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Page Curve and Entanglement Dynamics in an Interacting Fermionic Chain

T0 review · 5 major / 5 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read For an interacting fermionic chain, entanglement entropy follows a Page curve, with a min-entropy kink marking a quantum phase transition in the entanglement Hamiltonian before the Page time.

desk verdict Solid MPS numerics for the interacting Page-curve model, but the headline thermodynamic-limit phase diagram is an artifact of a misread regression: the intercept used is not the limit of the critical time. read the letter →

arxiv 2502.03563 v3 pith:I3VAU44K submitted 2025-02-05 quant-ph cond-mat.str-elhep-th

Rishabh Jha , Salvatore R. Manmana , Stefan Kehrein This is my paper · ORCID
classification quant-phcond-mat.str-elhep-th
keywords Pagecurveentanglemententropymin-entropyHamiltonianquantumphasetransitionRényimatrixproductstatesfermionicchain
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

This paper studies a one-dimensional chain of spinless fermions split into a filled interacting system and a large empty non-interacting environment, connected by a weak hopping link. It claims that after the quench the von Neumann entanglement entropy between system and environment follows a Page curve—linear growth, a peak at a Page time $t_{\text{Page}}$, then a bend down toward zero—even when interactions are present. The central new result is that the infinite-Rényi (min-)entropy develops a non-analyticity at a critical time $t_c < t_{\text{Page}}$, coinciding with a level crossing in the entanglement Hamiltonian that signals a quantum phase transition in time. Extrapolating to the thermodynamic limit, the critical time stays finite for weak interactions at weak system-environment coupling, but shrinks to zero for stronger interactions or stronger coupling, so the transition becomes instantaneous. If correct, Page-like bending is tied to a critical point in the modular Hamiltonian, and conserved-charge models should show the same temporal transition.

What carries the argument

The central object is the time-dependent entanglement (modular) Hamiltonian $H_E(t)=-\ln\rho_s(t)$, whose eigenvalues are $\epsilon_i=-\ln\lambda_i$ with $\lambda_i$ the Schmidt coefficients of the bipartite state; the Rényi index $n$ acts as inverse fictitious temperature, so the min-entropy $S_{\min}=-\ln\lambda_{\max}$ is the ground-state energy and the first excited state is $-\ln\lambda_2$. The argument runs on a level crossing of these two eigenvalues: because $H_E$ conserves particle number, the ground state sits in the $M$-particle sector before $t_c$ and in the $M-1$-particle sector after, and the crossing produces the min-entropy kink. A finite-size scaling ansatz $t_c\sim M^{-1/c}$ with $a,b,c\approx 1$ and an order-parameter fit $(1-m(t)/M)=A(t-t_c)^\beta$ with $\beta\approx 1/2$ carries the thermodynamic-limit extrapolation.

What would settle it

Compute the first min-entropy kink for $M=8$, $9$, and $10$ at the same couplings and check whether the critical decayed fraction continues to fall on the same straight line in $1/M$; any curvature or an intercept that moves outside the published error bars would overturn the thermodynamic-limit conclusion. A cheaper check is to compare the $M=7$ prediction with a single larger tensor-network run at $V/t_h=0.8$ and $g=0.25$, where the extrapolated critical fraction should stay at $\sim 0.21$.

Watch

Extended reading notes

Core claim

For a spinless fermionic chain with nearest-neighbor interaction $V$ in the system, starting from a fully filled system and an empty environment, the paper finds a Page curve in the von Neumann entropy for every interaction strength tested. It identifies the mechanism as a conserved-charge level crossing: the entanglement Hamiltonian $H_E=-\ln\rho_s$ conserves particle number, and as one particle leaves the system the ground state switches from the $M$-particle sector to the $M-1$-particle sector. At the crossing time $t_c$, the largest and second-largest Schmidt coefficients become equal, so the min-entropy $S_{\min}=-\ln\lambda_{\max}$ shows a kink. The paper interprets this as a continuous quantum phase transition in the entanglement Hamiltonian, with the decayed particle fraction as order parameter and critical exponent $\beta\approx 1/2$ in the same universality class as the non-interacting chain. A scaling analysis of system sizes $M=3$ through $7$ then gives the thermodynamic-limit statement: for weak interaction $V/t_h<1$ and weak system-environment coupling the nonanalyticity survives at a nonzero $t_c$, whereas for $V/t_h\gtrsim 1$ (or for stronger coupling) the extrapolated $t_c$ vanishes, which the paper explains by a domain-wall picture in which strong interactions freeze the system after a single particle decays.

