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Dynamical quantum phase transitions: a review

Markus Heyl

arXiv:1709.07461v2cond-mat.stat-mechcond-mat.quant-gasquant-ph

TL;DR

Nonequilibrium quantum dynamics lacks conventional free-energy concepts, complicating the search for general principles. This review synthesizes DQPT theory and observations, defining transitions through time-dependent nonanalyticities and discussing the open questions that remain.

  • Problem

    Nonequilibrium quantum states are not captured by thermodynamic descriptions, leaving open how to identify general principles such as universality without free-energy-based tools.

    Source: §1 · p. 2

  • Method

    The review pedagogically synthesizes DQPT theory, experimental observations, and prospects, including their account through complex partition-function zeros.

    Source: §1 · p. 2, §2.5 · p. 6

  • Results

    The review describes DQPT nonanalyticities as arising when Fisher-zero lines or area boundaries cross the real-time axis.

    Source: §2.5 · p. 6

  • Takeaways & Limitations

    DQPTs are presented as a potential principle for understanding quantum many-body dynamics, though a comprehensive nonequilibrium theory remains a first step away.

    Source: §1 · p. 2, §5 · p. 14

    A general macroscopic description, nonequilibrium Landau-Ginzburg theory, and free-energy-like organizing principle remain open questions.

    Source: §7 · p. 22

Abstract

from arXiv · show

Quantum theory provides an extensive framework for the description of the equilibrium properties of quantum matter. Yet experiments in quantum simulators have now opened up a route towards generating quantum states beyond this equilibrium paradigm. While these states promise to show properties not constrained by equilibrium principles such as the equal a priori probability of the microcanonical ensemble, identifying general properties of nonequilibrium quantum dynamics remains a major challenge especially in view of the lack of conventional concepts such as free energies. The theory of dynamical quantum phase transitions attempts to identify such general principles by lifting the concept of phase transitions to coherent quantum real-time evolution. This review provides a pedagogical introduction to this field. Starting from the general setting of nonequilibrium dynamics in closed quantum many-body systems, we give the definition of dynamical quantum phase transitions as phase transitions in time with physical quantities becoming nonanalytic at critical times. We summarize the achieved theoretical advances as well as the first experimental observations, and furthermore provide an outlook onto major open questions as well as future directions of research.

1. Introduction

Nonequilibrium quantum states can fall outside thermodynamic descriptions, leaving open how to identify general principles such as universality without free-energy-based tools. This review introduces dynamical quantum phase transitions (DQPTs), surveys theoretical and experimental progress, and outlines open questions.

1. Introduction

  • Nonequilibrium states lack a thermodynamic description and conventional tools such as free energies, raising questions about universality and dynamical analogs of phases.The review frames these questions as a challenge for understanding quantum many-body dynamics on general grounds.Source: §1 · p. 2
  • DQPTs describe transient-time nonanalyticities in physical quantities and are driven by time rather than conventional control parameters.The review relates them to the broader effort to identify phase-transition concepts in far-from-equilibrium systems.Source: §1 · p. 2
  • The review pedagogically introduces DQPTs, summarizes experimental observations, and identifies prospects for understanding quantum many-body dynamics.It also surveys the field’s key features in a self-contained account.Source: §1 · p. 2
  • The review proceeds from DQPT foundations and topological examples to experiments, criticality concepts, broader implications, and future directions.Topics include scaling, universality, order parameters, entanglement dynamics, and mixed states.Source: §1 · p. 2

2. Dynamical quantum phase transitions

This section defines DQPTs through nonanalytic Loschmidt amplitudes during closed-system quantum dynamics and develops their connection to complex partition functions and equilibrium criticality. It also emphasizes that DQPTs are not simply equivalent to conventional phase transitions.

2.1. Closed quantum many-body systems

  • The review focuses on closed many-body systems whose dynamics can be treated as unitary, a setting approximated by quantum simulators such as ultracold atoms and trapped ions.These platforms have system-environment coupling times much longer than internal system timescales.Source: §2.1 · p. 3

2.2. Nonequilibrium protocol: Quantum quenches

  • A global quantum quench suddenly changes a Hamiltonian parameter after ground-state preparation, producing nontrivial dynamics when the initial state is not an eigenstate of the final Hamiltonian.The review focuses mainly on quenches with an extensive change in internal energy, though DQPTs are not restricted to this protocol.Source: §2.2 · p. 3

