Symmetry decomposition of negativity of massless free fermions
SSymmetry decomposition of negativity of massless freefermions
Sara Murciano , Riccarda Bonsignori , and Pasquale Calabrese , SISSA and INFN Sezione di Trieste, via Bonomea 265, 34136 Trieste, Italy. International Centre for Theoretical Physics (ICTP), Strada Costiera 11, 34151Trieste, Italy.
Abstract.
We consider the problem of symmetry decomposition of the entanglementnegativity in free fermionic systems. Rather than performing the standard partialtranspose, we use the partial time-reversal transformation which naturally encodes thefermionic statistics. The negativity admits a resolution in terms of the charge imbalancebetween the two subsystems. We introduce a normalised version of the imbalanceresolved negativity which has the advantage to be an entanglement proxy for eachsymmetry sector, but may diverge in the limit of pure states for some sectors. Our mainfocus is then the resolution of the negativity for a free Dirac field at finite temperatureand size. We consider both bipartite and tripartite geometries and exploit conformalfield theory to derive universal results for the charge imbalance resolved negativity. Tothis end, we use a geometrical construction in terms of an Aharonov-Bohm-like fluxinserted in the Riemann surface defining the entanglement. We interestingly find thatthe entanglement negativity is always equally distributed among the different imbalancesectors at leading order. Our analytical findings are tested against exact numericalcalculations for free fermions on a lattice. a r X i v : . [ c ond - m a t . s t a t - m ec h ] F e b ymmetry decomposition of negativity of massless free fermions Contents1 Introduction 22 Charge imbalance resolved negativity 5 ρ A and ρ R A
37C Lattice-dependent terms and Fisher-Hartwig conjecture 39D Twisted partial transpose 421. Introduction
The Rényi entanglement entropies are the most successful way to characterise the bipartiteentanglement of a subsystem A in a pure state of a many-body quantum system [1–4],also from the experimental perspective [5–9]. Given the reduced density matrix (RDM) ρ A of a subsystem A , obtained after tracing out the rest of the system B as ρ A ≡ Tr ρ ,the Rényi entropies are defined as S n = 11 − n log Tr ρ nA . (1) ymmetry decomposition of negativity of massless free fermions n → of Eq. (1) and alsothe entire spectrum of ρ A can be reconstructed [10]. The essence of the replica trick isthat for integer n , in the path-integral formalism, Tr ρ nA is the partition function on an n -sheeted Riemann surface R n obtained by joining cyclically the n sheets along the region A [11, 12]. Furthermore with the experimental settings developed so far [5–9], only Rényientropies with integer n are accessible.For a mixed state, the entanglement entropies are no longer good measures ofentanglement since they mix quantum and classical correlations (e.g. in a hightemperature state, S gives the extensive result for the thermal entropy that has nothingto do with entanglement). The Peres criterion [13, 14] is a very powerful starting point toquantify mixed state entanglement: it states that given a system described by the densitymatrix ρ A , a sufficient condition for the presence of entanglement between two subsystems A and A (with A = A ∪ A ) is that the partial transpose ρ T A with respect to the degreesof freedom in A (or equivalently A ) has at least one negative eigenvalue. Starting fromthis criterion a computable measure of the bipartite entanglement for a general mixedstate can be naturally defined as [15] N ≡ Tr | ρ T A | − , (2)which is known as negativity . Here Tr | O | := Tr √ O † O denotes the trace norm of theoperator O . Another equivalent measure, termed logarithmic negativity, has been alsointroduced in [15] and it is defined as E ≡ log Tr | ρ T A | , (3)whose advantage with respect to N is that it scales and behaves more similarly to theRényi entropies (indeed for pure states E = S / [15]). Both N and E are entanglementmonotones [15, 16]. It is also useful to define the moments of the partial transpose (a.k.a.the Rényi negativity, RN) as R n = Tr( ρ T A ) n . (4)Because of the non-positiveness of the spectrum of ρ T A , the Rényi negativities define twoseparate sequences for even and odd n . Then the natural way to exploit the replicatrick is to obtain the negativity by considering the analytic continuation of the evensequence of R n e at n e → [17, 18] (which is different from R = 1 ). The moments R n with integer n ≥ can also be measured in experiments [19–21], but they arenot entanglement monotones. The entanglement negativity and Rényi negativities havebeen used to characterise mixed states in various quantum systems such as in harmonicoscillator chains [22–30], quantum spin models [31–44], (1+1)d conformal and integrablefield theories [17, 18, 45–52], topologically ordered phases of matter in (2+1)d [53–57],out-of-equilibrium settings [20, 58–66], holographic theories [67–72].An interesting issue concerns the quantification of mixed state entanglement infermionic systems. In particular, it has been pointed out that also when ρ A is a Gaussian ymmetry decomposition of negativity of massless free fermions n e → isnot possible and hence, the negativity, i.e. the only genuine measure of entanglement,is not accessible. To overcome this problem, an alternative estimator of mixed stateentanglement for fermionic systems has been introduced based on the time-reversal (TR)partial transpose (a.k.a partial time reversal ) [81–88]. The new estimator has been dubbed fermionic negativity , although it is not related to negative eigenvalues of any matrix. Itturned out that not only the fermionic negativity is an entanglement monotone [84], butalso that it is able to detect entanglement in mixed states where the standard negativityvanishes. For both these reasons, throughout this work, we will mainly focus on thefermionic negativity.In this manuscript, we consider a many-body system with an internal globalsymmetry and address the question of how mixed state entanglement splits intocontributions arising from distinct symmetry sectors. The explicit idea of consideringgenerally the internal structure of entanglement associated with symmetry is rather recent(the interested readers can consult the comprehensive literature on the subject [9,89–120]).For pure states, it has been established that the symmetry resolution of entanglementfollows from the block diagonal form of the reduced density matrix [89, 90]; one of themain findings is that the entanglement entropy is equally distributed among the differentsectors [93]. For mixed states, the literature is limited to the pioneering work [91], whereit was proven that whenever there is a conserved extensive charge, the negativity admitsa resolution in terms of the charge imbalance between the two subsystems. Here, we firstpoint out that by properly normalising the imbalance sectors (as also done in Ref. [120])one obtains a clearer resolution of the entanglement in the imbalance; then we showthat the imbalance-decomposition of negativity also holds using the partial TR definitionfor free fermions. We then use such decomposition to study the symmetry resolution ofthe entanglement of free fermions at finite temperature, exploiting the same field theorymethods used for the total negativity [17, 85].The paper is organised as follows. In Section 2, we provide some basic definitionsand briefly review the fermionic partial TR, motivating our work by simple examples fora tripartite and a bipartite geometry. After a brief summary of the results found in [91],we proceed with the general definition of the imbalance operator and the consequentdecomposition of the negativity, taking into account the normalisation of each sector. InSection 3, we review a method based on the replica trick to derive the leading order termfor the Rényi entropy of massless Dirac fermions in (1+1)d when both the temperature T and the system size L are finite. As a warm-up, we use this method to compute the chargedRényi entropies. In Sections 4 and 5 we then provide results for charged and imbalanceresolved negativity for tripartite and bipartite settings, respectively. Numerical checks forfree fermions on the lattice are also presented as a benchmark of the analytical results.We draw our conclusions in Section 6. Four appendices are also included: they providedetails about the analytical and numerical computations but they also make connections ymmetry decomposition of negativity of massless free fermions
2. Charge imbalance resolved negativity
In this section, we briefly review the definition of partial time reversal for fermionic densitymatrices following Ref. [81]. Then we present the symmetry resolution of the standardpartial transpose and of the partial TR of the density matrix. Simple examples will lead toa general definition of the imbalance resolution of entanglement negativity, both fermionicand bosonic. We closely follow Ref. [91], but we normalise differently the partial transposein each symmetry sector, so that the symmetry resolved negativity is a genuine indicatorof entanglement in the sector.
Let us start our discussion by recapitulating the definition of the partial transpose and itsrelation to the time-reversal transformation. Consider a density matrix ρ A in which A ispartitioned into two subsystems A and A such that A = A ∪ A ( ρ A can either be thereduced density matrix of a larger pure system ρ A = Tr B ( ρ ) or a mixed density matrix,e.g. thermal, for an entire system). It can always be written as ρ A = (cid:88) ijkl (cid:104) e i , e j | ρ A | e k , e l (cid:105) | e i , e j (cid:105) (cid:104) e k , e l | , (5)where | e j (cid:105) and | e k (cid:105) are orthonormal bases in the Hilbert spaces H and H correspondingto the A and A regions, respectively. The partial transpose of a density matrix for thesubsystem A is defined by exchanging the matrix elements in the subsystem A , i.e. ( | e i , e j (cid:105) (cid:104) e k , e l | ) T ≡ | e k , e j (cid:105) (cid:104) e i , e l | . (6)In terms of its eigenvalues λ i , the trace norm of ρ T A can be written as Tr | ρ T A | = (cid:88) i | λ i | = (cid:88) λ i > | λ i | + (cid:88) λ i < | λ i | = 1 + 2 (cid:88) λ i < | λ i | , (7)where in the last equality we used the normalisation (cid:80) i λ i = 1 . This expression makesevident that the negativity measures “how much” the eigenvalues of the partial transposeof the density matrix are negative, a property which is the reason of the name negativity.Moreover, in the absence of negative eigenvalues, Tr | ρ T A | = 1 and the negativity vanishes.For a bosonic system, it is known [14] that the partial transpose is the same as partialtime reversal in phase space. This correspondence was exploited in harmonic chains tocalculate the negativity in terms of the covariance matrix [22]. However, this is no longertrue for fermions. To understand why, let us consider a single-site system described byfermionic operators f and f † which obey the anticommutation relation { f, f † } = 1 . Weintroduce the Grassmann variables ξ, ¯ ξ , and the fermionic coherent states | ξ (cid:105) = e − ξf † | (cid:105) and (cid:104) ¯ ξ | = (cid:104) | e − f ¯ ξ . In this basis, the time reversal transformation reads | ξ (cid:105) (cid:104) ¯ ξ | → | i ¯ ξ (cid:105) (cid:104) iξ | ≡ ( | ξ (cid:105) (cid:104) ¯ ξ | ) R . (8) ymmetry decomposition of negativity of massless free fermions i . In the last equality we defined the time-reversaltranspose, specified by the apex R so to distinguish it from the standard transposition forwhich we use the apex T . This transformation rule can be generalised to a many-particle(lattice) system with a partial TR only on the degrees of freedom within A and reads ( |{ ξ j } j ∈ A , { ξ j } j ∈ A (cid:105) (cid:104){ ¯ χ j } j ∈ A , { ¯ χ j } j ∈ A | ) R = |{ i ¯ χ j } j ∈ A , { ξ j } j ∈ A (cid:105) (cid:104){ iξ j } j ∈ A , { ¯ χ j } j ∈ A | , (9)where |{ ξ j }(cid:105) = e − (cid:80) j ξ j f † j | (cid:105) , (cid:104){ ¯ χ j }| = (cid:104) | e − (cid:80) j f j ¯ χ j are the many-particle fermioniccoherent states.In the normal-ordered occupation number basis, the partial TR is |{ n j } j ∈ A , { n j } j ∈ A (cid:105) = ( f † m ) n m . . . ( f † m (cid:96) ) n m(cid:96) ( f † m (cid:48) ) n m (cid:48) . . . ( f † m (cid:48) (cid:96) ) n m (cid:48) (cid:96) | (cid:105) , (10)where n j ’s are occupation numbers in the subsystems A and A , which have (cid:96) and (cid:96) sites respectively (in 1D they represent the lengths of intervals), and we use the indices { m , . . . , m (cid:96) } ∪ { m (cid:48) , . . . , m (cid:48) (cid:96) } to denote the sites within the subsystem. The definition(9) in the occupation number basis is ( |{ n j } A , { n j } A (cid:105) (cid:104){ ¯ n j } A , { ¯ n j } A | ) R = ( − φ ( { n j } , { ¯ n j } ) ( |{ ¯ n j } A , { n j } A (cid:105) (cid:104){ n j } A , { ¯ n j } A | ) , (11)and can be viewed as the analogue of partial transposition in Eq. (6), up to the phasefactor φ ( { n j } , { ¯ n j } ) = [( τ + ¯ τ )mod2]2 + ( τ + ¯ τ )( τ + ¯ τ ) , (12)in which τ s = (cid:80) j ∈ A s n j , ¯ τ s = (cid:80) j ∈ A s ¯ n j are the number of the occupied states in the A s intervals, s = 1 , .It is useful to rewrite the partial TR using the Majorana representation of the operatoralgebra. We introduce the Majorana operators as c j − = f j + f † j , c j = i ( f j − f † j ) . (13)The density matrix in the Majorana representation takes the form ρ A = (cid:88) κ,τ | κ | + | τ | =even w κ,τ c κ m m . . . c κ m(cid:96) m (cid:96) c τ m (cid:48) m (cid:48) . . . c τ m (cid:48) (cid:96) m (cid:48) (cid:96) . (14)Here, c x = I and c x = c x , κ i , τ j ∈ { , } and κ ( τ ) is a m (cid:96) -component vector ( m (cid:48) (cid:96) ) withnorm | κ | = (cid:80) j κ j ( | τ | = (cid:80) j τ j ). The constraint on the parity of | κ | + | τ | is due to the factthat the density matrix commutes with the total fermion-number parity operator, i.e. wefocus our attention on physical states. Using Eq. (14), the partial TR with respect to thesubsystem A is defined by ρ R A = (cid:88) κ,τ | κ | + | τ | =even i | κ | w κ,τ c κ m m . . . c κ m(cid:96) m (cid:96) c τ m (cid:48) m (cid:48) . . . c τ m (cid:48) (cid:96) m (cid:48) (cid:96) . (15) ymmetry decomposition of negativity of massless free fermions Tr ρ R A = 1 . Nevertheless, we canstill use Eq. (2) to define a negativity because the eigenvalues of the combined operator [ ρ R A ( ρ R A ) † ] are always real. Following Refs. [81, 85, 86], in the rest of the manuscript, weshall use the term negativity (or fermionic negativity) to refer to the quantity N ≡ Tr | ρ R A | −
