Quantum thermalization mechanism and the emergence of symmetry-breaking phases

\'Angel L. Corps; Armando Rela\~no; Sebasti\'an G\'omez

REVIEW 3 major objections 5 minor 104 references

Quantum thermalization mechanism and the emergence of symmetry-breaking phases

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 Thermalization in degenerate quantum systems reduces to two conditions: observable averages over each subspace match microcanonical values, and intra-subspace eigenvalue spreads vanish; failure of the second on an order parameter produces…

desk verdict A clean two-condition generalization of ETH with a sound proof, but the finite-size numerical support for the claimed symmetry-breaking phases relies on an unverified near-degeneracy assumption and an inference stronger than the theorem. read the letter →

arxiv 2506.13370 v1 pith:T7SBFUYN submitted 2025-06-16 quant-ph cond-mat.stat-mech

Sebastián Gómez , Ángel L. Corps , Armando Relaño This is my paper · ORCID

classification quant-phcond-mat.stat-mech PACS 05.30.-d05.45.Mt03.65.-w

keywords eigenstatethermalizationhypothesisdegenerateenergysubspacessymmetrybreakingquantumBose-Hubbardmodelorderparameterchaosdiscretesymmetries

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

Thermalization of an isolated quantum system is usually attributed to the eigenstate thermalization hypothesis, which assumes non-degenerate spectra. This paper removes that assumption and states a sharper criterion: a system with degenerate energy subspaces thermalizes if and only if, for every observable, the average over each degenerate subspace equals the microcanonical average, and the observable's eigenvalues within each subspace have vanishing spread. The two failures are physically distinct: the first means the system is not chaotic and needs extra conserved quantities, while the second, for an order parameter, means initial states can relax to symmetry-broken equilibrium values; if the second failure covers an entire spectral region, that region is a symmetry-breaking phase. The authors prove the equivalence and illustrate it on a three-site Bose-Hubbard model, where the two conditions separate thermal, rotation-broken, reflection-broken, and mixed spectral regions without running full dynamics.

What carries the argument

The degenerate subspace $H_n$ and the restricted observable's eigenvalues $\lambda^{(O)}_{n,\alpha}$ form the central object. Each energy shell is decomposed into subspaces spanned by Hamiltonian eigenstates of equal energy, and the observable is diagonalized inside every subspace; the long-time average of the observable is then written in this adapted eigenbasis, where cross-subspace terms are killed by exponential decay, leaving only the $\lambda$'s weighted by initial-state populations. This decomposition carries the argument: the trace condition controls the mean level, the spread condition controls the fluctuations, and order parameters are recognized by having zero trace but non-zero individual eigenvalues.

What would settle it

For the Bose-Hubbard trimer at $N \approx 320$, time-evolve a narrow energy shell initial state in the claimed thermalizing region ($-3.1 \lesssim E/N \lesssim -2.3$) and check whether the long-time average of $\hat{h}_{12}/N$ equals the classical microcanonical value; a persistent deviation beyond the reported power-law width would falsify Eq. (2). Alternatively, repeat the eigenvalue extraction while keeping the $r=1$ sector separate from the degenerate pair and see whether the symmetry-breaking regions persist; if they disappear, the pairing assumption is load-bearing and the phase classification collapses.

Watch

Extended reading notes

Core claim

The central claim is a generalization of the eigenstate thermalization hypothesis to Hamiltonians with degenerate energy subspaces. For any physical observable $\hat O$, let $H_n$ be the subspace of eigenstates sharing energy $E_n$, define $T^{(O)}_n$ as the trace of $\hat O$ restricted to $H_n$, and let $\lambda^{(O)}_{n,\alpha}$ be its eigenvalues there. Then every sufficiently narrow initial state thermalizes if and only if the trace per dimension approaches the microcanonical average (Eq. 2) and the eigenvalue spread inside each subspace vanishes (Eq. 3). Failure of the first condition signals non-chaotic dynamics and additional conserved quantities requiring a generalized Gibbs ensemble; failure of the second, applied to an order parameter, is exactly the appearance of symmetry-breaking equilibrium states, and when it covers a whole spectral region that region is a symmetry-breaking phase. The proof assumes only that off-diagonal matrix elements between different energy subspaces decay exponentially with system size, as in Srednicki's ansatz.

Load-bearing premise

The phase classification rests on treating near-degenerate states as exactly degenerate by pairing the $r=1$ eigenstate closest in energy to the exactly degenerate pair, and on assuming exponential decay of off-diagonal matrix elements between different subspaces; if either fails, the reported phase regions are not established.

Editorial extensions

If this is right

In any system whose degenerate subspaces satisfy both conditions, every sufficiently narrow initial state equilibrates to the microcanonical ensemble, so thermalization can be certified by diagonalizing observables inside degenerate subspaces rather than simulating full dynamics.
Analyzing an order parameter's eigenvalues inside degenerate subspaces gives a direct spectral diagnostic for symmetry-breaking phases: a region where no subspace satisfies the vanishing-spread condition is an ordered phase.
Symmetry-breaking equilibrium states need not fill a whole phase: the Bose-Hubbard results show they can also appear as isolated subspaces inside a chaotic sea, playing a role similar to many-body scars.
Non-chaotic regions are flagged by failure of the trace condition, which in the model coincides with extra integrals of motion and signals the need for a generalized Gibbs ensemble description.
Both conditions reduce exactly to standard ETH when degeneracies are absent, so the formalism is a strict generalization of the usual mechanism rather than a separate one.

Reading between the lines

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

The same two-condition test could map symmetry-breaking phases in other lattice models with non-commuting discrete symmetries, such as spin chains with translation and reflection, using only static diagonalization rather than time evolution.
A practical finite-size diagnostic suggested by the paper: in candidate thermal regions the width of the intra-subspace eigenvalue distribution should decay at least as $N^{-1/2}$; regions where it saturates or decays more slowly are symmetry-breaking candidates.
If failure of the vanishing-spread condition occurs only for a vanishing fraction of subspaces, the formalism predicts scar-like long-lived oscillations, giving a sharper, quantitative definition of many-body scars in symmetry-degenerate systems.
The near-degeneracy pairing assumption could be tested directly by comparing the eigenvalue extraction against full diagonalization in the $r=1$ sector; agreement would confirm the phase boundaries, disagreement would shift them with system size.

