systems sepa-rately. Below, we illustrate the dynamics that follows aspin flip in a small link with N = 7. We initially considera closed system, for which the link is decoupled from thereservoirs, and then an open system where the reservoirsare modelled using the Lindblad master equation. In allcalculations, we assume dimensionless time and tempera-ture units by writing J t and k B T /(cid:15) respectively, where J = α /g . The corresponding time scale for cold-atomexperiments can be tuned by changing the density of par-ticles and the value of g . To interpret our results in thissection, one could use values 0 . − µ s [20], which aretypical for cold alkali atoms, e.g., Li. FIG. 3. The contour plot presents (cid:104) S ↓ i ( t ) (cid:105) as a function oftime, t , and spin site, i . The data are for an N = 7 systemwith an impurity initialized at the center with a) k B ∆ T = 0and b) k B ∆ T = 2 (cid:15) . The presence of the temperature dif-ference across the system leads to a directional motion of theimpurity towards the high-temperature part of the system.All presented quantities are dimensionless. A. Closed system Following our discussion above, we consider the effec-tive model (10) with the exchange coefficients α i = (cid:40) α ( T ) , i < ( N + 1) / α ( T ) , i ≥ ( N + 1) / , where N = 7, 1 ≤ i ≤ N − 1, and we fix k B T = 0 and k B T = 2 (cid:15) . For convenience, we define the temperaturedifference as ∆ T = ( T − T ). For this particular systemsize, we have (cid:15) /(cid:15) F = 1 / 3, where (cid:15) F is the Fermi energy.For the sake of discussion, we take as the initial state | ψ (cid:105) = |↑↑↑↓↑↑↑(cid:105) , and then consider time evolution ofthis state under the effect of the temperature difference,i.e., we solve the Schr¨odinger equation iψ (cid:48) = hψ with ψ ( t = 0) = ψ . We start by calculating the averageprobability for the impurity to be found at a given site, (cid:104) S ↓ i ( t ) (cid:105) = (cid:104) ψ ( t ) | S ↓ i | ψ ( t ) (cid:105) , (16)where S ↓ i = ( − σ zi ) / 2. Figure 3 illustrates the expec-tation value of this observable for k B ∆ T = 2 (cid:15) , and thezero-temperature case. The temperature difference leadsto a higher probability for the motion of the impurity to-wards the edge with a higher temperature, in agreementwith our discussion for the thermodynamic case. FIG. 4. Time evolution of the probability of finding the im-purity at the sites adjacent to the center of the chain for k B ∆ T = 2 (cid:15) . The red solid (blue dotted) curve shows theresult obtained with Eq. (3) for (cid:104) S ↓ ( t ) (cid:105) ( (cid:104) S ↓ ( t ) (cid:105) ). The blackcurves show the predictions of Eq. (17) with α = α ( T ) (gray,dashed) and α = α ( T ) (black, dot-dashed) at i = 5 (the prob-ability (17) for i = 3 is the same). All presented quantitiesare dimensionless. At ∆ T = 0, the dynamics of (cid:104) S ↓ i ( t ) (cid:105) can be foundanalytically, e.g., by using the Bethe ansatz. The prob-ability of finding the impurity at a particular site for alarge system reads as [63] (cid:104) S ↓ i ( t ) (cid:105) = [ J i ( J t )] , (17)where i denotes the lattice site, and J is the Bessel func-tion of the first kind. This result is expected to describea finite system for short times, see also an experiment ofRef. [28]. In Fig. 4, we compare our result for (cid:104) S ↓ i ( t ) (cid:105) to the predictions of Eq. (17). For this comparison,we choose the sites adjacent to the center of the chain i = N +12 ± 1. Figure 4 shows that the temperature gradi-ent introduces an asymmetry in the motion of the impu-rity, which occurs predominantly towards the larger tem-perature. The figure compares exact results to those ofEq. (17) obtained with the coefficients α ( T ) and α ( T ).We conclude that Eq. (17) can be used to study initialdynamics also for ∆ T (cid:54) = 0. At later times the inhomo-geneity and finite-size effects start to play an importantrole, and Eq. (17) fails. B. Open system We now take into account the two reservoirs depictedin Fig. 1. To that end, we work with the master equationthat describes time evolution of the spin density matrix, ρ s ( t ), ∂ρ s ( t ) ∂t = − i ¯ h [ h, ρ s ( t )]+ γ (cid:88) i =1 ,N (cid:0) S − i ρ s ( t ) S + i − { ρ s ( t ) , S + i S − i } (cid:1) , (18)where [ · · · ] and {· · ·} denote the commutator and anti-commutator, respectively, and the jump operators aregiven by S + = ( σ x + iσ y ) / S − = ( σ x − iσ y ) / γ , whichwe shall always present in units of J . The parameter γ describes the rate at which spins |↓(cid:105) are flipped to |↑(cid:105) [64].Notice that the coupling to the reservoir occurs only atthe edges of the spin chain.We write the density matrix at t = 0 as ρ s (0) = | ψ (cid:105)(cid:104) ψ | , where | ψ (cid:105) is the initial state considered in theprevious section, that is | ψ (cid:105) = |↑↑↑↓↑↑↑(cid:105) . In this sec-tion, we consider a more realistic linear temperature gra-dient across the system, which within the local densityapproximation leads to a set of continuously increasingcoefficients α i . The parameter ∆ T = T − T specifies thedifference of the temperatures at the edges of the