12 = Tr (cid:113) ρ R A ( ρ R A ) † − , (16)where the trace norm of the operator ρ R A is the sum of the square roots of the eigenvaluesof the product operator ρ R A ( ρ R A ) † . Usually one also defines the (fermionic) Rényinegativities, as R n = (cid:40) Tr( ρ R A ( ρ R A ) † . . . ρ R A ( ρ R A ) † ) , n even , Tr( ρ R A ( ρ R A ) † . . . ρ R A ) , n odd , (17)from which N = 12 (cid:16) lim n e → R n e − (cid:17) , where n e denotes an even n = 2 m [81]. We stress thatthe fermionic negativity (16) is not related to the presence of negative eigenvalues in thespectrum of ρ R A . Sometimes, we will refer to the standard negativity (2) as the bosonicnegativity. In the presence of symmetries, the RDM has a block diagonal structure which allowsto identify contributions of the entanglement entropy from individual charge sectors.In order to understand how symmetry is reflected in a block structure of the densitymatrix after partial transpose, we start with a simple example, taken from Ref. [91].Consider a particle in one out of three boxes, A , A , B , described by a pure state | Ψ (cid:105) = α | (cid:105) + β | (cid:105) + γ | (cid:105) . The RDM of A = A ∪ A is ρ A = Tr B | ψ (cid:105)(cid:104) ψ | = | γ | | (cid:105) (cid:104) | + ( α | (cid:105) + β | (cid:105) )( α ∗ (cid:104) | + β ∗ (cid:104) | ) , i.e. ρ A = | γ | | β | α ∗ β β ∗ α | α |
00 0 0 0 , (18)in the basis {| (cid:105) , | (cid:105) , | (cid:105) , | (cid:105)} . This matrix is clearly block diagonal with respect tothe total occupation number N A = N + N , where N and N respectively denote theparticle number of the subsystem A and A . According to Eq. (6), the partial transposeof ρ A is ρ T A = | γ | αβ ∗ | β | | α | βα ∗ . (19) ymmetry decomposition of negativity of massless free fermions N = (cid:12)(cid:12)(cid:12) | γ | − (cid:113) | γ | + | αβ | (cid:12)(cid:12)(cid:12) . Once we reshuffle the elements ofrows and columns in the basis of {| (cid:105) , | (cid:105) , | (cid:105) , | (cid:105)} , we get ρ T A = | γ | αβ ∗ βα ∗ | α |
00 0 0 | β | , (20)which has a block structure where each block is labelled by the occupation imbalance q = N − N : ρ T A ∼ = (cid:16) | α | (cid:17) q = − ⊕ (cid:32) | γ | αβ ∗ βα ∗ (cid:33) q =0 ⊕ (cid:16) | β | (cid:17) q =1 . (21)The structure of the above example is easily generalised to a many-body ρ A withsubsystems A and A characterised by particle number operator ˆ N and ˆ N ; performinga partial transposition of the relation [ ρ A , ˆ N A ] = 0 yields [91] [ ρ T A , ˆ N − ˆ N T ] = 0 , (22)from which we can do a block matrix decomposition according to the eigenvalues q of theimbalance operator ˆ Q = ˆ N − ˆ N T . We recall that this operator ˆ Q is basis dependent, asstressed in [91]; it has the form of an imbalance in the Fock basis, while in others it canbe different (as, e.g., in the computational basis we employ, see Appendix A).Let P q denote the projector into the subspace of eigenvalue q of the operator ˆ Q . Wedefine the normalised charge imbalance partially transposed density matrix as ρ T A ( q ) = P q ρ T A P q Tr( P q ρ T A ) , Tr( ρ T A ( q )) = 1 , (23)such that ρ T A = ⊕ q p ( q ) ρ T A ( q ) . (24)Here, p ( q ) = Tr( P q ρ T A ) is the probability of finding q as the outcome of a measurement of ˆ Q and corresponds to the sum of the diagonal elements of ρ T A ( q ) . Although the eigenvaluesof ρ T A can be negative, all the diagonal elements in the Fock basis are ≥ because thepartial transpose leaves invariant all the elements on the diagonal and so they remain thesame as those of ρ A which are ≥ . This is evident in the example (20) and it is the samefor any particle number. Hence p ( q ) satisfies p ( q ) ≥ and (cid:80) q p ( q ) = 1 , as it should befor a probability measure. We can thus define the (normalised) charge imbalance resolvednegativity as N ( q ) = Tr | ( ρ T A ( q )) | − . (25)Differently from [91], we prefer to deal with normalised quantities to preserve the naturalmeaning of negativity as a measure of entaglement: if in the q sector there are no negativeeigenvalues, according to Eq. (25), N ( q ) = 0 . Hence, this definition not only provides ymmetry decomposition of negativity of massless free fermions
9a resolution of the negativity, but also tells us in which sectors the negative eigenvaluesare, i.e. where the entanglement is. The total negativity, N , is resolved into (normalised)contributions from distinct imbalance sectors as N = (cid:88) q p ( q ) N ( q ) . (26)For the example of Eq. (21), the imbalance negativities are N ( ±
1) = 0 and N (0) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − (cid:114) + (cid:12)(cid:12)(cid:12) αβ | γ | (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) with p (0) = | γ | ; the only negative eigenvalue is in the sector q = 0 .Eq. (26) gives back the total negativity. We stress that the imbalance decomposition ofthe negativity as in Eq. (26) cannot be performed for the logarithmic negativity in Eq.(3), because of the nonlinearity of the logarithm.We conclude this section by discussing the important “pathological” case when p ( q ) = 0 for some values of the imbalance q , but ρ T A ( q ) is non-zero and so the negativityof the sectors diverges, although the total one is finite. For example, this happens setting γ = 0 in Eq. (18); in this case ρ A corresponds to a pure state. Actually, it is obvious thatevery time that ρ A is a pure state there will be some p ( q ) = 0 because N + N is fixedand hence also the parity of N − N is (so all the p ( q ) ’s where q has a different parityvanish). In such case, the origin of the problem can be traced back to the fact that the(pure-state) entanglement (entropy) is better resolved in terms of N or N rather thanin the imbalance, i.e. the symmetries of ρ A and ρ T A are larger than in the standard mixedcase. However, mixed states with some zero p ( q ) can be also easily built, although theyare difficult to encounter as mixed states in physical settings (and they all correspond tostates in which there is more symmetry than the imbalance). To understand the situationbetter, let us recall that p ( q ) is always the sum of some diagonal elements of both ρ T A ( q ) and ρ A . For the latter, the diagonal elements are the populations of states in the Fockbasis. Hence, we need at least a few zero populations to have a vanishing p ( q ) (and,e.g., this will never happen in a Gibbs state at finite temperature). In the matrix ρ A ,if the populations in a given sector of the total charge are zero, the entire block is zero(and hence the entanglement entropy of the sector is zero). However, when taking thepartial transpose, the off-diagonal elements are reshuffled in the matrix and, after beingre-organised in terms of the imbalance, we can end up with some blocks with all zeroson the diagonal (and so p ( q ) = 0 ) but with non-zero off-diagonal elements. In theseinstances, we cannot normalise with p ( q ) . (Have always in mind the example of Eq. (18)with γ = 0 : there are two sectors in ρ A with zero populations, N + N = 0 , ; after thepartial transposition, they both end up in imbalance q = 0 , see Eq. (21) which has non-zero off-diagonal terms). Anyhow, it makes sense that the imbalance negativity divergesin these cases. We are indeed facing sectors that have exactly zero populations, but stillhave some quantum correlations. In practice, as we shall see in the next section, thesevanishing p ( q ) are encountered only in the limit of a pure state (e.g. for T → ) andso diverging imbalance negativity signals that the state is getting pure and that a betterresolution of the entanglement is in N or N rather than in the imbalance. ymmetry decomposition of negativity of massless free fermions As a first simple example to show the importanceof the normalisation p ( q ) in the definition of imbalance resolved negativity, we reanalyse asimple known result [91] for the ground state of a Luttinger liquid (with parameter K ) ina tripartite geometry. We focus on two adjacent intervals of length (cid:96) and (cid:96) respectivelyembedded in an infinite line.Following [91], we start with the computation of the charged moments of the partialtranspose N T n ( α ) ≡ Tr(( ρ T A ) n e i ˆ Qα ) = (cid:104)T n V α ( u ) T − n V − α ( v ) T n V α ( v ) (cid:105) , (27)where, in the rhs, we use the correspondence with the 3-point correlation function offluxed twist field T n V α with scaling dimension ∆ n ( α ) = 124 (cid:16) n − n (cid:17) + K n (cid:16) α π (cid:17) , ∆ T no = ∆ n o ∆ T ne = 2∆ n e / . (28)Using these scaling dimensions, one finds log N T n ( α ) = log R n − K n (cid:16) απ (cid:17) log (cid:104) (cid:96) (cid:96) ( (cid:96) + (cid:96) ) (cid:15) (cid:105) , (29)where R n are neutral Rényi negativities. Notice in Eq. (29) only R n does depend on theparity of n [17], while the α dependence is the same for even and odd n .Upon performing a Fourier transform of Eq. (29), we obtain, through the saddle-pointapproximation, the (normalised) charge imbalance RN R n ( q ) = R n (cid:82) π − π dα π e − i ( q − ¯ q ) α e − α b n / [ (cid:82) π − π dα π e − i ( q − ¯ q ) α e − α b / ] n (cid:39) R n (cid:115) (2 πb ) n πb n e − ( q − ¯ q )22 ( bn − nb ) , (30)where b n = 1 π n log (cid:104) (cid:96) (cid:96) ( (cid:96) + (cid:96) ) (cid:15) (cid:105) . (31)The replica limit n e → is easily taken since there is no parity dependence in theimbalance part. For large (cid:96) , (cid:96) → ∞ (hence b n → ∞ ), we get N ( q ) = N + o (1) , (32)i.e. we found the equipartition of negativity in the different imbalance sectors atleading order for large subsystems. This behaviour is reminiscent of the equipartition ofentanglement entropy in a pure quantum system that possesses an internal symmetry [93].It is clear that negativity equipartition can be shown up only by properly normalising thepartial transpose in each sector as done here. As an important difference compared tothe entanglement entropies, we do not have additional log log (cid:96) [96] corrections to thesymmetry resolved quantities. ymmetry decomposition of negativity of massless free fermions Now we are ready to understand the block structure of the partial TR density matrixand how the fermionic negativity splits according to the symmetry. We first revisit thesimple example of the previous section in Eq. (18) for fermions. According to Eq. (8),the partial TR of ρ A in Eq. (18) is ρ R A = | γ | iαβ ∗ | β | | α | iβα ∗ , (33)i.e. the partial TR transformation does not spoil the block matrix structure according tothe occupation imbalance q = N − N : ρ R A ∼ = (cid:16) | α | (cid:17) q = − ⊕ (cid:32) | γ | iαβ ∗ iβα ∗ (cid:33) q =0 ⊕ (cid:16) | β | (cid:17) q =1 . (34)The (total) fermionic negativity is N = | γ | − (cid:118)(cid:117)(cid:117)(cid:116)
12 + | αβ | | γ | + (cid:115)
14 + | αβ | | γ | + (cid:118)(cid:117)(cid:117)(cid:116)
12 + | αβ | | γ | − (cid:115)
14 + | αβ | | γ | . (35)For a many-body state, the analogue of the commutation relation in Eq. (22) nowreads [ ρ R A , ˆ N − ˆ N R ] = 0 , (36)while the (normalised) charge imbalance resolved negativity is given by N ( q ) = Tr | ( ρ R A ( q )) | − , ρ R A ( q ) = P q ρ R A P q Tr( P q ρ R A ) . (37)We also define the charge imbalance resolved RN R n ( q ) = (cid:40) Tr( ρ R A ( q ) ρ R A ( q ) † . . . ρ R A ( q ) ρ R A ( q ) † ) , n even , Tr( ρ R A ( q ) ρ R A ( q ) † . . . ρ R A ( q )) , n odd , (38)from which N ( q ) = 12 (cid:16) lim n e → R n e ( q ) − (cid:17) . It is important to stress that the diagonalelements of ρ R A are the same as ρ T A (the TR operation does not touch the diagonalelements) and so the probabilities p ( q ) are identical for both the standard and the TRpartial transpose. Thus, all the considerations for the vanishing of p ( q ) in the previoussubsection apply also here. For the example of Eq. (34), the imbalance negativities are N ( ±
1) = 0 and N (0) = (cid:32) − (cid:114) + | αβ | | γ | + (cid:113) + | αβ | | γ | + (cid:114) + | αβ | | γ | − (cid:113) + | αβ | | γ | (cid:33) ymmetry decomposition of negativity of massless free fermions p (0) = | γ | . As a further check, (cid:80) q p ( q ) N ( q ) gives back the total negativity in Eq.(35).We think it is beneficial to give another basic example (taken from Ref. [85]) ofimbalance resolution with free fermions on a two-site lattice model described by theHamiltonian ˆ H = − ∆( f † f + f † f ) , (39)where ∆ is a tunnelling amplitude. In the basis {| (cid:105) , | (cid:105) , | (cid:105) , | (cid:105)} , the thermal densitymatrix is ρ = e − β ˆ H Tr( e − β ˆ H ) = 12 + 2 cosh( β ∆) β ∆) sinh( β ∆) 00 sinh( β ∆) cosh( β ∆) 00 0 0 1 . (40)Let us take the partial TR ρ R = 12 + 2 cosh( β ∆) i sinh( β ∆)0 cosh( β ∆) 0 00 0 cosh( β ∆) 0 i sinh( β ∆) 0 0 1 . (41)By reshuffling the elements of rows and columns in the basis of {| (cid:105) , | (cid:105) , | (cid:105) , | (cid:105)} , ρ R has a block matrix structure in the occupation imbalance between the subsystem andthe rest of the system that we can write explicitly as ρ R ∼ = (cid:16) cosh( β ∆)2+2 cosh( β ∆) (cid:17) q = − ⊕ (cid:32) β ∆) i sinh( β ∆)2+2 cosh( β ∆) i sinh( β ∆)2+2 cosh( β ∆) 12+2 cosh( β ∆) (cid:33) q =0 ⊕ (cid:16) cosh( β ∆)2+2 cosh( β ∆) (cid:17) q =1 . (42)When the state becomes pure, i.e. β ∆ (cid:29) , p ( q = 0) → . The interpretation is thesame as the one for the bosonic negativity: when the state is pure, the operator to resolvethe symmetry is ˆ N (or ˆ N ), rather than the imbalance. For completeness, we report thefermionic negativity N = 12 tanh (cid:16) β ∆2 (cid:17) , (43)and its splitting in the imbalance sectors: N ( ±
1) = 0 and N (0) = (cosh( β ∆) − with p (0) = β ∆)+1 , so that (cid:80) q p ( q ) N ( q ) = p (0) N (0) = N . Notice that as β → ∞ , p (0) → , N (0) → ∞ , but their product stays finite and tends to / .
