Spin transport through a spin-1/2 XXZ chain contacted to fermionic leads
We employ matrix-product state techniques to numerically study the zero-temperature spin transport in a finite spin-1/2 XXZ chain coupled to fermionic leads with a spin bias voltage. Current-voltage characteristics are calculated for parameters corresponding to the gapless XY phase and the gapped Néel phase. In both cases, the low-bias spin current is strongly suppressed unless the parameters of the model are fine-tuned. For the XY phase, this corresponds to a conducting fixed point where the conductance agrees with the Luttinger-liquid prediction. In the Néel phase, fine-tuning the parameters similarly leads to an unsuppressed spin current with a linear current-voltage characteristic at low bias voltages. However, with increasing the bias voltage, there occurs a sharp crossover to a region where a current-voltage characteristic is no longer linear and the smaller differential conductance is observed. We furthermore show that the parameters maximizing the spin current minimize the Friedel oscillations at the interface, in agreement with the previous analyses of the charge current for inhomogeneous Hubbard and spinless fermion chains.
Besides the more usual semiconductor- and metal-based spintronics, there have been proposals to use magnetic insulators in spin-based devices Kajiwara et al. (2010); Uchida et al. (2010); Hirobe et al. (2016). An advantage of these systems would be the absence of scattering due to conduction electrons, which may allow spin-current transmission over longer distances. Experiments have demonstrated the possibility to electrically induce a magnon spin current at a Pt/YFeO interface by using the spin-Hall effect Kajiwara et al. (2010). More recently, a spin current has been driven through the spin-1/2-chain material SrCuO by applying a temperature gradient Hirobe et al. (2016). This was interpreted as a spinon spin current induced by the spin-Seebeck effect.
A lot of research has been reported on the spin transport in the antiferromagnetic spin-1/2 XXZ chain, especially concerning the question whether the dynamics are ballistic or diffusive in the linear-response regime. At zero temperature, it is known from the exact Bethe-ansatz calculations that the spin transport is ballistic in the gapless phase and diffusive in the gapped phase Shastry and Sutherland (1990). There is considerable analytical and numerical evidence that this also holds true at any finite temperature Zotos (1999); Heidrich-Meisner et al. (2003); Žnidarič (2011); Karrasch et al. (2013a); Prosen (2011). A possible exception is the isotropic point for which differing results have been obtained.
Here, we study the finite-bias spin transport for a specific setup with fermionic leads at zero temperature. To this end, we employ the density-matrix renormalization group (DMRG) White (1992) and the real-time evolution of matrix-product states (MPS) via the time-evolving block decimation (TEBD) Vidal (2003). The difference from previous studies of transport in finite spin chains is our choice of the leads. In Refs. Prosen and Žnidarič, 2009; Benenti et al., 2009a, b, boundary driving modeled by a Lindblad equation was considered, which allows the direct calculation of the non-equilibrium steady state with matrix-product-operator techniques. Interestingly, a negative differential conductance was observed for strong driving in the gapped phase. Other studies have explored the transport in inhomogeneous XXZ chains van Hoogdalem and Loss (2011) and fermionic quantum wires coupled to non-interacting leads, which map to an XXZ chain through a Jordan-Wigner transformation Schmitteckert (2004); Bohr et al. (2006); Branschädel et al. (2010); Sedlmayr et al. (2012); Ponomarenko and Nagaosa (1999).
In setups with leads, the transport may be influenced by backscattering at the interfaces which, for repulsive interactions, can completely inhibit transport at low voltages and temperatures Kane and Fisher (1992a, b). In general, the strength of the backscattering will depend in a non-trivial way on the parameters on either side of the interface. In particular, it has been shown for typical models of fermionic chains that conducting fixed points with perfect conductance existSedlmayr et al. (2012, 2014); Morath et al. (2016).
