[Paper Review] Dynamical deconfinement transition driven by density of excitations

Nishan Ranabhat, Alessandro Santini+2 more|arXiv (Cornell University)|Oct 3, 2023

Quantum, superfluid, helium dynamicsPhysics and Astronomy3 citations

TL;DR

This paper investigates a dynamical deconfinement transition in long-range Ising spin models driven by thermal excitation density, using tensor network simulations to track real-time evolution of kink density and fluctuations. It identifies a transition from localized (confined) to delocalized (deconfined) kinks as temperature increases, with kink fluctuations serving as a key non-equilibrium signature of the transition, offering insights into confinement mechanisms in quantum spin systems and lattice gauge theories.

ABSTRACT

We investigate the deconfinement transition driven by excitations in long-range spin models. At low temperatures, these models exhibit a confined phase where domain-wall (or kinks) are localized. As temperature increases, kinks interact and propagate, leading to a transition to a de-confined phase. This transition is influenced by the interplay between thermal energy and interaction effects, resulting in extended, de-confined regions. Although kinks density is dynamically stable, non-equilibrium changes in their fluctuations characterize the transition. Our findings provide insights into the mechanisms of confinement and deconfinement in long-range spin models, with implications for both condensed matter physics and lattice gauge theories. Bridging these fields, this study sheds light on the universal aspects of confinement and opens avenues for further exploration and experimental verification.

Motivation & Objective

• To understand the mechanism of confinement and deconfinement transitions in long-range spin models driven by thermal excitation density.
• To explore the role of kink density and fluctuations in characterizing the transition from a confined to a deconfined phase.
• To simulate real-time dynamics of thermal states in the long-range Ising model using matrix product density operators (MPDO) and tensor network methods.
• To establish a connection between spin models and lattice gauge theories by demonstrating analogous confinement behavior.
• To provide a theoretically and experimentally accessible framework for probing non-equilibrium confinement transitions.

Proposed method

• The study employs a ferromagnetic long-range Ising model (LRIM) with power-law decaying interactions, governed by Hamiltonian $\hat{H}(J,\alpha,h) = -J\sum_{i<j} \frac{\hat{\sigma}^x_i \hat{\sigma}^x_j}{|i-j|^\alpha} - h\sum_i \hat{\sigma}^z_i $, with $J=1$.
• Thermal initial states are prepared as density matrices $\rho_\beta \propto e^{-\beta \hat{H}_0}$, with $\beta = 1/T$, and evolved in real time under a final Hamiltonian using the Time-Dependent Variational Principle (TDVP) algorithm with $dt = 0.05$.
• The kink density $\langle \hat{k} \rangle$ and kink fluctuation $\text{Var}(\hat{k})$ are computed at each time step to detect the transition.
• Full counting statistics of kink density are calculated via Fourier transformation of the trace $\text{Tr}[\rho e^{i\theta \hat{O}}]$, enabling reconstruction of the kink distribution.
• A semi-classical effective model with a single kink is used to theoretically predict the observed transition behavior.
• Convergence of simulations is verified using increasing bond dimensions ($\chi_{\text{max}} = 80, 100, 128$), with relative errors in kink density below $O(10^{-3})$.

Figure 1: Schematic diagram of the quench protocol: 1) Initial states are prepared as thermal density matrices of the initial Hamiltonian, $\propto e^{-\beta\hat{H}_{0}}$ at varying temperatures ( $\beta=1/T$ ). 2) The initial thermal density matrix is evolved in real time with a final Hamiltonian $

Experimental results

Research questions

• RQ1 How does the density of thermal excitations (kinks) drive a dynamical deconfinement transition in long-range spin models?
• RQ2 What role do kink fluctuations play as a non-equilibrium signature of the confinement-deconfinement transition?
• RQ3 How does the interplay between thermal energy and long-range interactions affect kink localization and propagation?
• RQ4 To what extent do tensor network simulations accurately capture the real-time dynamics of thermal states in long-range Ising models?
• RQ5 Can the observed transition be described by a semi-classical effective model of a single kink?

Key findings

• The average kink density $\langle \hat{k} \rangle$ monotonically decreases with increasing inverse temperature $\beta$, indicating a thermal transition into the ferromagnetic phase where $\langle \hat{k} \rangle = 0$.
• Kink fluctuations exhibit strong signatures of the deconfinement transition, with a sharp increase at intermediate temperatures, signaling the onset of delocalized, propagating kinks.
• For $\alpha \leq 2$, the thermal phase transition is robust; for $\alpha = 2.3$, the observed transition is an artifact of finite-size effects.
• The real-time evolution of the thermal density matrix shows richer dynamics than pure states, with kink density and fluctuation both serving as sensitive probes of the transition.
• Convergence tests confirm that bond dimensions up to $\chi_{\text{max}} = 128$ are sufficient, with relative errors in kink density below $O(10^{-3})$ across all tested parameters.
• The semi-classical model with a single kink successfully predicts the observed transition behavior, supporting the validity of the effective description.

Figure 2: (a) Time evolution of kink density following a quench to $\alpha=1.8$ , $h_{f}=0.4$ , system is initialized at $\alpha=1.8$ , $h_{i}=0.0$ at different temperatures. The horizontal dashed lines represent the kink densities of the thermal states corresponding to the quenches. The results are