system.As before, we fix T = 0. Our main focus is on the totalspin current j ( t ) = N − (cid:88) i (cid:104) ψ ( t ) | (cid:0) σ iy σ i +1 x − σ ix σ i +1 y (cid:1) | ψ ( t ) (cid:105) (19)which expresses the net spin motion in the system, andthe total magnetization m ( t ) = 12 N (cid:88) i (cid:104) ψ ( t ) | σ zi | ψ ( t ) (cid:105) . (20)The quantity m registers the effect of the loss terms con-tained in Eq. (18).Figure 5 shows time evolution of j ( t ) and m ( t ). Thespin current occurs in the presence of a finite temperaturegradient, see panel a). The amplitude of this current iscontrolled by ∆ T . For t → ∞ the current vanishes dueto the effects of the losses at the edges. Such an effectcan also be detected through the total magnetization:we find that m ( t → ∞ ) → N/ 2, which indicates that theimpurity is completely lost to the reservoirs, and the linkbecomes fully polarized.Next, we consider a scenario where, instead of a single-impurity state at t = 0, the initial state state is fullypolarized: | ψ ( t = 0) (cid:105) = |↑↑↑↑↑↑↑(cid:105) . The dynamics isinitiated by adding the spin-flip term to Eq. (18) P ( t ) = γ I (cid:16) S + N +12 ρ s ( t ) S − N +12 − { ρ s ( t ) , S − N +12 S + N +12 } (cid:17) . (21) P ( t ) acts only at the center of the chain and can be in-terpreted as a constant spin ‘pump’ that introduces spin-down spins in the system at a rate given by γ I , which weshall present in units of J .Figure 6 a) presents time evolution of the total currentfor different choices of ∆ T . As in Fig. (5) a), no currentis generated if ∆ T = 0. For a finite gradient, however,we observe a transient regime for small t , evolving to-wards a steady state at longer times. In the steady state,the effect of the losses at the edges matches the spin-flip term in the center. In the inset, we also show the FIG. 5. Time evolution [according to Eq. (18)] of a) the to-tal spin current and b) the total magnetization with γ/J =3 . γ is rather arbitrary here. It does not change the overall dy-namics, but only the time scale for reaching equilibrium. Allpresented quantities are dimensionless. behavior of the magnetization, which, contrarily to theprevious case, now drops from the fully polarized value toa constant determined by the parameters of the masterequation. It is worthwhile analyzing the behavior of thesteady-state current (i.e., j ( t → ∞ )) for different temper-ature gradients: we observe an increase in this quantitywith ∆ T in all cases, see Fig. 6 b). For T → 0, this in-crease is quadratic in temperature, which is in agreementwith our previous remark regarding the low- T limit. Wefind a noticeable sensitivity of the values of the steady-state current on the parameters γ I and γ . For example,for very large values of γ I , the system becomes saturatedwith spin-down particles, which reduces the total current.Our conclusion however always holds: a temperature gra-dient leads to an overall spin current for all consideredparameters and protocols. V. SUMMARY We study the dynamics of spin impurities placed ina small link between two reservoirs of different temper-atures. The link here is a one-dimensional system of FIG. 6. a) Time evolution [according to Eqs. (18) and (21)] ofthe total spin current for different values of the temperaturegradient with γ/J = 3 . 102 and γ I /J = 0 . γ I . The black dashed curve shows aquadratic fit for k B ∆ T /(cid:15) < 1. All presented quantities aredimensionless. strongly interacting cold atoms. The reservoirs are sim-ulated using an open-system approach. We show thatthe dynamics of the system can be obtained by consid-ering a spin chain whose exchange coefficients dependon temperature. Our argument is based upon the Bose- Fermi correspondence and the local density approxima-tion. Having established the effective spin-chain Hamil-tonian, we consider the motion of a single spin impurityinitialized at the center of the link. We observe thatthe motion of the impurity is highly influenced by thetemperature gradient. The impurity moves towards thehighest-temperature reservoir, leading to a spin currentin the system. Next, we consider a spin pump at thecenter of the system. In this case, the system evolvestowards a steady-state regime with a non-vanishing spincurrent whose strength depends on the temperature gra-dient. The formalism presented here can be applied todifferent atomic models, which can be mapped onto spin-chain Hamiltonians. For instance, the same formalismcan be applied to study a one-dimensional bosonic systemwith strong interactions, which realizes a XXZ Hamilto-nian [20, 22]. The spectrum of magnons in such a systemcan be modified by manipulating the boson-boson inter-actions, and we expect that the dynamics of an impuritycan be made different from that presented here.Our study provides a microscopic description of thecoupling between the spin degree of freedom and temper-ature for strongly interacting one-dimensional systems.It paves the way for studying spin caloritronics (and re-lated quantum technologies) with quantum simulators,in particular, cold-atom simulators. VI. 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