The primary concern of this paper is to numerically explore the possibility of such conducting fixed points for our specific setup of the junction. We consider both the gapless XY and the gapped Néel phase of the spin-1/2 XXZ chain. In the latter case, the energy gap leads to insulating behavior at zero temperature. One may then ask, how the insulating state breaks down at finite bias voltage and how the transport depends on the length of the chain. The charge transport in a similar setup with a Mott-insulating Hubbard chain has been addressed, e.g., in Ref. Heidrich-Meisner et al., 2010. Here, we show that conducting fixed points exist not only for gapless but, in as sense, also for gapped spin chains. However, beyond a low-bias region with nearly ideal conductance the current-voltage curves at these fixed points are qualitatively different in the two regimes, with a smaller conductance in the gapped phase.
The rest of this paper is organized as follows. In Sec. II, we introduce the model and describe the numerical method employed. In Sec. III, we recapitulate some relevant field-theoretical results for the transport in inhomogeneous quantum wires. We then demonstrate in Sec. IV the existence of non-trivial conducting fixed points. To this end, we calculate steady-state spin currents and Friedel oscillations at the interface. In Sec. V, current-voltage curves for the gapless and the gapped regime are examined. Finally, Sec. VI summarizes our main results.
Ii Model and method
We consider a spin-1/2 XXZ chain that is coupled via an exchange interaction to fermionic leads modeled by uniform tight-binding chains at half-filling. For simplicity, the exchange coupling between the spin chain and the leads is set to be identical to the exchange interaction inside the spin chain. The resulting Hamiltonian is
describes the spin-1/2 XXZ chain containing sites. As usual, is the component of the spin-1/2 operator at site , and . The left (right) tight-binding lead is described by
where is the annihilation operator of an electron at site with spin . By defining the spin operators , and at tight-binding sites , the couplings between the spin chain and the left and right leads can be written as
We now calculate the steady-state spin current that is generated by applying a spin bias voltage . As in Ref. Heidrich-Meisner et al., 2010, the potential is assumed to drop off linearly in the spin chain, which adds the following term in the Hamiltonian (see also Fig.1):
The operator of the local spin-current is defined as
where and is the component of Pauli matrices Bari et al. (1970). Our transport simulations are carried out in the zero-temperature limit. Then the system is initially in the ground state at time . More precisely, the time evolution is started from the ground state of , where the spin chain and the leads are already coupled, and the spin bias voltage is applied at . As discussed in Refs. Branschädel et al., 2010; Einhellinger et al., 2012, other setups are possible. For example, if one starts with the two leads decoupled from the spin chain and turns on the coupling, the transient behavior is different but the same steady-state properties are obtained. If, however, the system is in the ground state with a finite value of spin bias voltage and the couplings and are switched on, different steady-state currents are expected for large Einhellinger et al. (2012).
For the numerical calculation of the steady-state current, we mostly follow the MPS-based approach of Refs. Branschädel et al., 2010; Heidrich-Meisner et al., 2010; Einhellinger et al., 2012. The DMRG and parallel TEBD are used, respectively, to calculate the ground state of and simulate the time-evolution after the spin bias (described by ) is switched on at . We employ a standard Suzuki-Trotter approximation where the Hamiltonian is decomposed into terms acting on even and odd bonds. Specifically, a second-order decomposition with time step is used. The leads have to be truncated to finite length , which gives rise to a discretization in the energy spectrum. The error due to this may be reduced by choosing appropriate boundary conditions with bond-dependent hopping strength that increase the energy resolution in the relevant energy region Bohr et al. (2006); Branschädel et al. (2010). Here, however, we find the leads with uniform hopping to be sufficient.
In our calculation of the steady-state current, the accuracy is mainly limited by the accessible time scale. The finite size of the leads obviously restricts the simulations to the time until the current reflected at the open boundaries of the leads returns to the spin chain. Additionally, the entanglement growth of an out-of-equilibrium state requires an increase of the bond dimension during the course of the time evolution, which eventually makes an accurate MPS representation of the state too costly. In the current setup, the von-Neumann entanglement entropy of the state after the perturbation grows linearly with the time Heidrich-Meisner et al. (2010), which requires an exponential increase of the bond dimension for a fixed truncation error. The rate of the entanglement growth depends strongly on the applied voltage . Simulation for large are typically more expensive. We fix the truncation error to a maximum discarded weight , which, in the worst cases, requires bond dimensions as large as .
