On the Möbius transformation in the entanglement entropy of fermionic chains
OOn the M¨obius transformation in the entanglement entropy offermionic chains
Filiberto Ares ∗† Departamento de F´ısica Te´orica, Universidad de Zaragoza, 50009 Zaragoza, Spain
Jos´e G. Esteve ‡ and Fernando Falceto § Departamento de F´ısica Te´orica, Universidad de Zaragoza, 50009 Zaragoza, Spain andInstituto de Biocomputaci´on y F´ısica de Sistemas Complejos (BIFI), 50009 Zaragoza, Spain
Amilcar R. de Queiroz ¶ Departamento de F´ısica Te´orica, Universidad de Zaragoza, 50009 Zaragoza, Spain andInstituto de Fisica, Universidade de Brasilia,Caixa Postal 04455, 70919-970, Bras´ılia, DF, Brazil
There is an intimate relation between entanglement entropy and Riemann surfaces.This fact is explicitly noticed for the case of quadratic fermionic Hamiltonians withfinite range couplings. After recollecting this fact, we make a comprehensive analysisof the action of the M¨obius transformations on the Riemann surface. We are thenable to uncover the origin of some symmetries and dualities of the entanglemententropy already noticed recently in the literature. These results give further supportfor the use of entanglement entropy to analyse phase transitions.
I. INTRODUCTION
In the present work we make a comprehensive analysis of the symmetries and thedualities of the R´enyi entanglement entropy for the ground state of quadratic, translationalinvariant fermionic Hamiltonians with finite range couplings in a one-dimensional chain. ∗ Corresponding author. † Electronic address: [email protected] ‡ Electronic address: [email protected] § Electronic address: [email protected] ¶ Electronic address: [email protected] a r X i v : . [ m a t h - ph ] A p r We here focus on the case of non-critical, that is, gapped Hamiltonians with reflectionand charge conjugation symmetry.For such Hamiltonians, the entanglement entropy associated with a partial observa-tion on an interval of contiguous sites is a functional of the determinant of the 2-pointcorrelation matrix. This matrix is of the block Toeplitz type [1]. It was shown [2, 3] thatby using Riemann-Hilbert problem techniques one can obtain the asymptotics of this de-terminant. Furthermore, in the process of computing the asymptotics of the determinantone is led to cast the moduli of Hamiltonians through their dispersion relations in termsof a compact Riemann surface. The branch points of the hyperelliptic curve which definesthis Riemann surface depend on the coupling constants of the Hamiltonian.Our analysis of the symmetries and the dualities of the entanglement entropy is basedon the study of the action of M¨obius transformations on the above Riemann surface,which moves the branch points and, therefore, the coupling constants but leaves invariantthe asymptotic expansion of the determinant of 2-point correlations and, therefore, theentanglement entropy. We then obtain the following collection of remarkable results:1. We uncover the fact that the entanglement entropy only depends on the M¨obiustransformation invariants which are constructed with the branch points of the hy-perelliptic curve associated with the couplings of the model. This fact explains theellipses of constant entropy first noticed for the XY model in [4].2. We analyse which M¨obius transformations map the original couplings to other phys-ically allowed, i.e. to other that preserve the hermiticity of the Hamiltonian. Wefind that these M¨obius transformations are inversions and the 1+1 Lorentz group, SO (1 , SO (1 ,
1) the couplings change suchthat the dispersion relation behaves as a homogeneous field with scaling dimensionequal to the maximum range L of the coupling.3. We also note that the asymptotic expansion of the determinant is also invariantunder certain permutations of the branch points which induce a modular transfor-mation of the non contractible cycles of our Riemann surface. This fact, combinedwith the previous invariance under M¨obius transformations, can be useful to findand analyse symmetries and dualities between different Hamiltonians in terms ofentanglement entropy. As an example, we apply this idea to the XY spin chain.In particular, the duality noticed in [4] is discussed in terms of a Dehn twist ofthe underlying Riemann surface which in this XY case is a torus. Recall that twodistinct Dehn twists, which are two special transformations of the moduli of thetorus, generate the whole modular group in that case. The Dehn twists play animportant role in string theory, specially in the analysis of its many dualities. Seefor instance [5, 6]. Another suggestive work is [7] where the modular group is alsoused to understand the dualities of the XYZ spin chain.4. This framework also allows to extend the relation between the entanglement en-tropy of two Kramers-Wannier dual Ising models and its corresponding XX model,discussed by Igloi and Juhasz in [8], to general XY models.We organize this paper as follows. In section II we first present the most generalquadratic fermionic Hamiltonian with long range couplings, reflection and charge conju-gation symmetry, in a chain with N sites. We next obtain the corresponding dispersionrelation and then write a formula for the R´enyi entanglement entropy in terms of thedeterminant of the 2-point correlation matrix for the ground state. Following the noto-rious works [2, 3], we take the asymptotics of this determinant (13) by constructing acompact Riemann surface from the coupling constants of the model. It is this Riemannsurface that will allow us to analyse the symmetries and dualities of the entanglemententropy based on its behaviour under the M¨obius transformations. In section III we makea comprehensive analysis of the M¨obius transformations on the Riemann surface associ-ated with the moduli of coupling constants. We obtain that the entanglement entropyis left invariant, which allows us to obtain entropy preserving dualities between Hamil-tonians. Furthermore, the only M¨obius transformations that preserve the properties ofthe Riemann surface imposed by the couplings are inversion and the 1+1 Lorentz group.A beautiful outcome of this result is that under the latter the dispersion relation itselfbehaves as a homogeneous field with scaling dimension associated with the maximumrange L of the couplings. In section IV we apply our general results of previous sectionto the XY model. We are then able to uncover the geometrical origin of some dualitiesalready noticed in [4, 8] and to generalize them. Finally, we conclude in section V bysummarizing our findings and discussing some prospects for the future.This paper also contains two appendices. In appendix A, we show that the determinantof our block Toeplitz matrix is invariant under permutation of the branch points definingthe Riemann surface. In appendix B, we recollect the important facts concerning therepresentation of SL (2 , C ) on the space of homogeneous polynomials of two complexvariables. This plays a crucial role in the understanding how the Hamiltonian, and inparticular its couplings constants, change under M¨obius transformations. II. FINITE RANGE HAMILTONIANS, COMPACT RIEMANN SURFACESAND ENTANGLEMENT ENTROPY
