Multi-charged moments of two intervals in conformal field theory

As we already pointed out in Sec. 1, we take a spatial bipartition $A\cup B$ of an extended quantum system in a pure state $|\Psi\rangle$, with $A$ made of two disconnected regions, $A=A_{1}\cup A_{2}$. We assume that the system is endowed with a global $U(1)$ symmetry generated by a local charge $Q$. Given the partition of the system in different subsets, we can consider the charge operator in each of them; for example, in region $A$, it can be obtained as $Q_{A}=\mathop{\mbox{Tr}}\nolimits_{B}(Q)$. If $|\Psi\rangle$ is an eigenstate of $Q$, the density matrix $\rho=\ket{\Psi}\bra{\Psi}$ commutes with $Q$, i.e. $[Q,\rho]=0$, and, by taking the trace over $B$, we find that $[Q_{A},\rho_{A}]=0$. This implies that the reduced density matrix $\rho_{A}$ presents a block diagonal structure, in which each block corresponds to an eigenvalue $q\in\mathbb{Z}$ of $Q_{A}$. That is,

$$\rho_{A}=\bigoplus_{q}\Pi_{q}\rho_{A}=\bigoplus_{q}\left[p(q)\rho_{A}(q)\right],$$

(4)

where $\Pi_{q}$ is the projector onto the eigenspace associated to the eigenvalue $q$ and $p(q)=\mathop{\mbox{Tr}}\nolimits\left(\Pi_{q}\rho_{A}\right)$ is the probability of obtaining $q$ as the outcome of a measurement of $Q_{A}$. Notice that Eq. (4) guarantees the normalisation $\mathrm{Tr}[\rho_{A}(q)]=1$ for any $q$.

The amount of entanglement between $A$ and $B$ in each symmetry sector can be quantified by the symmetry-resolved Rényi entropies, defined as

$$S_{n}^{A}(q)\equiv\frac{1}{1-n}\ln\mathrm{Tr}[\rho_{A}(q)^{n}].$$

(5)

Taking the limit $n\to 1$ in this expression, we obtain the symmetry-resolved entanglement entropy,

$$S_{1}^{A}(q)\equiv-\mathrm{Tr}[\rho_{A}(q)\ln\rho_{A}(q)].$$

(6)

According to the decomposition of Eq. (4), the total entanglement entropy in Eq. (2) can be written as nc-10

$$S_{1}^{A}=\sum_{q}p(q)S_{1}^{A}(q)-\sum_{q}p(q)\ln p(q)\equiv S_{\textrm{c}}^{A}+S_{\textrm{num}}^{A},$$

(7)

where $S_{\textrm{c}}$ is known as configurational entropy and quantifies the average contribution to the total entanglement of all the charge sectors fis ; wv-03 ; bhd-18 ; bcd-19 , while $S_{\textrm{num}}$ is called number entropy and takes into account the entanglement due to the fluctuations of the value of the charge within the subsystem $A$ fis ; kusf-20 ; kusf-20b ; ms-20 ; kufs-21 ; zshgs-20 ; kufs-21b .

Since the total charge in $A$ is the sum of the charge in $A_{1}$ and $A_{2}$, $Q_{A}=Q_{A_{1}}+Q_{A_{2}}$, then the reduced density matrices $\rho_{A_{1}}$, $\rho_{A_{2}}$ of $A_{1}$ and $A_{2}$ can be independently decomposed in charged sectors as we did for $\rho_{A}$ in Eq. (4). Therefore, we can define the symmetry-resolved entropies $S_{n}^{A_{1}}(q_{1})$, $S_{n}^{A_{2}}(q_{2})$ for the regions $A_{1}$ and $A_{2}$ analogous to Eq. (5) for $A$, with $q=q_{1}+q_{2}$. In Ref. pbc-21 , it has been proposed to define the symmetry-resolved mutual information as

$$I^{A_{1}:A_{2}}(q)=\sum_{q_{1}=0}^{q}p(q_{1},q-q_{1})\left[S_{1}^{A_{1}}(q_{1})+S_{1}^{A_{2}}(q-q_{1})\right]-S_{1}^{A}(q).$$

(8)