In principle, an MPS representation with one tensor for each site in Eq. (1) could be used for all of our simulations. However, for small , where larger lead sizes are necessary to get accurate results, we find it advantageous to split the tight-binding leads into two branches with different component of the spin and employ a tree-tensor-network description Murg et al. (2010) analogous to Ref. Holzner et al., 2010. This algorithm scales as at the interfaces, instead of , but the representation of the tight-binding leads becomes much more efficient, allowing us to simulate larger leads. In addition, the worse scaling of the bond dimension is softened by the fact that the entanglement entropy at equilibrium is smallest at the interfaces, as already observed in Ref. Heidrich-Meisner et al., 2010.
Iii Luttinger liquid description
The low-energy physics of the spin-1/2 XXZ chain in the XY phase are described by the Luttinger-liquid (LL) model Giamarchi (2003)
where the bosonic fields obey the commutation relations and the LL parameter and the spin velocity are known from the Bethe-ansatz solution Luther and Peschel (1975). The charge transport in a system of spinless fermions with a nearest-neighbor interaction corresponds directly to the spin transport in the spin-1/2 XXZ chain since the models are related by a Jordan-Wigner transformation. For an infinite homogeneous chain, the spin conductance is given by Kane and Fisher (1992b)
In general, however, this expression is no longer valid when leads are taken into account. The effective low-energy Hamiltonian of the tight-binding leads in our setup described by Eq. (1) consists of two components of the form of Eq. (9) for the charge and spin sectors. Focusing only on the spin part, the naive field-theoretical description of our system becomes an inhomogeneous LL with the position-dependent LL parameter and spin velocity . It has been shown that the conductance of such a system is obtained by replacing the LL parameter in Eq. (10) with its asymptotic value in the leads Maslov and Stone (1995); Safi and Schulz (1995). For the non-interacting leads, the spin conductance therefore is , independent of the parameters in the spin chain.
By using an inhomogeneous LL model to describe a one-dimensional junction one assumes that backscattering at the interfaces can be neglected. This is justified for adiabatic contacts but not for the abrupt transition between the spin chain and the lead described in Eq. (1). For a chain of spinless fermions with uniform LL parameter , the effect of backscattering at an inhomogeneity on the conductance is well-known Kane and Fisher (1992a, b): At zero temperature, vanishes if the interactions are repulsive (i.e., ), while is not reduced for attractive interactions (i.e., ). An abrupt change in the system parameters of a quantum wire has a similar impact on the conductance, as has been studied for both spinless Sedlmayr et al. (2012, 2014) and spinful Morath et al. (2016) fermions using bosonization and quantum Monte Carlo methods. In those cases, whether the transport is suppressed at low temperatures depends on the LL parameters on each side of the interface. For the spinless model, the zero-temperature conductance vanishes for , where , and and are the LL parameters on the left and right sides of the interface Sedlmayr et al. (2012). Assuming that this result holds for the spin part of our model in Eq. (1), the backscattering at the interface between the spin chain and the lead should be relevant for all values of anisotropy , including negative values that correspond to . However, it was also shown that, even for abrupt junctions, conducting fixed points may be obtained by tuning certain system parameters such as the hopping and interaction strengths Sedlmayr et al. (2012); Morath et al. (2016). At these conducting fixed points, the backscattering amplitude becomes zero and thus the ideal conductance determined by the LL parameters of the leads is recovered. We will show in the following section that similar conducting fixed points with ideal spin transport exist for the spin-chain junction described in Eq. (1). In fact, our model at the isotropic point with corresponds to a strong-coupling limit of the inhomogeneous Hubbard chain for which such a conducting fixed point has been reported in Ref. Morath et al., 2016.
