Exactly conserved quasilocal operators for the XXZ spin chain
EExactly conserved quasilocal operators for the XXZspin chain
R. G. Pereira
Instituto de F´ısica de S˜ao Carlos, Universidade de S˜ao Paulo, C.P. 369, S˜ao Carlos,SP, 13560-970, Brazil
V. Pasquier
Institut de Physique Th´eorique, DSM, CEA, URA2306 CNRS, Saclay, F-91191Gif-sur-Yvette, France
J. Sirker
Department of Physics and Research Center OPTIMAS, Technical UniversityKaiserslautern, D-67663 Kaiserslautern, GermanyDepartment of Physics and Astronomy, University of Manitoba, Winnipeg,Manitoba, Canada R3T 2N2
I. Affleck
Department of Physics and Astronomy, University of British Columbia, Vancouver,B.C., Canada, V6T 1Z1
Abstract.
We extend T. Prosen’s construction of quasilocal conserved quantitiesfor the XXZ model [Phys. Rev. Lett. , 217206 (2011)] to the case of periodicboundary conditions. These quasilocal operators stem from a two-parameter transfermatrix which employs a highest-weight representation of the quantum group algebrainherent in the Yang-Baxter algebra. In contrast with the open chain, where theconservation law is weakly violated by boundary terms, the quasilocal operators inthe periodic chain exactly commute with the Hamiltonian and other local conservedquantities. a r X i v : . [ c ond - m a t . s t a t - m ec h ] S e p xactly conserved quasilocal operators for the XXZ spin chain
1. Introduction
Although a precise definition of quantum integrability is yet to be formulated, thecommon view is that quantum integrable models are characterized by a macroscopicnumber of local conserved quantities [1]. The most familiar examples are Bethe ansatzsolvable models in which a family of commuting operators can be derived by takinglogarithmic derivatives of the transfer matrix with respect to a spectral parameter [2, 3].Recently the physical consequences of nontrivial conservation laws have become relevantfor nonequilibrium dynamics of cold atomic gases confined to one-dimensional geometries[4, 5]. It is generally believed that the long time average of local observables in integrablesystems after a quantum quench is described by a generalized Gibbs ensemble (GGE)which incorporates conserved quantities besides the Hamiltonian [6, 7]. However, it isnot clear what are all the conserved quantities that need to be included in the densitymatrix of the GGE. While the set of all projection operators onto eigenstates of theHamiltonian — whose number increases exponentially with system size — is definitelymore than necessary to describe the equilibration of local observables, the family oflocal conserved quantities derived from the transfer matrix — whose number scalesonly linearly with system size — may not be sufficient for this purpose [8, 9]. Infact, recent results indicate that the GGE that includes only local conserved quantitiesfails to describe the steady state after a quench in the XXZ model [10, 11, 12]. GGEexpectation values for the XXZ model had been computed before in [13], where the smalldeviations from numerics were attributed to large relaxation times. New integrals ofmotion have also been constructed to explain why many-body localized systems (whichare not integrable in the usual sense) do not thermalize [14, 15].The question of additional conserved quantities beyond the local ones usuallyassociated with integrability has become even more pertinent since the discovery ofa conserved quasilocal operator for the XXZ model [16]. Using a matrix product ansatz,Prosen constructed a non-Hermitean operator that commutes with the Hamiltonianof an open XXZ chain up to boundary terms. The operator Z in Ref. [16] is notlocal in the usual sense because it cannot be written in the form Q n = (cid:80) Nj =1 q nj ,where q nj is a local density acting on sites j + 1 , . . . , j + n with n finite. Nevertheless,it is quasilocal in the sense that the operator norm defined at infinite temperatureas (cid:104) Z † Z (cid:105) = 2 − N Tr { Z † Z } grows linearly with system size, as it does for any localoperator. This quasilocal operator cannot be written as a linear combination of thelocal conserved quantities obtained from the transfer matrix because it has differentsymmetry properties. In particular, its imaginary part changes sign under spin inversion σ zj → − σ zj , σ ± j → σ ∓ j , whereas the local conserved quantities are all invariant under thesame transformation. This symmetry is important because it implies that, unlike thelocal conserved quantities, the quasilocal operator has an overlap with the spin currentoperator and provides a nonzero Mazur bound [17, 18] for the spin Drude weight athigh temperatures [19, 20, 21, 22, 23, 24, 25, 26]. This result establishes ballistic spintransport in the critical phase of the XXZ model at zero magnetic field, except at the xactly conserved quasilocal operators for the XXZ spin chain q [SU(2)] [30]. The latter arises naturally in the quantum inverse scatteringmethod, where it is convenient to view the XXZ model as the integrable q -deformationof the Heisenberg model [31, 32, 33, 34]. More recently, Prosen and Ilievski [35] madean explicit connection with integrability using a highest-weight Yang-Baxter transferoperator to derive a continuous family of quasilocal operators Z ( ϕ ) labeled by a complexparameter ϕ . This family contain the previously found operator as the particular choice ϕ = π/ ‡ The paper is organized as follows. In section 2, we review the derivation of thelocal conserved quantities of the XXZ model within the standard approach of takinglogarithmic derivatives of the transfer matrix. In section 3, we construct a family oftwo-parameter conserved quantities using an auxiliary transfer matrix with a highest-weight representation in the auxiliary space. The expansion of the conserved quantitiesabout special values of the representation parameter, along with a discussion of theconditions that lead to quasilocality, is presented in section 4. Section 5 makes thepoint that quasilocal operators are obtained only at first order in the expansion aboutthe special representation, as higher order operators are strictly nonlocal. Section 6contains the calculation of the Mazur bound for the spin Drude weight using a singlequasilocal conserved quantity. In section 7, we discuss the family of quasilocal operatorsobtained by varying the spectral parameter continuously. Finally, section 8 presents theconclusions.