The quantity $p(q_{1},q-q_{1})$, normalised as

$$\sum_{q_{1}=0}^{q}p(q_{1},q-q_{1})=1,$$

(9)

is the probability that a simultaneous measurement of the charges $Q_{A_{1}}$ and $Q_{A_{2}}$ yields $q_{1}$ and $q-q_{1}$, respectively, while the charge of the whole system $A$ is fixed to $q$. Although Eq. (8) is a natural definition, $I^{A_{1}:A_{2}}(q)$ is not in general a good measure of the total correlations between $A_{1}$ and $A_{2}$ within each charge sector since, in some cases, it can be negative pbc-21 . Nevertheless, we find interesting to investigate this quantity given that it provides a decomposition for the total mutual information (8) similar to the one reported in Eq. (7) for the entanglement entropy,

$$I^{A_{1}:A_{2}}=\sum_{q}p(q)I^{A_{1}:A_{2}}(q)+I^{A_{1}:A_{2}}_{\textrm{num}},$$

(10)

where $I^{A_{1}:A_{2}}_{\textrm{num}}\equiv S^{A_{1}}_{\textrm{num}}+S^{A_{2}}_{\textrm{num}}-S^{A}_{\textrm{num}}$ is the number mutual information.

The computation of the symmetry-resolved entanglement entropies and mutual information from the definitions (5) and (8) requires the knowledge of the entanglement spectra of $\rho_{A}$, $\rho_{A_{1}}$ and $\rho_{A_{2}}$ and their symmetry resolution. However, this is usually a very difficult task, in particular if one is interested in analytical expressions. Alternatively, one can employ the charged moments of the reduced density matrices. For $\rho_{A}$, they are defined as

$$Z_{n}^{A}(\alpha)=\mathop{\mbox{Tr}}\nolimits[\rho_{A}^{n}e^{i\alpha Q_{A}}].$$

(11)

Similar quantities can also be introduced for the two subsystems $A_{1}$ and $A_{2}$ that constitute $A$ by replacing $\rho_{A}$ and $Q_{A}$ by $\rho_{A_{p}}$ and $Q_{A_{p}}$, $p=1,2$. If we take now their Fourier transform,

$$\mathcal{Z}_{n}^{A}(q)=\int_{-\pi}^{\pi}\frac{{\rm d}\alpha}{2\pi}e^{-i\alpha q}Z_{n}^{A}(\alpha),$$

(12)

the symmetry-resolved entanglement entropies of the subsystem $A$ are given by goldstein ; xavier

$$S_{n}^{A}(q)=\frac{1}{1-n}\ln\left[\frac{\mathcal{Z}_{n}^{A}(q)}{\left(\mathcal{Z}_{1}^{A}(q)\right)^{n}}\right].$$

(13)

In a similar manner, replacing $A$ with $A_{1}$ and $A_{2}$, we can obtain the symmetry-resolved entanglement entropies of the two components of $A$.

Notice that, for computing the symmetry-resolved mutual information of Eq. (8), we need to determine $p(q_{1},q-q_{1})$, i.e. the probability that a measurement of $Q_{A_{1}}$ and $Q_{A_{2}}$ gives $q_{1}$ and $q-q_{1}$ respectively, with $Q_{A}$ fixed to $q$. In order to calculate it, we consider the generalisation of the charged moments in Eq. (11) introduced for the first time in Ref. pbc-21 ,

$$Z_{n}^{A_{1}:A_{2}}(\alpha,\beta)=\mathrm{Tr}\left[\rho_{A}^{n}e^{i\alpha Q_{A_{1}}+i\beta Q_{A_{2}}}\right].$$

(14)

We refer to them as multi-charged moments. When $\alpha=\beta$, Eq. (14) reduces to the charged moments of $A=A_{1}\cup A_{2}$ of Eq. (11). If we take the Fourier transform of Eq. (14),

$$\mathcal{Z}_{n}^{A_{1}:A_{2}}(q_{1},q_{2})=\int_{-\pi}^{\pi}\frac{{\rm d}\alpha}{2\pi}\frac{{\rm d}\beta}{2\pi}e^{-i\alpha q_{1}-i\beta q_{2}}Z_{n}^{A_{1}:A_{2}}(\alpha,\beta),$$

(15)

then $\mathcal{Z}_{1}^{A_{1}:A_{2}}(q_{1},q_{2})$ can be interpreted as the probability of having $q_{1}$ and $q_{2}$ as outcomes of a measurement of $Q_{A_{1}}$ and $Q_{A_{2}}$ respectively, independently of the value of $Q_{A}$. Therefore, it satisfies the normalisation

$$\sum_{q_{1},q_{2}}\mathcal{Z}_{1}^{A_{1}:A_{2}}(q_{1},q_{2})=1,$$

(16)

and $p(q_{1},q-q_{1})$ can be calculated as the conditional probability

$$p(q_{1},q-q_{1})=\frac{\mathcal{Z}_{1}^{A_{1}:A_{2}}(q_{1},q-q_{1})}{p(q)},$$

(17)

which fulfills Eq. (9).