3. Replica approach
In this section, we first review the replica approach to the charged entropies [118] andapply it to their calculation for a massless Dirac fermion at finite temperature, a resultthat was not yet obtained so far. Then we adapt the method to the charged Rényinegativities. Its applications will be presented in the successive sections. ymmetry decomposition of negativity of massless free fermions We start by recalling the symmetry resolution of the entanglement entropy. As alreadymentioned, in the presence of a U (1) symmetry ˆ Q , ρ A admits a charge decompositionaccording to the local charge ˆ Q A , where each block corresponds to different eigenspacesof ˆ Q A , which we can label as ˜ q ∈ Z , i.e. ρ A = ⊕ ˜ q ˜ p (˜ q ) ρ A (˜ q ) , ˜ p (˜ q ) = Tr( P ˜ q ρ A ) . (44)Here we use ˜ q for the eigenvalues of ˆ Q A to make a clear distinction with the eigenvalues ofthe imbalance q . Unless differently specified, A is a generic subsystem made of p intervals [ u i , v i ] , i.e A = ∪ pi =1 [ u i , v i ] . The symmetry resolved Rényi entropies are then defined as [93] S n (˜ q ) ≡ − n log Tr [ ρ A (˜ q )] n . (45)The direct use of the above definition to evaluate the symmetry resolved entropyrequires the knowledge of the spectrum of the RDM and its resolution in ˜ q , that isa nontrivial problem, especially for analytic computations. However, we can use theFourier representation of the projection operator and focus on the charged moments of ρ A , Z n ( α ) ≡ Tr [ ρ nA e iα ˆ Q A ] [90]. Their Fourier transforms Z n (˜ q ) = (cid:90) π − π dα π e − i ˜ qα Z n ( α ) , (46)are related to the entropies of the sector of charge ˜ q as S n (˜ q ) = 11 − n log (cid:20) Z n (˜ q ) Z (˜ q ) n (cid:21) . (47)We exploit the framework of the replica trick to evaluate the charged moments, which arethe main object of interest in this section.In a generic quantum field theory, the replica trick for computing Z n ( α ) can beimplemented by inserting an Aharonov-Bohm flux through a multi-sheeted Riemannsurface R n , such that the total phase accumulated by the field upon going through theentire surface is α [90]. The result is that Z n ( α ) is the partition function on such modifiedsurface, that, following Ref. [90], we dub R n,α . Here we focus on a massless Dirac fermiondescribed by the Lagrangian density L = ¯Ψ γ µ ∂ µ Ψ , (48)where ¯Ψ = Ψ † γ , γ = σ , γ = σ . Rather than dealing with fields defined on a nontrivial manifold R n,α , it is more convenient to work on a single plane with a n -componentfield Ψ = ψ ψ ... ψ n , (49) ymmetry decomposition of negativity of massless free fermions ψ j is the field on the j -th copy. Upon crossing the cut A , the vector field Ψ transforms according to the twist matrix T α T α = e iα/n e iα/n . . . . . . ( − n − . (50)The idea of using the twist matrix for the Dirac fermions at α = 0 was originally suggestedin [121] (see also [122]). The matrix T α has eigenvalues λ k = e i αn e πi kn , k = − n − , . . . , n − . (51)By diagonalising T α with a unitary transformation, the problem is reduced to n decoupledand multi-valued fields ψ k in a two dimensional spacetime. In particular, the chargedmoments become Z n ( α ) = ( n − / (cid:89) k = − ( n − / Z k,n ( α ) , (52)where Z k,n ( α ) is the partition function for a Dirac field that along A picks up a phaseequal to e i αn e πi kn , or equivalently the phase picked up going around one of the entanglingpoints u i , v i is e i αn e πi kn and e − i αn e − πi kn , respectively. The main difference with respectto the standard computation for the Rényi entropies is that, for a charged quantity, theboundary conditions of the multivalued fields along A depend also on the flux α andnot only on the replica index. This multivaluedness can be circumvented with the sametrick used for α = 0 [121], i.e. by absorbing it in an external gauge field coupled to asingle-valued fields ˜ ψ k . Indeed, the singular gauge transformation ψ k ( x ) = e i (cid:72) C dy µ A µk ( y ) ˜ ψ k ( x ) , (53)allows us to absorb the phase along A into the gauge field at the price of changing theLagrangian density into L k = ¯˜ ψ k γ µ ( ∂ µ + iA kµ ) ˜ ψ k . (54)The actual value of A kµ in Eq. (53) is fixed by requiring that, for any loop C , the originalboundary conditions for the multivalued field ψ k are reproduced. This is achieved with (cid:73) C ui dx µ A kµ = − πkn − αn , (cid:73) C vi dx µ A kµ = + 2 πkn + αn , (55)where C u i and C v i are circuits around left and right endpoints of the i -th interval. If thecircuit C does not encircle any endpoint, (cid:72) C dx µ A kµ = 0 . If more endpoints are encircled ymmetry decomposition of negativity of massless free fermions (cid:15) µν ∂ ν A kµ ( x ) = 2 π (cid:18) kn + α πn (cid:19) p (cid:88) i =1 [ δ ( x − u i ) − δ ( x − v i )] , (56)where p is the number of intervals.After the transformation (53), the desired charged partition sum Z k,n ( α ) is writtenas Z k,n ( α ) = (cid:104) e i (cid:82) d xA kµ j µk (cid:105) , (57)where j µk = ¯˜ ψ k γ µ ˜ ψ k is the Dirac current and A kµ satisfies Eq. (55) or, equivalently, (56).Eq. (57) is more easily calculated by bosonisation, which maps the Dirac current to thederivative of a scalar field and the Lagrangian of the k -th fermion to that of a real masslessscalar field φ k , L k = π ∂ µ φ k ∂ µ φ k (here we work with the normalisation of the boson fieldsuch that the Dirac fermion corresponds to a compactified boson with radius R = 2 , asin [123]). Therefore we can evaluate Z k,n ( α ) as the correlation function of the vertexoperators V a ( x ) = e − iaφ k ( x ) , i.e. Z k,n ( α ) = (cid:104) p (cid:89) i =1 V kn + α πn ( u i ) V − kn − α πn ( v i ) (cid:105) . (58)An important observation about Eq. (55) is that we can arbitrarily add πm phaseshifts, with m an integer, to the right hand side without affecting the total phase factoralong the circuits. This ambiguity leads to inequivalent different representations of thepartition function Z k,n ( α ) in Eq. (52), which in turn must be written as a summation overall allowed representations. The asymptotic behaviour of each term for large subsystemsize, (cid:96) , is a power law (cid:96) − α m and the leading term corresponds to the one with the smallestexponent α m . For the charged moments, the leading order is given by m = 0 , but thisis not the case for the entanglement negativity. See Appendix B for a more detaileddiscussion of this issue.Let us now apply this machinery to study the charged moments of a free Diracfermion on a torus with multiple intervals ( u a , v a ) , ( a = 1 , . . . , p ) . To have more compactformulas, we rescale the spatial coordinates by the system size L . The torus is definedby two periods which, in our units, are 1 and τ = iβ/L , where β = 1 /T is the inversetemperature. The partition function depends on the boundary conditions along the twocycles, which specify the spin structure of the fermion on the torus. Let z be a holomorphiccoordinate on the torus: it has the periodicities z = z + 1 and z = z + τ . The holomorphiccomponent of the fermion on the torus satisfies four possible boundary conditions ˜ ψ k ( z + 1) = e πiν ˜ ψ k ( z ) , ˜ ψ k ( z + τ ) = e πiν ˜ ψ k ( z ) , (59)where ν and ν take the values or . The anti-holomorphic component is a functionof ¯ z and satisfies the same boundary conditions as the holomorphic part. We denote the ymmetry decomposition of negativity of massless free fermions ν = ( ν , ν ) sector where ν = 1 , , , corresponds to (0 , , (0 , / , (1 / , / , (1 / , ,respectively (for standard fermions, the physical boundary conditions are anti-periodicalong both cycles and so ν = 3 , but the other spin structures have important applicationstoo). Hence, we just need the correlation function of the vertex operators V e ( z, ¯ z ) = e ieφ ( z, ¯ z ) on the torus with boundary conditions corresponding to the sector ν . These canbe found in Ref. [123] and read (cid:104) V e ( z , ¯ z ) V e ( z , ¯ z ) . . . V e N ( z N , ¯ z N ) (cid:105) ν = (cid:12)(cid:12)(cid:12) (cid:89) i 12 log(2 π B ) + O (1) . (70)In order to make contact with some known results in literature [93, 96, 98], we report theexplicit expression for B in the low and high temperature limit for p = 1 B = π log (cid:16) Lπ(cid:15) sin (cid:96)πL (cid:17) , LT (cid:28) , π log (cid:16) βπ(cid:15) sinh (cid:96)πβ (cid:17) , LT (cid:29) . (71)The result in Eq. (70) has been dubbed equipartition of entanglement [93]: at leadingorder the entanglement is the same in the different charge sectors. As a side result, herewe showed that entanglement equipartition holds also at finite size and temperature, atleast for a free Dirac field. The above procedure is easily adapted to the computation of the charged moments of thepartial TR defined as N n ( α ) = (cid:40) Tr( ρ R A ( ρ R A ) † . . . ρ R A ( ρ R A ) † e i ˆ Q A α ) , n even , Tr( ρ R A ( ρ R A ) † . . . ρ R A e i ˆ Q A α ) , n odd . (72)Hence, in order to compute the imbalance resolved negativity, we need to study thecomposite operator ρ R A ( ρ R A ) † . The charged moments in Eq. (72) are defined for two ymmetry decomposition of negativity of massless free fermions u v v ‘ ‘ e i(cid:25) ( kn (cid:0) ’ (cid:25) + (cid:11) (cid:25)n ) k e (cid:0) i(cid:25) ( kn (cid:0) ’ (cid:25) + (cid:11) (cid:25)n ) k e (cid:0) i(cid:25) ( kn + (cid:11) (cid:25)n ) k e i(cid:25) ( kn + (cid:11) (cid:25)n ) k Figure 1: Tripartite geometry for two adjacent intervals. In the plot the interval A is theblue one on the left and A the grey one on the right, of length (cid:96) and (cid:96) respectively. B isthe reminder. The partial transpose is taken on A . For each branch point, we report thephase taken by the field ψ k going around it. subsystems A and A with different twist matrices respectively denoted by T R α and T α .The new twist matrix T R α for the transposed time reversed subsystem is given by T R α = . . . ( − n − e − iα/n e − iα/n e − iα/n . . .. . . . . . . (73)The two matrices, T α and T R α , are simultaneously diagonalisable. Consequently, we candecompose our problem into n decoupled copies in which the fields have different twistphases along the two subsystems. As a result, N n ( α ) is decomposed as N n ( α ) = ( n − / (cid:89) k = − ( n − / Z R ,k ( α ) , (74)where Z R ,k ( α ) is the partition function for fields with twist phases equal to e − πi ( kn + α πn ) and e πi ( kn + α πn − ϕn π ) , respectively along A and A . Here ϕ n = π for n = n e even and ϕ n = n − n π for n = n o odd (as follows from the diagonalisation of Eq. (73)). In particular,the probability p ( q ) is the Fourier transform of N ( α ) = Tr[ ρ R e i ˆ Qα ] , that, with a minorabuse of terminology, we dub charged probability . In this case, the twist matrices alongthe two intervals are just phases given by T α = e iα and T R α = T − α = e − iα .A Fourier transform leads us to the imbalance resolved negativities (38) Z R ,n ( q ) = (cid:90) π − π dα π N n ( α ) , p ( q ) = (cid:90) π − π dα π N ( α ) , (75)from which R n ( q ) = Z R ,n ( q ) p n ( q ) , N ( q ) = 12 (cid:16) lim n e → R n e ( q ) − (cid:17) . (76)Let us stress the replica limit for R n e ( q ) is lim n e → Z R ,n e ( q ) p ( q ) , i.e. while it is sufficient to set n = 1 in the denominator, the numerator requires an analytic continuation from the evensequence at n e → , in agreement with the definition in Eq. (16). In the following section,we compute the imbalance resolved entanglement negativity for two geometries. ymmetry decomposition of negativity