R\'enyi entanglement entropies for the compactified massless boson with open boundary conditions

Alvise Bastianello

REVIEW 2 major objections 4 minor 97 references

R\'enyi entanglement entropies for the compactified massless boson with open boundary conditions

T0 review · 2 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The integer-index Rényi entropies of the open-boundary compact boson are fixed by linear integral equations and ratios of Riemann theta functions.

desk verdict A genuinely new derivation of OBC compact-boson Rényi entropies with a nontrivial free-fermion benchmark, held back mainly by two explicitly unproven regularity assumptions that the paper itself flags. read the letter →

arxiv 1909.00806 v2 pith:T6GVKVRP submitted 2019-09-02 cond-mat.stat-mech cond-mat.str-elhep-th

Alvise Bastianello This is my paper · ORCID

classification cond-mat.stat-mechcond-mat.str-elhep-th MSC 81P4082B20 PACS 03.65.Ud05.30.-d11.25.Hf

keywords RényientanglemententropycompactifiedbosonLuttingerliquidopenboundaryconditionsRiemannthetafunctionFredholmdeterminantconformalfieldtheory

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

Rényi entanglement entropies are notoriously hard to compute beyond the universal conformal scaling term, and for a finite interval in the middle of an open system even the operator content of the conformal field theory does not fix them. This paper claims to close that gap for the compactified massless boson (the Luttinger liquid), giving explicit formulas for all integer Rényi indices in terms of Fredholm determinants made from the density two-point correlator, and, in the homogeneous case, reducing the answer to linear integral equations plus a ratio of Riemann theta functions. The ratio encodes the compactification radius K, so the entanglement beyond the conformal part is determined by K alone. The claim is anchored by the exact recovery of the free-fermion result at K=1 and by the explicit prediction that the weak–strong-coupling duality K↔1/(4K) of the periodic case is absent under open boundary conditions.

What carries the argument

The machinery is a free-boson wavefunction written as a Gaussian in the derivative ∂θ of the phase field, whose kernel is the inverse of the density correlator Φ(x,y)=⟨φ(x)φ(y)⟩−⟨φ(x)⟩⟨φ(y)⟩. Twisting the correlator by a phase 2πΩ on the subsystem, Φ_Ω(x,y)=Φ(x,y)$e^{{i2πΩ(χ_B(y)−χ_B(x))}}$, and taking ratios of Fredholm determinants det((1+$Φ^{{-1}}$Φ_Ω)/2) implements the replica gluing of the fields modulo 2π. The compactification radius enters through integer winding sums, organized into a Riemann $\theta$ function with matrix M_{ab} defined in Eq. (9); the coefficients I_{l/N} come from the linear integral equation (10). In the homogeneous case K factorizes out of the determinant ratio, leaving the $\theta$-function ratio F_N(X) as the object carrying all dependence on K and on the four-point ratio X.

What would settle it

Compute the right-hand side of Eq. (18) for two different intervals with the same four-point ratio X and the same K and N: if F_N(X) differs between the two, the claimed universality fails and the theta-function formula is wrong. A second decisive test is exact diagonalization of a microscopic lattice model whose Luttinger parameter K is known: at K≠1, the Rényi entropy of a middle interval should match Eq. (17) up to a geometry-independent constant, and a mismatch beyond that constant would refute the claim.

Watch

Extended reading notes

Core claim

The paper's central claim is Eq. (7): for a system with open boundary conditions, the N-th Rényi entropy of an interval [x1,x2] in the middle is a product of Fredholm determinants built from the twisted density correlator Φ_Ω(x,y) (obtained by multiplying Φ(x,y) by a phase 2πΩ on the interval's boundaries), times a Riemann $\theta$ function whose matrix M is assembled from coefficients I_{l/N} that solve the linear integral equation (10). In the homogeneous case, conformal symmetry fixes the scaling part, and the paper derives the simplified forms (17) and (18), where the only K-dependent piece beyond the boundary term log g is the universal function F_N(X)=$K^{{(N−1)/2}}$ times a ratio of Riemann $\theta$ functions. The result is universal in the CFT sense: F_N depends on the geometry only through the four-point ratio X=x1(L−x2)/(L(x2−x1)). At K=1 the formula reproduces the known free-fermion entropies, and for general K it predicts that the compactification radius shapes the finite part of the entanglement and that the periodic-case duality K→1/(4K) is broken by the open boundaries. The same construction extends to arbitrary multi-interval bipartitions through Eq. (79).

Load-bearing premise

The argument depends on assuming a mathematical regularity property of the solution to the central integral equation (an integrable power-law divergence at the interval edges) and that taking the lattice spacing to zero leaves only a constant offset that is independent of the interval position and of K. The paper verifies this numerically but states it did not prove it.