Iv Conducting fixed point
iv.1 Spin current
To search for conducting fixed points, we simulate the spin transport at finite spin bias for the two-lead setup described in Eq. (1). Let us first illustrate the procedure used to obtain the steady-state spin current. Figure 2(a) shows the spin current profile for different time after the spin bias is switched on at . The current starts to flow in the spin chain and spreads over to the leads, where the wavefront moves with the Fermi velocity . While the spin current in the spin chain becomes position and time independent in a true steady state, we find it fluctuating even at the maximum simulated time. Therefore, here the steady-state value is estimated from the time dependence of the spin current between the spin chain and the lead, as demonstrated in Fig. 2(b) for the isotropic chain. After a transient time of , the spin current oscillates around its steady-state value with a period of approximately . This kind of oscillation has been explained as a Josephson current that arises because of the finite size of the leads and the corresponding gap between the single-particle energy levels Branschädel et al. (2010). We calculate the steady-state value of the spin current either by simply averaging over multiple periods of the oscillation or by adapting it to
where and are fit parameters Branschädel et al. (2010).
The spin current generally depends on both the anisotropy and the ratio of the exchange interaction in the spin chain and the hopping amplitude in the leads. For most of the parameter space, the spin current is expected to be strongly suppressed because of the backscattering at the interfaces. As we will show, however, the system can be tuned to a conducting fixed point for each by varying . In the isotropic chain considered in Fig. 2, for example, the corresponding value is . The current there is much larger than for the other values shown, and , which lie away from the conducting fixed point.
The ratio affects not only the steady-state value of the spin current but also the oscillation of the current as a function of time . For a fixed size of the leads with , the current oscillation at the interface is strongest at where it appears nearly undamped [see Fig. 2(b)]. For either larger or smaller value of , on the other hand, the oscillation decays relatively quickly with increasing . By using the tree-tensor-network method, we also consider a junction with much larger leads of sites. In this case, the current oscillation at the conducting fixed point becomes significantly smaller, as shown in Fig. 2(b), which confirms that it is mostly caused by the discretization of the single-particle energy levels in the leads Branschädel et al. (2010). The steady-state values of the spin current estimated from the simulations are the same for and .
Figure 3 shows the dependence of the steady-state spin current on the ratio at the fixed spin bias voltage for several values of . In each case, a clear maximum of the spin current appears. We first address the gapless phase for , 0, and where the LL description is applicable. For parameters in this regime, the maximum current obtained is close to that for a LL with adiabatic contacts, which indicates that a conducting fixed point with ideal linear conductance exists at the corresponding ratio . As is decreased, becomes larger. In addition, the current peak as a function of broadens, which suggests that the backscattering becomes less relevant. A current maximum remains, however, even for negative .
Figure 3 also shows the results for in the gapped phase. While a sharp peak is still observed, the maximum value of the spin current does not reach the ideal value in this case. How the spin transport differs in the gapped and gapless phases of the antiferromagnetic XXZ chain will be analyzed in Sec. V.
iv.2 Friedel oscillations
Besides its effect on the transport, the backscattering at inhomogeneities is known to induce characteristic Friedel oscillations of the local density or magnetization with twice the Fermi wavenumber Rommer and Eggert (2000). The Friedel oscillations at the interface vanish, however, if the backscattering amplitude is tuned to zero. The calculation of the magnetization profile therefore constitutes a different, perhaps more efficient way to search a conducting fixed point Sedlmayr et al. (2012). As a consistency check for the results of the spin-transport simulations above, we now investigate the dependence of the Friedel oscillations on for fixed with no spin bias applied. Since the magnetization is uniform in the spin-flip symmetric case, we examine the local susceptibility Sedlmayr et al. (2012) instead by adding a small uniform magnetic field described by . For these calculations, we consider a single interface between the tight-binding lead and the spin chain because the Friedel oscillations typically decay over a distance longer than the spin-chain length accessible in our transport simulations. Furthermore, we consider finite temperatures by using the grand-canonical purification method Verstraete et al. (2004), which avoids problems in the convergence of the DMRG ground-state calculations. The purification method allows us to keep track of the growth of the Friedel oscillations starting from the interface and the open ends of the system as the temperature is lowered successively. We terminate the simulations when the finite system size begins to affect the results. The finite-temperature calculations also allow us to study the gapped phase of the spin chain where the ground state is antiferromagnetically long-range ordered.
