Entanglement of Inhomogeneous Free Bosons and Orthogonal Polynomials

David Raveh; Eric Ragoucy; Gilles Parez; Luc Vinet; Pierre-Antoine Bernard; Rafael I. Nepomechie

REVIEW 3 major objections 5 minor 64 references

Entanglement of Inhomogeneous Free Bosons and Orthogonal Polynomials

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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

Pith's one-line read The leading ground-state entanglement entropy of an inhomogeneous free-boson chain is governed by the width of the locally massless region around the potential's minimum, with coefficient $a^*/6$ where $a^*$ is read off from the…

desk verdict Bosonic extension of the orthogonal-polynomial entanglement method works in the solvable examples, but the general scaling law is an acknowledged ansatz and the interpolating model's solvability is not established. read the letter →

arxiv 2505.15610 v2 pith:Z2ER5XVK submitted 2025-05-21 cond-mat.stat-mech hep-thmath-phmath.MPquant-ph

Pierre-Antoine Bernard , Rafael I. Nepomechie , Gilles Parez , Eric Ragoucy , David Raveh , Luc Vinet This is my paper · ORCID

classification cond-mat.stat-mechhep-thmath-phmath.MPquant-ph MSC 81P4033C4582B20

keywords inhomogeneousfreebosonsentanglemententropyorthogonalpolynomialsAskeyschemeeffectivecentralchargeharmonicchainmasslessregionKrawtchouk

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper establishes that for one-dimensional inhomogeneous free-boson chains, the leading ground-state entanglement entropy is controlled entirely by how the local potential $V_x$ vanishes near its minimum. It defines the exponent $a^*$ as the largest scaling window $N^{a^*}$ around the minimum in which the potential tends to zero, and argues that entanglement grows as $S_A \sim (a^*/6)\log N$: the same logarithmic law as a uniform massless chain, but with a reduced prefactor. This matters because it turns a many-body entanglement calculation into a short analytic check on the potential, and it predicts a continuum of effective central charges rather than only the CFT values. The paper verifies the claim in exactly solvable chains based on Krawtchouk polynomials ($a^*=1/2$), dual Hahn polynomials ($a^*=1$), and a one-parameter interpolating family with $c_{\rm eff}(b)=(2-b)/2$ for $0\le b<1$ and $c_{\rm eff}(b)=1/2$ for $b\ge1$.

What carries the argument

The central object is the inhomogeneous potential $V_x$ in the harmonic-chain Hamiltonian, together with the exponent $a^*$ extracted from the limit $V_{x_0\pm yN^a}\to0$. The argument works by identifying the largest region of effective size $N_{\rm eff}\sim N^{a^*}$ around the potential's minimum where the theory is locally massless, and then borrowing the homogeneous massless-boson result $S_A\sim\frac{1}{6}\log N_{\rm eff}$. On the exact-solvability side, the models are engineered so that the tridiagonal matrix $K$ in the Hamiltonian has eigenvectors given by orthogonal polynomials of the Askey scheme, the standard classification of hypergeometric orthogonal polynomials, with Krawtchouk and dual Hahn polynomials as examples; this supplies closed-form spectra and enables direct numerical checks through the correlation matrix.

What would settle it

Construct a smooth free-boson chain whose potential has a cubic minimum, for example $V_x=|x-x_0|^3/N$; the paper's definition gives $a^*=1/3$ and predicts $S_A\sim\frac{1}{18}\log N$ for the bipartition at $x_0$. Exact diagonalization of the correlation matrix at several $N$ would either confirm the slope $\frac{1}{18}$ or refute the central claim if the observed logarithmic coefficient differs.

Watch

Extended reading notes

Core claim

On its own terms, the paper's discovery is a scaling law for inhomogeneous free-boson chains. If the inhomogeneous potential $V_x$ has a zero at $x_0$ in the thermodynamic limit, and $a^*$ is the largest number such that $V_{x_0 \pm y N^a}\to 0$ for every $a<a^*$, then the entanglement entropy of the bipartition through $x_0$ behaves as $S_A \sim (a^*/6)\log N$. The prefactor $a^*$ is called the effective central charge $c_{\rm eff}$; it is not an actual conformal charge, but it plays that role in the logarithmic growth. For the Krawtchouk chain the potential is quadratic at its minimum and gives $a^*=1/2$; for the dual Hahn chain the potential is quadratic but its curvature decays with system size, giving $a^*=1$; and the interpolating family with $\gamma=pN^b$, $\delta=(1-p)N^b$ yields $c_{\rm eff}(b)=(2-b)/2$ for $0\le b<1$ and $c_{\rm eff}(b)=1/2$ for $b\ge1$. Exact diagonalization matches all these predictions.

Load-bearing premise

The paper assumes that the locally massless region around $x_0$ behaves like a uniform massless free-boson chain, whose logarithmic prefactor is known to be $1/6$, so the entropy there is $\frac{1}{6}\log N_{\rm eff}$ plus a constant.