Load-bearing premise

The thermodynamic-limit results assume that the critical decayed fraction and the critical time are exactly linear in $1/M$ for $M=3$ to $7$, so a linear regression to $M\to\infty$ is valid, but the paper offers neither data beyond $M=7$ nor an analytic finite-size formula to justify that linearity.

Editorial extensions

If this is right

  • If the central claim is correct, the bending down of the Page curve is not a smooth saturation but the finite-size echo of a genuine critical point in the entanglement Hamiltonian at $t_c<t_{\text{Page}}$.
  • All Rényi entropies inherit the Page-curve shape, but only the min-entropy is sharp; finite-$n$ Rényi entropies show a crossover, not a kink, because the fictitious temperature is nonzero.
  • The temporal quantum phase transition persists when interactions are switched on in the tested range, with the order-parameter exponent $\beta\approx 1/2$ unchanged, so the interacting chain belongs to the same universality class as the free chain.
  • In the thermodynamic limit the critical time is nonzero for weak interactions at weak coupling, and vanishes for intermediate-to-strong interactions or stronger coupling, so the entanglement transition becomes effectively instantaneous.

Reading between the lines

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

  • One can test the conserved-charge mechanism directly by exact diagonalization of a small two-dimensional charge-conserving cluster: if a min-entropy kink appears without fine-tuning, the phenomenon is not special to one dimension.
  • In an ultracold-atom implementation, the strong-interaction freeze predicts both a nearly vanishing particle leakage at large system size and a Schmidt gap that closes on a time scale $\sim 1/V$; observing either would confirm the thermodynamic-limit scenario.
  • The strong dependence on the system-environment coupling $g$ suggests that $g$ is a second control axis: tuning $g$ at fixed $V$ might turn the continuous temporal transition first-order, which the paper explicitly leaves open.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

5 major / 5 minor

Summary. The paper studies the quench dynamics of a one-dimensional spinless fermionic chain in which a small interacting 'system' (M sites, filled initially) is coupled by a weak hopping g to a large non-interacting 'environment' (N sites, empty initially). Using matrix-product-state time evolution (TDVP, bond dimension up to 5000, discarded weight 1e-12) for total L=50 and system sizes M=3,...,7, the authors report: (i) Page-curve-like von Neumann entanglement entropy that peaks at a Page time and then decays; (ii) a non-analyticity in the min-entropy at a critical time tc before the Page time, attributed to a level crossing in the associated entanglement Hamiltonian; (iii) finite-size scaling collapses with exponents a,b,c ≈ 1 and an order-parameter exponent β ≈ 0.5 across interaction strengths, matching the free-case universality; and (iv) a thermodynamic-limit extrapolation claiming that a non-zero critical time persists for weak-to-intermediate interactions and vanishes (instantaneous dynamics) for stronger interactions, with an interpretation based on hole domain walls in the strongly interacting regime. The paper extends the solvable free-fermion model of Ref. [27] to interacting systems and is accompanied by publicly deposited data and simulation code.

Significance. If the central thermodynamic-limit claim were sound, the paper would provide a valuable extension of Page-curve entanglement dynamics to a non-integrable interacting chain, with a robust temporal 'quantum phase transition' in the entanglement Hamiltonian and a universal order-parameter exponent. The numerical work is careful and reproducible: the data are deposited on Zenodo, the MPS parameters are well documented, and the finite-size collapse and β ≈ 0.5 extraction are plausible and internally consistent. However, the headline thermodynamic-limit phase diagram is based on a regression analysis whose interpretation is mathematically incorrect (as detailed in the major comments), and the 'one-to-one correspondence' between the min-entropy non-analyticity and the entanglement-Hamiltonian level crossing is definitional rather than an independent finding. The finite-size data, the Page curve, and the critical exponents retain value, but the abstract's main physical claim is not supported by the analysis as presented.

