Entanglement R\'enyi entropy and boson-fermion duality in massless Thirring model

We investigate the second R\'enyi entropy of two intervals in the massless Thirring model describing a self-interacting Dirac fermion in two dimensions. Boson-fermion duality relating this model to a free compact boson theory enables us to simplify the calculation of the second R\'enyi entropy, reducing it to the evaluation of the partition functions of the bosonic theory on a torus. We derive exact results on the second R\'enyi entropy, and examine the dependence on the sizes of the intervals and the coupling constant of the model both analytically and numerically. We also explore the mutual R\'enyi information, a measure quantifying the correlation between the two intervals, and find that it generally increases as the coupling constant of the Thirring model becomes larger.


I. INTRODUCTION
Quantum entanglement, a fundamental phenomenon in modern physics characterized by non-local correlations between quantum states separated by distance, has garnered extensive attention not only in quantum information theory but also in high energy physics, condensed matter theory, and gravitation theory.In quantum field theory (QFT), various measures of quantum entanglement have proven to be indispensable tools for characterizing and classifying different phases of matter, particularly topological phases [1,2], while also capturing universal scaling behaviors in critical systems [3][4][5][6].Moreover, quantum entanglement has found an unexpected connection with gravitational physics via the holographic principle [7,8], yielding fresh perspectives on the intricate structures of spacetime, including those governing black hole physics, as well as non-perturbative aspects of QFT.(See e.g., [9][10][11][12][13] for reviews.) The entanglement Rényi entropy (ERE) is one of the principal measures that quantify the amount of quantum entanglement shared between different parts of a quantum system.They are a generalization of the more familiar measure known as entanglement entropy, defined by where ρ V is the reduced density matrix of a subsystem V and n is a parameter that can take on any positive integer.In QFT, the subsystem V is given by a subregion on a time slice, and the reduced density matrix ρ V can be represented by the Euclidean path integral on a manifold with a cut along V .Hence, the calculation of the ERE amounts to that of the partition function Z n on the replica manifold, which is an n-sheeted manifold branched over V .The codimension-two conical singularity around ∂V often prevents us from performing the path integral analytically on the replica manifold.The ERE has had exact results for general n only in simple cases to date, such as an interval in two-dimensional conformal field theories (CFTs) [3,5], multiple intervals in free massless fermion theory [14], and spherical region in (d ≥ 3)-dimensional free massless scalar and fermion theories [15].Another exactly calculable case is the second ERE (n = 2) of two intervals in a two-dimensional free compact boson theory [16][17][18].However, when it comes to incorporating interactions, the calculation of the ERE proves to be more challenging, and only a few approximate results have been obtained, such as in the O(N ) models in the large-N limit [19], so far [20].
In this paper, we calculate the ERE exactly in the massless Thirring model describing a self-interacting Dirac fermion in two dimensions [21]; This model can be seen as a marginal deformation of a free massless Dirac fermion and enjoys conformal symmetry with central charge c = 1.Moreover, it possesses a duality with a free boson theory of a compact radius.We can leverage the boson-fermion duality to reduce the formidable calculation of the ERE in the interacting fermionic theory to a more manageable task in the free bosonic theory.Nevertheless, a subtlety arises when evaluating the ERE with the boson-fermion duality [22].Since the replica manifold Σ n,N associated with the n-th ERE of N intervals is a Riemann surface of genus g = (n − 1)(N − 1), fermionic theories depend on the spin structures along 2g one-cycles on Σ n,N .On the other hand, the dual bosonic theories can be defined without specifying the spin structures, so their EREs do not appear to match unless n = 1 or N = 1.This apparent paradox is resolved by taking into account the coupling of the dual bosonic theory to a topological field theory (the Kitaev Majorana chain [23]), and the gauging of a non-anomalous Z 2 symmetry [24][25][26][27][28].
We employ this refined perspective on the boson-fermion duality and derive the exact results on the second ERE of two intervals (n = N = 2) in the massless Thirring model.This paper is structured as follows.In section II, we provide a concise review of the modern formulation of boson-fermion duality for bosonic theories with a nonanomalous Z 2 symmetry in two dimensions.While the duality extends to higher-genus Riemann surfaces, we concentrate on the torus case.We express the partition functions of fermionic theories with four distinct spin structures (periodic/anti-periodic boundary conditions along the two one-cycles) in terms of the linear combinations of four bosonic partition functions which incorporate the background Z 2 gauge fields.This sets the stage for the application of the Thirring/compact boson duality in the later sections.Section III begins with a brief account of the path integral formulation of the ERE in fermionic theories.We then proceed to derive exact results on the second ERE of two intervals as well as the mutual Rényi information, a measure quantifying the correlation between these intervals.We also examine their dependence on both interval sizes and the coupling constant analytically in some limits and numerically in broad parameter regions.Finally, section IV is dedicated to discussion and potential directions for future works.

