Modular Hamiltonian and Typicality of Degenerate States

Goncalo C. Oliveira; Gregory C. Levine

REVIEW

Modular Hamiltonian and Typicality of Degenerate States

Quantum Physics · Submitted 6 Nov 2026 · 34 references

Pith's verdict: The two-state modular-Hamiltonian formula is a genuine, checkable result; the Section V many-state extension rests on a spectral-sharing assumption that Eigenstate Typicality does not provide.

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arXiv ID2611.09595 (version 1)

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CitationGoncalo C. Oliveira, Gregory C. Levine. "Modular Hamiltonian and Typicality of Degenerate States." arXiv:2611.09595 (2026). https://pith.science/paper/DLAPJ5A5

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Claim paper

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Editor's note

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The paper's load-bearing claim holds for the two-state case, and the explicit first-order formula in Eq. (30) is the genuinely new thing. The many-state argument in Section V does not carry the same weight; it rests on a spectral-sharing assumption that nothing in the paper proves.

What's actually new is Eq. (30), the perturbative modular Hamiltonian h_theta = h_a - (theta/4) sum[(E_alpha' - E_alpha)/sinh((E_alpha' - E_alpha)/2)] |alpha>

The stress-test is right about where the paper is soft. The two-state result is not the weak point; Section V is. Eq. (57) defines a separate Schmidt basis for each member of the degenerate set, then assumes rho_T|alpha_a> ~ d^2_alpha|alpha_a> (b156) with the phrase "not unreasonable." But Eigenstate Typicality controls local expectation values, not the eigenvectors of reduced density matrices. For free fermions the Schmidt basis is fixed by the single-particle correlation matrix m_a, and m_a - m_b is small only for the neighboring-microstate geometry with Delta k << 1/L_A used in Eqs. (39)-(40). For generic pairs in an extensive manifold, Delta k is not small and the eigenvalues of rho_a need not match rho_T. If spectral sharing fails, the residual rho_a - rho_T is not parametrically smaller than the off-diagonal term in (53), the first-order correction (55) loses its small parameter, and the O(N(L_A/L)^delta) estimate cannot support the FD-deviation conclusion. The authors flag it as an assumption, which is honest, but it is load-bearing and untested.

Two smaller items. The abstract's Fermi-Dirac deviation claim (b5) is asserted with no quantitative calculation of the single-particle distribution; the body gives the modular-Hamiltonian formula and a magnitude estimate, but nothing that converts the non-Gaussianity into a measurable deviation. And the O((L_A/L)^4) estimate is restricted to the O_AAbar sector and Delta k << 1/L_A, so it concerns a special geometry, not the general case the abstract implies. The unpublished self-citation [30] cannot anchor the CFT extension.

Who this is for: people working on modular Hamiltonians, Gaussian-state theory, or eigenstate typicality in free-fermion systems. The two-state formula deserves a serious referee. Section V should be either proved or explicitly hived off as conjecture. I'd send it out.

What would settle it

Compute the modular Hamiltonian for the same two-state superposition in a larger chain and fit the largest two-body matrix element as a function of $L_A/L$ at fixed mixing angle: Eq. (30) predicts a linear-in-$\theta$ coefficient with magnitude scaling as $(L_A/L)^4$, so any clearly different exponent would falsify the estimate. A sharper test uses a degenerate pair whose connecting operator acts only inside $A$, making $O_{\bar A\bar A} = 0$ and predicting a vanishing linear correction; a nonzero $O(\theta)$ two-body term in that case would disprove the claim that the complement piece dominates. A third check is to extract the single-particle momentum distribution of subsystem $A$ in the (12/6/6) system and test whether its deviation from Fermi-Dirac is linear in the mixing angle $\theta$ near the pure points.

Referee report

Significance

If the two-state result holds — and my reading of Eqs. (21)–(30), cross-checked against the exact numerics of Section III, is that it does — it is a clean demonstration that a superposition of degenerate free-fermion eigenstates violates Wick's theorem, producing a non-Gaussian reduced density matrix and an interacting modular Hamiltonian even though the underlying Hamiltonian is free. The paper's strengths are concrete: an exact finite-size many-body computation with nontrivial internal consistency checks (two-body terms vanish at θ_pure and return away from it; one-body elements agree with the single-particle correlation-function computation), a closed-form first-order perturbation theory whose matrix-logarithm expansion is derived in Appendix A and whose sign and prefactor conventions are internally consistent, and an honest account of the numerical sensitivity of the matrix logarithm near singular ρ. No numerical constants are fitted; θ is the physical mixing angle.

