Pith. sign in

REVIEW 3 major objections 5 minor 149 references

Geometry of Noisy Quantum Many-Body Dynamics with Continuous Symmetries: Entanglement and Correlations

T0 review · 3 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Late-time dynamics of noisy many-body systems with continuous symmetries — Rényi entanglement growth and non-hydrodynamic correlator decay — is controlled by the quantum geometry of k-commutant manifolds and their singularities.

desk verdict A serious geometric-framework paper with a solid U(1) core and an intriguing but unverified SU(2) log-t prediction that rides on a continuous equal-superposition postulate. read the letter →

arxiv 2608.11297 v1 pith:W64QQVWV submitted 2026-08-11 cond-mat.stat-mech cond-mat.str-elhep-thmath-phmath.MPquant-ph

Marco Lastres , Sanjay Moudgalya This is my paper · ORCID
classification cond-mat.stat-mechcond-mat.str-elhep-thmath-phmath.MPquant-ph
keywords Browniancircuitsk-commutantmanifoldsreplicaHamiltoniansRényientanglemententropynon-hydrodynamiccorrelatorscontinuoussymmetriestime-dependentvariationalprinciplevoidstates
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper claims that in noisy (Brownian) quantum circuits carrying a continuous global symmetry, the average late-time dynamics of Rényi entanglement entropies and of non-hydrodynamic correlation functions is set by the quantum geometry of $k$-commutant manifolds — the ground-state manifolds of the effective replica Hamiltonians that describe the noise-averaged evolution. In generic interacting systems these manifolds are fixed by the symmetry group alone and contain singularities, generated by frozen states, where different replica-symmetry branches touch. A domain wall of replicas (the object that computes entanglement) or a non-hydrodynamic local operator relaxes by driving the field through such a singularity, nucleating a diffusively melting void; the void produces sub-ballistic $S_2\sim\sqrt{\kappa t}$ Rényi entropy growth in one dimension and stretched-exponential decay $e^{-\sqrt{\kappa t}}$ of non-hydrodynamic correlators, with exponential decay in $d\ge 3$ and $e^{-\kappa t/\log t}$ in $d=2$. For non-Abelian symmetries the singular set is itself a continuous manifold, so a continuum of degenerate semiclassical trajectories contributes and the entanglement inside the void grows logarithmically. The authors calibrate these geometric rules on exactly solvable toy models and check them against tensor-network simulations of the effective models.

What carries the argument

The $k$-commutant manifold: the set of fully polarized product states spanning the ground-state space of the effective replica Hamiltonian $P^{(k)}$, equivalently the symmetry algebra of $k$ replicas of the noisy circuit. In interacting systems it is a complex projective variety whose singularities sit at replicated frozen states, and its Fubini-Study metric turns the time-dependent variational principle into a diffusive heat equation $\partial_t v^\gamma = 2J(\nabla^2 v^\gamma + \Gamma^\gamma_{\alpha\beta}\nabla v^\alpha\cdot\nabla v^\beta)$ for the semiclassical field. The machinery is completed by two posited rules: branch-switching points act as Dirichlet boundaries that drift to equalize the field pressure $|\partial_x\mathbf{n}(x^-)|^2 = |\partial_x\mathbf{n}(x^+)|^2$, and when several lowest-energy trajectories exist the evolved state is their equal superposition. These ingredients convert observable dynamics into pure geometry: entanglement growth and correlator decay are fixed by how fields on the manifold traverse its singularities.

What would settle it

Simulate the $k=2$ effective Hamiltonian of the $U(1)$ Brownian model (Eqs. 124–129) and measure the spatial profile of the purity weight function of an evolved half-chain domain wall at long times: the paper predicts a purity decay $e^{-\sqrt{\kappa t}}$ with the error-function weight $r(x,t)=\tan(\pi/4\,(1-|\mathrm{erf}(x/\sqrt{4\kappa t})|))$ and a cusp at the interface, whereas the alternative large-$q$ membrane picture (Ref. [31], Eq. 150) predicts a Gaussian weight at the same exponent; observing a Gaussian profile, or a one-dimensional purity exponent different from $1/2$, would falsify the singularity-driven void mechanism.

Watch

Extended reading notes

Core claim

The central claim is that the late-time physics of symmetric Brownian circuits — for both entanglement and correlation observables — reduces to the imaginary-time dynamics of smooth field configurations on ferromagnetic ground-state manifolds of the effective replica Hamiltonians $P^{(k)}$, the $k$-commutant manifolds. For interacting systems that form symmetric $k$-designs these manifolds are determined by the symmetry alone: for $U(1)$ the $k=2$ manifold is two copies of $S^2\times S^2$ touching at the two frozen states, and for $SU(2)$ it is two copies of $\mathrm{CP}^3\times \mathrm{CP}^3$ intersecting on a continuous $(S^2)^4$ submanifold. The authors show that the low-energy excitations obey TDVP equations that reduce to a heat equation along the manifold's Fubini-Study metric, and they add two calibrated prescriptions: singularities act as Dirichlet boundaries whose position drifts to equalize field pressure, and equally energetic trajectories superpose with equal weights. Applying these rules, one-dimensional Rényi entanglement grows as $\sqrt{\kappa t}$ because the domain wall melts through the singularity into a void, and the product of suppressed local weights inside the void makes the purity decay as $e^{-\sqrt{\kappa t}}$; non-hydrodynamic operators generate voids of the same structure, giving the same stretched-exponential decay of their squared correlators, with exponential decay in $d\ge 3$ and $e^{-\kappa t/\log t}$ in $d=2$. For non-Abelian symmetries the void is threaded by a continuous family of degenerate trajectories, giving logarithmic entanglement growth within the void and a dynamical distinction between Abelian and non-Abelian continuous symmetries.

Load-bearing premise

The load-bearing premise is that the semiclassical TDVP dynamics on the $k$-commutant manifold — including the posited singularity rules (Dirichlet pinning with pressure equalization) and equal-weight superposition of degenerate trajectories — captures the leading late-time imaginary-time evolution of the replicated boundary states; these prescriptions are calibrated on the $\mathrm{es}\eta$ toy model and then applied to the $U(1)$ and $SU(2)$ effective Hamiltonians without a derivation from the microscopic models, so if quantum fluctuations around the trajectories are not subleading, the predicted exponents and weight functions fail.