Editorial extensions

If this is right

For any inhomogeneous free-boson chain with a smooth potential that vanishes at one point, the leading entanglement scaling is obtained from a short Taylor expansion of $V_x$ near that point, with no exact solvability required.
In the Krawtchouk chain the effective central charge is $c_{\rm eff}=1/2$, so the half-chain entropy grows only as $\frac{1}{12}\log N$.
In the dual Hahn chain $c_{\rm eff}=1$, so the entropy grows as $\frac{1}{6}\log N$, the same rate as the homogeneous massless chain.
The interpolating family realizes a continuum of values $c_{\rm eff}(b)$ between $1/2$ and $1$, plateauing at $1/2$ for $b\ge1$, and specific parameter choices reproduce the central charges of minimal models.
When the cut point $x$ lies farther than $N_{\rm eff}$ from the potential minimum, the entanglement entropy saturates to a constant in $N$; it grows logarithmically only inside the effective massless region.

Reading between the lines

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

The same potential-analyticity rule should apply to inhomogeneous free-boson chains that are not exactly solvable: one could test it on a generic smooth potential with a non-quadratic minimum and compare exact diagonalization to the predicted coefficient.
A natural extension is to ask whether the exponent $a^*$ also controls the leading entanglement scaling of inhomogeneous free-fermion chains, where curved-space CFT already exists; if so, the two species would share a common geometric criterion.
Because the paper identifies no underlying conformal symmetry, the effective central charge $c_{\rm eff}=a^*$ is a scaling coefficient rather than a CFT datum; computing subleading constants would clarify how much of the homogeneous CFT structure survives.
The plateau $c_{\rm eff}=1/2$ for all $b\ge1$ suggests that once the curvature of the potential minimum decays fast enough, the entropy saturates at the Krawtchouk value; testing finite-$b$ corrections could reveal whether the approach to saturation is monotone.

Share X Bluesky LinkedIn Reddit HN

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

Here is the short version: the Krawtchouk and dual Hahn bosonic chains are clean, new examples, and the numbers are convincing; the general scaling law (Eq. 25) is an unproved guess that the authors themselves concede in Sec. V, so treat the broad 'applicable to a broad class' claim with caution.

What is genuinely new: the orthogonal-polynomial approach, previously used for free fermions (Refs. [44,47]), is extended to free bosons, and the paper works out three families—Krawtchouk, dual Hahn, and an interpolating family—with exact diagonalization matching the predicted leading log coefficient. The potential-scaling heuristic (a* from Vx ~ 0 on scales N^a) is a nice practical shortcut, and in these models it is checked independently rather than fitted; only additive constants are fitted.

The soft spots are real but mostly self-acknowledged. Eq. (25) is asserted from a 'locally massless region behaves like the homogeneous c=1 chain' picture. The stress-test note is right that this ignores the inhomogeneous Jx; in the solvable models Jx is strongly N-dependent, and the paper's own Sec. V says a curved-space CFT for bosons would require Jx=1. So the a*/6 prefactor is an expectation, not a derivation. Because the paper says this explicitly, I do not see it as a deception—just an overbroad abstract.

The weakest section is the interpolating family (Sec. IV.D). Modifying the dual Hahn potential by N^{-b} and scaling gamma, delta with N changes the tridiagonal K, and the authors do not show that the resulting model remains exactly solvable; they give no eigenvectors or eigenvalues. The numerics match the predicted ceff(b), but the model's status as an exactly solvable point is unclear.

The citation pattern is fine: heavy self-citation but the prior fermionic papers are genuinely the direct predecessors. Nothing circular in the leading coefficient—it comes from the potential, not from the entropy data.

Bottom line: this is a useful, honest contribution to the entanglement-in-solvable-lattices subfield. It deserves a serious referee, but I would ask the referee to push for a sharper statement of what is proven versus conjectured, and to clarify the interpolating model.

Referee Report

3 major / 5 minor

Summary. The paper studies ground-state entanglement entropy in one-dimensional inhomogeneous free-boson chains. It proposes a general method: for a smooth potential V_x with a minimum at x0, one defines a* as the largest scaling exponent such that V_{x0 ± y N^a} tends to zero for all a < a*, and predicts SA ~ (a*/6) log N. The method is illustrated on three exactly solvable families obtained from orthogonal polynomials of the Askey scheme: the Krawtchouk chain (a* = 1/2), the dual Hahn chain (a* = 1), and an interpolating family with a continuum of effective central charges ceff(b). In each case, the prediction is compared with exact numerical diagonalization using the correlation-matrix formalism, and the leading logarithmic coefficient is not fitted. The paper explicitly states in Sec. V that extending curved-space CFT to bosons is nontrivial and that consistency in the continuum limit requires Jx = 1.