Major objections (3)

[Section V, after Eq. (57)] The many-state extension rests on the spectral-sharing assumption ρ_T|α_a⟩ ≈ d²_α|α_a⟩ (sentence b156). This is not a consequence of Eigenstate Typicality: ET controls expectation values of local operators, not the eigenvectors of reduced density matrices. For free-fermion eigenstates the Schmidt basis is fixed by the eigenvectors of the single-particle correlation matrix m_a; the control on ‖m_a − m_b‖ obtained in Eqs. (39)–(40) is established only for the neighboring-microstate geometry with Δk ≪ 1/L_A (b119). For generic pairs in an extensive degenerate manifold Δk is not small, the coherent-antipodal phase argument fails, and the spectra and eigenvectors of ρ_a need not coincide with those of ρ_T. If spectral sharing fails, the residual (ρ_a − ρ_T) in Eq. (53) is not parametrically smaller than the off-diagonal term, so the first-order expansion (55)–(56) has no controlled small parameter and the O(N(L_A/L)^δ) estimate (b158) cannot support the FD-deviation and initial-condition-encoding conclusions. The authors acknowledge b156 as an assumption, but it is load-bearing and is neither derived from the correlation-matrix structure nor tested numerically; a direct finite-size test comparing the eigenvectors of ρ_a for several members of a degenerate manifold with those of ρ_T would settle the issue, and without such support the conclusions should be restricted to the two-state setting. The authors acknowledge this: “This is not an unreasonable assumption in that one expectsd 2α = exp (−E α) to reflect the energies of the fermi sea subsystem independent of the typical degenerate state chosen.”
[Section V, Eq. (53)] A second uncontrolled step is the treatment of the off-diagonal block (1/N)Σ_{a≠b} ϵ_ab tr_Ā(|ψ_a⟩⟨ψ_b|) as a perturbation. Each term has trace norm at most 1 and there are N² terms, so with c_a ∼ N^{−1/2} the trace norm of this block can be O(1) relative to ρ_T for generic coefficients; unlike in the two-state calculation there is no tunable θ that makes it small. The approximation in (53) is therefore an assumption about the off-diagonal block, not a consequence of ET or of normalization, and the O(N(L_A/L)^δ) estimate (b158) does not repair this: even for (L_A/L)^δ ≪ 1 the prefactor N is unbounded, a point the paper itself concedes when it notes that N ∼ L/L_A saturates perturbation theory. The many-state corrections and the conclusions drawn from them (FD deviations, encoding of initial conditions) are consequently not established by the present argument.
[Section IV, Eqs. (39)–(41)] The magnitude estimate ⟨α′|O†_{ĀĀ}|α⟩ ∼ O((L_A/L)^4) in Eq. (41) is derived under two stated restrictions — O confined to the ĀĀ part and Δk ≪ 1/L_A (b111, b119) — and the derivation is heuristic rather than a bound: the antipodal-coherence argument of Fig. 5(a) and the adjacency appeal in Eq. (39) read the scaling off the sums by inspection. The exact computations of Section III cover a single system size (12/6/6), so the (L_A/L)^δ dependence is never verified numerically; repeating the exact computation for a second size (for example, (16/8/8)) would test it. Since this estimate is the quantitative basis for the O(N(L_A/L)^δ) correction in Section V (b158), the magnitude claim is unsupported even apart from the two issues above; the two-state qualitative claim (interacting h_θ) does not need it and is independently confirmed by the numerics.

Minor objections (5)

[Eqs. (24)–(28)] The symbol G is used inconsistently: in Eq. (24) G_{αα′} denotes the dyadic operator including |α⟩⟨α′|, while in Eqs. (27)–(28) G_{α′α} is used as a scalar coefficient with the index order reversed. The final result is presumably unaffected, but the convention should be stated explicitly.
[Paragraph after Eq. (15)] The sentence claiming that cos²(θ/2)ρ_a + sin²(θ/2)ρ_b ≈ ρ_{a,b} 'for all θ' up to O(θ²) corrections is internally inconsistent: the approximation is a small-θ statement. Since Eq. (30) is a first-order result, the paper should state explicitly that the exact numerics in Figs. 1–3, not the perturbation theory, cover the large-θ regime.
[Fig. 1 caption] The caption labels |θ = π/2⟩ as pure and |θ = π⟩ as impure, which appears to invert the natural reading of Eq. (11) (at θ = π the state is simply |e_k⟩). The explanation — that the numerically diagonalized eigenstates are not the momentum-ordered determinants and that specific linear combinations at θ_pure are — appears only in the main text and should be in the caption.
[Abstract and Section V (sentence b5)] The abstract states as a finding that single-particle measurements would exhibit non-thermal deviations from the Fermi–Dirac distribution, but the first-order correction to the momentum occupation n(k) = tr(ρ_θ c†_k c_k) is never computed. Since this is a quantitative prediction, the authors should either compute it from Eqs. (15) and (30) or explicitly label it a conjecture.
[Section IV, opening paragraph] The convention for the mixing angle changes between Sections III and IV (θ_pure = ±π/2 in Section III, and 'now θ_pure = 0, π' in Section IV). This is presumably a shift of the expansion origin and should be stated explicitly.

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The paper

The paper claims

A superposition of degenerate eigenstates makes the modular Hamiltonian of a free-fermion subsystem interacting, so its reduced density matrix is no longer Gaussian.