2. Local conserved quantities
Our goal is to derive generating functions of operators that commute with the XXZHamiltonian [2] H = N (cid:88) j =1 (cid:0) σ xj σ xj +1 + σ yj σ yj +1 + ∆ σ zj σ zj +1 (cid:1) , (2.1)where σ x,y,z denote the standard Pauli matrices. The Hamiltonian acts on the tensorproduct of vector spaces V ⊗ N ≡ V ⊗ V ⊗ . . . ⊗ V N , where V j = C is the quantum space ‡ After this work had been submitted, a new paper by Prosen [36] appeared on the arXiv which alsodiscusses the exact conservation of quasilocal operators for the periodic chain. xactly conserved quasilocal operators for the XXZ spin chain j .In the general scheme of the quantum inverse scattering method, one starts byintroducing an R matrix that depends on a complex spectral parameter z and satisfiesthe Yang-Baxter equation [2, 3] R ( zw − ) R Q ( z ) R Q ( w ) = R Q ( w ) R Q ( z ) R ( zw − ) . (2.2)Here R ( z ) acts nontrivially on V ⊗ V and as the identity on a third, auxiliary space Q with dimension d Q , to be specified below. For the XXZ model (or six-vertex model),we can write R ( z ) = R ( z ) ⊗ with the R matrix R ( z ) = a cz − bb cz a , (2.3)with a = zq − z − q − , b = z − z − , c = q − q − . The parameter q is related to theanisotropy ∆ in Eq. (2.1) by ∆ = ( q + q − ) /
2. We use the following notation formatrices that act on the tensor product of two spaces [3] : R = R ijkl e V ij ⊗ e V kl . (2.4)Here e V ij are matrices acting on V defined by e V ij = ˆ e i ⊗ ˆ e j , where the set of vectors { ˆ e i } forms an orthonormal basis of V . In the example of the tensor product of two-dimensional spaces ( i, j = 1 , R matrix in Eq. (2.3) is written in the form R ( z ) = R R R R R R R R R R R R R R R R . (2.5)The product of e V ij matrices has the property e V ij e V kl = δ jk e V il . Using this propertywe can write down each element of the Yang-Baxter equation (2.2) corresponding to e V ij ⊗ e V kl ⊗ e Q mn , with i, j, k, l = 1 , m, n = 1 , . . . , d Q : (cid:88) a,b =1 d Q (cid:88) c =1 [ R ( zw − )] iakb [ R Q ( z )] ajmc [ R Q ( w )] blcn = (cid:88) a,b =1 d Q (cid:88) c =1 [ R Q ( w )] kbmc [ R Q ( z )] iacn [ R ( zw − )] ajbl . (2.6)The Lax operator associated with a given site j can be introduced as an R matrixthat acts on V j ⊗ Q L j ( z ) = R j Q ( z ) . (2.7)Then Eq. (2.2) implies the quadratic relation for Lax operators involving the R matrixin Eq. (2.3) R ( zw − ) L ( z ) L ( w ) = L ( w ) L ( z ) R ( zw − ) . (2.8) xactly conserved quasilocal operators for the XXZ spin chain K, S + , S − : L j ( z ) = 12 (cid:20) ( z − z − )2 j ⊗ ( K + K − ) + ( z + z − )2 σ zj ⊗ ( K − K − )+( q − q − )( zσ + j ⊗ S − + z − σ − j ⊗ S + ) (cid:3) . (2.9)Written as a matrix in V j (with entries that act on Q ), the Lax operator is L j ( z ) = 12 (cid:32) zK − z − K − z ( q − q − ) S − z − ( q − q − ) S + zK − − z − K (cid:33) . (2.10)The Yang-Baxter equation (2.8) is then satisfied provided that the operators S ± , K acting on Q obey the quantum group algebra U q [SU(2)][34] KS + = qS + K, (2.11) KS − = q − S − K, (2.12) (cid:2) S + , S − (cid:3) = K − K − q − q − . (2.13)Choosing Q = C , we can use the spin-1/2 representation K = q τ z / , S ± = τ ± , (2.14)where τ x,y,z are Pauli matrices in the auxiliary space. The monodromy matrix for N spins (acting on V ⊗ V ⊗ . . . V N ⊗ Q ) is defined as T Q ( z ) = L N ( z ) L N − ( z ) . . . L ( z ) . (2.15)The transfer matrix is t Q ( z ) = tr Q { T Q ( z ) } , (2.16)where tr Q denotes the trace over the auxiliary space Q . It can be shown [2, 3]that the transfer matrix forms a one-parameter family of commuting operators in V ⊗ V ⊗ . . . ⊗ V N :[ t Q ( z ) , t Q ( w )] = 0 , ∀ z, w ∈ C . (2.17)The local conserved quantities Q n are given by [2] Q n +1 = d n dz n ln t Q ( z ) (cid:12)(cid:12)(cid:12)(cid:12) z =1 , n ≥ . (2.18)The operator Q is proportional to the XXZ Hamiltonian in Eq. (2.1). The firstnontrivial conserved quantity, Q , coincides with the energy current operator [37]. Ingeneral, each Q n can be written as a sum of operators that act on n neighbouring spinsand is therefore local.Consider the spin inversion transformation C defined in the quantum space as C − σ zj C = − σ zj , C − σ ± j C = σ ∓ j , ∀ j . The Lax operator in Eq. (2.9) transforms as˜ L j ( z ) = C − L j ( z ) C = 12 (cid:20) ( z − z − )2 j ⊗ ( K + K − ) − ( z + z − )2 σ zj ⊗ ( K − K − )+( q − q − )( zσ − j ⊗ S − + z − σ + j ⊗ S + ) (cid:3) . (2.19) xactly conserved quasilocal operators for the XXZ spin chain Q : W ( z ) = [ W ( z )] − = (cid:32) zz − (cid:33) . (2.20)This transformation is such that[ W ( z )] − q τ z / W ( z ) = q − τ z / , (2.21)[ W ( z )] − τ + W ( z ) = z τ − , (2.22)[ W ( z )] − τ − W ( z ) = z − τ + . (2.23)It follows that[ W ( z )] − ˜ L j ( z ) W ( z ) = L j ( z ) . (2.24)As a result, ˜ t Q ( z ) = C − t Q ( z ) C = tr Q { ˜ L N ( z ) ˜ L N − ( z ) . . . ˜ L ( z ) } = tr Q { W − ˜ L N W W − ˜ L N − W . . . W − ˜ L W } = tr Q { L N ( z ) L N − ( z ) . . . L ( z ) } = t Q ( z ) . (2.25)Since Eq. (2.25) is verified for all z (cid:54) = 0 , ∞ , we conclude that all the Q n ’s derived byexpanding t Q ( z ) about z = 1 are invariant under spin inversion. For instance, this isclearly the case for the XXZ Hamiltonian at zero magnetic field in Eq. (2.1).