Significance. If the proposed scaling law Eq. (25) were established, the paper would provide a simple and powerful recipe: the leading entanglement scaling of an inhomogeneous free-boson chain would be read directly from the analytic behavior of the potential. The paper's strengths are the exact solvability of the models, the explicit orthogonal-polynomial diagonalization, and the numerical verification of the leading coefficient for three families without fitting that coefficient. The interpolating family is a nice touch, producing a continuous ceff(b) that includes familiar rational values. However, the central relation Eq. (25) is an assumption rather than a derivation, and the paper's own conclusion acknowledges that the bosonic curved-space CFT machinery is not available. The general claim of applicability to arbitrary smooth inhomogeneities is therefore stronger than what is demonstrated.

major comments (3)

[Sec. IIIB, Eq. (25)] The central scaling law SA ~ (a*/6) log N is not derived from the Hamiltonian. The argument replaces the locally massless region, defined solely by V_{x0 ± y N^a} -> 0, with a homogeneous massless free-boson chain of length Neff and central charge 1, yielding the prefactor 1/6. However, the Hamiltonian (18) also contains Jx, and in the solvable examples Jx scales nontrivially with N (Krawtchouk: Jx ~ N; dual Hahn: Jx ~ N^2). The paper's Sec. V explicitly concedes that a curved-space CFT treatment for bosons is nontrivial and that consistency in the continuum limit requires Jx = 1. Thus Eq. (25) is an ansatz, not a consequence of the model. To justify the general method, the authors should either provide a derivation or controlled approximation from the Hamiltonian, or clearly reframe Eq. (25) as a conjecture and restrict the claims of generality accordingly.
[Sec. IVB, Eq. (30)] In the Krawtchouk case with mK = 1/sqrt(2), the single-particle spectrum is Lambda_k = k + 1/2, so the many-body gap is sqrt(1/2), independent of N. This is not a gapless system in the standard sense, yet the region where V_x vanishes is interpreted as "locally massless" and the massless c = 1 result is invoked to predict SA ~ (1/12) log N. The paper should explain how a gapped spectrum can nevertheless produce logarithmic entanglement growth and why the homogeneous massless prefactor 1/6 applies. Without such an explanation, the identification of ceff = a* with a central charge is not established.
[Sec. IVD, Eq. (44)] The expansion leading to the piecewise effective length Neff ~ N^{(2-b)/2} for b < 1 and Neff ~ N^{1/2} for b >= 1 is not shown. Since this interpolating family is the main evidence for the continuous ceff(b), the derivation of Eq. (44) should be verifiable from the definitions in Eqs. (35) and (42). In addition, the rescaling tild(V)_x = V_x N^{-b} with gamma = p N^b and delta = (1-p) N^b changes both V_x and J_x through Eq. (35), so the prediction again relies on the unproved prefactor in Eq. (25). At minimum, the explicit leading-order terms and the location of the crossover at b = 1 should be presented.

minor comments (5)

[Sec. IIIB, Eq. (24)] The definition of I = [0, a*) is circular because a* appears on both sides before the supremum is specified. It would be clearer to define a* = sup{ a : lim_{N->∞} V_{x0 ± y N^{a'}} = 0 for all a' < a }.
[Sec. IVB and IVC, Figs. 3 and 5] The captions should state unambiguously that only the additive constants (and not the logarithmic slopes) are obtained from the numerical fits, so that the reader can verify the leading coefficient is not a fitted parameter.
[Sec. IVD, text after Eq. (46)] The sentence beginning "for b = 1/2 q(q+1)" uses q both as an integer in the minimal-model context and as a parameter elsewhere; please use a different symbol (e.g., r) for the integer to avoid confusion.
[References] Reference [13] is listed as "P. Francesco" but the correct author is "P. Di Francesco"; please correct the citation.
[Sec. V, first paragraph] The conclusion states that the method "applies to arbitrary inhomogeneous models with smooth potential," which is stronger than what is demonstrated, especially given the caveat in the same section about the bosonic continuum limit. Please soften this claim to apply to the class of models where the assumptions behind Eq. (25) are justified.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the leading coefficient a*/6 is computed from the potential's large-N behavior and then verified against exact diagonalization; Eq. (25) is an acknowledged expectation, not a fitted or self-referential output.

full rationale

The paper's central scaling law, Eq. (25), is not derived by fitting: a* is defined from the potential in Eq. (24) and computed analytically from the expansions of V_x in Eqs. (32), (39), and (44) before the entropy is computed. The entanglement entropy is then obtained independently by exact diagonalization through the correlation-matrix formulas of Sec. II B. Only additive constants and subleading terms in the fits are numerically adjusted; the leading coefficient a*/6 is a genuine prediction. The use of Ref. [44] for the J_x and B_x parameters of the orthogonal-polynomial chains is a citation to prior exact results, and the fermionic results cited there do not determine the bosonic entropy scaling; the bosonic potential analysis and numerical entropy computation in this paper are new. Section V candidly states that a curved-space CFT treatment for bosons is nontrivial and would require J_x = 1 in the continuum limit; this is an acknowledged limitation of the argument, not a circular step. No fitted parameter is renamed as a prediction, no uniqueness theorem is imported from the authors' prior work to forbid alternatives, and no self-citation chain forces the result. I therefore find no circularity.