Figure 4 shows the magnetization profile around the interface for the magnetic field strength . Here, we fix instead of because for the values of the anisotropy considered, the Friedel oscillations are much stronger in the spin chain than in the lead. Since the spin chain without magnetic field corresponds to a half-filled chain of fermions, the local magnetization oscillates with wavenumber . As expected, the effect is larger at low temperatures. For the fixed exchange anisotropy, the strength of the Friedel oscillations has a minimum as a function of . This behavior can be observed in both the gapless and gapped regimes.
To measure the overall strength of the Friedel oscillations, we introduce a quantity
where is chosen so that the Friedel oscillations due to the open boundary at the end of the spin chain are excluded. By calculating , we search for a value of that minimizes the Friedel oscillations for a fixed anisotropy .
The results for and are shown in Fig. 5. In all cases studied, including the gapped regime, we find a clear minimum of the Friedel oscillation strength where approximately . When the temperature is lowered, the position of the minimum moves to smaller . The temperature dependence seems to be stronger for small . By identifying the position of the minimum for as the conducting fixed point, we obtain for . This value agrees with the results of the spin-transport simulations for , despite the fact that we now consider the limit of a large spin chain. Identifying similarly in the gapped phase, we obtain for , which also coincides with the value of where the spin current becomes maximum in Fig. 3. When calculating as a function of the anisotropy , we find no qualitative difference across the phase boundary at .
V Current-voltage characteristics
Having established the existence of conducting fixed points with a finite linear conductance in the previous section, we now turn our attention to the spin bias dependence of the spin current. To examine how the current-voltage curve is modified by the backscattering at the interfaces and the presence of a finite energy gap, the system parameters at and away from the line of conducting fixed points are considered for both the gapless and gapped phases of the antiferromagnetic spin-1/2 XXZ chain.
v.1 Gapless regime
First, we study the gapless XY phase where the spin chain can be described by a LL model. As mentioned in Sec. III, a spin conductance is expected unless the transport is hindered by the backscattering at the interfaces. We have already confirmed that this ideal value can be obtained approximately at low spin bias by tuning to a conducting fixed point for a given anisotropy . By calculating the current-voltage curve, we can determine at what energy scale the LL description becomes invalid and the linear behavior breaks down. Figure 6 shows the results for the isotropic spin chain and the XX spin chain where the conducting fixed points are and , respectively (see Fig. 3 and Fig. 5). In both cases, the current-voltage curve for shows good agreement with the LL prediction up to at least , despite the strong inhomogeneity at the interfaces. For , increasing the length of the spin chain to leads to stronger deviations at large while the currents for remain nearly unchanged. Possible length-dependent corrections to the conductance have been considered, for example, in Refs. Sedlmayr et al., 2013; Matveev and Andreev, 2011.
Away from the conducting fixed points, the low-bias conductance is strongly reduced by backscattering. This is demonstrated in Fig. 6 for an isotropic chain and values and that are significantly smaller or larger than . In a LL with an impurity, the differential conductance eventually approaches the ideal value with a power law as the bias in increased Fisher and Glazman (1997). This is consistent with our results for where an approximately linear current-voltage relation is restored for . For , on the other hand, the differential conductance drops off again at , likely because the bias voltage considered is already comparable or larger than the exchange constant . In any case, the current should vanish in the large- limit because of the finite bandwidth of the leads Einhellinger et al. (2012); Branschädel et al. (2010).
v.2 Gapped regime
In Sec. IV, it was shown that the finite temperature Friedel oscillations around the interfaces can be tuned to zero by varying even in the gapped phase. Therefore, a fixed point with vanishing relevant backscattering seems to exist in this regime as well. One may then ask how the current-voltage curve there differs from that at a conducting fixed point in the gapless phase. In the following, we examine this for the anisotropy parameter where the Friedel oscillations disappear at .