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.

Desk Editor's Note

read the letter

You should know: the two-condition generalized ETH (Eqs. 2-3) is a clean and useful formulation, and the Appendix A proof is right under the stated Srednicki ansatz. The numerical model is a nice illustration, but the phase-level conclusions rest on two assumptions the paper doesn't check.

What's new: the iff characterization. For degenerate subspaces, thermalization of all narrow initial states requires both the trace condition (2) and the intra-subspace eigenvalue spread condition (3). That is a genuine generalization of ETH, cleanly separating two failure modes. The authors correctly credit the NATS literature (Refs 75-82) and their own prior work (Ref 74); the incremental contribution is the compact two-condition criterion and its application to a model with two noncommuting discrete symmetries. The proof is logically sound: the forward direction is straightforward, and the backward direction correctly constructs non-thermalizing initial states for each violated condition.

Main soft spot is the construction of H_n in Appendix C. The theorem requires exact degeneracy within each subspace. The numerics build triples by taking the exactly degenerate r=e^{2πi/3} and r=e^{4πi/3} pair and adding the r=1 eigenstate with closest energy. At finite N these three states are not degenerate, and the paper gives no estimate of the splitting Δ_n, no scaling test, and no argument that Δ_n is dynamically irrelevant. For the imbalance I, coherences between the r=1 state and the complex pair oscillate at Δ_n and average to zero, so the eigenvalues of the 3x3 restriction are not obviously the stationary long-time values. The paper's own assumption "for all practical purposes" is plausible for large N in a symmetry-broken phase, but N=320 is not the thermodynamic limit, and the claim that no subspaces fulfill Eq. (3) is only as good as this pairing.

Second soft spot: the theorem's backward direction only guarantees the existence of non-thermalizing initial states when Eq. (3) fails. The paper then calls a whole spectral region a symmetry-breaking phase when Eq. (3) fails everywhere. That inference is stronger than the theorem: failure everywhere could still leave some eigenstates of the order parameter with zero eigenvalue, giving symmetric equilibrium states. The classical trajectories and the max-|λ| plots suggest all branches are nonzero in the fully broken region, but the paper does not show that every λ is bounded away from zero. So the phase-level "iff" is a conjecture supported by numerics, not a proven consequence.

These are not fatal to the theoretical result. The theorem stands; the numerical boundaries are provisional. The paper would be strengthened by reporting the typical splitting Δ_n as a function of N, showing the λ-eigenvalues converge to the exact-degeneracy values, and either proving or softening the phase-level claim.

Who is this for? People working on ETH, thermalization, symmetry breaking in isolated systems, and quantum simulators with discrete symmetries. The generalized ETH is a useful organizing criterion. The numerical example is a good testbed but not a conclusive phase diagram.

I would send this to a serious referee. The theorem deserves publication even if the numerical support requires revision. With scaling checks and a more careful phase claim, it would be solid.

Referee Report

3 major / 5 minor

Summary. The manuscript proposes a generalized eigenstate thermalization hypothesis (ETH) for isolated quantum systems with degenerate energy subspaces. It states that every sufficiently narrow initial state thermalizes if and only if two conditions hold for any physical observable: the trace of the observable over each degenerate subspace, divided by the subspace dimension, approaches the microcanonical average (Eq. (2)), and the spread of the observable's eigenvalues inside each subspace vanishes (Eq. (3)). The failure of Eq. (2) is attributed to non-chaotic behavior, while the failure of Eq. (3) for an order parameter is interpreted as symmetry breaking; if Eq. (3) fails on an entire spectral region, that region is identified as a symmetry-breaking phase. The proof is presented in Appendix A under a Srednicki-type exponential decay of off-diagonal matrix elements between different energy subspaces. The formalism is then applied numerically to the three-site Bose-Hubbard model with D3 symmetry, using exact diagonalization and classical trajectories to identify thermalizing, rotation-breaking, reflection-breaking, and mixed spectral regions.

Significance. If the proposed criteria are validated, the paper offers a clean, basis-independent framework for understanding thermalization in the presence of degeneracies and connects failure of the generalized ETH to symmetry breaking. The Appendix A theorem is simple and appears logically sound under its stated assumptions, and the model study is informative, with a useful classical-quantum comparison. The main value would be in providing a concrete diagnostic for where symmetry-breaking equilibrium states can arise. However, the phase-level claims go beyond what the theorem establishes, and the numerical construction of degenerate subspaces relies on an approximate degeneracy that is not validated; the current evidence is finite-size and does not yet justify the thermodynamic-limit phase classification.

major comments (3)

[Main text, after Eq. (3): phase criterion] The phase criterion 'a spectral region is a symmetry-breaking phase if and only if it has no subspaces fulfilling Eq. (3)' is not established by Appendix A. The backward direction of the theorem (Appendix A, Eqs. (A6)-(A7)) only proves that when Eq. (3) fails there exists at least one initial state, namely an eigenstate |~E_m,alpha>, whose long-time order-parameter average is non-thermal. It does not show that all, or typical, initial conditions in that spectral region equilibrate to one of the lambda branches; the statement that 'every initial condition equilibrates in one of these branches, or in a quantum superposition of them' requires an additional argument. As written, an initial state with comparable weights on several branches could have a vanishing long-time average even when each subspace violates Eq. (3). The numerical evidence in Fig. 3 concerns classical trajectories and eigenvalue distributions, not quantum quenches from generic narrow initial states.
[Appendix C, 'Eigenvalues lambda(O)_n,alpha and traces T(O)_n'] The construction of the three-dimensional subspaces H_n is not justified by the theorem. The exactly degenerate pair with r=e^{2pi i/3} and r=e^{4pi i/3} is combined with the r=1 eigenstate whose energy is closest to that pair, but at any finite N this r=1 state is not degenerate with the pair. The theorem in Appendix A applies to exactly degenerate subspaces; for a nonzero splitting Delta_n, coherences between the r=1 state and the complex pair oscillate at frequency Delta_n/hbar and vanish in the infinite-time average, so the eigenvalues lambda(I)_{n,alpha} obtained from the 3x3 matrix are not stationary long-time averages. The manuscript provides no estimate of Delta_n, no comparison of Delta_n with relevant dynamical scales, and no scaling test showing that the triples converge to exact degeneracy in the thermodynamic limit. Therefore the three-branch structure in Fig. 3(a) and the inferred rotation-breaking phase in Fig. 1(b) are not established by the numerical data.
[Figs. 1-3: finite-size evidence for phase regions] The phase classification is based on finite-size data without scaling analysis in the symmetry-breaking regions. Fig. 1 reports max_alpha |lambda(I)_{n,alpha}| and max_alpha |lambda(C)_{n,alpha}| for N=320 only, and Fig. 3 uses N=320; the finite-size scaling shown in Fig. 2 is limited to the thermalizing region. Since Eq. (3) is an asymptotic condition in the thermodynamic limit, observing that lambda is 'significantly different from zero' in some energy windows at N=320 does not distinguish a true symmetry-breaking phase from finite-size remnant order. The authors should provide scaling of the violations of Eq. (3) in each proposed phase, for example the maximum or the fraction of subspaces with |lambda| above a threshold as a function of N, or else state explicitly that the phase boundaries are finite-size observations rather than thermodynamic-limit conclusions.