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

The central prediction uses no fitted leading coefficient; a* is read from potential expansions. The main unproved input is the ansatz that a locally massless region yields the homogeneous prefactor 1/6. Several additive constants are fitted to numerics, which is normal for subleading terms but means the perfect match plots are not parameter-free.

free parameters (6)

m_K (Krawtchouk mass) = 1/sqrt(2)
Chosen so that the Krawtchouk potential vanishes at x0 = pN in the large-N limit; this choice is required for the massless-region argument.
m_H (dual Hahn mass) = sqrt((gamma+delta)/2)
Chosen so that the dual Hahn potential vanishes at x0 = N*gamma/(gamma+delta); required for the massless-region argument.
s0 (homogeneous entropy offset) = -0.0276
Fitted to the homogeneous massless chain numerics in Sec. IIB and Fig. 1; not a prediction.
Krawtchouk log-fit constants = 0.016 (p=1/2), -0.008 (p=1/4)
Additive constants in Eq. (33) obtained by numerical fit in Fig. 3; only the coefficient 1/12 is predicted.
dual Hahn log-fit constants = 0.009 and -0.009
Additive constants in Eq. (40) obtained by numerical fit in Fig. 5; only the coefficient 1/6 is predicted.
interpolating model subleading constants = not specified
Fig. 7 states solid lines use Eq. (46) with additional constants and subleading terms obtained from numerical fit.

assumptions (4)

domain assumption The homogeneous massless free-boson chain obeys the Calabrese-Cardy formula SA ~ (c/6) log[...] with c = 1.
Used as the input for the homogeneous case in Sec. IIB and as the template for Eq. (25).
ad hoc to paper A region where Vx vanishes behaves as an effectively massless free boson with the same central charge c = 1, yielding SA ~ (1/6) log Neff.
Introduced around Eq. (25); no derivation is given, and Sec. V notes that a curved-space CFT treatment is nontrivial.
ad hoc to paper The scaling of the effective massless region is fully controlled by the asymptotic behavior of the potential Vx near its minimum; the inhomogeneous couplings Jx do not independently affect the leading log coefficient.
The method in Sec. III.B uses only Vx in Eq. (24); Jx enters only through the definition of Vx in Eq. (21), not through an independent mechanism.
standard math Krawtchouk and dual Hahn polynomials provide the eigenvectors of the corresponding tridiagonal matrices.
Taken from Refs. [44,48]; used in Secs. IV.B and IV.C.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Entanglement of Inhomogeneous Free Bosons and Orthogonal Polynomials." pith.science (2026). https://pith.science/paper/Z2ER5XVK

@misc{pith2026250515610,
  author       = {Pith},
  title        = {Pith review of: Entanglement of Inhomogeneous Free Bosons and Orthogonal Polynomials},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Z2ER5XVK}},
  note         = {Machine review of arXiv:2505.15610}
}

read the original abstract

In this paper, we investigate the ground-state entanglement entropy in inhomogeneous free-boson models in one spatial dimension. We develop a powerful method to extract the leading term in the entanglement scaling, based on the analytic properties of the inhomogeneous potential. This method is applicable to a broad class of models with smooth spatial inhomogeneities. As a case study, we apply this approach for a family of exactly-solvable models characterized by orthogonal polynomials of the Askey scheme, finding a perfect match between the numerical and analytical results.

Figures

Figures reproduced from arXiv: 2505.15610 by the authors.

**Figure 2.** FIG. 2. Illustration of a generic smooth inhomogeneous potential ∗ [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]

**Figure 5.** Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]

**Figure 6.** Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]

**Figure 7.** Figure 7: FIG. 7. Entanglement entropy of the dual Hahn chain with parameters [PITH_FULL_IMAGE:figures/full_fig_p011_7.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

64 extracted references · 34 canonical work pages

[1]

Entanglement in many-body systems,

L. Amico, R. Fazio, A. Osterloh, and V. Vedral, “Entanglement in many-body systems,”Rev. Mod. Phys.80, 517 (2008), arXiv:quant-ph/0703044

arXiv 2008
[2]

To numerically investigate the entanglement entropy, we use the machinery introduced in Sec

This scaling thus corresponds to an effective central chargeceff = 1/2. To numerically investigate the entanglement entropy, we use the machinery introduced in Sec. IIB. In particular, we employ Eqs. (15), (16) and (17) with the matricesK and U pertaining to the Krawtchouk case. We show the comparison between the prediction of Eq. (33) and the exact diago...

work page
[3]

Quantum entanglement in condensed matter systems,

N. Laflorencie, “Quantum entanglement in condensed matter systems,”Phys. Rept.646, 1 (2016),arXiv:1512.03388 [cond-mat.str-el]

arXiv 2016
[4]

Scaling of Entanglement close to a Quantum Phase Transitions

A. Osterloh, L. Amico, G. Falci, and R. Fazio, “Scaling of entanglement close to a quantum phase transitions,”Nature 416, 608 (2002),arXiv:quant-ph/0202029

work page Pith review arXiv 2002
[5]