Let us consider a N -site chain of size (cid:96) with N spinless fermions described by aquadratic, translational invariant Hamiltonian with finite range couplings ( L < N/ H = 12 N (cid:88) n =1 L (cid:88) l = − L (cid:16) A l a † n a n + l + B l a † n a † n + l − B l a n a n + l (cid:17) . (1)Here a n and a † n represent the annihilation and creation operators at the site n . The onlynon-vanishing anticommutation relations are { a † n , a m } = δ nm . We assume periodic boundary conditions a n + N = a n in (1).We take A l and B l real, and in order that H is Hermitian, the hopping couplings mustbe symmetric A − l = A l . In addition, without loss of generality, we may take B l = − B − l .For a comprehensive analysis of more general choices of couplings A l and B l see [1].Proceeding like in [1] we introduce F k = L (cid:88) l = − L A l e iθ k l , θ k = 2 πkN , (2)and G k = L (cid:88) l = − L B l e iθ k l . (3)Note that in our case, F k is real and F − k = F k while G k is imaginary and G − k = − G k .The Hamiltonian can now be written in diagonal form by means of anticommutingannihilation, creation operators d k , d † k , the Bogoliubov transform of the Fourier modes, H = E + N − (cid:88) k =0 Λ k (cid:18) d † k d k − (cid:19) , where E = 12 N − (cid:88) k =0 F k . (4)The dispersion relation reads Λ k = (cid:112) ( F k + G k )( F k − G k ) , (5)which takes non-negative values. Hence the ground state of the theory | GS (cid:105) is the vacuumof the Fock space for the d k modes, i.e. d k | GS (cid:105) = 0, ∀ k .In this paper we focus our study in the R´enyi entanglement entropy for the groundstate of Hamiltonian (1).Given a subset X of contiguous sites of the fermionic chain, with complementary set Y we have the corresponding factorization of the Hilbert space H = H X ⊗ H Y . If thesystem is in the pure state | GS (cid:105) , the reduced density matrix for X is obtained by takingthe partial trace with respect to H Y , ρ X = Tr Y | GS (cid:105) (cid:104) GS | . The R´enyi entanglemententropy of X is S α ( X ) = 11 − α log Tr( ρ αX ) . (6)As it is well known [9–13], due to the fact that the n -point vacuum expectation valuesatisfies the Wick decomposition property, it is possible to derive the reduced densitymatrix from the two point correlation function. Namely, for any pair of sites n, m ∈ X ,we introduce the correlation matrix( V X ) nm = 2 (cid:42) a n a † n ( a † m , a m ) (cid:43) − δ nm I = δ nm − (cid:10) a † m a n (cid:11) (cid:104) a n a m (cid:105) (cid:10) a † n a † m (cid:11) (cid:10) a † n a m (cid:11) − δ nm . Following [9–11] one can show that the R´enyi entanglement entropy reads S α ( X ) = 12(1 − α ) Tr log (cid:20)(cid:18) I + V X (cid:19) α + (cid:18) I − V X (cid:19) α (cid:21) , (7)where I is the 2 | X | × | X | identity matrix and | X | denotes the size of X .The Cauchy’s residue theorem allows us to implement (7) as S α ( X ) = lim ε → + πi (cid:73) C f α (1 + ε, λ ) d log D X ( λ )d λ d λ, (8)where D X ( λ ) = det( λ I − V X ), f α ( x, y ) = 11 − α log (cid:20)(cid:18) x + y (cid:19) α + (cid:18) x − y (cid:19) α (cid:21) , (9) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) −1 +1 ε ε −1− v l (cid:0)(cid:1) (cid:0)(cid:1) (cid:0)(cid:1) (cid:0)(cid:1) (cid:0)(cid:1) FIG. 1: Contour of integration, cuts and poles for the computation of S α ( X ). The cuts for the function f α extend to ±∞ . and C is the contour depicted in Fig. 1 which surrounds the eigenvalues v l of V X , all ofthem lying in the real interval [ − , V X ) nm = 1 N N − (cid:88) k =0 M k e iθ k ( n − m ) , with M k = 1Λ k F k G k − G k − F k . We now consider the thermodynamic limit, N → ∞ , (cid:96) → ∞ with N/(cid:96) fixed. In thislimit the previous N -tuples, like Λ k = Λ( θ k ) , F k = F ( θ k ) , G k = G ( θ k ) and the others,with θ k = 2 πk/N , become 2 π -periodic functions of the continuous variable θ , that is,Λ( θ ) , F ( θ ) , G ( θ ) , . . . .In order to obtain the asymptotic behaviour of the entanglement entropy (8), we mustcompute D X ( λ ) ≡ det( λ I − V X ) when the size | X | of X is large.This task was solved in [2, 3] by reducing it to a particular Wiener-Hopf factorizationproblem. To describe the result we introduce the (complex) Laurent polynomialsΘ( z ) = L (cid:88) l = − L A l z l , Ξ( z ) = L (cid:88) l = − L B l z l , (10)that are related to F and G , from (2) and (3), by F ( θ ) = Θ(e iθ ) and G ( θ ) = Ξ(e iθ ). Interms of (10) we extend the symbol M ( θ ) to the complex plane M ( z ) = 1 (cid:112) Θ ( z ) − Ξ ( z ) Θ( z ) Ξ( z ) − Ξ( z ) − Θ( z ) = U g ( z ) g ( z ) − U − , with U = 1 √ − and g ( z ) = (cid:115) Θ( z ) + Ξ( z )Θ( z ) − Ξ( z ) . Now, as it is shown in [2, 3, 14], the asymptotic behaviour of the logarithmic derivativeof D X ( λ ) verifiesdd λ ln D X ( λ ) = 2 λλ − | X | ++ 12 π (cid:90) | z | =1 Tr (cid:2)(cid:0) u (cid:48) + ( z ) u − ( z ) + v − ( z ) v (cid:48) + ( z ) (cid:1) M − ( z ) (cid:3) d z + . . . (11)where the prime denotes the derivative with respect to z and u + , v + are the solution tothe following Wiener-Hopf factorization problem:i) M ( z ) = u + ( z ) u − ( z ) = v − ( z ) v + ( z ),ii) with u ± − ( z ) , v ± − ( z ) analytic outside the unit circle and u ± ( z ) , v ± ( z ) analytic inside.This problem has been solved in [2, 3] and the results, expressed in terms of thetafunctions associated to the Riemann surface determined by g ( z ), are described in thefollowing.First, consider the analytic structure of g ( z ). Actually it is a bivalued function in thecomplex plane, but it is a single valued meromorphic function in the Riemann surfacedetermined by the complex curve w = P ( z ) ≡ z L (Θ( z ) + Ξ( z ))(Θ( z ) − Ξ( z )) , w, z ∈ C , (12)where C denotes the Riemann sphere.Here we shall assume that the polynomial P ( z ) has 4 L different simple roots andtherefore (12) defines a two-sheet Riemann surface of genus g = 2 L −