of massless free fermions 4. Charged and symmetry resolved negativities in a tripartite geometry Let us study the negativity of two subsystems consisting of two adjacent intervals A , A ,of lengths (cid:96) , (cid:96) out of a system of length L , as depicted in Fig 1. We place the branchpoints at u = − (cid:96) /L = − r , v = u = 0 , and v = (cid:96) /L = r and the multivalued fields ψ k take up a phase e πi ( kn + α πn − ϕn π ) at u , e − πi ( kn + α πn − ϕn π ) e − πi ( kn + α πn ) at v and e πi ( kn + α πn ) at v . By introducing a gauge field A kµ , as explained in Sec. 3.1, we have to impose propermonodromy conditions such that the field is almost pure gauge except at the branchpoints, where delta function singularities are necessary to recover the correct phases ofthe multivalued fields. Hence, the flux of the gauge fields is given by π (cid:15) µν ∂ ν A kµ ( x )= (cid:18) kn + α πn − ϕ n π (cid:19) δ ( x − u ) − (cid:18) kn + απn − ϕ n π (cid:19) δ ( x − v ) + (cid:18) kn + α πn (cid:19) δ ( x − v ) . (77)Through bosonisation, Z ( ν ) R ,k ( α ) can be written as a correlation function of vertex operators V a ( x ) = e − iaφ k ( x ) as Z ( ν ) R ,k ( α ) = (cid:10) V kn + α πn − ϕn π ( u ) V − kn − α πn + ϕn π ( v ) V − kn − α πn ( v ) V kn + α πn ( v ) (cid:11) = (cid:10) V kn + α πn − ϕn π ( u ) V − kn − απn + ϕn π ( v ) V kn + α πn ( v ) (cid:11) . (78)Using the correlation function in Eq. (60), the final result is ‡ Z ( ν ) R ,k ( α ) = | θ ( r | τ ) | − kn + α πn − ϕn π )( kn + απn − ϕn π ) | θ ( r | τ ) | − kn + α πn )( kn + απn − ϕn π ) | θ ( r + r | τ ) | kn + α πn )( kn + α πn − ϕn π ) × (cid:12)(cid:12)(cid:12) (cid:15)L ∂ z θ (0 | τ ) (cid:12)(cid:12)(cid:12) − ∆ k ( α ) (cid:12)(cid:12)(cid:12) θ ν (( kn + α πn )( r − r ) + ϕ n π r | τ ) θ ν (0 | τ ) (cid:12)(cid:12)(cid:12) , (79)where ∆ k ( α ) = − k n − kαn π − α n π +3 k ϕ n nπ +3 αϕ n nπ − ϕ n π − θ ( − k )(1+ 3 kn + 3 α nπ − ϕ n π ) , (80)and θ ( x ) is the step function. It is important to note that for k < , we have to modify theflux at u and v , ϕ n , by inserting an additional π and − π fluxes. Essentially, we needto find the dominant term with the lowest scaling dimension in the mode expansion, asdiscussed in Appendix B. Moreover, the case of odd n = n o requires particular attention:as | α | > / π , also the mode k = 0 requires an additional π and − π fluxes at u and v , respectively. Putting together the various pieces and using Eq. (74), the logarithm ofthe charged moments of ρ R A are given by log N ( ν ) n ( α ) = log N n, ( α ) + log N ( ν ) n, ( α ) , (81) ‡ Differently from Eq. (41) in [85] or Eq. (80) in [81], rather then using the absolute values we explicitlychange ϕ n → ϕ n − π for k < . ymmetry decomposition of negativity of massless free fermions log N n, ( α ) = log R n − α π n log (cid:12)(cid:12)(cid:12) θ ( r | τ ) θ ( r | τ ) θ ( r + r | τ ) − (cid:16) (cid:15)L ∂ z θ (0 | τ ) (cid:17) − (cid:12)(cid:12)(cid:12) , log R n o = − (cid:16) n o − n o (cid:17) log (cid:12)(cid:12)(cid:12) θ ( r | τ ) θ ( r | τ ) θ ( r + r | τ ) (cid:16) (cid:15)L ∂ z θ (0 | τ ) (cid:17) − (cid:12)(cid:12)(cid:12) , log R n e = − (cid:16) n e − n e (cid:17) log (cid:12)(cid:12)(cid:12) θ ( r | τ ) θ ( r | τ ) (cid:16) (cid:15)L ∂ z θ (0 | τ ) (cid:17) − (cid:12)(cid:12)(cid:12) − (cid:16) n e + 212 n e (cid:17) log (cid:12)(cid:12)(cid:12) θ ( r + r | τ ) (cid:16) (cid:15)L ∂ z θ (0 | τ ) (cid:17) − (cid:12)(cid:12)(cid:12) . (82)The first equation for N n, ( α ) is always valid for any alpha for n = n e even, but only inthe region | α | < / π for n = n o odd; otherwise it must be modified as log N n o , ( α ) = log R n o − α π n o log (cid:12)(cid:12)(cid:12) θ ( r | τ ) θ ( r | τ ) θ ( r + r | τ ) − (cid:16) (cid:15)L ∂ z θ (0 | τ ) (cid:17) − (cid:12)(cid:12)(cid:12) + | α | n o π log (cid:12)(cid:12)(cid:12) θ ( r | τ ) θ ( r | τ ) θ ( r + r | τ ) − (cid:16) (cid:15)L ∂ z θ (0 | τ ) (cid:17) − (cid:12)(cid:12)(cid:12) − n log (cid:12)(cid:12)(cid:12) θ ( r | τ ) (cid:16) (cid:15)L ∂ z θ (0 | τ ) (cid:17) − (cid:12)(cid:12)(cid:12) , for | α | > / π. (83)Hence for odd n = n o , the exponent of the charged moments N ( ν ) n o ( α ) has a discontinuityas a function of α for | α | = π . This singular behaviour in α is reminiscent of whatfound for the negativity spectrum of free fermions in [86]. Let us also note that the aboveresult does not hold for n o = 1 , for which we will provide an analytical expression in thefollowing. The spin structure dependent term is log N ( ν ) n, ( α ) = 2 ( n − / (cid:88) k = − ( n − / log (cid:12)(cid:12)(cid:12) θ ν (( kn + α πn )( r − r ) + ϕ n π r | τ ) θ ν (0 | τ ) (cid:12)(cid:12)(cid:12) . (84)Although our main focus is the state with ν = 3 , we notice that N (1) n, ( α ) above is strictlyinfinite because θ (0 | τ ) = 0 . This is related to the fermion zero mode in this sector andis not a prerogative of the charged quantities.In the case of intervals of equal lengths (cid:96) = (cid:96) = (cid:96) the charged logarithmic negativity(i.e., E ν ( α ) ≡ lim n e → log N ( ν ) n e ( α ) ) simplifies as E ν ( α ) = E ( ν ) − α π log (cid:12)(cid:12)(cid:12) θ ( r | τ ) θ (2 r | τ ) − (cid:16) (cid:15)L ∂ z θ (0 | τ ) (cid:17) − (cid:12)(cid:12)(cid:12) , with E ( ν ) = 14 log (cid:12)(cid:12)(cid:12) θ ( r | τ ) θ (2 r | τ ) − (cid:16) (cid:15)L ∂ z θ (0 | τ ) (cid:17) − (cid:12)(cid:12)(cid:12) + 2 log (cid:12)(cid:12)(cid:12) θ ν ( r | τ ) θ ν (0 | τ ) (cid:12)(cid:12)(cid:12) , (85)where r = (cid:96)/L .Eq. (85) represents our final field theoretical result for the charged logarithmicnegativities in a tripartite geometry with two equal intervals. We now test this predictionagainst exact lattice computations obtained with the techniques reported in Appendix A. ymmetry decomposition of negativity of massless free fermions (cid:15) in (85) that does depend also on α , but not onthe size and temperature. We can exploit the latter property to deduce its exact valuefrom the knowledge of the lattice negativities at T = 0 in the thermodynamic limit thatcan be determined via Fisher-Hartwig techniques, as reported in Appendix C, cf. Eq.(178). The numerical results for the charged negativities are shown in Fig. 2, where fourpanels highlight the dependence on (cid:96), T , α , and (cid:96)/L , respectively. The agreement withthe parameter-free asymptotic results (85) is always excellent. Let us critically discussthese results. First, it is known for α = 0 , the logarithmic negativity saturates at finitetemperature once (cid:96)T (cid:29) [85], i.e., obeys an area law; conversely the top-left panel of Fig.2 shows that E ( α ) follows a volume law. This scaling can be also inferred analytically fromthe high-temperature limit reported in the following subsection. In the top-right panelof the same figure, we observe that E ( α ) has a plateau at low temperatures, i.e when T (cid:28) /L so that the temperature is smaller than the energy finite-size gap (of order /L ); consequently the system behaves as if it is at zero temperature with exponentiallysmall corrections in T L . For larger T a linear decrease sets up for low enough T , beforean exponential high temperature behaviour takes place (this is not shown in the picture,but see next subsection). In the bottom-left panel of Fig. 2, we analyse the α dependencefixing (cid:96) and L for a few values of β . We observe a fairly good agreement between latticeand field theory, although when α gets closer to ± π the agreement gets worse. This isnot surprising because charged quantities exactly at ± π are known to be singular [98] andconsequently finite (cid:96) effects are more severe. Moreover, the plot clearly shows that E ( α ) has a differentiable maximum in α = 0 (that we need for the saddle point approximation).In the bottom-right panel, we show that the difference E ( α, T ) − E ( α, is a universalfunction of β/L and (cid:96)/L : we verify this behaviour by looking at various system sizes, L , and showing that they all collapse on the same curve. The agreement also slightlyimproves as L increases, as it should.Let us conclude this subsection reporting the result for the charged probability N ( α ) = Tr[ ρ R e i ˆ Qα ] that requires to specialise the above discussion to the case n = 1 .Hence, N ( ν )1 ( α ) reduces to one mode, k = 0 , and Eq. (112) becomes π (cid:15) µν ∂ ν A µ ( x ) = (cid:16) α π (cid:17) δ ( x − u ) − (cid:16) απ (cid:17) δ ( x − v ) + (cid:16) α π (cid:17) δ ( x − v ) . (86)As detailed in the last part of Appendix B, we need to find the dominant term, i.e. withthe lowest scaling dimension, in the mode expansion. In particular, it turns out that for | α/π | > / an additional − π flux has to be inserted at v while an additional π has tobe added at u or, equivalently, at v . This is the only difference with respect to n o (cid:54) = 1 , ymmetry decomposition of negativity of massless free fermions ‘ . . . . . E ( α = . ) β = 20 β = 50 β = 100GSTL . 002 0 . 004 0 . 006 0 . 008 0 . T − . − . − . − . − . − . . E ( α , T ) − E ( α , T = ) α = 1 , ‘ = 20 α = 1 , ‘ = 40 α = 1 , ‘ = 60 α = 2 , ‘ = 20 α = 2 , ‘ = 40 α = 2 , ‘ = 60 − − − α − − − − E ( α ) β = 10 β = 20 β = 100 . . . . . . ‘/L − . − . − . − . − . . E ( α = . , T ) − E ( α = . , T = ) β = 10 , L = 40 β = 20 , L = 80 β = 25 , L = 40 β = 50 , L = 80 Figure 2: Charged negativity E ( α ) in a tripartite torus with subsystem length (cid:96) = (cid:96) = (cid:96) .CFT results (85), lines, against numerics on the lattice, symbols. Top-left: E ( α ) as afunction of (cid:96) for α = 0 . . We consider different values of β = 1 /T : in particular, GS standsfor ground state, i.e. T = 0 while TL refers to the thermodynamic limit T = 0 , L → ∞ .System size is fixed to L = 200 sites, except for the TL curve. Top-right: E ( α ) as a functionof the temperature T for different values of α and (cid:96) , with L = 200 . The subtraction ofthe value E ( α, T = 0) cancels the dependence on the cutoff and the resulting curves areuniversal. Bottom-left: E ( α ) as a function α for L = 200 and (cid:96) = 20 for a few β . Theagreement is perfect away from the boundaries α = ± π . Bottom-right: Scaling collapse ofthe charged negativity as a function of β/L and (cid:96)/L . We fix α = 0 . . when the π flux has to be inserted only in u . Hence, the final expression is given by N ( ν )1 ( α ) = | θ ( r | τ ) | − α π | θ ( r | τ ) | − α π | θ ( r + r | τ ) | α π | (cid:15) N /L∂ z θ (0 | τ ) | − α π (cid:12)(cid:12)(cid:12) θ ν ( | α π | ( r − r ) | τ ) θ ν (0 | τ ) (cid:12)(cid:12)(cid:12) | α | ≤ π f ( r , r ; | α | ) θ ( r + r | τ ) | | απ | ( | α π |− | (cid:15) N /L∂ z θ (0 | τ ) | − | α | ( −| α | +2 π )2 π − (cid:12)(cid:12)(cid:12) θ ν ( | α π | ( r − r )+ r | τ ) θ ν (0 | τ ) (cid:12)(cid:12)(cid:12) | α | > π (87)where f ( x, y ; q ) = [ x q − − q +1) y q ( − q +1) + x ↔ y ] . The cutoff related to the chargedprobability is denoted as (cid:15) N . Its explicit expression, for a lattice regularisation of the ymmetry decomposition of negativity of massless free fermions − − − α − − − − − N ( ) ( α ) α = − / π α = 2 / π ‘ = 30 , GS‘ = 20 , β = 100 ‘ = 30 , β = 50 ‘ = 20 , β = 20 − − − α − − − − − N ( ) ( α ) α = − π/ α = π/ ‘ = 20 , GS‘ = 50 , β = 100 ‘ = 50 , β = 50 ‘ = 20 , β = 20 Figure 3: The charged probability N (3)1 ( α ) for tripartite (left) and bipartite (right)geometry as a function of α . We set L = 100 . Analytical predictions in Eqs. (87) and (119)are compared with the exact lattice computations at different β . Notice the