Editorial extensions

If this is right

  • In one-dimensional interacting noisy $U(1)$-symmetric systems, the annealed second Rényi entropy of a half-chain domain wall grows as $\sqrt{\kappa t}$ with the explicit weight function $r(x,t)$ of Eq. (147), and the entanglement-membrane picture of sharp domain walls is replaced by diffusively melting voids.
  • Squared non-hydrodynamic autocorrelators decay as $e^{-\sqrt{\kappa t}}$ in $d=1$, as $e^{-\kappa t}$ in $d\ge 3$, and as $e^{-\kappa t/\log t}$ in $d=2$, while hydrodynamic correlators decay algebraically ($\sim 1/(\kappa t)$ in $d=1$) because replicas decouple within a single branch.
  • For $SU(2)$ and more generally any non-Abelian continuous symmetry with multiplicity-free irreps, the intersection of the $e$- and $\eta$-branches is a continuous manifold, so the melting domain wall passes through a continuum of degenerate solutions and the void acquires logarithmic entanglement growth, unlike the finite two-branch 'cat' entanglement of the $U(1)$ case.
  • Because the $k$-commutant manifolds depend only on the symmetry whenever the generators form a symmetric $k$-design, the same exponents and weight functions apply to any isotropic interacting noisy model with the same symmetry — including Haar-random circuits — independent of microscopic details of the noise.
  • Extrapolating to $k\ge 3$, the paper conjectures that the two shortest trajectories through the replicated frozen states still dominate, so all Rényi entropies grow diffusively as $\sqrt{\kappa t}$, while the von Neumann limit remains open and subtle.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the manifold geometry is the controlling datum, the same TDVP machinery should give the exponents for observables the paper lists but does not compute — symmetry-resolved entropies, entanglement asymmetry, and OTOCs — by dressing the $\eta$-branch states with charged operators, with no new replica calculation required.
  • The numerically observed $t^{1/3}$ growth of the sink region attributes the leading correction to quantum fluctuations of the singularity; checking this exponent at longer times in the full $U(1)$ model would test whether the fluctuation picture survives beyond the $\mathrm{es}\eta$ toy model.
  • The Abelian/non-Abelian difference in trajectory degeneracy implies a measurable fingerprint: $U(1)$ voids should plateau at $\log 2$ entanglement from the two-branch cat state, while $SU(2)$ voids should show persistent logarithmic growth, a contrast that is sharper than the shared $\sqrt{t}$ entropy exponent.
  • Multipole-conserving and fragmented systems are a natural next testbed: the paper expects their commutant manifolds to be non-ferromagnetic, but if an analogous geometric structure exists, a void-like mechanism might survive with modified exponents, linking the geometric picture to constrained dynamics.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. This paper develops a geometric framework for the averaged imaginary-time dynamics of replica Hamiltonians that arise in noisy Brownian circuits with continuous symmetries. The authors show that the ground-state manifolds of the effective replica Hamiltonians are controlled by the geometry of k-commutants, and that in interacting systems these manifolds are unions of smooth branches (e.g., (S^2)^k for U(1)) that intersect at "frozen" states, producing singularities. Using the time-dependent variational principle (TDVP) on these manifolds, they derive diffusive equations of motion for low-energy field configurations, with additional prescriptions for branch-switching singularities (Dirichlet boundaries plus pressure equalization) and for equal superpositions of degenerate semiclassical trajectories. The framework is first calibrated on the ferromagnetic Heisenberg model and on a newly introduced three-state "esη" toy model, then applied to U(1)-symmetric models, where it yields sub-ballistic Rényi entanglement growth (~√t in 1D) and stretched-exponential decay of non-hydrodynamic correlators (~e^{-√κt} in 1D), and to SU(2)-symmetric models, where a continuous family of degenerate trajectories is predicted to produce logarithmic entanglement growth within the void. The paper also contrasts these interacting cases with free-fermion systems, whose commutant manifolds are smooth, and discusses higher replica numbers, higher dimensions, and extensions to other entropic quantities.

Significance. If the central claims hold, this paper provides a unified geometric explanation for several previously disparate phenomena: the diffusive growth of Rényi entropies in charge-conserving systems, the stretched-exponential decay of non-hydrodynamic correlators, the role of void states in rigorous bounds, and the difference between Abelian and non-Abelian symmetry structures. The TDVP equations are derived carefully and the analytic solutions are compared with MPS numerics in the Heisenberg and esη models, with partial checks in the U(1) sector; the predictions are concrete and falsifiable. The paper also clearly credits prior work on void arguments and on membrane pictures, and it is honest about the status of several key ingredients. The main significance risk is that the non-Abelian logarithmic-growth claim rests on an equal-superposition prescription for a continuum of trajectories that is not derived from the microscopic model and is not numerically checked in the SU(2) effective Hamiltonian.

major comments (3)
  1. [Sec. VII B 2-3, Eqs. (181)-(187)] The SU(2) logarithmic-entanglement prediction rests entirely on the equal-superposition prescription for a continuum of degenerate TDVP trajectories. The paper itself labels this a postulate in Sec. IV B 4, and it is calibrated only for the smooth Heisenberg domain wall and the discrete two-trajectory U(1) case. For SU(2), the degenerate family is parametrized by (S^2)^4, the measure dμ(n) in Eq. (181) is not derived, and no simulation of the SU(2) effective model is presented to check even the existence of the cat state. If the true replicated state is not an equal-weight coherent superposition---for example, if fluctuation determinants around different saddles are n-dependent or the saddles decohere---the log t growth could be replaced by saturation or a different rate. Because the abstract's non-Abelian claim depends on this, the gap is load-bearing and needs either a microscopic derivation or a numerical test.
  2. [Sec. III C and VI A 2, Eq. (128)] The exact determination of the ferromagnetic ground-state spaces and the manifold identifications, including Eq. (49) for U(1) and Eq. (168) for SU(2), is asserted to be proven in the in-preparation Ref. [75], with the text stating at one point "This is rigorously proven in Ref. [75]." These manifolds are the foundation of the TDVP analysis: if the ferromagnetic ground-state structure or the k-design property fails, the whole geometric picture changes. Since Ref. [75] is not available to the reader, the exactness of these central mathematical inputs cannot be checked from the manuscript alone. The authors should either include self-contained proofs, state the precise conditions under which the k-design/ferromagnetism holds, or explicitly mark these statements as conjectures for the purposes of this paper.
  3. [Sec. VII B 3 and Appendix E, Eq. (187)] The predicted log(κt) growth is computed for the entanglement of the variational state |A:\bar A(t)⟩ itself, whereas the physical annealed Rényi entropy is defined through the replicated boundary-state overlap in Eq. (26), whose U(1) analog is evaluated in Eq. (147). The paper does not show how logarithmic entanglement growth of the variational state translates into logarithmic growth of the overlap observable that defines the Rényi entropy. Without such a mapping, the SU(2) logarithmic-growth claim is not quantitatively connected to the quantity originally defined; the authors should either spell out this connection or restrict the claim to the variational state.
minor comments (5)
  1. [Sec. VI B 4 and Fig. 8] The numerical comparison for the U(1) purity prediction is restricted to short times, and the text notes that the predictions with the cusped weight function r(x,t) and the Gaussian weight function ř(x,t) are almost indistinguishable for the tested initial states. This limitation should be stated more prominently, since the cusp is a distinctive qualitative prediction of the present framework.
  2. [Abstract and Sec. VIII] The abstract mentions Haar-random circuits as part of the framework's scope, but the body explicitly treats Brownian models and only argues that the results are expected to generalize to other circuits forming symmetric k-designs. A more precise wording in the abstract would avoid overstating the scope.
  3. [Sec. V D 2 and VI B 2] The t^{1/3} growth of the sink region is reported numerically but no derivation is given. Since this exponent is used as evidence that fluctuations beyond TDVP are subleading, a short analytic argument or a reference to a derivation would strengthen the claim.
  4. [Sec. VI A 2, Eq. (126)] The notation |σ_p⟩ with p∈{0,1} is introduced in Eq. (126), but the meaning of the subscript p and its relation to the local parity sector defined in Eq. (125) could be stated more explicitly for readers not familiar with the replica basis.
  5. [Eq. (150) and Sec. VI B 3] The discussion of the breakdown of the entanglement membrane picture is clear, but the relationship between the present cusped weight function and the large-q extrapolation of Ref. [31] would benefit from a more explicit statement that the two predictions also differ in their functional form, not only in their derivation.

Circularity Check

1 steps flagged · score 4.0 of 10

Central manifold geometry is deferred to the authors' own companion proof; the exponent predictions themselves are derived, not fitted.

  1. self citation load bearing [Sec. III C (page 8, second column); Sec. VI A 2, footnote 3 (page 24); Sec. VII A]
    "In an upcoming work Ref. [75], we provide a more general algebraic framework to rigorously prove the forms of the ferromagnetic ground states and the associated manifolds starting from the Hamiltonians, which also applies to the Hamiltonians studied in this work. [Footnote 3: This is rigorously proven in Ref. [75].]"