3. Conserved quantities from two-parameter transfer matrix
The idea to obtain a generating function of conserved quantities which are not invariantunder spin inversion is to introduce an auxiliary transfer matrix that commutes with t Q ( z ) but employs a different representation of the quantum group algebra. Let usconsider an auxiliary space A with dimension d A . We denote the Lax operator definedin V j ⊗ A by L j ( z ) = R j A ( z ) . (3.1)By analogy with Eq. (2.15), we can define the corresponding monodromy matrix T A ( z ) = L N ( z ) L N − ( z ) . . . L ( z ) , (3.2)as well as the auxiliary transfer matrix t A ( z ) = tr A { T A ( z ) } . (3.3) xactly conserved quasilocal operators for the XXZ spin chain R matrixin Q ⊗ A as follows: T Q ( z ) T A ( w ) R QA ( w/z ) = R N Q ( z ) . . . R Q ( z ) R N A ( w ) . . . R A ( w ) R QA ( w/z )= R N Q . . . R Q R N A . . . R A R Q R A R QA = R N Q . . . R Q R N A . . . R A R QA R A R Q = R QA R N A . . . R A R N Q . . . R Q = R QA ( w/z ) T A ( w ) T Q ( z ) . (3.4)Taking the trace of Eq. (3.4) over Q and A , we obtain[tr Q { T Q ( z ) } , tr A { T A ( w ) } ] = 0 , (3.5)thus [ t Q ( z ) , t A ( w )] = 0 ∀ z, w ∈ C . (3.6)Therefore, since the XXZ Hamiltonian is among the operators generated by t Q ( z ), wecan use t A ( z ) as a generating function of conserved quantities.We shall work with the highest weight representation of U q [SU(2)]: K | r (cid:105) = uq r | r (cid:105) , (3.7) S + | r (cid:105) = − a r | r + 1 (cid:105) , (3.8) S − | r (cid:105) = b r | r − (cid:105) , (3.9)where u ∈ C is arbitrary. The index r can be interpreted as positions in a lattice inthe auxiliary space, and the operators S + and S − perform hopping between nearest-neighbour sites. Eq. (2.13) imposes the relation a r b r +1 − a r − b r = u q r − u − q − r q − q − , (3.10)which is satisfied by the choice a r = v u q r − u − q − r q − q − , (3.11) b r = v − q r − q − r q − q − , (3.12)where v is another arbitrary parameter which we set to 1 hereafter. In this representationthe Casimir operator is a function of the parameter u : C = ( q − q − ) S + S − + q − K + qK − = u q − + u − q. (3.13)The dimension of the auxiliary space depends on the value of ∆ = ( q + q − ) / q is a root of unity, i.e. q = e iλ with λ = lπ/m and l, m ∈ Z coprimes, we have b = b m = 0. In these cases we can restrict the auxiliary space index r to 0 ≤ r ≤ m − d A = m . Notice that for q = e iπl/m we have∆ = cos( πl/m ), hence | ∆ | ≤
1, which corresponds to the gapless phase of the XXZmodel. xactly conserved quasilocal operators for the XXZ spin chain u . The Lax operator defined in Eq. (3.1) is a function of both u and the spectralparameter z . Similarly to Eq. (2.9), we can write L j ( z, u ) = i [ j ⊗ A ( z, u ) + σ zj ⊗ A z ( z, u )+ σ + j ⊗ A + ( z, u ) + σ − j ⊗ A − ( z, u )] , (3.14)where A ( z, u ) = ( z − z − )4 i [ K ( u ) + K − ( u )] , (3.15) A z ( z, u ) = ( z + z − )4 i [ K ( u ) − K − ( u )] , (3.16) A + ( z, u ) = z i ( q − q − ) S − ( u ) , (3.17) A − ( z, u ) = z − i ( q − q − ) S + ( u ) . (3.18)In this notation, the conserved quantity defined in Eq. (3.3) reads (hereafter we omitthe index A in tr A ) t A ( z, u ) = i N (cid:88) { α j } tr { A α N . . . A α A α } N (cid:89) j =1 σ α j j , (3.19)where the sum is over all α j ∈ { , z, + , −} and we use the notation σ j ≡ j .The operator in Eq. (3.19) is translationally invariant due to the cyclic propertyof the trace. On the other hand, it is not necessarily invariant under spin reversalfor general u . (The similarity between matrices used in Eq. (2.24) is not verified forarbitrary values of u .) Moreover, t A ( z, u ) is not invariant under parity transformation P , which we can define as the reflection about the link between sites j = 1 and j = N : P − σ α j j P = σ α j N +1 − j . We have P − t A ( z, u ) P = i N (cid:88) { α j } tr { A α N . . . A α A α } N (cid:89) j =1 σ α j N +1 − j = i N (cid:88) { α j } tr { A α . . . A α N − A α N } N (cid:89) j =1 σ α j j . (3.20)We note that t A in Eq. (3.19) can also be written as t A ( z, u ) = i N (cid:88) { α j } tr { A tα A tα . . . A tα N } N (cid:89) j =1 σ α j j , (3.21)where A tα denotes the transpose of A α . We define the two-parameter conserved quantitywhich is odd under parity as I ( z, u ) = ( − i ) N [ P − t A ( z, u ) P − t A ( z, u )] . (3.22) xactly conserved quasilocal operators for the XXZ spin chain I ( z, u ) = (cid:88) { α j } tr { A α . . . A α N − A tα . . . A tα N } N (cid:89) j =1 σ α j j . (3.23)For reference, let us comment on the particular cases ∆ = 0 and ∆ = ±
1. For ∆ = 0the XXZ model is equivalent to free fermions via a Jordan-Wigner transformation. Thispoint corresponds to m = 2, q = i ; in this case the generators of the quantum groupalgebra become K = u (cid:32) i (cid:33) , S + = u − − u i σ − , S − = σ + . (3.24)Note that, although the auxiliary space is two-dimensional A = C , the representationdiffers from Eq. (2.14) for general u . Only for u = e − iπ/ do we recover a parity-invariantrepresentation. On the other hand, at the ferromagnetic SU(2) point ∆ = − q = − m = 1), the A α matrices reduce to numbers and the conserved quantity in Eq. (3.23)vanishes identically. At the antiferromagnetic SU(2) point ∆ = 1 ( q = 1, m → ∞ ) theoperator is not identically zero but the representation becomes infinite dimensional.
4. Quasilocal conserved quantities