Entanglement in quantum critical phenomena,

G. Vidal, J. I. Latorre, E. Rico, and A. Kitaev, “Entanglement in quantum critical phenomena,”Phys. Rev. Lett.90, 227902 (2003), arXiv:quant-ph/0211074

arXiv 2003
[6]

Topological entanglement entropy,

A. Kitaev and J. Preskill, “Topological entanglement entropy,”Phys. Rev. Lett.96, 110404 (2006), arXiv:hep-th/0510092

arXiv 2006
[7]

Detecting topological order in a ground state wave function,

M. Levin and X.-G. Wen, “Detecting topological order in a ground state wave function,”Phys. Rev. Lett.96, 110405 (2006), arXiv:cond-mat/0510613

arXiv 2006
[8]

Evolution of entanglement entropy in one-dimensional systems,

P. Calabrese and J. L. Cardy, “Evolution of entanglement entropy in one-dimensional systems,”J. Stat. Mech.P04010 (2005), arXiv:cond-mat/0503393

arXiv 2005

Show all 64 references

[9]

Entanglement and thermodynamics after a quantum quench in integrable systems,

V. Alba and P. Calabrese, “Entanglement and thermodynamics after a quantum quench in integrable systems,”Proc. Nat. Acad. Sci.114, 7947 (2017),arXiv:1608.00614 [cond-mat.stat-mech]

2017 arXiv
[10]

Entanglement spreading in non-equilibrium integrable systems,

P. Calabrese, “Entanglement spreading in non-equilibrium integrable systems,”SciPost Phys. Lect. Notes20 (2020), arXiv:2008.11080 [cond-mat.stat-mech]

2020 arXiv
[11]

An area law for one-dimensional quantum systems,

M. B. Hastings, “An area law for one-dimensional quantum systems,”J. Stat. Mech.P08024 (2007), arXiv:0705.2024 [quant-ph]

2007 arXiv
[12]

Area laws in quantum systems: mutual information and correlations,

M. M. Wolf, F. Verstraete, M. B. Hastings, and J. I. Cirac, “Area laws in quantum systems: mutual information and correlations,” Phys. Rev. Lett.100, 070502 (2008),arXiv:0704.3906 [quant-ph]

2008 arXiv
[13]

Colloquium: Area laws for the entanglement entropy,

J. Eisert, M. Cramer, and M. B. Plenio, “Colloquium: Area laws for the entanglement entropy,”Rev. Mod. Phys.82, 277 (2010), arXiv:0808.3773 [quant-ph]

2010 arXiv
[14]

Francesco, P

P. Francesco, P. Mathieu, and D. Sénéchal,Conformal field theory. Springer Science & Business Media, 2012

2012
[15]

Entanglement entropy and quantum field theory,

P. Calabrese and J. L. Cardy, “Entanglement entropy and quantum field theory,”J. Stat. Mech.P06002 (2004), arXiv:hep-th/0405152

2004 arXiv
[16]

Entanglement entropy and conformal field theory,

P. Calabrese and J. Cardy, “Entanglement entropy and conformal field theory,”J. Phys. A42, 504005 (2009), arXiv:0905.4013 [cond-mat.stat-mech] . 13

2009 arXiv
[17]

Conformal field theory for inhomogeneous one-dimensional quantum systems: the example of non-interacting Fermi gases,

J. Dubail, J.-M. Stéphan, J. Viti, and P. Calabrese, “Conformal field theory for inhomogeneous one-dimensional quantum systems: the example of non-interacting Fermi gases,”SciPost Phys.2, 002 (2017),arXiv:1606.04401 [cond-mat.str-el]

2017 arXiv
[18]

Calculation of reduced density matrices from correlation functions,

I. Peschel, “Calculation of reduced density matrices from correlation functions,”J. Phys. A: Math. Gen.36, L205 (2003), arXiv:cond-mat/0212631

2003 arXiv
[19]

Reduced density matrices and entanglement entropy in free lattice models,

I. Peschel and V. Eisler, “Reduced density matrices and entanglement entropy in free lattice models,”J. Phys. A: Math. Theor.42, 504003 (2009),arXiv:0906.1663 [cond-mat.stat-mech]

2009 arXiv
[20]

Entanglement entropy in free quantum field theory,

H. Casini and M. Huerta, “Entanglement entropy in free quantum field theory,”J. Phys. A: Math. Theor.42, 504007 (2009), arXiv:0905.2562 [hep-th]

2009 arXiv
[21]

Entanglement entropy with interface defects,

I. Peschel, “Entanglement entropy with interface defects,”J. Phys. A: Math. Gen.38, 4327 (2005), arXiv:cond-mat/0502034 [cond-mat.stat-mech]

2005 arXiv
[22]

Evolution of entanglement entropy following a quantum quench: Analytic results for the XY chain in a transverse magnetic field,

M. Fagotti and P. Calabrese, “Evolution of entanglement entropy following a quantum quench: Analytic results for the XY chain in a transverse magnetic field,”Phys. Rev. A78, 010306 (2008),arXiv:0804.3559 [cond-mat.stat-mech]

