Asymmetry and dynamical criticality

Abstract: Symmetries play a central role in both equilibrium and nonequilibrium phase transitions, yet their quantitative characterization in dynamical quantum phase transitions (DQPTs) remains an open challenge. In this work, we establish a direct connection between symmetry properties of a many-body model and measures of quantum asymmetry, showing that asymmetry monotones provide a robust and physically transparent indicator of dynamical quantum criticality. Focusing on the quenched Lipkin-Meshkov-Glick model, we demonstrate that asymmetry measures associated with collective spin generators faithfully capture the onset of DQPTs, reflecting the dynamical restoration or breaking of underlying symmetries. Remarkably, the time-averaged asymmetry exhibits clear signatures of the dynamical critical point, in close correspondence with both the dynamical order parameter and the behavior of entropy production. We further uncover a quantitative link between asymmetry generation and thermodynamic irreversibility, showing that peaks in asymmetry coincide with maximal entropy production across the transition. Our results position asymmetry as a unifying concept bridging symmetry, information-theoretic quantifiers, and nonequilibrium thermodynamics in dynamical quantum phase transitions, providing a powerful framework for understanding critical dynamics beyond traditional order parameters.

• The paper presents a novel method using asymmetry monotones to diagnose dynamical quantum criticality in the Lipkin-Meshkov-Glick (LMG) model.
• Asymmetry measures, quantified by the $\ell_1$-norm, track the coherence in eigenbases of spin generators and correlate with entropy production in dynamical quantum phase transitions (DQPTs).
• The study shows that these monotones identify dynamical critical points, especially in anisotropic regimes, with implications for the irreversibility of quantum transitions.

Overview and motivation

This paper by Nascimento and Céleri establishes a connection between the resource-theoretic notion of quantum asymmetry and dynamical quantum phase transitions (DQPTs), using the quenched Lipkin–Meshkov–Glick (LMG) model as a testbed (2602.00900). The central claim is that asymmetry monotones—quantities that measure how strongly a state breaks a given symmetry—serve as robust, physically transparent indicators of dynamical criticality, complementing the two standard diagnostics: nonanalyticities in the Loschmidt echo rate function (type-II DQPTs) and time-averaged magnetization order parameters (type-I DQPTs).

The motivation stems from a recognized gap in the DQPT literature. Conserved quantities such as the moments $\Tr(\rho L^k)$ of a symmetry generator are blind to coherence between eigenstates of $L$, and hence cannot distinguish symmetric from asymmetric states with identical spectral distributions. Asymmetry measures, developed in the resource theory of quantum reference frames, close this gap by quantifying exactly this coherence. The paper's contribution is to show that these information-theoretic quantifiers track dynamical critical points and correlate quantitatively with entropy production.

Model and quench protocol

The LMG Hamiltonian describes $N$ spin-$1/2$ particles with all-to-all interactions,

$$H = -\frac{J}{j}\left(J_z^2 + \gamma J_y^2\right) - 2hJ_x,$$

with $j = N/2$ and anisotropy parameter $\gamma \in [0,1]$. The model possesses permutation symmetry and a discrete $\mathbb{Z}_2$ parity symmetry ($J_y \to -J_y$, $J_z \to -J_z$), whose spontaneous breaking separates the paramagnetic and ferromagnetic equilibrium phases. For quenches from $L$0, the dynamical critical point in the anisotropic case is $L$1; for $L$2, $L$3. In the isotropic limit $L$4, the Hamiltonian reduces to a function of $L$5 alone, $L$6, and DQPTs are suppressed because the return probability amplitude never vanishes. This suppression plays a key role in interpreting the numerical results below.

The asymmetry measure

The authors adopt the $L$7-norm-based asymmetry monotone

$L$8

which vanishes if and only if $L$9 commutes with the generator $N$0 and otherwise quantifies coherence in the eigenbasis of $N$1. Because the physically relevant $N$2 parity is discrete while $N$3 is formulated for continuous Lie-group generators, the authors recast parity inversion as a $N$4-rotation about the $N$5-axis within the $N$6 subgroup generated by $N$7, and compute $N$8 for all three collective spin generators $N$9, $1/2$0, $1/2$1. This choice is pragmatic rather than fundamental: the analysis probes the parity-related symmetry through continuous generators, and the paper is explicit that the formulation does not correspond to a genuine rotational symmetry of the model.

Main results

Three independent quantities are compared across the $1/2$2 plane at fixed $1/2$3: the time-averaged asymmetry $1/2$4, the time-averaged entropy production lower bound $1/2$5 based on the Bures angle between the evolved state and an equilibrium reference state, and the dynamical order parameter $1/2$6.

Transient dynamics. Quenches crossing the critical point ($1/2$7) produce rapid growth of $1/2$8 followed by fast saturation, whereas subcritical quenches ($1/2$9) show slow growth and persistence near the initial asymmetry. This mirrors earlier findings that DQPTs accelerate the saturation of entropy production bounds, and indicates that crossing the critical point rapidly generates coherence in the eigenbases of all three generators.