minor comments (5)

[Appendix A, Eq. (A1)] The proof assumes that the observable can be diagonalized within each degenerate subspace by a transformation that preserves an orthonormal eigenbasis; for the non-Hermitian but normal operator hat I this is valid, but the manuscript should state this, since Eq. (3) is applied to a non-Hermitian order parameter.
[S1, after Eq. (2)] The statement that full chaos 'immediately' implies Eq. (2) should be qualified: ETH per symmetry sector refers to the sector-resolved microcanonical average, and one must assume that the relevant sector averages coincide with the global O_ME(E) for the trace condition to follow. The numerical test in Fig. 1(a) is therefore important and should be emphasized.
[Fig. 2(a)] The fitted exponent for T(h12) is -0.419, which is not very close to the Srednicki expectation of -1/2; the sentence calling this 'close to Srednicki's ansatz' should quantify the expected finite-size corrections or be softened.
[Fig. 2 caption and text] The caption of Fig. 2(b)-(d) says the number of subspaces is counted for differences 'below a given arbitrary bound', while the text says 'larger than several bounds'; these should be made consistent.
[Introduction, generalized ETH setup] The phrase 'we rule out remaining degeneracies by assuming that off-diagonal matrix elements between them fulfill Srednicki's ansatz' is ambiguous; it should say explicitly that the exponential decay assumption applies to matrix elements between different energy subspaces En != Em.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the generalized-ETH theorem is proven from stated assumptions; the only self-citation (Ref. 74) is interpretive and not load-bearing.

full rationale

The central derivation (Eqs. (2)-(3)) is proved in Appendix A directly from the stated assumptions: exact degeneracy within each H_n and exponential decay of off-diagonal couplings between different subspaces. The forward direction combines Eqs. (2) and (3) by the triangle inequality to obtain Eq. (A3), and the backward direction explicitly constructs non-thermalizing initial conditions (a uniform subspace superposition and an O-eigenstate) when either condition fails. No fitted parameter is later renamed as a prediction: the trace and eigenvalue data in Figs. 1-3 are computed from the Hamiltonian and compared to independent classical microcanonical averages. The S2 interpretation uses the fact that order parameters have zero trace in symmetric multiplets and cites the authors' prior work (Ref. 74) for the 'branches' language, but the load-bearing statement — that a nonzero λ can be realized as a stationary long-time average — is independently proved in Appendix A. The closest-energy pairing of r=1 with the exactly degenerate complex pair (Appendix C) is an approximation whose dynamical validity is not established, but it is an assumption about finite-size numerics, not a circular reduction. Overall, the paper is self-contained in its formal derivation; the score of 2 reflects only the minor, non-load-bearing self-citation in the interpretive discussion.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central theorem is self-contained given the Srednicki-ansatz assumption, but the numerical phase classification rests on hand-picked thresholds, a closest-energy pairing rule for H_n, and the classical-limit correspondence. No new particles or forces are introduced.

free parameters (3)

Symmetry-breaking classification threshold = |I| > 0.1 and |C| > sqrt(3)/10
Appendix C: a classical trajectory is counted as symmetry-breaking if the absolute long-time average of the order parameter exceeds one tenth of its maximum possible value. This hand-chosen cutoff determines the percentages in Fig. 1(b,c).
Chaos classification threshold = time-averaged trajectory separation > 0.3
Appendix C: a trajectory is called chaotic if the time average of the distance between two trajectories computed with 20 and 25 significant figures exceeds 0.3. The threshold is arbitrary and affects Fig. 1(a).
Energy-region boundaries = thermalizing -3.1 to -2.3; rotation SB <= -3.9; reflection SB >= -0.9; mixed windows
The spectral regions in Fig. 1 are identified visually from the data; the boundary values are not derived from the generalized ETH and influence how the phase diagram is reported.

assumptions (5)

domain assumption Remaining degeneracies beyond each H_n are absent, and off-diagonal matrix elements between different energy subspaces decay exponentially with system size (Srednicki's ansatz).
Invoked in the main text before Eq. (2) and used in Appendix A to drop n != m terms in the long-time average; without it the equivalence between Eqs. (2)-(3) and thermalization is not established.
domain assumption For order parameters, the trace over a symmetry multiplet vanishes: T_n^{(M)}=0 for all n.
Used in scenario S2 to convert failure of Eq. (3) into nonzero order-parameter branches; stated with reference [58] but not proved in this paper.
domain assumption The classical limit of the Bose-Hubbard model (a_j -> sqrt(N/2)(q_j+i p_j), N -> infinity with hbar_eff proportional to 1/N) faithfully represents the thermodynamic limit and provides the microcanonical reference O_ME(E).
Appendix B; energies and order parameters are compared against classical microcanonical averages, so the classical-limit correspondence is load-bearing for the numerical tests.
standard math Berry-Robnik distribution describes level spacing for mixed regular-chaotic spectra and is used to estimate the quantum chaos fraction rho.
Appendix C; standard result, but fitting the one-parameter curve to numerical level spacings is an interpretive step needed for the chaos percentages in Fig. 1(a).
ad hoc to paper No exact degeneracies in the r=1 sector; the closest r=1 eigenstate is the correct partner for the exactly degenerate rotation doublet when building H_n.
Appendix C; pairing by closest energy is a modeling choice specific to this paper and is needed to build H_n. If wrong, the computed eigenvalues lambda and the phase classification are not physically meaningful.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Quantum thermalization mechanism and the emergence of symmetry-breaking phases." pith.science (2026). https://pith.science/paper/T7SBFUYN