In the rest of the paper, we will analyse the previous quantities in $1+1$-dimensional CFTs with a global $U(1)$ symmetry. We will assume that the entire system is in the ground state and that the spatial dimension is an infinite line which we will divide into two parts $A$ and $B$, with $A$ made up of two disjoint intervals, namely $A=A_{1}\cup A_{2}=[u_{1},v_{1}]\cup[u_{2},v_{2}]$. If we denote by $\ell_{1}$ and $\ell_{2}$ the lengths of the two intervals and $d$ their separation, we have

$$\ell_{1}=|v_{1}-u_{1}|,\qquad\ell_{2}=|v_{2}-u_{2}|,\qquad d=|u_{2}-v_{1}|,\qquad x=\frac{\ell_{1}\ell_{2}}{(d+\ell_{1})(d+\ell_{2})},$$

(18)

where we have also introduced the cross ratio $x$ of the four end-points, which takes values between $0$ and $1$.

As explained in detail in Refs. cc-04 ; cc-09 , using the path integral representation of $\rho_{A}$, the moments $Z_{n}^{A}(0)=\mathop{\mbox{Tr}}\nolimits[\rho_{A}^{n}]$ are equal to the partition function of the CFT on a Riemann surface, which we call $\Sigma_{n}$, obtained as follows. We take the complex plane where the CFT is originally defined and we perform two cuts along the intervals $A_{1}=(u_{1},v_{1})$ and $A_{2}=(u_{2},v_{2})$. Then we replicate $n$ times the cut plane and we glue the copies together along the cuts in a cyclical way as we illustrate in Fig. 1. We eventually obtain an $n$-sheeted Riemann surface of genus $n-1$, which is symmetric under the $\mathbb{Z}_{n}$ cyclic permutation of the sheets.

Alternatively, instead of replicating the space-time where the CFT is initially defined, one can take $n$ copies of the CFT on the complex plane and quotient it by the $\mathbb{Z}_{n}$ symmetry under the cyclic exchange of the copies. We then get the orbifold theory ${\rm CFT}^{\otimes n}/\mathbb{Z}_{n}$. The moments $Z_{n}^{A}(0)$ are equal to the four-point correlation function on the complex plane cc-04 ; ccd-08 ,

$$Z_{n}^{A}(0)=\langle\tau_{n}(u_{1})\tilde{\tau}_{n}(v_{1})\tau_{n}(u_{2})\tilde{\tau}_{n}(v_{2})\rangle,$$

(19)

where $\tau_{n}$ and $\tilde{\tau}_{n}$ are dubbed as twist and anti-twist fields k-87 ; dixon ; cc-04 ; ccd-08 . They implement in the orbifold the multivaluedness of the correlation functions on the surface $\Sigma_{n}$ when we go around its branch points. In fact, the winding around the point where $\tau_{n}$ ($\tilde{\tau}_{n}$) is inserted maps a field $\mathcal{O}_{k}$ living in the copy $k$ of the orbifold into the copy $k+1$ ($k-1$), that is

$$\tau_{n}(u)\mathcal{O}_{k}(e^{2\pi i}(z-u))=\mathcal{O}_{k+1}(z-u)\tau_{n}(u).$$

(20)

The twist and anti-twist fields are spinless primaries with conformal weight

$$h_{n}^{\tau}=\frac{c}{24}\left(n-\frac{1}{n}\right),$$

(21)

where $c$ is the central charge of the initial CFT.

The charged moments of $\rho_{A}$ can also be computed employing the previous frameworks. As argued in Ref. goldstein , the operator $e^{i\alpha Q_{A}}$ can be interpreted as a magnetic flux between the sheets of the surface $\Sigma_{n}$, such that a charged particle moving along a closed path that crosses all the sheets acquires a phase $e^{i\alpha}$. For the multi-charged moments introduced in Eq. (14), we have to insert two different magnetic fluxes $\alpha$ and $\beta$ at the cuts $A_{1}$ and $A_{2}$ respectively, as we pictorially show in Fig. 1. They can be implemented by a local $U(1)$ operator $\mathcal{V}_{\alpha}(x)$ that generates a phase shift $e^{i\alpha}$ along the real interval $[x,\infty)$. Then the charged moments are equal to the four-point correlation function on the surface $\Sigma_{n}$

$$Z_{n}^{A_{1}:A_{2}}(\alpha,\beta)=Z_{n}^{A}(0)\langle\mathcal{V}_{\alpha}(u_{1})\mathcal{V}_{-\alpha}(v_{1})\mathcal{V}_{\beta}(u_{2})\mathcal{V}_{-\beta}(v_{2})\rangle_{\Sigma_{n}}.$$