    The paper's late-time predictions for U(1) and SU(2) are all obtained by applying TDVP on the exact k=2 commutant manifolds M^(2)_U(1) and M^(2)_SU(2). Those exact manifold forms are not proved in this paper; they are asserted and deferred to the authors' own forthcoming companion work [75]. The TDVP equations, void solutions, and the resulting sqrt(t) Rényi growth, e^{-sqrt(kappa t)} correlator decay, and log(t) entanglement growth are all consequences of this asserted geometry. Thus the load-bearing premise of the derivation rests on an unverified self-citation, rather than on an independent, machine-checked, or externally reproduced theorem.

full rationale

Most of the derivation is not circular in the reduction-by-construction sense. The sqrt(t) Rényi growth and e^{-sqrt(kappa t)} non-hydrodynamic correlator decay follow from solving the TDVP heat-type equations (Eqs. 96, 139, 159, 185) on the assumed manifold, and the paper provides independent numerical calibration in the Heisenberg model, the es-eta toy model, and the U(1) model; no target exponent is fitted from the data. The U(1) log(2) entanglement offset is a direct consequence of the two-trajectory superposition and is compared with MPS data (Fig. 7c), not imposed as a fit. The equal-superposition prescription for multiple TDVP solutions is explicitly introduced as a postulate (Sec. IV B 4) and then applied to the SU(2) continuum with a uniform measure d mu(n) that is not derived from the microscopic Brownian model; this is a missing-derivation or correctness gap, but it is not circular, because the log(t) growth in Eq. (187) is a mathematical consequence of the posited superposition rather than identical to it. The most substantial circularity-adjacent defect is the load-bearing self-citation: the exact forms of M^(2)_U(1) and M^(2)_SU(2), which are the central geometric inputs, are deferred to the authors' own unpublished [75] rather than proved or independently verified here. Because the exponent derivation itself is self-contained once the manifold is granted, and the paper contains independent numerical checks, the score is 4 rather than 6 or higher.

Assumptions & free parameters 1 free parameters · 10 assumptions · 1 invented entities

The central framework rests on the k-design/ferromagnetism assumption, the low-energy continuum form of the effective Hamiltonian, and several ad hoc TDVP prescriptions (singularity dynamics, equal superpositions). No new physical particles or forces are introduced; the esη model is a mathematical toy. The only explicit free parameter is the UV cutoff b for d≥3 predictions.

free parameters (1)
  • UV cutoff b for d≥3 η-particle dynamics = of order lattice spacing (not fixed)
    Introduced in Sec. V E 3 to regularize the Dirichlet boundary condition at the η-particle. It controls the stationary energy E_st and hence the exponential decay rate e^{-κt} in d≥3. The prediction is not parameter-free.
assumptions (10)
  • standard math Brownian average of replicated unitaries equals imaginary-time evolution under P^(k) = Σ_α (L^(k)_{h_α})^2 (Eqs. 2, 4).
    Standard result in Brownian circuit literature (Refs [16,17,20,21,23,45]); used to define all quantities studied.
  • standard math Ground states of P^(k) are in one-to-one correspondence with the k-commutant of the Brownian generators (Eqs. 15-16).
    Follows from P^(k) being a sum of positive-semidefinite terms; the paper uses this to equate the ground-state manifold with the k-commutant manifold.
  • domain assumption The Brownian gate sets in the interacting U(1) and SU(2) models form symmetric k-designs for k=2, i.e., their k-commutants are exactly the operator algebras generated by the on-site symmetry group G.
    Stated in Sec. III D 2 and Sec. VI A 4. The U(1) case is claimed to be rigorously proven in Ref [75] (footnote 3); the SU(2) case is stated as proven in Ref [75] (Sec. VII A). Not proven in this paper.
  • domain assumption For the models studied, the ground state space of the effective Hamiltonian is ferromagnetic: ker(P^(k)) = span{|v>^⊗L} for |v> in a local manifold M_GS (Eq. 30).
    Assumed for the physical models on the basis of the k-design property; the paper says the general proof is in Ref [75] (Sec. III C). This is load-bearing because the entire geometric framework requires the ground states to be fully polarized product states.
  • domain assumption The low-energy continuum Hamiltonian for a ferromagnetic effective model has the form H[v] = J ∫ d^d x g_{αβ}(v) ∇v^α · ∇v^β (Eq. 68).
    Used to write TDVP equations as a heat equation on the manifold. A derivation is sketched around Eq. (B35) under fairly general assumptions, but it inherits the ferromagnetism/k-design assumptions.
  • ad hoc to paper In the TDVP treatment of singular manifolds, branch-switching points act as stationary Dirichlet boundaries for the fields, with a pressure equalization condition on their motion (Eqs. 86-87).
    Posited in Sec. V B; higher-dimensional generalization in Appendix B 6 b. It is calibrated on the esη toy model and assumed to hold for U(1) and SU(2). Not derived from the microscopic Hamiltonian.
  • ad hoc to paper When multiple energy-degenerate TDVP solutions exist, the true evolved state is approximated by an equal superposition of all such solutions (Sec. IV B 4).
    Introduced to explain the log(2) half-chain entanglement in the U(1) domain wall (Sec. VI B 2) and the log(κt) growth for SU(2) (Sec. VII B 3). This is an extension of standard TDVP without a derivation.
  • domain assumption Replica decoupling: for k=2, away from the singularities, the energy functional factorizes into two independent copies, each governed by the k=1 effective dynamics (Appendix B 6 c; Eqs. 132, 175).
    Used to reduce the two-field dynamics to two independent Heisenberg problems. Holds to leading order away from the singular points.
  • domain assumption In the U(1) qubit model, the local parity sector Z_i=+1 contains all relevant boundary states, and the |ξ_p> states are gapped and can be neglected for low-energy dynamics (Sec. VI A 1).
    Restricts the local Hilbert space to 6 states; the gap of |ξ_p> is due to the one-site generators Z_i I_j and I_i Z_j. This is needed to map the effective Hamiltonian to the esη form.
  • ad hoc to paper For d≥3 single-particle (η) dynamics, the Dirichlet condition is regularized with a finite radius b, leading to a stationary harmonic profile θ_st(x) (Eqs. 116-119).
    The UV cutoff b is introduced to define the boundary condition; the asymptotic decay rate in d≥3 depends on b through E_st ∝ κ (b/a)^{d-2}. The value of b/a is not determined.
invented entities (1)
  • esη toy model
    purpose: Three-state ferromagnetic model with a point-like singularity in its ground state manifold, used to develop and numerically test the TDVP framework for singular manifolds.
    This is a constructed model, not a physical entity; it has no falsifiable handle outside the paper's own simulations.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Geometry of Noisy Quantum Many-Body Dynamics with Continuous Symmetries: Entanglement and Correlations." pith.science (2026). https://pith.science/paper/W64QQVWV

@misc{pith2026260811297,
  author       = {Pith},
  title        = {Pith review of: Geometry of Noisy Quantum Many-Body Dynamics with Continuous Symmetries: Entanglement and Correlations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/W64QQVWV}},
  note         = {Machine review of arXiv:2608.11297}
}
abstract

We study unitary quantum dynamics in noisy Brownian models with global continuous symmetries, such as $U(1)$ and $SU(2)$, focusing on R\'enyi entanglement entropies and hydrodynamic and non-hydrodynamic correlators. By mapping the averaged late-time dynamics to the low-energy physics of effective replica Hamiltonians, we find that the evolution is controlled by the quantum geometry of their ground-state manifolds, which is directly related to the geometry of $k$-commutants---the symmetry algebra of $k$ replicas of the system. In interacting systems, these $k$-commutants are generically determined solely by the symmetries of the system, independent of microscopic details of the noisy evolution. This allows us to use the time-dependent variational principle (TDVP) to provide simple geometric explanations for the sub-ballistic R\'enyi entanglement growth and the anomalous decay of non-hydrodynamic correlators in interacting systems with continuous symmetries. We find this behavior to be intimately connected to singularities within the $k$-commutant manifolds, arising from frozen ``void'' states in the Hilbert space that exist due to continuous on-site symmetries. This also demystifies the important role of voids in the dynamics of these observables, previously identified in $U(1)$ symmetric systems. We compare these behaviors in interacting systems with Abelian and non-Abelian continuous symmetries and in free-fermion systems, which differ in the geometry of their $k$-commutants. Ultimately, this work provides a general geometric framework for systematically studying observables in noisy systems with continuous symmetries, including Haar-random circuits.