II. BOSON-FERMION DUALITY
Boson-fermion duality makes it much easier for us to go back and forth between a bosonic theory and a fermionic one.In particular, we can construct a two-dimensional fermionic theory T F from a bosonic one T B via the fermionization dictionary [27,28]; where Z B 2 is a non-anomalous global symmetry in the bosonic theory.In this section, we firstly expand on the procedure incorporating the relation (3).We next rewrite the partition function of the massless Thirring model in terms of those of a free compact boson model by applying the fermionization dictionary (3).

A. Fermionization on a torus
Given a bosonic theory T B with a Z 2 symmetry, the construction of a corresponding fermionic theory T F proceeds through the following two steps: • Couple the bosonic theory T B with the spin topological QFT (TQFT) Kitaev.
• Gauge the diagonal Z B 2 symmetry of the theory T B × Kitaev.
The fermionization procedure is applicable on any Riemann surface, but for the sake of simplicity, we will focus our attention on the case of a torus T below.
First of all, Kitaev is the fermionic spin TQFT associated with a given Z B 2 global symmetry, defined by the partition function where is the background gauge field associated with Z B 2 symmetry, and characterized by holonomy around the two one-cycles γ 1 and γ 2 of T; Also, ρ is the spin structure in the fermionic theory T F .On a torus, there are four choices of the spin structure depending on whether the fermionic field subjects to the periodic (P) or anti-periodic (A) boundary condition for each one-cycle on T: ρ = PP, AP, PA, AA.In addition, Arf [ρ] is the Arf invariant, which takes its value in zero or one depending on the spin structure ρ; We should note that the insertion of a non-trivial background Z B 2 gauge field T along a one-cycle γ changes the spin structure associated with γ.For instance, the holonomy of the Z B 2 gauge field with its value (1, 0) shifts the spin structures as follows: and thereby the Arf invariant Arf [(1, 0) + ρ] is given by Similar considerations apply straightforwardly to the other gauge configurations.At the level of the partition function, coupling the bosonic theory T B to the TQFT Kitaev amounts multiplying the bosonic torus partition function Z B [T ] with the TQFT defined by (4); Next, we proceed to perform the gauging of the Z B 2 global symmetry.This means that we promote the background gauge field T to the dynamical one t, and sum over all configurations of the gauge field t.This summation is a finite dimensional analog of the usual path integral, and the resulting gauged theory (T B × Kitaev)/Z B 2 can be written as follows: 1 2 The fermionization dictionary (3) asserts that (10) is exactly the same as the fermionic torus partition function, For instance, let us consider the case where ρ = AA.By performing the summation in (11) with ( 6) and ( 8), we find where we use a shorthand notation Z B [ab] for the partition function Z B [(a, b)].

