Multi-charged moments and symmetry-resolved Rényi entropy of free compact boson for multiple disjoint intervals

The phenomenon of entanglement has proved to be a cornerstone of quantum theories. Entanglement has been at the centre of the recent development in many areas of physics, especially in quantum computation nielsen2010quantum , critical quantum many-body systems amico2008entanglement , gauge-gravity duality ryu2006holographic ; ryu2006aspects and black-hole entropy solodukhin2011entanglement . In $1$d critical systems, entanglement for a single interval is sensitive to the central charge of the corresponding conformal field theory vidal2003entanglement . To study entanglement in the state $|{\Psi}\rangle$ of a quantum system, the system is first partitioned into two subsystems, subsystem $A$ and its complement $B$, such that the Hilbert space is $\mathcal{H}=\mathcal{H}_{A}\otimes\mathcal{H}_{B}$. Among several measures of entanglement, the Rényi entropies are the most prominent. Rényi entropies $S_{n}$ are given by

where $n$ is a positive integer, and the reduced density matrix $\rho_{A}$ is given by $\mathrm{Tr}_{B}|{\Psi}\rangle\langle{\Psi}|$. The $n\to 1$ limit of eq.(1) is just the entanglement entropy. The Rényi entropies for a $2$d conformal field theories with a central charge $c$ are found to be proportional to the central charge $c$ and scales as the logarithm of length of the subsystem $A$ holzhey1994geometric ; calabrese2004entanglement ; calabrese2009entanglement .

We now consider the scenario when the subsystem $A$ is composed of multiple disjoint intervals $A_{i}$, $A=\cup_{i}A_{i}$. Entanglement in $2$d conformal field theories in this scenario now also becomes sensitive to the local conformal operator content as well calabrese2009entanglement2 ; coser2014renyi . The evaluation of Rényi entropies in field theories involves computing the partition function on the replica surface. In the present case, the replica surface is a Riemann surface with a genus $g>0$. The partition function of modular invariant conformal field theories on such surfaces shows interesting behaviour and thus so does entanglement. Entanglement measures have been substantially investigated for disjoint intervals in critical systems furukawa2009mutual ; calabrese2009entanglement2 ; calabrese2011entanglement ; rajabpour2012entanglement ; coser2014renyi ; alba2011entanglement ; coser2016spin ; fagotti2010entanglement ; calabrese2012entanglement ; coser2016towards ; headrick2013bose ; casini2005entanglement .

Entanglement studies for quantum systems with global internal symmetries can be made more refined. The entanglement measures of such systems for certain states decompose into the local charge sectors of the subsystem $A$ corresponding to the global symmetry. The studies which resolve entanglement into the local charge sectors have been termed symmetry-resolved entanglement. Since the seminal work of goldstein2018symmetry , similar studies on symmetry-resolved entanglement have been made in different contexts. In critical systems, symmetry resolution of entanglement entropies xavier2018equipartition ; turkeshi2020entanglement ; bonsignori2019symmetry ; fraenkel2020symmetry ; ares2022symmetry ; jones2022symmetry ; 2023 ; barghathi2019operationally ; barghathi2018renyi ; ghasemi2023universal ; murciano2020entanglement ; murciano2020symmetry ; murciano2020symmetry1 ; ares2022multi ; foligno2023entanglement ; capizzi2022entanglement ; horvath2021u ; capizzi2022renyi ; capizzi2022renyi2 ; capizzi2023full ; parez2021exact ; parez2021quasiparticle ; estienne2021finite , relative entropies capizzi2021symmetry ; chen2021symmetry , operator entanglement rath2023entanglement ; murciano2023more and negativity cornfeld2018imbalance ; murciano2021symmetry ; gaur2023charge ; chen2022charged ; chen2022dynamics ; chen2023dynamics ; feldman2019dynamics ; parez2022dynamics ; berthiere2023reflected have been studied largely for $U(1)$ symmetry. Symmetry resolved entanglement of Wess-Zumino-Witten models have also been studied in calabrese2021symmetry . Symmetry resolved entanglement of $2$d CFTs have also been shown to have an interesting relation with the boundary conformal field theory di2023boundary ; kusuki2023symmetry ; PhysRevLett.131.151601 . Similar studies have been made in the context of gravity as well zhao2021symmetry ; weisenberger2021symmetry ; zhao2022charged ; belin2013holographic ; milekhin2021charge ; gaur2023symmetry . The protocols for measuring symmetry resolved entanglement have been discussed in neven2021symmetry ; cornfeld2018imbalance .

In the present work, we study the symmetry-resolved Rényi entropies of free compact boson for an arbitrary number of disjoint intervals. We also evaluate the multi-charged moments of the free compact boson in the same settings. The multi-charged moments obtained here are essential for studying symmetry-resolved mutual information. Mutual information measures quantify the entanglement between the intervals themselves, however, they are not a pure measure of correlations only and hence can be negative sometimes kormos2017temperature .

To set the stage for calculations later, we consider $N$ disjoint intervals $A_{i}$ whose boundary points are denoted $u_{i}$ and $v_{i}$, see Figure 1. The length of each interval $A_{i}$ is denoted $\ell_{i}$.