The reading

The paper asks what happens to eigenstate typicality when the global pure state is not a single free-fermion eigenstate but a superposition of degenerate ones, and it focuses on the modular Hamiltonian — the operator $h = -\log\rho$ built from the reduced density matrix of a small subsystem. For a single typical excited eigenstate of a one-dimensional free-fermion lattice, that operator is quadratic (free) and reproduces Fermi-Dirac statistics; the paper argues that any genuine superposition of degenerate eigenstates violates Wick's theorem and makes the modular Hamiltonian interacting, so the reduced density matrix is not Gaussian. A first-order perturbation formula, Eq. (30), ties the interacting matrix elements inside the subsystem to transition amplitudes of the degenerate-state operator acting in the complementary region, and exact computation on chains of up to twelve sites shows two-body terms appear for every mixing angle except the special values where the superposition is a pure eigenstate. The interaction strength scales as $O(L_A^4/L^4)$ in one dimension, and the authors argue that mixtures of enough degenerate states saturate the perturbation theory. If correct, the result implies that non-thermal deviations from the Fermi-Dirac distribution are observable in a small subsystem and that a fermion system's initial condition is encoded in the interacting part of its modular Hamiltonian.

What carries the argument

Three ingredients carry the argument. First, Wick's theorem and Gaussianity: for a single pure eigenstate of the noninteracting lattice Hamiltonian, the reduced density matrix is Gaussian, $\rho_a = \exp(-\sum_{i,j\in A} h^a_{ij}\,c^\dagger_i c_j)$, with $\mathbf{h}^a = \log(\mathbf{m}_a^{-1} - \mathbf{I})$ determined by the single-particle correlation matrix $\mathbf{m}_a$; all two-body and higher terms vanish. Second, a matrix-logarithm perturbation theory: using the integral representation of the logarithm, the paper expands $\log(\rho_a + (\theta/2)M_{ab})$ to first order in $\theta$, and evaluates the integral in the Schmidt basis of $\rho_a$, producing the $\sinh$-weighted energy kernel of Eq. (28). Third, the connecting operator $O$ with $|e_b\rangle = O^\dagger|e_a\rangle$, split into pieces acting within $A$, between $A$ and $\bar A$, and within $\bar A$; for $L_A \ll L_{\bar A}$ only the $\bar A$-$\bar A$ piece contributes at leading order, so each interacting matrix element inside $A$ becomes a product of two projected contractions in $\bar A$, each of size $O(L_A^2/L^2)$ for momentum transfers $\Delta k \ll 1/L_A$ in one dimension.

Extended reading notes

Core claim

The central claim is contained in Eq. (30). For the state $|\theta\rangle = \cos(\theta/2)|e_a\rangle + \sin(\theta/2)|e_b\rangle$ built from two degenerate excited eigenstates of a one-dimensional free-fermion lattice, the modular Hamiltonian of a subsystem $A$ is, to first order in the mixing angle, $$h_\$\theta$ = h_a - \frac{\$\theta$}{4}\sum_{\$\alpha$,\$\alpha$'} \frac{E_{\$\alpha$'}-E_\$\alpha$}{\$\sinh$[(E_{\$\alpha$'}-E_\$\alpha$)/2]}\,\langle\$\alpha$'|O_{\bar A\bar A}+O^\dagger_{\bar A\bar A}|\$\alpha$\rangle\,|\$\alpha$\rangle\langle\$\alpha$'| + O(\$theta^{2}$),$$ where $|\alpha\rangle$ are the Schmidt states of $A$ for the unperturbed eigenstate, $E_\alpha = -2\log d_\alpha$ are the modular energies, and $O^\dagger$ maps $|e_a\rangle$ to $|e_b\rangle$. Because the correction is off-diagonal in the Schmidt basis and does not take a one-body form, $h_\theta$ is interacting and $\rho_\theta = e^{-h_\theta}$ is non-Gaussian; the mechanism is the failure of Wick's theorem for the cross term $\mathrm{tr}_{\bar A}(|e_a\rangle\langle e_b| + |e_b\rangle\langle e_a|)$. Exact diagonalization of systems up to twelve sites confirms the picture: two-body and four-body matrix elements of $h_\theta$ vanish precisely at $\theta = \pm\pi/2$, where the superposition is a pure eigenstate, and are nonzero for every other angle, vanishing linearly or quadratically as $\theta$ approaches the pure points. The authors further estimate each projected contraction contributing to the matrix element as $O(L_A^2/L^2)$, giving two-body modular matrix elements of order $(L_A/L)^4$ in one dimension, and argue that a mixture of $N$ degenerate states yields corrections of order $N(L_A/L)^\delta$ with $\delta = 4$, so perturbation theory saturates once $N \sim L/L_A$.