Now we turn to the task of extracting quasilocal operators from I ( z, u ) in Eq. (3.23).In order to calculate the Mazur bound for the Drude weight at high temperatures [18],it is convenient to define the inner product between two operators A and B acting on V ⊗ N based on the thermal average at infinite temperature: (cid:104) A † B (cid:105) = 2 − N Tr { A † B } , (4.1)where Tr denotes the trace over the quantum space V ⊗ N . From Eq. (4.1) it can beshown that the norm of I ( z, u ) reduces to (cid:104)I † ( z, u ) I ( z, u ) (cid:105) = 2 tr A⊗A { [ T ( z, u, u )] N − [ T ( z, u, u )] N } . (4.2)Here T ( z, u, ¯ u ) and T ( z, u, ¯ u ) are transfer matrices in A ⊗ A T ( z, u, ¯ u ) = (cid:88) α =0 ,z, ± C α A ∗ α ( z, u ) ⊗ A α ( z, ¯ u ) , (4.3) T ( z, u, ¯ u ) = (cid:88) α =0 ,z, ± C α A ∗ α ( z, u ) ⊗ A tα ( z, ¯ u ) , (4.4)where C α = 12 Tr (cid:8) σ α ( σ α ) † (cid:9) . (4.5)In contrast with Prosen’s construction for the open chain [16], where the norm iscomputed from the matrix element between boundary states, Eq. (4.2) involves thetrace over the auxiliary space. The analogy with the open chain can be explored furtherif we notice that, by setting the spectral parameter to be z = i , the matrix A z inEq. (3.16) vanishes and the conserved quantity does not contain any σ zj operators, as xactly conserved quasilocal operators for the XXZ spin chain z = i . We return to the question of general values of z in section7. For z = i the nonvanishing operators in auxiliary space simplify to A ( z = i, u ) = m − (cid:88) r =0 uq r + u − q − r | r (cid:105)(cid:104) r | , (4.6) A + ( z = i, u ) = m − (cid:88) r =0 q r +1 − q − r − | r (cid:105)(cid:104) r + 1 | , (4.7) A − ( z = i, u ) = m − (cid:88) r =0 u q r − u − q − r | r + 1 (cid:105)(cid:104) r | . (4.8)After fixing the value of the spectral parameter, we are still free to choose the valueof u in the representation of the quantum group algebra. We notice that the condition u = 1 is special because in this case a = 0, then the state | r = 0 (cid:105) is annihilated by A ± and decouples from the other states. Hereafter we choose u = 1, but the result forthe other roots is equivalent. For u = 1 the Casimir operator becomes C = q + q − .Interestingly, a similar kind of special representation appears in open spin chains wherethe quantum group is an actual symmetry commuting with the Hamiltonian [33]. Inthat case, the Casimir for a spin-1 / q N − + q − N +1 andbecomes special if q N = − i.e. , for values of q that obey a “root of unity condition”depending on the chain length [38].Let us then analyze the operator I ≡ I ( z = i, u = 1) . (4.9)Setting u = 1 in Eqs. (4.6) through (4.8), we obtain (recall q = e iλ ) A (1) ≡ A ( z = i, u = 1) = m − (cid:88) r =0 cos( λr ) | r (cid:105)(cid:104) r | , (4.10) A + (1) ≡ A + ( z = i, u = 1) = i m − (cid:88) r =0 sin[ λ ( r + 1)] | r (cid:105)(cid:104) r + 1 | , (4.11) A − (1) ≡ A − ( z = i, u = 1) = − i m − (cid:88) r =0 sin( λr ) | r + 1 (cid:105)(cid:104) r | . (4.12)The transfer matrices in Eqs. (4.3) and (4.4) become T (1) ≡ T ( z = i, u = 1 , ¯ u = 1)= m − (cid:88) r,s =0 cos( λr ) cos( λs ) | r, s (cid:105)(cid:104) r, s | + 12 m − (cid:88) r,s =0 sin[ λ ( r + 1)] sin[ λ ( s + 1)] | r, s (cid:105)(cid:104) r + 1 , s + 1 | + 12 m − (cid:88) r,s =0 sin( λr ) sin( λs ) | r + 1 , s + 1 (cid:105)(cid:104) r, s | , (4.13) xactly conserved quasilocal operators for the XXZ spin chain T (1) ≡ T ( z = i, u = 1 , ¯ u = 1)= m − (cid:88) r,s =0 cos( λr ) cos( λs ) | r, s (cid:105)(cid:104) r, s | + 12 m − (cid:88) r,s =0 sin[ λ ( r + 1)] sin[ λ ( s + 1)] | r, s + 1 (cid:105)(cid:104) r + 1 , s | + 12 m − (cid:88) r,s =0 sin( λr ) sin( λs ) | r + 1 , s (cid:105)(cid:104) r, s + 1 | . (4.14)The transfer matrices are block diagonal in subspaces of Kronecker states {| r, ( r + k )(mod m ) (cid:105)} with fixed k = 0 , . . . , m − T (1), or Kroneckerstates {| r, ( − r + k )(mod m ) (cid:105)} in the case of T (1). Since we are interested in the scalingof the operator norm in Eq. (4.2) with system size N as N → ∞ , we may restrictourselves to the subspace in which the transfer matrices have their largest eigenvalue.This happens when k = 0 for both T (1) and T (1). Within the k = 0 subspace wedenote | r, ± r (cid:105) → | r (cid:105) and obtain the reduced transfer matrices T = m − (cid:88) r =0 cos ( λr ) | r (cid:105)(cid:104) r | + 12 m − (cid:88) r =0 sin [ λ ( r + 1)] | r (cid:105)(cid:104) r + 1 | + 12 m − (cid:88) r =0 sin ( λr ) | r + 1 (cid:105)(cid:104) r | , (4.15) T = m − (cid:88) r =0 cos ( λr ) | r (cid:105)(cid:104) r | − m − (cid:88) r =0 sin( λr ) sin[ λ ( r + 1)] ×× [ | r (cid:105)(cid:104) r + 1 | + | r + 1 (cid:105)(cid:104) r | ] . (4.16)It is useful to note that T = − B + 12 ∆ B , (4.17) T = − B − B ∆ B, (4.18)where B is the diagonal matrix B = (cid:80) m − r =0 sin( rλ ) | r (cid:105)(cid:104) r | and ∆ is the uniform hoppingmatrix on an open chain with length m ∆ = m − (cid:88) r =0 ( | r (cid:105)(cid:104) r + 1 | + | r + 1 (cid:105)(cid:104) r | ) . (4.19)The matrix T is symmetric, thus its eigenvalues are all real. Since B | (cid:105) = 0, wefind that | r = 0 (cid:105) is an eigenvector of T with eigenvalue 1. It is easy to verify that all theother eigenvalues are smaller than 1. § On the other hand, T is not symmetric. However,in Appendix A we show that T and T are similar and have exactly the same spectrum(see also Fig. 1). It also follows from Eq. (4.17) that | r = 0 (cid:105) is the right eigenvector of § For λ ∈ R , i.e. | ∆ | ≤
1, we can show that the largest eigenvalue of T is 1 using the Gershgorin circletheorem. In the gapped Neel phase ∆ > xactly conserved quasilocal operators for the XXZ spin chain u Figure 1.