(22)

In the orbifold theory, the magnetic flux can be incorporated by considering the composite twist field $\tau_{n,\alpha}\equiv\tau_{n}\cdot\mathcal{V}_{\alpha}$. Thus, if we take a field $\mathcal{O}_{k}$ in the copy $k$ of the orbifold, then the winding $(z-u)\mapsto e^{2\pi i}(z-u)$ around the point $u$ where $\tau_{n,\alpha}$ is inserted gives rise to a phase $e^{i\alpha/n}$,

$$\tau_{n,\alpha}(u)\mathcal{O}_{k}(e^{2\pi i}(z-u))=e^{i\alpha/n}\mathcal{O}_{k+1}(z-u)\tau_{n,\alpha}(u).$$

(23)

The same applies to the composite anti-twist field $\tilde{\tau}_{n,\alpha}\equiv\tilde{\tau}_{n}\cdot\mathcal{V}_{\alpha}$, which takes a field from the copy $k$ to $k-1$ adding a phase $e^{i\alpha/n}$. Therefore, Eq. (22) can be re-expressed as the four-point function on the complex plane

$$Z_{n}^{A_{1}:A_{2}}(\alpha,\beta)=\langle\tau_{n,\alpha}(u_{1})\tilde{\tau}_{n,-\alpha}(v_{1})\tau_{n,\beta}(u_{2})\tilde{\tau}_{n,-\beta}(v_{2})\rangle.$$

(24)

In Ref. goldstein , it is shown that, if $\mathcal{V}_{\alpha}$ is a spinless primary operator with conformal weight $h_{\alpha}^{\mathcal{V}}$, then so are the composite twist and anti-twist fields, with conformal weights

$$h_{n,\alpha}=h_{n}^{\tau}+\frac{h_{\alpha}^{\mathcal{V}}}{n}.$$

(25)

One can further consider other configurations for the magnetic fluxes between the sheets of the Riemann surface $\Sigma_{n}$. In general, if we assume that a particle gets a different phase $e^{i\alpha_{j}}$ when it goes around each branch point, provided they satisfy the neutrality condition $\alpha_{1}+\alpha_{2}+\alpha_{3}+\alpha_{4}=0$, then the partition function of this theory is given by

$$Z_{n}^{A_{1}:A_{2}}(\{\alpha_{j}\})=Z_{n}^{A}(0)\langle\mathcal{V}_{\alpha_{1}}(u_{1})\mathcal{V}_{\alpha_{2}}(v_{1})\mathcal{V}_{\alpha_{3}}(u_{2})\mathcal{V}_{\alpha_{4}}(v_{2})\rangle_{\Sigma_{n}},$$

(26)

or, in terms of the composite twist fields, by

$$Z_{n}^{A_{1}:A_{2}}(\{\alpha_{j}\})=\langle\tau_{n,\alpha_{1}}(u_{1})\tilde{\tau}_{n,\alpha_{2}}(v_{1})\tau_{n,\alpha_{3}}(u_{2})\tilde{\tau}_{n,\alpha_{4}}(v_{2})\rangle.$$

(27)

Then the multi-charged moments $Z_{n}^{A_{1}:A_{2}}(\alpha,\beta)$ can be treated as the particular case in which $\alpha_{1}=-\alpha_{2}=\alpha$ and, due to the neutrality condition, $\alpha_{3}=-\alpha_{4}=\beta$.

In Sec. 3, we will compute the charged moments $Z_{n}^{A_{1}:A_{2}}(\alpha,\beta)$—and more in general the partition functions $Z_{n}^{A_{1}:A_{2}}(\{\alpha_{j}\})$—for the massless Dirac fermion using the orbifold theory ${\rm CFT}^{\otimes n}/\mathbb{Z}_{n}$. On the other hand, in Sec. 4, we will adopt a geometric approach to obtain the multi-charged moments of the compact boson from the correlation function on the Riemann surface $\Sigma_{n}$ of Eq. (26).

