---
title: Higher-Order Symmetric Quantum Mpemba Effect
url: https://www.emergentmind.com/topics/higher-order-symmetric-quantum-mpemba-effect
type: topic
---

# Higher-Order Symmetric Quantum Mpemba Effect

Searching arXiv for recent 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 [2606.06653]. Closely related work frames the quantum Mpemba effect through entanglement asymmetry and charge variance in symmetry-preserving dynamics [2310.04419, 2507.02301], 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 [2509.08078].

## 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 \(\sigma(0)\), the quantum Mpemba criterion for \(\mathcal O\) is satisfied when
\[
\Delta S^{(n)}_{\mathcal O}\bigl(\rho_A(0)\bigr)>
\Delta S^{(n)}_{\mathcal O}\bigl(\sigma_A(0)\bigr),
\]
but at some time \(t_M\),
\[
\Delta S^{(n)}_{\mathcal O}\bigl(\rho_A(t_M)\bigr)<
\Delta S^{(n)}_{\mathcal O}\bigl(\sigma_A(t_M)\bigr).
\]
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 [2606.06653].

In the broader symmetry-based literature, the same underlying logic is formulated for a single conserved \(U(1)\) 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 [2507.02301]. A distinct but compatible generalization defines “higher-order” through successive symmetry projections in Krylov space, producing a hierarchy of \(k\)th-order symmetric complexities and \(k\)th-order Mpemba inversions [2509.08078].

## 2. Symmetry-resolved diagnostics

The central diagnostic in the symmetry perspective is entanglement asymmetry. For a subsystem \(A\) with reduced density matrix \(\rho_A(t)\) and its \(U(1)\)-symmetrized version
\[
\rho_{A,Q}(t)=\sum_q \Pi_q\,\rho_A(t)\,\Pi_q,
\]
the von Neumann entanglement asymmetry is
\[
\Delta S_A(t)=\mathrm{Tr}\Bigl[\rho_A(t)\bigl(\ln\rho_A(t)-\ln\rho_{A,Q}(t)\bigr)\Bigr].
\]
It satisfies \(\Delta S_A(t)\ge 0\), vanishes iff \([\rho_A(t),Q_A]=0\), 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 \(\Delta S_A(0)\), becomes less asymmetric after a finite Mpemba time \(t_M\) [2310.04419].

Charge variance provides a second symmetry-based distance from equilibrium. For \(Q_A=\sum_{i\in A}\sigma_i^z\),
\[
\sigma_Q^2(t)=\langle Q_A^2\rangle_t-\langle Q_A\rangle_t^2.
\]
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 [2507.02301].

In integrable systems, the microscopic criterion has a specific structure. In the large-\(\ell\) 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 [2310.04419].

A different diagnostic framework is complexity-theoretic. Starting from a Krylov basis \(\{|K_n\rangle\}\), the spread-complexity operator is
\[
\hat C_K=\sum_{n=0}^{D-1} n\,|K_n\rangle\langle K_n|,
\]
and with symmetry-sector projectors \(\{\Pi_q\}\), the symmetric and asymmetric projective complexities are
\[
\hat C_{K,S}=\sum_q\sum_{n=0}^{D-1} n\,\Pi_q|K_n\rangle\langle K_n|\Pi_q,\qquad
\hat C_{K,A}=\hat C_K-\hat C_{K,S}.
\]
At \(t=0\), the asymmetric complexity is strictly negative and \(C_{K,S}(0)=|C_{K,A}(0)|\). The structural complexity
\[
C_{\mathrm{structural}}=|C_{K,A}(0)|=C_{K,S}(0)
\]
was reported to predict whether two tilted states will exhibit a Mpemba inversion under time evolution. The same work proposes a hierarchy
\[
C_K=C_{K,S}^{(0)}\ge C_{K,S}^{(1)}\ge C_{K,S}^{(2)}\ge\cdots,
\]
with \(k\)th-order symmetric complexity operator
\[
\hat C_{K,S}^{(k)}
=\sum_{n=0}^{D-1}n\sum_{q_1,\dots,q_k}
P_{q_1\cdots q_k}|K_n\rangle\langle K_n|P_{q_1\cdots q_k}^\dagger,
\]
where \(P_{q_1\cdots q_k}=\Pi_{q_1}\Pi_{q_2}\cdots\Pi_{q_k}\). A \(k\)th-order Mpemba inversion occurs when the ordering of \(C_{K,S}^{(k)}\) between two initial states reverses at some \(t_M\) [2509.08078].

## 3. Fragmented charge–dipole systems

The paradigmatic many-body setting for the higher-order symmetric effect is a system with simultaneous conservation of charge \(Q\) and dipole \(P\), where the Hilbert space fragments into exponentially many disconnected Krylov sectors [2606.06653]. Two models were used.