1. On the otherhand, since the real coupling constants of (1) satisfy A l = A − l and B − l = − B l , we haveΘ( z − ) = Θ( z ) , Θ( z ) = Θ( z ) , Ξ( z − ) = − Ξ( z ) , Ξ( z ) = Ξ( z ) . Therefore if z j is a root of P ( z ) then z j and z − j are roots as well. Actually the rootsof P ( z ) coincide with the zeros and poles of the rational function g ( z ) and the previousrelation specializes as follows: if z j is a zero (pole) of g ( z ) then z j is also a zero (pole)while z − j is a pole (zero).We fix an order in the roots of P ( z ) with the only requierement that the first half ofthem is inside the unit circle and the other half outside, i.e. | z j | < , j = 1 , . . . , L and | z j | > , j = 2 L + 1 , . . . , L (notice that if all roots are simple, then necessarily | z j | (cid:54) = 1).Equipped with the previous data we fix the branch cuts of g ( z ), which are 2 L nonintersecting curves Σ ρ , ρ = 0 , . . . , g that join z ρ +1 and z ρ +2 . Notice that it is alwayspossible to choose them such that they do not cross the unit circle.Associated to these cuts we have a canonical homology basis of cycles in the Riemannsurface a r , b r , r = 1 , . . . , g where a r , in the upper Riemann sheet, surrounds Σ r anticlock-wise and the dual cycle b r surrounds the branch points z , z , · · · , z r +1 clockwise. In Fig.2 we depict a possible arrangement of the branch points and cuts for g = 3 ( L = 2) aswell as cycles a and b . (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) z z z z zzz z z b a |z|=1L=2 g= FIG. 2: Possible arrangement of the branch points and cuts of w = (cid:112) P ( z ) for genus g = 3 ( L = 2).Note that we must have z = z − , z = z − , z = z = z − = z − . The branch points in black ( • ) arezeros of g ( z ) ( (cid:15) j = 1) while those in white ( ◦ ) are poles ( (cid:15) j = − a and b are twoof the basic cycles. The canonical basis of holomorphic formsd ω r = ϕ r ( z ) (cid:112) P ( z ) d z, with ϕ r ( z ) a polynomial of degree smaller than g, is chosen so that (cid:82) a r d ω r (cid:48) = δ rr (cid:48) . Theg × g symmetric matrix of periods Π = (Π rr (cid:48) ) is given byΠ rr (cid:48) = (cid:90) b r d ω r (cid:48) . Associated with this matrix of periods we can now define the Riemann theta-functionwith characteristics (cid:126)p, (cid:126)q ∈ R g , ϑ (cid:104) (cid:126)p(cid:126)q (cid:105) : C g → C , given by ϑ (cid:104) (cid:126)p(cid:126)q (cid:105) ( (cid:126)s ) ≡ ϑ (cid:104) (cid:126)p(cid:126)q (cid:105) ( (cid:126)s | Π) = (cid:88) (cid:126)n ∈ Z g e πi ( (cid:126)n + (cid:126)p )Π · ( (cid:126)n + (cid:126)p )+2 πi ( (cid:126)s + (cid:126)q ) · ( (cid:126)n + (cid:126)p ) , and its normalized version (cid:98) ϑ (cid:104) (cid:126)p(cid:126)q (cid:105) ( (cid:126)s ) ≡ (cid:98) ϑ (cid:104) (cid:126)p(cid:126)q (cid:105) ( (cid:126)s | Π) = ϑ (cid:104) (cid:126)p(cid:126)q (cid:105) ( (cid:126)s | Π) ϑ (cid:104) (cid:126)p(cid:126)q (cid:105) (0 | Π) . Finally, one has the following expression for the asymptotic value of the determinant(adapted from Proposition 1 of [3]):log D X ( λ ) = | X | log( λ −
1) + log (cid:16) (cid:98) ϑ (cid:2) (cid:126)µ(cid:126)ν (cid:3) ( β ( λ ) (cid:126)e ) (cid:98) ϑ (cid:2) (cid:126)µ(cid:126)ν (cid:3) ( − β ( λ ) (cid:126)e ) (cid:17) + · · · , (13)where the dots represent terms that vanish in the large | X | limit. The argument of thetheta function contains β ( λ ) = 12 πi log λ + 1 λ − , (14)and (cid:126)e ∈ Z g , a g dimensional vector whose first L − L are 1.The characteristics of the theta function are half integer vectors (cid:126)µ, (cid:126)ν ∈ ( Z / g which aredetermined as follows: first assign an index (cid:15) j to every root z j of P ( z ), so that (cid:15) j = 1( (cid:15) j = −
1) if z j is a zero (pole) of g ( z ); then the vectors are µ r = 14 ( (cid:15) r +1 + (cid:15) r +2 ) ,ν r = 14 r +1 (cid:88) j =2 (cid:15) j , r = 1 , . . . , g . (15)Note that the indices must satisfy (cid:15) j = − (cid:15) j (cid:48) whenever z j = z − j (cid:48) and (cid:15) j = (cid:15) j (cid:48) if z j = z j (cid:48) .In order to give an explicit expression for the determinant we have fixed an order of theroots with the only requirement that the 2 L first ones are inside the unit circle and thelast ones outside. Of course, for consistency (13) should not depend on the chosen order.Actually it is an instructive exercise to show that the value of the determinant (in thethermodynamic limit) is in fact invariant under the transposition of two roots, providedthey sit at the same side of the unit circle. In appendix A we outline a proof of this fact.0 III. M ¨OBIUS TRANSFORMATION
If we now perform a holomorphic bijective transformation in the Riemann sphere, z (cid:48) = f ( z ), we move the branch points and cuts, thus modifying the holomorphic forms.But it is clear that the matrix of periods is unchanged. Therefore the theta function doesnot change and the entropy derived from (13) is left invariant.It is a mathematical fact that the only holomorphic one-to-one maps of the Riemannsphere into itself are the M¨obius transformations z (cid:48) = az + bcz + d , a bc d ∈ SL (2 , C ) . They act, see Appendix B, on the Laurent polynomial Θ byΘ (cid:48) ( z (cid:48) ) = ( az + b ) − L ( dz − + c ) − L Θ( z ) , (16)which is again a Laurent polynomial with monomials of degree between L and − L . Hence,the M¨obius transformation can be seen as a change of the couplings from A l to A (cid:48) l . Inexactly the same way we transform Ξ with coefficients B l to a new Ξ (cid:48) with coefficients B (cid:48) l .Notice that if we use the new Laurent polynomials to get the new rational function g (cid:48) = Θ (cid:48) + Ξ (cid:48) Θ (cid:48) − Ξ (cid:48) , we have g (cid:48) ( z (cid:48) ) = g ( z ).But this is not the end of the story because as we mentioned in the previous section, theroots of P ( z ) (12) satisfy certain properties, namely they come in quartets z j , z j , z − j and z − j . In order to preserve this property, we must require that the M¨obius transformationcommute with the complex conjugation and inversion.The commutation with conjugation restricts SL (2 , C ) to the semidirect product SL (2 , R ) (cid:111) { I, iσ x } . These are the transformations that preserve the real line. Note thatthe second factor in the semidirect product is related to the inversion z (cid:48) = 1 /z and mapsthe upper half plane into the lower one while the first factor contains the transformationsthat map the upper half plane to itself.If we further impose that the allowed transformations must commute with the inversion,we are finally left with the group generated by the reflection z (cid:48) = 1 /z and the 1+1 Lorentz1group SO (1 ,