2008 arXiv
[23]

Analytical results for the entanglement dynamics of disjoint blocks in the XY spin chain,

G. Parez and R. Bonsignori, “Analytical results for the entanglement dynamics of disjoint blocks in the XY spin chain,” J. Phys. A: Math. Theor.55, 505005 (2022),arXiv:2210.03637 [cond-mat.stat-mech]

2022 arXiv
[24]

Boundary-corner entanglement for free bosons,

C. Berthiere, “Boundary-corner entanglement for free bosons,”Phys. Rev. B99, 165113 (2019),arXiv:1811.12875 [cond-mat.str-el]

2019 arXiv
[25]

Symmetry resolved entanglement in two-dimensional systems via dimensional reduction,

S. Murciano, P. Ruggiero, and P. Calabrese, “Symmetry resolved entanglement in two-dimensional systems via dimensional reduction,” J. Stat. Mech.083102 (2020), arXiv:2003.11453 [cond-mat.stat-mech]

2020 arXiv
[26]

Entanglement of inhomogeneous free fermions on hyperplane lattices,

P.-A. Bernard, N. Crampé, R. I. Nepomechie, G. Parez, L. Poulain d’Andecy, and L. Vinet, “Entanglement of inhomogeneous free fermions on hyperplane lattices,”Nucl. Phys. B984, 115975 (2022),arXiv:2206.06509 [cond-mat.stat-mech]

2022 arXiv
[27]

Entanglement negativity between separated regions in quantum critical systems,

G. Parez and W. Witczak-Krempa, “Entanglement negativity between separated regions in quantum critical systems,” Phys. Rev. Res.6, 023125 (2024),arXiv:2310.15273 [cond-mat.str-el]

2024 arXiv
[28]

Symmetry resolved entanglement in free fermionic systems,

R. Bonsignori, P. Ruggiero, and P. Calabrese, “Symmetry resolved entanglement in free fermionic systems,”J. Phys. A: Math. Theor.52, 475302 (2019),arXiv:1907.02084 [cond-mat.stat-mech]

2019 arXiv
[29]

Symmetry resolved entanglement in gapped integrable systems: a corner transfer matrix approach,

S. Murciano, G. Di Giulio, and P. Calabrese, “Symmetry resolved entanglement in gapped integrable systems: a corner transfer matrix approach,”SciPost Phys.8, 046 (2020),arXiv:1911.09588 [cond-mat.stat-mech]

2020 arXiv
[30]

Quasiparticle dynamics of symmetry-resolved entanglement after a quench: Examples of conformal field theories and free fermions,

G. Parez, R. Bonsignori, and P. Calabrese, “Quasiparticle dynamics of symmetry-resolved entanglement after a quench: Examples of conformal field theories and free fermions,”Phys. Rev. B103, L041104 (2021),arXiv:2010.09794 [cond-mat.stat-mech]

2021 arXiv
[31]

Analytical results for the entanglement Hamiltonian of a free-fermion chain,

V. Eisler and I. Peschel, “Analytical results for the entanglement Hamiltonian of a free-fermion chain,”J. Phys. A: Math. Theor.50, 284003(Jun, 2017) ,arXiv:1703.08126 [cond-mat.stat-mech]

2017 arXiv
[32]

On entanglement hamiltonians of an interval in massless harmonic chains,

G. Di Giulio and E. Tonni, “On entanglement hamiltonians of an interval in massless harmonic chains,”J. Stat. Mech. 033102 (2020), arXiv:1911.07188 [cond-mat.stat-mech]

2020 arXiv
[33]

Entanglement Hamiltonian for inhomogeneous free fermions,

R. Bonsignori and V. Eisler, “Entanglement Hamiltonian for inhomogeneous free fermions,”J. Phys. A: Math. Theor.57, 275001 (2024), arXiv:2403.14766 [cond-mat.stat-mech]

2024 arXiv
[34]

Entanglement negativity in a two dimensional harmonic lattice: Area law and corner contributions,

C. De Nobili, A. Coser, and E. Tonni, “Entanglement negativity in a two dimensional harmonic lattice: Area law and corner contributions,”J. Stat. Mech.083102 (2016), arXiv:1604.02609 [cond-mat.stat-mech]

2016 arXiv
[35]

Partial time-reversal transformation and entanglement negativity in fermionic systems,

H. Shapourian, K. Shiozaki, and S. Ryu, “Partial time-reversal transformation and entanglement negativity in fermionic systems,” Phys. Rev. B95, (2017), arXiv:1611.07536 [cond-mat.str-el]

2017 arXiv
[36]

Twisted and untwisted negativity spectrum of free fermions,

H. Shapourian, P. Ruggiero, S. Ryu, and P. Calabrese, “Twisted and untwisted negativity spectrum of free fermions,” SciPost Phys.7, (2019), arXiv:1906.04211 [cond-mat.stat-mech]

2019 arXiv
[37]

