Higher-Order Symmetric Quantum Mpemba Effect

• The higher-order symmetric quantum Mpemba effect is defined by a symmetry-resolved generalization where states with greater initial asymmetry restore charge and dipole symmetry faster than less broken states.
• Evidence in charge–dipole conserving many-body systems reveals distinct Mpemba crossing times for charge and dipole asymmetries, emphasizing the role of Hilbert-space fragmentation.
• Key diagnostics include entanglement asymmetry, charge variance, and Krylov complexity frameworks, which together characterize multiple layers of symmetry restoration dynamics.

Searching arXiv for papers on higher-order symmetric quantum Mpemba effect and closely related symmetry-based QME work. The higher-order symmetric quantum Mpemba effect is a symmetry-resolved generalization of the quantum Mpemba effect in which a state that initially breaks one or more symmetries more strongly can restore them faster than a less broken state, with the inversion detected by symmetry-projected observables rather than by a single scalar distance to equilibrium. In the most explicit realization to date, closed many-body systems with simultaneous charge and dipole conservation exhibit Mpemba-like crossings in both charge and dipole entanglement asymmetries, and these crossings occur on parametrically distinct timescales despite strong Hilbert-space fragmentation (Aditya et al., 4 Jun 2026). Closely related work frames the quantum Mpemba effect through entanglement asymmetry and charge variance in symmetry-preserving dynamics (Rylands et al., 2023, Yu et al., 3 Jul 2025), while a separate line of work defines a hierarchy of projective symmetric complexities whose crossings realize a $k$th-order Mpemba inversion in Krylov space (Beetar et al., 9 Sep 2025).

1. Definition and scope

For a closed many-body system with global $U(1)$ charge and dipole moment,

$$Q = \sum_{x=1}^L S^z_x,\qquad P = \sum_{x=1}^L x\,S^z_x,\qquad S^z_x = \tfrac12\sigma^z_x,$$

the reduced state of a subsystem $A$ is

$\rho_A(t)=\mathrm{tr}_{B}[\rho(t)],$

and the symmetry-projected reduced state associated with an observable $\mathcal O\in\{Q,P\}$ is

$$\rho_{A,\mathcal O}(t)=\sum_o\Pi^{\mathcal O_A}_o\,\rho_A(t)\,\Pi^{\mathcal O_A}_o.$$

The $n$-th entanglement asymmetry is then

$$\Delta S^{(n)}_{\mathcal O}(\rho_A) = S_n\bigl(\rho_{A,\mathcal O}\bigr)-S_n(\rho_A), \qquad S_n(\rho)=\frac{1}{1-n}\ln\mathrm{tr}(\rho^n).$$

Given two initial states $\rho(0)$ and $U(1)$0, the quantum Mpemba criterion for $U(1)$1 is satisfied when

$U(1)$2

but at some time $U(1)$3,

$U(1)$4

When both charge and dipole asymmetries are tracked, the effect is termed a higher-order symmetric quantum Mpemba effect because it involves two independent symmetry constraints rather than a single symmetry-resolved observable (Aditya et al., 4 Jun 2026).

In the broader symmetry-based literature, the same underlying logic is formulated for a single conserved $U(1)$5 charge using subsystem symmetry restoration. There, a more asymmetric initial state can restore symmetry faster than a less asymmetric one, and the crossings are diagnosed by symmetry-sensitive observables such as entanglement asymmetry or charge variance (Yu et al., 3 Jul 2025). A distinct but compatible generalization defines “higher-order” through successive symmetry projections in Krylov space, producing a hierarchy of $U(1)$6th-order symmetric complexities and $U(1)$7th-order Mpemba inversions (Beetar et al., 9 Sep 2025).