The organisation of this work is as follows. In Section 2 we discuss the symmetry-resolved Rényi entropy, and the multi-charged moments of theories with a global $U(1)$ symmetry and also discuss the replica method for multi-charged moments in CFTs. In Section 3, we discuss the theory of free compact boson and review the Rényi entropy results in the theory. In Section 4, we evaluate the charged and multi-charged moments of free compact boson for multiple disjoint intervals. In Section 5, we evaluate the symmetry resolved Rényi entropy of free compact boson in the same settings. Finally, in Section 6 we give a conclusion of the present work. We also have four appendices. In Appendices A, B, and D some necessary computations and calculations are discussed. In Appendix C we evaluate the multi-charged moments of massless Dirac fermion for multiple disjoint intervals.

In this section we discuss the symmetry resolution of Rényi entropy for theories with a global $U(1)$ symmetry goldstein2018symmetry . We also discuss the charged and the multi-charged moments in the same context. Finally, we revisit the replica trick for evaluating the multi-charged moments in CFTs.

In this section, we discuss the theory of $2\mathrm{d}$ free compact boson. We also revisit the results for the Rényi entropy of free compact boson when $N\geq 2$.

Free compact boson in $2\mathrm{d}$ is a conformally invariant theory with central charge $c=1$. It is also the continuum theory of Luttinger liquid. The theory is described by the Lagrangian

The target space of the theory is a circle of circumference $2\pi R$, where the compactification radius $R$ is related to the Luttinger parameter $K$ by $K=\sqrt{\frac{2}{R}}$. The theory has a duality under $R\to{2}/{R}$ and is self-dual at $R=\sqrt{2}$. This duality is known as the T-duality.

Free compact boson possesses $U(1)\times U(1)$ symmetry. The theory is invariant under the translations in the target space $\varphi\to\varphi+a$, this corresponds to a $U(1)$ symmetry, sometimes denoted $U_{shift}(1)$. The second $U(1)$ symmetry is due to the conserved current $j^{\mu}=\frac{1}{2\pi}\epsilon^{\mu\nu}\partial_{\nu}\varphi$ in $2\mathrm{d}$. We denote this second $U(1)$ symmetry by $U_{wind}(1)$. Under the T-duality $(R\to 2/R)$ we have the correspondence between the symmetries: $U_{shift}(1)\to U_{wind}(1)$ and $U_{wind}(1)\to U_{shift}(1)$. In the present work we focus on $U_{wind}(1)$ at arbitrary compactification radius $R$.

The $n=2$ Rényi entropy for two disjoint intervals case was first studied in furukawa2009mutual , and this result was generalised to arbitrary integer values of $n$ in calabrese2009entanglement2 . The Rényi entropies for multi-interval case was studied in coser2014renyi ; headrick2013bose . The $n^{th}$ moment of the reduced density matrix for arbitrary integer values of $n$ and $N$ is given by the generalised expression

where $c_{N,n}$ is a non-universal constant and $y_{ij}$ are the cross-ratios

The factor $\mathcal{F}_{N,n}$ comes from the local conformal algebra (i.e. it is not fixed by the global conformal invariance) and is a function of the cross-ratios $x_{i}$ (introduced later in eq.(25)). It is given by

In the above equation, $\Theta$ is the Riemann Siegel theta function. A $k-$dimensional theta function is defined as

where the characteristics $\boldsymbol{\varepsilon}$, $\boldsymbol{\delta}\in\left(\mathbb{Z}/2\right)^{k}$ and $\boldsymbol{u}\in\mathbb{C}^{k}$. In eq.(19) $\Omega$ is a $k\times k$ symmetric matrix with a positive definite imaginary part. When $\varepsilon=\delta=0$, the theta function in eq.(19) is denoted $\Theta(\boldsymbol{u}|\Omega)$.

In eq.(18), $\tau$ is the Riemann period matrix associated with the Riemann surface of the $n^{th}$ moment of the reduced density matrix. It is a $g\times g$ matrix, where $g$ is the genus of the Riemann surface. Let us write $\tau$ as

where $\mathcal{R}$ and $\mathcal{I}$ are $g\times g$ real symmetric matrices. Then the matrix $T$ in eq.(18) is given by

where $K$ is the Luttinger parameter introduced earlier. We note from eq.(16), that $Z_{N,n}$ is invariant under $K\to 1/K$, as we would expect from the T-duality of compact boson. It has been noted in headrick2013bose , that $Z_{N,n}$ at self-dual radius matches the corresponding $Z_{N,n}$ of the Dirac fermion when $\mathcal{R}$ in eq.(20) vanishes. This happens in particular for all values of $n$ when $N=2$, and similarly for all values of $N$ when $n=2$. It should also be noted that the factor $\mathcal{F}_{N,n}$ becomes unity at self-dual radius when $\mathcal{R}$ vanishes.

In this section, we obtain the multi-charged and charged moments of free compact boson for an arbitrary number of disjoint intervals.