Load-bearing premise

The load-bearing premise is that the many nearly degenerate eigenstates behind a realistic Fermi system's approach to equilibrium can be treated as exactly degenerate states that all share the same Schmidt spectrum, so one typical reduced density matrix describes every member of the set, and that the leading correction is dominated by scattering inside the complement with momentum transfer small compared to $1/L_A$; if the Schmidt spectra differ from state to state, the size and sign of the correction terms become uncontrolled.

Editorial extensions

If this is right

The reduced density matrix of a small subsystem of a noninteracting fermion chain is non-Gaussian whenever the global state is a genuine superposition of degenerate eigenstates, so single-particle measurements in the subsystem deviate from the Fermi-Dirac distribution even though the fermions do not interact.
The Gaussian, free modular Hamiltonian realized by a pure eigenstate is a measure-zero special case: any admixture of degenerate states, however small, turns on two-body and higher interaction terms in $h_\theta$.
For weak mixtures the correction is linear in the mixing angle with matrix elements of order $(L_A/L)^4$ in one dimension; with $N$ degenerate states the correction grows as $N(L_A/L)^\delta$ and perturbation theory saturates when $N \sim L/L_A$.
The interacting corrections depend on the specific coefficients of the superposition, so the modular Hamiltonian and multiparticle correlations within $A$ carry information about how the state was prepared that the diagonal ensemble alone discards.
The authors conjecture that for a large superposition with very small $A$, the modular Hamiltonian resembles a random interacting matrix of the Sachdev-Ye-Kitaev type, with the caveat that the relevant dynamics is modular flow rather than real-time evolution.

Reading between the lines

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

A quantitative test the authors did not run: compute $h_\theta$ for eigenstates related by a two-body momentum move at several subsystem sizes and directly measure the $(L_A/L)^4$ scaling exponent at fixed mixing angle; the same numerics would reveal at which $\theta$ the quadratic corrections (visible in Fig. 3) overtake the linear term, delimiting the perturbative regime.
The $\sinh$-weighted kernel $(E_{\alpha'}-E_\alpha)/\sinh[(E_{\alpha'}-E_\alpha)/2]$ is the same weight that appears in first-order entanglement perturbations around a reference state; treating the degenerate mixture as a coherent excitation of the modular Hamiltonian could connect this construction to modular-flow and relative-entropy results, and would turn the paper's SYK speculation into a que
The mechanism is probably not specific to fermions: any superposition of degenerate Gaussian states is non-Gaussian, so superpositions of degenerate Gaussian states of free bosons or of quadratic spin models should produce interacting modular Hamiltonians as well, and a free-boson lattice calculation would be a cheap check of that transferability.

Original abstract

Free fermions exhibit Eigenstate Typicality (ET) meaning that the reduced density matrix, $\rho$, of a small spatial subsystem of a {\sl single}, typical, excited energy eigenstate is gaussian with a modular hamiltonian, $h \equiv -\log \rho$, asymptotically proportional to the free fermion hamiltonian. In effect, the ET may be thought of as providing an independent route to establishing the Fermi-Dirac (FD) distribution for free fermions. We revisit this argument for fermions belonging to a superposition of degenerate eigenstates. Specifically, we show that the modular hamiltonian corresponding to a superposition of degenerate eigenstates in a 1-d free fermion lattice system is interacting, and therefore the reduced density matrix for the subsystem is not gaussian. Measurements of the single particle distribution of a small subsystem would then exhibit non-thermal deviations from the FD distribution. These features may be traced to a violation of Wick's theorem for the chosen excited state. We develop a perturbation theory for the modular hamiltonian corresponding to a weak mixture of degenerate states and study the modular hamiltonian (exactly) computationally. Finally, we speculate that initial conditions in a realistic fermi system evolving towards equilibrium might be encoded in the interaction terms of the modular hamiltonian and consequent correlations of fermions.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-11-09: 1 request, 6,461 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.36 · human participants 0.01 · controlled comparison 0.14 · randomized assignment 0.01 · states quantitative result 0.07 · compares with prior results 0.90 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