Editorial extensions

If this is right

For any homogeneous open-boundary compact boson, integer-index Rényi entropies of a middle interval are computable semi-analytically: solving the linear integral equation (10) and evaluating the theta-function ratio (18) replaces costly numerical entanglement calculations.
The free-fermion point K=1 must be exactly reproduced, so the formula supplies a benchmark for interacting lattice models whose low-energy description is a Luttinger liquid with K≠1.
The universal function F_N(X) is fixed by the compactification radius alone, meaning the difference between two systems with the same geometry but different K is entirely captured by the theta-function ratio, not by the scaling term.
For multi-interval bipartitions, the same replica construction yields Rényi entropies via theta functions with (N−1)(NI−2) summation variables, so the approach covers arbitrary bipartitions, not just single intervals.
Because the input is the density correlator, the Fredholm-determinant form (7) applies also to inhomogeneous Luttinger liquids with space-dependent velocity and K, where conformal symmetry is not available.

Reading between the lines

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

If F_N(X) is truly universal in X and K, the same theta-function structure should control other boundary-condition choices (e.g., Neumann/free phase) obtained by exchanging φ and θ, suggesting a one-parameter family of universal functions for different boundary conditions.
The explicit but only numerically accessible matrix M invites asymptotic expansions in X→0 and X→1; comparing those limits with known CFT/OPE predictions would test the theta-function form and could produce closed-form small-distance expansions.
Equation (7) is built entirely from the ground-state density correlator, so the same determinant machinery could be applied to time-dependent or steady-state situations where a CFT description is absent, provided the correlator is known.
The paper leaves the ground-state degeneracy g undetermined in the inhomogeneous case; a regularization of the target-space sum in Eq. (53) would turn the present formulas into fully parameter-free predictions for inhomogeneous systems.

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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

The thing to know: this paper supplies the missing open-boundary counterpart to the known periodic-boundary compact-boson formulas. That is real. The central result, Eq. (7) and its homogeneous reduction to Eqs. (17)-(18), is a new derivation, not a rehash, and it is carefully built on the Casini-Huerta free-system formalism plus the theta-function structure already used for PBC. The K=1 free-fermion check in Fig. 1 is a legitimate nontrivial benchmark, and the breakdown of the K -> 1/(4K) duality under OBC is a nice, correct observation. I found no internal contradiction in the main line of the derivation; the Gaussian manipulations and replica Fourier transform are coherent, and the reduction of the determinant ratio to a K-independent object is argued cleanly.

The paper is honest about its own limits, which is exactly where the soft spots are. The continuum limit of the discretized Fredholm determinant (Appendix A) and the integrability of the endpoint singularity of s_{l/N}(x) in Eq. (10) are both asserted on numerical evidence, not proven. The paper says so explicitly, and the stress-test note is right that these are load-bearing for Eq. (18): if the additive constant in Eq. (17) carried K or endpoint dependence, the theta-function expression for F_N would not be universal. I would add that the boundary constant for inhomogeneous systems is genuinely undetermined in this approach, so the inhomogeneous generalization is more a recipe than a closed result. The absence of shipped code and data makes the numerical checks harder to audit, but the checks themselves are simple enough to reproduce and the K=1 comparison is convincing for the structure.

Do I think the central claim holds? Yes, with the caveat that "holds" here means "provides a correct semi-analytic scheme, modulo two plausible regularity assumptions." Those assumptions could fail, but nothing in the paper suggests they do; the numerics are consistent and the free-fermion point anchors the overall normalization. The paper deserves a serious referee: the derivation is technical, the result is useful for DMRG benchmarking, and the open questions it poses (analytic continuation in N, the boundary term) are the right ones. The main work for the author is either proving the regularity assumptions or at least sharpening the numerical evidence into a well-documented package with code.

For peer review: accept it, with the expectation of heavy revision plus a request for data/code and an explicit discussion of what would make the determinant limit rigorous. The reader's conditional verdict is about right.

Referee Report

2 major / 4 minor

Summary. The paper derives expressions for the integer-index Rényi entanglement entropies of the compactified massless boson (Luttinger liquid) on a finite interval with open boundary conditions, allowing for spatial inhomogeneity of the couplings. The central result is Eq. (7), which expresses the Rényi entropy for a middle interval in terms of Fredholm determinants of density correlators, coefficients I_{ℓ/N} obtained from linear integral equations, and a Riemann theta function associated with the compactification. In the homogeneous case, the paper reduces this to Eq. (17), with the universal finite-size function F_N(X) written in Eq. (18) as a ratio of Riemann theta functions depending on the Luttinger parameter K. The derivation is based on a coherent-state path integral and gaussian integrations; the free-fermion point K=1 is used as a benchmark to fix additive constants and verify the structure. Generalizations to arbitrary bipartitions are given in Eq. (79).