1) whose elements act on z by z (cid:48) = z cosh ζ + sinh ζz sinh ζ + cosh ζ , ζ ∈ R . (17)This is precisely the subgroup of the M¨obius tranformations that preserve the unit circleand the real line. Its connected component maps the upper half plane and the unit discinto themselves.We unraveled the symmetry by considering the final expression for the asymptotic de-terminant (13), but one can trace back its origin to the Wiener-Hopf factorization problemdescribed in the previous section. In fact, assume we perform a M¨obius transformationthat preserve the unit circle mapping its exterior into its exterior and the interior intothe interior. Then, the property of being analytic outside (inside) the circle is preservedwhich implies that the solution to the Wiener-Hopf factorization for M ( z ) is transformedinto that for M (cid:48) ( z ). Inserting it into (11) one immediately sees that the logarithmicderivative of D X ( λ ) and hence the entanglement entropy, are unchanged, at least in theasymptotic limit. Actually, a similar reasoning shows that the determinant is invariant forany M¨obius transformation provided it keeps inside (outside) the unit circle the branchingpoints that were originally inside (outside).As for the physical interpretation of the transformations, the inversion correspondsto reverting the orientation in the chain n ↔ N − n which, clearly, does not affect theentropy of the system. The coupling constants transform in a very simple way, namely, A (cid:48) l = A l , B (cid:48) l = − B l . The symmetries in the Lorentz group, however, have a rather nontrivial implementation. In Appendix B we shall discuss in full generality how the couplingconstants of the Hamiltonian behave under such Lorentz transformations.On the other hand, it is interesting to observe that these particular M¨obius transforma-tions leave invariant not only the von Neumann entropy but also the R´enyi entanglemententropy (6) for any value of α . Actually, all the spectral properties of the two-pointcorrelation function are preserved, at least in the large | X | limit.A very important observation is that all the previous does not apply for the critical,gapless theories, i.e. when couples of roots of P ( z ) coincide at the unit circle. Thisimportant case will be studied separately in a future publication [15].With respect to the dynamical aspects of these transformations, we may say thatthey are not a symmetry of the Hamiltonian. In the thermodynamic limit they act as a2rescaling of the spectrum. Actually, as the unit circle is mapped into itself, we may viewthe transformation as a change in the momentum of the modes together with a rescalingof its energy. More concretely, adopting again the active view point we haveΛ (cid:48) ( θ (cid:48) ) = (cid:18) ∂θ (cid:48) ∂θ (cid:19) L Λ( θ ) , where θ (cid:48) is the image of θ under the M¨obius transformation,e iθ (cid:48) = e iθ cosh ζ + sinh ζ e iθ sinh ζ + cosh ζ , and, therefore, ∂θ (cid:48) ∂θ = 1sinh 2 ζ cos θ + cosh 2 ζ . Interestingly enough under the above transformations the dispersion relation Λ( θ ) behavesas a homogeneous field of dimension L . Moreover, the dimension is directly associatedwith the range of the coupling. We recall that such a transformation actually correspondsto a change of the coupling constants of the theory.The action of SO (1 ,
1) in C has two fixed points in z = ±
1. In particular, 1 is stableand − (cid:45) FIG. 3: Flow of SO (1 ,
1) in C . Note that it preserves the unit circle and the real line and maps the unitdisc and the upper/lower half plane into themeselves. The points z = ± − L + 1 fixed points whose associated complex curves are w = P ( z ) ≡ ( z − L − j ( z + 1) j , j = 0 , . . . , L. The only stable one corresponds to j = 0. The other fixed points have a j -dimensionalunstable manifold. Of course, all these Hamiltonians correspond to critical theories.Summarizing, we have found a transformation of the coupling constants of a freefermionic chain with finite range coupling such that the spectral properties of the two-pointcorrelation function are left invariant in the thermodynamic limit, provided the theoryhas a mass gap. As a byproduct we show that the asymptotic behaviour of the R´enyi en-tanglement entropy is unchanged, whenever it is finite. On the other hand, the modes inthe one particle energy spectrum are rescaled with a dimension given by the range of thecouplings of the theory. The connected component of the symmetry group is composedby the M¨obius transformations associated to the 1+1 Lorentz subgroup SO (1 , X ismade of several disjoint intervals. IV. THE XY SPIN CHAIN
Our first application of the transformations introduced in the previous section is tounderstand the different dualities and invariances of the entanglement entropy in the XYmodel. This has been studied in [4]. Here we rederive their results and find some newones using the ideas of the previous section.The Hamiltonian reads H XY = 12 N (cid:88) n =1 (cid:2) (1 + γ ) σ xn σ xn +1 + (1 − γ ) σ yn σ yn +1 − hσ zn (cid:3) . The coupling constants h and γ are assumed to be real and positive.4A Jordan-Wigner transformation allows us to write this Hamiltonian in terms offermionic operators, namely H XY = N (cid:88) n =1 (cid:104) a † n a n +1 + a † n +1 a n + γ ( a † n a † n +1 − a n a n +1 ) − ha † n a n (cid:105) + N h , (18)which is a particular case of the Hamiltonian of (1) with nearest neighbours coupling, L = 1. We are now set to apply the previous results.The Laurent polynomials are in this caseΘ( z ) = z − h + z − , Ξ( z ) = γ ( z − z − ) , and the dispersion relation isΛ( θ ) = (cid:113) ( h − θ ) + 4 γ sin θ. (19)The theory is gapless for h = 2 (Ising universality class) or γ = 0 , h < g ( z ) = (Θ( z ) + Ξ( z )) / (Θ( z ) − Ξ( z )) are given by z ± = h/ ± (cid:112) ( h/ + γ −
11 + γ , (20)and its poles are the inverses z − ± . Note that the critical theories correspond to the valueof the couplings for which some of the zeros are in the complex unit circle. Here we donot discuss the case of critical theories, which deserves a special treatment [15]. Our focushere is on the non-critical theories, its symmetries, dualities and other properties.If we now apply the transformations (17) the couplings of the theory change as γ (cid:48) = γ ( h/
