Phantom Bethe roots in the integrable open spin 1/2 XXZ chain
PPhantom Bethe roots in the integrable open spin / XXZ chain
Xin Zhang, Andreas Kl¨umper, and Vladislav Popkov Department of Physics, University of Wuppertal, Gaussstraße 20, 42119 Wuppertal, Germany
We investigate special solutions to the Bethe Ansatz equations (BAE) for open integrable
XXZ
Heisenbergspin chains containing phantom (infinite) Bethe roots. The phantom Bethe roots do not contribute to the energyof the Bethe state, so the energy is determined exclusively by the remaining regular excitations. We rederive thephantom Bethe roots criterion and focus on BAE solutions for mixtures of phantom roots and regular (finite)Bethe roots. We prove that in the presence of phantom Bethe roots, all eigenstates are split between two invariantsubspaces, spanned by chiral shock states. Bethe eigenstates are described by two complementary sets of BetheAnsatz equations for regular roots, one for each invariant subspace. The respective “semi-phantom” Bethevectors are states of chiral nature, with chirality properties getting less pronounced when more regular Betheroots are added. For the easy plane case “semi-phantom” Bethe states carry nonzero magnetic current, and arecharacterized by quasi-periodic modulation of the magnetization profile, the most prominent example being thespin helix states (SHS). We illustrate our results investigating “semi-phantom” Bethe states generated by oneregular Bethe root (the other Bethe roots being phantom), with simple structure of the invariant subspace, in alldetails. We obtain the explicit expressions for Bethe vectors, and calculate the simplest correlation functions,including the spin-current for all the states in the single particle multiplet.
PACS numbers:
INTRODUCTION
Contemporary experimental techniques allow to realize almost perfect one-dimensional
XXZ spin chains with adjustableanisotropy [1]. The spin-1/2
XXZ chain, being an integrable interacting many-body system, is one of the best studied paradig-matic models in quantum statistical mechanics [2]. Ongoing advances make the
XXZ model a source of inspiration andfascinating new discoveries, such as finding a set of quasilocal conservation laws [3], calculating finite temperature correlationfunctions [4, 5], and giving a major contribution to the theory of finite-temperature quantum transport [6].In our letter [7] we have shown that SHS correspond to a novel type of Bethe roots, phantom “singular” Bethe roots in theBethe Ansatz equations. These exotic Bethe roots describe excitations, which do not contribute to the energy of the system.We show that the presence of such excitations leads to highly atypical chiral
XXZ eigenstates carrying nonzero spin currentsand exhibiting periodic modulations of the magnetization density profile. We have established a criterion for the presence ofphantom Bethe roots, for both periodic and open boundaries, and investigated phantom Bethe states formed exclusively fromphantom Bethe excitations.In the present manuscript we treat the most challenging case of an open
XXZ
Hamiltonian with non-diagonal boundaryfields. The Bethe Ansatz equations for the spectrum have been formally constructed [8, 9] for the general integrable open chain.The separation of variables method has been applied to this model in [10–12]. And under a certain compatibility condition theso-called alternative modified Algebraic Bethe Ansatz approach [13, 14] was successful. Despite these works little is knownabout the solutions to the eigenvalue equations and even less is known about the structure of the eigenstates. Here we demonstratethat in presence of any number of phantom Bethe roots the Hilbert space of the system is split into two blocks, invariant withrespect to the action of the Hamiltonian. The condition for the splitting coincides with the condition needed for the applicabilityof the alternative modified Algebraic Bethe Ansatz [13, 14].We refer to our finding as the “splitting theorem” which gives us a tool to study the structure of phantom Bethe states, be-longing to each invariant subspace, and to show that “semi-phantom” Bethe states retain the chiral character of “fully-phantom”spin-helix states, as long as the number of regular Bethe roots involved remains small in comparison to the system size. Inspecial cases the explicit form of phantom Bethe states and various observables including spin magnetization current can becalculated analytically.The plan of the paper is the following. After introducing the model we remind of the concept of phantom Bethe roots, andderive the phantom Bethe roots existence criterion. Next, we prove the theorem about the splitting of the Hilbert space into twoinvariant chiral subspaces, and describe the basis states spanning them. In the final part of the manuscript we use the gainedknowledge to investigate the phantom Bethe states belonging to the invariant subspace with dimension N + 1 where N is thelength of the XXZ spin chain. a r X i v : . [ c ond - m a t . s t a t - m ec h ] F e b PHANTOM BETHE ROOTS IN THE OPEN