Figures

Figures reproduced from arXiv: 2608.11297 by the authors.

Figure 1
Figure 1. FIG. 1. Pictorial representations of the vectorization of the [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Illustration of the ferromagnetic ground state manifolds of the models studied here. The lines show paths between the [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. (a) Illustration of the ground state manifold and diffusive relaxation for the Heisenberg model imaginary-time dynamics. [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: FIG. 4. (a) Illustration of a field configuration for the [PITH_FULL_IMAGE:figures/full_fig_p017_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Comparison between solution to the TDVP equa [PITH_FULL_IMAGE:figures/full_fig_p019_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Comparison with numerical data of the TDVP pre [PITH_FULL_IMAGE:figures/full_fig_p021_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. (a) Illustration of the two symmetry-related semiclassical trajectories that contribute to the formation of the void in [PITH_FULL_IMAGE:figures/full_fig_p024_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Comparison with numerical data of the TDVP pre [PITH_FULL_IMAGE:figures/full_fig_p027_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. Comparison of the weight functions appearing in the [PITH_FULL_IMAGE:figures/full_fig_p028_9.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

149 extracted references · 37 canonical work pages

  1. [75]

    Lastres and S

    M. Lastres and S. Moudgalya, in preparation, (2026)

    2026

  2. [1]

    D’Alessio, Y

    L. D’Alessio, Y. Kafri, A. Polkovnikov, and M. Rigol, From quantum chaos and eigenstate thermalization to statistical mechanics and thermodynamics, Advances in Physics65, 239 (2016)

    2016

  3. [2]

    T. Mori, T. N. Ikeda, E. Kaminishi, and M. Ueda, Thermalization and prethermalization in isolated quan- tum systems: a theoretical overview, Journal of Physics B: Atomic, Molecular and Optical Physics51, 112001 (2018)

    2018

  4. [3]

    Rigol, V

    M. Rigol, V. Dunjko, and M. Olshanii, Thermalization and its mechanism for generic isolated quantum sys- tems, Nature452, 854 (2008)

    2008

  5. [4]

    Polkovnikov, K

    A. Polkovnikov, K. Sengupta, A. Silva, and M. Ven- galattore, Colloquium: Nonequilibrium dynamics of closed interacting quantum systems, Reviews of Mod- ern Physics83, 863 (2011)

    2011

  6. [5]

    Nahum, J

    A. Nahum, J. Ruhman, S. Vijay, and J. Haah, Quantum entanglement growth under random unitary dynamics, Phys. Rev. X7, 031016 (2017)

    2017

  7. [6]

    Nahum, J

    A. Nahum, J. Ruhman, S. Vijay, and J. Haah, Operator spreading in random unitary circuits, Physical Review X8, 021014 (2018)

    2018

  8. [7]

    C. W. von Keyserlingk, T. Rakovszky, F. Pollmann, and S. L. Sondhi, Operator hydrodynamics, otocs, and entanglement growth in systems without conservation laws, Phys. Rev. X8, 021013 (2018)

    2018

Show all 149 references
  1. [8]

    A. Chan, A. De Luca, and J. T. Chalker, Solution of a minimal model for many-body quantum chaos, Phys. Rev. X8, 041019 (2018)

    2018

  2. [9]

    M. P. Fisher, V. Khemani, A. Nahum, and S. Vi- jay, Random quantum circuits, Annual Review of Con- densed Matter Physics14, 335 (2023)

    2023

  3. [10]

    Bertini, Non-equilibrium quantum many-body physics with quantum circuits, SciPost Physics Lecture Notes , 124 (2026)

    B. Bertini, Non-equilibrium quantum many-body physics with quantum circuits, SciPost Physics Lecture Notes , 124 (2026)

    2026

  4. [11]

    Collins and I

    B. Collins and I. Nechita, Random quantum channels i: Graphical calculus and the bell state phenomenon, Communications in Mathematical Physics297, 345–370 (2010)

    2010

  5. [12]

    Hunter-Jones, Unitary designs from statistical me- chanics in random quantum circuits, arXiv e-prints (2019), arXiv:1905.12053 [quant-ph]

    N. Hunter-Jones, Unitary designs from statistical me- chanics in random quantum circuits, arXiv e-prints (2019), arXiv:1905.12053 [quant-ph]

    2019 arXiv

  6. [13]

    Nahum, S

    A. Nahum, S. Roy, S. Vijay, and T. Zhou, Real-time cor- relators in chaotic quantum many-body systems, Phys. Rev. B106, 224310 (2022)

    2022

  7. [14]

    Zhou and A

    T. Zhou and A. Nahum, Emergent statistical mechan- ics of entanglement in random unitary circuits, Physical Review B99, 10.1103/physrevb.99.174205 (2019)

    2019 doi

  8. [15]

    Zhou and A

    T. Zhou and A. Nahum, Entanglement Membrane in Chaotic Many-Body Systems, Physical Review X10, 031066 (2020), arXiv:1912.12311 [cond-mat.str-el]

    2020 arXiv

  9. [16]

    Lashkari, D

    N. Lashkari, D. Stanford, M. Hastings, T. Osborne, and P. Hayden, Towards the fast scrambling conjecture, Journal of High Energy Physics2013, 22 (2013)

    2013

  10. [17]

    S¨ underhauf, L

    C. S¨ underhauf, L. Piroli, X.-L. Qi, N. Schuch, and J. I. Cirac, Quantum chaos in the Brownian SYK model with large finite N : OTOCs and tripartite information, Jour- nal of High Energy Physics2019, 38 (2019)

    2019

  11. [18]

    Agarwal and S

    L. Agarwal and S. Xu, Emergent symmetry in brownian syk models and charge dependent scrambling, Journal of High Energy Physics2022, 10.1007/jhep02(2022)045 (2022)

    2022 doi

  12. [19]

    Zhou and X

    T. Zhou and X. Chen, Operator dynamics in a Brownian quantum circuit, Physical Review E99, 052212 (2019)

    2019

  13. [20]

    Xu and B

    S. Xu and B. Swingle, Locality, Quantum Fluctuations, and Scrambling, Phys. Rev. X9, 031048 (2019)

    2019

  14. [21]

    Bauer, D

    M. Bauer, D. Bernard, and T. Jin, Stochastic dissipative quantum spin chains (I) : Quantum fluctuating discrete hydrodynamics, SciPost Phys.3, 033 (2017)

    2017

  15. [22]

    Bernard and L

    D. Bernard and L. Hruza, Exact entanglement in the driven quantum symmetric simple exclusion process, SciPost Phys.15, 175 (2023)

    2023

  16. [23]

    Vardhan and S

    S. Vardhan and S. Moudgalya, Entanglement dynam- ics from universal low-lying modes, Phys. Rev. B113, 014308 (2026)

    2026

  17. [24]

    Rakovszky, F

    T. Rakovszky, F. Pollmann, and C. W. von Keyserlingk, Diffusive Hydrodynamics of Out-of-Time-Ordered Cor- relators with Charge Conservation, Physical Review X 8, 031058 (2018)

    2018

  18. [25]