@misc{pith2026250613370,
  author       = {Pith},
  title        = {Pith review of: Quantum thermalization mechanism and the emergence of symmetry-breaking phases},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/T7SBFUYN}},
  note         = {Machine review of arXiv:2506.13370}
}

read the original abstract

We propose a generalization of the eigenstate thermalization hypothesis accounting for the emergence of symmetry-breaking phases. It consists of two conditions that any system with a degenerate spectrum must fulfill in order to thermalize. The failure of each of them generates a different non-thermalizing scenario. One is due to the absence of chaos and may indicate that extra constants of motion are required to describe equilibrium states. The other one implies the existence of initial conditions evolving towards symmetry-breaking equilibrium states. If it spreads across an entire spectral region, then this region gives rise to a symmetry-breaking phase. We explore the applicability of this formalism by means of numerical experiments on a three-site Bose-Hubbard model with two non-commuting discrete symmetries.

Figures

Figures reproduced from arXiv: 2506.13370 by the authors.

**Figure 3.** FIG. 3. (a)-(b) Complex eigenvalues of [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Review history (2 revisions) →

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

104 extracted references · 60 canonical work pages

[1]

First, spectral statistics and classical trajectories (see Appendix C) indicate that there is full chaos

Our main result consists in identifying the following spectral regions: Thermalizing region, with −3.1 ≲ E/N ≲ −2.3. First, spectral statistics and classical trajectories (see Appendix C) indicate that there is full chaos. According to S1, this implies that Eq. ( 2) is necessarily fulﬁlled. In Fig. 1(a) we show that the trace of ˆh12 always ﬂuctuates arou...
[2]

A Way of Making Europe

and ( 3) are fulﬁlled within this region. Thus, the standard mi- crocanonical ensemble is expected to hold, and therefore no symmetry breaking can be observed. Rotation-breaking region, with E/N ≲ −3.9. Fig. 1(b) shows that none of the subspaces fulﬁll Eq. ( 3) for ˆI. From S2, we conclude that this is an ordered phase in which the rotational symmetry is ...
[3]

A deeper test is shown in Fig

(see Appendix B for details). A deeper test is shown in Fig. 2. In panel (a), we can see that the width of the distribution of T (ˆh12) n /dn − h12(E) decreases as a power law, σ ∝ N − 0. 419. This decreasing behavior is close to Srednicki’s ansatz, in which diagonal expectation values ﬂuctuate around the microcanonical with a width proportional to e− S(E...
[4]

commutes with the total number of particles, ˆN = ∑ i ˆa† i ˆai. The symmetries of this model are described by the dihedral group D3, which has two non-commuting dis- crete generators: a 2 π/3 rotation, ˆR |n1, n2, n3⟩ = |n3, n1, n2⟩, where ni indicates the number of parti- cles in the site i; and a reﬂection, ˆS |n1, n2, n3⟩ = |n3, n2, n1⟩ (see Appendix ...
[5]

and ( 3) hold if and only if all the initial conditions narrow enough in energy ther- malize. Forward implication.- Let us consider that the system starts in an initial condition, |Ψ(0) ⟩ = ∑ n,α cn,α |En,α ⟩, where cn,α ⁄= 0 only in a subextensive interval [ E − δE, E + δE], and let us calculate the resulting long- time average of a physical observable ˆ...
[6]

(A3) And therefore, the long-time average given in Eq

and ( 3) are satisﬁed, then maxn,α |λ(O) n,α − O(E)| η ˆO → 0, N → ∞. (A3) And therefore, the long-time average given in Eq. ( A2) coincides with the microcanonical average. Backward implication.- This is equivalent to show- ing that if (a) max n |T (O) n /dn − OME(E)| ↛ 0 or (b) maxn,α |λ(O) n,α − T (O) n /dn| ↛ 0 when N → ∞ , then there exists at least ...
[7]

Thus, the long-time average of the expectation value of ˆO is simply ˆO = ⟨ ~Em,α ⏐ ⏐ ⏐ ˆO |Em,α ⟩ = λ(O) m,α (A7) And this initial state does not thermalize

Let λ(m) α be the eigenvalue of ˆO(m) such that⟨ ~Em,β ⏐ ⏐ ⏐ ˆO ⏐ ⏐ ⏐~Em,α ⟩ = λm,α and let us consider an initial 6 condition |Ψ(0) ⟩ = ⏐ ⏐ ⏐~Em,α ⟩ . Thus, the long-time average of the expectation value of ˆO is simply ˆO = ⟨ ~Em,α ⏐ ⏐ ⏐ ˆO |Em,α ⟩ = λ(O) m,α (A7) And this initial state does not thermalize. Appendix B: Some details of the model Classica...
[8]

(B4) Symmetries.– Eq

+ e4πi/ 3(q2 3 + p2 3) ] , (B2) C N = ∑ j (pjqj+1 − pj+1qj), (B3) and h12 N = q1q2 + p1p2. (B4) Symmetries.– Eq. ( 4) is invariant under 2π/3 and 4π/3 rotations, and under the reﬂection around any of its sym- metry axes. This set of transformations constitutes the dihedral D3 group, whose generators are the operators ˆR and ˆS. Appendix C: Details of nume...

Show all 104 references

[9]

von Neumann, Beweis des ergodensatzes und desh- theorems in der neuen mechanik, Z

J. von Neumann, Beweis des ergodensatzes und desh- theorems in der neuen mechanik, Z. Phys. 57, 30 (1929)

1929
[10]

J. M. Deutsch, Quantum statistical mechanics in a closed system, Phys. Rev. A 43, 2046 (1991)

1991
[11]

Srednicki, Chaos and quantum thermalization, Phys

M. Srednicki, Chaos and quantum thermalization, Phys. Rev. E 50, 888 (1994)

1994
[12]

Rigol, V

M. Rigol, V. Dunjko, and M. Olshanii, Thermalization and its mechanism for generic isolated quantum systems, Nature 452, 854 (2008)

2008
[13]

Polkovnikov, K

A. Polkovnikov, K. Sengupta, A. Silva, and M. Vengalat- tore, Colloquium: Nonequilibrium dynamics of closed in- teracting quantum systems, Rev. Mod. Phys. 83, 863 (2011)