XXZ
CHAIN
We consider the
XXZ spin-1/2 chain with open boundaries. The Hamiltonian reads H = N − (cid:88) n =1 h n,n +1 + h + h N , (1)with h n,n +1 = σ xn σ xn +1 + σ yn σ yn +1 + cosh ησ zn σ zn +1 − cosh η I, (2) h = sinh η sinh( α − )cosh( β − ) (cosh( θ − ) σ x + i sinh( θ − ) σ y + cosh( α − )sinh( β − ) σ z ) , (3) h N = sinh η sinh( α + )cosh( β + ) (cosh( θ + ) σ xN + i sinh( θ + ) σ yN − cosh( α + )sinh( β + ) σ zN ) . (4)Here α ± , β ± and θ ± are boundary parameters. The system (1) is integrable [8] and its exact solutions are given by the off-diagonal Bethe Ansatz (ODBA) method [8, 9, 15] and the separation of variables method [10–12]. All eigenvalues of theHamiltonian (1) are classified by different sets of N Bethe roots { µ j } , as E = N (cid:88) j =1 η cosh(2 µ j ) − cosh η + E , (5) E = − sinh η (coth( α − ) + tanh( β − ) + coth( α + ) + tanh( β + )) , (6)where the roots satisfy the BAE of the form [8, 15] G ( − µ j − η ) Q ( − µ j − η )sinh(2 µ j + η ) sinh N ( µ j + η ) + 2 G ( µ j − η ) Q ( µ j − η )sinh(2 µ j − η ) sinh N ( µ j − η ) = c sinh(2 µ j ) , j = 1 , . . . , N, (7) Q ( u ) = N (cid:89) k =1 sinh( u − µ k − η u + µ k − η ,G ( u ) = (cid:89) σ = ± sinh( u − α σ ) cosh( u − β σ ) ,c = cosh [( N + 1) η + α − + β − + α + + β + ] − cosh( θ − − θ + ) . (8)Note that unlike the periodic chain, here each eigenstate and the corresponding eigenvalue are characterized by a set of Betheroots { µ j } with strictly N members. Typically, it is taken for granted that all { µ j } Nj =1 are bounded, so that every Bethe rootgives a nonzero contribution to the energy (5). However, it was pointed out by us [7], that unbounded “phantom” solutions ofBAE (7) do exist, which lead to “phantom” excitations not contributing to the energy. For completeness, below we give thedefinition and the derivation of the phantom Bethe roots existence criterion. Definition.
We shall call a Bethe root µ p satisfying (7), a phantom Bethe root, if it does not give a contribution to the respectiveenergy eigenvalue (5) i.e. if
Re[ µ p ] = ±∞ . (9)We assume that, out of N Bethe roots, N − M roots are phantom, µ p = ∞ + γ p , p = 1 , , . . . N − M, (10)where γ p are some finite imaginary constants. The more precise formulation of (10) is µ p = µ ∞ + γ p with µ ∞ → ∞ . Theremaining M Bethe roots µ N − M +1 , µ N − M +2 , . . . µ N are supposed to remain finite. In this situation, the BAE decouple for thephantom roots and the regular roots. Inserting (10) into (7), for ≤ p ≤ N − M we obtain e W N − M (cid:89) k =1 e γ k − γ p sinh( γ p − γ k + η ) + e − W N − M (cid:89) k =1 e γ k − γ p sinh( γ p − γ k − η ) = 2 c, (11) W = ( M + 1) η + α − + β − + α + + β + . Let us use the Ansatz γ k = i πk/ ( N − M ) , (12)and denote ω = e i π/ ( N − M ) , so that ω N − M = − and e γ k = ω k . Then we can rewrite the first term on the LHS of (11) as N − M (cid:89) k =1 e γ k − γ p (2 sinh( γ p − γ k + η )) = e ( N − M ) η N − M (cid:89) n =1 (cid:0) − ω n − e − η (cid:1) = e ( N − M ) η (cid:16) − e − η ( N − M ) (cid:17) = 2 sinh(( N − M ) η ) , (13)where we used the identity N − M (cid:89) n =1 (cid:0) − ω n − z (cid:1) = 1 − z N − M , (14)as both sides are polynomials of degree N − M in z , share the same zeros and have identical -th order coefficient. For z = e − η the RHS of (14) reduces to the term in brackets of line (13). Analogously, we obtain N − M (cid:89) k =1 e γ k − γ j sinh( γ j − γ k − η ) = − N − M ) η ) . (15)The LHS of (11) can thus be rewritten as W sinh(( N − M ) η ) = 2 cosh(( N − M ) η + W ) − N − M ) η − W ) = 2 c. (16)Recalling the definition of W in (11) and c in (8) we note that c = 2 cosh(( N − M ) η + W ) − θ − − θ + ) . In order tosatisfy (16) we must require cosh( θ − − θ + ) = cosh(( N − M ) η − W ) , i.e. ± ( θ + − θ − ) = (2 M − N + 1) η + α − + β − + α + + β + mod 2 π i . (17)Therefore, under