    Khemani, A

    V. Khemani, A. Vishwanath, and D. A. Huse, Opera- tor Spreading and the Emergence of Dissipative Hydro- dynamics under Unitary Evolution with Conservation Laws, Physical Review X8, 031057 (2018)

    2018

  19. [26]

    Crossley, P

    M. Crossley, P. Glorioso, and H. Liu, Effective field the- ory of dissipative fluids, Journal of High Energy Physics 2017, 95 (2017)

    2017

  20. [27]

    Liu and P

    H. Liu and P. Glorioso, Lectures on non-equilibrium effective field theories and fluctuating hydrodynamics, PoSTASI2017, 008 (2018)

    2018

  21. [28]

    L. V. Delacretaz, Boulder Lectures on Thermal Dynam- ics and Hydrodynamic EFTs, arXiv e-prints (2026), arXiv:2606.02391 [hep-th]

    2026 arXiv

  22. [29]

    Jonay, D

    C. Jonay, D. A. Huse, and A. Nahum, Coarse-grained dynamics of operator and state entanglement, arXiv e- prints (2018), arXiv:1803.00089 [cond-mat.stat-mech]

    2018 arXiv

  23. [30]

    Mezei, Membrane theory of entanglement dynam- ics from holography, Phys

    M. Mezei, Membrane theory of entanglement dynam- ics from holography, Phys. Rev. D98, 106025 (2018), arXiv:1803.10244 [hep-th]

    2018 arXiv

  24. [32]

    Huang, Dynamics of r´ enyi entanglement entropy in diffusive qudit systems, IOP SciNotes1, 035205 (2020)

    Y. Huang, Dynamics of r´ enyi entanglement entropy in diffusive qudit systems, IOP SciNotes1, 035205 (2020)

    2020

  25. [33]

    Zhou and A

    T. Zhou and A. W. W. Ludwig, Diffusive scaling of R\’enyi entanglement entropy, Physical Review Re- search2, 033020 (2020)

    2020

  26. [34]

    Han and X

    Y. Han and X. Chen, Entanglement dynamics in U(1) symmetric hybrid quantum automaton circuits, Quan- tum7, 1200 (2023), 2305.18141v2

    2023 arXiv

  27. [35]

    Turkeshi, P

    X. Turkeshi, P. Calabrese, and A. De Luca, Quantum mpemba effect in random circuits, Phys. Rev. Lett.135, 040403 (2025)

    2025

  28. [36]

    Majidy, U

    S. Majidy, U. Agrawal, S. Gopalakrishnan, A. C. Pot- ter, R. Vasseur, and N. Y. Halpern, Critical phase and spin sharpening in su(2)-symmetric monitored quantum circuits, Phys. Rev. B108, 054307 (2023)

    2023

  29. [37]

    Z. Li, H. Zheng, Y. Wang, L. Jiang, Z.-W. Liu, and J. Liu, SU(d)-symmetric random unitaries: quantum scrambling, error correction, and machine learning, npj Quantum Information11, 158 (2025). 39

    2025

  30. [38]

    Zanardi, Virtual Quantum Subsystems, Phys

    P. Zanardi, Virtual Quantum Subsystems, Phys. Rev. Lett.87, 077901 (2001)

    2001

  31. [39]

    S. D. Bartlett, T. Rudolph, and R. W. Spekkens, Ref- erence frames, superselection rules, and quantum infor- mation, Rev. Mod. Phys.79, 555 (2007)

    2007

  32. [40]

    Read and H

    N. Read and H. Saleur, Enlarged symmetry algebras of spin chains, loop models, and S-matrices, Nuclear Physics B777, 263 (2007)

    2007

  33. [41]

    Moudgalya and O

    S. Moudgalya and O. I. Motrunich, Hilbert Space Frag- mentation and Commutant Algebras, Phys. Rev. X12, 011050 (2022)

    2022

  34. [42]

    Moudgalya and O

    S. Moudgalya and O. I. Motrunich, From symmetries to commutant algebras in standard Hamiltonians, Annals of Physics455, 169384 (2023)

    2023

  35. [43]

    Moudgalya and O

    S. Moudgalya and O. I. Motrunich, Exhaustive Charac- terization of Quantum Many-Body Scars using Commu- tant Algebras, Physical Review X14, 041069 (2024)

    2024

  36. [44]

    Moudgalya and O

    S. Moudgalya and O. I. Motrunich, Numerical meth- ods for detecting symmetries and commutant algebras, Phys. Rev. B107, 224312 (2023)

    2023

  37. [45]

    Moudgalya and O

    S. Moudgalya and O. I. Motrunich, Symmetries as Ground States of Local Superoperators: Hydrodynamic Implications, PRX Quantum5, 040330 (2024)

    2024

  38. [46]

    Lastres, F

    M. Lastres, F. Pollmann, and S. Moudgalya, Nonuniver- sality from conserved superoperators in unitary circuits, Phys. Rev. B113, 014310 (2026)

    2026

  39. [47]

    Gross, K

    D. Gross, K. Audenaert, and J. Eisert, Evenly dis- tributed unitaries: On the structure of unitary designs, Journal of Mathematical Physics48, 052104 (2007)

    2007

  40. [48]

    Dankert, R

    C. Dankert, R. Cleve, J. Emerson, and E. Livine, Exact and approximate unitary 2-designs, Phys. Rev. A80, 012304 (2009)

    2009

  41. [49]

    A. W. Harrow and R. A. Low, Random quantum circuits are approximate 2-designs, Communications in Mathe- matical Physics291, 257 (2009)

    2009

  42. [50]

    S. N. Hearth, M. O. Flynn, A. Chandran, and C. R. Laumann, Unitaryk-designs from random number- conserving quantum circuits, Phys. Rev. X15, 021022 (2025)

    2025

  43. [51]

    H. Liu, A. Hulse, and I. Marvian, Unitary Designs from Random Symmetric Quantum Circuits, arXiv e-prints (2024), arXiv:2408.14463 [quant-ph]

    2024 arXiv

  44. [52]

    Mitsuhashi, R

    Y. Mitsuhashi, R. Suzuki, T. Soejima, and N. Yosh- ioka, Unitary designs of symmetric local random cir- cuits, Phys. Rev. Lett.134, 180404 (2025)

    2025

  45. [53]

    Y. Bao, S. Choi, and E. Altman, Symmetry enriched phases of quantum circuits, Annals of Physics435, 168618 (2021), special issue on Philip W. Anderson

    2021

  46. [54]

    Z. Li, H. Zheng, J. Liu, L. Jiang, and Z.-W. Liu, De- signs from Local Random Quantum Circuits with SU(d) Symmetry, PRX Quantum5, 040349 (2024)

    2024

  47. [55]

    Ogunnaike, J

    O. Ogunnaike, J. Feldmeier, and J. Y. Lee, Unify- ing Emergent Hydrodynamics and Lindbladian Low- Energy Spectra across Symmetries, Constraints, and Long-Range Interactions, Phys. Rev. Lett.131, 220403 (2023)

    2023

  48. [56]

    Lastres and S

    M. Lastres and S. Moudgalya, Geometry of free fermion commutants (2026), arXiv:2604.05031 [quant-ph]

    2026 arXiv

  49. [57]

    McCulloch, J

    E. McCulloch, J. A. Jacoby, C. von Keyserlingk, and S. Gopalakrishnan, Subexponential decay of local cor- relations from diffusion-limited dephasing, Physical Re- view Letters136, 10.1103/393g-z21y (2026)

    2026 doi

  50. [58]

    McCulloch, J

    E. McCulloch, J. A. Jacoby, and S. Gopalakrishnan, Long-lived local quantum coherences from hydrody- namic large deviations (2026), arXiv:2604.27074 [quant- ph]

    2026 arXiv

  51. [59]