Significance. If the central claim holds, this is a significant step: it provides the first explicit semi-analytic treatment of Rényi entropies for the compactified boson with open boundary conditions, going beyond the free-fermion case and complementing existing periodic-boundary results. The reduction to linear integral equations and Riemann theta functions is elegant and computationally practical, and the paper's honest discussion of its own limitations is a strength. The nontrivial K=1 benchmark and the numerical checks of the endpoint singularities and determinant continuum limits support the plausibility of the result. However, the two regularity assumptions underlying the continuum limit and the theta-function reduction are acknowledged by the author to be unproven; these are load-bearing for the universal claims, so the result is not yet fully established.

major comments (2)

[Section II, item 5; Eqs. (10)-(11)] The derivation of Eq. (7) and, in particular, the theta-function reduction to Eq. (18) require that for every ℓ=1,...,N-1 the solution s_{ℓ/N}(x) of the integral equation (10) has an integrable power-law singularity at the interval endpoints, so that I_{ℓ/N} is finite and the theta series converges. The paper explicitly states that this property could not be proven analytically and is supported only by numerical evidence (Appendix A, Fig. 6). This is load-bearing: if any exponent μ(ℓ/N) in Eq. (11) reached 1, I_{ℓ/N} would diverge and Eq. (18) would not be the universal function F_N(X). I ask the author to provide an analytical argument for the integrability, at least in the homogeneous case, or to clearly restate the final formula as a conjecture with a quantitative numerical characterization covering the full range of ℓ and geometries used in Figs. 1 and 2.
[Appendix A, Eqs. (A4)-(A5); Fig. 5] The identification of the 'const.' term in Eq. (17) as independent of K and of the interval endpoints relies on the assertion that the discretized Fredholm determinant ratio in Eq. (A5) has a continuum limit up to a UV-divergent additive constant with no geometry- or K-dependence. The numerical evidence in Fig. 5 is obtained at K=1 only and verifies a power-law divergence, but it does not establish the required universality of the subtracted constant. Since Eq. (18) isolates F_N(X) from this constant, this gap directly affects the central claim. Please provide a proof or a much stronger justification of the continuum limit, or explicitly restrict the universal statement to the discretized level.

minor comments (4)

[Section II A, Eqs. (17)-(18)] The notation 'const.' in Eq. (17) is used differently from Eq. (13): in Eq. (13) it is the UV cutoff contribution, while in Eq. (17) it also absorbs the free-fermion parts of Eq. (7). Please clarify this to avoid the impression that the constant is being fitted.
[Figure 1 caption] The caption states that the numerical curves are shifted by a constant offset to match the free-fermion result at K=1; this should be stated explicitly in the main text before Eq. (17), because it is essential for understanding how the additive constant is fixed.
[Appendix A, Eq. (A7)] The exponent notation appears inconsistent: the text defines the divergence exponent as ν(Ω), but Eq. (A7) writes the sum as ν(N/ℓ) rather than ν(ℓ/N). Please check the argument of ν in this equation.
[Various] There are several typographical errors: 'coﬃcients' in Appendix A, 'conlcusions' and 'wee commented' in the caption of Fig. 6, and 'Aﬄeck-Boundary boundary entanglement' in Section II, item 4. These should be corrected.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the K-dependent universal function is derived algebraically, not fitted or imported by self-citation.

full rationale

The paper's central claim is a self-contained derivation: Eq. (7) follows from a Gaussian path-integral computation of Tr(rho_B^N), and the homogeneous specialization Eq. (17) is obtained by factoring the constant Luttinger parameter K out of the density correlator (Eq. (15)), so that K cancels from the Fredholm-determinant ratio and survives only inside the Riemann theta functions. The universal function F_N(X) in Eq. (18) is not defined as the entanglement entropy nor fitted to it; it is the ratio of theta sums that emerges from the derived expression once the CFT scaling part, Eq. (13), is subtracted. The free-fermion point K=1 is used only to fix the additive UV-cutoff constant and as a numerical benchmark; the K- and X-dependence of F_N is not adjusted. No load-bearing step reduces to its own input by construction, and no central premise is justified exclusively by a self-citation: the cited PBC theta-function results [57-59] and the boundary degeneracy g=K^{1/4} from Ref. [79] are external inputs, and g is a geometry-independent constant, not part of the K-dependent universal function. The paper openly flags unresolved analytic issues (the unproven integrable power-law singularity of s_{l/N}, the continuum limit of the discretized determinants, and the regularization of g), but these are correctness or rigor concerns, not circularity. The derivation remains self-contained against the free-fermion benchmark, so the appropriate circularity score is 0.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