B. Compact boson/massless Thirring duality
In this subsection, we apply the fermionization dictionary (11) to the massless Thirring model whose partition function Z F [ρ, λ, τ ] on a torus T with a modulus τ is given by where I[ρ, λ, τ ] is the massless Thirring action defined by [21] Here, / D ρ is the Dirac operator with a spin structure ρ and λ is the coupling constant, which will be referred to as the Thirring coupling in this paper.
The bosonic counterpart is a free compact boson theory whose torus partition function where ϕ is the compact boson field with the periodicity 2π, and T is the Z B 2 gauge field.Also, R is the compact radius, which is related to the Thirring coupling λ as follows [29]: When the modulus is set to be pure-imaginary, τ = i ℓ, the torus partition function Z B [T, R, i ℓ] is given by the following simple form: where η is the Dedekind eta function [30] η(τ ) = q and ϑ i (i = 2, 3, 4) are the Jacobi theta functions [30] ϑ 2 (τ ) = The fermionization dictionary (11) allows us to write the fermionic partition function Z F [ρ, λ, iℓ] as the summation of the bosonic ones.In what follows, we focus on the spin structure ρ = AA on the torus, which will be relevant to the calculation of the ERE in section III B. In the case of the compact boson/massless Thirring duality, the formula (12) becomes; where Thirring coupling λ and compact radius R are related as (16).By plugging ( 17)-( 20) into ( 23) with help of identities in Appendix B, we find where Ξ j (j = 2, 3, 4) are defined by Before closing this section, we make a comment on the action of the T-duality in the compact boson theory on the massless Thirring model.We should first note that the reflection positivity restricts the coefficient of the kinetic term in the compact boson theory to be positive, and hence ( 16) leads to a valid range for the Thirring coupling: λ > −1.This is however not the end of the story, since there are equivalent points under the T-duality.Indeed, the T-duality in the compact boson theory maps the radius R to its inverse (see section 3.5 in the literature [27] for detail) as which acts as the T-duality transformation on the Thirring coupling via (16); It follows that the T-duality swaps the two coupling regions: (−1, 0] and [0, ∞), hence the fundamental domain of the Thirring coupling is either FIG. 1.The regions V and V in two dimensions for two intervals case.The entangling regions V is shown in red, whose end points are denoted by u1, v1, u2 and v2.

III. ENTANGLEMENT RÉNYI ENTROPY IN MASSLESS THIRRING MODEL
We are now in position to derive the ERE and the mutual Rényi information (MRI) in the massless Thirring model.In section III A, we present a brief review on the replica method for calculating ERE [10].In section III B, we derive the ERE in the massless Thirring model by utilizing the boson-fermion duality introduced in section II.In section III C, we examine the ERE and the MRI in the massless Thirring model by varying both the crossratio and the Thirring coupling.In section III D, we consider the tripartite information for three intervals, two of which are adjacent to each other.