2. Symmetry-resolved diagnostics

The central diagnostic in the symmetry perspective is entanglement asymmetry. For a subsystem $U(1)$8 with reduced density matrix $U(1)$9 and its $$Q = \sum_{x=1}^L S^z_x,\qquad P = \sum_{x=1}^L x\,S^z_x,\qquad S^z_x = \tfrac12\sigma^z_x,$$0-symmetrized version

$$Q = \sum_{x=1}^L S^z_x,\qquad P = \sum_{x=1}^L x\,S^z_x,\qquad S^z_x = \tfrac12\sigma^z_x,$$1

the von Neumann entanglement asymmetry is

$$Q = \sum_{x=1}^L S^z_x,\qquad P = \sum_{x=1}^L x\,S^z_x,\qquad S^z_x = \tfrac12\sigma^z_x,$$2

It satisfies $$Q = \sum_{x=1}^L S^z_x,\qquad P = \sum_{x=1}^L x\,S^z_x,\qquad S^z_x = \tfrac12\sigma^z_x,$$3, vanishes iff $$Q = \sum_{x=1}^L S^z_x,\qquad P = \sum_{x=1}^L x\,S^z_x,\qquad S^z_x = \tfrac12\sigma^z_x,$$4, and tends to zero whenever the subsystem locally restores the symmetry. Two initial symmetry-broken states exhibit a quantum Mpemba effect when the more asymmetric one, as measured by $$Q = \sum_{x=1}^L S^z_x,\qquad P = \sum_{x=1}^L x\,S^z_x,\qquad S^z_x = \tfrac12\sigma^z_x,$$5, becomes less asymmetric after a finite Mpemba time $$Q = \sum_{x=1}^L S^z_x,\qquad P = \sum_{x=1}^L x\,S^z_x,\qquad S^z_x = \tfrac12\sigma^z_x,$$6 (Rylands et al., 2023).

Charge variance provides a second symmetry-based distance from equilibrium. For $$Q = \sum_{x=1}^L S^z_x,\qquad P = \sum_{x=1}^L x\,S^z_x,\qquad S^z_x = \tfrac12\sigma^z_x,$$7,

$$Q = \sum_{x=1}^L S^z_x,\qquad P = \sum_{x=1}^L x\,S^z_x,\qquad S^z_x = \tfrac12\sigma^z_x,$$8

In the review formulation, both entanglement asymmetry and charge variance diagnose symmetry breaking and restoration, and either observable can display crossing dynamics characteristic of the quantum Mpemba effect (Yu et al., 3 Jul 2025).

In integrable systems, the microscopic criterion has a specific structure. In the large-$$Q = \sum_{x=1}^L S^z_x,\qquad P = \sum_{x=1}^L x\,S^z_x,\qquad S^z_x = \tfrac12\sigma^z_x,$$9 limit, the initial ordering of entanglement asymmetry is equivalent to the more asymmetric state having larger charge fluctuations in the subsystem, while the late-time inversion requires the slowest, minimal-velocity modes to carry less of the total charge fluctuation in the more asymmetric state. This links the effect to how symmetry-breaking weight is distributed among slow quasiparticles rather than only to the total amount of asymmetry (Rylands et al., 2023).

A different diagnostic framework is complexity-theoretic. Starting from a Krylov basis $A$0, the spread-complexity operator is

$A$1

and with symmetry-sector projectors $A$2, the symmetric and asymmetric projective complexities are

$A$3

At $A$4, the asymmetric complexity is strictly negative and $A$5. The structural complexity

$A$6

was reported to predict whether two tilted states will exhibit a Mpemba inversion under time evolution. The same work proposes a hierarchy

$A$7

with $A$8th-order symmetric complexity operator

$A$9

where $\rho_A(t)=\mathrm{tr}_{B}[\rho(t)],$0. A $\rho_A(t)=\mathrm{tr}_{B}[\rho(t)],$1th-order Mpemba inversion occurs when the ordering of $\rho_A(t)=\mathrm{tr}_{B}[\rho(t)],$2 between two initial states reverses at some $\rho_A(t)=\mathrm{tr}_{B}[\rho(t)],$3 (Beetar et al., 9 Sep 2025).

3. Fragmented charge–dipole systems

The paradigmatic many-body setting for the higher-order symmetric effect is a system with simultaneous conservation of charge $\rho_A(t)=\mathrm{tr}_{B}[\rho(t)],$4 and dipole $\rho_A(t)=\mathrm{tr}_{B}[\rho(t)],$5, where the Hilbert space fragments into exponentially many disconnected Krylov sectors (Aditya et al., 4 Jun 2026). Two models were used.