major comments (5)
  1. [Appendix B, Table IV and Section VI] The regression of Y = tc/M against X = 1/M does not have the Y-intercept as the thermodynamic-limit value of tc; the intercept is lim_{M→∞} tc/M, i.e., the limit of the intensive variable. A finite TL critical time would appear in the slope (since tc/M ≈ tc∞/M for large M), not in the intercept. The reported slopes are never discussed. For example, for V=0, g=0.5, Table IV gives intercept −0.0005 and slope 2.2669, implying tc ≈ 2.27 − 0.0005 M, which becomes unphysical for large M; in no case do the fits imply tc → 0. Consequently, the abstract's claim of 'non-zero critical time in the thermodynamic limit only for weak to intermediate interaction strengths, while the dynamics becomes instantaneous for interactions large enough' is not supported by the presented regression. The authors should instead regress tc against 1/M (or analyze tc/M with the correct interpretation) and reconcile the result with the scaling ansatz of Eq. (9).
  2. [Section V, Eq. (9) and Section VI] The scaling ansatz tc ∼ M^{-1/c} assumes from the outset that tc vanishes in the thermodynamic limit for g=0.5 — precisely the result that the regression analysis in Section VI is meant to establish. Using Eq. (9) to justify the linear extrapolation is therefore circular. The finite-size collapse in Fig. 8 and the exponents in Table I only demonstrate that the data for M=3,...,7 obey a scaling form with the finite-size tc(M); they do not provide independent evidence about the M→∞ behavior of tc. The extrapolation procedure should be justified separately from the collapse, or the claim about the thermodynamic limit should be substantially weakened.
  3. [Section III and Appendix A, Eq. (A5)] Because Smin ≡ −ln λmax, as stated in Eq. (A5), a non-analyticity in Smin is by definition the event at which the largest Schmidt coefficient is overtaken by the second-largest one, i.e., a crossing of the ground and first-excited eigenvalues of the entanglement Hamiltonian. The 'one-to-one correspondence' emphasized in Fig. 6 and Section III is therefore a definitional identity, not an empirically discovered relation. The physical content lies in the existence, universality, and scaling of the crossing, not in the 'quantum phase transition' label itself. The manuscript should explicitly separate the definitional relation from the quantitative results so that the interpretive claim is not presented as an independent finding.
  4. [Abstract vs Section VI and Tables III–IV] The abstract states a generic dichotomy between weak-to-intermediate interactions (non-zero TL critical time) and large interactions (instantaneous dynamics). Section VI, however, finds that for the standard coupling g=0.5 the thermodynamic critical value of the decayed fraction is zero for all interaction strengths, and a non-vanishing TL value appears only for the weaker coupling g=0.25 (and only for V≲1 according to the text). The abstract should either be qualified to make the explicit g-dependence clear, or the g=0.5 data should be re-analyzed to show in what precise sense the claimed dichotomy holds. As written, the abstract overgeneralizes the presented simulation results.
  5. [Section II and Section VI] The thermodynamic extrapolation is performed over M=3,...,7 at fixed total size L=50, so the environment size N=50−M changes from 47 to 43 as M increases. This is a narrow system-size range, and M→∞ at fixed L is not the same as the thermodynamic limit of a small system coupled to an infinite environment. The authors state that time scales before particles reach the right edge are considered, but for M=7 and tmax=40 with group velocity 2, the right edge would be reached at t≈21.5, so this statement is not generally true. The analysis should specify the time window used, and ideally include a check that the first non-analyticity and its scaling are insensitive to N or to the open boundary for at least one parameter set.
minor comments (5)
  1. [Section VI, Eq. (13)] Clarify the meaning of λ(t) in Eq. (13): the expression uses weights (1−λ)^2 and λ^2, which are the squares of the Schmidt coefficients if λ is the probability of the second sector; the text calls 1−λ(t) and λ(t) the 'Schmidt coefficients', which is inconsistent with their sum being 1. The derivation of the ansatz from the Schmidt decomposition should be made explicit.
  2. [Appendix B, Table IV caption] The caption states that the 'Y−intercept is the extrapolation of the critical time in the thermodynamic limit,' but since Y is tc/M, this statement is incorrect for tc itself; it is the limit of tc/M. This wording is likely the source of the interpretive error in Section VI and should be corrected.
  3. [Fig. 5 caption] The phrase 'plotted agains the decayed fraction' contains a typo ('agains'), and Appendix B uses 'interacting strengths' where 'interaction strengths' is intended.
  4. [Section II] The statement that the environment is 'much larger' than the system is an overstatement for L=50 and M=7 (N=43); recommend stating the actual ratios or using 'larger'.
  5. [Section VI, after Table III] The sentence 'This is also true at V = 0, in agreement with Ref. [27]' is ambiguous: Table III shows a clearly non-zero intercept for g=0.25, V=0.0, so the sentence needs to state explicitly which coupling and which quantity (decayed fraction or critical time) is meant.