A. Replica method
To introduce an ERE, we first divide the total space into V and V .Here, V is the disjoint union of N intervals: and V is the complementary region to V .(See Fig. 1 for N = 2 case.)Throughout this paper, we assume that the state associated with the total space is the vacuum |0⟩, then the density matrix of the total space is The reduced density matrix of the region V is defined by tracing ρ tot over all degree of freedom in V ; The n-th ERE S n (V ) of the region V is defined by We should note that, by analytically continuing a positive integer n to a real number and taking the limit n → 1, ERE reduces to the entanglement entropy S(V ): We can also introduce the n-th where V 1 and V 2 are disjoint unions of the entangling region While ERE depends on the choice of the ultraviolet (UV) cutoff in QFT, MRI is known to be finite and independent of the regularization scheme [18].Intuitively, MRI quantifies how the two regions V 1 and V 2 are entangled with each other.
In order to derive ERE and MRI, we employ the (Euclidean) path integral representation of Tr V [ρ n V ] [5, 10]: where, Z n,N is the partition function on the n-sheeted manifold Σ n,N which consists of n-copies of the original manifold with an appropriate gluing conditions along with the entangling region V (see Fig. 2).Also, I[Ψ, Ψ] is the Euclidean action where Ψ denotes the n-vector whose component is the field living on each copy: Furthermore, Σ n,N means the restricted path integral with the following gluing condition: where T is an n × n matrix and represents the boundary condition of region V .For a fermionic system, the matrix T is given by [31] T ≡ By substituting (33) into the definition of the ERE (30), we can express the n-th ERE in terms of the partition function Z n,N on the n-sheeted manifold Σ n,N : We record a few exact results of EREs derived by the replica method below: • Two-dimensional CFT: In the case of one interval, the n-th ERE in an arbitrary CFT is given by [3][32] where ϵ is UV cutoff length and c is the central charge in CFT.We emphasize that the coefficient in front of the logarithm depends only on the central charge and the number of sheets, hence is schemeindependent.
The sketch of the n-sheeted manifold Σn,N .Each sheet represents the original spacetime and the sheets are sewed together at the entangling region • Free massless Dirac fermion in two dimensions: The n-th ERE of arbitrary N intervals in the free massless Dirac fermion is calculated in [14].In particular, in the case (n, N ) = (2, 2), the ERE is given by where ϵ is the UV cutoff length and x is the crossratio defined by The exact results (38) and (39) will be used later.Finally, we comment on the spin structure on the twosheeted manifold Σ 2,2 with two intervals (n = N = 2).This manifold Σ 2,2 has two non-contractable cycles on which the spin structure ϱ is defined (see the left panel of Fig. 3).Specifying the spin structure ϱ is equivalent to determining whether the fermionic field is subject to the periodic (P) or anti-periodic (A) boundary condition around each cycle.Thus, the four spin structures ϱ =PP, AP, PA, AA are obtained on Σ 2,2 by assigning one of the two choices along the two cycles.Among the four spin structures, however, only ϱ = PP is compatible with the gluing condition (35) [18].For (n, N ) = (2, 2), the gluing condition (35) reads One can check that the sign of the fermionic field does not change when it moves around each cycle as illustrated on the left side in Fig. 3. Hence, we will deal with the spin structure ϱ = PP on the manifold Σ 2,2 in the rest of the paper.

B. Exact results on ERE and MRI
Now, we turn to the derivation of the ERE with (n, N ) = (2, 2) in the massless Thirring model (2) on the two-dimensional plane R 2 .In general, it is challenging to calculate the ERE analytically in an interacting system by the replica trick as one cannot make the n fields independent by diagonalizing the gluing condition (35) by a change of variable.In what follows, we circumvent this issue by mapping the massless Thirring model (2) to a free compact boson by the boson-fermion duality.
It follows from the formula (37) that the ERE with (n, N ) = (2, 2) for the massless Thirring model of the coupling λ is given by where Z 1 and Z 2,2 [PP, V, λ] are the partition functions on the two-dimensional plane R 2 and the two-sheeted manifold Σ 2,2 with the spin structure ϱ = PP, respectively.The partition function Z 2,2 on Σ 2,2 appears to be difficult to evaluate, but one can map Σ 2,2 to a torus T by an appropriate conformal transformation [33] and relate Z 2,2 to the torus partition function Z F as follows [34]: where ϵ is the UV cutoff length and Z F [AA, λ, i ℓ] is the torus partition function of the massless Thirring with the complex structure modulus τ = i ℓ defined in (13).Note that the spin structure ρ = AA on T is different from the one ϱ = PP on Σ 2,2 as the periodicity for the fermionic field changes under the conformal transformation [35].Also, x is the cross-ratio defined by (40) and it can be also written in terms of the modulus i ℓ: where ϑ j (τ ) (j = 2, 3, 4) are Jacobi theta functions (22).By substituting ( 43) into (42), we can rewrite the ERE with (n, N ) = (2, 2) in the massless Thirring model as By substituting the boson-fermion duality (24) to (45), we obtain the exact formula of the ERE in the massless V can be written as the path integral over the two-sheeted manifold Σ2,2 with the proper gluing condition (35)  Thirring model: where S 2 (V, 0) is given by We should note that S 2 (V, 0) obtained via the bosonfermion duality is consistent with the known result S Free 2 (V ) in (39).For later convenience, we define the deviation ∆S 2 of the ERE from the free part as follows: Notice that ∆S 2 only depends on the cross-ratio x and the Thirring coupling λ.Then, we can easily check that ∆S 2 is invariant under the modular S transformation: which ensures that ∆S 2 takes the local maximum or minimum at x = 1/2.Furthermore, the deviation ∆S 2 is invariant under the T-duality transformation ( 27): Also, ∆S 2 reduces to a simpler form in special cases as shown below: • λ = 1 : In this case, we can explicitly write down ∆S 2 (x, λ) in term of x; . (51) • λ → −1 + δ : At this point, the system becomes unstable and ∆S 2 (x, λ) shows a logarithmic divergence; • x → 0 : When the cross-ratio x is close to zero (or, equivalently one), the ERE S 2 (V, λ) approaches the free one S 2 (V, 0) for any λ; • |λ| ≪ 1 : When the coupling λ is quite small, we can expand the analytic formula (46) in powers of λ.
After some manipulations, we arrive at the following result; We should remark that the leading term starts at the second order of λ.This can be understood as follows.In the small coupling region, the duality relation ( 27) is approximated by λ dual ≃ −λ, and the duality invariance (50) implies which excludes a linear term in λ in the expansion.
Finally, we mention the MRI with (n, N ) = (2, 2) defined by where V 1 and V 2 are disjoint unions of the entangling region . The first and second terms are nothing but the EREs with (n, N ) = (2, 1) given by (38).The third term is the ERE with (n, N ) = (2, 2), and the analytic result has already been obtained in (46) by the boson-fermion duality.Therefore, by substituting (38) and ( 46) into (56), we get the MRI with (n, N ) = (2, 2) in the massless Thirring model; where is the contribution from the free theory: In particular, when the cross-ratio x is very small and the Thirring coupling λ is in the range 1 − √ 3 < λ < 1 + √ 3, the MRI (57) can be approximated as follows; and clearly takes the positive value [36].We emphasize that the above result ( 59) is perfectly consistent with the universal asymptotic behavior of the MRI [18, equation (4.26)].