2011
[14]

Gogolin and J

C. Gogolin and J. Eisert, Equilibration, thermalisatio n, and the emergence of statistical mechanics in closed quantum systems, Rep. Prog. Phys. 79, 056001 (2016)

2016
[15]

D’Alessio, Y

L. D’Alessio, Y. Kafri, A. Polkovnikov, and M. Rigol, From quantum chaos and eigenstate thermalization to statistical mechanics and thermodynamics, Adv. Phys, 65, 239 (2016)

2016
[16]

Reimann, Generalization of von Neumman’s Approach to Thermalization, Phys

P. Reimann, Generalization of von Neumman’s Approach to Thermalization, Phys. Rev. Lett. 115, 010403 (2015)

2015
[17]

Yoshizawa, E

T. Yoshizawa, E. Iyoda, and T. Sagawa, Numerical Large Deviation Analysis of the Eigenstate Thermalization Hy- pothesis, Phys. Rev. Lett. 120, 200604 (2018)

2018
[18]

Sugimoto, R

S. Sugimoto, R. Hamazaki, and M. Ueda, Test of the Eigenstate Thermalization Hypothesis Based on Local Random Matrix Theory, Phys. Rev. Lett. 126, 120602 (2021)

2021
[19]

Sugimoto, R

S. Sugimoto, R. Hamazaki, and M. Ueda, Eigen- state Thermalization in Long-Range Interacting Systems, Phys. Rev. Lett. 129, 030602 (2022)

2022
[20]

P. Reimann. Equilibration of isolated macroscopic qua n- tum systems under experimentally realistic conditions. Phys. Scr. 86 (2015) 058512

2015
[21]

P. Reimann. Foundation of Statistical Mechanics under Experimentally Realistic Conditions. Phys. Rev. Lett. 101, 190403 (2008)

2008
[22]

Linden, S

N. Linden, S. Popescu, A. J. Short, and A. Winter. Quan- tum mechanical evolution towards thermal equilibrium. Phys. Rev. E 79, 061103 (2009)

2009
[23]

Linden, S

N. Linden, S. Popescu, A. J. Short, and A. Winter. On the speed of ﬂuctuations around thermodynamic equilib- rium. New. J. Phys. 12 055021 (2010)

2010
[24]

P. Reimann. Canonical thermalization. New J. Phys. 12 055027 (2010)

2010
[25]

A. J. Short. Equilibration of quantum systems and sub- systems. New J. Phys. 13 053009 (2011)

2011
[26]

A. J. Short and T. C. Farrelly. Quantum equilibration in ﬁnite time. New J. Phys. 14 013063 (2012)

2012
[27]

Trotzky, Y-A

S. Trotzky, Y-A. Chen, A. Flesch, I. P. McCulloch, U. Schollw¨ ock, J. Eisert, and I. Bloch, Probing the relax- ation towards equilibrium in an isolated strongly corre- lated one-dimensional gas, Nat. Phys. 8, 325 (2012)

2012
[28]

A. M. Kaufman, M. Eric Tai, A. Lukin, M. Rispoli, R. Schittko, P. M. Preiss, M. Greiner, Quantum thermal- ization through entanglement in an isolated many-body syste,m, Science 353, 794 (2016)

2016
[29]

G. Clos, D. Porras, U. Warring, and T. Schaetz, Time- Resolved Observation of Thermalization in an Isolated Quantum System, Phys. Rev. Lett. 117, 170401 (2016)

2016
[30]

Y. Tang, W. Kao, K-Y. Li, S. Seo, K. Mallayya, M. Rigol, S. Gopalakrishnan, and B. L. Lev, Thermaliza- tion near Integrability in a Dipolar Quantum Newton’s Cradle, Phys. Rev. X 8, 021030 (2018)

2018
[31]

V. K. B. Kota, A. Rela˜ no, J. Retamosa, and M. Vyas, Thermalization in the two-body random ensemble, J. Stat. Mech. P10028 (2011)

2011
[32]

Rigol and M

M. Rigol and M. Srednicki, Alternatives to Eigenstate Thermalization, Phys. Rev. Lett. 108, 110601 (2012)

2012
[33]

T. N. Ikeda, Y. Watanabe, and M. Ueda, Finite-size scal- ing analysis of the eigenstate thermalization hypothesis in a one-dimensional interacting Bose gas, Phys. Rev. E 87, 012125 (2013)

2013
[34]

Beugeling, R

W. Beugeling, R. Moessner, and M. Haque, Finite-size scaling of eigenstate thermalization, Phys. Rev. E 89, 042112 (2014)

2014
[35]

Steinigeweg, A

R. Steinigeweg, A. Khodja, H. Niemeyer, C. Gogolin, and J. Gemmer, Pushing the Limits of the Eigenstate Thermalization Hypothesis towards Mesoscopic Quan- tum Systems, Phys. Rev. Lett. 112, 130403 (2014)

2014
[36]

H. Kim, T. N. Ikeda, D. A. Huse, Testing whether all eigenstates obey the eigenstate thermalization hypothe- sis, Phys. Rev. E 90, 052105 (2014)

2014
[37]

Jansen, J

D. Jansen, J. Stoplp, L. Vidmar, and F. Heidrich- Meisner, Eigenstate thermalization and quantum chaos in the Holstein polaron model, Phys. Rev. B 99, 155130 (2019)

2019
[38]

Brenes, T

M. Brenes, T. LeBlond, J. Goold, and M. Rigol, Eigen- state Thermalization in a Locally Perturbed Integrable System, Phys. Rev. Lett. 125, 070605 (2020)

2020
[39]

Richter, A

J. Richter, A. Dymarsky, R. Steinigeweg, and J. Gem- mer, Eigenstate thermalization hypothesis beyond stan- dard indicators: Emergence of random-matrix behavior at small frequencies, Phys. Rev. E 102, 042127 (2020)

2020
[40]

J. Wang, M. H. Lamann, J. Richter, R. Steinigeweg, A. Dymasrky, and J. Gemmer, Eigenstate Thermaliza- tion Hypothesis and Its Deviations from Random-Matrix Theory beyond Thermalization Time, Phys. Rev. Lett. 128, 180601 (2022)

2022
[41]