In Sec. 2, we explained that the partition function $Z_{n}^{A_{1}:A_{2}}(\{\alpha_{j}\})$ can be obtained either by considering the theory on a complicated Riemann surface or by replicating it $n$-times and working with the orbifold on the complex plane. For the massless Dirac field theory, the latter approach is more convenient. Thus, let us take the $n$-component field

$$\Psi=\begin{pmatrix}\psi_{1}\\ \psi_{2}\\ \vdots\\ \psi_{n}\end{pmatrix},$$

(30)

where $\psi_{j}$ is the Dirac field on the $j$-th copy of the system. Eq. (23) describes the effect of the composite twist fields on the components of $\Psi$ when going around the end-points of the subsystem $A=A_{1}\cup A_{2}$. This transformation can be encoded in the matrix

$$T_{a}=\begin{pmatrix}0&e^{ia/n}&&\\ &0&e^{ia/n}&\\ &&\ddots&\ddots\\ (-1)^{n-1}e^{ia/n}&&0\end{pmatrix}.$$

(31)

In the general case of Eq. (27), $\Psi$ transforms according to $T_{\alpha_{2p-1}}$ when winding around the point $u_{p}$ and to the transpose matrix $T_{\alpha_{2p}}^{t}$ when going around the point $v_{p}$, with $p=1,2$. The matrix $T_{a}$ in Eq. (31), sometimes called twist matrix, was introduced for the case $a=0$ in Refs. CFH ; ccd-08 and for general $a$ in Ref. MDC-20 . Its eigenvalues are of the form

$$t_{k}=e^{ia/n}e^{2\pi ik/n},\qquad k=-\frac{n-1}{2},\dots,\frac{n-1}{2}.$$

(32)

By simultaneously diagonalising all the $T_{\alpha_{j}}$ with a unitary transformation (which is independent of $\alpha_{j}$), we can recast the replicated theory in $n$ decoupled fields $\psi_{k}$ on the plane, which are multi-valued,

	$\displaystyle\psi_{k}(e^{i2\pi}(z-u_{p}))$	$\displaystyle=$	$\displaystyle e^{i\alpha_{2p-1}/n}e^{2\pi ik/n}\psi_{k}(z-u_{p}),$
	$\displaystyle\psi_{k}(e^{i2\pi}(z-v_{p}))$	$\displaystyle=$	$\displaystyle e^{i\alpha_{2p}/n}e^{-2\pi ik/n}\psi_{k}(z-v_{p}).$		(33)

Notice that this technique, known as diagonalisation in the replica space, can be applied only to free theories since, otherwise, the $k$-modes do not decouple. For the free massless Dirac theory, this allows us to write the Lagrangian of the replicated theory as

$$\mathcal{L}_{{\rm D},n}=\sum_{k}\mathcal{L}_{k},\qquad\mathcal{L}_{k}=\bar{\psi}_{k}\gamma^{\mu}\partial_{\mu}\psi_{k}.$$

(34)

Following this approach, the partition function of Eq. (27) factorises into

$$Z_{n}^{A_{1}:A_{2}}(\{\alpha_{j}\})=\prod_{k=-\frac{n-1}{2}}^{k=\frac{n-1}{2}}Z_{k,n}^{A_{1}:A_{2}}(\{\alpha_{j}\}),$$

(35)

where $Z_{k,n}^{A_{1}:A_{2}}(\{\alpha_{j}\})$ is the partition function for a Dirac field $\psi_{k}$ with the boundary conditions of Eq. (3.1).

The main difference between the partition functions $Z_{k,n}^{A_{1}:A_{2}}(\{\alpha_{j}\})$ and the standard computation of Ref. CFH for Rényi entropies is that the boundary conditions of the multi-valued fields around the branch points now depend on the phases $\alpha_{j}$. This multivaluedness can be removed, as done in CFH for $\alpha_{j}=0$, by introducing an external $U(1)$ gauge field $A_{\mu}^{k}$ coupled to single-valued fields $\tilde{\psi}_{k}$. In fact, if we apply the singular gauge transformation

$$\psi_{k}(x)=e^{i\int_{x_{0}}^{x}\mathrm{d}y^{\mu}A_{\mu}^{k}}\tilde{\psi}_{k}(x),$$

(36)

then the Lagrangian for the $k$-th mode can be rewritten as

$$\mathcal{L}_{k}=\bar{\tilde{\psi}}_{k}\gamma^{\mu}\left(\partial_{\mu}+iA_{\mu}^{k}\right)\tilde{\psi}_{k},$$