2) sinh 2 ζ + cosh 2 ζ , (21) h (cid:48) / h/
2) cosh 2 ζ + sinh 2 ζ ( h/
2) sinh 2 ζ + cosh 2 ζ , (22)while the zeros change as z (cid:48)± = z ± cosh ζ + sinh ζz ± sinh ζ + cosh ζ , and similarly for the poles.Actually, as it was discussed in the previous section, the entropy derived from (8) and(13) is invariant under any M¨obius transformation of the Riemann surface. Therefore it5only depends on the zeros and poles of g ( z ) through the M¨obius invariants. For the XYmodel in which we have just four such points (two zeros z + , z − and two poles z − , z − − )the only invariants are functions of the cross ratio x ≡ ( z + , z − ; z − , z − − ) = ( z + − z − )( z − − z − − )( z + − z − − )( z − − z − ) . (23)If we use (20), this cross ratio can be written in terms of the couplings as x = 1 − ( h/ γ . (24)The explicit form of the von Neumann entanglement entropy for non-critical XY modelwas computed in [2] and it was rewritten in [1, 4] in terms of the parameter x . See also[22, 23]. We express the von Neumann entropy in three distinct regions, namely 1a, 1band 2: • Region 1a: 0 < x < S = 16 (cid:20) log (cid:18) − x √ x (cid:19) + 2(1 + x ) π I ( √ − x ) I ( √ x ) (cid:21) + log 2 . (25) • Region 1b: x > S = 16 (cid:34) log (cid:32) − x − √ x − (cid:33) + 2(1 + x − ) π I ( (cid:112) − x − ) I ( √ x − ) (cid:35) + log 2 . (26) • Region 2: x < S = 112 (cid:34) log (cid:16) − x − x − ) (cid:17) + 4( x − x − ) π (2 − x − x − ) I (cid:18) √ − x (cid:19) I (cid:18) √ − x − (cid:19)(cid:35) . Here I ( z ) is the complete elliptic integral of the first kind I ( z ) = (cid:90) d y (cid:112) (1 − y )(1 − z y ) . As it was discussed before, the fact that the entropy depends solely of the parameter x can be derived as a consequence of the M¨obius invariance that we uncover in this paper.A consequence of such invariance is that the R´enyi entropy is a function of x as well. Inthe following we will use the M¨obius transformations to study some dualities and otherrelations that occur between theories in the different regions we introduce above.6 A. Duality between regions 1a and 1b
Examining the expressions of the entropy in the region 1, (25) and (26), it is clear thatthe entropy is invariant under the change of x to x − . We would like to understand thisproperty in the light of the symmetries discussed before.In fact we may derive the duality in the following way. Assume we start with a theorywith couplings γ a > , h a in the region 1a, i.e., γ a > − ( h a / >
0. Therefore, the points z a + and z a − are two real zeros inside the unit circle. Imagine now that we permute themwithout permuting their inverses. As it was discussed in section II, see also AppendixA, the permutation between zeros on the same side of the unit circle does not affect theentropy, however the cross ratio (23) is now inverted: ( z a − , z a + ; z − a + , z − a − ) = x − . Notethat from the point of view of the corresponding Riemann surface, which in this case isa torus, this permutation of zeros is equivalent to cut it along the a -cycle, perform a 2 π rotation of one of the borders, and glue them again. This is precisely one of the two Dehntwists that generate the modular group SL (2 , Z ) of the torus.One may object that the roots in the new pairs z a − , z − a + and z a + , z − a − are not relatedby inversion any more, as it should be in the XY model. Here is where the M¨obiustransformations come to the rescue. By suitably chosing a SL (2 , C ) transformation thatdoes not belong to SO (1 , z b + , z − b + and z b − , z − b − with the additional property z b + = z b − . Considering now that the M¨obiustransformations leave the entropy invariant, we can explain the duality between (25) and(26). In particular, one may take as M¨obius transformation that for which z b + − z b − z b + z b − = − i z a + − z a − z a + z a − . These roots actually correspond to a particular choice of the couplings for the XYmodel, h b , γ b , belonging to region 1b, which can be related to the original ones by γ b = (cid:115) − (cid:18) h a (cid:19) , h b (cid:112) − γ a . In figure 4, we depict graphically in the plane ( γ, h ) this choice: the point (cid:52) has coordi-nates ( γ a , h a ) and (cid:53) , with coordinates ( γ b , h b ), is its dual.It should be noticed that the duality, that is manifest for von Neumann also holds forthe R´enyi entropy (8), as it is based on the equality of the determinants D X ( λ ) for the7two values of the couplings. On the other hand, in order to establish the duality we mustperform a M¨obius transformation, which implies that the dispersion relation has changedas a homogeneous field of dimension L = 1. Therefore, in this case the spectrum of theHamiltonian transforms non trivialy.Before proceeding we examine for the XY model how the expressions for a generalRiemann surface specializes to one with genus 1. In this case the Riemann theta functionin g complex variables reduces to the elliptic theta function with characteristics in onevariable ϑ [ µν ]( s | τ ). The period matrix Π is replaced by the modulus τ defined by τ = i I ( ξ ) I ( (cid:112) − ξ ) , (27)with ξ = √ x, < x < , case 1a , √ x − , x > , case 1b , √ − x − , x < , case 2 , and the order chosen for the zeros and poles of g ( z ) is z a + , z a − , z − a + , z − a − , for case 1a ,z b + , z b − , z − b + , z − b − , for case 1b ,z − , z − , z − − , z , for case 2 , that fulfills in all cases the requirement of the previous section, i. e. the first two branchpoints are inside the unit circle and the last two outside. Consequently, the assignementof indices is (+1 , +1 , − , −
1) for the cases 1a, 1b and ( − , +1 , − , +1) for the case 2.Now, one can easily compute the characteristics to give µ a = µ b = − / , ν a = ν b = 0 forcases 1a, 1b and µ = 0 , ν = 0 in case 2. B. Duality between regions 1a and 2: Kramers-Wannier duality