Kinoshita, T

T. Kinoshita, T. Wenger, and D. S. Weiss, A quantun Netwon’s cradle, Nature 440, 900 (2006)

2006
[42]

Gring, M

M. Gring, M. Kuhnert, T. Langen, T. Kitagawa, B. Rauer, M. Schreitl, I. Mazets, D. Adu Smith, E. Dem- ler, and J. Schmiedmayer, Relaxation and Prethermaliza- tion in an Isolated Quantum System, Science 337, 1318 (2012)

2012
[43]

Langen, S

T. Langen, S. Erne, R. Geiger, B. Rauer, T. Schweigler, W. Rohringer, I. E. Mazets, T. Gasenzer, and J. Schmied- mayer, Experimental observation of a generalized Gibbs 8 ensemble, Science, 348, 207 (2015)

2015
[44]

Rigol, Breakdown of Thermalization in Finite One- Dimensional Systems, Phys

M. Rigol, Breakdown of Thermalization in Finite One- Dimensional Systems, Phys. Rev. Lett. 103, 100403 (2009)

2009
[45]

F. H. L. Essler and M. Fagotti, Quench dynamics and relaxation in isolated integrable quantum spin chains, J. Stat. Mech. 064002 (2016)

2016
[46]

Mierzejewski and L

M. Mierzejewski and L. Vidmar, Quantitative Impact of Integrals of Motion on the Eigenstate Thermalization Hy- pothesis, Phys. Rev. Lett. 124 040603 (2020)

2020
[47]

J.-y. Choi, S. Hild, J. Zeiher, P. Schauß, A. Rubio- Abadal, T. Yefsah, V. Khemani, D. A. Huse, I. Bloch, and C. Gross, Exploring the many-body localization transition in two dimensions, Science 352, 1547 (2016)

2016
[48]

Smith, A

J. Smith, A. Lee, P. Richerme, B. Neyenhuis, P. W. Heiss, P. Hauke, M. Heyl, D. A. Huse, and C. Monroe, Many-body localization in a quantum simulator with pro- grammable random disorder, Nat. Phys. 12, 907 (2016)

2016
[49]

Oganesyan and D

V. Oganesyan and D. A. Huse, Localization of interactin g fermions at high temperature, Phys. Rev. B 75, 155111 (2007)

2007
[50]

Nandkishore and D

R. Nandkishore and D. A. Huse, Many-body localiza- tion and thermalization in quantum statistical mechan- ics, Annu. Rev. Condens. Matter Phys. 6, 15 (2015)

2015
[51]

A. L. Corps, R. A. Molina, and A. Rela˜ no, Thouless En- ergy Challenges Thermalization on the Ergodic Side of the Many-Body Localization Transition, Phys. Rev. B 102, 014201 (2020)

2020
[52]

A. L. Corps, R. A. Molina, and A. Rela˜ no, Signatures of a critical point in the many-body localization transition, SciPost Phys. 10, 107 (2021)

2021
[53]

C. J. Turner, A. A. Michailidis, D. A. Abanin, M. Serbyn, and Z. Papi´ c, Weak ergodicity breaking from quantum many-body scars, Nat. Phys. 14, 745 (2018)

2018
[54]

C. J. Turner, A. A. Michailidis, D. A. Abanin, M. Serbyn, and Z. Papi´ c, Quantum scarred eigenstates in a Rydberg atom chain: Entanglement, breakdown of thermalization, and stability to perturbations, Phys. Rev. B 98, 235155 (2018)

2018
[55]

Schecter and T

M. Schecter and T. Iadecola, Weak ergodicity breaking and Quantum Nany-Body Scars in Spin-1 xy Magnets, Phys. Rev. Lett. 123, 147201 (2019)

2019
[56]

Serbyn, D

M. Serbyn, D. A. Abanin, and Z. Papi´ c, Quantum many- body scars and weak breaking of ergodicity, Nat. Phys. 17, 675 (2021)

2021
[57]

J. O. Austin-Harris, I. Rana, S. E. Begg, C. Binegar, T. Bilitewski, and Y. Liu, Observation of Ergodicity Break- ing and Quantum Many-Body Scars in Spinor Gases, Phys. Rev. Lett. 134, 113401 (2025)

2025
[58]

Rela˜ no, Thermalization in an interacting spin sys- tem in the transition from integrability to chaos, J

A. Rela˜ no, Thermalization in an interacting spin sys- tem in the transition from integrability to chaos, J. Stat. Mech. P07016 (2010)

2010
[59]

Systematic Construction of Counterexamples to the Eigenstate Thermalization Hypothesis

N. Shiraishi and T. Mori, Systematic Construction of Counterexamples to the Eigenstate Thermalization Hy- pothesis, Phys. Rev. Lett. 119, 030601 (2017); R. Mondi- ani, K. Mallayya, L. F. Santos, and M. Rigol, Comment on “Systematic Construction of Counterexamples to the Eigens...

2017
[60]

Srednicki, The approach to thermal equilibrium in quantized chaotic systems, J

M. Srednicki, The approach to thermal equilibrium in quantized chaotic systems, J. Phys. A: Math. Gen. 32, 1163 (1999)

1999
[61]

M. L. Mehta, Random matrices, Academic Press (2004)

2004
[62]

Beugeling, R, Moessner, and M

W. Beugeling, R, Moessner, and M. Haque, Oﬀ-diagonal matrix elements of local operators in many-body quan- tum systems, Phys. Rev. E 91, 012144 (2015)

2015
[63]

Mondiani and M

R. Mondiani and M. Rigol, Eigenstate thermalization in the two-dimensional transverse ﬁeld Ising model. II. Oﬀ- diagonal matrix elements of observables, Phys. Rev. E 96, 012157 (2017)

2017
[64]

Nation and D

C. Nation and D. Porras, Oﬀ-diagonal observable ele- ments from random matrix theory: distributions, ﬂuctu- ations, and eigenstate thermalization, New. J. Phys. 20, 103003 (2018)

2018
[65]

LeBlond and M

T. LeBlond and M. Rigol, Eigenstate thermalization for observables that break Hamiltonian symmetries and its counterpart in interacting integrable systems, Phys. Rev. E 102, 062113 (2020)

2020
[66]

A. J. Beekman, L. Rademaker, J. van Wezel, An in- troduction to spontaneous symmetry breaking, SciPost Phys. Lect. Notes 11 (2019)

2019
[67]