Finite-temperature entanglement negativity of free fermions,

H. Shapourian and S. Ryu, “Finite-temperature entanglement negativity of free fermions,”J. Stat. Mech.(2019), arXiv:1807.09808 [cond-mat.stat-mech]

2019 arXiv
[38]

Logarithmic negativity in out-of-equilibrium open free-fermion chains: An exactly solvable case,

V. Alba and F. Carollo, “Logarithmic negativity in out-of-equilibrium open free-fermion chains: An exactly solvable case,” SciPost Phys.15, 124 (2023),arXiv:2205.02139 [cond-mat.stat-mech]

2023 arXiv
[39]

Logarithmic negativity in quantum Lifshitz theories,

J. Angel-Ramelli, C. Berthiere, V. G. M. Puletti, and L. Thorlacius, “Logarithmic negativity in quantum Lifshitz theories,” J. High Energ. Phys.2020, 11 (2020),arXiv:2002.05713 [hep-th]

2020 arXiv
[40]

Reflected entropy and computable cross-norm negativity: Free theories and symmetry resolution,

C. Berthiere and G. Parez, “Reflected entropy and computable cross-norm negativity: Free theories and symmetry resolution,” Phys. Rev. D108, 054508 (2023),arXiv:2307.11009 [hep-th]

2023 arXiv
[41]

Reflected entropy and Markov gap in Lifshitz theories,

C. Berthiere, B. Chen, and H. Chen, “Reflected entropy and Markov gap in Lifshitz theories,”J. High Energ. Phys. 2023, 160 (2023),arXiv:2307.12247 [hep-th]

2023 arXiv
[42]

Entanglement in spin chains with gradients,

V. Eisler, F. Iglói, and I. Peschel, “Entanglement in spin chains with gradients,”J. Stat. Mech.P02011 (2009), arXiv:0810.3788 [cond-mat.stat-mech]

2009 arXiv
[43]

Entanglement over the rainbow,

G. Ramírez, J. Rodríguez-Laguna, and G. Sierra, “Entanglement over the rainbow,”J. Stat. Mech.1506, P06002 (2015), arXiv:1503.02695 [quant-ph]

2015 arXiv
[44]

More on the rainbow chain: entanglement, space-time geometry and thermal states,

J. Rodríguez-Laguna, J. Dubail, G. Ramírez, P. Calabrese, and G. Sierra, “More on the rainbow chain: entanglement, space-time geometry and thermal states,”J. Phys. A50, 164001 (2017),arXiv:1611.08559 [cond-mat.str-el]

2017 arXiv
[45]

Free-Fermion entanglement and orthogonal polynomials,

N. Crampé, R. I. Nepomechie, and L. Vinet, “Free-Fermion entanglement and orthogonal polynomials,”J. Stat. Mech. 1909, 093101 (2019),arXiv:1907.00044 [cond-mat.stat-mech]

2019 arXiv
[46]

Inhomogeneous XX spin chains and quasi-exactly solvable models,

F. Finkel and A. González-López, “Inhomogeneous XX spin chains and quasi-exactly solvable models,”J. Stat. Mech. 2009, 093105 (2020),arXiv:2007.00369 [cond-mat.str-el] . 14

2020 arXiv
[47]

Entanglement entropy of inhomogeneous XX spin chains with algebraic interactions,

F. Finkel and A. González-López, “Entanglement entropy of inhomogeneous XX spin chains with algebraic interactions,” J. High Energ. Phys.2021, 184 (2021),arXiv:2107.12200 [cond-mat.str-el]

2021 arXiv
[48]

Entanglement in fermionic chains and bispectrality,

N. Crampé, R. I. Nepomechie, and L. Vinet, “Entanglement in fermionic chains and bispectrality,”Rev. Math. Phys.33, 2140001 (2021), arXiv:2001.10576 [math-phys]

2021 arXiv
[49]

The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue,

R. Koekoek and R. F. Swarttouw, “The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue,” tech. rep., 1996.arXiv:math/9602214 [math.CA]

1996 arXiv
[50]

Entanglement of free-fermion systems, signal processing and algebraic combinatorics,

P.-A. Bernard, N. Crampé, R. Nepomechie, G. Parez, and L. Vinet, “Entanglement of free-fermion systems, signal processing and algebraic combinatorics,”arXiv:2401.07150 [quant-ph]

arXiv
[51]

Fermionic logarithmic negativity in the Krawtchouk chain,

G. Blanchet, G. Parez, and L. Vinet, “Fermionic logarithmic negativity in the Krawtchouk chain,”J. Stat. Mech.113101 (2024), arXiv:2408.16531 [cond-mat.stat-mech]

2024 arXiv
[52]

Distinctive features of inhomogeneous spin chains,

P.-A. Bernard, G. Parez, and L. Vinet, “Distinctive features of inhomogeneous spin chains,”arXiv:2411.09487 [quant-ph]

arXiv
[53]

Entanglement Hamiltonian and orthogonal polynomials,

P.-A. Bernard, R. Bonsignori, V. Eisler, G. Parez, and L. Vinet, “Entanglement Hamiltonian and orthogonal polynomials,” arXiv:2412.12021 [cond-mat.stat-mech]
[54]

