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Stochastic-geometrical analysis to investigate critical behavior for statistical-mechanical models

悟 半田

Published Mar 25, 2019• 0 citations
Publisher

Abstract

This work is based on the paper [49].In Chapter 4, we will introduce the quantum Ising model, especially focusing on the graphical representations of it.By using the Suzuki-Trotter transformation and a graphical representation based on the classical Ising model, we will analyze the susceptibility of the quantum Ising model.This is an ongoing work with Kamijima and Sakai and based on the preprint [50].

DOI
10.14943/doctoral.k13554

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This work is based on the paper [49].In Chapter 4, we will introduce the quantum Ising model, especially focusing on the graphical representations of it.By using the Suzuki-Trotter transformation and a graphical representation based on the classical Ising model, we will analyze the susceptibility of the quantum Ising model.This is an ongoing work with Kamijima and Sakai and based on the preprint [50].
Driven-dissipative many-body systems are difficult to analyze analytically due to their nonequilibrium dynamics, dissipation, and many-body interactions. In this paper, we consider a driven-dissipative infinite-range Ising model with local spontaneous emission, which naturally emerges from the open Dicke model in the large-detuning limit. Utilizing an adaptation of the Suzuki-Trotter quantum-to-classical mapping, we develop an exact field-theoretical analysis and a diagrammatic representation of the spin model that can be understood from a simple scattering picture. With this representation, we are able to analyze critical behavior, finite-size scaling, and the effective temperature near the respective phase transition. Our formalism further allows a detailed study of the ordered phase where we find a ``heating'' region within which the effective temperature becomes negative, thereby exhibiting a truly nonequilibrium behavior. At the phase transition, we find two distinct critical behaviors with overdamped and underdamped critical dynamics at generic and weakly dissipative critical points, respectively. We further show that the underdamped critical behavior is robust against short-range perturbations and is not an artifact of the mean-field nature of the model. To treat such perturbations, we extend our diagrammatic representation to include the coupling to spin waves due to the short-range interactions. The field-theoretical approach and the diagrammatics developed in this work should prove useful in applications to generic short-range driven-dissipative spin systems.
The Suzuki-Trotter transformation has been used to transform NNN-spin S=12S=\frac{1}{2}S=21​ Heisenberg and XY\mathrm{XY}XY chains into two-dimensional ($N\ifmmode\times\else\texttimes\fi{}2m$) classical Ising models with complicated interactions. A Monte Carlo method using multispin flipping techniques is then used to study these models for $\frac{\mathrm{kT}}{|J|}\ensuremath{\ge}0.0125$ and $m\ensuremath{\le}10$. Finite-size scaling is used to analyze the low-temperature susceptibility of the ferromagnetic Heisenberg chain, and we find a power-law divergence $\ensuremath{\chi}={T}^{\ensuremath{-}\ensuremath{\gamma}}$ with $\ensuremath{\gamma}=1.32\ifmmode\pm\else\textpm\fi{}0.13$. Difficulties in the method, particularly due to singular bonding, are discussed.
Simulating an open quantum system on a digital quantum computer often involves the use of Trotter-Suzuki (TS) Product Formulas (PF) to approximate the system's time evolution. Precise estimates for the required number of Trotter steps (and hence the overall gate count) can be crucial for minimising the computational cost of these methods. Building on established theoretical guarantees, we derive analytic bounds for the First- and Second-Order Deterministic and Randomised TS-PF, directly relating the number of Trotter steps to the model parameters, evolution time and precision. These bounds enable concrete resource estimation for each method. We then present a computationally efficient classical algorithm that uses diamond norm estimates of individual Liouvillian terms and a binary search to significantly reduce the Trotter steps required for a target precision. Our numerical results on two prototypical models - an XX-Spin Chain with boundary driving and local dephasing, and a Transverse-Field Ising Model - show that the theoretical (analytic) bounds are often overly conservative, whereas the empirical (optimised) bounds yield a significantly smaller number of Trotter steps for the same precision. Among the methods investigated, the Second-Order Randomised TS-PF typically achieves the lowest resource demands, especially for larger systems. These findings emphasise the significance of empirical bounding strategies in achieving more resource-efficient simulations of Markovian open quantum systems.