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Energy transport in chiral clock model

Naveen Nishad, G. J. Sreejith

Published Jan 1, 2022• 5 citations
Publisher

Abstract

We characterize the energy transport in a one dimensional ${\mathbb{Z}}_{3}$ chiral clock model. The model generalizes the ${\mathbb{Z}}_{2}$ 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 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.

DOI
10.1088/1367-2630/ac4736

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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.
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 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 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.
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.
Recent studies report on anomalous spin transport for the integrable Heisenberg spin chain at its isotropic point. Anomalous scaling is also observed in the time-evolution of non-equilibrium initial conditions, the decay of current-current correlations, and non-equilibrium steady state averages. These studies indicate a space-time scaling with x∼t2/3x \sim t^{2/3}x∼t2/3 behaviour at the isotropic point, in sharp contrast to the ballistic form x∼tx \sim tx∼t expected for integrable systems. In our contribution we study the scaling behaviour for the Faddeev-Takhtajan model, which is an integrable classical spin chain with the same symmetries as the quantum Heisenberg model. We report on equilibrium time correlations and the evolution with step initial conditions. Remarkably, the scaling function is identical to the one obtained from Kardar-Parisi-Zhang equation. In addition, we present results for the easy-plane and easy-axis regimes for which, repectively, ballistic and diffusive spin transport is observed, whereas the energy remains ballistic over the entire parameter regime.
We numerically investigate the heat conduction in a random-exchange Ising spin chain by using the quantum master equation. The chain is subject to a uniform transverse field hhh, while the exchange couplings Qn{{Q}_{n}}Qn​ between the nearest neighbor spins are random; the largest size we simulate is up to 10. This model is integrable; i.e., the nearest neighbor level spacing distribution is Poissonian. However, we find clear evidence of nonballistic transport. In the small coupling regime $({Q}_{n}\ensuremath{\ll}h)$, an energy and/or temperature gradient in the bulk of the system is observed and the energy current appears to be proportional to the inverse of the system size. Moreover, we find that in the low and high temperature regimes, the thermal conductivity $\ensuremath{\kappa}$ and the specific heat Cv{C}_{v}Cv​ have the same dependence on temperature. The large coupling case $({Q}_{n}\ensuremath{\sim}h)$ is also discussed.