Absolute value of the eigenvalues of the transfer matrices T ( z = i, u, u )(solid blue lines) and T ( z = i, u, u ) (dashed red lines) as a function of u ∈ R for q = e iπ/ . The quasilocal conserved quantity is obtained by expanding about u = 1,where T and T have the same spectrum and their largest eigenvalue is normalized to1. T with eigenvalue 1. We will also need the left eigenvector of T with eigenvalue 1. InAppendix B we show that the solution to the eigenvalue equation (cid:104) L |T = (cid:104) L | yields (cid:104) L | = m − (cid:88) r =0 (1 − r/m ) (cid:104) r | . (4.20)The left eigenvector (cid:104) L | is not normalized to unity but is such that (cid:104) L | (cid:105) = 1.When calculating the norm of the conserved quantity using Eq. (4.2), we can usethe macroscopic number of transfer matrices to project the auxiliary space into theeigenvectors of T or T with eigenvalue 1. In Appendix C we show thatlim n →∞ T n = lim n →∞ T n/ (cid:88) r | r (cid:105)(cid:104) r |T n/ = | (cid:105)(cid:104) L | , (4.21)lim n →∞ T n = lim n →∞ T n/ (cid:88) r | r (cid:105)(cid:104) r |T n/ = | (cid:105)(cid:104) | . (4.22)In particular, the projection allows us to compute traces involving an arbitrary matrix M lim n →∞ tr {T n M } = (cid:104) L | M | (cid:105) , (4.23)lim n →∞ tr {T n M } = (cid:104) | M | (cid:105) . (4.24)It turns out that we do not get a quasilocal operator by simply setting z = i, u = 1.The reason is that, since T and T are related by a similarity transformation, theoperator I in Eq. (4.9) actually has zero norm: (cid:104)I † I (cid:105) = tr {T N − T N } = 0 . (4.25)Nevertheless, the properties of the transfer matrices suggest that quasilocal operatorscan be generated by expanding I ( z = i, u ) about u = 1: I ( z = i, u = 1 + ε ) = ε I + O ( ε ) . (4.26) xactly conserved quasilocal operators for the XXZ spin chain I = ∂ I ( z, u ) ∂u (cid:12)(cid:12)(cid:12)(cid:12) z = i,u =1 = (cid:88) { α j } tr (cid:26) ∂∂u [ A α ( u ) . . . A α N ( u ) − A tα ( u ) . . . A tα N ( u )] (cid:27) N (cid:89) j =1 σ α j j , (4.27)with matrices A α ( u ) given in Eqs. (4.6), (4.7) and (4.8). Using the transfer matrices inEqs. (4.3) and (4.4), we can express the norm of I as follows: (cid:104)I † I (cid:105) = 2 tr (cid:26) ∂ ∂u∂ ¯ u [ T ( z, u, ¯ u )] N (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) z = i,u =¯ u =1 − (cid:26) ∂ ∂u∂ ¯ u [ T ( z, u, ¯ u )] N (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) z = i,u =¯ u =1 . (4.28)The operators inside the trace in Eq. (4.28) contain a macroscopic number of transfermatrices. Once again, this allows us to restrict to the Kronecker spaces which containeigenvectors with eigenvalue 1. Let us introduce a shorthand notation for the derivativesof the reduced transfer matrices: T ( n,n (cid:48) )1 ≡ ∂ n ∂u n ∂ n (cid:48) ∂ ¯ u n (cid:48) T ( z = i, u, ¯ u ) (cid:12)(cid:12)(cid:12)(cid:12) u =¯ u =1 , (4.29)and likewise for T ( n,n (cid:48) )2 . The derivatives in Eq. (4.28) yield (cid:104)I † I (cid:105) N = 2tr (cid:110) ( T ) N − T (1 , + ( T ) N − T (1 , (cid:111) + 2 N − (cid:88) n =0 tr (cid:110) T (1 , ( T ) n T (0 , ( T ) N − − n (cid:111) + 2 N − (cid:88) n =0 tr (cid:110) T (1 , ( T ) n T (0 , ( T ) N − − n (cid:111) . (4.30)In order for I to be quasilocal, the righthand side of Eq. (4.30) must approach afinite value in the limit N → ∞ . First consider the last two terms in Eq. (4.30). Thederivatives of the A α matrices at u = 1 are A (cid:48) (1) ≡ ∂A ∂u (cid:12)(cid:12)(cid:12)(cid:12) u =1 = i m − (cid:88) r =0 sin( λr ) | r (cid:105)(cid:104) r | , (4.31) A (cid:48) + (1) ≡ ∂A + ∂u (cid:12)(cid:12)(cid:12)(cid:12) u =1 = 0 , (4.32) A (cid:48)− (1) ≡ ∂A − ∂u (cid:12)(cid:12)(cid:12)(cid:12) u =1 = − m − (cid:88) r =0 cos( λr ) | r + 1 (cid:105)(cid:104) r | . (4.33)Thus, T (0 , = T (1 , xactly conserved quasilocal operators for the XXZ spin chain i m − (cid:88) r =0 sin(2 λr ) | r (cid:105)(cid:104) r | + i m − (cid:88) r =0 sin(2 λr ) | r + 1 (cid:105)(cid:104) r | , (4.34) T (0 , = i m − (cid:88) r =0 sin( λr ) cos( λr ) | r (cid:105)(cid:104) r | + i m − (cid:88) r =0 sin( λr ) cos[ λ ( r + 1)] | r + 1 (cid:105)(cid:104) r | , (4.35) T (1 , = i m − (cid:88) r =0 sin( λr ) cos( λr ) | r (cid:105)(cid:104) r |− i m − (cid:88) r =0 cos( λr ) sin[ λ ( r + 1)] | r + 1 (cid:105)(cid:104) r | . (4.36)We then notice that T (0 , | (cid:105) = T (1 , | (cid:105) = 0 , (4.37) T (0 , | (cid:105) = (cid:104) |T (1 , = 0 . (4.38)These relations are a result of the decoupling of state | (cid:105) from the other states at u = 1(see comment around Eq. (4.9)). Together with the projection in Eqs. (4.21) and (4.22),these relations imply that the last two terms in Eq. (4.30) vanish in the thermodynamiclimit.We are left with the contributions in the first line of Eq. (4.30), which givelim N →∞ (cid:104)I † I (cid:105) N = 2 (cid:104) L |T (1 , | (cid:105) + 2 (cid:104) |T (1 , | (cid:105) . (4.39)The derivatives of the reduced transfer matrices in Eq. (4.39) are T (1 , = m − (cid:88) r =0 sin ( λr ) | r (cid:105)(cid:104) r | + 2 m − (cid:88) r =0 cos ( λr ) | r + 1 (cid:105)(cid:104) r | , (4.40) T (1 , = m − (cid:88) r =0 sin ( λr ) | r (cid:105)(cid:104) r | + 2 m − (cid:88) r =0 cos( λr ) cos[ λ ( r + 1)] | r + 1 (cid:105)(cid:104) r | . (4.41)The only nonzero matrix element that contributes to the norm of I is (cid:104) L |T (1 , | (cid:105) . Using Eq. (4.20), we obtainlim N →∞ (cid:104)I † I (cid:105) N = 4 (cid:18) − m (cid:19) . (4.42)This proves I is quasilocal for m > u = 1 in our case,see Eq. (4.9)) and the subrepresentation obtained in this case (the single state | r = 0 (cid:105) ) isparity invariant. It is then clear that the two terms in Eq. (4.2) are equal to Λ N , whereΛ is the largest eigenvalue in the parity-invariant subspace (assuming it dominates the xactly conserved quasilocal operators for the XXZ spin chain δu = ε (cid:28)
1, we findthat the transfer matrices in Eq. (4.2) behave as (here ν = 1 , T ν (1 + (cid:15) ) = (cid:32) Λ εB ν εC ν D + εF ν (cid:33) , (4.43)where D, F ν are matrices in the subspace orthogonal to the parity invariant subspace.As a result, the eigenvalues behave as Tr { T Nν } = (Λ + ε A ν ) N ≈ Λ N + N ε A ν . Thenorm will then be linear in N as long as A (cid:54) = A .