    Hackl, T

    L. Hackl, T. Guaita, T. Shi, J. Haegeman, E. Demler, and J. I. Cirac, Geometry of variational methods: dy- namics of closed quantum systems, SciPost Phys.9, 048 (2020)

    2020

  52. [60]

    Bengtsson and K

    I. Bengtsson and K. ˙Zyczkowski,Geometry of Quan- tum States: An Introduction to Quantum Entanglement (Cambridge University Press, Cambridge, England, UK, 2017)

    2017

  53. [61]

    Bernard and T

    D. Bernard and T. Jin, Solution to the quantum symmetric simple exclusion process: The continuous case, Communications in Mathematical Physics384, 1141–1185 (2021)

    2021

  54. [62]

    Swann, D

    T. Swann, D. Bernard, and A. Nahum, Spacetime picture for entanglement generation in noisy fermion chains, Phys. Rev. B112, 064301 (2025)

    2025

  55. [63]

    Huang, M

    X. Huang, M. Qi, J.-H. Zhang, and A. Lucas, Hy- drodynamics as the effective field theory of strong-to- weak spontaneous symmetry breaking, Physical Review B111, 125147 (2025)

    2025

  56. [64]

    L. Chen, N. Sun, and P. Zhang, Strong-to-weak sym- metry breaking and entanglement transitions, Physical Review B111, L060304 (2025)

    2025

  57. [65]

    L. A. Lessa, R. Ma, J.-H. Zhang, Z. Bi, M. Cheng, and C. Wang, Strong-to-Weak Spontaneous Symmetry Breaking in Mixed Quantum States, PRX Quantum6, 010344 (2025)

    2025

  58. [66]

    P. Sala, S. Gopalakrishnan, M. Oshikawa, and Y. You, Spontaneous strong symmetry breaking in open sys- tems: Purification perspective, Physical Review B110, 155150 (2024)

    2024

  59. [67]

    M. Fava, L. Piroli, D. Bernard, and A. Nahum, Mon- itored fermions with conservedU(1) charge, Physical Review Research6, 10.1103/physrevresearch.6.043246 (2024)

    2024 doi

  60. [68]

    Sierant, X

    P. Sierant, X. Turkeshi, and P. S. Tarabunga, Theory of the matchgate commutant (2026), arXiv:2603.12392 [quant-ph]

    2026

  61. [69]

    Braccia, N

    P. Braccia, N. L. Diaz, M. Larocca, M. Cerezo, and D. Garc´ ıa-Mart´ ın, The commutant of fermionic gaussian unitaries (2026), arXiv:2603.19210 [quant-ph]

    2026

  62. [70]

    J. M. Radcliffe, Some properties of coherent spin states, Journal of Physics A: General Physics4, 313 (1971)

    1971

  63. [71]

    Fradkin,Field theories of condensed matter physics (2013)

    E. Fradkin,Field theories of condensed matter physics (2013)

    2013

  64. [72]

    Altland and B

    A. Altland and B. D. Simons,Condensed Matter Field Theory, 2nd ed. (Cambridge University Press, 2010)

    2010

  65. [73]

    E. A. Kochetov, Su(2) coherent-state path integral for the heisenberg ferromagnet, Phys. Rev. B52, 4402 (1995)

    1995

  66. [74]

    Balakrishnan and A

    R. Balakrishnan and A. R. Bishop, Nonlinear excita- tions on a quantum ferromagnetic chain, Phys. Rev. Lett.55, 537 (1985)

    1985

  67. [76]

    McCulloch, J

    E. McCulloch, J. De Nardis, S. Gopalakrishnan, and R. Vasseur, Full counting statistics of charge in chaotic many-body quantum systems, Physical Review Letters 131, 10.1103/physrevlett.131.210402 (2023)

    2023 doi

  68. [77]

    Zeier and T

    R. Zeier and T. Schulte-Herbr¨ uggen, Symmetry princi- ples in quantum systems theory, Journal of Mathemat- ical Physics52, 113510 (2011). 40

    2011

  69. [78]

    M. Fava, L. Piroli, T. Swann, D. Bernard, and A. Nahum, Nonlinear sigma models for monitored dy- namics of free fermions, Phys. Rev. X13, 041045 (2023)

    2023

  70. [79]

    E. A. Kochetov, Su(2) coherent-state path integral, Journal of Mathematical Physics36, 4667 (1995)

    1995

  71. [80]

    A. M. Perelomov, Coherent states for arbitrary lie groups, Commun. Math. Phys.26, 222 (1972)

    1972

  72. [81]

    A. M. Perelomov,Generalized coherent states and their applications, Texts and monographs in physics (Springer-Verlag, 1986)

    1986

  73. [82]

    E. A. Kochetov, Path integral over the general- ized coherent states, Journal of Mathematical Physics 10.1063/1.531078 (1995)

    1995 doi

  74. [83]

    F. A. Berezin, Models of Gross-Neveu Type as Quan- tization of Classical Mechanics With Nonlinear Phase Space, Commun. Math. Phys.63, 131 (1978)

    1978

  75. [84]

    J. P. Zima, E. M. Stoudenmire, S. R. White, O. Par- collet, and J. Kaye, Fast Tensor Network Imaginary Time Evolution by Implicit Stepping on Logarith- mic Grids, ArXiv e-prints 10.48550/arXiv.2606.02930 (2026), 2606.02930

    2026 doi arXiv

  76. [85]

    Aharoni,Introduction to the Theory of Ferromag- netism, International series of monographs on physics (Oxford University Press, 2000)

    A. Aharoni,Introduction to the Theory of Ferromag- netism, International series of monographs on physics (Oxford University Press, 2000)

    2000

  77. [86]

    Masaoka, T

    R. Masaoka, T. Soejima, and H. Watanabe, Quadratic dispersion relations in gapless frustration-free systems, Physical Review B110, 195140 (2024)

    2024

  78. [87]

    J.-M. St´ ephan, Return probability after a quench from a domain wall initial state in the spin-1/2 XXZ chain, Journal of Statistical Mechanics: Theory and Experi- ment2017, 103108 (2017)

    2017

  79. [88]

    R. A. Blythe and A. J. Bray, Survival probability of a diffusing particle in the presence of poisson-distributed mobile traps, Physical Review E67, 10.1103/phys- reve.67.041101 (2003)

    2003 doi

  80. [89]

    Agarwal, S

    L. Agarwal, S. Sahu, and S. Xu, Charge transport, in- formation scrambling and quantum operator-coherence in a many-body system with U(1) symmetry, Journal of High Energy Physics2023, 37 (2023)

    2023

  81. [90]

    Barratt, U

    F. Barratt, U. Agrawal, S. Gopalakrishnan, D. A. Huse, R. Vasseur, and A. C. Potter, Field theory of charge sharpening in symmetric monitored quantum cir- cuits, ArXiv e-prints 10.1103/PhysRevLett.129.120604 (2021), 2111.09336

    2021 arXiv

  82. [91]

    Agrawal, A

    U. Agrawal, A. Zabalo, K. Chen, J. H. Wilson, A. C. Potter, J. H. Pixley, S. Gopalakrishnan, and R. Vasseur, Entanglement and charge-sharpening tran- sitions in U(1) symmetric monitored quantum circuits, ArXiv e-prints 10.1103/PhysRevX.12.041002 (2021), 2107.10279

    2021 arXiv

  83. [92]

    Goldstein and E

    M. Goldstein and E. Sela, Symmetry-Resolved Entan- glement in Many-Body Systems, Physical Review Let- ters120, 200602 (2018)

    2018

  84. [93]

    Fraenkel and M

    S. Fraenkel and M. Goldstein, Symmetry resolved en- tanglement: exact results in 1D and beyond, Journal of Statistical Mechanics: Theory and Experiment2020, 033106 (2020)

    2020

  85. [94]