discontinuitiesat α = ± / π (left) and α = ± π/ (right). Dirac field, is given in Eq. (180a) and (180b) for | α | ≤ / π and | α | > / π , respectively.This introduction of a new symbol (cid:15) N is necessary in order to avoid confusion with thecutoff (cid:15) obtained in the replica limit as n e → , given explicitly for the lattice model inEq. (179) (and it is different from n o = 1 ). In this section we report the low and high temperature limits of the charged Rényinegativity. Actually, the results that we are going to derive in the following for thetripartite geometry can be much more easily deduced by mapping the results in thecomplex plane (29) (i.e. both L, β → ∞ ) to a cylinder periodic in either space or time(obtaining the forthcoming Eqs. (92) and (99), respectively). It is however a highly nontrivial check for the correctness of our formulas that these results are re-obtained in theproper limits. For sake of conciseness, we focus on even n = n e and on the ν = 3 sector,but similar formulas hold for all other cases.In the low temperature limit where τ = iβ/L → i ∞ , we can take advantage of therelation lim β →∞ θ ( z | iβ/L ) = 2 e − πβ/ (4 L ) sin πz + O ( e − πβ/L ) . (88)In this way we obtain for the spin-independent part log N n e , ( α ) = log R n e − α π n e ln (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:18) Lπ(cid:15) (cid:19) sin (cid:0) π(cid:96) L (cid:1) sin (cid:0) π(cid:96) L (cid:1) sin (cid:16) π ( (cid:96) + (cid:96) ) L (cid:17) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + O ( e − π/ ( LT ) ) , (89)while using the product representation of the theta function (65), the spin structure ymmetry decomposition of negativity of massless free fermions log N (3) n e , ( α ) == 2 ∞ (cid:88) j =1 ( − j +1 j πjβ/L ) (cid:18) cos( j ( r − r ) α/n e ) sin( πjr ) − sin( πjr )sin( πj ( r − r ) /n e ) − n e (cid:19) . (90)Thus, at the leading order, Eq. (89) is the whole story at zero temperature, since in thereplica limit the above expression contributes to the charged negativity as E (3)1 ( α ) = lim n e → log N (3) n e , ( α ) = 4 e − π/ ( LT ) (cid:18) cos(( r − r ) α ) cos( π ( r + r ) / π ( r − r ) / − (cid:19) . (91)Putting everything together, in the low temperature limit the logarithmic chargednegativity of two adjacent intervals for spatially antiperiodic fermions is given by E ( α, LT (cid:28) 1) = E − α π n e log (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:18) Lπ(cid:15) (cid:19) sin (cid:0) π(cid:96) L (cid:1) sin (cid:0) π(cid:96) L (cid:1) sin (cid:16) π ( (cid:96) + (cid:96) ) L (cid:17) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + O ( e − π/ ( LT ) ) , (92)where E ( LT (cid:28) 1) = log (cid:12)(cid:12)(cid:12) ( Lπ(cid:15) ) sin( π(cid:96) L ) sin( π(cid:96) L )sin( π ( (cid:96) (cid:96) L ) (cid:12)(cid:12)(cid:12) . We can also study the low-temperaturebehaviour of Eq. (87), which reads N ( α, LT (cid:28) (cid:39) − α π log | L π (cid:15) N sin ( π(cid:96) L ) sin ( π(cid:96) L )sin( π ( (cid:96) (cid:96) L ) | , | α | ≤ π π −| α | ) | α | π log | L π (cid:15) N sin ( π(cid:96) L ) sin ( π(cid:96) L )sin( π ( (cid:96) (cid:96) L ) | − log | L π (cid:15) N sin( π(cid:96) L ) sin( π(cid:96) L ) | . | α | > π . (93)To investigate the high temperature behaviour, τ = iβ/L → , we can use themodular transformation rules for the theta functions: θ ( z | τ ) = − ( − iτ ) − / e − iπz /τ θ ( z/τ | − /τ ) ,θ ( z | τ ) =( − iτ ) − / e − iπz /τ θ ( z/τ | − /τ ) , (94)and the asymptotic form of the θ function in the small β limit θ ( z/τ | − /τ ) = − ie − πL β sinh( πzLβ ) + O ( e πLβ ( z − / ) , ≤ z ≤ / . (95)Therefore, the leading terms of the spin-independent part of the charged negativities canbe written as log N n e , ( α ) = log R n e + ( (cid:96) − (cid:96) ) α πn e βL − α π n e ln (cid:12)(cid:12)(cid:12)(cid:16) βπ(cid:15) (cid:17) sinh (cid:16) π(cid:96) β (cid:17) sinh (cid:16) π(cid:96) β (cid:17) sinh (cid:16) π ( (cid:96) + (cid:96) ) β (cid:17) (cid:12)(cid:12)(cid:12) + O ( e − πLT ) , (96) ymmetry decomposition of negativity of massless free fermions log N (3) n e , ( α ) = − π βL (cid:20)(cid:18) n e − n e (cid:19) ( (cid:96) − (cid:96) ) + n e (cid:96) ( (cid:96) − (cid:96) ) + n e (cid:96) (cid:21) − ( (cid:96) − (cid:96) ) α πLβn e + − ∞ (cid:88) j =1 ( − j j πjLβ ) cosh (cid:16) j ( (cid:96) − (cid:96) ) αβn e (cid:17) sinh( π(cid:96) j/β ) − sinh( π(cid:96) j/β )sinh (cid:16) π ( (cid:96) − (cid:96) ) jn e β (cid:17) − n e . (97)For fixed (cid:96) , /β and τ = iβ/L → we get E (3)1 ( α ) = − π(cid:96) (cid:96) βL − ( (cid:96) − (cid:96) ) α πLβ , (98)and therefore, E ( α, LT (cid:29) 1) = E − α π n e log (cid:12)(cid:12)(cid:12)(cid:16) βπ(cid:15) (cid:17) sinh (cid:16) π(cid:96) β (cid:17) sinh (cid:16) π(cid:96) β (cid:17) sinh (cid:16) π ( (cid:96) + (cid:96) ) β (cid:17) (cid:12)(cid:12)(cid:12) + O ( e − πLT ) , (99)where E ( LT (cid:29) 1) = log (cid:12)(cid:12)(cid:12) ( βπ(cid:15) ) sinh( π(cid:96) β ) sinh( π(cid:96) β )sinh( π ( (cid:96) (cid:96) β ) (cid:12)(cid:12)(cid:12) . This limit confirms analytically thevolume law behaviour observed in Fig. 2.The high-temperature limit of the charged probability N ( α ) in Eq. (87) is N ( α, LT (cid:29) (cid:39) − α π log | β π (cid:15) N sinh ( π(cid:96) L ) sinh ( π(cid:96) L )sinh( π ( (cid:96) (cid:96) L ) | | α | ≤ π , (2 π −| α | ) | α | π log | β π (cid:15) N sinh ( π(cid:96) L ) sinh ( π(cid:96) L )sinh( π ( (cid:96) (cid:96) L ) | − log | β π (cid:15) N sinh( π(cid:96) L ) sinh( π(cid:96) L ) | . | α | > π (100)Let us conclude the subsection comparing these new results with those for thestandard (bosonic) charged negativity reported in Eq. (29). At zero temperature andin the thermodynamic limit (cid:96) i (cid:28) L , Eq. (89) matches exactly the bosonic negativity (29)(at K = 1 to describe free fermions) obtained in the same limit. As discussed deeplyin Ref. [85] for the Rényi negativity (at α = 0) , this shows that the choice of chargedmoments of the partial TR we made in Eq. (72) provides a partition function evaluatedon the same worldsheet R n,α as the one for the moments of the standard charged partialtranspose in [91]. Again for conciseness of the various formulas, in this subsection we focus on the case (cid:96) = (cid:96) = (cid:96) (when also a closed-form expression for the spin-dependent part is easier towrite), but more general formulas are similarly derived. Since we are ultimately using ymmetry decomposition of negativity of massless free fermions − − q . . . . . . . . N ( q ) β = 20 β = 50 GS − − q . . . . . . . . R ( q ) β = 20 β = 50 β = 100 GS . . . . . . T . . . . . N ( q ) q = 0 , L = 100 , ‘ = 40 q = 1 , L = 100 , ‘ = 40 q = 0 , L = 200 , ‘ = 40 q = 1 , L = 200 , ‘ = 40 N Figure 4: Imbalance resolved negativities for a few different values of q , L = 200 , (cid:96) = (cid:96) = (cid:96) = 30 , with n e → (left-panel) and n = 2 (middle-panel). The dashed black linesare the truly asymptotic result (108) showing equipartition, while the solid lines include thefirst correction due to the cutoffs as in Eq. (107). The dashed coloured lines are the ratiobetween the Fourier transforms without exploiting the saddle point approximation. Forsmall q , the field theory prediction (in which the lattice cutoffs are included) well describesthe numerical data. In the right-panel, (cid:96) = 40 is fixed, we report two system sizes andtwo values of q and plot N ( q ) as a function of T . The coloured lines are Eq. (107) whilethe dashed one represents Eq. (108). The plot confirms the equipartition of negativity.Moreover, for large T , N ( q ) becomes a universal function of π(cid:96)T . a saddle point approximation to make the Fourier transform (75), the charged moments(72) can be truncated at Gaussian level in α as N ( ν ) n ( α ) = R ( ν ) n e − b n α / , (101)where b n = 1 π n log (cid:12)(cid:12)(cid:12) θ ( r | τ ) θ (2 r | τ ) − (cid:16) (cid:15)L ∂ z θ (0 | τ ) (cid:17) − (cid:12)(cid:12)(cid:12) . (102)The Fourier transform reads Z ( ν ) R ,n ( q ) = R ( ν ) n (cid:90) π − π dα π e − iqα e − α b n / , (103)where we used that the expectation value of the charge imbalance operator ˆ Q A for a freeDirac field is ¯ q = 0 at any temperature. In the saddle point approximation the integrationdomain is extended to the whole real line and we end up in a simple Gaussian integral,obtaining Z ( ν ) R ,n ( q ) (cid:39) R ( ν ) n √ πb n e − q bn . (104)Through a similar analysis, we compute p ( ν ) ( q ) = (cid:90) π − π dα π e − iqα N ( ν )1 ( α ) , (105)which through the saddle-point approximation reads p ( q ) (cid:39) e − q bN √ πb N b N = 1 π log (cid:12)(cid:12)(cid:12) θ ( r | τ ) θ (2 r | τ ) − (cid:16) (cid:15) N L ∂ z θ (0 | τ ) (cid:17) − (cid:12)(cid:12)(cid:12) . (106) ymmetry decomposition of negativity of massless free fermions α ∈ [ − π, π ] , the quantity N ν ( α ) has a global maximum for α = 0 andtwo local maxima for α = ± π , see Fig. 3 (left). However, since N ν ( ± π ) < N ν (0) , we canneglect the contributions to the integral coming from the regions close to the extrema at α = ± π . A similar reasoning applies to all odd charged moments R n o ( α ) . Once again, letus stress the difference between the cutoff (cid:15) N and the cutoff (cid:15) obtained in the replica limit b = lim n e → b n e , whose lattice expression is given in Eqs. (180a) and (179), respectively.Putting everything together, we obtain R ( ν ) n ( q ) = R ( ν ) n (cid:115) (2 πb N ) n πb n e − q ( bn − nbN ) , N ( ν ) ( q ) = 12 (cid:16) e E ( ν ) (cid:114) b N b e − q ( b − bN ) − (cid:17) . (107)When the O (1) terms are negligible with respect to the leading order ones in the variance, b N (cid:39) b , hence N ( ν ) ( q ) (cid:39) N ( ν ) , (108)i.e. exact equipartition of negativity in the different imbalance sectors at leading order,as shown for the bosonic negativity in section (2.2). A similar result holds even if (cid:96) (cid:54) = (cid:96) in the low/high temperature limits, it would be sufficient to modify the expression of thevariances in Eqs. (102) and (106).It is instructive to explicitly write down the first term breaking the equipartition.For large L , we can expand the exponential in Eq. (107) as e − q ( b − bN ) (cid:39) − q log( | (cid:15)/(cid:15) N | ) π L ) ≡ − γ (log L ) q , (109)and (cid:114) b N b (cid:39) γ (cid:48) log L , (110)where γ and γ (cid:48) are implicitly defined, also in terms of the cutoffs (cid:15) and (cid:15) N in AppendixC. To sum up, we get N ( ν ) ( q ) (cid:39) N ( ν ) (cid:16) γ (cid:48) log L − γ (log L ) q + . . . (cid:17) , (111)where we have derived the leading q -dependent contributions and shown that theequipartition is broken at order / (log L ) .In Fig. 4 we test the accuracy of our predictions against exact lattice numericalcalculations. It is evident that equipartition is broken for all the values of (cid:96), T, L weconsidered and the effect is more pronounced as | q | is increased. However, the mainsmooth part of corrections to the scaling is captured by Eq. (107), see the full line in theplots, and does not come as a surprise. Also the presence of further subleading oscillating(in q ) corrections have been observed for the resolved entropies [96] and were expected.In our case, such corrections are enhanced by the presence of the maxima at α = ± π in N ( α ) , see Fig. 4, that provide large corrections to the scaling in p ( q ) . Indeed, takingthe Fourier transforms without making the saddle-point approximation, the agreementbetween numerics and field theory is perfect. As (cid:96) (cid:29) /T , all these corrections becomesmaller and imbalance resolved negativity flattens in q , mainly as a consequence of thelowering of the maxima at α = ± π in N ( α ) , see Fig. 4. ymmetry decomposition of negativity of massless free fermions u v v ‘ e i(cid:25) ( kn (cid:0) ’ (cid:25) + (cid:11) (cid:25)n ) k e (cid:0) i(cid:25) ( kn (cid:0) ’ (cid:25) + (cid:11) (cid:25)n ) k e (cid:0) i(cid:25) ( kn + (cid:11) (cid:25)n ) k e i(cid:25) ( kn + (cid:11) (cid:25)n ) k ‘ ‘ u ‘ ‘ ! L (cid:0) ‘ e (cid:0) i(cid:25) ( kn + (cid:11) (cid:25)n ) k e i(cid:25) ( kn + (cid:11) (cid:25)n ) k Figure 5: The tripartite geometry (left) of an interval of length (cid:96) symmetrically embeddedinside another subsystem of total length (cid:96) . A single interval in a chain of total length L (right) is obtained taking the limit (cid:96) → L − (cid:96) of this tripartite geometry. The bottompanel shows the phase taken by the field ψ k going around each branch point. The reduceddensity matrix corresponds to the union of the coloured regions, and the partial transposeis applied to the blue region. 