Free fermions exhibit Eigenstate Typicality (ET) meaning that the reduced density matrix, $\rho$, of a small spatial subsystem of a {\sl single}, typical, excited energy eigenstate is gaussian with a modular hamiltonian, $h \equiv -\log \rho$, asymptotically proportional to the free fermion hamiltonian. In effect, the ET may be thought of as providing an independent route to establishing the Fermi-Dirac (FD) distribution for free fermions. We revisit this argument for fermions belonging to a superposition of degenerate eigenstates. Specifically, we show that the modular hamiltonian corresponding to a superposition of degenerate eigenstates in a 1-d free fermion lattice system is interacting, and therefore the reduced density matrix for the subsystem is not gaussian. Measurements of the single particle distribution of a small subsystem would then exhibit non-thermal deviations from the FD distribution. These features may be traced to a violation of Wick's theorem for the chosen excited state. We develop a perturbation theory for the modular hamiltonian corresponding to a weak mixture of degenerate states and study the modular hamiltonian (exactly) computationally. Finally, we speculate that initial conditions in a realistic fermi system evolving towards equilibrium might be encoded in the interaction terms of the modular hamiltonian and consequent correlations of fermions.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Free fermions exhibit Eigenstate Typicality (ET) meaning that the reduced density matrix, $\rho$, of a small spatial subsystem of a {\sl single}, typical, excited energy eigenstate is gaussian with a modular hamiltonian, $h \equiv -\log \rho$, asymptotically proportional to the free fermion hamiltonian.	background 1.00	0.08	0.04	asserted 1.00
s1	In effect, the ET may be thought of as providing an independent route to establishing the Fermi-Dirac (FD) distribution for free fermions.	interpretation 0.52	0.07	0.03	hedged possibility 0.94
s2	We revisit this argument for fermions belonging to a superposition of degenerate eigenstates.	contribution 0.89	0.57	0.04	asserted 0.67
s3	Specifically, we show that the modular hamiltonian corresponding to a superposition of degenerate eigenstates in a 1-d free fermion lattice system is interacting, and therefore the reduced density matrix for the subsystem is not gaussian.	result 0.99	0.95	0.81	asserted 1.00
s4	Measurements of the single particle distribution of a small subsystem would then exhibit non-thermal deviations from the FD distribution.	interpretation 0.71	0.57	0.17	asserted 0.64
s5	These features may be traced to a violation of Wick's theorem for the chosen excited state.	interpretation 0.98	0.64	0.08	hedged possibility 0.96
s6	We develop a perturbation theory for the modular hamiltonian corresponding to a weak mixture of degenerate states and study the modular hamiltonian (exactly) computationally.	method 0.85	0.96	0.06	asserted 0.99
s7	Finally, we speculate that initial conditions in a realistic fermi system evolving towards equilibrium might be encoded in the interaction terms of the modular hamiltonian and consequent correlations of fermions.	future work 0.52	0.11	0.07	speculative 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.88.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** FIG. 1: Modular hamiltonian,

**Figure 2.** FIG. 2: Modular hamiltonian

**Figure 3.** FIG. 3: Three representative two-body (interacting) matrix elements of the modular hamiltonian

**Figure 4.** FIG. 4: Right moving branches of two degenerate states

**Figure 5.** FIG. 5: (a) Depiction of Kronecker delta function sum with region

Figures reproduced from arXiv:2611.09595 by the authors.

References

Works cited (34)

[30]

Lashkari, N., Liu, H., & Rajagopal, S. ”Perturbation theory for the logarithm of a positive operator,” Journal of High Energy Physics 2023, 97 (2023)

View on Pith 2023
[1]

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

View on Pith 1991
[2]

Srednicki, M. ”Chaos and quantum thermalization,” Phys. Rev. E 50, 888 (1994) [arXiv:cond- mat/9403051]

View on Pith 1994
[3]

Deutsch, J. M. ”Eigenstate thermalization hypothesis,” Reports on Progress in Physics 81, 082001 (2018) [arXiv:1805.01616]

View on Pith Pith review arXiv 2018
[4]

D’Alessio, L., Kafri, Y., Polkovnikov, A., & Rigol, M. ”From quantum chaos and eigenstate thermalization to statistical mechanics and thermodynamics,” Advances in Physics 65, 239 (2016) [arXiv:1509.06411]. 23

View on Pith Pith review arXiv 2016
[5]

Rigol, M., Dunjko, V., & Olshanii, M. ”Thermalization and its mechanism for generic isolated quantum systems,” Nature (London) 452, 854 (2008) [arXiv:0708.1324]

View on Pith Pith review arXiv 2008
[6]

is not strictly satisfied. AllM A initial and final elements were present in the modular hamiltonian consistent 9 θ/2π h |100111⟩⟨011101| |101010⟩⟨010110| |001101⟩⟨100011| -0.4 -0.2 0.0 0.2 0.4 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 FIG. 3: Three representative two-body (interacting) matrix elements of the modular hamiltonian corresponding to the mixture (11) of...

View on Pith
[7]

Popescu, S., Short, A. J., & Winter, A. ”Entanglement and the foundations of statistical mechanics,” Nature Physics 2, 754 (2006) [arXiv:quant-ph/0511225]

View on Pith Pith review arXiv 2006
[8]

Goldstein, S., Lebowitz, J. L., Tumulka, R., & Zangh ` ı, N. ”Canonical Typicality,” Phys. Rev. Lett. 96, 050403 (2006) [arXiv:cond-mat/0511091]

View on Pith Pith review arXiv 2006
[9]

Garrison, J. R., & Grover, T. ”Does a Single Eigenstate Encode the Full Hamiltonian?,” Physical Review X 8, 021026 (2018) [arXiv:1503.00729]

View on Pith Pith review arXiv 2018
[10]

Lai, H.-H. & Yang, K. ”Entanglement entropy scaling laws and eigenstate typicality in free fermion systems,” Physical Review B 91, 081110 (2015) [arXiv:1409.1224]

View on Pith Pith review arXiv 2015
[11]