(37)

with the advantage of absorbing the phase around the end-points of $A_{1}\cup A_{2}$ into the gauge field. The only requirement that $A_{\mu}^{k}$ in Eq. (36) must satisfy is that, integrated along any closed curve $\mathcal{C}$ that encircles the end-points of $A$, the boundary conditions of Eq. (3.1) for $\psi_{k}$ must be reproduced. For this purpose, we require

$$\oint_{\mathcal{C}_{u_{p}}}{\rm d}y^{\mu}A_{\mu}^{k}=-\frac{2\pi k}{n}-\frac{\alpha_{2p-1}}{n},\qquad\oint_{\mathcal{C}_{v_{p}}}{\rm d}y^{\mu}A_{\mu}^{k}=\frac{2\pi k}{n}-\frac{\alpha_{2p}}{n},$$

(38)

where $\mathcal{C}_{u_{p}}$ and $\mathcal{C}_{v_{p}}$ are closed contours around the end-points of the $p$-th interval. Moreover, we have to impose that, if $\mathcal{C}$ does not enclose any end-point, then $\oint_{\mathcal{C}}{\rm d}y^{\mu}A_{\mu}^{k}=0$. Applying the Stoke’s theorem, the conditions of Eqs. (38) can be expressed in differential form,

$$\frac{1}{2\pi}\epsilon^{\mu\nu}\partial_{\nu}A_{\mu}^{k}(x)=\sum_{p=1}^{2}\left[\left(\frac{\alpha_{2p-1}}{2\pi n}+\frac{k}{n}\right)\delta(x-u_{p})+\left(\frac{\alpha_{2p}}{2\pi n}-\frac{k}{n}\right)\delta(x-v_{p})\right].$$

(39)

Once the transformation of Eq. (36) is performed, the partition function $Z_{k,n}^{A_{1}:A_{2}}(\{\alpha_{j}\})$ of the $k$-th mode is equal to the vacuum expectation value

$$Z_{k,n}^{A_{1}:A_{2}}(\{\alpha_{j}\})=\left\langle e^{i\int d^{2}xj^{\mu}_{k}A_{\mu}^{k}}\right\rangle,$$

(40)

where $j^{\mu}_{k}\equiv\bar{\tilde{\psi}}_{k}\gamma^{\mu}\tilde{\psi}_{k}$ is the conserved Dirac current for each mode. Eq. (40) can be easily computed via bosonisation CFH , which allows us to write the current in terms of the dual scalar field $\phi_{k}$ such that $j^{\mu}_{k}=\epsilon^{\mu\nu}\partial_{\nu}\phi_{k}/\sqrt{\pi}$. If we use this result in Eq. (40), and we apply Eq. (39), then $Z_{k,n}^{A_{1}:A_{2}}(\{\alpha_{j}\})$ is equal to the following correlation function of vertex operators $V_{a}(y)=e^{-ia\phi_{k}(y)}$

$$Z_{k,n}^{A_{1}:A_{2}}(\{\alpha_{j}\})=\langle V_{\frac{k}{n}+\frac{\alpha_{1}}{2\pi n}}(u_{1})V_{-\frac{k}{n}+\frac{\alpha_{2}}{2\pi n}}(v_{1})V_{\frac{k}{n}+\frac{\alpha_{3}}{2\pi n}}(u_{2})V_{-\frac{k}{n}+\frac{\alpha_{4}}{2\pi n}}(v_{2})\rangle.$$

(41)

Notice that the neutrality condition $\alpha_{1}+\alpha_{2}+\alpha_{3}+\alpha_{4}=0$ ensures that the latter correlator does not vanish. The correlation function of vertex operators in the complex plane is well-known (see, for instance, Ref. difrancesco ) and, therefore, Eq. (41) can be easily calculated. Plugging the result into Eq. (35) and performing the product over $k$, we obtain

$$Z_{n}^{A_{1}:A_{2}}(\{\alpha_{j}\})\propto\\ \left[\ell_{1}\ell_{2}(1-x)\right]^{\frac{1-n^{2}}{6n}}\left[d^{\alpha_{2}\alpha_{3}}\ell_{1}^{\alpha_{1}\alpha_{2}}\ell_{2}^{\alpha_{3}\alpha_{4}}(d+\ell_{1})^{\alpha_{1}\alpha_{3}}(d+\ell_{2})^{\alpha_{2}\alpha_{4}}(d+\ell_{1}+\ell_{2})^{\alpha_{1}\alpha_{4}}\right]^{\frac{1}{2\pi^{2}n}},$$