The first is a random brickwork circuit on spin-$\rho_A(t)=\mathrm{tr}_{B}[\rho(t)],$6 degrees of freedom with depth-$\rho_A(t)=\mathrm{tr}_{B}[\rho(t)],$7 Floquet operator

$\rho_A(t)=\mathrm{tr}_{B}[\rho(t)],$8

where each four-site gate conserves both $\rho_A(t)=\mathrm{tr}_{B}[\rho(t)],$9 and $\mathcal O\in\{Q,P\}$0. The only non-trivial move is the pair hop

$\mathcal O\in\{Q,P\}$1

Each gate is block-diagonal in the $\mathcal O\in\{Q,P\}$2 sectors, acts as a Haar-random $\mathcal O\in\{Q,P\}$3 on the two-dimensional pair-hop subspace, and contributes random phases on frozen basis states (Aditya et al., 4 Jun 2026).

The second is a coherent pair-hopping Hamiltonian with the same symmetries,

$\mathcal O\in\{Q,P\}$4

with open boundary conditions (Aditya et al., 4 Jun 2026).

To study symmetry restoration, the work focused on the annealed Rényi-2 asymmetry

$\mathcal O\in\{Q,P\}$5

In the circuit setting, $\mathcal O\in\{Q,P\}$6 is computed by a two-replica tensor network whose local transfer tensor is

$\mathcal O\in\{Q,P\}$7

with $\mathcal O\in\{Q,P\}$8 labeling $\mathcal O\in\{Q,P\}$9 blocks. In the Hamiltonian setting, the pure state is propagated by Chebyshev expansion $$\rho_{A,\mathcal O}(t)=\sum_o\Pi^{\mathcal O_A}_o\,\rho_A(t)\,\Pi^{\mathcal O_A}_o.$$0, then projected into sectors to evaluate both $$\rho_{A,\mathcal O}(t)=\sum_o\Pi^{\mathcal O_A}_o\,\rho_A(t)\,\Pi^{\mathcal O_A}_o.$$1 and the von Neumann asymmetry $$\rho_{A,\mathcal O}(t)=\sum_o\Pi^{\mathcal O_A}_o\,\rho_A(t)\,\Pi^{\mathcal O_A}_o.$$2 (Aditya et al., 4 Jun 2026).

4. Evidence for higher-order crossings

In the random circuit, system sizes up to $$\rho_{A,\mathcal O}(t)=\sum_o\Pi^{\mathcal O_A}_o\,\rho_A(t)\,\Pi^{\mathcal O_A}_o.$$3 were reached. For two tilted-ferromagnet states $$\rho_{A,\mathcal O}(t)=\sum_o\Pi^{\mathcal O_A}_o\,\rho_A(t)\,\Pi^{\mathcal O_A}_o.$$4, so that $$\rho_{A,\mathcal O}(t)=\sum_o\Pi^{\mathcal O_A}_o\,\rho_A(t)\,\Pi^{\mathcal O_A}_o.$$5 is initially more asymmetric, both $$\rho_{A,\mathcal O}(t)=\sum_o\Pi^{\mathcal O_A}_o\,\rho_A(t)\,\Pi^{\mathcal O_A}_o.$$6 and $$\rho_{A,\mathcal O}(t)=\sum_o\Pi^{\mathcal O_A}_o\,\rho_A(t)\,\Pi^{\mathcal O_A}_o.$$7 exhibit crossings at intermediate time $$\rho_{A,\mathcal O}(t)=\sum_o\Pi^{\mathcal O_A}_o\,\rho_A(t)\,\Pi^{\mathcal O_A}_o.$$8. This demonstrates a Mpemba effect for both charge and dipole. At later times, both asymmetries plateau at a nonzero value because frozen fragments obstruct complete symmetry restoration (Aditya et al., 4 Jun 2026).

The first crossing times scale differently with subsystem size $$\rho_{A,\mathcal O}(t)=\sum_o\Pi^{\mathcal O_A}_o\,\rho_A(t)\,\Pi^{\mathcal O_A}_o.$$9: $n$0 This is the clearest quantitative signature that the charge and dipole sectors support distinct higher-order symmetry-restoration timescales rather than a single universal Mpemba time (Aditya et al., 4 Jun 2026).