We derived above a duality for the entanglement entropy that was based in the ex-change of the two zeros on the same side of the unit circle. This does not change theRiemann surface, nor the characteristics, and leaves the entropy invariant. Another dual-ity that preserves the Riemann surface is the exchange of a real root with its inverse. This8establishes a relation between the regions 1a and 2. This corresponds to z a + = ( z ) − .The relation between the coupling constants is then, h h a ; γ − h = γ a − h a . In figure 4, the point (cid:72) in region 2 is the dual of (cid:52) .One immediately obtains that under this duality, the dispersion relation changes asΛ ( θ ) = 2 h a Λ a ( θ ) , which implies that up to a trivial rescaling the spectrum of the Hamiltonian is unchanged.For the particular value γ a = 1, in which XY reduces to the Ising model, the dual theoryis also the Ising model ( γ = 1) with a different magnetic field. In this case this dualitycoincides with the Kramers-Wannier duality [24] which, as it is notoriously known, is veryuseful for determining the critical point of the Ising model [25, 26].If we now consider the behaviour of the entropy under the duality, we observe that theRiemann surface does not change and henceforth the modulus τ is invariant. However,as it was discussed above, the characteristics are transformed from µ a = − / µ = 0and therefore D X ( λ ) and the entropy are modified.This duality can be generalized to Hamiltonians of higher range L . We can exchangeone real zero of g ( z ) by its inverse, which is a pole of g ( z ), or one complex zero of g ( z ) and its complex conjugate by their inverses, which are poles of g ( z ). Then boththe Riemann surface and the spectrum of the Hamiltonian remain invariant. However,the characteristics change and then entanglement entropy varies too. Observe that tworegions of the space of couplings related by one of these dualities are separated by a criticalhypersurface.Note that the two dualities between 1a and 1b regions and 1a and 2 are of differentnature. In the former the entanglement entropy is invariant while the spectrum of theHamiltonian changes. On the contrary, in the latter the entanglement entropy varies andthe spectrum of the Hamiltonian is preserved. C. Relation between the dual theories in 1a and 2, with region 1b
A beautiful result is that although the two dual theories in regions 1a and 2 have differ-ent entanglement entropies, they can be combined to obtain the entropy of a Hamiltonian9in the region 1b. This result was first noticed in [8] for the Ising line ( γ = 1). Our resultsextend this property to other points in the coupling space.In order to proceed, recall the useful identity of theta functions [27] that combines thetwo dual theories, (cid:98) ϑ [ µ +1 / ν ]( s | τ ) (cid:98) ϑ [ µν ]( s | τ ) = (cid:98) ϑ [ µ +1 / ν ]( s | τ / . If we apply this identity for µ = 0 , ν = 0 to the expression (13) for the determinant wededuce that by summing up the R´enyi entanglement entropy for two dual theories, likethose with coupling constants γ a , h a and γ , h which (see (27) and the previous section)correspond to the same modulus τ , we can obtain the R´enyi entropy of a third theory inthe region 1 b with coupling constants γ T , h T such that its modulus is τ / h T / (cid:112) − γ a , γ T = γ a − √ − x a √ − x a , with x a = 1 − ( h a / γ a . Then if we introduce y = √ − x a , by application of (24), (27) we obtain τ ≡ τ a = i I ( (cid:112) − y ) I ( y ) and τ T = i I (cid:18) − y y (cid:19) I (cid:18) √ y y (cid:19) , and using the following Landen identities for elliptic functions [28], I (cid:18) √ y y (cid:19) = (1 + y ) I ( y ) , and I (cid:18) − y y (cid:19) = 1 + y I ( (cid:112) − y ) , we obtain τ T = τ / S aα + S α = S Tα , (28)where the superindices obviously refer to the theories in the corresponding regions withthe coupling constants defined above. For example, in figure 4, the point (cid:52) is at ( γ a , h a )and (cid:72) at ( γ , h ); then we have depicted (cid:7) at ( γ T , h T ), so its entanglement entropy is thesum of entanglement entropies of the two other points according to (28).0The relation is particularly simple when γ a = γ = 1 (Ising model) in which case thefinal theory corresponds to the XY model with zero transverse magnetic field h T = 0 andanisotropy parameter γ T = 1 − h a /
21 + h a / . A peculiarity of this case is that the relation (28) for the entanglement entropy holds notonly in the asymptotic limit but also with finite size | X | , in which case it reads S aα ( | X | ) + S α ( | X | ) = S Tα (2 | X | ) . This relation has been known for some time [8] and, in particular, it has been used toderive the entanglement entropy for the critical Ising model using the known results forXX.Another simple instance is when we consider the critical lines of the Ising and XXuniversality classes, h a = 2 and γ b = 0 respectively. In this case, we can establish aduality between the R´enyi entropies of the models. If we take h a = 2 and | γ a | ≤ h = 2, γ = γ a , γ T = 0 and h T / (cid:112) − γ a . Before writing explicitly the relation of the entanglement entropies wemust consider that we are dealing with critical theories, which implies that the entropyof an interval X scales logarithmically with its length | X | . As we did for the finite sizecase, in order to recover the additive relation between the entropies in the critical case,we must take different length intervals for the different theories, namely S Tα (2 | X | ) = 2 S aα ( | X | ) . This is an interesting relation because, combined with the results of [11, 29] for theXX model, it allows to compute the R´enyi entanglement entropy for the critical Isinguniversality class ( h a = 2). The final result is S aα ( | X | ) = α + 112 α log( | X || γ a | ) + U α Ising , (29)where U α Ising = α + 16 α log 2 + 12 πi (cid:90) − d f α (1 , λ )d λ log Γ(1 / − β ( λ ))Γ(1 / β ( λ )) d λ, with Γ the Gamma function. We remark that the expression (29) for the entanglemententropy of the critical Ising line is a new result. Only the case | γ a | = 1 was previously1known in the literature, see [8, 30]. In Fig. 5 we check that this formula matches withnumeric computations. Note that, since it is an asymptotic result, we have finite sizeeffects which are more important when the branch points approach the unity, i.e. γ a → h γ XX critical line Ising critical line γ + (cid:0) h (cid:1) = 1 x, τ − x − , τx − , τ − (cid:20) √ − x − √ − x (cid:21) τ
1b 2 1a
FIG. 4: Plane of couplings ( γ, h ) for the XY Hamiltonian. According to the expression for theentanglement entropy, we distinguish three different regions: 1a, 1b and 2. Dashed curves represent theinduced flow of SO (1 , (cid:77) in 1b is (cid:79) , S (cid:79) α = S (cid:77) α . The dual theory of (cid:77) in 2 is (cid:72) and, although the corresponding tori have thesame modulus, S (cid:72) α (cid:54) = S (cid:77) α . The entanglement entropy of the theory (cid:7) is given by (28), S (cid:7) α = S (cid:77) α + S (cid:72) α . . . . . . . . . . . . . . . . . . . . .