    Bonsignori, P

    R. Bonsignori, P. Ruggiero, and P. Calabrese, Symmetry resolved entanglement in free fermionic systems, Journal of Physics A: Mathematical and Theoretical52, 475302 (2019)

    2019

  86. [95]

    Murciano, G

    S. Murciano, G. Di Giulio, and P. Calabrese, Entangle- ment and symmetry resolution in two dimensional free quantum field theories, Journal of High Energy Physics 2020, 73 (2020)

    2020

  87. [96]

    Murciano, P

    S. Murciano, P. Calabrese, and L. Piroli, Symmetry- resolved Page curves, Physical Review D106, 046015 (2022)

    2022

  88. [97]

    F. Ares, S. Murciano, and P. Calabrese, Entanglement asymmetry as a probe of symmetry breaking, Nature Communications14, 2036 (2023)

    2023

  89. [98]

    F. Ares, S. Murciano, P. Calabrese, and L. Piroli, En- tanglement asymmetry dynamics in random quantum circuits, Physical Review Research7, 033135 (2025)

    2025

  90. [99]

    Fossati, F

    M. Fossati, F. Ares, J. Dubail, and P. Calabrese, En- tanglement asymmetry in CFT and its relation to non- topological defects, Journal of High Energy Physics 2024, 59 (2024)

    2024

  91. [100]

    Lastres, S

    M. Lastres, S. Murciano, F. Ares, and P. Calabrese, En- tanglement asymmetry in the critical XXZ spin chain, Journal of Statistical Mechanics: Theory and Experi- ment2025, 013107 (2025)

    2025

  92. [101]

    Murciano, F

    S. Murciano, F. Ares, I. Klich, and P. Calabrese, Entan- glement asymmetry and quantum Mpemba effect in the XY spin chain, Journal of Statistical Mechanics: Theory and Experiment2024, 013103 (2024)

    2024

  93. [102]

    Rakovszky, F

    T. Rakovszky, F. Pollmann, and C. von Keyserlingk, Entanglement growth in diffusive systems with large spin, Communications Physics4, 91 (2021)

    2021

  94. [103]

    ˇZnidariˇ c, Entanglement growth in diffusive systems, Communications Physics3, 100 (2020)

    M. ˇZnidariˇ c, Entanglement growth in diffusive systems, Communications Physics3, 100 (2020)

    2020

  95. [104]

    Hulse, H

    A. Hulse, H. Liu, and I. Marvian, Qudit circuits with SU(d) symmetry: Locality imposes additional conser- vation laws, arXiv preprint arXiv:2105.12877 (2021), arXiv:2105.12877 [quant-ph]

    2021 arXiv

  96. [105]

    Hulse, H

    A. Hulse, H. Liu, and I. Marvian, A frame- work for semi-universality: Semi-universality of 3-qudit SU(d)-invariant gates, ArXiv e-prints 10.48550/arXiv.2407.21249 (2024), 2407.21249

    2024 doi arXiv

  97. [106]

    Wu and J

    Y. Wu and J. F. Rodriguez-Nieva, Quantum state randomization constrained by non-Abelian symme- tries, ArXiv e-prints 10.48550/arXiv.2604.05043 (2026), 2604.05043

    2026 doi arXiv

  98. [107]

    Hall,Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Graduate Texts in Math- ematics (Springer International Publishing, 2015)

    B. Hall,Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Graduate Texts in Math- ematics (Springer International Publishing, 2015)

    2015

  99. [108]

    N. P. Landsman, Lecture notes on C*-algebras, Hilbert C*-modules, and quantum mechanics, arXiv preprint math-ph/9807030 (1998)

    1998 arXiv

  100. [109]

    Aditya, E

    S. Aditya, E. Tirrito, P. Sierant, and X. Turkeshi, Coherence dynamics in quantum many-body systems with conservation laws, ArXiv e-prints 10.48550/arXiv.2604.23192 (2026), 2604.23192

    2026 doi arXiv

  101. [110]

    Y. Li, P. Sala, and F. Pollmann, Hilbert space fragmen- tation in open quantum systems, Phys. Rev. Res.5, 043239 (2023)

    2023

  102. [111]

    Y. Li, F. Pollmann, N. Read, and P. Sala, Highly en- tangled stationary states from strong symmetries, Phys. Rev. X15, 011068 (2025)

    2025

  103. [112]

    Y. Li, P. Sala, F. Pollmann, S. Moudgalya, and O. Motrunich, Dynamics in the presence of local symmetry-breaking impurities, Phys. Rev. B112, 155108 (2025)

    2025

  104. [113]

    C. Wang, S. Balasubramanian, Y. Han, E. Lake, X. Chen, and Z.-C. Yang, Exponentially slow ther- malization in 1D fragmented dynamics, arXiv e-prints (2025), arXiv:2501.13930 [quant-ph]. 41

    2025 arXiv

  105. [114]

    Y. Han, X. Chen, and E. Lake, Exponentially slow ther- malization and the robustness of Hilbert space fragmen- tation, arXiv e-prints (2024), arXiv:2401.11294 [quant- ph]

    2024 arXiv

  106. [115]

    Feldmeier, P

    J. Feldmeier, P. Sala, G. De Tomasi, F. Pollmann, and M. Knap, Anomalous diffusion in dipole- and higher-moment-conserving systems, Phys. Rev. Lett. 125, 245303 (2020)

    2020

  107. [116]

    Moudgalya, A

    S. Moudgalya, A. Prem, D. A. Huse, and A. Chan, Spectral statistics in constrained many-body quantum chaotic systems, Phys. Rev. Res.3, 023176 (2021)

    2021

  108. [117]

    P. Sala, T. Rakovszky, R. Verresen, M. Knap, and F. Pollmann, Ergodicity Breaking Arising from Hilbert Space Fragmentation in Dipole-Conserving Hamiltoni- ans, Phys. Rev. X10, 011047 (2020)

    2020

  109. [118]

    Khemani, M

    V. Khemani, M. Hermele, and R. Nandkishore, Local- ization from Hilbert space shattering: From theory to physical realizations, Phys. Rev. B101, 174204 (2020)

    2020

  110. [119]

    Moudgalya, A

    S. Moudgalya, A. Prem, R. Nandkishore, N. Regnault, and B. A. Bernevig, Thermalization and Its Absence within Krylov Subspaces of a Constrained Hamiltonian, inMemorial Volume for Shoucheng Zhang, Chap. 7, pp. 147–209

  111. [120]

    Moudgalya, B

    S. Moudgalya, B. A. Bernevig, and N. Regnault, Quan- tum many-body scars and Hilbert space fragmentation: a review of exact results, Reports on Progress in Physics 85, 086501 (2022)

    2022

  112. [121]

    Yang, Distinction between transport and r´ enyi en- tropy growth in kinetically constrained models, Phys

    Z.-C. Yang, Distinction between transport and r´ enyi en- tropy growth in kinetically constrained models, Phys. Rev. B106, L220303 (2022)

    2022

  113. [122]

    Z.-C. Yang, F. Liu, A. V. Gorshkov, and T. Iadecola, Hilbert-space fragmentation from strict confinement, Phys. Rev. Lett.124, 207602 (2020)

    2020

  114. [123]

    Rakovszky, P

    T. Rakovszky, P. Sala, R. Verresen, M. Knap, and F. Pollmann, Statistical localization: From strong frag- mentation to strong edge modes, Phys. Rev. B101, 125126 (2020)

    2020

  115. [124]

    Moitra and R

    S. Moitra and R. Sensarma, Building entanglement entropy out of correlation functions for interacting fermions, Phys. Rev. B108, 174309 (2023)

    2023

  116. [125]

    Swann and A

    T. Swann and A. Nahum, Continuum mechanics of en- tanglement in noisy interacting fermion chains (2026), arXiv:2601.21134 [cond-mat.stat-mech]