No new particles, forces, or conserved quantities are introduced. The replica space, the winding integers, and the theta functions are computational devices. The two free inputs beyond the model parameters are the undetermined UV constant and the boundary degeneracy g, both disclosed.

free parameters (2)

UV/constant additive offset in Eqs. (7), (17) = Not determined; matched numerically to the free-fermion K=1 result in Fig. 1
The Rényi entropy is defined up to a divergent, interval-independent constant; the theory cannot fix it and the plots set it by hand.
Ground state degeneracy g = K^{1/4} for homogeneous OBC (Ref. [79]); undetermined for inhomogeneous systems
g enters Eq. (7) as an undetermined multiplicative constant. The homogeneous value is imported from Oshikawa and Affleck; the conclusions state that inhomogeneous applications need a direct computation.

assumptions (5)

domain assumption The ground state wavefunction of the compactified boson is Gaussian and fixed by the two-point connected correlator Phi (Section III A).
Free-boson theory; compactification is inserted later through field identifications modulo 2 pi.
domain assumption Fields in different replicas are identified modulo 2 pi in connected regions, leading to integer summations over m_j and z_j (Section III C, Eqs. (54)-(57)).
This encodes the circle compactification; a different identification would change the theta-function sums.
domain assumption Boundary fields obey phi_left - phi_right in pi Z (Eq. (46)), otherwise the trace vanishes.
U(1) charge conservation in the microscopic model; standard for Luttinger liquids.
ad hoc to paper The auxiliary solution s of Eq. (10) has an integrable endpoint singularity, and the discretized Fredholm determinants converge in the continuum up to a geometry-independent constant (Section II, item 5; Appendix A, Figs. 5-6).
The paper states it could not prove these properties analytically and relies on numerical evidence.
domain assumption The CFT result (13) fixes the scaling part of the entropy up to a universal function F_N(X) of the four-point ratio.
Standard boundary-conformal-invariance input used to organize Eq. (17).

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Pith. "Pith review of R\'enyi entanglement entropies for the compactified massless boson with open boundary conditions." pith.science (2026). https://pith.science/paper/T6GVKVRP

@misc{pith2026190900806,
  author       = {Pith},
  title        = {Pith review of: R\'enyi entanglement entropies for the compactified massless boson with open boundary conditions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/T6GVKVRP}},
  note         = {Machine review of arXiv:1909.00806}
}

read the original abstract

We investigate the R\'enyi entanglement entropies for the one-dimensional massless free boson compactified on a circle, which describes the low energy sector of several interacting many-body 1d systems (Luttinger Liquid). We focus on systems on a finite segment with open boundary conditions and possible inhomogeneities in the couplings. We provide expressions for the R\'enyi entropies of integer indices in terms of Fredholm determinant-like expressions. Within the homogeneous case, we reduce the problem to the solution of linear integral equations and the computation of Riemann Theta functions. We mainly focus on a single interval in the middle of the system, but results for generic bipartitions are given as well.

Figures

Figures reproduced from arXiv: 1909.00806 by the authors.

**Figure 2.** FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]

**Figure 5.** Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p017_5.png]

**Figure 6.** Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p018_6.png]

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Works this paper leans on

97 extracted references · 76 canonical work pages

[1]

Φ is a linear operator constructed from the two-point connected correlator of the density ﬁeld Φ( x,y ) = ⟨ˆφ(x)ˆφ(y)⟩−⟨ ˆφ(x)⟩⟨ˆφ(y)⟩. We then deﬁne Φ Ω starting from Φ and adding a twist with a phase 2 πΩ at the boundaries of B ΦΩ(x,y ) = Φ(x,y )ei2πΩ(χB(y)−χB(x)), (8) with χB(x) the characteristic function of the subsystem B, being 1 if x∈B and zero otherwise
[2]

det ( 1+Φ−1Φ𝓁/N 2 ) is understood as a Fredholm determinant, being the domain of the operator the whole system [0,L ]. From a practical point of view, the two-point correlator is discretized on a proper lattice and the determi- nants are computed, then the limit of inﬁnitesimal lattice spacing is considered (see Appendix A). While taking this limit, the l...
[3]

(7) is over all the possible integers ma∈ Z and is an example of a Riemann Theta function

The summation in Eq. (7) is over all the possible integers ma∈ Z and is an example of a Riemann Theta function. Its appearance is due to the non-trivial compactiﬁcation radius and similar functions are known to be present in the R´ enyi entropy with PBC [57–59] as well
[4]