The multi-charged moments of free compact boson in two disjoint intervals case for Rényi entropy was first obtained in ares2022multi , these moments were then used to study the symmetry resolved Rényi entropy and mutual information. In gaur2023charge , these quantities were obtained for Rényi negativity to study the symmetry-resolved Rényi negativity in the same settings.

The flux generating vertex operators $\mathcal{V}_{\alpha}$ in (13) for free compact boson are just the boson vertex operators goldstein2018symmetry

with the scaling dimension $h_{\alpha}^{\mathcal{V}}$

To simplify our calculations we first use the global conformal invariance to map the points $u_{1}\to 0$, $u_{N}\to 1$, and $v_{N}\to\infty$ using the map

Let’s denote the image of all the boundary points under this map as

where $j\in\{1,2,\cdots N\}$. We have the order $0<x_{1}<x_{2}\cdots<x_{2N-3}<1$ and we also keep in mind the notations $x_{0}=0$, $x_{2N-2}=1$, and $x_{2N-1}=\infty$. We must now compute the correlation function for the Vertex operators at $x_{i}$ on the Riemann surface $\Sigma_{N,n}(\boldsymbol{x})$ (see Appendix A), i.e. $\left\langle\prod_{i=1}^{N}\mathcal{V}_{\alpha_{j}}(x_{2j-2})\mathcal{V}_{-\alpha_{j}}(x_{2j-1})\right\rangle$. The Riemann surface $\Sigma_{N,n}(\boldsymbol{x})$ has the genus $g=(N-1)\times(n-1)$. The correlation functions of vertex operators have been studied extensively in the string theory literature verlinde1987chiral ; eguchi1987chiral . A general $M-$point correlation function is given by

The map $\boldsymbol{w}$ is a $g$ dimensional vector and is known as the Abel-Jacobi map. It is defined from the Riemann surface $\Sigma_{N,n}(\boldsymbol{x})$ to its Jacobian torus $J\left(\Sigma_{N,n}(\boldsymbol{x})\right)$. For the Riemann surface $\Sigma_{N,n}(\boldsymbol{x})$, we define its Jacobian lattice $\Lambda=\mathbb{Z}^{g}+\tau(\boldsymbol{x})\mathbb{Z}^{g}$, where $\tau(\boldsymbol{x})$ (given by eq.(79)) is the period matrix of $\Sigma_{N,n}(\boldsymbol{x})$. The Jacobian torus is then defined as the quotient space $\mathbb{C}^{g}/\Lambda$. The Abel-Jacobi map $\boldsymbol{w}$ is given by

where $j\in(1,2,\cdots,(N-1))$, and $k\in(1,2,\cdots,(n-1))$ and we have chosen $z=0$ as the reference point. The quantities $\nu_{j,r}(z)$ are the normalised holomorphic differentials given by eq.(78) in Appendix A.

The quantity $E(z_{i},z_{i^{\prime}})$ is the Prime form of the Riemann surface $\Sigma_{n}(\boldsymbol{x})$ and is given by fay2006theta ; mumford2007tata

where ${\boldsymbol{\Delta}}=({\boldsymbol{\varepsilon}},{\boldsymbol{\delta}})$, where $\boldsymbol{\varepsilon}$, and $\boldsymbol{\delta}$ were defined below eq.(19). However, here $\boldsymbol{\Delta}$ must be a non-singular odd half characteristic. The prime form $E(z_{i},z_{i^{\prime}})$ is independent of the choice of $\boldsymbol{\Delta}$. The quantity $h_{\boldsymbol{\Delta}}(z_{i})$ is a holomorphic 1-form and is given by

In our case, $z_{i}$ are the branch points and the normalised holomorphic differentials in eq.(29) are singular. This makes the vertex operator correlation function in eq.(26) ill-defined. This issue is resolved by introducing the regularised vertex operators $\mathcal{V}^{(*)}_{\alpha_{i}}(z_{i})$ ares2022multi . The normalised holomorphic differentials $\nu_{j,r}(z_{i})$ exhibit the leading order singular behaviour as $z\to z_{i}$

where $\nu^{(*)}_{j,r}(z_{i})$ is non-singular at the branch points. Using $\nu^{(*)}_{j,r}(z_{i})$ in eq.(29), and eq.(28) we may define the regularised holomorphic $1-$forms and regularised prime forms

This leads us to define the regularised Vertex operators $\mathcal{V}^{(*)}_{\alpha_{i}}(z_{i})$

where $\kappa_{n}$ is a surface-dependent constant. The constant $\kappa_{n}$ will be fixed later, but we will find that it is independent of $N$. The correlation function of the regularised vertex operators at branch points is well defined, and is given by

where $z_{i}=x_{i}$. The flux is to be identified as $\beta_{2j-2}=\alpha_{j}$, and $\beta_{2j-1}=-\alpha_{j}$ in the equation above, where $j\in\{1,2,\cdots,N\}$.

In the rest of this section, we will evaluate this correlation function for arbitrary values of $N$ and $n$. We first revisit the two disjoint intervals case, we will use a different canonical homology basis than ares2022multi . We then extend these results to $N>2$.