Figure 7 displays the current-voltage curve of a spin chain with sites for as well as for smaller and larger values of . For , the conductance appears to approach as the voltage is decreased to zero, suggesting that almost ideal spin transport can indeed be achieved at low energy. At larger voltage, on the other hand, the differential conductance drops off sharply, which is not observed in the LL phase. This crossover occurs approximately at . As in the LL regime, the spin current at small bias voltage is strongly reduced away from . Since the XXZ spin chain with is a spin insulator in the thermodynamic limit, the spin transport should become increasingly suppressed with increasing the system size . The effect of on the current-voltage curve for is shown in Fig. 8. As expected, the spin current becomes smaller when is increased. There is still a crossover below which the perfect spin conductance seems to be approached. However, this crossover is shifted to smaller bias voltage with .
Similar behavior, i.e, unsuppressed current for small systems at low energy, occurs in the charge transport through Hubbard chains with an odd number of sites Oguri (2001). Perhaps more relevant to our model, such effect has been predicted for one-dimensional charge-density-wave insulators adiabatically contacted to non-interacting leads, by using field theoretical methods Ponomarenko and Nagaosa (1999). This model may be interpreted as a XXZ spin chain with the anisotropy set to zero outside a finite region with that corresponds to the charge-density-wave part. In contrast to our results, a negative differential conductance was obtained. However, this may be related to the different choice of the leads.
For sufficiently long spin chains, we observe an upturn of the spin current at large spin bias. A setup analogous to ours has been considered in the calculation of the charge current through a Mott-insulating Hubbard chain connected to non-interacting leads Heidrich-Meisner et al. (2010). It was shown that the current-voltage curve can be described by a function where and are constants. In particular, is approximately proportional to the square of the charge gap of the disconnected Hubbard chain. This relation was previously obtained for the current in a periodic chain and explained in terms of a Landau-Zener mechanism Oka et al. (2003). The upturn observed for in Fig. 8 suggests that a similar activated behavior occurs in our model for long enough chains where the low-voltage transport is suppressed. However, our available data is not sufficient to check the specific functional form and the dependence on the spin gap of the isolated spin chain.
We have numerically studied the finite-bias spin transport in a spin-1/2 XXZ chain connected to half-filled tight-binding leads at zero temperature, focusing on the effect of scattering at the interfaces. By calculating the steady-state spin current and the Friedel oscillations, it was shown that in the LL regime, conducting fixed points with the ideal linear conductance exist, similarly as in related models for inhomogeneous quantum wires. Our results furthermore indicate that conducting fixed points also appear in the gapped phase. There, however, the nearly ideal spin transport can only be observed in a small bias voltage region, which shrinks when the length of the spin chain is increased.
More difficult to treat numerically, but closer to actual experiments, is the finite temperature case. For the finite-temperature simulations, one could employ a similar TEBD method where the MPS describes a purification of the density matrix instead of a pure state. With the approach in Ref. Karrasch et al., 2013b, it may also be possible to study a setup where a spin current is driven by a temperature gradient, mimicking the experiment in Ref. Hirobe et al., 2016.
Here, we have only considered junctions composed of spin-1/2 chains. A possible extension would be to study analogous systems for spin ladders or chains with higher local spin. The spin-1 Heisenberg chain, for example, might be interesting since it is experimentally realizable and differs from the spin-1/2 chain in several aspects: Its elementary excitations are magnons instead of spinons, it is non-integrable, and it exhibits symmetry-protected edge states at open boundaries. For a setup with leads, the question then arises how the contact is affected by these edge states.
DMRG simulations were performed using the ITensor library ITe (). F. L. was supported by Deutsche Forschungsgemeinschaft through project FE 398/8-1 and by the International Program Associate (IPA) program in RIKEN. This work was supported by Grant-in-Aid for Scientific Research from MEXT Japan under the Grant No. 17K05523 and also in part by HPCI Strategic Programs for Innovative Research (SPIRE) (Nos. hp120246, hp140130, and hp150140). T. S. acknowledges the Simons Foundation for funding.
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