81 0 0 .
02 0 . S ( X ) γ a | X | = | X | = | X | = | X | =
20 00 . . . . . . . . . . . . . . . . . . .
05 0 . S ( X ) γ a | X | = | X | = | X | = | X | = FIG. 5: Numerical check of expression (29) for the entanglement entropy along the critical Ising line.We have computed the von Neumann entropy ( α →
1, left panel) and the R´enyi entanglement entropywith α = 2 (right panel) for different lengths of X , varying γ a . The insets are a zoom of the plot forsmall values of γ a where the finite size effects are more relevant, especially in the case of the R´enyientanglement entropy. V. CONCLUSION
We have seen in this work the importance of casting the space of coupling constants ofa model in terms of a Riemann surface. This simple fact shows up when we are computingthe asymptotics of the determinant of the 2-point correlation matrix of the model. Thenext observation is that this determinant is kept invariant under M¨obius transformationsacting on the Riemann surface that, on the other hand, change the value of the couplings.Therefore we can have changes in the Hamiltonian that leave the determinant invariant.This fact allows us to uncover the origin and set up in a unified language the symmetriesand dualities of such models.Now, the entanglement entropy is a functional of this determinant. Therefore the manysuggestions that the entanglement entropy is the right tool to study phase transitions findtheir explicit realization in the kind of analysis we perform in this work.As a next step, we will extend the analysis of the M¨obius transformations to the caseof critical Hamiltonians [15]. This requires some other techniques since, because of theconfluence of couples of branch points in the unit circle, new singularities appear.3There are several directions to generalize our results. One could consider the case ofseveral disjoint intervals, as mentioned at the end of section III, or take complex couplingconstants that break reflection and charge conjugation symmetry [1, 24]. It would bealso interesting to apply our analysis to excited states [29, 31] and to the evolution of theentanglement entropy after a quantum quench [32–34] in order to determine whether theM¨obius symmetry and the different dualities hold also in these cases.
Acknowledgments:
Research partially supported by grants 2014-E24/2, DGIID-DGAand FPA2015-65745-P, MINECO (Spain). FA is supported by FPI Grant No. C070/2014,DGIID-DGA/European Social Fund. ARQ is supported by CAPES process number BEX8713/13-8 and by CNPq under process number 305338/2012-9.
Appendix A: Determinant is invariant under permutations of branch points
In this appendix we will check the consistency of the expression for log D X ( λ ) that weintroduced in (13). In particular we will show that it does not depend on the order inwhich we choose the roots of P ( z ), provided we take the first half inside and the last halfoutside of the unit circleSuppose that we exchange the order of two roots z j and z j , with j κ = 2 r κ + 1 + u κ , κ =1 , r κ = 1 , . . . , g and u κ = 0 ,
1. If we follow the prescriptions of section II this inducesa change in the fundamental cycles which are transformed into a (cid:48) r , b (cid:48) r , such that a r = a (cid:48) r , r (cid:54) = r , r ,a (cid:48) r + ∆ , r = r ,a (cid:48) r − ∆ , r = r , b r = b (cid:48) r , r < r + u ,b (cid:48) r + ∆ , r + u ≤ r ≤ r − u ,b (cid:48) r , r > r − u (A1)where ∆ = b (cid:48) r − b (cid:48) r + r − u (cid:88) t = r + u a (cid:48) t . This transformation is a particular instance of the most general change of basis of cyclesgiven by a modular transform [27] ba = A BC D b (cid:48) a (cid:48) , A BC D ∈ Sp ( Z ) . (cid:48) = ( A − Π C ) − (Π D − B ) . While the normalized theta functions are related by (cid:98) ϑ (cid:104) (cid:126)p (cid:48) (cid:126)q (cid:48) (cid:105) ( (cid:126)s (cid:48) | Π (cid:48) ) = e − πi(cid:126)sC · (cid:126)s (cid:48) (cid:98) ϑ (cid:104) (cid:126)p(cid:126)q (cid:105) ( (cid:126)s | Π)where (cid:126)s (cid:48) = (cid:126)s ( C Π (cid:48) + D ) , (A2)and the characteristics verify( (cid:126)p (cid:48) , (cid:126)q (cid:48) ) b (cid:48) a (cid:48) = ( (cid:126)p, (cid:126)q ) ba − (cid:0) diag( C T A ) , diag( D T B ) (cid:1) b (cid:48) a (cid:48) . In the particular case of the transformation (A1), after a straightforward calculation,one obtains( (cid:126)p (cid:48) , (cid:126)q (cid:48) ) b (cid:48) a (cid:48) = ( (cid:126)p, (cid:126)q ) ba + 12 ( u + u )( b (cid:48) r + b (cid:48) r ) + 12 ( u + u − r − u (cid:88) t = r + u a (cid:48) t . (A3)We shall examine now how the arguments of the theta functions in (13), (cid:126)s = ± β ( λ ) (cid:126)e ,are modified by the transposition. Taking into account the definition of (cid:126)e and the formof the matrices C and D one has (cid:126)e C = 0 and (cid:126)e D = (cid:126)e if and only if the two roots z j and z j that we exchange verify j , j ≤ L or j , j > L , which means that both roots sit atthe same side of the unit circle. In this case applying (A2) one has (cid:126)s (cid:48) = (cid:126)s = ± β ( λ ) (cid:126)e .In (15) we gave a prescription to obtain