Within our approach,g emerges as an ill-deﬁned constant (i.e

g is the ground state degeneracy, giving rise to the Aﬄeck-Boundary boundary entanglement [64, 65]. Within our approach,g emerges as an ill-deﬁned constant (i.e. independent onx1 andx2) whose computation requires a proper regularization scheme yet to be devised. For the homogeneous compact boson with open boundary con- ditions, i.e. Dirichlet boundary con...
[5]

(10) The solution s𝓁/N(x) displays a power-law singularity near the endpoint x1 s𝓁/N(x)∼ ⏐⏐⏐⏐ 1 x−x1 ⏐⏐⏐⏐ µ(𝓁/N) , (11) and similarly at x2

The matrix M is deﬁned as Mab = N−1∑ 𝓁=1 e−i2π𝓁(a−b)/N NI𝓁/N (9) 4 and lastly the real coeﬃcients I𝓁/N are obtained from the solution of a linear integral equation I𝓁/N = ∫ x2 x1 dxs𝓁/N(x), ∫ L 0 dy (Φ(x,y ) + Φ𝓁/N(x,y ))s𝓁/N(y) =χB(x). (10) The solution s𝓁/N(x) displays a power-law singularity near the endpoint x1 s𝓁/N(x)∼ ⏐⏐⏐⏐ 1 x−x1 ⏐⏐⏐⏐ µ(𝓁/N) , (11) ...
[6]

(6), which therefore do not aﬀect the R´ enyi entropies

We stress that the connected correlator Φ(x,y ) is independent from the precise choice of the boundary ﬁelds in Eq. (6), which therefore do not aﬀect the R´ enyi entropies. Eq. (7) requires as an input the two-point connected correlation function of the density ﬁeld. For the general case where both the sound velocity and the Luttinger parameter are space ...
[7]

Amico, R

L. Amico, R. Fazio, A. Osterloh, and V. Vedral, Rev. Mod. Phys. 80 517 (2008)

2008
[8]

Calabrese, J

P. Calabrese, J. Cardy, and B. Doyon, J. Phys. A 42 500301 (2009)

2009

Show all 97 references

[9]

Laﬂorencie, Physics Report 643, 1 (2016)

N. Laﬂorencie, Physics Report 643, 1 (2016)

2016
[10]

Calabrese, J

P. Calabrese, J. Cardy J. Stat. Mech. (2016) 064003

2016
[11]

Calabrese, J

P. Calabrese, J. Cardy J. Stat. Mech. (2004) P06002

2004
[12]

Calabrese, J

P. Calabrese, J. Cardy, J. Phys. A: Math. Theor. 42 504005 (2009)

2009
[13]

Holzhey, F

C. Holzhey, F. Larsen, F. Wilczek, Nucl. Phys. B 424, 3 443-477 (1994)

1994
[14]

Schuch, M

N. Schuch, M. M. Wolf, F. Verstraete, and J. I. Cirac, Phys. Rev. Lett. 100, 030504 (2008)

2008
[15]

Schuch, M

N. Schuch, M. M. Wolf, K. G. H. Vollbrecht, and J. I. Cirac, New J. Phys. 10, 033032 (2008)

2008
[16]

Perales and G

A. Perales and G. Vidal, Phys. Rev. A 78, 042337 (2008)

2008
[17]

Hauke, F

P. Hauke, F. M. Cucchietti, L. Tagliacozzo, I. Deutsch, and M. Lewenstein, Rep. Prog. Phys. 75 082401 (2012)

2012
[18]

Dubail, J

J. Dubail, J. Phys. A 50, 234001 (2017)

2017
[19]

Leviatan, F

E. Leviatan, F. Pollmann, J. H. Bardarson, D. A. Huse, and E. Altman, arXiv:1702.08894 (2017)

2017 arXiv
[20]

Islam, R

R. Islam, R. Ma, P. Preiss, M. Tai, A. Lukin, M. Rispoli, M. Greiner, Nature, 528, 77-83 (2015)

2015
[21]

A. M. Kaufman, M. Tai, A. Lukin, A. Rispoli, R. Schittko, P. Preiss, M. Greiner, Science 353, 794800 (2016)

2016
[22]

A. J. Daley, H. Pichler, J. Schachenmayer, P. Zoller, Phys. Rev. Let.. 109, 020505 (2012)

2012
[23]

Unmuth-Yockey, J

J. Unmuth-Yockey, J. Zhang, P. Preiss, L.-P. Yang, S.-W. Tsai, Y. Meurice, Phys. Rev. A 96, 023603 (2017)

2017
[24]