Circularity Check

1 steps flagged · score 4.0 of 10

The min-entropy kink and the entanglement-Hamiltonian level crossing are the same event by the paper's own Eq. (A5), so the 'temporal quantum phase transition' is partly a relabeling; the independent MPS scaling results and free-case benchmark keep the core derivation non-circular.

  1. self definitional [Sec. III 'Quantum phase transition in the entanglement Hamiltonian' and Appendix A, Eq. (A5)]
    "We find a one-to-one correspondence between Figs. 5 and 6 where non-analyticity in the min-entropy appears exactly when there is a crossing of the ground state with the first excited state of the entanglement Hamiltonian. ... ϵGS = − ln λmax = Sn→∞ = Smin. (A5)"

    Appendix A defines Smin ≡ lim_{n→∞} S_n = −ln λ_max and simultaneously defines the ground-state eigenvalue of H_E = −ln ρ_s as ϵ_GS = −ln λ_max. Therefore Smin is the ground-state energy of H_E by construction (Eq. A5). A kink in Smin is literally the event where the largest and second-largest Schmidt coefficients become degenerate, i.e., the 'ground-state/first-excited-state crossing' of H_E. The claimed one-to-one correspondence is thus an identity following from Eqs. (4), (A4) and (A5), not an independent empirical finding. The independent content is the numerical observation that such crossings do occur and the thermodynamic-limit scaling of their position; the label 'quantum phase transition in the entanglement Hamiltonian' is a repackaging of this Schmidt-crossing event.

full rationale

The paper's core numerical program is self-contained: time-dependent MPS data produce the Page curve, the min-entropy kink, the Schmidt spectra, and a finite-size collapse with exponents a, b, c near unity and order-parameter exponent β≈0.5. These results do not reduce to fits of the quantities they claim to predict. The main genuine circular step is the interpretation layer: because Eq. (A5) defines the min-entropy as the ground-state eigenvalue of H_E, the 'one-to-one correspondence' between the min-entropy non-analyticity and the ground-state/first-excited-state crossing is an identity, not a discovery. This does not destroy the paper's independent content, because the existence, location, and scaling of the crossings are numerical findings; however, the 'quantum phase transition' language is partly a renaming of the largest-Schmidt-coefficient crossover. The heavy reliance on Ref. [27], by one of the present authors, is not penalized as load-bearing self-citation: that free-fermion solution is exactly solvable, parameter-free, and used as an external benchmark, so it qualifies as independent evidence under the rules. The thermodynamic-limit regression in Table IV regresses the normalized quantity tc/M against 1/M; this is a valid extrapolation for the normalized time used throughout the paper, though the abstract's unqualified phrase 'non-zero critical time' is ambiguous and the small intercepts for g=0.5 would also be consistent with zero at the paper's own conservative 3σ error estimate. That is a clarity/correctness concern, not a circular reduction. Overall, the central phase diagram claim retains independent numerical support, so the circularity score is moderate rather than severe.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The paper's claims rest on the accuracy of MPS/TDVP numerics, the correctness of the free-case solution in the self-cited Ref. [27], and the validity of linear-in-1/M extrapolation from M=3..7 to the thermodynamic limit. The exponents a, b, c and beta are fitted outputs from the numerical data; they are listed as free parameters because the universality claim depends on their fitted values. No new physical entities are introduced. The 'quantum phase transition' interpretation is a mapping from a level crossing in the entanglement spectrum (which is definitional for the min-entropy kink, see Eq. A5) to the language of continuous phase transitions.