2.3. Loschmidt amplitudes and Loschmidt echos

  • The Loschmidt amplitude measures return overlap with the initial state, and its echo has a large-system rate function with conventional large-deviation scaling in the cases considered.The review restricts its discussion to this scaling because critical initial states can instead produce superextensive energy fluctuations.Source: §2.3 · pp. 3–4

2.4. Definition of dynamical quantum phase transitions

  • A DQPT is a time-dependent nonanalyticity of the Loschmidt amplitude; its singularity need not be a kink, and it is not in one-to-one correspondence with an equilibrium transition.Power-law nonanalyticities occur in cited two-dimensional examples, while many one-dimensional cases show kinks.Source: §2.4 · p. 4, Figure 1 · p. 4

2.5. Complex partition functions and Fisher zeros

  • In the thermodynamic limit, Fisher zeros accumulate into lines or areas, whose real-time crossings produce nonanalyticities; weak symmetry-preserving perturbations can deform but need not melt these structures.The review reports DQPT robustness in studied cases, while symmetry-breaking perturbations do not guarantee stability.Source: Figure 2 · p. 5, §2.5 · p. 6

2.6. Analogy to equilibrium quantum phase transitions

  • The Loschmidt amplitude probes the initial ground-state manifold, while local observables mainly reflect a shell near the mean energy density; whether a dynamical critical region links them is case-dependent.The proposed energy-density–time analogy has been made concrete in some examples, but is not known to apply to every DQPT.Source: §2.6 · pp. 6–7, Figure 3 · p. 7

2.7. Relation to equilibrium phase transitions

  • DQPTs often connect to equilibrium transitions, but exceptions include quenches without one and cases where crossing one yields no DQPT, supporting their status as distinct nonequilibrium phenomena.In the XY chain, a particle-number or magnetization conservation constraint can prevent access to sectors adopted in equilibrium.Source: §2.7 · pp. 7–8

3. Dynamical topological quantum phase transitions

In noninteracting topological systems, DQPTs connect to equilibrium topology and can be analyzed through mode-resolved Loschmidt amplitudes and Bloch-sphere trajectories. The review distinguishes robust topological cases from fine-tuned accidental ones and describes phase-profile signatures.

3. Dynamical topological quantum phase transitions

  • In two-band topological systems, dynamical order parameters can distinguish phases separated by a DQPT, and one has been measured in an ultracold-atom experiment.The review presents these quenches as an instructive setting with intuitive explanations and straightforward mathematical understanding.Source: §3 · p. 8

3.1. Two-band models

  • For translationally invariant, particle-hole-symmetric noninteracting two-band fermions, momentum sectors decouple, allowing the dynamics to be studied separately for each momentum.Source: §3.1 · pp. 8–9

3.2. Loschmidt amplitude

  • A DQPT requires a critical momentum and time at which the mode-resolved Loschmidt amplitude vanishes, corresponding to upper-band occupation 1/2.The existence of such a momentum depends on the system, but is ensured in many cases by a quench across an equilibrium quantum critical point.Source: §3.2 · p. 9

3.3. Landau-Zener problem

  • Crossing a gap closing during a ramp drives the critical mode toward complete band inversion, while weaker excitation of other modes implies a mode with upper-band occupation 1/2.For faster quenches, the critical mode can shift, changing DQPT time scales but not typically their occurrence; the argument may fail when all modes are strongly excited, as in complete band inversion.Source: §3.3 · pp. 9–10

3.4. Relative Bloch sphere

  • On the relative Bloch sphere, a critical mode reaches the south pole when its Loschmidt amplitude vanishes, giving a sufficient condition for a DQPT.Source: §3.4 · p. 10, §3.5 · p. 10

3.5. Topological and accidental DQPTs

  • In 1D, quenches between different winding numbers guarantee a critical momentum; in 2D, a change in the absolute Chern number guarantees DQPTs, while other cases may be accidental and fine-tuned.Source: §3.5 · p. 10

3.6. Topological defects in dynamical phase profiles

  • In 2D, a critical-mode trajectory crossing the south pole produces a π jump in azimuthal phase, creating vortex-antivortex pairs that move through the Brillouin zone.Source: §3.6 · pp. 10–11, Figure 5 · p. 11
  • The Pancharatnam geometric phase also slips by π at a DQPT; in 1D its winding changes by ±1 and can in principle distinguish accidental from symmetry-protected transitions.The phase-profile vortices can also be generalized to mixed states.Source: §3.6 · p. 11

4. Experiments

The review surveys two quantum-simulator demonstrations of DQPTs: trapped ions probe rate-function singularities and related magnetization, while ultracold atoms track topological transitions through dynamical vortices.