(42)

where $x$ is the cross-ratio defined in Eq. (18). When we take $\alpha_{1}=-\alpha_{2}=\alpha$ and $\alpha_{3}=-\alpha_{4}=\beta$ in this expression, we get the multi-charged moments in Eq. (14) as

$\displaystyle Z_{n}^{A_{1}:A_{2}}(\alpha,\beta)$

$\displaystyle=$

$\displaystyle c_{n;\alpha,\beta}\left[\ell_{1}\ell_{2}(1-x)\right]^{\frac{1-n^{2}}{6n}}\left[\left(1-x\right)^{-\alpha\beta}\ell_{1}^{-\alpha^{2}}\ell_{2}^{-\beta^{2}}\right]^{\frac{1}{2\pi^{2}n}}.$

(43)

We assume that all the length scales in this formula have been regularised through a UV cutoff which is included in the multiplicative constant $c_{n;\alpha,\beta}$.

An interesting case to analyse is when the two intervals $A_{1}$ and $A_{2}$ become adjacent; that is, when $d\to 0$. In that limit, the cross-ratio $x$ tends to one such that

$$1-x=d\frac{\ell_{1}+\ell_{2}}{\ell_{1}\ell_{2}}+O(d^{2}),$$

(44)

and Eq. (43) vanishes. Nevertheless, in this regime, the distance $d$ must be regarded as another UV cutoff, which can be absorbed in the multiplicative constant $c_{n;\alpha,\beta}$. Therefore, from Eqs. (44) and (43), one expects

$$Z_{n}^{A_{1}:A_{2}}(\alpha,\beta)=\tilde{c}_{n;\alpha,\beta}(\ell_{1}+\ell_{2})^{\frac{1-n^{2}}{6n}}\left[\frac{\ell_{1}^{\alpha\beta-\alpha^{2}}\ell_{2}^{\alpha\beta-\beta^{2}}}{(\ell_{1}+\ell_{2})^{\alpha\beta}}\right]^{\frac{1}{2\pi^{2}n}},$$

(45)

which agrees with the fact that, according to Eq. (24), the multi-charged moments $Z_{n}^{A_{1}:A_{2}}(\alpha,\beta)$ must tend to the three-point function of primaries $\langle\tau_{n,\alpha}(u_{1})\mathcal{V}_{-\alpha+\beta}(v_{1})\tilde{\tau}_{n,-\beta}(v_{2})\rangle$, in the limit $d\to 0$.

In Fig. 2, we check the expressions obtained in Eqs. (43) and  (45) for $Z_{n}^{A_{1}:A_{2}}(\alpha,\beta)$ with exact numerical calculations performed in the tight-binding model, which is a chain of non-relativistic free fermions whose scaling limit is described by the massless Dirac field theory. The details of the numerical techniques employed are given in Appendix A. In order to compare Eqs. (43) and (45) with the numerical data, we need the concrete expression of the non-universal factors $c_{n;\alpha,\beta}$ and $\tilde{c}_{n;\alpha,\beta}$ for this particular model. When the two intervals $A_{1}$ and $A_{2}$ are far away, that is in the limit $d\to\infty$, the multi-charged moments of $A_{1}\cup A_{2}$ factorise into those of $A_{1}$ and $A_{2}$,

$$\lim_{d\to\infty}Z_{n}^{A_{1}:A_{2}}(\alpha,\beta)=Z_{n}^{A_{1}}(\alpha)Z_{n}^{A_{2}}(\beta).$$

(46)

Therefore, one expects $c_{n;\alpha,\beta}$ to be the product of the two non-universal constants associated to $A_{1}$ and $A_{2}$ as single intervals. The latter were obtained for the tight-binding model in Ref. riccarda by exploiting the asymptotic properties of Toeplitz determinants. We can also apply here those results, taking into account that each interval is associated to a different flux, either $\alpha$ or $\beta$. Then we have

$$c_{n;\alpha,\beta}=e^{\left[-\frac{1}{3}\left(n-\frac{1}{n}\right)-\frac{\alpha^{2}}{2\pi^{2}n}-\frac{\beta^{2}}{2\pi^{2}n}\right]\log 2+\Upsilon(n,\alpha)+\Upsilon(n,\beta)},$$

(47)

where riccarda

$$\Upsilon{(n,\alpha)}={ni}\int_{-\infty}^{\infty}{\rm d}w[\tanh(\pi w)-\tanh(\pi nw+i\alpha/2)]\ln\frac{\Gamma(\frac{1}{2}+iw)}{\Gamma(\frac{1}{2}-iw)}.$$