Elben, B

A. Elben, B. Vermersch, M. Dalmonte, J. I. Cirac, P. Zoller, Phys. Rev. Lett. 120, 050406 (2018). 19

2018
[25]

Lukin, M

A. Lukin, M. Rispoli, R. Schittko, M. E. Tai, A. M. Kaufman, S. choi, V. Khemani, J. Leonard, M. Greiner, Science 364, 256-260 (2019)

2019
[26]

Brydges, A

T. Brydges, A. Elben, P. Jurcevic, B. Vermersch, C. Maier, B. P. Lanyon, P. Zoller, R. Blatt, C. F. Roos, Science 364, 260-263 (2019)

2019
[27]

Calabrese, A

P. Calabrese, A. Lefevre, Phys. Rev. A 78, 032329 (2008)

2008
[28]

V. Alba, P. Calabrese, E. Tonni, J. Phys. A 51, 024001 (2018)

2018
[29]

H. Li, F. D. M. Haldane, Phys. Rev. Lett. 101, 010504 (2008)

2008
[30]

A. A. Belavin, A. M. Polyakov, A. B. Zamolodchikov, Nucl. Phys. B 241, 333 (1984)

1984
[31]

Di Francesco, P

P. Di Francesco, P. Mathieu, D. S´ en´ echalConformal ﬁeld theory (Springer Science & Business Media, 2012)

2012
[32]

Haldane, Physics Letters A 81, 153 (1981)

F. Haldane, Physics Letters A 81, 153 (1981)

1981
[33]

Haldane, Journal of Physics C: Solid State Physics 14, 2585 (1981)

F. Haldane, Journal of Physics C: Solid State Physics 14, 2585 (1981)

1981
[34]

Haldane, Physical Review Letters 47,1840 (1981)

F. Haldane, Physical Review Letters 47,1840 (1981)

1981
[35]

Cazalilla, Journal of Physics B: Atomic, Molecular and Optical Physics 37,S1 (2004)

M. Cazalilla, Journal of Physics B: Atomic, Molecular and Optical Physics 37,S1 (2004)

2004
[36]

Giamarchi, Quantum physics in one dimension , vol

T. Giamarchi, Quantum physics in one dimension , vol. 121 (Oxford university press, 2004)

2004
[37]

A. M. Tsvelik, Quantum ﬁeld theory in condensed matter physics (Cambridge university press, 2007)

2007
[38]

Cazalilla, R

M. Cazalilla, R. Citro, T. Giamarchi, E. Orignac, and M. Rigol, Reviews of Modern Physics 83 1405 (2011)

2011
[39]

Ruggiero, E

P. Ruggiero, E. Tonni, P. Calabrese, J. Stat. Mech. (2018) 113101

2018
[40]

M. A. Rajabpour, F. Gliozzi, J. Stat. Mech. P02016 (2012)

2012
[41]

Caraglio, F

M. Caraglio, F. Gliozzi, JHEP 11 (2008) 076

2008
[42]

Furukawa, V

S. Furukawa, V. Pasquier, J. Shiraishi, Phys. Rev. Lett. 102, 170602 (2009)

2009
[43]

Headrick, Phys

M. Headrick, Phys. Rev. D 82, 126010 (2010)

2010
[44]

Headrick, A

M. Headrick, A. Lawrence, M. Roberts, J. Stat. Mech. P02022 (2013)

2013
[45]

Casini, M

H. Casini, M. Huerta, JHEP 03 (2009) 048

2009
[46]

Facchi, G

P. Facchi, G. Florio, G. Parisi, S. Pascazio, Phys. Rev. A 77 060304 (2008)

2008
[47]

V. Alba, L. Tagliacozzo, P. Calabrese, Phys. Rev. B 81 060411 (2010)

2010
[48]

V. Alba, L. Tagliacozzo, P. Calabrese, J. Stat. Mech. (2011) P06012

2011
[49]

Igloi, I

F. Igloi, I. Peschel, EPL 89 (2010) 40001

2010
[50]

Fagotti, P.Calabrese, J

M. Fagotti, P.Calabrese, J. Stat. Mech. (2010) P04016

2010
[51]

Calabrese, J

P. Calabrese, J. Stat. Mech. (2010) P09013

2010
[52]

Fagotti, EPL 97 (2012) 17007

M. Fagotti, EPL 97 (2012) 17007

2012
[53]

B. Chen, J. Zhang, JHEP 1311 (2013) 164

2013
[54]

B. Chen, J. Long, J. Zhang, JHEP 1404 (2014) 041

2014
[55]

F. Ares, J. G. Esteve, F. Falceto, Phys. Rev. A 90, 062321 (2014)

2014
[56]

