[Paper Review] Dynamical quantum phase transitions in many-body localized systems

Jad C. Halimeh, Nikolay Yegovtsev+1 more|arXiv (Cornell University)|Mar 7, 2019

Cold Atom Physics and Bose-Einstein Condensates4 citations

TL;DR

This paper investigates dynamical quantum phase transitions (DQPTs) in many-body localized (MBL) systems using the l-bit formalism and exact diagonalization. It demonstrates that DQPT singularities can emerge in MBL phases under specific conditions, linking them to the structure of l-bits and non-thermalizing dynamics, with experimental relevance in ultracold atoms and ion traps.

ABSTRACT

We investigate dynamical quantum phase transitions in disordered quantum many-body models that can support many-body localized phases. Employing $l$-bits formalism, we lay out the conditions for which singularities indicative of the transitions appear in the context of many-body localization. Using the combination of the mapping onto $l$-bits and exact diagonalization results, we explicitly demonstrate the presence of these singularities for a candidate model that features many-body localization. Our work paves the way for understanding dynamical quantum phase transitions in the context of many-body localization, and elucidating whether different phases of the latter can be detected from analyzing the former. The results presented are experimentally accessible with state-of-the-art ultracold-atom and ion-trap setups.

Motivation & Objective

• To determine whether dynamical quantum phase transitions (DQPTs) can occur in many-body localized (MBL) systems, which do not thermalize after a quench.
• To investigate the conditions under which singularities in the Loschmidt return rate—signatures of DQPTs—arise in MBL phases.
• To establish a connection between the emergent l-bit structure in MBL systems and the presence or absence of DQPT singularities.
• To provide a theoretical and numerical framework for detecting MBL phases through dynamical signatures rather than equilibrium properties.
• To assess the experimental feasibility of observing these DQPTs in state-of-the-art ultracold-atom and ion-trap quantum simulators.

Proposed method

• Employing the l-bit formalism to map the MBL Hamiltonian into a quasi-local basis of conserved operators (l-bits), enabling analysis of non-thermal dynamics.
• Using perturbation theory to construct l-bits τj as corrections to spin operators σxj, up to second order in the disorder strength h relative to exchange couplings Jn.
• Deriving effective Hamiltonian terms in the l-bit basis, including new three-spin interactions K(3)n,n+2,n+4 ∝ hδj/2, which emerge at second order in perturbation theory.
• Computing the Loschmidt amplitude and return rate using the mapped Hamiltonian to detect non-analyticities indicative of DQPTs.
• Combining analytical perturbative results with exact diagonalization to validate the presence and absence of DQPT singularities in finite-size systems.
• Analyzing the spectral form factor and level correlations to connect DQPTs with long-range energy-level correlations in integrable-like MBL systems.

Experimental results

Research questions

• RQ1 Under what conditions do dynamical quantum phase transitions (DQPTs) manifest in many-body localized systems?
• RQ2 How do the emergent l-bits in MBL systems influence the appearance of singularities in the Loschmidt return rate?
• RQ3 Can DQPTs serve as a dynamical probe for distinguishing MBL phases from thermalizing phases?
• RQ4 What is the role of higher-order perturbative corrections in the l-bit construction in enabling or suppressing DQPTs?
• RQ5 Are the DQPT singularities in MBL systems robust to disorder and finite-size effects, and how do they compare to those in clean integrable systems?

Key findings

• DQPT singularities are found in the Loschmidt return rate for a candidate MBL model, indicating that dynamical phase transitions can occur in non-thermalizing systems.
• The presence of DQPTs is linked to the existence of long-range, quasi-local l-bits that emerge in the MBL phase, particularly through second-order perturbative corrections.
• A new three-spin interaction term K(3)n,n+2,n+4 ∝ hδj/2 appears in the effective Hamiltonian at second order in perturbation theory, contributing to the dynamics that may support DQPTs.
• In the regime h/J₀ ≪ 1, no singularities are observed in the return rate, suggesting a suppression of DQPTs at weak disorder, consistent with the breakdown of dynamical criticality.
• The spectral form factor and level correlations in the MBL regime suggest strong energy-level correlations, which are necessary for DQPTs, supporting the connection between DQPTs and non-thermalizing dynamics.
• The results are experimentally accessible in ultracold-atom and ion-trap platforms, where the Loschmidt return rate and level statistics can be measured following a quantum quench.