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arXiv:2311.00046 · Physical review. B./Physical review. B

Temperature dependence of energy transport in the Z 3 chiral clock model

Yongchan Yoo, Brian G. Swingle

Published Jun 3, 2024• 1 citation
arXivHTMLPDF

Abstract

We employ matrix product state simulations to study energy transport within the nonintegrable regime of the one-dimensional ${\mathbb{Z}}_{3}$ 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.

DOI
10.1103/physrevb.109.235104

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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.
We study the temperature dependence of energy diffusion in two chaotic gapped quantum spin chains, a tilted-field Ising model and an XZ model, using an open-system approach. We introduce an energy imbalance by coupling the chain to thermal baths at its boundary and study the nonequilibrium steady states of the resulting Lindblad dynamics using a matrix product operator ansatz for the density matrix. We define an effective local temperature profile by comparing local reduced density matrices in the steady state with those of a uniform thermal state. We then measure the energy current for a variety of driving temperatures and extract the temperature dependence of the energy diffusion constant. For the Ising model, we are able to study temperatures well below the energy gap and find a regime of dilute excitations where rare three-body collisions control energy diffusion. A kinetic model correctly predicts the observed exponential increase of the energy diffusion constant at low temperatures. For the XZ model, we are only able to access intermediate to high temperatures relative to the energy gap, and we show that the data are well described by an expansion around the infinite temperature limit. We also discuss the limitations of the particular driving scheme and suggest that lower temperatures can be accessed using larger baths.
In this work we analyze the simultaneous emergence of diffusive energy transport and local thermalization in a nonequilibrium one-dimensional quantum system, as a result of integrability breaking. Specifically, we discuss the local properties of the steady state induced by thermal boundary driving in a XXZ spin chain with staggered magnetic field. By means of efficient large-scale matrix product simulations of the equation of motion of the system, we calculate its steady state in the long-time limit. We start by discussing the energy transport supported by the system, finding it to be ballistic in the integrable limit and diffusive when the staggered field is finite. Subsequently, we examine the reduced density operators of neighboring sites and find that for large systems they are well approximated by local thermal states of the underlying Hamiltonian in the nonintegrable regime, even for weak staggered fields. In the integrable limit, on the other hand, this behavior is lost, and the identification of local temperatures is no longer possible. Our results agree with the intuitive connection between energy diffusion and thermalization.
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 develop a general framework for studying the equilibrium and non-equilibrium properties of arbitrary networks of Sachdev-Ye-Kitaev clusters coupled to thermal baths. We proceed to apply this technique to the problem of energy transport, which is known to be diffusive due to the strange metal behavior of these models. We use the external baths to impose a temperature gradient in the system and study the emerging non-equilibrium steady state using the Schwinger-Keldysh formalism. We consider two different configurations for the baths, implementing either a boundary or bulk driving, and show that the latter leads to a significantly faster convergence to the steady state. This setup allows us to compute both the temperature and frequency dependence of the diffusion constant. At low temperatures, our results agree perfectly with the previously known values for diffusivity in the conformal limit. We also establish a relationship between energy transport and quantum chaos by showing that the diffusion constant is upper bounded by the chaos propagation rate at all temperatures. Moreover, we find a simple analytical form for the non-equilibrium Green's functions in the linear response regime and use it to derive exact closed-form expressions for the diffusion constant in various limits. We mostly focus on uniform one-dimensional chains, but we also discuss higher-dimensional generalizations.
We consider charge and spin transport in the one-dimensional Hubbard model at infinite temperature, half-filling, and zero magnetization. Implementing matrix-product-operator simulations of the nonequilibrium steady states of boundary-driven open Hubbard chains for up to 100 sites we find clear numerical evidence of diffusive transport for weak driving and any (nonzero and finite) value of the interaction UUU. In addition, nonequilibrium steady states are characterized by nonzero spin-up spin-down density-density correlations. For maximal driving, on the other hand, we find an insulating behavior and cosine-shaped density profiles.