A. Coser,E. Tonni, P. Calabrese, J. Stat. Mech. (2016) 053109

2016
[57]

Z. Li, J. Zhang, JHEP 1605 (2016) 130

2016
[58]

F. Liu, X. Liu, JHEP 01 (2016) 058

2016
[59]

Belin, C

A. Belin, C. A. Keller, I. G. Zadeh, J. Phys. A 50, 435401 (2017)

2017
[60]

Mukhi, S

S. Mukhi, S. Murthy, J.-Q. Wu, JHEP 01 (2018) 005

2018
[61]

Dupic, B

T. Dupic, B. Estienne, Y. Ikhlef, SciPost Phys. 4, 031 (2018)

2018
[62]

J. C. Xavier, F. C. Alcaraz, G. Sierra, Phys. Rev. B 98, 041106(R) (2018)

2018
[63]

Calabrese, J

P. Calabrese, J. Cardy, E. Tonni, J. Stat. Mech. (2009) P11001

2009
[64]

Calabrese, J

P. Calabrese, J. Cardy, E. Tonni, J. Stat. Mech. (2011) P01021

2011
[65]

Coser, L

A. Coser, L. Tagliacozzo, E. Tonni, J. Stat. Mech. (2014) P01008

2014
[66]

Cardy, arXiv: hep-th/0411189, (2008)

J. Cardy, arXiv: hep-th/0411189, (2008)

2008 arXiv
[67]

Kondo, Progr

J. Kondo, Progr. Theoret. Phys., 32 (1964), 37-49

1964
[68]

A. C. Hewson, (1997). The Kondo problem to heavy fermions (Vol. 2). Cambridge university press

1997
[69]

Aﬄeck, N

I. Aﬄeck, N. Laﬂorencie, E. S Sørensen, J. Phys. A: Math. Theor. 42 504009 (2009)

2009
[70]

H.-Q. Zhou, T. Barthel, J. O. Fjærestad, U. Schollw¨ ock, Phys. Rev. A 74, 050305(R) (2006)

2006
[71]

Aﬄeck, A

I. Aﬄeck, A. W. W. Ludwig, Phys. Rev. Lett. 67, 161 (1991)

1991
[72]

Casini, C

H. Casini, C. D. Fosco, M. Huerta, J. Stat. Mech. (2005) P07007

2005
[73]

Fagotti, P

M. Fagotti, P. Calabrese J. Stat. Mech. (2011) P01017

2011
[74]

Calabrese, M

P. Calabrese, M. Mintchev, E. Vicari, Phys. Rev. Lett. 107, 020601 (2011)

2011
[75]

Calabrese, M

P. Calabrese, M. Mintchev, E. Vicari, J. Stat. Mech. P09028 (2011)

2011
[76]

Schollwck, Matrix product state algorithms: DMRG, TEBD and relatives

U. Schollwck, Matrix product state algorithms: DMRG, TEBD and relatives. In Strongly Correlated Systems (pp. 67-98). (Springer, Berlin, Heidelberg 2013)

2013
[77]

Casini, M

H. Casini, M. Huerta 2009 J. Phys. A: Math. Theor. 42 504007

2009
[78]

Dubail, J.-M

J. Dubail, J.-M. St´ ephan, J. Viti, P. Calabrese, SciPost Physics2, 002 (2017)

2017
[79]

Dubail, J.-M

J. Dubail, J.-M. St´ ephan, P. Calabrese, SciPost Physics3, 019 (2017)

2017
[80]

Y. Brun, J. Dubail, SciPost Phys. 4, 037 (2018)

2018
[81]

Y. Brun, J. Dubail, SciPost Phys. 2, 012 (2017)

2017
[82]

Eisler, D

V. Eisler, D. Bauernfeind, Phys. Rev. B 96, 174301 (2017)

2017
[83]

Ruggiero, Y

P. Ruggiero, Y. Brun, J. Dubail, SciPost Phys. 6, 051 (2019)

2019
[84]

Murciano, P.Ruggiero, P

S. Murciano, P.Ruggiero, P. Calabrese, J. Stat. Mech. (2019) 034001

2019
[85]

Oshikawa, I

M. Oshikawa, I. Aﬄeck, Nucl. Phys. B 495 533-582 (1997). 20

1997
[86]

Aﬄeck, J

I. Aﬄeck, J. Phys. A: Math. Gen. 31 2761 (1998)

1998
[87]

C. M. Agon, M. Headrick, D. L. Jaﬀeris, S. Kasko, Phys. Rev. D 89, 025018 (2014)

2014
[88]

De Nobili, A

C. De Nobili, A. Coser, and E. Tonni, J. Stat. Mech. (2015) P06021

2015
[89]