The first is a random brickwork circuit on spin-\(\tfrac12\) degrees of freedom with depth-\(t\) Floquet operator
\[
\mathcal U = \prod_{a=1}^4
\prod_{j\equiv a\;(\bmod\,4)}
U_{j,j+1,j+2,j+3},
\]
where each four-site gate conserves both \(Q\) and \(P\). The only non-trivial move is the pair hop
\[
|{+}{-}{-}{+}\rangle\;\leftrightarrow\;|{-}{+}{+}{-}\rangle.
\]
Each gate is block-diagonal in the \((Q,P)\) sectors, acts as a Haar-random \(U(2)\) on the two-dimensional pair-hop subspace, and contributes random phases on frozen basis states [2606.06653].

The second is a coherent pair-hopping Hamiltonian with the same symmetries,
\[
H
= J\sum_{i=1}^{L-3}
\Bigl(
S^+_iS^-_{i+1}S^-_{i+2}S^+_{i+3}
+\mathrm{h.c.}
\Bigr),
\qquad
[H,Q]=[H,P]=0,
\]
with open boundary conditions [2606.06653].

To study symmetry restoration, the work focused on the annealed Rényi-2 asymmetry
\[
\Delta S^{(2)}_{\mathcal O}
= -\ln
\frac{
\mathcal P^{(2)}_{A,\mathcal O}(t)
}{
\mathcal P^{(2)}_{A}(t)
},
\qquad
\mathcal P^{(2)}_{A}=\mathbb E[\mathrm{tr}\rho_A^2],
\qquad
\mathcal P^{(2)}_{A,\mathcal O}
=\mathbb E[\mathrm{tr}\rho_{A,\mathcal O}^2].
\]
In the circuit setting, \(\mathcal P^{(2)}\) is computed by a two-replica tensor network whose local transfer tensor is
\[
\mathcal T_{i:i+3}
= \sum_{\alpha_1,\alpha_2}
\frac{|I_{\alpha_1\alpha_2}\rrangle\!\llangle I_{\alpha_1\alpha_2}|}{d_{\alpha_1}d_{\alpha_2}}
+
\frac{|J_{\alpha_1\alpha_2}\rrangle\!\llangle J_{\alpha_1\alpha_2}|}{d_{\alpha_1}d_{\alpha_2}-\delta_{\alpha_1\alpha_2}},
\]
with \(\alpha\) labeling \((Q,P)\) blocks. In the Hamiltonian setting, the pure state is propagated by Chebyshev expansion \(e^{-iHt}\), then projected into sectors to evaluate both \(\Delta S_{\mathcal O}^{(2)}\) and the von Neumann asymmetry \(\Delta S_{\mathcal O}^{v}\) [2606.06653].

## 4. Evidence for higher-order crossings

In the random circuit, system sizes up to \(L=128\) were reached. For two tilted-ferromagnet states \(\theta_1<\theta_2\), so that \(\theta_2\) is initially more asymmetric, both \(\Delta S_Q^{(2)}(t)\) and \(\Delta S_P^{(2)}(t)\) exhibit crossings at intermediate time \(t_M\). 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 [2606.06653].

The first crossing times scale differently with subsystem size \(L_A\):
\[
t^Q_M \propto L_A^{2.09},
\qquad
t^P_M \propto L_A^{1.49}
\qquad (L_A\lesssim 10).
\]
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 [2606.06653].

The pair-hopping Hamiltonian shows the same qualitative structure under coherent evolution. Exact vector simulations up to \(L=20\) found similar Mpemba crossovers in both \(\Delta S_Q^v(t)\) and \(\Delta S_P^v(t)\) for boundary and bulk subsystems with \(L_A\approx 3\). Bulk subsystems exhibit smaller late-time plateaux than boundary subsystems, consistent with weaker edge localization [2606.06653].

These results extend the earlier symmetry-based QME literature, where the effect was already established for \(U(1)\)-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 \(t_{\mathrm{QME}}\sim \ell/v_{\max}\), and many-body localized systems display crossings on a timescale that grows exponentially with subsystem size even though the system does not thermalize [2507.02301]. 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 [2606.06653]. Charge–dipole constraints generate two extreme classes of fragments.

Frozen fragments are single product states containing no movable pattern \(\ket{+ - - +}\); they have dimension \(1\) 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
\[
P_{\rm fr}=\sum_{\alpha\in\mathrm{frozen}}P_\alpha,\qquad
P_{\rm act}=\sum_{\alpha\in\mathrm{active}}P_\alpha,\qquad
P_{\rm fr}+P_{\rm act}=I,
\]
a pure state splits into \(\psi_{\rm fr}+\psi_{\rm act}\) with constant weights \(w_{\rm fr/act}\). The normalized sector states are
\[
\tilde\rho_{\rm fr}=\frac{P_{\rm fr}\rho P_{\rm fr}}{w_{\rm fr}},
\qquad
\tilde\rho_{\rm act}=\frac{P_{\rm act}\rho P_{\rm act}}{w_{\rm act}}.
\]
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 [2606.06653].