5. Charged and symmetry resolved negativities in a bipartite geometry In this section we move to the imbalance resolved negativity of a single interval atfinite temperature. This geometry can be studied more effectively by considering atripartite geometry where an interval of length (cid:96) is symmetrically embedded insideanother subsystem of total length (cid:96) , as depicted in Fig 5. Eventually, we take the limit (cid:96) → L − (cid:96) in our calculations, where L is the total length of the chain, in such a waythat the part B becomes the empty set and consequently the system becomes bipartite.Choosing the locations of the branch points at u = − (cid:96) / (2 L ) = − r / , v =( (cid:96) / (cid:96) ) /L = r / r , u = (cid:96) /L = r , v = 0 , the multivalued field ψ k takes upa phase e πi ( kn + α πn − ϕn π ) , e − πi ( kn + α πn − ϕn π ) going around v and u , respectively while goingaround v and u picks up a phase e πi ( kn + α πn ) , e − πi ( kn + α πn ) , respectively. As repeatedlyused, this multivaluedness of the field ψ k can be removed by introducing a single-valuedfield coupled to an external gauge field A kµ , as in Eq. (53). In order to recover the correctphases of the multivalued fields, A kµ has to satisfy π (cid:15) µν ∂ ν A kµ ( x ) = (cid:18) kn + απn − ϕ n π (cid:19) ( δ ( x − v ) − δ ( x − u )) + (cid:18) kn + α πn (cid:19) ( δ ( x − v ) − δ ( x − u )) . (112)Therefore, Z ( ν ) R ,k ( α ) can be expressed as the following correlation function of vertexoperators Z ( ν ) R ,k ( α ) = (cid:104) V − kn − α πn ( u ) V kn + απn − ϕn π ( v ) V − kn − απn + ϕn π ( u ) V kn + α πn ( v ) (cid:105) . (113) ymmetry decomposition of negativity of massless free fermions Z ( ν ) R ,k ( α ) = | θ ( r | τ ) | − kn + απn − ϕn π ) (cid:12)(cid:12)(cid:12) θ ( r | τ ) θ ( r + r | τ ) (cid:12)(cid:12)(cid:12) − kn + α πn )( kn + απn − ϕn π ) | θ ( r + r | τ ) | − kn + α πn ) × (cid:12)(cid:12)(cid:12) (cid:15)L ∂ z θ (0 | τ ) (cid:12)(cid:12)(cid:12) − ∆ k ( α ) | θ ν (( kn + α πn )( r − r ) + ϕ n π r | τ ) θ ν (0 | τ ) | , (114)where ∆ k ( α ) = − k n − kαn π − α n π + 4 k ϕ n nπ + 2 αϕ n nπ − ϕ n π − θ ( − k )(1 + 4 kn + 2 αnπ − ϕ n π ) . (115)Also in this case we fix the value of ϕ n for k < according to the discussion in AppendixB, taking care of the mode k = 0 for n = n o . This leads to the spin-independent terms log N n, ( α ) = log R n − α π n log (cid:12)(cid:12)(cid:12) θ ( r | τ ) θ (cid:0) r | τ (cid:1) θ ( r + r | τ ) θ (cid:0) r + r | τ (cid:1) ( (cid:15)L ∂ z θ (0 | τ )) (cid:12)(cid:12)(cid:12) , log N n o , ( | α | > π/ 2) = log R n o − α π n o log (cid:12)(cid:12)(cid:12) θ ( r | τ ) θ (cid:0) r | τ (cid:1) θ ( r + r | τ ) θ (cid:0) r + r | τ (cid:1) ( (cid:15)L ∂ z θ (0 | τ )) (cid:12)(cid:12)(cid:12) + 2 | α | πn o log (cid:12)(cid:12)(cid:12) θ ( r | τ ) θ (cid:0) r | τ (cid:1) θ (cid:0) r + r | τ (cid:1) ( (cid:15)/L∂ z θ (0 | τ )) (cid:12)(cid:12)(cid:12) − n log (cid:12)(cid:12)(cid:12) θ ( r | τ )( (cid:15)L ∂ z θ (0 | τ )) (cid:12)(cid:12)(cid:12) , log R n o = − (cid:16) n o − n o (cid:17) log (cid:12)(cid:12)(cid:12) θ ( r | τ ) θ (cid:0) r | τ (cid:1) θ ( r + r | τ ) θ (cid:0) r + r | τ (cid:1) ( (cid:15)L ∂ z θ (0 | τ )) (cid:12)(cid:12)(cid:12) , log R n e = − (cid:16) n e − n e (cid:17) log (cid:12)(cid:12)(cid:12) θ ( r | τ ) θ (cid:0) r | τ (cid:1) θ (cid:0) r + r | τ (cid:1) ( (cid:15)L ∂ z θ (0 | τ )) (cid:12)(cid:12)(cid:12) − (cid:16) n e − n e (cid:17) log (cid:12)(cid:12)(cid:12) θ ( r + r | τ )( (cid:15)L ∂ z θ (0 | τ )) − (cid:12)(cid:12)(cid:12) , (116)and log N ( ν ) n, ( α ) = 2 ( n − / (cid:88) k = − ( n − / log | θ ν (( kn + α πn )( r − r ) + ϕ n π r | τ ) θ ν (0 | τ ) | . (117)for the spin structure dependent term. In this geometry, N ( ν ) n o ( α ) presents a discontinuityfor | α | = π , as shown for n o = 1 in Fig. 3.At this point, we derived all the needed formulas to take the limit r → − r and toreproduce the bipartite geometry in which we are interested. Using that θ ( z + 1) = θ ( z ) ,the spin-independent part of the charged logarithmic negativity (for even n and for oddand α < | π/ | ) becomes log N n, ( α ) = log R n − α π n log (cid:12)(cid:12)(cid:12) Lθ ( r | τ ) (cid:15)∂ z θ (0 | τ ) (cid:12)(cid:12)(cid:12) , (118)while the spin-dependent ones are just given by (117) without major simplifications. ymmetry decomposition of negativity of massless free fermions N ( α ) = Tr[ ρ R e i ˆ Qα ] . The finalexpression can be read off from Eq. (116) and, after some standard manipulations, canbe put in the form N ( ν )1 ( α ) = (cid:16) θ ( r | τ ) (cid:15)NL ∂ z θ (0 | τ ) (cid:17) − απ ) (cid:12)(cid:12)(cid:12) θ ν ( | α π | (1 − r ) | τ ) θ ν (0 | τ ) (cid:12)(cid:12)(cid:12) | α | ≤ π , (cid:16) θ ( r | τ ) (cid:15)NL ∂ z θ (0 | τ ) (cid:17) − | απ |− (cid:12)(cid:12)(cid:12) θ ν ( | α π | (1 − r )+ r | τ ) θ ν (0 | τ ) (cid:12)(cid:12)(cid:12) , | α | > π . (119)The cutoff for the charged probability is denoted by (cid:15) N and its explicit expression is givenin Eq. (182a) and (182b) for | α | ≤ π/ and | α | > π/ , respectively. We recall that, as inthe tripartite case, (cid:15) N is different from the cutoff (cid:15) obtained in the replica limit as n e → and explicitly given in Eq. (181). In this section we report the low and high temperature limits of the charged Rényinegativity, focussing, once again, on even n = n e and on the ν = 3 sector. The lowtemperature limits of the spin-independent part, Eq. (116), can be obtained through therelation (88), finding log N n e , ( α ) = log R n e − α π n e (cid:104) log (cid:12)(cid:12)(cid:12) Lπ(cid:15) sin (cid:16) π(cid:96) L (cid:17)(cid:12)(cid:12)(cid:12)(cid:105) + O ( e − π/ ( LT ) ) , (120)while the low temperature limit of the spin-dependent term, Eq. (117), can be obtainedthrough the product representation (65) of the theta functions and it reads E (3)1 ( α ) = lim n e → log Z (3) n e , ( α ) = 4 e − π/ ( LT ) (cid:18) cos(( r − r ) α ) cos( π ( r + r ) / π ( r − r ) / − (cid:19) . (121)Therefore, as τ = iβ/L → and r → − r , we get in the replica limit E ( LT (cid:28) α ) = (cid:16) − α π (cid:17) log (cid:12)(cid:12)(cid:12) Lπ(cid:15) sin (cid:16) π(cid:96) L (cid:17)(cid:12)(cid:12)(cid:12) + O ( e − π/ ( LT ) ) . (122)We notice that Eq. (122) coincides with ρ / A e i ˆ Q A α ) , as it should for pure states(and mentioned in the introduction for α = 0 ).The low-temperature limit of Eq. (119) is log N ( α ) (cid:39) − α π log (cid:12)(cid:12)(cid:12) Lπ(cid:15) N sin( π(cid:96) L ) (cid:12)(cid:12)(cid:12) , | α | ≤ π , − | α |− π ) π log (cid:12)(cid:12)(cid:12) Lπ(cid:15) N sin( π(cid:96) L ) (cid:12)(cid:12)(cid:12) , | α | > π . (123)Interestingly, the previous expansion shows that its Fourier transform vanishes for oddvalues of the imbalance q . This agrees with the discussion at the end of Section 2.3: as τ = iβ/L → , the state becomes pure and the vanishing of p ( q ) occurs because the parityof the imbalance is fixed by the conservation of ˆ N + ˆ N . This reflects the fact that theentanglement is better resolved in subsystem charges rather than in the imbalance. ymmetry decomposition of negativity of massless free fermions log N n e , ( α ) = log R n e − α π n e (cid:104) − π (cid:96) βL + log (cid:12)(cid:12)(cid:12) β(cid:15)π sinh (cid:16) π(cid:96) β (cid:17)(cid:12)(cid:12)(cid:12)(cid:105) + O ( e − πLT ) . (124)The spin-dependent term in Eq. (116) can be evaluated as follows log N (3) n e , ( α ) = − π βL (cid:20)(cid:18) n e − n e (cid:19) ( L − (cid:96) ) + n e (cid:96) ( L − (cid:96) ) + n e (cid:96) (cid:21) − ( L − (cid:96) ) α πLβn e + − ∞ (cid:88) j =1 ( − j j πjLβ ) (cid:16) cosh (cid:16) j ( L − (cid:96) ) αβn e (cid:17) sinh( π ( L − (cid:96) ) j/β ) − sinh( ϕ n e (cid:96) j/β )sinh (cid:16) π ( L − (cid:96) ) jn e β (cid:17) − n e (cid:17) , (125)which gives in the replica limit, for β/L → and (cid:96) /β fixed, E (3)1 ( α ) = π(cid:96) ( (cid:96) − L )2 βL − ( L − (cid:96) ) α πβL + O ( e − πLT ) . (126)To sum up, we find in the high temperature regime E ( LT (cid:29) α ) = (cid:16) − α π (cid:17)(cid:104) log (cid:12)(cid:12)(cid:12) βπ(cid:15) sinh( π(cid:96) β ) (cid:12)(cid:12)(cid:12) − π(cid:96) β (cid:105) − α L πβ + O ( e − πLT ) . (127)Given the result found in Eq. (122), one could be tempted to do a conformal mappingto a cylinder periodic in time, to get E ( LT (cid:29) α ) naive = (cid:16) − α π (cid:17) log (cid:12)(cid:12)(cid:12) β(cid:15)π sinh (cid:16) π(cid:96) β (cid:17)(cid:12)(cid:12)(cid:12) , (128)which is nothing but the finite temperature logarithmic charged entropy of order / .This naive derivation provides a wrong result whose origin has been extensively discussedin [46] and it remains the right interpretation also for α (cid:54) = 0 . Indeed, for pure states (i.e. T → ), the n e − sheeted Riemann surface, R n e ,α , decouples in two independent ( n e / -sheeted surfaces characterised by the parity of the sheets and therefore E ( α )( LT (cid:28) 1) =2 log Tr( ρ / A e i ˆ Q A α ) . Conversely, this decoupling of the sheets does not occur at finitetemperature. The lack of decoupling is manifested in the presence of the linear terms (cid:96)/β and L/β in Eq. (127) which cannot be derived through a simple conformal mapping.We also present the high-temperature limit of the charged probability in Eq. (119),that is log N ( α ) (cid:39) − α π (cid:104) log (cid:12)(cid:12)(cid:12) βπ(cid:15) N sinh( π(cid:96) β ) (cid:12)(cid:12)(cid:12) − π(cid:96) β (cid:105) − α L πβ , | α | ≤ π , − α − π ) π log (cid:12)(cid:12)(cid:12) βπ(cid:15) N sinh( π(cid:96) β ) (cid:12)(cid:12)(cid:12) − α(cid:96) β + α (cid:96) πβ − α L πβ , | α | > π . (129) ymmetry decomposition of negativity of massless free fermions . . . . . . . . . πℓT − − − − − − − E ( α , T ) − E ( α , T = ) α = 0 . α = 1 α = 2 . . . . . . . . . πℓT − . − . − . − . − . − . − . . E ( α , T ) − E ( α , T = ) α = 0 . α = 1 α = 2 Figure 6: The charged logarithmic negativity for a bipartite geometry in the infinite-line(left) or semi-infinite (right). We set L = 200 . Analytical prediction in Eqs. (127) and(131). A simple generalisation of the previous calculation concerns the charged logarithmicnegativity for a semi-infinite system. For free fermions, the semi-infinite geometry isobtained from the infinite one by cutting the interval A in half. Because of the structureof the vertex operators correlations in Eq. (113), the