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arXiv:2109.11230 · New Journal of Physics

Energy transport in Z3 chiral clock model

Naveen Nishad, G. J. Sreejith

Published Dec 30, 2021• 0 citations
arXivHTMLPDF

Abstract

Abstract We characterize the energy transport in a one dimensional Z 3 chiral clock model. The model generalizes the Z 2 symmetric Transverse Field Ising Model (TFIM). The model is parametrized by a chirality parameter θ, in addition to f and J which are analogous to the transverse field and the nearest neighbor spin coupling in the TFIM. Unlike the well-studied TFIM and XYZ models, this model does not transform to a fermionic system. We use a matrix product states implementation of the Lindblad master equation to obtain the Non-Equilibrium Steady State (NESS) in systems of sizes up to 48. We present the estimated NESS current and its scaling exponent γ as a function of θ at different f/J. The estimated γ(f/J, θ) points to ballistic energy transport along a line of integrable points f = J cos 3θ in the parameter space; all other points deviate from ballistic transport. Analysis of finite size effects within the available system sizes suggest diffusive behavior away from the integrable points.

DOI
10.1088/1367-2630/ac4736

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Abstract We characterize the energy transport in a one dimensional Z 3 chiral clock model. The model generalizes the Z 2 symmetric Transverse Field Ising Model (TFIM). The model is parametrized by a chirality parameter θ, in addition to f and J which are analogous to the transverse field and the nearest neighbor spin coupling in the TFIM. Unlike the well-studied TFIM and XYZ models, this model does not transform to a fermionic system. We use a matrix product states implementation of the Lindblad master equation to obtain the Non-Equilibrium Steady State (NESS) in systems of sizes up to 48. We present the estimated NESS current and its scaling exponent γ as a function of θ at different f/J. The estimated γ(f/J, θ) points to ballistic energy transport along a line of integrable points f = J cos 3θ in the parameter space; all other points deviate from ballistic transport. Analysis of finite size effects within the available system sizes suggest diffusive behavior away from the integrable points.
We characterize the energy transport in a one dimensional Z3{\mathbb{Z}}_{3}Z3​ chiral clock model. The model generalizes the Z2{\mathbb{Z}}_{2}Z2​ symmetric Transverse Field Ising Model (TFIM). The model is parametrized by a chirality parameter θ , in addition to f and J which are analogous to the transverse field and the nearest neighbor spin coupling in the TFIM. Unlike the well-studied TFIM and XYZ models, this model does not transform to a fermionic system. We use a matrix product states implementation of the Lindblad master equation to obtain the Non-Equilibrium Steady State (NESS) in systems of sizes up to 48. We present the estimated NESS current and its scaling exponent γ as a function of θ at different f / J . The estimated γ ( f / J , θ ) points to ballistic energy transport along a line of integrable points f = J cos 3 θ in the parameter space; all other points deviate from ballistic transport. Analysis of finite size effects within the available system sizes suggest diffusive behavior away from the integrable points.
We employ matrix product state simulations to study energy transport within the nonintegrable regime of the one-dimensional Z3{\mathbb{Z}}_{3}Z3​ chiral clock model. To induce a nonequilibrium steady state throughout the system, we consider open system dynamics with boundary driving featuring jump operators with adjustable temperature and footprint in the system. Given a steady state, we diagnose the effective local temperature by minimizing the trace distance between the true local state and the local state of a uniform thermal ensemble. Via a scaling analysis, we extract the transport coefficients of the model at relatively high temperatures above both its gapless and gapped low-temperature phases. In the medium-to-high temperature regime we consider, diffusive transport is observed regardless of the low-temperature physics. We calculate the temperature dependence of the energy diffusion constant as a function of model parameters, including in the regime where the model is quantum critical at the low temperature. Notably, even within the gapless regime, an analysis based on power series expansion implies that intermediate-temperature transport can be accessed within a relatively confined setup. Although we are not yet able to reach temperatures where quantum critical scaling would be observed, our approach is able to access the transport properties of the model over a broad range of temperatures and parameters. We conclude by discussing the limitations of our method and potential extensions that could expand its scope, for example, to even lower temperatures.
The heat conductivity of nonperiodic quantum Ising chains whose ends are connected with heat baths at different temperatures are studied numerically by solving the Lindblad master equation. The chains are subjected to a uniform transverse field h, while the exchange coupling J{m} between the nearest-neighbor spins takes the two values J{A} and J{B} arranged in Fibonacci, generalized Fibonacci, Thue-Morse, and period-doubling sequences. We calculate the energy-density profile and energy current of the resulting nonequilibrium steady states to study the heat-conducting behavior of finite but large systems. Although these nonperiodic quantum Ising chains are integrable, it is clearly found that energy gradients exist in all chains and the energy currents appear to scale as the system size ~N{α}. By increasing the ratio of couplings, the exponent α can be modulated from α > -1 to α < -1 corresponding to the nontrivial transition from the abnormal heat transport to the heat insulator. The influences of the temperature gradient and the magnetic field to heat conduction have also been discussed.
The heat conduction in an open transverse-field Ising chain is studied by using quantization in the Fock space of operators in the weak-coupling regimes, i.e. the coupling is much smaller than the transverse field. The non-equilibrium steady state is obtained for large-size systems coupled to Markovian baths at its ends. The ballistic transport is observed in the uniform chain and normal diffusion in the random exchange chain. In addition, the ballistic-diffusive transition is found at the intermediate disorder regime. The thermal conductivity κ is also calculated in the low- and high-temperature regimes. It is shown that κ decays as κ∼ T −2 at high temperatures.
Anomalous transport in low dimensional spin chains is an intriguing topic that can offer key insights into the interplay of integrability and symmetry in many-body dynamics. Recent studies have shown that spin-spin correlations in spin chains, where integrability is either perfectly preserved or broken by symmetry-preserving interactions, fall in the Kardar-Parisi-Zhang (KPZ) universality class. Similarly, energy transport can show ballistic or diffusive-like behavior. Although such behavior has been studied under equilibrium conditions, no results on nonequilibrium spin transport in classical spin chains has been reported so far. In this work, we investigate both spin and energy transport in classical spin chains (integrable and nonintegrable) when coupled to two reservoirs at two different temperatures/magnetizations. In both the integrable case and the broken-integrability (but spin-symmetry preserving) case, we report anomalous scaling of spin current with system size (J^{s}∝L^{-μ}), with an exponent value that, within error bars, is close to the KPZ universality class value μ≈2/3. On the other hand, it is noteworthy that the energy current remains ballistic (J^{e}∝L^{-η} with η≈0) in the purely integrable case while there is departure from ballistic behavior (η>0) when integrability is broken regardless of spin-symmetry. We also present results on other interesting observables in the nonequilibrium steady state such as the spatial profiles of magnetization and energy, and spin-spin correlations.
In this work we study the heat transport in an XXZ spin-1/2 Heisenberg chain with homogeneous magnetic field, incoherently driven out of equilibrium by reservoirs at the boundaries. We focus on the effect of bulk dephasing (energy-dissipative) processes in different parameter regimes of the system. The non-equilibrium steady state of the chain is obtained by simulating its evolution under the corresponding Lindblad master equation, using the time evolving block decimation method. In the absence of dephasing, the heat transport is ballistic for weak interactions, while being diffusive in the strongly-interacting regime, as evidenced by the heat-current scaling with the system size. When bulk dephasing takes place in the system, diffusive transport is induced in the weakly-interacting regime, with the heat current monotonically decreasing with the dephasing rate. In contrast, in the strongly-interacting regime, the heat current can be significantly enhanced by dephasing for systems of small size.