An exactly solvable dissipative toy model sharpens this picture. The model uses the symmetric pair-flip Hamiltonian
\[
H
= J\sum_{x=1}^{L/2}
\bigl(
S^+_{j_0+x}S^+_{j_0-x}
+\mathrm{h.c.}
\bigr),
\]
which conserves dipole exactly and preserves only charge parity, together with on-site dephasing,
\[
\dot\rho = -i[H,\rho]
+\gamma\sum_{x=1}^L
\Bigl(
S^z_x\rho S^z_x
-\tfrac12\{(S^z_x)^2,\rho\}
\Bigr).
\]
Because the dynamics factorize into independent mirror pairs, the reduced single-site density matrix takes the form
\[
\rho_{\rm red}(t)
=
\begin{pmatrix}
\tfrac{1+z(t)}2 & \kappa(t)\\[6pt]
\kappa^*(t) & \tfrac{1-z(t)}2
\end{pmatrix},
\]
with
\[
z(t)
= e^{-\gamma t/2}\,\cos\theta\Bigl[
\cos(2\Delta t)+\tfrac{\gamma/4}{\Delta}\,\sin(2\Delta t)
\Bigr],
\]
\[
\kappa(t)
= \tfrac{\sin\theta}{2}\,e^{-\gamma t/2}\!\bigl(\cos t + i\cos\theta\sin t\bigr),
\qquad
\Delta=\sqrt{1-(\tfrac\gamma4)^2}.
\]
The corresponding charged moment is
\[
g(\alpha)
=
\mathrm{tr}\!\bigl[
\rho_{\rm red}(t)e^{-i\alpha S^z}\rho_{\rm red}(t)e^{+i\alpha S^z}
\bigr]
=
\mathcal A(t)+\mathcal B(t)\cos(m\alpha),
\]
where \(m=1\) for charge and \(m\) is the site position for dipole, with
\[
\mathcal A(t)=\frac{1+z(t)^2}{2},
\qquad
\mathcal B(t)=2|\kappa(t)|^2.
\]

For a left-half subsystem,
\[
Z_A(\alpha,t)=\prod_{\ell=1}^{L_A}\bigl[\mathcal A(t)+\mathcal B(t)\cos(\ell\alpha)\bigr],
\]
and
\[
\Delta S_O^{(2)}(t)=
-\ln
\frac{\int_{-\pi}^{\pi}\frac{d\alpha}{2\pi}\,Z_A(\alpha,t)}{Z_A(0,t)}.
\]
In the dissipative regime \(\gamma>0\), if \(\eta(t)=\mathcal B(t)/\mathcal A(t)\ll 1\) at late times, then
\[
\Delta S^{(2)}_{A,O}(t)=L_A\,\eta(t)+O\bigl(L_A^2\eta(t)^2\bigr),
\qquad O\in\{Q,P\}.
\]
For two tilts \(\theta_1<\theta_2\), the first Mpemba crossing solves \(\eta(\theta_1,t_M)=\eta(\theta_2,t_M)\), giving
\[
t_M=\arcsin\!\Bigl[
\tfrac1{\sqrt{\sin^2\theta_1+\sin^2\theta_2}}
\Bigr].
\]
This first crossing time is independent of subsystem size \(L_A\) and independent of whether one probes charge or dipole. The same model also shows that dephasing is essential: without \(\gamma>0\), there is no monotonic symmetry restoration or bona fide Mpemba effect [2606.06653].

## 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 [2606.06653]. Another route, proposed in the Krylov-complexity framework, builds a hierarchy of successive symmetry projections and defines \(k\)th-order symmetric complexity as the part of the spread complexity invariant under \(k\) 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 \(k\) 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 \(n\) [2509.08078].

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 \(U(1)\times U(1)\), one may track a joint entanglement asymmetry on sectors \((q_1,q_2)\); for measurement-induced and Floquet settings, one may seek a cascade of crossings associated with successive restoration of translation, \(U(1)\), inversion, or other symmetries. The same review suggests that an \(n\)-th-order symmetric QME could be diagnosed by an \(n\)-component order parameter \(O=(O_1,\dots,O_n)\) whose components cross in a non-trivial sequence during relaxation [2507.02301].

Several open problems remain explicit. Measuring \(\Delta S_A\) requires tomography plus dephasing or randomized measurement protocols with exponential overhead in \(|A|\). 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 [2507.02301].

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 [2606.06653]. 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.

Source: https://www.emergentmind.com/topics/higher-order-symmetric-quantum-mpemba-effect