5. Nonlocal operators generated in the expansion of I ( z, u )The expansion in Eq. (4.26) to higher orders in ε = u − I (cid:96) = ∂ (cid:96) ∂u (cid:96) I ( z = i, u ) (cid:12)(cid:12)(cid:12)(cid:12) u =1 . (5.1)But I is the only quasilocal operator in this series because in general the norm of I (cid:96) scales like N (cid:96) for N → ∞ . To see this, consider the case of I : (cid:104)I † I (cid:105) = 2 tr (cid:26) ∂ ∂u ∂ ¯ u [ T ( z, u, ¯ u )] N (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) z = i,u =¯ u =1 − (cid:26) ∂ ∂u ∂ ¯ u [ T ( z, u, ¯ u )] N (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) z = i,u =¯ u =1 . (5.2)When applying derivatives in Eq. (5.2), we can discard terms which contain T (0 , , T (1 , , T (0 , , T (1 , , since their contribution vanishes in the thermodynamic limit.The result for large N is (cid:104)I † I (cid:105) N = 2tr (cid:104) T (2 , ( T ) N − (cid:105) + 4 N − (cid:88) n =0 tr (cid:104) T (1 , ( T ) n T (1 , ( T ) N − − n (cid:105) + 2 N − (cid:88) n =0 tr (cid:104) T (2 , ( T ) n T (0 , ( T ) N − − n (cid:105) − ( T → T ) . (5.3)In contrast with Eq. (4.30), the terms on the righthand side of Eq. (5.3) that involvesums do not vanish identically because the matrices T (1 , , T (2 , , T (0 , do not annihilatethe state | (cid:105) (and likewise for T ). In fact, the result of the sum increases linearly with N since the trace does not decay with the number n of T ’s between the derivatives.The coefficient of the O ( N ) term in the norm of I stems from terms in the sumswith n ∼ N . In order to extract this coefficient, we insert another projection onto theeigenvectors of T or T with eigenvalue 1 and obtainlim N →∞ (cid:104)I † I (cid:105) N = 4 (cid:104) L |T (1 , | (cid:105) − (cid:104) |T (1 , | (cid:105) + 2 (cid:104) L |T (2 , | (cid:105)(cid:104) L |T (0 , | (cid:105)− (cid:104) |T (2 , | (cid:105)(cid:104) |T (0 , | (cid:105) . (5.4) xactly conserved quasilocal operators for the XXZ spin chain N →∞ (cid:104)I † I (cid:105) N = 16 (cid:18) − m (cid:19) . (5.5)For general (cid:96) ≥
1, the expression for the norm of I (cid:96) contains terms in which anumber (cid:96) of matrices T (1 , are distributed over the N sites of the chain. Since thecontribution in the trace does not decrease with the separation between the T (1 , ’s, thenorm grows with the number of ways to choose the positions of these matrices whenthey are far apart, therefore (cid:104)I † (cid:96) I (cid:96) (cid:105) ∝ N (cid:96) .
6. Mazur bound
Within linear response theory, the optical conductivity for a given model is related tothe dynamical current-current correlation function via the Kubo formula. The real partof the optical conductivity can be written as σ (cid:48) ( ω ) = 2 πDδ ( ω ) + σ reg ( ω ) , (6.1)where D is the Drude weight and σ reg ( ω ) is the regular part. A nonzero Drudeweight implies infinite dc conductivity, i.e. , ballistic transport. The connection betweenintegrability and transport is made particularly clear by means of the Mazur bound [18]for the Drude weight at finite temperature TD ≥ LT (cid:88) k |(cid:104) J Q k (cid:105)| (cid:104) Q † k Q k (cid:105) . (6.2)Here L is the system size, (cid:104) (cid:105) denotes the thermal average, J is the current operator and { Q k } is a set of operators that commute with the Hamiltonian and are orthogonalized inthe form (cid:104) Q † k Q l (cid:105) = δ kl (cid:104) Q † k Q k (cid:105) . Although integrable models possess an infinite numberof conserved quantities in the thermodynamic limit, it suffices to find one single operatorthat gives a nonzero contribution to the right hand side of Eq. (6.2) in order to establishballistic transport.The current operator is obtained from the continuity equation for the density ofthe conserved charge. The spin current operator for the XXZ model (2.1) reads J = i (cid:88) j ( σ + j σ − j +1 − σ − j σ + j +1 ) . (6.3)This operator is odd under spin inversion: C − J C = − J. (6.4)As discussed in the section 2, all the local conserved quantities derived from the transfermatrix t Q are invariant under spin inversion. This includes the XXZ Hamiltonian atzero magnetic field. As a result, (cid:104) J Q n (cid:105) = 0 for all the local Q n ’s.Let us now show that the quasilocal operator I in Eq. (4.27) provides a nonzeroMazur bound at zero magnetic field. Notice that, since [ I , H ] = 0 exactly, there is no xactly conserved quasilocal operators for the XXZ spin chain (cid:225)(cid:225)(cid:225)(cid:225)(cid:225)(cid:225)(cid:225)(cid:225)(cid:225)(cid:225)(cid:225)(cid:225)(cid:225)(cid:225)(cid:225)(cid:225)(cid:225)(cid:225)(cid:225)(cid:225)(cid:225)(cid:225)(cid:225)(cid:225)(cid:225)(cid:225)(cid:225)(cid:225)(cid:225)(cid:225)(cid:225)(cid:225)(cid:225)(cid:225)(cid:225)(cid:225)(cid:225)(cid:225)(cid:225)(cid:225)(cid:225)(cid:225)(cid:225)(cid:225)(cid:225)(cid:225)(cid:225)(cid:225)(cid:225)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)
10 20 30 40 50 N D I Figure 2.
Mazur bound from I ( z = i, u ) as a function of chain length N for q = e iπ/ and three different values of the parameter u : u = 1 .
01 (red circles), u = 1 . u = 1 . u → N → ∞ in Eq. (6.7). issue with the violation of the conservation law by boundary terms as in the open chain[29]. We need to calculate the overlap between J and I (cid:104) J I (cid:105) = 2 − N Tr { J I } = N i (cid:88) { α j } ∂∂u tr (cid:8) A α ( u ) . . . A α N ( u ) − A tα ( u ) . . . A tα N ( u ) (cid:9)(cid:12)(cid:12) u =1 ×× − N Tr { N (cid:89) j =1 σ α j j ( σ +1 σ − − σ − σ +2 ) } = N i ∂∂u tr { [ A − ( u ) , A + ( u )]( A ( u )) N − } (cid:12)(cid:12)(cid:12)(cid:12) u =1 . (6.5)For N → ∞ , the factors of ( A ) N project the auxiliary space into | (cid:105) , which is theeigenvector of A with eigenvalue 1. The nonzero contribution stems from applying toderivative to A − ( u ): (cid:104) J I (cid:105) = − N i (cid:104) | A + (1) A (cid:48)− (1) | (cid:105) = − N sin λ. (6.6)Using the result for the norm in Eq. (4.42), we find that the contribution from I to the Mazur bound in the high temperature limit is of the form D ≥ D I / T with D I = lim N →∞ (cid:104) J I (cid:105) N (cid:104)I † I (cid:105) = sin λ mm − . (6.7)This result agrees with the bound obtained from the quasi conserved operator for theopen chain [16].Since I = I ( z = i, u = 1) vanishes, the result in Eq. (6.7) can also be written as D I ( z = i ) = lim N →∞ lim u → |(cid:104) J I ( z = i, u ) (cid:105)| N (cid:104)I † ( z = i, u ) I ( z = i, u ) (cid:105) , (6.8)where the order of the limits matters. The role of the limit u → N → ∞ isillustrated in Fig. 2, where we calculate the Mazur bound for finite chains numerically xactly conserved quasilocal operators for the XXZ spin chain u (cid:54) = 1, theconserved quantity I ( z = i, u ) is nonlocal and its contribution to the Mazur bounddecreases exponentially with system size. For small | u − | and large finite N , the Mazurbound approaches a plateau that agrees with the analytical result in the thermodynamiclimit.