P. W. Anderson, An Approximate Quantum Theory of the Antiferromagnetic Ground State , Phys. Rev. 86, 694 (1952)

1952
[68]

Bernu, C

B. Bernu, C. Lhuillier, and L. Pierre, Signature of N´ eel Order in Exact Spectra of Quantum Antiferromagnets on Finite Lattices, Phys. Rev. Lett. 69, 2590 (1992)

1992
[69]

Azaria, B

P. Azaria, B. Delamotte, and D. Mouhanna, Sponta- neous Symmetry Breaking in Quantum Frustrated Anti- ferromagnets, Phys. Rev. Lett. 70, 2483 (1993)

1993
[70]

Bernu, P

B. Bernu, P. Lecheminant, C. Lhuillier, and L. Pierre, Exact spectra, spin susceptibilities, and order parameter of the quantum Heisenberg antiferromagnet on a triangu- lar lattice , Phys. Rev. B 50, 10048 (1994)

1994
[71]

Koma and H

T. Koma and H. Tasaki, Symmetry Breaking and Finite Size Eﬀects in Quantum Many-Body Systems , J. Stat. Phys. 76, 745 (1994)

1994
[72]

Tasaki, Long-Range Order, ”Tower” of States, and Symmetry Breaking in Lattice Quantum Systems , J

H. Tasaki, Long-Range Order, ”Tower” of States, and Symmetry Breaking in Lattice Quantum Systems , J. Stat. Phys. 174, 735 (2019)

2019
[73]

In the case of discrete symmetries both decays are exponential

The scenario becomes even more complicated when con- sidering that the gap between the states belonging to the tower decays as 1 /N , being N the number of particles, whereas the decay of the mean level spacing is exponen- tial. In the case of discrete symmetries both decays a...
[74]

L. F. Santos and M. Rigol, Onset of quantum chaos in one-dimensional bosonic and fermionic systems and its relation to thermalization, Phys. Rev. E 81, 036206
[75]

L. F. Santos and M. Rigol, Localization and the eﬀects of symmetries in the thermalization properties of one- dimensional quantum systems, Phys. Rev. E 82, 031130 (2010)

2010
[76]

J. M. G. G´ omez, K. Kar. V. K. B. Kota, R. A. Molina, A. Rela˜ no, and J. Retamosa, Many-body quantum chaos: Recent developments and applications to nuclei, Phys. Rep. 499, 103 (2011)

2011
[77]

Gubin and L

A. Gubin and L. F. Santos, Quantum chaos: An intro- duction via chains of interacting spins 1/2, Am. J. Phys 80, 246 (2012)

2012
[78]

Bohigas, M

O. Bohigas, M. J. Giannoni, and C. Schmit, Characteri- zation of Chaotic Quantum Spectra and Universality of Level Fluctuation Laws, Phys. Rev. Lett. 52, 1 (1984)

1984
[79]

E. T. Jaynes, Information theory and statistical mecha n- ics, Phys. Rev. 106, 620 (1957)

1957
[80]

E. T. Jaynes, Information theory and statistical mecha n- ics. II, Phys. Rev. 108, 171 (1957). 9

1957
[81]

Rigol, A

M. Rigol, A. Muramatsu, and M. Olshanii, Hard-core bosons on optical superlattices: Dynamics and relaxation in the superﬂuid and insulating regimes, Phys. Rev. A 74, 053616 (2006)

2006
[82]

A. L. Corps and A. Rela˜ no, General theory for dis- crete symmetry-breaking equilibrium states, Phys. Rev. E 110, 034137 (2024)

2024
[83]

Guryanova, S

Y. Guryanova, S. Popescu, A. J. Short, R. Silva, and P. Skrzypczyk, Thermodynamics of quantum systems with multiple conserved quantities, Nat. Commun. 7, 12049 (2016)

2016
[84]

N. Y. Halpern, P. Faist, J. Oppenheim, and A. Winter, Microcanonical and resource-theoretic derivations of the thermal state of a quantum system with noncommuting charges, Nat. Commun. 7, 12051 (2016)

2016
[85]

Yunger Halpern, M

N. Yunger Halpern, M. E. Beverland, and A. Kalev, Non- commuting conserved charges in quantum many-body thermalization, Phys. Rev. E 101, 042117 (2020)

2020
[86]

Kranzl, A

F. Kranzl, A. Lasek, M. K. Joshi, A. Kalev, R. Blatt, C. F. Roos, and N. Yunger Halper, Experimental observa- tion of thermalization with noncommuting charges, PRX Quantum 4, 020318 (2023)

2023
[87]

Majidy, W

S. Majidy, W. F. Braasch, A. Lasek, T. Upadyaya, A. Kalev, and N. Yunger Halpern, Noncommuting con- served charges in quantum thermodynamics and beyond, Nature Reviews Physics 5, 689 (2023)

2023
[88]

Murthy, A

C. Murthy, A. Babakhani, F. Iniguez, M. Srednicki, and N. Yunger Halpern, Non-abelian eigenstate thermaliza- tion hypothesis, Phys. Rev. Lett. 130, 140402 (2023)

2023
[89]

Lasek, J

A. Lasek, J. D. Noh, J. LeSchack, and N. Y. Halpern, Nu- merical evidence for the non-abelian eigenstate thermal- ization hypothesis, arXiv:2412.07838 [quant-ph] (2024)

2024 arXiv
[90]

Patil and M

R. Patil and M. Rigol, Eigenstate thermalization in spi n- 1/2 systems with SU(2) symmetry. arXiv:2503.01846 [quant-ph] (2025)

2025 arXiv
[91]

Tsubota and K

M. Tsubota and K. Kasamatsu, Josephson Current Flow- ing in Cyclically Coupled Bose-Einstein Condensates, J. Phys. Soc. Jpn 69, 1942 (2000)

2000
[92]

D. R. Scherer, C. N. Weiler, T. W. Neely, and B. P. Anderson, Vortex Formation by Merging of Multiple Trapped Bose-Einstein Condensates, Phys. Rev. Lett. 98, 110402 (2007)

2007
[93]

Gallemi, M

A. Gallemi, M. Guilleumas, J. Martorell, R. Mayol, A. Polls, and B. Juli´ a-D ´ ıaz, Fragmented condensation in Bose-Hubbard trimes with tunable tunneling, New J. Phys. 17, 073014 (2015)