free parameters (4)
  • scaling ansatz exponent a = 0.97-0.99 (Table I)
    Fitted by finite-size data collapse of min-entropy near tc in Section V; presented as output, not input, but the universality claim depends on this fitted value.
  • scaling ansatz exponent b = 0.98-0.99 (Table I)
    Fitted by finite-size data collapse of the order parameter in Section V; used to claim universal scaling.
  • scaling ansatz exponent c = 1.01-1.02 (Table I)
    Fitted by finite-size data collapse and used in the tc ~ M^{-1/c} relation of Section V.
  • critical exponent beta = 0.51-0.54 (Table II)
    Extracted from power-law fit of the order parameter versus time near tc in Section V; used to claim the transition is in the free-case universality class.
assumptions (6)
  • domain assumption MPS/TDVP simulations with bond dimension chi=5000 and discarded weight 10^-12 accurately capture the quench dynamics for the studied times and sizes.
    Section IIA states the MPS parameters but provides no convergence check against exact results for the interacting case.
  • domain assumption The free-case exact solution and its thermodynamic-limit results from Ref. [27] are correct and carry over to interacting systems for the non-analyticity structure.
    Ref. [27] is an analytically solvable free-fermion model; the paper uses its prediction beta=0.5 and the expectation that later non-analyticities smear out in the TL. One of the present authors is the sole author of Ref. [27].
  • ad hoc to paper The first non-analyticity in min-entropy persists in the thermodynamic limit and the later ones smear out.
    Section III: 'We refrain from studying the later non-analyticities, since we expect that, similar to the findings in the free case [27], in the TL they will smear out, so that only the first non-analyticity will persist.'
  • ad hoc to paper The critical decayed fraction and critical time are linear functions of 1/M, so linear regression over M=3..7 extrapolates to the thermodynamic limit.
    Section VI and Fig. 10 use a linear fit of 1-m(tc)/M versus 1/M with five data points to obtain the TL intercept in Tables III and IV.
  • domain assumption The level crossing of the two largest Schmidt coefficients in the finite system becomes a genuine non-analyticity (quantum phase transition) in the thermodynamic limit.
    Section IV asserts conserved-charge sectors of HE lead to level crossings that become a depletion transition in the TL; no rigorous proof is given for the interacting case.
  • ad hoc to paper For strong interactions, only the largest and second-largest Schmidt coefficients matter, so the two-state ansatz Eq. (13) describes the dynamics.
    Appendix C shows numerical Schmidt coefficients for V=3.0 and V=5.0 supporting this, but the ansatz is not derived from the Hamiltonian.

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Pith. "Pith review of Page Curve and Entanglement Dynamics in an Interacting Fermionic Chain." pith.science (2026). https://pith.science/paper/I3VAU44K

@misc{pith2026250203563,
  author       = {Pith},
  title        = {Pith review of: Page Curve and Entanglement Dynamics in an Interacting Fermionic Chain},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/I3VAU44K}},
  note         = {Machine review of arXiv:2502.03563}
}
abstract

Generic non-equilibrium many-body systems display a linear growth of bipartite entanglement entropy in time, followed by a volume law saturation. In stark contrast, the Page curve dynamics of black hole physics shows that the entropy peaks at the Page time $t_{\text{Page}}$ and then decreases to zero. Here, we investigate such Page-like behavior of the von Neumann entropy in a model of strongly correlated spinless fermions in a typical system-environment setup, and characterize the properties of the Page curve dynamics in the presence of interactions using numerically exact matrix product states methods. The two phases of growth, namely the linear growth and the bending down, are shown to be separated by a non-analyticity in the min-entropy before $t_{\text{Page}}$, which separates two different quantum phases, realized as the respective ground states of the corresponding entanglement (or equivalently, modular) Hamiltonian. We confirm and generalize, by introducing interactions, the findings of \href{https://journals.aps.org/prb/abstract/10.1103/PhysRevB.109.224308}{Phys. Rev. B 109, 224308 (2024)} for a free spinless fermionic chain where the corresponding entanglement Hamiltonian undergoes a quantum phase transition at the point of non-analyticity. However, in the presence of interactions, a scaling analysis gives a non-zero critical time for the non-analyticity in the thermodynamic limit only for weak to intermediate interaction strengths, while the dynamics leading to the non-analyticity becomes \textit{instantaneous} for interactions large enough. We present a physical picture explaining these findings.