Mark, D. K., Surace, F., Elben, A., et al. ”Maximum Entropy Principle in Deep Thermalization and in Hilbert-Space Ergodicity,” Physical Review X 14, 041051 (2024) [arXiv:2403.11970]

View on Pith Pith review arXiv 2024
[12]

Lucas, M., Piroli, L., De Nardis, J., & De Luca, A. ”Generalized deep thermalization for free fermions,” Phys. Rev. A 107, 032215 (2023) [arXiv:2207.13628]

View on Pith Pith review arXiv 2023
[13]

Tian, C., Yang, K., Fang, P., Zhou, H.-J., & Wang, J. ”Hidden thermal structure in Fock space,” Phys. Rev. E 98, 060103 (2018) [arXiv:1808.00784]

View on Pith Pith review arXiv 2018
[14]

Lian, B. ”Conserved quantities from entanglement Hamiltonian,” Phys. Rev. B 105, 035106 (2022) [arXiv:2104.11753]

View on Pith Pith review arXiv 2022
[15]

Brand˜ ao, F. G. S. L., Crosson, E., S ¸ahinoˇ glu, M. B., & Bowen, J. ”Quantum Error Correcting Codes in Eigenstates of Translation-Invariant Spin Chains,” Phys. Rev. Lett. 123, 110502 (2019) [arXiv:1710.04631]

View on Pith Pith review arXiv 2019
[16]

Bao, N., & Cheng, N. ”Eigenstate thermalization hypothesis and approximate quantum error correction,” Journal of High Energy Physics 2019, 152 (2019) [arXiv:1906.03669]

View on Pith Pith review arXiv 2019
[17]

Almheiri, A., Dong, X., & Harlow, D. ”Bulk locality and quantum error correction in AdS/CFT,” Journal of High Energy Physics 2015, 163 (2015) [arXiv:1411.7041]

View on Pith Pith review arXiv 2015
[18]

Jafferis, D. L., Lewkowycz, A., Maldacena, J., & Suh, S. J. ”Relative entropy equals bulk relative entropy,” Journal of High Energy Physics 2016, 4 (2016) [arXiv:1512.06431]

View on Pith Pith review arXiv 2016
[19]

Dong, X., Harlow, D., & Wall, A. C. ”Reconstruction of Bulk Operators within the Entanglement Wedge in Gauge-Gravity Duality,” Phys. Rev. Lett. 117, 021601 (2016) [arXiv:1601.05416]. 24

View on Pith Pith review arXiv 2016
[20]

Faulkner, T., & Lewkowycz, A. ”Bulk locality from modular flow,” Journal of High Energy Physics 2017, 151 (2017) [arXiv:1704.05464]

View on Pith Pith review arXiv 2017
[21]

Holzhey, C., Larsen, F., & Wilczek, F. ”Geometric and renormalized entropy in conformal field theory,” Nuclear Physics B 424, 443 (1994) [arXiv:hep-th/9403108]

View on Pith Pith review arXiv 1994
[22]

Casini, H., Huerta, M., & Myers, R. C. ”Towards a derivation of holographic entanglement entropy,” Journal of High Energy Physics 2011, 36 (2011) [arXiv:1102.0440]

View on Pith Pith review arXiv 2011
[23]

Dymarsky, A., Lashkari, N., & Liu, H. ”Subsystem eigenstate thermalization hypothesis,” Phys. Rev. E 97, 012140 (2018) [arXiv:1611.08764]

View on Pith Pith review arXiv 2018
[24]

Lashkari, N., Dymarsky, A., & Liu, H. ”Eigenstate thermalization hypothesis in conformal field theory,” Journal of Statistical Mechanics: Theory and Experiment 2018, 033101 (2018)

View on Pith 2018
[25]

Burke, P. C., Nakerst, G., & Haque, M. ”Assigning temperatures to eigenstates,” Phys. Rev. E 107, 024102 (2023) [arXiv:2111.05083]

View on Pith Pith review arXiv 2023
[26]

Casini, H., & Huerta, M. ”Reduced density matrix and internal dynamics for multicomponent regions,” Classical and Quantum Gravity 26, 185005 (2009) [arXiv:0903.5284]

View on Pith Pith review arXiv 2009
[27]

Lashkari, N. ”Modular Hamiltonian for Excited States in Conformal Field Theory,” Phys. Rev. Lett. 117, 041601 (2016) [arXiv:1508.03506]

View on Pith Pith review arXiv 2016
[28]

Arias, R., Botta-Cantcheff, M., Martinez, P. J., & Zarate, J. F. ”Modular Hamiltonian for holographic excited states,” Phys. Rev. D 102, 026021 (2020) [arXiv:2002.04637]

View on Pith Pith review arXiv 2020
[29]

Peschel, I. ”Letter to the Editor: Calculation of reduced density matrices from correla- tion functions,” Journal of Physics A Mathematical General 36, L205 (2003) [arXiv:cond- mat/0212631]

View on Pith 2003
[31]

G. Levine and G. Oliveira, in preparation

View on Pith
[32]

