[Paper Review] Dynamical and excited-state quantum phase transitions in collective systems

Ángel L. Corps, A. Relaño|arXiv (Cornell University)|May 23, 2022

Quantum many-body systems99 references39 citations

TL;DR

This paper establishes a unified theoretical framework linking dynamical quantum phase transitions (DPTs-I and DPTs-II) in infinite-range interacting many-body systems to excited-state quantum phase transitions (ESQPTs). It shows that DPTs-I emerge from symmetry restoration after quenches from broken-symmetry states, with critical behavior demarcated by an ESQPT, while DPTs-II are forbidden in one ESQPT phase due to conserved charges. The key contribution is a generalized microcanonical ensemble incorporating two conserved charges that quantifies information loss and explains non-analytic dynamics.

ABSTRACT

We study dynamical phase transitions (DPTs) in quantum many-body systems with infinite-range interaction, and present a theory connecting the two kinds of known DPTs (sometimes referred to as DPTs-I and DPTs-II) with the concept of excited-state quantum phase transition (ESQPT), traditionally found in collective models. We show that DPTs-I appear as a manifestation of symmetry restoration after a quench from the broken-symmetry phase, the limits between these two phases being demarcated precisely by an ESQPT. We describe the order parameters of DPTs-I with a generalization of the standard microcanonical ensemble incorporating the information of an additional conserved charge identifying the corresponding phase. We also show that DPTs-I are linked to a mechanism of information erasure brought about by the ESQPT, and quantify this information loss with the statistical ensemble that we propose. Finally, we show analytically that DPTs-II are forbidden in these systems for quenches leading a broken-symmetry initial state to the same broken-symmetry phase, on one side of the ESQPT, and we provide a formulation of DPTs-II depending on the side of the ESQPT where the quench ends. We analyze the connections between various indicators of DPTs-II. Our results are numerically illustrated in the infinite-range transverse-field Ising model and are applicable to a large class of collective quantum systems satisfying a set of conditions.

Motivation & Objective

• To unify the understanding of two distinct types of dynamical quantum phase transitions (DPTs-I and DPTs-II) in many-body systems.
• To identify the role of excited-state quantum phase transitions (ESQPTs) as the critical boundary separating distinct dynamical phases.
• To develop a generalized microcanonical ensemble incorporating two conserved charges to describe long-time dynamics in DPT-I regimes.
• To explain the mechanism of information erasure during adiabatic crossing of the ESQPT critical energy.
• To clarify the conditions under which DPTs-II are allowed or forbidden, depending on the quench endpoint relative to the ESQPT.

Proposed method

• Analytical derivation of the role of a conserved operator ˆC that commutes with energy projectors only below the ESQPT critical energy.
• Construction of a generalized microcanonical ensemble incorporating two conserved charges to describe long-time averages of order parameters in DPT-I.
• Use of semiclassical analysis to understand the emergence of non-analytic behavior in return probabilities and order parameters.
• Numerical validation using the infinite-range transverse-field Ising model, with focus on quench dynamics and return probability non-analyticities.
• Formulation of DPTs-II in terms of the quenched state's energy relative to the ESQPT critical energy, distinguishing between normal and anomalous phases.
• Application of statistical mechanics and large-N limits to derive the thermodynamic behavior of collective systems with Z2 symmetry.

Experimental results

Research questions

• RQ1 How are DPTs-I and DPTs-II connected to the excited-state quantum phase transition (ESQPT) in collective quantum systems with infinite-range interactions?
• RQ2 Why is the main mechanism for DPTs-II forbidden in one phase separated by the ESQPT, particularly when quenching to a broken-symmetry state?
• RQ3 What is the role of conserved charges in determining the dynamical behavior and information loss during quench dynamics across the ESQPT?
• RQ4 How does the generalized microcanonical ensemble, incorporating two conserved charges, describe the long-time dynamics of DPT-I order parameters?
• RQ5 What is the physical origin of the distinction between normal and anomalous dynamical phases in DPTs-II, and how is it determined by the quench endpoint relative to the ESQPT?

Key findings

• DPTs-I arise as a manifestation of symmetry restoration after a quench from a broken-symmetry state, with the critical point precisely located at the ESQPT energy.
• The order parameter in DPTs-I is non-zero only below the ESQPT critical energy, where the conserved operator ˆC commutes with energy projectors in the thermodynamic limit.
• The generalized microcanonical ensemble incorporating two conserved charges successfully describes the long-time oscillatory behavior of DPT-I order parameters.
• Information erasure during adiabatic crossing of the ESQPT critical energy is quantified by the proposed statistical ensemble, linking it to non-analytic dynamics.
• DPTs-II are forbidden in the broken-symmetry phase when the quench ends in the same phase, due to the non-commutation of ˆC with energy projectors above the ESQPT.
• The sum of return probabilities to parity-broken ground states equals the survival probability in the same ESQPT region, providing a key consistency check for DPTs-II.