Emergence of curved light-cones in a class of inhomogeneous Luttinger liquids,

J. Dubail, J.-M. Stéphan, and P. Calabrese, “Emergence of curved light-cones in a class of inhomogeneous Luttinger liquids,” SciPost Phys.3, 019 (2017),arXiv:1705.00679 [cond-mat.str-el]

2017 arXiv
[55]

One-particle density matrix of trapped one-dimensional impenetrable bosons from conformal invariance,

Y. Brun and J. Dubail, “One-particle density matrix of trapped one-dimensional impenetrable bosons from conformal invariance,” SciPost Phys.2, 012 (2017),arXiv:1701.02248 [cond-mat.stat-mech]

2017 arXiv
[56]

The Inhomogeneous Gaussian Free Field, with application to ground state correlations of trapped 1d Bose gases,

Y. Brun and J. Dubail, “The Inhomogeneous Gaussian Free Field, with application to ground state correlations of trapped 1d Bose gases,”SciPost Phys.4, 037 (2018),arXiv:1712.05262 [cond-mat.stat-mech]

2018 arXiv
[57]

J. C. Mason and D. C. Handscomb,Chebyshev Polynomials. Chapman and Hall/CRC, 2002

2002
[58]

Classical Sturmian sequences,

A. Zhedanov, “Classical Sturmian sequences,”arXiv:1904.03789 [math.CA]

1904 arXiv
[59]

Theory of complex spectra. II,

G. Racah, “Theory of complex spectra. II,”Phys. Rev.62, 438 (1942)

1942
[60]

Nature of the symmetry group of the 6j-symbol,

A. S. Zhedanov and Y. A. Granovskiı, “Nature of the symmetry group of the 6j-symbol,”Zh. Eksp. Teor. Fiz.94, 49 (1988)

1988
[61]

“Hidden symmetry

A. S. Zhedanov, ““Hidden symmetry” of Askey-Wilson polynomials,”Theor. Math. Phys.89, 1146 (1991)

1991
[62]

Parameters of association schemes that are both P-and Q-polynomial,

D. A. Leonard, “Parameters of association schemes that are both P-and Q-polynomial,”J. Comb. Theory Ser. A.36, 355 (1984)

1984
[63]

Orthogonal polynomials, duality and association schemes,

D. A. Leonard, “Orthogonal polynomials, duality and association schemes,”SIAM J. Numer. Anal.13, 656 (1982)

1982
[64]

J. A. Wilson,Hypergeometric series recurrence relations and some new orthogonal functions. The University of Wisconsin-Madison, 1978

1978

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Entanglement in many-body systems,

L. Amico, R. Fazio, A. Osterloh, and V. Vedral, “Entanglement in many-body systems,”Rev. Mod. Phys.80, 517 (2008), arXiv:quant-ph/0703044

arXiv 2008

[2] [2]

To numerically investigate the entanglement entropy, we use the machinery introduced in Sec

This scaling thus corresponds to an effective central chargeceff = 1/2. To numerically investigate the entanglement entropy, we use the machinery introduced in Sec. IIB. In particular, we employ Eqs. (15), (16) and (17) with the matricesK and U pertaining to the Krawtchouk case. We show the comparison between the prediction of Eq. (33) and the exact diago...

work page

[3] [3]

Quantum entanglement in condensed matter systems,

N. Laflorencie, “Quantum entanglement in condensed matter systems,”Phys. Rept.646, 1 (2016),arXiv:1512.03388 [cond-mat.str-el]

arXiv 2016

[4] [4]

Scaling of Entanglement close to a Quantum Phase Transitions

A. Osterloh, L. Amico, G. Falci, and R. Fazio, “Scaling of entanglement close to a quantum phase transitions,”Nature 416, 608 (2002),arXiv:quant-ph/0202029

work page Pith review arXiv 2002

[5] [5]

Entanglement in quantum critical phenomena,

G. Vidal, J. I. Latorre, E. Rico, and A. Kitaev, “Entanglement in quantum critical phenomena,”Phys. Rev. Lett.90, 227902 (2003), arXiv:quant-ph/0211074

arXiv 2003

[6] [6]

Topological entanglement entropy,

A. Kitaev and J. Preskill, “Topological entanglement entropy,”Phys. Rev. Lett.96, 110404 (2006), arXiv:hep-th/0510092

arXiv 2006

[7] [7]

Detecting topological order in a ground state wave function,

M. Levin and X.-G. Wen, “Detecting topological order in a ground state wave function,”Phys. Rev. Lett.96, 110405 (2006), arXiv:cond-mat/0510613

arXiv 2006

[8] [8]

Evolution of entanglement entropy in one-dimensional systems,

P. Calabrese and J. L. Cardy, “Evolution of entanglement entropy in one-dimensional systems,”J. Stat. Mech.P04010 (2005), arXiv:cond-mat/0503393

arXiv 2005