    2026

  117. [126]

    Zerba, S

    C. Zerba, S. Gopalakrishnan, and M. Knap, Strong- to-weak symmetry breaking in monitored dipole con- serving quantum circuits, arXiv e-prints (2025), arXiv:2512.14830 [quant-ph]

    2025

  118. [127]

    Nahum and J

    A. Nahum and J. L. Jacobsen, Bayesian critical points in classical lattice models, Phys. Rev. B112, 235113 (2025)

    2025

  119. [128]

    Gopalakrishnan, E

    S. Gopalakrishnan, E. McCulloch, and R. Vasseur, Mon- itored fluctuating hydrodynamics, Phys. Rev. X16, 011024 (2026)

    2026

  120. [129]

    Hauser, K

    J. Hauser, K. Su, H. Ha, J. Lloyd, T. G. Kiely, R. Vasseur, S. Gopalakrishnan, C. Xu, and M. P. A. Fisher, Strong-to-Weak Symmetry Breaking in Open Quantum Systems: From Discrete Particles to Continuum Hydrodynamics, arXiv e-prints (2026), arXiv:2602.16045 [quant-ph]

    2026

  121. [130]

    J. R. Klauder, Path integrals and stationary-phase ap- proximations, Phys. Rev. D19, 2349 (1979)

    1979

  122. [131]

    Klauder and B

    J. Klauder and B. Skagerstam,Coherent States : Ap- plications In Physics And Mathematical Physics(World Scientific Publishing Company, 1985)

    1985

  123. [132]

    Schliemann and F

    J. Schliemann and F. G. Mertens, Semiclassical descrip- tion of heisenberg models via spin-coherent states, Jour- nal of Physics: Condensed Matter10, 1091 (1998)

    1998

  124. [133]

    Kerin and D

    M. Kerin and D. Wraith,Homogeneous metrics on spheres, Master’s thesis, National University of Ireland, Maynooth (2003)

    2003

  125. [134]

    L. E. Thomas, Quantum heisenberg ferromagnets and stochastic exclusion processes, Journal of Mathematical Physics21, 1921 (1980)

    1980

  126. [135]

    Sch¨ utz, 1 - exactly solvable models for many-body systems far from equilibrium (Academic Press, 2001) pp

    G. Sch¨ utz, 1 - exactly solvable models for many-body systems far from equilibrium (Academic Press, 2001) pp. 1–251

    2001

  127. [136]

    Kipnis and C

    C. Kipnis and C. Landim,Scaling Limits of Interacting Particle Systems(Springer, Berlin, Germany, 1999)

    1999

  128. [137]

    Anton, R

    L. Anton, R. A. Blythe, and A. J. Bray, Spatial fluctu- ations of a surviving particle in the trapping reaction, Journal of Physics A: Mathematical and General38, 133–144 (2004)

    2004

  129. [138]

    Bravyi, Monte Carlo simulation of stoquastic Hamil- tonians, Quant.Inf.Comput15, 1122 (2015)

    S. Bravyi, Monte Carlo simulation of stoquastic Hamil- tonians, Quant.Inf.Comput15, 1122 (2015). 42 Appendix A: Review of the Spin Coherent Path Integral In this appendix, we provide a self-contained review of theSU(2) spin coherent state path integral [73, 79], including its ...

    2015

  130. [139]

    By design, the expectation value of the Pauli vector recovers the unit vector:⟨⃗ n|⃗ σ|⃗ n⟩=⃗ n

    Coherent State Representation and the Path Integral Action A spin-coherent state|⃗ n⟩[70] for a spin-1 2 degree of freedom is parameterized by a three-dimensional unit vector⃗ n defined as |⃗ n⟩= cosθ 2|↑⟩+e iφ sinθ 2|↓⟩, ⃗ n:= (sinθcosφ,sinθsinφ,cosθ) T (A1) which can be thou...

  131. [140]

    originate

    Subtleties on the Boundary Conditions of the Path Integral The prescription for the boundary conditions of the fieldsz,¯zin Eq. (A5) deserves some clarifications. In the usual derivation of the path integral action, one would expect that the boundary conditions for all fields ...

  132. [141]

    Example: A single spin underH=Jσ z To illustrate the crucial role of the boundary conditions and the boundary actionS bd, we consider a single spin governed by the HamiltonianH=Jσ z. In terms of coherent states, the Hamiltonian term and the variation of energy action read ⟨z|H...

  133. [142]

    (32) in the continuum limit, the energy functionalH[z,¯z] is [79] H[z,¯z] = 2J Z ddx ∇¯z·∇z (1 + ¯zz)2.(A17) This functional form is equivalent to the expression Eq

    Path Integral Formulation of the Ferromagnetic Heisenberg Model For the Heisenberg model of Eq. (32) in the continuum limit, the energy functionalH[z,¯z] is [79] H[z,¯z] = 2J Z ddx ∇¯z·∇z (1 + ¯zz)2.(A17) This functional form is equivalent to the expression Eq. (66) [derived l...

  134. [143]

    Generalization to Matrix Coherent States and Coset Manifolds The construction detailed above forSU(2) spin- 1 2 coherent states can be extended to broader classes of quantum systems where the manifold of states is as a homogeneous coset spaceG/H[80, 131] for a Lie groupGand a ...

  135. [144]

    These states are identified up to normalization and phase factors, so that|ψ(θ)⟩∼λ|ψ(θ)⟩forλ∈C

    Geometric Formulation on K¨ ahler Manifolds LetM⊆P(H) be a variational manifold of quantum states|ψ(θ)⟩parameterized by a set of real coordinatesθ α. These states are identified up to normalization and phase factors, so that|ψ(θ)⟩∼λ|ψ(θ)⟩forλ∈C. WhenMis a K¨ ahler manifold of ...

  136. [145]

    Application to spatially varying fields While Eq. (B5) is the general expression for TDVP on a variational manifold of finite dimension, we are interested in applying this framework to the case of a variational family of spatially varying fieldsv(x) which take values on a mani...

  137. [146]

    Semiclassical Trajectories in TDVP vs Path Integral We remark that the real time TDVP equation in Eq. (B5) can be derived as the Euler-Lagrange equation of the Lagrangian: L(θ,∂tθ;t) = Re ⟨ψ(θ)|i∂t−H|ψ(θ)⟩ ⟨ψ(θ)|ψ(θ)⟩ (B13) 47 If we assume that the coordinatesθ α are holomorph...

  138. [147]

    (32)], similar to Ref

    Continuum Evolution for the Heisenberg Model We now specialize this framework to the continuum limit of the spin- 1 2 ferromagnetic Heisenberg model [Eq. (32)], similar to Ref. [132]. Our variational manifold consists of a continuous tensor product of independent single-site c...

  139. [148]

    Let Π ψ be the projector onto the tangent space of the manifold at the point|ψ⟩; then the imaginary-time TDVP equations of Eq

    Quantifying Error via Manifold Leakage One way to assess the validity of the semi-classical continuum limit, is to measure the “leakage” rateϵ 2 out of the variational manifold. Let Π ψ be the projector onto the tangent space of the manifold at the point|ψ⟩; then the imaginary...

  140. [149]

    unbalanced

    Continuum Expansion of Effective Hamiltonians In this section, we provide a general geometric derivation for the leading order of the continuum limit of effective Hamiltonians with continuous symmetries. We first treat the ferromagnetic case associated to standard on-site symm...

  141. [150]

    Ferromagnetic Heisenberg model The mapping for the Heisenberg model is the well-known mapping between imaginary-time ferromagnetic Heisenberg evolution, and the Symmetric Simple Exclusion Process (SSEP) [134, 135]. Consider a state on a spin- 1 2 many-body quantum system, and ...

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.