Time-averaged signatures. The quantity $$H = -\frac{J}{j}\left(J_z^2 + \gamma J_y^2\right) - 2hJ_x,$$0 exhibits the sharpest transition line in the $$H = -\frac{J}{j}\left(J_z^2 + \gamma J_y^2\right) - 2hJ_x,$$1 plane, rising abruptly near $$H = -\frac{J}{j}\left(J_z^2 + \gamma J_y^2\right) - 2hJ_x,$$2 for $$H = -\frac{J}{j}\left(J_z^2 + \gamma J_y^2\right) - 2hJ_x,$$3 and $$H = -\frac{J}{j}\left(J_z^2 + \gamma J_y^2\right) - 2hJ_x,$$4 for $$H = -\frac{J}{j}\left(J_z^2 + \gamma J_y^2\right) - 2hJ_x,$$5. The authors are careful to attribute this enhanced sensitivity to the kinematic relation between $$H = -\frac{J}{j}\left(J_z^2 + \gamma J_y^2\right) - 2hJ_x,$$6 and the magnetization order parameter, not to any claim that rotations about $$H = -\frac{J}{j}\left(J_z^2 + \gamma J_y^2\right) - 2hJ_x,$$7 define the transition's symmetry—an important conceptual distinction between sensitivity to criticality and sensitivity to symmetry restoration.

Parity-specific behavior. The generator $$H = -\frac{J}{j}\left(J_z^2 + \gamma J_y^2\right) - 2hJ_x,$$8 shows qualitatively different behavior depending on anisotropy. For $$H = -\frac{J}{j}\left(J_z^2 + \gamma J_y^2\right) - 2hJ_x,$$9, critical quenches enhance $j = N/2$0 as expected; for $j = N/2$1, the pattern inverts, which the authors explain via the proximity to the isotropic regime where the Hamiltonian becomes $j = N/2$2-symmetric and stronger transverse fields preserve rather than break the symmetry. Despite reduced contrast, the peak of $j = N/2$3 coincides with the region where the dynamical order parameter decays to zero, identifying it as a signature of the dynamical restoration of $j = N/2$4 parity.

Anisotropy dependence. Both $j = N/2$5 and $j = N/2$6 show critical features whose location shifts with $j = N/2$7 and progressively disappear as $j = N/2$8, consistent with the exact suppression of DQPTs in the isotropic limit. The agreement among asymmetry, entropy production, and the order parameter across the full $j = N/2$9 plane constitutes the paper's strongest evidence that time-averaged asymmetry captures essential features of dynamical criticality.

Relation to irreversibility

A notable result is the quantitative correspondence between peaks in time-averaged asymmetry and maxima in entropy production. Since $\gamma \in [0,1]$0 lower-bounds thermodynamic irreversibility through the Bures-angle geometric construction, the coincidence of asymmetry maxima with maximal entropy production supports a thermodynamic interpretation of asymmetry generation: DQPTs are accompanied by enhanced irreversibility and rapid redistribution of coherence across symmetry sectors. Prior work had established this correspondence only at a single anisotropy value ($\gamma \in [0,1]$1); extending it to $\gamma \in [0,1]$2 and $\gamma \in [0,1]$3 strengthens the claim considerably.

Limitations and open questions

Several caveats qualify the results. All computations are performed at finite $\gamma \in [0,1]$4; convergence to the thermodynamic limit is asserted from the order-parameter scaling but not systematically demonstrated for the asymmetry measures themselves. The mapping of discrete parity onto a continuous $\gamma \in [0,1]$5 generator is an interpretive device, and the paper acknowledges that $\gamma \in [0,1]$6 probes the relevant symmetry only indirectly. The entropy-production comparison relies on a specific geometric lower bound relative to an assumed equilibrium reference state, so the claimed asymmetry–irreversibility link inherits the assumptions of that construction. Finally, the analysis is restricted to pure-state unitary quenches in a single mean-field model.

The authors identify concrete extensions: Holevo-type asymmetry measures with direct entropic interpretation, mixed-state and finite-temperature DQPTs, open-system dynamics where dissipation competes with symmetry-driven coherence generation, connections to quantum speed limits via generalized relative entropies, and experimental accessibility in cold-atom and trapped-ion platforms. Whether asymmetry-based signatures are universal across models and symmetry classes remains unanswered.

Conclusion

The paper demonstrates that $\gamma \in [0,1]$7-norm asymmetry monotones with respect to collective spin generators faithfully locate the dynamical critical point of the quenched LMG model, agree with independent diagnostics from entropy production and the dynamical order parameter, and distinguish symmetry-restoration physics ($\gamma \in [0,1]$8) from mere criticality-sensitive observables ($\gamma \in [0,1]$9). Anisotropy emerges as a control parameter that shifts and ultimately extinguishes dynamical criticality toward the isotropic limit. The work positions asymmetry as a unifying quantity linking symmetry breaking, quantum coherence, and nonequilibrium thermodynamics in dynamical phase transitions, while leaving universality and open-system generalizations as explicit open problems.