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arXiv:1810.11807 · Europhysics Letters (EPL)

Driving XXZ spin chains: Magnetic-field and boundary effects

F. Lange, Satoshi Ejima, H. Fehske

Published Jan 11, 2019• 5 citations
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Abstract

Using the time-evolving block decimation, we study the spin transport through spin-(1/2) and spin-1 XXZ chains subjected to an external magnetic field and contacted to noninteracting fermionic leads. For generic system-lead couplings, the spin conductance exhibits several resonances as a function of the magnetic-field strength. In the spin-(1/2) but not the spin-1 case, the coupling to the leads can be fine-tuned to reach a conducting fixed point, where the peak structure is washed out and the spin conductance is large throughout the gapless Luttinger-liquid phase. For the Haldane phase of the spin-1 chain, we analyse how the spin transport is affected by spin-(1/2) edge states, and argue that two-impurity Kondo physics is realised.

DOI
10.1209/0295-5075/125/17001

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Using the time-evolving block decimation, we study the spin transport through spin-(1/2) and spin-1 XXZ chains subjected to an external magnetic field and contacted to noninteracting fermionic leads. For generic system-lead couplings, the spin conductance exhibits several resonances as a function of the magnetic-field strength. In the spin-(1/2) but not the spin-1 case, the coupling to the leads can be fine-tuned to reach a conducting fixed point, where the peak structure is washed out and the spin conductance is large throughout the gapless Luttinger-liquid phase. For the Haldane phase of the spin-1 chain, we analyse how the spin transport is affected by spin-(1/2) edge states, and argue that two-impurity Kondo physics is realised.
We employ matrix-product state techniques to numerically study the zero-temperature spin transport in a finite spin-12\frac{1}{2}21​ 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 N\'eel 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 N\'eel 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 the current-voltage characteristic is no longer linear and a 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.
Using a one-dimensional spin-orbital model as a typical example of quantum spin systems with richer symmetries, we study the effect of an isolated impurity on its low-energy dynamics in the gapless phase through bosonization and renormalization-group methods. In the case of internal impurities, depending on the symmetry, the boundary fixed points can be either an open chain with a residual spin or (and) orbital triplet left behind, or a periodic chain. However, these two fixed points are indistinguishable in the sense that in both cases, the lead-correction-to-scaling boundary operators (LCBO) only show Fermi-liquid-like corrections to thermodynamical quantities (except the possible Curie-like contributions from the residual moments in the former cases). In the case of external (Kondo) impurities, the boundary fixed points, depending on the sign of orbital couplings, can be either an open chain with an isolated orbital doublet due to Kondo screening or an intermediate fixed point with the same LCBO as that of the two-channel Kondo problem. Comparison with the Kondo effect in one-dimensional Heisenberg spin chain and multiband Hubbard models is also made.
We study the real-time dynamics of the Kondo effect after a quantum quench in which a magnetic impurity is coupled to two metallic Hubbard chains. Using an effective field theory approach, we find that for noninteracting electrons the charge current across the impurity is given by a scaling function that involves the Kondo time. In the interacting case, we show that the Kondo time decreases with the strength of the repulsive interaction and the time dependence of the current reveals signatures of the Kondo effect in a Luttinger liquid. In addition, we verify that the relaxation of the impurity magnetization does not exhibit universal scaling behavior in the perturbative regime below the Kondo time. Our results highlight the role of nonequilibrium dynamics as a valuable tool in the study of quantum impurities in interacting systems.
Spin-1/2 chains demonstrate some of the striking effects of interactions and quantum fluctuations in one-dimensional systems. The XXZ model has been used to study the unusual properties of anisotropic spin chains in an external magnetic field. The zero temperature phase diagram for this model exhibits a critical or quasi-long-range-ordered phase which is a realization of a Luttinger liquid. While many static properties of spin-1/2 chains have been explained by combinations of analytical techniques such as bosonization and Bethe ansatz, the standard approach fails in the calculation of some time-dependent correlation functions. I present a study of the longitudinal dynamical structure factor for the XXZ model in the critical regime. I show that an approximation for the line shape of the dynamical structure factor in the limit of small momentum transfer can be obtained by going beyond the Luttinger model and treating irrelevant operators associated with band curvature effects. This approach is able to describe the width of the on-shell peak and the high-frequency tail at finite magnetic field. Integrability is shown to affect the low-energy effective model at zero field, with consequences for the line shape. The power-law singularities at the thresholds of the particle-hole continuum are investigated using an analogy with the X-ray edge problem. Using methods of Bethe ansatz and conformal field theory, I compute the exact exponents for the edge singularities of the dynamical structure factor. The same methods are used to study the long-time asymptotic behavior of the spin self-correlation function, which is shown to be dominated by a high-energy excitation.
We study the behavior of Heisenberg, antiferromagnetic, integer-spin chains in the presence of a magnetic field exceeding the attendant spin gap. For temperatures much smaller than the gap, the spin chains exhibit Luttinger liquid behavior. We compute exactly both the corresponding Luttinger parameter and the Fermi velocity as a function of magnetic field. This enables the computation of a number of correlators from which we derive the spin conductance, the expected form of the dynamic structure factor relevant to inelastic neutron-scattering experiments, and NMR relaxation rates. We also comment upon the robustness of the magnetically induced gapless phase both to finite temperature and finite couplings between neighboring chains.
We investigate the real-time dynamics of a critical spin-1/2 chain (XXZ model) prepared in an inhomogeneous initial state with different magnetizations on the left and right halves. Using the time-evolving block decimation method, we follow the front propagation by measuring the magnetization and entanglement entropy profiles. At long times, as in the free fermion case [Antal et al., Phys. Rev. E 59, 4912 (1999)], a large central region develops where correlations become time independent and translation invariant. The shape and speed of the fronts is studied numerically and we evaluate the stationary current as a function of initial magnetic field and as a function of the anisotropy $\ensuremath{\Delta}$. We compare the results with the conductance of a Tomonaga-Luttinger liquid, and with the exact free-fermion solution at $\ensuremath{\Delta}=0$. We also investigate the two-point correlations in the stationary region and find a good agreement with the ``twisted'' form obtained by Lancaster and Mitra [Phys. Rev. E 81, 061134 (2010)] using bosonization. Some deviations are nevertheless observed for strong currents.