C. Parameter dependence of ERE and MRI
In section III B, we obtained the exact results of the ERE and MRI with (n, N ) = (2, 2) in the massless Thirring model.To grasp the physical meanings of these results, we explore the dependence of the ERE and MRI on the cross-ratio in section III C 1 and the Thirring coupling in section III C 2.

Cross-ratio dependence
We begin with examining the cross-ratio dependence of the ERE in (46) and the MRI in (57).In Fig. 4, we plot the deviation ∆S 2 by varying the cross-ratio x while fixing the Thirring coupling λ .Notice that both the upper and lower panels in Fig. 4 are symmetric under the map x → 1 − x.Also, ∆S 2 takes the maximum or minimum value at x = 1/2.These behaviors are in accordance with the modular S symmetry (49).Furthermore, we can readily check that ∆S 2 becomes zero at x = 0 (or, equivalently x = 1) for any values of λ.
To elucidate the physical meaning of this phenomenon, we rewrite the cross-ratio x defined by (40) as follows; where ℓ 1 , ℓ 2 and ℓ 3 are the lengths of the interval defined by (see also Fig. 5) From this expression, we can easily see that the limit x → 0 is equivalent to the following limits of the interval lengths; In the following, we concentrate on the two kinds of limits: ℓ 1 → 0 and ℓ 3 → ∞.
• ℓ 1 → 0: In the limit ℓ 1 → 0, the entangling region V becomes the single interval [u 2 , v 2 ], and the ERE with (n, N ) = (2, 2) in the massless Thirring model becomes the ERE with (n, N ) = (2, 1) in the free theory,[37] In this figure, we fix the interval lengths ℓ1 and ℓ2, hence the limits x → 0 and x → 1 are equivalent to the ones ℓ3 → ∞ and ℓ3 → 0, respectively.When x = 0, the two intervals are decoupled with each other, and the ERE can be written as the sum of the ones on the single interval (64).The contribution of the Thirring interaction to the ERE is locally maximized (or minimized) at x = 1/2.When x = 1, the two intervals are merged into the single interval with the length ℓ1 + ℓ2, and the ERE is given by (66).
hence the deviation ∆S 2 must vanish under the limit ℓ 1 → 0 by definition.
• ℓ 3 → ∞: In the limit ℓ 3 → ∞, the two intervals are decoupled to each other, and the correlation between these intervals tends to vanish.Therefore, the ERE with (n, N ) = (2, 2) is reduced to the sum of the EREs with (n, N ) = (2, 1) in the free theory as follows; That explains why ∆S 2 goes to zero in the limit ℓ 3 → ∞.
Here, it is instructive to compare the ERE in the limit x → 1 with (64).In a similar manner to x → 0, the limit x → 1 can be rephrased as We should notice that we can realize the limit x → 1 by putting the two intervals together: ℓ 3 → 0 with the interval lengths ℓ 1 and ℓ 2 fixed.In this case, the ERE with (n, N ) = (2, 2) in the massless Thirring model reduces to the ERE with (n, N ) = (2, 1) of the interval length ℓ 1 + ℓ 2 ; Figure 6 illustrates the behaviors of the ERE in varying the cross-ratio x (or the distance ℓ 3 between two intervals) while keeping the interval lengths ℓ 1 and ℓ 2 .Finally, we plot the MRI with (n, N ) = (2, 2) in (57) by changing the cross-ratio x in Fig. 7.In particular, from (64) and (66), the MRI with (n, N ) = (2, 2) behaves under the limits ℓ 3 → ∞ and ℓ 3 → 0 as follows; which are completely consistent with the plot in Fig. 7.