entanglement in the semi-infinitesystem is equal to half of that of the infinite one in Eq. (116). Therefore, the chargedlogarithmic negativity of a finite interval with length (cid:96) = r L is given by log N ( ν ) n ( α ) = log R n − α π n log (cid:12)(cid:12)(cid:12) θ (2 r | τ )( (cid:15)L ∂ z θ (0 | τ )) (cid:12)(cid:12)(cid:12) ++ ( n − / (cid:88) k = − ( n − / log (cid:12)(cid:12)(cid:12) θ ν (( kn + α πn )(1 − r ) + ϕ n π r | τ ) θ ν (0 | τ ) (cid:12)(cid:12)(cid:12) . (130)The low and high temperature limits of this expression are obtained in analogy with theinfinite line case, e.g. as τ → we get E ( α ) = (cid:16) − α π (cid:17)(cid:104) log (cid:12)(cid:12)(cid:12) βπ(cid:15) sinh( 2 π(cid:96) β ) (cid:12)(cid:12)(cid:12) − π(cid:96) β (cid:105) − α L πβ + O ( e − πLT ) . (131)The correctness of these CFT charged negativities is tested against lattice calculationsin Fig. 6. Here we plot E ( α, T ) − E ( α, T = 0) that turns out to be a universal function of π(cid:96)T and πLT , in agreement with Eqs. (127) and (131). As for the tripartite case, in orderto test our final field theoretical results, we have taken into account the explicit expressionfor the cutoff (cid:15) . Since it does not depend on the temperature, it can be extracted fromthe knowledge of the lattice charged moments Tr( ρ / A e i ˆ Q A α ) at T = 0 , derived in [96] andexplicitly reported in the Appendix, see Eq. (181). ymmetry decomposition of negativity of massless free fermions As usual, the Fourier transform (75) is performed in the scaling regime with a saddlepoint approximation. Hence, the charged moments (72) can be truncated at Gaussianlevel in α as N ( ν ) ( α ) n = R ( ν ) n e − α b n , (132)where b n = 4 π n log (cid:12)(cid:12)(cid:12) A (1) n Lθ ( r | τ ) (cid:15)∂ z θ (0 | τ ) (cid:12)(cid:12)(cid:12) , (133)and we used a quadratic approximation for log N ( ν ) n, ( α ) = A (0) n − α π n log A (1) n . (134)The RN in the sector q are R ( ν ) n ( q ) = Z R ,n ( q )[ p ( q )] n = R ( ν ) n (cid:90) π − π dα π e − iqα e − α b n / (cid:104) (cid:90) π − π dα π e − iqα N ( α ) (cid:105) n , (135)and, through the saddle point approximation, R ( ν ) n (cid:39) R ( ν ) n (cid:115) (2 πb N ) n πb n e − q ( bn − nbN ) , N ( ν ) ( q ) (cid:39) (cid:16) e E ( ν ) (cid:114) πb N πb e − q ( b − bN ) − (cid:17) , (136)with b = lim n e → b n e , b N = 4 π log (cid:12)(cid:12)(cid:12) A (1) N Lθ ( r | τ ) (cid:15) N ∂ z θ (0 | τ ) (cid:12)(cid:12)(cid:12) . (137)The explicit expressions for the cutoff (cid:15) and (cid:15) N can be found in Eqs. (181) and (182a),respectively. When the O (1) terms are negligible with respect to the leading order ones, b (cid:39) b N and we find the exact equipartition of negativity in the different charge sectors atleading order, i.e. N ( ν ) ( q ) (cid:39) N ( ν ) , on the same lines as for the tripartite case. We usedthat the leading contribution to the integral p ( q ) comes from the region near the saddlepoint α = 0 , despite the presence of two local maxima at α = ± π . This is possible aslong as T > : when T → , the secondary maxima become degenerate with the one in and they cannot be neglected. Their degeneracy is indeed related to the fact that p ( q ) becomes zero for all odd q .Finally, it is worth reporting the high temperature limits of the variances in Eq.(137), that following the steps in Section 5.1, simplify as b = lim n e → π n e (cid:104) log (cid:12)(cid:12)(cid:12) βπ(cid:15) sinh( π(cid:96) β ) (cid:12)(cid:12)(cid:12) − π(cid:96) β + πL β (cid:105) ,b N = 4 π (cid:104) log (cid:12)(cid:12)(cid:12) βπ(cid:15) N sinh( π(cid:96) β ) (cid:12)(cid:12)(cid:12) − π(cid:96) β + πL β (cid:105) . (138) ymmetry decomposition of negativity of massless free fermions − − − − q . . . . . N ( q ) β = 20 β = 30 β = 50 β = 70 . . . . . . T . . . . . . p ( q ) q = 0 , L = 2 ‘ = 40 q = 1 , L = 2 ‘ = 40 q = 0 , L = 2 ‘ = 100 q = 0 , L = 2 ‘ = 200 q = 1 , L = 2 ‘ = 200 Figure 7: Left panel: The imbalance-resolved negativity as a function of q in a bipartitegeometry. The subsystem size is fixed, (cid:96) = 40 , the total system sizes is also fixed, L = 200 ,while T is varied. The coloured full lines represent Eq. (136) while the dashed ones representEq. (135) in the replica limit. Right panel: The probability of finding q as outcome of ameasurement of ˆ Q . As the state becomes pure (i.e. T → ), p ( q ) → for odd q , as explainedin Sec. 2.3. The full lines correspond to the Fourier transforms of the charged probabilityin (119) without saddle-point approximation. The symmetry resolved negativities for a single interval embedded inside a semi-infinite chain or in the low-temperature regime are straightforwardly derived with minormodifications of the above calculations.Also for this bipartite geometry it is instructive to identify the first term breakingequipartition. Expanding to order O ((log L ) − ) the above expressions, we get N ( ν ) ( q ) (cid:39) N ( ν ) (cid:16) − ˜ γ log | θ ( r | τ )( ∂ z θ (0 | τ ) /L ) | ) − q ˜ γπ | θ ( r | τ )( ∂ z θ (0 | τ ) /L ) − | ) (cid:17) , (139)where ˜ γ = π log( (cid:15) N A / ( (cid:15) A N )) . Since A / A N (cid:39) , ˜ γ can be explicitly computed throughthe results for (cid:15) and (cid:15) N found in Appendix C.Our analytic results for the symmetry resolved negativity are compared with thenumerical data in the left panel of Figure 7. The equipartition of negativity is brokenfor all the considered values of (cid:96), T, L and the effect is more evident as | q | is increased.However, Eq. (136) can capture the smooth part of these corrections to the scaling, asshown by the full line in the plots. As (cid:96) (cid:29) /T , the corrections due to the presence ofthe maxima at α = ± π in N ( α ) become smaller and the imbalance resolved negativityflattens in q , mainly as a consequence of the lowering of the maxima at α = ± π in N ( α ) ,see Fig. 4. We also check the correctness of our prediction for p ( q ) in the right-panel ofthe same figure: we can observe that as the state becomes pure (i.e. T → ), p ( q ) → for odd q , as explained above. As already stressed many times, the divergent behaviourof negativity in the same charge sector is a consequence of the fact that the imbalance isno longer the right quantum number to resolve the entanglement. ymmetry decomposition of negativity of massless free fermions 6. Conclusions We studied the entanglement negativity in systems with a conserved local charge and wefound it to be decomposable into symmetry sectors. The partial TR operation does notspoil the result found for the standard partial transposition operation [91]: the resultingoperator that commutes with the partial TR density matrix is not the total charge, butrather an imbalance operator, which is essentially the difference operator between thecharge in the two regions. We introduced a normalised version of the charge imbalanceresolved negativity (both fermionic and bosonic) which has the great advantage to bean entanglement proxy also for the symmetry sectors, e.g. it vanishes if the standardpartial transpose has only positive eigenvalues in the sector. The price to pay is that thenormalised symmetry resolved negativity diverges (for some sectors) in the limit of purestates, as a consequence of the fact that the imbalance is no longer the best quantityto resolve the entanglement. Another interesting property of this normalisation for thesector partial transpose is the negativity equipartition , i.e. the entanglement is the samein all imbalance sectors, in full analogy to entropy equipartition for pure states [93].We then considered the (1+1)d CFT corresponding to free massless Dirac fermionsat finite temperature T and finite size L. We derived field theory predictions for thedistribution of negativity in both tripartite and bipartite settings (i.e. the entanglementbetween two adjacent intervals and the one between one interval and the remainder,respectively). We tested our prediction against numerical computations for a latticeversion of the Dirac field, in which the non-universal terms are fixed from exact analyticcomputations. In both geometries, we find that, at leading order, the charge imbalanceresolved negativity satisfies entanglement equipartition. We identify the subleading termsresponsible for the breaking of equipartition in the lattice model.There are different aspects that our manuscript leaves open for further study. Thefirst one concerns the calculation of the time evolution of the charged and imbalanceresolved negativity to understand if and how the quasiparticle picture remains true withinthe sectors of an internal local symmetry of a quantum many-body system, as recentlydone for the resolved entropies [95]. Secondly, one may use the corner transfer matrix toinvestigate the symmetry decomposition of negativity in gapped one-dimensional modelsby combining former studies of the total negativity [42] with those for symmetry resolution[99]. Eventually, the generalisation of one-dimensional results to higher dimensions can bedone using the dimensional reduction approach, as already done for the total negativityin [85]. Decoupling the initial d-dimensional problem into one-dimensional ones in amixed space-momentum representation [100] would allow to generalise the above resultsto higher dimensional Fermi surfaces. Acknowledgments We thank Marcello Dalmonte, Giuseppe Di Giulio, Moshe Goldstein, and Vittorio Vitalefor useful discussions. The authors acknowledge support from ERC under Consolidator ymmetry decomposition of negativity of massless free fermions AppendicesA. Numerical methods In this first appendix, we report how to numerically calculate the charged negativityassociated with the partial TR (15) for free fermions on a lattice described by the hoppingHamiltonian on a chain H F F = − (cid:88) j ( f † j +1 f j + H . c . ) . (140)The technique is a straightforward generalisation to α (cid:54) = 0 of the one presented in [85].This method is used throughout the main text to obtain all lattice numerical results.Let Γ = I − C be the covariance matrix where C ij = Tr( ρf † i f j ) is the correlationmatrix. For a thermal state, the single-particle correlator reads C ij = (cid:88) k u ∗ k ( i ) u k ( j ) e βω k + 1 , (141)where ω k and u k ( i ) are the single-particle eigenvalues and eigenvectors of the Hamiltonian(140). For a bipartite Hilbert space H A ⊗ H B where A = A ∪ A , the covariance matrixtakes a block form Γ = (cid:32) Γ Γ Γ Γ (cid:33) , (142)where Γ and Γ are the reduced covariance matrices of the two subsystems A and A ,respectively, while Γ and Γ † contain the cross correlations between them. By simpleGaussian states’ manipulations, the correlation matrices associated with ρ R A , ( ρ R A ) † canbe written as [85] Γ ± = (cid:32) − Γ ± i Γ ± i Γ Γ (cid:33) . (143)The objects we are interested in are N n e = Tr[( ρ R ( ρ R ) † ) n e / e i ˆ Qα ] and N ( α ) =Tr[ ρ R e i ˆ Qα ] . The imbalance of the relativistic Dirac field corresponds to the discretisedoperator ˆ Q = ˆ N A ∪ A − / (cid:96) + (cid:96) ) . Notice that in this basis, ˆ Q is not the difference,but the sum of the number operators. Furthermore, it presents a shift compared to thenumber operator of the non-relativistic fermions. The single particle correlation matrixassociated to the normalised composite density operator ρ x = ρ R ( ρ R ) † /Z x is [85, 124] Γ x = ( + Γ + Γ − ) − (Γ + + Γ − ) , (144)where the normalisation factor is Z x = Tr(Γ x ) = Tr( ρ A ) . In terms of eigenvalues of ymmetry