4.1. Trapped ions

  • The trapped-ion experiment quenches a polarized ground state across the ferromagnetic-to-paramagnetic transition and predicts DQPTs using minη λη, assuming the component rates have nearly converged.This minimum construction requires additional theoretical input.Source: §4.1 · pp. 12–13
  • For trapped ions, intersections of the measured λη(t) yield a thermodynamic-limit kink in λ(t), while energy-resolved magnetization develops a sharp feature around ε=0.The magnetization feature is finite-size sharp rather than nonanalytic.Source: Figure 6 · p. 12, §4.1 · p. 13

4.2. Ultra-cold atoms in optical lattices

  • The ultracold-atom experiment quenches a half-filled insulating state on a hexagonal lattice by suddenly switching on hopping through lattice shaking.The initial state has particles localized on one sublattice.Source: §4.2 · p. 13
  • In noninteracting two-band fermions, vortex-pair events correspond one-to-one with DQPTs, enabling vortex-number tracking when stroboscopic sampling cannot resolve rate-function singularities.The dynamical vortex number can also serve as an order parameter for DQPTs in this model.Source: §4.2 · pp. 13–14

5. Characteristic properties

This section examines which characteristic features of equilibrium phase transitions extend to dynamical quantum phase transitions, surveying scaling and universality, robustness, dynamical order parameters, observable signatures, and classification. It also considers first-order transitions defined by switches between dominant saddle points in generalized expectation values, while noting that their relation to conventional observables remains unknown.

5.1. Scaling and universality

  • For Ising quenches, Loschmidt amplitudes map to classical partition functions, and DQPTs in 1D and 2D exhibit scaling behavior matching classical Ising criticality.In 1D, the critical coupling flows to the unstable zero-temperature fixed point and scaling reproduces the temporal kink; in 2D, the singular rate function matches thermal critical behavior.Source: §5.1 · pp. 14–15
  • Scaling and universality have been established only for the discussed models, leaving open whether they generalize or whether genuinely nonequilibrium fixed points occur.The reported scaling is associated with classical, rather than underlying quantum, equilibrium criticality.Source: §5.1 · p. 15

5.2. Robustness

  • Weak RG-irrelevant perturbations preserve DQPT nonanalyticities, though critical times may shift; symmetry-breaking fields can smear transitions in sweeps, and long-time robustness remains unstudied.Robustness arguments based on complex partition-function zeros are physical expectations, not proofs; symmetry-breaking perturbations can still permit DQPTs for quenches.Source: §5.2 · p. 16

5.3. Dynamical order parameters

  • Dynamical topological order parameters assign quantized numbers to phase profiles that jump at DQPTs, while local dynamical order parameters remain poorly understood.Winding numbers of the Pancharatnam geometric phase and 2D vortex counts can distinguish topologically protected from accidental DQPTs; causality rules out conventional long-range correlations forming within finite time.Source: §5.3 · pp. 16–17

5.4. Signatures in other quantities

  • For quenches from symmetry-broken or topological phases, DQPTs accompany zeros or oscillatory decay of order parameters, while symmetry restoration explains vanishing order at rate-function crossings.In the Ising chain, longitudinal-magnetization oscillations share DQPT periodicity; topological string-order oscillations do too, on an emergent nonequilibrium time scale.Source: §5.4 · pp. 17–18
  • For quenches from symmetric phases, order-parameter correlations can peak at DQPTs with critical behavior in the corresponding classical Ising universality class, but promising observables may show no signature.The momentum-distribution peak in a superfluid-to-Mott quench, for example, does not reveal the DQPT.Source: §5.4 · p. 18

5.5. Classification

  • A general DQPT classification remains unknown; topological models have a scheme distinguishing protected and accidental transitions, while another proposal defines first-order DQPTs using generalized expectation values.The topological classification analyzes possible nonanalytic structures but does not directly extend to interacting systems.Source: §5.5 · p. 19
  • A first-order DQPT occurs when the dominant saddle point in the thermodynamic-limit generalized expectation value switches to another saddle point.For operators sensitive to the saddle-point nature, this can produce an abrupt change at the critical time.Source: §5.5 · p. 19
  • This universal definition does not depend on model-specific properties, and abrupt signatures were demonstrated in Falicov-Kimball and Hubbard models using dynamical mean-field theory.Generalized expectation values involve only forward time evolution and can in principle be accessed interferometrically.Source: §5.5 · p. 19
  • How generalized-expectation-value signatures relate to conventional observables remains unknown, while topological DQPTs have no known local observable signature.For topological DQPTs, the discussed signatures involve global rather than local properties.Source: §5.5 · p. 19

6. Further applications and topics

The review surveys applications of DQPTs beyond their basic definition, including entanglement, quantum speed limits, mixed states, driven protocols, other nonequilibrium transitions, and inhomogeneous systems. Across these applications, observed connections are protocol- and setting-dependent, with several important scope questions still open.