Coupling dependence
We move on to study the Thirring coupling dependence of the ERE in (46) and the MRI in (57) with keeping the cross-ratio x = 1/2.In Fig. 8, we plot ∆S 2 (x = 1/2, λ) by changing λ.Notice that the positive coupling λ ≥ 0 can be identified with the negative one λ dual = −λ/(1+λ) ≤ 0 via the T-duality (27).Therefore, it is sufficient to consider only the positive coupling region λ ≥ 0. Let λ max > 0 be the Thirring coupling that maximizes ∆S 2 (x = 1/2, λ).Then, we observe that ∆S 2 (x = 1/2, λ) monotonically increases in the range 0 ≤ λ ≤ λ max , and decreased in λ ≥ λ max .Also, we find out that there is a special point λ * ∼ = 1.244where ∆S 2 (x = 1/2, λ) = 0, namely the ERE in the massless Thirring model is equal to the one in the free theory.Such a special point λ * also appear for another value of cross-ratio x.Finally, we also plot the Thirring coupling dependence of the MRI in Fig. 8.The MRI in the massless Thirring model is always positive for any value of λ and shows a logarithmic divergence at λ = ∞.

D. Tripartite information
Let the entangling region V consist of three intervals A, B and C.Then, the n-th tripartite information In topological field theory, the tripartite information I 1 (A, B, C) has been studied as topological entanglement entropy [1].In holographic theory, I 1 (A, B, C) is shown to be negative [38,39].In contrast to I 1 (A, B, C), the n-th tripartite information for n ̸ = 1 has, however, not been well-investigated.In what follows, we consider the n = 2 case and examine the coupling dependence of the second tripartite information I 2 (A, B, C) in the massless Thirring model.We focus on the case where the two intervals B and C are adjacent, namely, the region B ∪ C is just a single interval (see Fig. 9).In this case, the second tripartite information I 2 (A, B, C) becomes a linear combination of the three mutual Rényi informations.
For the massless Thirring model, we have already derived the mutual Rényi information (57), thus we can calculate the second tripartite information I 2 (A, B, C) for B and C being adjacent.In Fig. 10, we plot the λ dependence of the second tripartite information for several configurations of the three intervals.We find that the second tripartite information vanishes at λ = 0, I 2 (A, B, C) = 0, which is consistent with the extensive properties of ERE for the free massless fermion [40].We also note that the the sign of the second tripartite information is indefinite, i.e., it takes negative value for small λ while it becomes positive for large λ.It may be intriguing to compare our result with the one obtained by the holographic formula [41].