7. Continuous family of quasilocal operators
The choice of the spectral parameter z = i in Eq. (4.27) is not required to derive aquasilocal conserved quantity. Generalizing the results of section 4 to arbitrary valuesof z , we find that the norm of I ( z ) = ∂∂u I ( z, u ) (cid:12)(cid:12)(cid:12)(cid:12) u =1 (7.1)can be computed from the reduced transfer matrices ( cf. Eqs. (4.15) and (4.16)) T ( z ) = m − (cid:88) r =0 (cid:8)(cid:2) (Im z ) cos ( λr ) + (Re z ) sin ( λr ) (cid:3) | r (cid:105)(cid:104) r | + | z | [ λ ( r + 1)] | r (cid:105)(cid:104) r + 1 | + | z | − ( λr ) | r + 1 (cid:105)(cid:104) r | (cid:27) , (7.2) T ( z ) = m − (cid:88) r =0 (cid:8)(cid:2) (Im z ) cos ( λr ) + (Re z ) sin ( λr ) (cid:3) | r (cid:105)(cid:104) r |−
12 sin( λr ) sin[ λ ( r + 1)] (cid:2) | z | | r (cid:105)(cid:104) r + 1 | + | z | − | r + 1 (cid:105)(cid:104) r | (cid:3)(cid:27) . (7.3)The state | r = 0 (cid:105) is an eigenvector of T ( z ) and a right eigenvector of T ( z ),with eigenvalue Λ = (Im z ) . As discussed at the end of section 4, the condition forquasilocality is that Λ be the largest eigenvalue of both transfer matrices. As shown in[35], this condition is satisfied by a continuous set of values of z in the complex plane.As a measure of quasilocality, we use the Mazur bound in Eq. (6.8) generalized toarbitrary z . A nonzero value of D I ( z ) implies that the norm of I ( z ) is extensive. Fig. 3illustrates the magnitude of the Mazur bound D I ( z ) in the complex z plane for q = e iπ/ (∆ = 1 / D I ( z ) is maximum at z = ± i , where it assumes the valuepredicted by Eq. (6.7). The domain where D I ( z ) > z = | z | e iθ , the conserved operator I ( z ) is quasilocal inside the cone || θ | − π | < π m .Therefore, the Mazur bound obtained from a single quasilocal operator ismaximized by the choice z = ± i . However, the entire continuous family {I ( z ) } canbe used to raise the bound. The idea is to replace the sum on the rhs of Eq. (6.2)by an integral over the spectral parameter z , using the orthogonality between differentelements in the family, as done in [35]. xactly conserved quasilocal operators for the XXZ spin chain Figure 3.
Magnitude of the Mazur bound D I ( z ) calculated using a single quasilocaloperator I ( z ), as a function of the spectral parameter z , for q = e iπ/ . Brighterregions correspond to larger values of D I ( z ).
8. Conclusion
We have described a method to derive quasilocal operators which commute with theHamiltonian of the XXZ chain with periodic boundary conditions. The key to thisprocedure is to introduce an auxiliary transfer matrix t A ( z, u ) that depends on twoparameters, namely the usual spectral parameter z and the representation parameter u . The latter is a parameter of the highest-weight representation for the quantumgroup algebra that arises in the Yang-Baxter relation for the Lax operator. For valuesof anisotropy ∆ = cos( πl/m ), with l, m integers, the highest-weight representationhas finite dimension m . The two-parameter conserved operator I ( z, u ) that has anonzero overlap with the spin current operator is defined from a linear combinationof the auxiliary transfer matrix and its conjugate under parity. The norm of I ( z, u ) canbe calculated using the transfer matrices T ( z, u ) and T ( z, u ), which are related by asimilarity transformation. A quasilocal operator is obtained by expanding I ( z, u ) aboutthe special value u = 1 where the highest-weight representation becomes reducible andthe eigenvector of T ( z, u = 1) and T ( z, u = 1) with the largest eigenvalue decouplesfrom the other states. The remaining spectral parameter z labels a continuous familyof quasilocal conserved quantities {I ( z ) } . This is in contrast with the usual discreteset of local conserved quantities which are obtained by taking logarithmic derivatives ofthe transfer matrix t Q ( z ) (defined with a spin-1 / / xactly conserved quasilocal operators for the XXZ spin chain {I ( z ) } has a fractal ∆ dependence [16, 29]which is perhaps absent in the actual Drude weight at high temperatures [25].We note that, while here we have focused on the periodic chain, it should be possibleto apply the same techniques to integrable models with open boundaries, taking intoaccount reflection operators at the boundaries. In fact, in [39] a two-parameter familyof transfer matrices has been constructed for the open asymmetric simple exclusionprocess (ASEP) (see Eqs. (47) and (48) of [39], which are the generalization of theconserved quantities to the open case). The effect of the boundary parameters ontransport properties is an interesting open question.The role of quasilocal conserved quantities in the GGE also remains to be clarified[10, 11, 12]. Remarkably, there is evidence that expectation values of local observablesin post-quench steady states deviate from the predictions of the GGE even for ∆ > i.e. in the gapped N´eel phase, where the method described here does not yieldany quasilocal operators. Acknowledgements
We thank Fabian Essler, Andreas Kl¨umper, and Gr´egoire Misguich for discussions. Thiswork is supported by CNPq (R.G.P.), the SFB/TR 49 (J.S.), NSERC (J.S., I.A.), andCIfAR (I.A.).
Appendix A. Similarity between reduced transfer matrices T and T The transfer matrices defined in section 4 are T = sin λ . . .
00 cos λ sin λ . . . sin λ cos λ sin λ . . . . . . cos ( m − λ , (A.1) T = . . .