Peres, Phys

A. Peres, Phys. Rev. Lett. 77, 1413 (1996)

1996
[90]

Zyczkowski, P

K. Zyczkowski, P. Horodecki, A. Sanpera, M. Lewenstein, Phys. Rev. A 58, 883 (1998)

1998
[91]

Zyczkowski, Phys

K. Zyczkowski, Phys. Rev. A 60, 3496 (1999)

1999
[92]

J. Lee, M. S. Kim, Y. J. Park, S. Lee, J. Mod. Opt. 47, 2151 (2000)

2000
[93]

Eisert, M

J. Eisert, M. B. Plenio, J. Mod. Opt. 46, 145 (1999)

1999
[94]

Vidal, R

G. Vidal, R. F. Werner, Phys. Rev. A 65, 032314 (2002)

2002
[95]

M. B. Plenio, Phys. Rev. Lett. 95, 090503 (2005)

2005
[96]

Calabrese, J

P. Calabrese, J. Cardy, E. Tonni Phys. Rev. Lett. 109, 130502 (2012)

2012
[97]

Calabrese, J.Cardy, E

P. Calabrese, J.Cardy, E. Tonni, J. Stat. Mech. (2013) P02008

2013

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Φ is a linear operator constructed from the two-point connected correlator of the density ﬁeld Φ( x,y ) = ⟨ˆφ(x)ˆφ(y)⟩−⟨ ˆφ(x)⟩⟨ˆφ(y)⟩. We then deﬁne Φ Ω starting from Φ and adding a twist with a phase 2 πΩ at the boundaries of B ΦΩ(x,y ) = Φ(x,y )ei2πΩ(χB(y)−χB(x)), (8) with χB(x) the characteristic function of the subsystem B, being 1 if x∈B and zero otherwise

[2] [2]

det ( 1+Φ−1Φ𝓁/N 2 ) is understood as a Fredholm determinant, being the domain of the operator the whole system [0,L ]. From a practical point of view, the two-point correlator is discretized on a proper lattice and the determi- nants are computed, then the limit of inﬁnitesimal lattice spacing is considered (see Appendix A). While taking this limit, the l...

[3] [3]

(7) is over all the possible integers ma∈ Z and is an example of a Riemann Theta function

The summation in Eq. (7) is over all the possible integers ma∈ Z and is an example of a Riemann Theta function. Its appearance is due to the non-trivial compactiﬁcation radius and similar functions are known to be present in the R´ enyi entropy with PBC [57–59] as well

[4] [4]

Within our approach,g emerges as an ill-deﬁned constant (i.e

g is the ground state degeneracy, giving rise to the Aﬄeck-Boundary boundary entanglement [64, 65]. Within our approach,g emerges as an ill-deﬁned constant (i.e. independent onx1 andx2) whose computation requires a proper regularization scheme yet to be devised. For the homogeneous compact boson with open boundary con- ditions, i.e. Dirichlet boundary con...

[5] [5]

(10) The solution s𝓁/N(x) displays a power-law singularity near the endpoint x1 s𝓁/N(x)∼ ⏐⏐⏐⏐ 1 x−x1 ⏐⏐⏐⏐ µ(𝓁/N) , (11) and similarly at x2

The matrix M is deﬁned as Mab = N−1∑ 𝓁=1 e−i2π𝓁(a−b)/N NI𝓁/N (9) 4 and lastly the real coeﬃcients I𝓁/N are obtained from the solution of a linear integral equation I𝓁/N = ∫ x2 x1 dxs𝓁/N(x), ∫ L 0 dy (Φ(x,y ) + Φ𝓁/N(x,y ))s𝓁/N(y) =χB(x). (10) The solution s𝓁/N(x) displays a power-law singularity near the endpoint x1 s𝓁/N(x)∼ ⏐⏐⏐⏐ 1 x−x1 ⏐⏐⏐⏐ µ(𝓁/N) , (11) ...

[6] [6]

(6), which therefore do not aﬀect the R´ enyi entropies

We stress that the connected correlator Φ(x,y ) is independent from the precise choice of the boundary ﬁelds in Eq. (6), which therefore do not aﬀect the R´ enyi entropies. Eq. (7) requires as an input the two-point connected correlation function of the density ﬁeld. For the general case where both the sound velocity and the Luttinger parameter are space ...

[7] [7]

Amico, R

L. Amico, R. Fazio, A. Osterloh, and V. Vedral, Rev. Mod. Phys. 80 517 (2008)

2008

[8] [8]

Calabrese, J

P. Calabrese, J. Cardy, and B. Doyon, J. Phys. A 42 500301 (2009)

2009