Essler, F. H. L., & Fagotti, M. ”Quench dynamics and relaxation in isolated integrable quan- tum spin chains,” Journal of Statistical Mechanics: Theory and Experiment 6, 064002 (2016) [arXiv:1603.06452]

View on Pith Pith review arXiv 2016
[33]

Chowdhury, D., Georges, A., Parcollet, O., & Sachdev, S. ”Sachdev-Ye-Kitaev models and beyond: Window into non-Fermi liquids,” Reviews of Modern Physics 94, 035004 (2022) [arXiv:2109.05037]

View on Pith Pith review arXiv 2022
[34]

See related work: Pourjafarabadi, M., Najafzadeh, H., Vaezi, M.-S., & Vaezi, A. ”Entangle- 25 ment Hamiltonian of Interacting Systems: Local Temperature Approximation and Beyond,” arXiv e-prints arXiv:2012.05248 (2020) [arXiv:2012.05248]. 26

View on Pith Pith review arXiv 2020

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [30]

Lashkari, N., Liu, H., & Rajagopal, S. ”Perturbation theory for the logarithm of a positive operator,” Journal of High Energy Physics 2023, 97 (2023)

View on Pith 2023

[2] [1]

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

View on Pith 1991

[3] [2]

Srednicki, M. ”Chaos and quantum thermalization,” Phys. Rev. E 50, 888 (1994) [arXiv:cond- mat/9403051]

View on Pith 1994

[4] [3]

Deutsch, J. M. ”Eigenstate thermalization hypothesis,” Reports on Progress in Physics 81, 082001 (2018) [arXiv:1805.01616]

View on Pith Pith review arXiv 2018

[5] [4]

D’Alessio, L., Kafri, Y., Polkovnikov, A., & Rigol, M. ”From quantum chaos and eigenstate thermalization to statistical mechanics and thermodynamics,” Advances in Physics 65, 239 (2016) [arXiv:1509.06411]. 23

View on Pith Pith review arXiv 2016

[6] [5]

Rigol, M., Dunjko, V., & Olshanii, M. ”Thermalization and its mechanism for generic isolated quantum systems,” Nature (London) 452, 854 (2008) [arXiv:0708.1324]

View on Pith Pith review arXiv 2008

[7] [6]

is not strictly satisfied. AllM A initial and final elements were present in the modular hamiltonian consistent 9 θ/2π h |100111⟩⟨011101| |101010⟩⟨010110| |001101⟩⟨100011| -0.4 -0.2 0.0 0.2 0.4 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 FIG. 3: Three representative two-body (interacting) matrix elements of the modular hamiltonian corresponding to the mixture (11) of...

View on Pith

[8] [7]

Popescu, S., Short, A. J., & Winter, A. ”Entanglement and the foundations of statistical mechanics,” Nature Physics 2, 754 (2006) [arXiv:quant-ph/0511225]

View on Pith Pith review arXiv 2006

[9] [8]

Goldstein, S., Lebowitz, J. L., Tumulka, R., & Zangh ` ı, N. ”Canonical Typicality,” Phys. Rev. Lett. 96, 050403 (2006) [arXiv:cond-mat/0511091]

View on Pith Pith review arXiv 2006

[10] [9]

Garrison, J. R., & Grover, T. ”Does a Single Eigenstate Encode the Full Hamiltonian?,” Physical Review X 8, 021026 (2018) [arXiv:1503.00729]

View on Pith Pith review arXiv 2018

[11] [10]

Lai, H.-H. & Yang, K. ”Entanglement entropy scaling laws and eigenstate typicality in free fermion systems,” Physical Review B 91, 081110 (2015) [arXiv:1409.1224]

View on Pith Pith review arXiv 2015

[12] [11]

Mark, D. K., Surace, F., Elben, A., et al. ”Maximum Entropy Principle in Deep Thermalization and in Hilbert-Space Ergodicity,” Physical Review X 14, 041051 (2024) [arXiv:2403.11970]

View on Pith Pith review arXiv 2024

[13] [12]

Lucas, M., Piroli, L., De Nardis, J., & De Luca, A. ”Generalized deep thermalization for free fermions,” Phys. Rev. A 107, 032215 (2023) [arXiv:2207.13628]

View on Pith Pith review arXiv 2023

[14] [13]

Tian, C., Yang, K., Fang, P., Zhou, H.-J., & Wang, J. ”Hidden thermal structure in Fock space,” Phys. Rev. E 98, 060103 (2018) [arXiv:1808.00784]

View on Pith Pith review arXiv 2018

[15] [14]

Lian, B. ”Conserved quantities from entanglement Hamiltonian,” Phys. Rev. B 105, 035106 (2022) [arXiv:2104.11753]

View on Pith Pith review arXiv 2022

[16] [15]

Brand˜ ao, F. G. S. L., Crosson, E., S ¸ahinoˇ glu, M. B., & Bowen, J. ”Quantum Error Correcting Codes in Eigenstates of Translation-Invariant Spin Chains,” Phys. Rev. Lett. 123, 110502 (2019) [arXiv:1710.04631]