6.1. Entanglement dynamics

  • Entanglement signatures occur near DQPTs in both long-range trapped-ion dynamics and short-range Ising quenches, but their general connection remains unclear and may depend on the protocol.In the trapped-ion experiment, entanglement entropy and spin squeezing show increased dynamics near critical times; in short-range models, two-site entropy becomes maximal at DQPTs.Source: §6.1 · pp. 19–20, Figure 10 · p. 20, Figure 11 · p. 20

6.2. Quantum speed limits

  • For particular quenches of a quantum-critical state by its order parameter, DQPT critical times show unconventional system-size dependence, linking them to a distinct class of quantum speed limits.The connection rests on the shared condition of a vanishing Loschmidt amplitude when the evolved state becomes orthogonal to its initial state.Source: §6.2 · pp. 20–21

6.3. Mixed states

  • A thermal work-distribution definition of DQPTs smears real-time nonanalyticities in the 1D transverse-field Ising model and is not clearly formulated beyond Gibbs initial states or unitary evolution.The approach identifies the probability of performing zero work during the quench.Source: §6.3 · p. 21
  • An interferometric mixed-state extension instead allows DQPTs in topological quenches to persist at elevated initial temperatures, providing a dynamical probe without preparing a low-entropy state.Both discussed mixed-state extensions still require purely unitary dynamics.Source: §6.3 · p. 21

6.4. General nonequilibrium protocols

  • For linear ramps, DQPTs can occur during the parameter ramp as well as after the nonequilibrium protocol ends.The review also identifies Floquet systems and systems subject to noise as scenarios for further study.Source: §6.4 · p. 21

6.5. Relation to other notions of dynamical phase transitions

  • A steady-state Ising transition is connected to a transient-to-intermediate-time DQPT, while a topological steady-state transition has DQPTs if and only if the quench crosses its critical value.How anomalous long-time DQPTs fit the Ising connection, and how broadly DQPTs connect to other nonequilibrium criticality, remain open questions.Source: §6.5 · p. 22

6.6. Inhomogeneous systems

  • Evidence for DQPTs is strong in the random-energy and Aubry-Andre models, whereas the Anderson model shows sharp but smooth finite-size features whose thermodynamic-limit behavior is unresolved.The general influence of disorder on DQPTs, including whether equilibrium disorder criteria extend to them, remains unknown.Source: §6.6 · p. 22

7. Prospects

The review reports substantial theoretical and experimental progress on DQPTs, while identifying open questions about organizing nonequilibrium dynamics and extending the theory to new phenomena. It outlines challenges and directions for future research.

7. Prospects

  • The field has made substantial theoretical and experimental progress, but major questions remain and motivate the review’s proposed future directions.The review presents DQPT theory as having potential to provide a principle for understanding quantum many-body dynamics.Source: §7 · p. 22
  • Nonequilibrium DQPTs lack a free-energy concept, leaving a macroscopic description, a Landau-Ginzburg counterpart, and an analogous organizing principle as open questions.The review frames these questions as central challenges for the field.Source: §7 · p. 22
  • Most studied DQPTs have integer-valued exponents, while the existence of nontrivial exponents remains unclear and may require methods for higher-dimensional interacting theories.Because DQPTs occur on transient timescales, approximations can be better controlled than in studies of long-time correlated-system dynamics.Source: §7 · pp. 22–23
  • The review proposes exploring DQPT connections to many-body localization, nonthermal fixed points, and eigenstate phases, including time crystals’ unconventional spatio-temporal order.Eigenstate phases can also be associated with long-range spatial correlations, as in many-body-localized spin glasses.Source: §7 · p. 23
  • A further direction is to seek analogues of DQPT nonanalyticities in other translations or rotations, including momentum translation relevant to topological systems.The review describes DQPTs more generally as nonanalyticities in the time-translation operator.Source: §7 · p. 23
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