The pair-hopping Hamiltonian shows the same qualitative structure under coherent evolution. Exact vector simulations up to $n$1 found similar Mpemba crossovers in both $n$2 and $n$3 for boundary and bulk subsystems with $n$4. Bulk subsystems exhibit smaller late-time plateaux than boundary subsystems, consistent with weaker edge localization (Aditya et al., 4 Jun 2026).

These results extend the earlier symmetry-based QME literature, where the effect was already established for $n$5-symmetric random circuits and Hamiltonian evolution using entanglement asymmetry. In that setting, crossings occur in integrable and non-integrable chains, the characteristic time in Hamiltonian systems scales as $n$6, and many-body localized systems display crossings on a timescale that grows exponentially with subsystem size even though the system does not thermalize (Yu et al., 3 Jul 2025). The fragmented charge–dipole setting therefore does not replace the standard symmetry-restoration picture; it refines it by adding a second conserved moment and by forcing the dynamics to occur inside a fragmented Hilbert space.

5. Mechanism: fragmentation, frozen memory, and active relaxation

The mechanism identified in fragmented systems is a decomposition into frozen and active Krylov sectors (Aditya et al., 4 Jun 2026). Charge–dipole constraints generate two extreme classes of fragments.

Frozen fragments are single product states containing no movable pattern $n$7; they have dimension $n$8 and retain exact memory of the initial asymmetry. Active fragments are larger orbits linked by pair hops and therefore support relaxation.

With coarse-grained projectors

$n$9

a pure state splits into $$\Delta S^{(n)}_{\mathcal O}(\rho_A) = S_n\bigl(\rho_{A,\mathcal O}\bigr)-S_n(\rho_A), \qquad S_n(\rho)=\frac{1}{1-n}\ln\mathrm{tr}(\rho^n).$$0 with constant weights $$\Delta S^{(n)}_{\mathcal O}(\rho_A) = S_n\bigl(\rho_{A,\mathcal O}\bigr)-S_n(\rho_A), \qquad S_n(\rho)=\frac{1}{1-n}\ln\mathrm{tr}(\rho^n).$$1. The normalized sector states are

$$\Delta S^{(n)}_{\mathcal O}(\rho_A) = S_n\bigl(\rho_{A,\mathcal O}\bigr)-S_n(\rho_A), \qquad S_n(\rho)=\frac{1}{1-n}\ln\mathrm{tr}(\rho^n).$$2

The sector-resolved asymmetries then separate the full dynamics into two contributions: frozen-sector asymmetry remains at its initial value, active-sector asymmetry decays and displays Mpemba crossings, and the full asymmetry inherits both a transient crossing and a late-time plateau (Aditya et al., 4 Jun 2026).

An exactly solvable dissipative toy model sharpens this picture. The model uses the symmetric pair-flip Hamiltonian

$$\Delta S^{(n)}_{\mathcal O}(\rho_A) = S_n\bigl(\rho_{A,\mathcal O}\bigr)-S_n(\rho_A), \qquad S_n(\rho)=\frac{1}{1-n}\ln\mathrm{tr}(\rho^n).$$3

which conserves dipole exactly and preserves only charge parity, together with on-site dephasing,

$$\Delta S^{(n)}_{\mathcal O}(\rho_A) = S_n\bigl(\rho_{A,\mathcal O}\bigr)-S_n(\rho_A), \qquad S_n(\rho)=\frac{1}{1-n}\ln\mathrm{tr}(\rho^n).$$4

Because the dynamics factorize into independent mirror pairs, the reduced single-site density matrix takes the form

$$\Delta S^{(n)}_{\mathcal O}(\rho_A) = S_n\bigl(\rho_{A,\mathcal O}\bigr)-S_n(\rho_A), \qquad S_n(\rho)=\frac{1}{1-n}\ln\mathrm{tr}(\rho^n).$$5

with

$$\Delta S^{(n)}_{\mathcal O}(\rho_A) = S_n\bigl(\rho_{A,\mathcal O}\bigr)-S_n(\rho_A), \qquad S_n(\rho)=\frac{1}{1-n}\ln\mathrm{tr}(\rho^n).$$6

$$\Delta S^{(n)}_{\mathcal O}(\rho_A) = S_n\bigl(\rho_{A,\mathcal O}\bigr)-S_n(\rho_A), \qquad S_n(\rho)=\frac{1}{1-n}\ln\mathrm{tr}(\rho^n).$$7