(48)

Expanding $\Upsilon(n,\alpha)$ up to quadratic order in $\alpha$, then Eq. (47) can be approximated as

$$c_{n;\alpha,\beta}\approx e^{-\frac{1}{3}\left(n-\frac{1}{n}\right)\ln 2+2\Upsilon(n,0)-\frac{\zeta_{n}}{2\pi^{2}n}(\alpha^{2}+\beta^{2})},$$

(49)

where $\zeta_{n}=\ln 2-2\pi^{2}n\gamma_{2}(n)$ and

$$\gamma_{2}(n)=\frac{1}{2}\left.\frac{\partial^{2}\Upsilon(n,\alpha)}{\partial\alpha^{2}}\right|_{\alpha=0}=\frac{ni}{4}\int_{0}^{\infty}{\rm d}w[\tanh^{3}(\pi nw)-\tanh(\pi nw)]\log\frac{\Gamma(\frac{1}{2}+iw)}{\Gamma(\frac{1}{2}-iw)}.$$

(50)

In the limit of adjacent intervals, given by Eq. (45), the multiplicative constant $\tilde{c}_{n;\alpha,\beta}$ cannot be determined using the known results for Toeplitz determinants. However, we can conjecture an analytical approximation for it at quadratic order in $\alpha$ and $\beta$. When $d\to 0$, we can associate to each end-point of the intervals $u_{1}$, $v_{1}=u_{2}$ and $v_{2}$ the fluxes $\alpha$, $\beta-\alpha$ and $-\beta$ respectively. From the results for one interval of Ref. riccarda , one can conjecture that each end-point with flux $\alpha$ contributes with a factor $e^{-\frac{\zeta_{n}}{4\pi^{2}n}\alpha^{2}}$ to the constant $\tilde{c}_{n;\alpha,\beta}$, if we restrict to terms up to order $\alpha^{2}$. Therefore, the combination of all the fluxes in our case should contribute with a total factor $e^{-\frac{\zeta_{n}}{4\pi^{2}n}(\alpha^{2}+(\beta-\alpha)^{2}+\beta^{2})}$. We then expect that $\tilde{c}_{n;\alpha,\beta}$ should be well be approximated by

$$\tilde{c}_{n;\alpha,\beta}=e^{-\frac{1}{6}\left(n-\frac{1}{n}\right)\ln 2+\Upsilon(n,0)-\frac{\zeta_{n}}{2\pi^{2}n}(\alpha^{2}+\beta^{2}-\alpha\beta)}.$$

(51)

When $\alpha=\beta=0$, the expression above simplifies to the multiplicative constant for the moments of a single interval obtained in Ref. JK04 . In spite of the heuristic reasoning of this result, in Fig. 2, we check its validity by comparing it against exact numerical data.

In the top and middle panels of Fig. 2, we study $Z_{1}^{A_{1}:A_{2}}(\alpha,\beta)$ as function of $\alpha$, for various values of $\beta$, $\ell_{1}$, $\ell_{2}$ and $d>0$. The points correspond to the exact numerical values obtained as we described in Appendix A, while the solid curves are the analytic prediction of Eq. (43), taking as multiplicative constant $c_{n,\alpha,\beta}$ that in Eq. (47). We find an excellent agreement. In the bottom left panel, we analyse the case of adjacent intervals ($d=0$). The numerical data for $Z_{1}^{A_{1}:A_{2}}(\alpha,\beta)$ are in very good agreement with the analytical expression (solid curves) of Eq. (45), using Eq. (51) as multiplicative constant. Finally, in the bottom right panel, we plot $Z_{1}^{A_{1}:A_{2}}(\alpha,\beta)$ as function of the cross-ratio $x$, for various values of $\alpha$, $\beta$, $\ell_{1}$ and $\ell_{2}$. The curves represent the prediction of Eq. (43). The continuous ones correspond to take as non-universal constant $c_{n;\alpha,\beta}$ the full expression of Eq. (47), while for the dashed ones we have used the quadratic approximation of Eq. (49). The agreement between the analytic prediction and the numerical data is extremely good, even considering the quadratic approximation for the non-universal constant. As expected, this agreement is better for small values of $\alpha$ and $\beta$, while, around $\pm\pi$, we need to take larger subsystem sizes $\ell_{1}$, $\ell_{2}$ in order to suppress the finite-size corrections which are well-known and characterised for the charged moments with a single flux insertion riccarda .