decomposition of negativity of massless free fermions log N n ( α ) = − iα (cid:96) + (cid:96) N (cid:88) j =1 log (cid:34)(cid:18) − ν x j (cid:19) n/ + e iα (cid:18) ν x j (cid:19) n/ (cid:35) ++ n N (cid:88) j =1 log (cid:2) ζ j + (1 − ζ j ) (cid:3) , (145)where ν x j and ζ j are eigenvalues of the matrices Γ x (144) and C (141), respectively. Interms of the eigenvalues ν ’s of Γ ± (143) ( Γ + and Γ − have the same spectrum), the chargednormalisation N ( α ) is log N ( α ) = − iα (cid:96) + (cid:96) N (cid:88) j =1 log (cid:20)(cid:18) − ν j (cid:19) + e iα (cid:18) ν j (cid:19)(cid:21) . (146)Taking the Fourier transform of the numerical data for N n e ( α ) and N ( α ) , we finallyobtain the imbalance resolved negativities. B. Mode expansion of charged moments of ρ A and ρ R A Following Ref. [121], we report some details about the transformation of the traceformulas into a product of n decoupled partition functions for non-interacting systemswith conserved U (1) charge. As mentioned in the main text, after diagonalising the twistmatrices a partition function on a multi-sheet geometry can be decomposed as Z n ( α ) = (cid:89) k Z k,n ( α ) , Z k,n ( α ) = (cid:104) e i (cid:82) d xA kµ j µk (cid:105) , (147)in which (cid:15) µν ∂ ν A k,µ ( x ) = 2 π p (cid:88) i =1 ν k,i ( α ) δ ( x − u i ) , (148)where πν k,i ( α ) is the vorticity of gauge flux determined by the eigenvalues of the twistmatrix, p are the intervals defined between a pair of points u i − and u i . The vorticitiessatisfy the neutrality condition (cid:80) i ν k,i ( α ) = 0 for every k . As already stressed, there areseveral representations of the partition function Z k,n ( α ) . In order to obtain the asymptoticbehaviour, one needs to take the sum over all the representations ˜ Z k,n ( α ) = (cid:88) { m i } Z ( m ) k,n ( α ) , Z ( m ) k,n ( α ) = (cid:104) e i (cid:82) d xA ( m ) k,µ j µk (cid:105) ,(cid:15) µν ∂ ν A ( m ) k,µ ( x ) = 2 π p (cid:88) i =1 ˜ ν k,i ( α ) δ ( x − u i ) , (149)where m i is a set of integers and ˜ ν k,i ( α ) = ν k,i ( α ) + m i are shifted flux vorticities obeying (cid:80) i m i = 0 because of neutrality condition. By the bosonisation technique, we may write ˜ Z k,n ( α ) = E { m i } (cid:89) i We focus on the evaluation of the explicit cutoffs induced by the lattice for the chargedRényi negativity, for the ground state of the Hamiltonian (140). Analogous results forthe charged Rényi and entanglement entropies have already been worked out in [96].The evaluation of the charged negativity relies on the Fisher-Hartwig conjecture for thedeterminant of Toeplitz matrices. Here we closely follow the derivation for the negativityat T = 0 [81]. Denoting by u i ( r j ) = (cid:104) r j | u i (cid:105) the single particle eigenstate, the wave functiondescribing the ground state of the Hamiltonian has the form of a Slater determinant (cid:104){ r j }| Ψ (cid:105) = det[ u i ( r j )] . (159)Hence, the partition function Z R ,k ( α ) in Eq. (74) is Z R ,k ( α ) = det M R ,kmm (cid:48) ( α ) = det (cid:0) (cid:104) u m | T R α,k | u m (cid:48) (cid:105) (cid:1) , (160)where T R α is the matrix T R α,k = diag (cid:104) , ..., , e iϕ − πi kn − i αn , ..., e iϕ − πi kn − i αn , e i π kn + i αn , ..., e i π kn + i αn , , ..., (cid:105) . (161)Writing explicitly the single particle eigenstates as plane waves u i ( r j ) = √ L e i πmL j , m = ± , ± , ..., ± ( L/ − , we obtain M R ,kmm (cid:48) ( α ) = (cid:104) u m | T R α,k | u m (cid:48) (cid:105) = 1 L L − (cid:88) j =0 e − i πjL ( m − m (cid:48) ) T R α,k ( j ) == 12 π (cid:90) π dθe − iθ ( m − m (cid:48) ) / T R α,k (cid:18) Lθ π (cid:19) , (162)where we used the notation T R α,k ( j ) ≡ [ T R α,k ] jj for the diagonal entries of the matrix. Thelast identity in (162) is obtained in the scaling regime L → ∞ , (cid:96) → ∞ and (cid:96)/L fixed.Hence, in this basis, the matrix M R ,kmm (cid:48) ( α ) is a Toeplitz matrix.The asymptotic evaluation of the determinant of a Toeplitz matrix is based on aspecial standard structure that we now describe. In general, a Toeplitz matrix has theform T L [ φ ] = ( φ i − j ) , where φ k is the k -th Fourier coefficient of the symbol φ ( θ ) . TheFisher-Hartwig conjecture gives the asymptotic behaviour of the determinant of Toeplitzmatrices whose symbol admits a canonical factorisation as φ ( θ ) = ψ ( θ ) R (cid:89) r =1 t β r ,θ r ( θ ) u α r ,θ r ( θ ) , (163)where t β r ,θ r ( θ ) = exp[ − iβ r ( π − θ + θ r )] , θ r < θ < π + θ r , (164) u α r ,θ r ( θ ) = (2 − θ − θ r )) α r , Re [ α r ] > − , (165) ymmetry decomposition of negativity of massless free fermions ψ ( θ ) is a smooth vanishing function with zero winding number and R is the number ofdiscontinuities of φ ( θ ) . For L → ∞ , the Fisher-Hartwig formula gives det T L [ φ ] = ( F [ ψ ]) L (cid:32) R (cid:89) r =1 L α i − β i (cid:33) E FH , (166)where F [ ψ ] = exp (cid:18) π (cid:90) π ln ψ ( θ ) dθ (cid:19) , (167)and E FH = E [ ψ ] R (cid:89) i =1 G (1 + α i + β i ) G (1 + α i − β i ) G (1 + 2 α i ) (cid:89) ≤ i (cid:54) = j ≤ R (cid:0) − e i ( θ i − θ j ) (cid:1) − ( α i + β i )( α j − β j ) × R (cid:89) i =1 (cid:0) ψ − (cid:0) e iθ i (cid:1)(cid:1) − α i − β i (cid:0) ψ + (cid:0) e − iθ i (cid:1)(cid:1) − α i + β i , (168)assuming that ψ ( θ ) admits the Wiener-Hopf factorisation ψ ( θ ) = F [ ψ ] ψ + (exp( iθ )) ψ − (exp( − iθ )) . (169)Here E [ ψ ] = exp ( (cid:80) ∞ k =1 ks k s − k ) , with s k corresponding to the k -th Fourier coefficient of ln ψ ( θ ) , and G are the Barnes G -function G (1 + z ) = (2 π ) z/ e − ( z +1) z/ − γ E z / ∞ (cid:89) n =1 [(1 + z/n ) n e − z + z / (2 n ) ] , (170)and γ E the Euler constant.In the case of the negativity for two adjacent intervals and even n = n e , the symbol φ ( θ ) is given by φ ( θ ) = e iπ − πi ( k/n + α/ (2 πn )) , − πr < θ < ,e πi ( k/n + α/ (2 πn )) , < θ < πr , , πr < θ < π − πr , (171)with r i = 2 (cid:96) i /L . Therefore, it has three discontinuities and admits the following canonicalfactorization: φ ( θ ) = ψ ( θ ) t β ( k ) , − πr ( θ ) t β ( k ) ,πr ( θ ) t β ( k ) , ( θ ) , (172)where ψ ( θ ) = e iπr / − iπ ( kn + α πn )( r − r ) , (173) β ( k ) = kn + α πn = β ( k ) + 12 = β ( k ) , (174) β ( k ) = − β ( k ) + 12 , (175) ymmetry decomposition of negativity of massless free fermions R = 3 and α i = 0 . When β < , we have | β | > / and the FH conjecture in its original form breaks down (we can also use the mostgeneral hypothesis in which the Fisher-Hartwig conjecture works, i.e. if we introducethe seminorm ||| β ||| = max j,k | β k − β k | , where ≤ j, k ≤ , the conjecture has beenverified for ||| β ||| < [125]). In this case we should use the generalised Fisher-Hartwigconjecture, see e.g. [125, 126] in which one sums over all the inequivalent representationsof the symbol, i.e. summing over all possible { ˆ β i } such that ˆ β i = β i + n i , R (cid:88) i =1 n i = 0 . (176)The leading terms is given by the set(s) of integers { n i } that minimises the function (cid:80) Ri =1 ˆ β i . This is identical to the condition derived in Sec. B. We denote the correspondingset of solutions by M β . The Toeplitz determinant is sum of the standard form (166)corresponding to these solutions. Then, the asymptotic behaviour of the Toeplitzdeterminant in (160) is Z R ,k ( α ) = (2 − π(cid:96) /L )) − ( | k/n + α/ (2 πn ) |− / | k/n + α/ (2 πn ) |− / × (2 − π(cid:96) /L )) −| k/n + α/ (2 πn ) | (2 | k/n + α/ (2 πn ) |− / × (2 − π ( (cid:96) + (cid:96) ) /L )) | k/n + α/ (2 πn ) | ( | k/n + α/ (2 πn ) |− / × J α ( k )( L ) ∆ k ( α ) , (177)where ∆ k ( α ) is given by Eq. (80) and J α ( k ) = (cid:81) i =1 G (1 + β i ( k )) G (1 − β i ( k )) . Usingstandard manipulation of the Barnes G-function, we can rewrite the O (1) terms as (cid:16) n − n + 3 α π n (cid:17) log (cid:15) = k = ( n − (cid:88) k = − ( n − log 2 k J α ( k ) = − log 2 (cid:16) n − n + 3 α π n (cid:17) − (1 + γ E ) (cid:16) n − n + 3 α nπ (cid:17) + ∞ (cid:88) m =1 − π + n π + 6 α mnπ + ∞ (cid:88) m =1 m log ( m n ) n Γ[ n − nm − α π ]Γ[ n − nm + α π ]Γ[ n +2 nm − α π ]Γ[ − n +2 nm − α π ]Γ[ − n − nm − α π ]Γ[ − n − nm + α π ]Γ[ − n +2 nm + α π ]Γ[ − n +2 nm − α π ] . (178)Eq. (178) is well approximated by the expansion at the second order in α since highercorrections O ( α ) are negligible for most practical purposes (this is in full analogy withthe charged entropies [96]). In particular, in the replica limit n e → we have − (cid:18) − α π (cid:19) log (cid:15) (cid:39) . − . α . (179)In conclusion, the Fisher-Hartwig technique allows us to re-derive the leading CFT terms(at T = 0) and also provides the cutoff due to lattice regularisation. ymmetry decomposition of negativity of massless free fermions (cid:15) N of the chargedprobability in Eq. (87) and we report the final result α π log (cid:15) N (cid:39) − . α | α | ≤ π/ (180a) − (cid:18) − | α | π (cid:19) | α | π log (cid:15) N (cid:39) − . . | α | − . α | α | > π/ (180b)For a bipartite geometry, the lattice cutoff (cid:15) can be computed using the equivalencebetween the (charged) negativity and the (charged) Rényi entropy for pure states,exploiting the results of Ref. [96], i.e. − ( 12 − α π ) log (cid:15) (cid:39) . − . α . (181)We also write down the final result for the cutoff (cid:15) N of the charged probability in Eq.(119) α π log (cid:15) N (cid:39) − . α , | α | ≤ π/ , (182a) ( π − | α | ) π log (cid:15) N (cid:39) − . | α | − π ) , | α | > π/ . (182b)As already discussed in the main text, the knowledge of the exact expression for this non-universal quantities is relevant in order to test our analytical predictions against numericaldata without any fitting parameter. D. Twisted partial transpose For T = 0 , the fermionic Rényi negativity in CFT (17) is equal [85] to the bosonic Rényinegativity for both even and odd values of n , also in the charged case [91]. The definition(17) has been employed since ρ R A is not necessarily Hermitian. Then, the trace norm interms of square root of the eigenvalues of the composite operator ρ x = ρ R A ( ρ R A ) † providesa well-defined entanglement negativity for fermions. However, one can also introducea hermitian partial transpose, which is suitable to define another fermionic negativitybecause of its real spectrum [86]. This is done by considering the composite operator ˜ ρ x = ( ρ ˜ R A ) , in terms of the twisted partial transpose ρ ˜ R A = ρ R A ( − F , with ( − F thefermion number parity in A . The associated charged moments are ˜ N n = Tr[( ρ ˜ R A ) n e i ˆ Qα ] .The use of the composite operator ˜ ρ x is distinct from the previous one. In fact, the T ˜ R α matrix which glues together ρ ˜ R A is T ˜ R α = . . . − e − iα/n e − iα/n e − iα/n . . .. . . . . . . (183) ymmetry decomposition of negativity of massless free fermions e πi ( kn + α πn − ϕn π ) and e − πi ( kn + α πn ) , with ϕ n = π for both n even and odd. This means that for n even thetwo Rényi negativity are equal (and indeed the negativity may be obtained from thereplica limit n e → of both). Hence, we report the final results for the two geometriesstudied in the main text for n = n o . 1. Adjacent intervals: The spin-independent part of the moments of negativityare given by log ˜ N n o , ( α ) = log ˜ R n o − α π n o log | θ ( r | τ ) θ ( r | τ ) θ ( r + r | τ ) − ( (cid:15)L ∂ z θ (0 | τ )) − | + | α | n o π log | θ ( r | τ ) θ ( r | τ ) θ ( r + r | τ ) − ( (cid:15)L ∂ z θ (0 | τ )) − | , (184)while the spin structure dependent term is the same as Eq. (84) with ϕ n o = π . 2. 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