The corresponding charged moment is

$$\Delta S^{(n)}_{\mathcal O}(\rho_A) = S_n\bigl(\rho_{A,\mathcal O}\bigr)-S_n(\rho_A), \qquad S_n(\rho)=\frac{1}{1-n}\ln\mathrm{tr}(\rho^n).$$8

where $$\Delta S^{(n)}_{\mathcal O}(\rho_A) = S_n\bigl(\rho_{A,\mathcal O}\bigr)-S_n(\rho_A), \qquad S_n(\rho)=\frac{1}{1-n}\ln\mathrm{tr}(\rho^n).$$9 for charge and $\rho(0)$0 is the site position for dipole, with

$\rho(0)$1

For a left-half subsystem,

$\rho(0)$2

and

$\rho(0)$3

In the dissipative regime $\rho(0)$4, if $\rho(0)$5 at late times, then

$\rho(0)$6

For two tilts $\rho(0)$7, the first Mpemba crossing solves $\rho(0)$8, giving

$\rho(0)$9

This first crossing time is independent of subsystem size $U(1)$00 and independent of whether one probes charge or dipole. The same model also shows that dephasing is essential: without $U(1)$01, there is no monotonic symmetry restoration or bona fide Mpemba effect (Aditya et al., 4 Jun 2026).

6. Alternative higher-order constructions and open problems

The literature uses more than one construction of “higher-order” symmetry-resolved Mpemba behavior. One route, realized in fragmented systems, employs multiple commuting conserved quantities, specifically charge and dipole, and tracks their distinct asymmetry crossings (Aditya et al., 4 Jun 2026). Another route, proposed in the Krylov-complexity framework, builds a hierarchy of successive symmetry projections and defines $U(1)$02th-order symmetric complexity as the part of the spread complexity invariant under $U(1)$03 nested sector projections. In that setting, first-order complexity tracks intra-sector diffusion, second-order complexity isolates processes involving two-sector coherence sequences, and higher $U(1)$04 probe progressively deeper layers of the thermalization pathway. The predicted hallmark is a cascade of Mpemba effects, with multiple crossings at progressively later times and in progressively narrower windows of Krylov index $U(1)$05 (Beetar et al., 9 Sep 2025).

The symmetry review outlines additional generalizations. For non-Abelian or discrete symmetries, one may define symmetry-resolved asymmetries by projecting onto group sectors; for multiple commuting symmetries such as $U(1)$06, one may track a joint entanglement asymmetry on sectors $U(1)$07; for measurement-induced and Floquet settings, one may seek a cascade of crossings associated with successive restoration of translation, $U(1)$08, inversion, or other symmetries. The same review suggests that an $U(1)$09-th-order symmetric QME could be diagnosed by an $U(1)$10-component order parameter $U(1)$11 whose components cross in a non-trivial sequence during relaxation (Yu et al., 3 Jul 2025).

Several open problems remain explicit. Measuring $U(1)$12 requires tomography plus dephasing or randomized measurement protocols with exponential overhead in $U(1)$13. Extending the framework to non-commuting symmetries is technically subtler because the projected density matrix is no longer straightforward to define. Large-scale scaling analyses beyond small-system exact diagonalization were identified as necessary for higher dimensions and long-range models. The relation of symmetry-based Mpemba effects to dynamical phase transitions, prethermalization plateaux, and Kibble–Zurek scaling also remains unresolved (Yu et al., 3 Jul 2025).

A common misconception is that strong fragmentation should eliminate anomalous relaxation altogether. The fragmented charge–dipole results show the opposite: fragmentation does not preclude the quantum Mpemba effect, but reshapes it into the coexistence of frozen memory and active-fragment relaxation (Aditya et al., 4 Jun 2026). Another source of ambiguity is terminological. In current usage, “higher-order symmetric quantum Mpemba effect” can refer either to multiple symmetry-resolved asymmetries with distinct crossing scales or to a hierarchy of successive symmetry projections in Krylov space. Both formulations retain the same core content: anomalous ordering reversal in symmetry restoration, resolved into finer structural layers than in the standard single-symmetry quantum Mpemba effect.