Figures

Figures reproduced from arXiv: 2502.03563 by the authors.

Figure 1
Figure 1. Illustration of the black hole information para [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Sketch of the setup studied in this paper, see [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. MPS results for the time evolution of the [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (20 more)
Figure 4
Figure 4. Figure 4: Same as in Fig [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 7
Figure 7. Figure 7: Schematic depicting the quantum phase transi [PITH_FULL_IMAGE:figures/full_fig_p007_7.png]
Figure 6
Figure 6. Figure 6: (a) MPS results for the time evolution of the [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]
Figure 8
Figure 8. Figure 8: Top row: raw data in the vicinity of the critical point where top left is for the decayed fraction of particles [PITH_FULL_IMAGE:figures/full_fig_p009_8.png]
Figure 9
Figure 9. Figure 9: We perform the scaling relation in Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p010_9.png]
Figure 10
Figure 10. Figure 10: Linear regression plot where we extrapolate [PITH_FULL_IMAGE:figures/full_fig_p011_10.png]
Figure 11
Figure 11. Figure 11: When a particle leaks to the environment, [PITH_FULL_IMAGE:figures/full_fig_p012_11.png]
Figure 13
Figure 13. Figure 13: Von Neumann bipartite entanglement entropy [PITH_FULL_IMAGE:figures/full_fig_p016_13.png]
Figure 15
Figure 15. Figure 15: Min-entropy density is plotted against the [PITH_FULL_IMAGE:figures/full_fig_p017_15.png]
Figure 16
Figure 16. Figure 16: The same behavior as plotted in Fig [PITH_FULL_IMAGE:figures/full_fig_p018_16.png]
Figure 17
Figure 17. Figure 17: We plot the crossing between the ground state [PITH_FULL_IMAGE:figures/full_fig_p019_17.png]
Figure 18
Figure 18. Figure 18: The same behavior as plotted in Fig [PITH_FULL_IMAGE:figures/full_fig_p020_18.png]
Figure 19
Figure 19. Figure 19: We plot the quantum phase transition that [PITH_FULL_IMAGE:figures/full_fig_p021_19.png]
Figure 20
Figure 20. Figure 20: The same behavior as plotted in Fig [PITH_FULL_IMAGE:figures/full_fig_p022_20.png]
Figure 21
Figure 21. Figure 21: Linear regression plot where we extrapolate [PITH_FULL_IMAGE:figures/full_fig_p023_21.png]
Figure 22
Figure 22. Figure 22: Results for strongly interacting cases for (a) V=3.0 and (b) V=5.0 where other parameter values are [PITH_FULL_IMAGE:figures/full_fig_p024_22.png]
Figure 23
Figure 23. Figure 23: The corresponding Schmidt coefficients for the graphs shown in Fig. [PITH_FULL_IMAGE:figures/full_fig_p024_23.png]
Figure 24
Figure 24. Figure 24: Parameters: V = 0.8, g = 0.5, ts = te = 1.0, L = 50 and M = {3, 4, 5, 6, 7} [PITH_FULL_IMAGE:figures/full_fig_p025_24.png]
Figure 25
Figure 25. Figure 25: Parameters: V = 0.4, g = 0.5, ts = te = 1.0, L = 50 and M = {3, 4, 5, 6, 7} [PITH_FULL_IMAGE:figures/full_fig_p025_25.png]
Figure 26
Figure 26. Figure 26: Parameters: V = 0.0, g = 0.5, ts = te = 1.0, L = 50 and M = {3, 4, 5, 6, 7}. This is also used for benchmarking as the free case is solvable analytically in the thermodynamic limit [27] [PITH_FULL_IMAGE:figures/full_fig_p026_26.png]

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