View on Pith Pith review arXiv 2019

[17] [16]

Bao, N., & Cheng, N. ”Eigenstate thermalization hypothesis and approximate quantum error correction,” Journal of High Energy Physics 2019, 152 (2019) [arXiv:1906.03669]

View on Pith Pith review arXiv 2019

[18] [17]

Almheiri, A., Dong, X., & Harlow, D. ”Bulk locality and quantum error correction in AdS/CFT,” Journal of High Energy Physics 2015, 163 (2015) [arXiv:1411.7041]

View on Pith Pith review arXiv 2015

[19] [18]

Jafferis, D. L., Lewkowycz, A., Maldacena, J., & Suh, S. J. ”Relative entropy equals bulk relative entropy,” Journal of High Energy Physics 2016, 4 (2016) [arXiv:1512.06431]

View on Pith Pith review arXiv 2016

[20] [19]

Dong, X., Harlow, D., & Wall, A. C. ”Reconstruction of Bulk Operators within the Entanglement Wedge in Gauge-Gravity Duality,” Phys. Rev. Lett. 117, 021601 (2016) [arXiv:1601.05416]. 24

View on Pith Pith review arXiv 2016

[21] [20]

Faulkner, T., & Lewkowycz, A. ”Bulk locality from modular flow,” Journal of High Energy Physics 2017, 151 (2017) [arXiv:1704.05464]

View on Pith Pith review arXiv 2017

[22] [21]

Holzhey, C., Larsen, F., & Wilczek, F. ”Geometric and renormalized entropy in conformal field theory,” Nuclear Physics B 424, 443 (1994) [arXiv:hep-th/9403108]

View on Pith Pith review arXiv 1994

[23] [22]

Casini, H., Huerta, M., & Myers, R. C. ”Towards a derivation of holographic entanglement entropy,” Journal of High Energy Physics 2011, 36 (2011) [arXiv:1102.0440]

View on Pith Pith review arXiv 2011

[24] [23]

Dymarsky, A., Lashkari, N., & Liu, H. ”Subsystem eigenstate thermalization hypothesis,” Phys. Rev. E 97, 012140 (2018) [arXiv:1611.08764]

View on Pith Pith review arXiv 2018

[25] [24]

Lashkari, N., Dymarsky, A., & Liu, H. ”Eigenstate thermalization hypothesis in conformal field theory,” Journal of Statistical Mechanics: Theory and Experiment 2018, 033101 (2018)

View on Pith 2018

[26] [25]

Burke, P. C., Nakerst, G., & Haque, M. ”Assigning temperatures to eigenstates,” Phys. Rev. E 107, 024102 (2023) [arXiv:2111.05083]

View on Pith Pith review arXiv 2023

[27] [26]

Casini, H., & Huerta, M. ”Reduced density matrix and internal dynamics for multicomponent regions,” Classical and Quantum Gravity 26, 185005 (2009) [arXiv:0903.5284]

View on Pith Pith review arXiv 2009

[28] [27]

Lashkari, N. ”Modular Hamiltonian for Excited States in Conformal Field Theory,” Phys. Rev. Lett. 117, 041601 (2016) [arXiv:1508.03506]

View on Pith Pith review arXiv 2016

[29] [28]

Arias, R., Botta-Cantcheff, M., Martinez, P. J., & Zarate, J. F. ”Modular Hamiltonian for holographic excited states,” Phys. Rev. D 102, 026021 (2020) [arXiv:2002.04637]

View on Pith Pith review arXiv 2020

[30] [29]

Peschel, I. ”Letter to the Editor: Calculation of reduced density matrices from correla- tion functions,” Journal of Physics A Mathematical General 36, L205 (2003) [arXiv:cond- mat/0212631]

View on Pith 2003

[31] [31]

G. Levine and G. Oliveira, in preparation

View on Pith

[32] [32]

Essler, F. H. L., & Fagotti, M. ”Quench dynamics and relaxation in isolated integrable quan- tum spin chains,” Journal of Statistical Mechanics: Theory and Experiment 6, 064002 (2016) [arXiv:1603.06452]

View on Pith Pith review arXiv 2016

[33] [33]

Chowdhury, D., Georges, A., Parcollet, O., & Sachdev, S. ”Sachdev-Ye-Kitaev models and beyond: Window into non-Fermi liquids,” Reviews of Modern Physics 94, 035004 (2022) [arXiv:2109.05037]

View on Pith Pith review arXiv 2022

[34] [34]

See related work: Pourjafarabadi, M., Najafzadeh, H., Vaezi, M.-S., & Vaezi, A. ”Entangle- 25 ment Hamiltonian of Interacting Systems: Local Temperature Approximation and Beyond,” arXiv e-prints arXiv:2012.05248 (2020) [arXiv:2012.05248]. 26

View on Pith Pith review arXiv 2020