the characteristics for the theta functionsinvolved in the computation of the entanglement entropy. They depend on the positionoccupied by the poles and zeros of g ( z ) which are labeled by a sign (cid:15) j . If we exchangetwo of them, the original characteristics (cid:126)µ , (cid:126)ν change into˜ µ r = µ r , r (cid:54) = r , r ,µ r + δ, r = r ,µ r − δ, r = r , ˜ ν r = ν r , r < r + u ,ν r + δ, r + u ≤ r ≤ r − u ,ν r , r > r − u , (A4)where δ = (cid:15) j − (cid:15) j . These, in general, are different from those obtained by the applicationof (A3) to (cid:126)µ and (cid:126)ν which we denote by (cid:126)µ (cid:48) and (cid:126)ν (cid:48) . After a somehow lengthy but direct5computation one obtains( (cid:126) ˜ µ − (cid:126)µ (cid:48) , (cid:126) ˜ ν − (cid:126)ν (cid:48) ) b (cid:48) a (cid:48) = (cid:18) u (cid:15) j − − u (cid:15) j + 12 (cid:19) b (cid:48) r − (cid:18) u (cid:15) j + 12 − u (cid:15) j − (cid:19) b (cid:48) r + (cid:18) (1 − u ) (cid:15) j − − (1 − u ) (cid:15) j + 12 (cid:19) r − u (cid:88) t = r + u a (cid:48) t . (A5)The important point to notice here is that, given that u κ = 0 , (cid:15) j = ±
1, onealways has (cid:126) ˜ µ − (cid:126)µ (cid:48) , (cid:126) ˜ ν − (cid:126)ν (cid:48) ∈ Z g . This implies that (cid:98) ϑ (cid:104) (cid:126) ˜ µ(cid:126) ˜ ν (cid:105) = (cid:98) ϑ (cid:104) (cid:126)µ (cid:48) (cid:126)ν (cid:48) (cid:105) , as one can easily check from the definitions.Finally, putting everything together one has (cid:98) ϑ (cid:104) (cid:126) ˜ µ(cid:126) ˜ ν (cid:105) ( ± β ( λ ) (cid:126)e | Π (cid:48) ) = (cid:98) ϑ (cid:104) (cid:126)µ (cid:48) (cid:126)ν (cid:48) (cid:105) ( ± β ( λ ) (cid:126)e | Π (cid:48) ) = (cid:98) ϑ (cid:2) (cid:126)µ(cid:126)ν (cid:3) ( ± β ( λ ) (cid:126)e | Π) , where for the second equality we assume that the two branch points, whose order wasexchanged, belong both to the first half of the ordering ( j , j ≤ L ) or both to the secondhalf ( j , j > L ).From the equality of the normalized theta-functions we deduce that a change in theorder in which we take the roots does not affect the expression for the determinant,provided we do not exchange a root inside the unit circle with one outside. Appendix B: M¨obius transformations on the Space of homogeneous polynomials
In order to obtain how Laurent polynomials (10) and, accordingly, our Hamiltonian (1)change under M¨obius transformations we can study the representations of SL (2 , C ) on thespace of homogeneous polynomials of two complex variables [35]. In general, adopting thepassive point of view, to each element V = a bc d ∈ SL (2 , C ) corresponds the lineartransformation in C ( z , z ) (cid:55)→ ( dz − bz , − cz + az ) . Associated with this transformation we have the operator T V which acts on the space offunctions f : C → C , such as T V f ( z , z ) = f ( dz − bz , − cz + az ) . T V is a reducible representation of SL (2 , C ) in the space of functions of twocomplex variables because this space contains an invariant subspace under T V : the space H L of homogeneous, two variable polynomials of degree 2 L , h ( z , z ) = L (cid:88) j = − L u j z L + j z L − j . The restriction of T V to H L , T H V , is an irreducible representation of SL (2 , C ) on thespace of two complex variable functions.We can realize the space of Laurent polynomials L L of degree L in one (complex)variable from H L . In fact, any Laurent polynomialΥ( z ) = L (cid:88) l = − L u l z l can be expressed like Υ( z ) = z − L h ( z, . Therefore, taking into account the homogeinity of h ( z , z ), we arrive at the representationof SL (2 , C ) in the space of Laurent polynomials T L V Υ( z ) = ( dz − b ) L ( − c + az − ) L Υ (cid:18) dz − b − cz + a (cid:19) , (B1)which is just the passive point of view of expression (16).Now let us choose as a basis of L L the monomials { z m } with − L ≤ m ≤ L , and the SU (2)-invariant scalar product for which( z m , z n ) = δ mn . (B2)The matrix elements of T L V in this basis are t ( V ) mn = ( z m , T L V z n ) . After a bit of algebra and using (B1), (B2) and Newton’s binomial theorem we arrive to t ( V ) mn = [( L + m )!( L − m )!( L + n )!( L − n )!] − / L − n,L + m ) (cid:88) j =max(0 ,m − n ) (cid:18) L − nj (cid:19)(cid:18) L + nL + m − j (cid:19) ( − n − m a L − n − j b j c n − m + j d L + m − j . z ) transform under V ∈ SL (2 , C ) as follows u (cid:48) l = L (cid:88) m = − L t ( V ) lm u m . Since the coefficients of our Laurent polynomials Θ( z ) and Ξ( z ) are precisely the couplingsof our Hamiltonian (1), we have just found their behaviour under M¨obius transformations, A (cid:48) l = L (cid:88) m = − L t ( V ) lm A m ; B (cid:48) l = L (cid:88) m = − L t ( V ) lm B m . Note that t ( V ) m,n = t ( V ) − m, − n when a = d and b = c , as it happens for SO (1 , A − l = A l and B − l = − B l . [1] F. Ares, J. G. Esteve, F. Falceto, A. R. de Queiroz, Entanglement in fermionic chainswith finite range coupling and broken symmetries , Phys. Rev. A 92, 042334 (2015), arXiv:1506.06665 [quant-ph][2] A. R. Its, B. Q. Jin, V. E. Korepin,
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