IV. CONCLUSION AND DISCUSSION
In this paper, we derived the exact results on the ERE and MRI with (n, N ) = (2, 2) in the massless Thirring model by using the replica method and the boson-fermion duality.We first rewrote the ERE (42) with (n, N ) = (2, 2) in terms of the torus partition function of the massless Thirring model via the conformal map (43).Then, we exploited the boson-fermion duality between the massless Thirring model and a free compact boson theory, and expressed the torus partition function of the former in terms of those of the latter as in (23).We also explored the physical properties of the ERE and MRI with (n, N ) = (2, 2) in the massless Thirring model by examining the dependence on the cross-ratio and Thirring coupling in section III B and III C. The ERE grows as λ increases in the small λ regions while it takes the maximum value at the critical point λ = λ max and starts to decrease monotonically in the large λ region.It would be interesting to check if this result can be reproduced in a perturbative calculation with respect to the Thirring coupling λ.On the other hand, we observe that the MRI is non-negative and increases monotonically in the large λ region.Since the MRI quantifies the correlation between two subsystems, this behavior is accordance with our intuition such that quantum entanglement between the two intervals becomes large as the coupling λ increases.
This work opens several interesting directions for further studies.While we only dealt with the massless Thirring model, this serves as the first step toward applying the boson-fermion duality to examine the ERE and MRI.We believe that the boson-fermion duality will become a fundamental tool for comprehending quantum information quantities and will contribute to a deeper understanding of the dynamics of QFTs.
We can generalize our analysis to include multiple replica sheets or intervals.As mentioned in Introduction, the n-sheeted manifold Σ n,N transforms into a Riemann surface of genus g = (n − 1)(N − 1) through a conformal map, and the boson-fermion duality also applies to such cases [27].Since the partition function of a free compact boson on a higher genus Riemann surface has been previously studied in [24,25,42], it is possible to extend our work to higher genus cases given a conformal map from a singular Riemann surface Σ n,N to a smooth one as in [34] for the n = N = 2 case.Furthermore, our work may be extended to the Thirring model with a mass term.Under the boson-fermion duality, the fermion mass term is known to be mapped to the sine-Gordon potential [29].When the mass term is small, we can take into account the mass-correction to the ERE and MRI as a relevant perturbation of CFT.(For the free massive fermion theory, a perturbative calculation has been performed in [14,43].)It is intriguing to apply the conformal perturbation theory to the massive Thirring model and derive the corrections to the ERE and MRI obtained in this paper.
In this paper, we delved into the specific spin structure ρ = PP on the two-sheeted manifold Σ 2,2 since it arises in the (n, N ) = (2, 2) ERE due to the gluing condition (35) in the replica method.Changing the spin structure ϱ = PP to another may be achieved by the insertion of the Z 2 fermion parity operator (−1) F C which acts on a fermionic field in a subregion C as (−1) F C : ψ(x) → −ψ(x) for x ∈ C, and quantum information measures associated with the other spin structures ϱ = AA, AP, PA on Σ 2,2 are discussed in [44,45].Our method can also be adapted to the calculations of such measures without difficulty.
In the moduli space of c = 1 bosonic theory, there are some special points where supersymmetries become enhanced.On the one hand, the n-th supersymmetric Rényi entropy [46] of arbitrary intervals in the N = (2, 2) superconformal field theory has been derived in [47,48].It may be interesting to compare the result obtained in this paper with the supersymmetric Rényi entropy.
[Left].The two-sheeted manifold Σ2,2 is conformally equivalent to the torus T [Right].Fermions are subject to periodic (anti-periodic) boundary conditions along the two cycles in the figure on Σ2,2 (T).

I
n (A, B, C) is defined by I n (A, B, C) ≡ I n (A, B) + I n (A, C) − I n (A, B ∪ C) .

15 FIG. 10 .
FIG. 10.The Thirring coupling dependence of the second tripartite information I2(A, B, C).We plot the three different configurations of the regions A, B and C.