2015
[94]

M. A. Garcia-March, S. van Frank, M. Bonneau, J. Schmiedmayer, M. Lewenstein, and L. F. Santos, New J. Phys. 20, 113039 (2018)

2018
[95]

de la Cruz, S

J. de la Cruz, S. Lerma-Hern´ andez, and J. G. Hirsch, Quantum chaos in a system with high degree of symme- tries, Phys. Rev. E 102, 032208 (2020)

2020
[96]

Wozniak, J

D. Wozniak, J. Kroha, and A. Posazhennikova, Chaos onset in large rings of Bose-Einstein condensates, Phys. Rev. A 106, 033316 (2022)

2022
[97]

Nakerst and M

G. Nakerst and M. Haque, Chaos in the three-site Bose- Hubbard model: Classical versus quantum, Phys. Rev. E 107, 024210 (2023)

2023
[98]

Arwas, A

G. Arwas, A. Vardi, and D. Cohen, Triangular Bose- Hubbard trimer as a minimal model for a superﬂuid cir- cuit, Phys. Rev. A 89, 013601 (2014)

2014
[99]

Peres, New Conserved Quantities and Test for Regular Spectra, Phys

A. Peres, New Conserved Quantities and Test for Regular Spectra, Phys. Rev. Lett. 53, 1711 (1984)

1984
[100]

M. V. Berry and M. Robnik, Semiclassical level spac- ings when regular and chaotic orbits coexist, J. Phys. A: Math. Gen. 17 2413 (1984)

1984
[101]

Berry and M

M. Berry and M. Tabor, Level clustering in the regular spectrum, Proc. R. Soc. A 356, 375 (1997)

1997
[102]

J. M. G. G´ omez, R. A. Molina, A. Rela˜ no and J. Re- tamosa, Misleading signatures of quantum chaos, Phys. Rev. E 66, 036209 (2002)

2002
[103]

A. L. Corps and A. Rela˜ no, Long-range level correlatio ns in quantum systems with ﬁnite Hilbert space dimension, Phys. Rev. E 103, 012208 (2021)

2021
[104]

M. C. Gutzwiller, Chaos in Classical and Quantum Me- chanics (Springer, New York, 1990)

1990

Pith tools

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

[1] [1]

First, spectral statistics and classical trajectories (see Appendix C) indicate that there is full chaos

Our main result consists in identifying the following spectral regions: Thermalizing region, with −3.1 ≲ E/N ≲ −2.3. First, spectral statistics and classical trajectories (see Appendix C) indicate that there is full chaos. According to S1, this implies that Eq. ( 2) is necessarily fulﬁlled. In Fig. 1(a) we show that the trace of ˆh12 always ﬂuctuates arou...

[2] [2]

A Way of Making Europe

and ( 3) are fulﬁlled within this region. Thus, the standard mi- crocanonical ensemble is expected to hold, and therefore no symmetry breaking can be observed. Rotation-breaking region, with E/N ≲ −3.9. Fig. 1(b) shows that none of the subspaces fulﬁll Eq. ( 3) for ˆI. From S2, we conclude that this is an ordered phase in which the rotational symmetry is ...

[3] [3]

A deeper test is shown in Fig

(see Appendix B for details). A deeper test is shown in Fig. 2. In panel (a), we can see that the width of the distribution of T (ˆh12) n /dn − h12(E) decreases as a power law, σ ∝ N − 0. 419. This decreasing behavior is close to Srednicki’s ansatz, in which diagonal expectation values ﬂuctuate around the microcanonical with a width proportional to e− S(E...

[4] [4]

commutes with the total number of particles, ˆN = ∑ i ˆa† i ˆai. The symmetries of this model are described by the dihedral group D3, which has two non-commuting dis- crete generators: a 2 π/3 rotation, ˆR |n1, n2, n3⟩ = |n3, n1, n2⟩, where ni indicates the number of parti- cles in the site i; and a reﬂection, ˆS |n1, n2, n3⟩ = |n3, n2, n1⟩ (see Appendix ...

[5] [5]

and ( 3) hold if and only if all the initial conditions narrow enough in energy ther- malize. Forward implication.- Let us consider that the system starts in an initial condition, |Ψ(0) ⟩ = ∑ n,α cn,α |En,α ⟩, where cn,α ⁄= 0 only in a subextensive interval [ E − δE, E + δE], and let us calculate the resulting long- time average of a physical observable ˆ...

[6] [6]

(A3) And therefore, the long-time average given in Eq

and ( 3) are satisﬁed, then maxn,α |λ(O) n,α − O(E)| η ˆO → 0, N → ∞. (A3) And therefore, the long-time average given in Eq. ( A2) coincides with the microcanonical average. Backward implication.- This is equivalent to show- ing that if (a) max n |T (O) n /dn − OME(E)| ↛ 0 or (b) maxn,α |λ(O) n,α − T (O) n /dn| ↛ 0 when N → ∞ , then there exists at least ...

[7] [7]

Thus, the long-time average of the expectation value of ˆO is simply ˆO = ⟨ ~Em,α ⏐ ⏐ ⏐ ˆO |Em,α ⟩ = λ(O) m,α (A7) And this initial state does not thermalize

Let λ(m) α be the eigenvalue of ˆO(m) such that⟨ ~Em,β ⏐ ⏐ ⏐ ˆO ⏐ ⏐ ⏐~Em,α ⟩ = λm,α and let us consider an initial 6 condition |Ψ(0) ⟩ = ⏐ ⏐ ⏐~Em,α ⟩ . Thus, the long-time average of the expectation value of ˆO is simply ˆO = ⟨ ~Em,α ⏐ ⏐ ⏐ ˆO |Em,α ⟩ = λ(O) m,α (A7) And this initial state does not thermalize. Appendix B: Some details of the model Classica...

[8] [8]

(B4) Symmetries.– Eq

+ e4πi/ 3(q2 3 + p2 3) ] , (B2) C N = ∑ j (pjqj+1 − pj+1qj), (B3) and h12 N = q1q2 + p1p2. (B4) Symmetries.– Eq. ( 4) is invariant under 2π/3 and 4π/3 rotations, and under the reﬂection around any of its sym- metry axes. This set of transformations constitutes the dihedral D3 group, whose generators are the operators ˆR and ˆS. Appendix C: Details of nume...