00 cos λ − sin λ sin 2 λ . . . − sin λ sin 2 λ cos λ . . . . . . cos ( m − λ . (A.2)First we note that the sign of the off-diagonal terms of T can be changed by applyingthe “ Z gauge transformation” S| r (cid:105) = ( − r | r (cid:105) . Defining ˜ T = S − T S , we obtain˜ T = − B + 12 B ∆ B, (A.3)which is to be compared to Eqs. (4.17) and (4.18). xactly conserved quasilocal operators for the XXZ spin chain T is similar to ˜ T . It suffices to show that they have thesame characteristic polynomial. The characteristic polynomial for T readsdet( T − x ) = (1 − x ) det (cid:20) (1 − x ) − ˜ B + 12 ˜∆ ˜ B (cid:21) , (A.4)where ˜ B is the diagonal matrix ˜ B = (cid:80) m − r =1 sin( rλ ) | r (cid:105)(cid:104) r | and ˜∆ is the uniform hoppingmatrix ˜∆ = (cid:80) m − r =1 ( | r (cid:105)(cid:104) r + 1 | + | r + 1 (cid:105)(cid:104) r | ) in the ( m − T (and also for T , given the similarity between them)is det( ˜ T − x ) = (1 − x ) det (cid:20) (1 − x ) − ˜ B + 12 ˜ B ˜∆ ˜ B (cid:21) . (A.5)Since ˜ B is invertible (all of its eigenvalues are nonzero for λ = lπ/m with l, m coprimes),we can apply a similarity transformation inside the determinant sign as follows:det( ˜ T − x ) = (1 − x ) det (cid:20) ˜ B − (cid:18) (1 − x ) − ˜ B + 12 ˜ B ˜∆ ˜ B (cid:19) ˜ B (cid:21) = (1 − x ) det (cid:20) (1 − x ) − ˜ B + 12 ˜∆ ˜ B (cid:21) = det( T − x ) . (A.6)This shows that T and T are similar. Appendix B. Left eigenvector of T with eigenvalue 1 Let | L (cid:105) denote the left eigenvectors of T with eigenvalue 1, which obeys (cid:104) L |T = (cid:104) L | . (B.1)We expand | L (cid:105) in the orthonormal basis of {| r (cid:105)} vectors | L (cid:105) = m − (cid:88) r =0 v r | r (cid:105) . (B.2)Our problem is then to find the coefficients v r . For short, we denote the matrix elementsof T as (cid:104) i |T | j (cid:105) = t i,j . A useful identity is t r,r +1 t r +1 ,r = 14 (1 − t r,r )(1 − t r +1 ,r +1 ) . (B.3)The eigenvalue equation (B.1) is satisfied identically for the column r = 0. Thiscorresponds to the freedom of choosing the value of v (or the normalization of | L (cid:105) ).Let us turn to the next simplest equation, the one stemming from the column r = m − v m − t m − ,m − + v m − t m − ,m − = v m − , (B.4)from which we get v m − = t m − ,m − (1 − t m − ,m − ) C v m − , (B.5) xactly conserved quasilocal operators for the XXZ spin chain C = 1. Next, the equation for column r = m − v m − t m − ,m − + v m − t m − ,m − + v m − t m − ,m − = v m − . (B.6)Using Eqs. (B.3) and (B.6), we obtain v m − = t m − ,m − (1 − t m − ,m − ) C v m − , (B.7)with C = 1 − / (4 C ). In general, we find for r = 1 , . . . , m − v m − r = t m − r − ,m − r (1 − t m − r,m − r ) C r − v m − r − , (B.8)with C = 1 and C r = 1 − C r − , ≤ r ≤ m − . (B.9)This relation express C r as a continued fraction and the solution can be readily seen tobe C r = r + 22 r + 2 . (B.10)In addition, we can use the explicit expression for the matrix elements of T in Eq.(A.1), which gives t r,r +1 − t r,r = sin λr − cos λr = 12 . (B.11)Thus Eq. (B.8) simplifies to v m − r = rr + 1 v m − r − , r = 2 , . . . , m. (B.12)Writing the coefficients v r , r = 1 , . . . , m −
1, in terms of v , we find v r = (cid:16) − rm (cid:17) v . (B.13)Finally, setting v = 1 we obtain the vector in Eq. (B.2) | L (cid:105) = m − (cid:88) r =0 (cid:16) − rm (cid:17) | r (cid:105) . (B.14) Appendix C. Calculating traces in the thermodynamic limit
Let M be an arbitrary m × m matrix (not necessarily Hermitean), where m is thedimension of the auxiliary space A . We want to compute tr { M } using the eigenvectorsof T . Since T is non-Hermitean, its right and left eignevectors are different: T | R j (cid:105) = λ j | R j (cid:105) , (C.1) (cid:104) L j |T = λ j (cid:104) L j | . (C.2)Nonetheless, the left and right eigenvalues are equivalent because T and ( T ) t have thesame characteristic polynomial [40]. xactly conserved quasilocal operators for the XXZ spin chain i.e. (cid:104) R j | R l (cid:105) (cid:54) = δ i,j . Let {| r (cid:105)} denote the orthonormal basis of vectors representing sites inthe auxiliary space. We can expand the vectors | r (cid:105) in the non-orthogonal eigenvectorbasis in the form | r (cid:105) = (cid:88) j V r,j | R j (cid:105) , (C.3) | r (cid:105) = (cid:88) j W r,j | L j (cid:105) . (C.4)The transpose of Eq. (C.4) yields (cid:104) r | = (cid:88) j (cid:104) L j | W r,j = (cid:88) j (cid:104) L j | W tj,r . (C.5)The inverse transformation reads | R i (cid:105) = (cid:88) r ( V − ) i,r | r (cid:105) , (C.6) | L i (cid:105) = (cid:88) j ( W − ) i,r | r (cid:105) . (C.7)It can be proved that the left eigenvectors are orthogonal to right eigenvectors withdifferent eigenvalues (and degenerate eigenvectors can be orthogonalized) [40], so that( W − ) t V − = D, (C.8)with D j,l = (cid:104) L j | R l (cid:105) = d j δ j,l . (C.9)Thus the inverse of D is also diagonal:( W t V ) j,l = 1 d j δ j,l . (C.10)The trace of M can be written asTr { M } = (cid:88) r (cid:104) r | M | r (cid:105) = (cid:88) r,j,l W tj,r V r,l (cid:104) L j | M | R l (cid:105) = (cid:88) j,l ( W t V ) j,l (cid:104) L j | M | R l (cid:105) = (cid:88) j (cid:104) L j | M | R j (cid:105) d j . (C.11)In our case M is a product of a large number ( ∼ O ( N )) of transfer matrices T onthe left and on the right. In the thermodynamic limit the trace is dominated by thecontributions from the eigenvector of T with eigenvalue λ j = 1: | R (cid:105) = | (cid:105) , (cid:104) L | = (cid:104) L | . (C.12) xactly conserved quasilocal operators for the XXZ spin chain N →∞ Tr { M } = (cid:104) L | M | R (cid:105) d = (cid:104) L | ˜ M | (cid:105)(cid:104) L | (cid:105) , (C.13)where ˜ M is obtained from M by dropping the factors of T N . Using eigenvectorsnormalized as in Eqs. (4.20), we can write simplylim N →∞ Tr { M } = (cid:104) L | ˜ M | (cid:105) . (C.14)The following relation is also useful: (cid:88) r | r (cid:105)(cid:104) r | = (cid:88) r,j,l V r,j W tl,r | R j (cid:105)(cid:104) L l | = (cid:88) j,l ( W t V ) l,j | R j (cid:105)(cid:104) L l | = (cid:88) j | R j (cid:105)(cid:104) L j | d j . (C.15)If there is a large number of T ’s on both sides we can project onto the eigenvectorswith eigenvalue λ j = 1lim n →∞ (cid:32) T n (cid:88) r | r (cid:105)(cid:104) r |T n (cid:33) = | (cid:105)(cid:104) L |(cid:104) L | (cid:105) = | (cid:105)(cid:104) L | . (C.16)We also need to calculate traces involving T . These are easier because T issymmetric and the eigenvector with eigenvalue 1 is simply | (cid:105) . Thus the trace of amatrix with O ( N ) factors of T can be reduced tolim N →∞ Tr { M } = (cid:104) | ˜ M | (cid:105) . (C.17)The equivalent of Eq. (C.16) for T islim n →∞ (cid:32) T n (cid:88) r | r (cid:105)(cid:104) r |T n (cid:33) = | (cid:105)(cid:104) | . (C.18) [1] B. Sutherland, Beautiful Models (World Scientific, Singapore, 2004).[2] V. E. Korepin, A. G. Izergin, and N. M. Bogoliubov,
Quantum inverse scattering method andcorrelation functions (Cambridge Univ. Press, Cambridge, 1993).[3] A. R. Chowdhury and A. G. Choudhury,