condition (17), N − M out of N Bethe roots in (7) can be chosen phantom. M naturally has the range ≤ M < N .To obtain the BAE for the M remaining finite roots x j = µ N − M + j , we substitute (10) into (7) and take j > N − M . TheLHS of (7) contains factoring divergent terms, so that the finite constant c on the RHS of (7) can be neglected. The leadingorder gives the final BAE [13, 16, 17] (cid:20) sinh( x j + η )sinh( x j − η ) (cid:21) N (cid:89) σ = ± sinh( x j − α σ − η )sinh( x j + α σ + η ) cosh( x j − β σ − η )cosh( x j + β σ + η )= M (cid:89) k (cid:54) = j sinh( x j − x k + η ) sinh( x j + x k + η )sinh( x j − x k − η ) sinh( x j + x k − η ) , j = 1 , . . . , M. (18)while the respective energy has contributions from the M finite Bethe roots only, E = M (cid:88) j =1 η cosh(2 x j ) − cosh η + E . (19)The Hamiltonian H is invariant upon the following substitutions α ± → − α ± β ± → − β ± (20) θ ± → i π + θ ± . Now Eq. (17) will be mapped onto itself under substitutions (20) and M → (cid:102) M = N − M − . Using substitutions (20) andletting M → (cid:102) M in (18) and (19), we obtain another set of BAE with (cid:102) M finite roots and the respective eigenvalues.Under condition (17) with ≤ M < N , there exist two sets of homogeneous BAE, one with N − M and the other with N − (cid:102) M phantom Bethe roots. The remaining M finite Bethe roots satisfy homogeneous BAE (18). The other homogeneousBAE with (cid:102) M finite Bethe roots can be obtained after the substitutions (20) and M → (cid:102) M [13, 18], namely, (cid:20) sinh( x j + η )sinh( x j − η ) (cid:21) N (cid:89) σ = ± sinh( x j + α σ − η )sinh( x j − α σ + η ) cosh( x j + β σ − η )cosh( x j − β σ + η )= (cid:102) M (cid:89) k (cid:54) = j sinh( x j − x k + η ) sinh( x j + x k + η )sinh( x j − x k − η ) sinh( x j + x k − η ) , j = 1 , . . . , (cid:102) M , (21)while the respective energy has contributions from the (cid:102) M = N − M − finite Bethe roots only, E = (cid:102) M (cid:88) j =1 η cosh(2 x j ) − cosh η − E . (22)We remark that by our initial assumption about the existence of some phantom Bethe roots among the total of N Bethe roots,the number M of regular roots in (17) naturally takes the values ≤ M < N . For condition (17) satisfied with M = N it hasbeen argued in [16] that the BAE set (18) alone yields the full spectrum (of course with all Bethe roots regular).For convenience, introduce the notation ∆ = cosh η = cos γ, η = i γ. When the constraint (17) holds, the hermiticity of the Hamiltonian requires in the case | ∆ | < (the easy plane regime) Re[ α ± ] = Re[ θ ± ] = Re[ η ] = 0 , Im[ β ± ] = 0 a nd β + = − β − , (23)and in the case ∆ > (the easy axis regime) Im[ α ± ] = Im[ β ± ] = Im[ η ] = 0 , Re[ θ ± ] = 0 a nd θ + = θ − mod 2i π. (24)Finally, for hermitian Hamiltonian (1) the sign on the left hand side of Eq. (17) can be switched, by a reparametrization α ± → − α ± , η → − η , which leaves the Hamiltonian invariant. Indeed, for | ∆ | < , the left hand side of Eq. (17) switches signunder the reparametrization using β + + β − = 0 , see (23). For | ∆ | > we have θ + = θ − from (24), so the sign on the left handside of Eq. (17) is irrelevant. Without loss of generality, we choose the “ + ” sign in (17), yielding θ + − θ − = (2 M − N + 1) η + α − + β − + α + + β + mod 2 π i . (25)Below we formulate our main result, demonstrating a splitting of the Hilbert space into two chiral invariant subspaces, if thecondition (25) is fulfilled. MIXTURE OF PHANTOM AND REGULAR ROOTS: SPLITTING OF THE HILBERT SPACE INTO TWO CHIRALINVARIANT SUBSPACES
Here we show that under the Phantom Bethe roots (PBR) criterion (25) the Hilbert space splits into two subspaces, invariantunder the action of the open
XXZ
Hamiltonian (1) and describe them.Introduce the parameters u = i π − θ + + 2 η η , v = 2 α + + 2 β + − i π η . (26)Define the following local vectors for each site nφ n ( x ) = (cid:16) , − e − ( u + x + n − N − η (cid:17) , (27) ˜ φ n ( x ) = (cid:18) e ( u + x + N − n − η (cid:19) . (28)Note the second component of these states depends on the position index n . Let us introduce two families of factorized statesparametrized by an integer number m : “descending” azimuthal shocks Φ + ( n , . . . n m ) = ⊗ n k =1 φ k ( v +2 m ) ⊗ n k = n +1 φ k ( v +2 m − . . . ⊗ Nk m +1 = n m +1 φ k m ( v ) , (29) ≤ n < n . . . < n m ≤ N, m = 0 , , . . . M, and “ascending” shocks Φ − ( n , . . . n m ) = ⊗ n k =1 ˜ φ k ( v − m ) ⊗ n k = n +1 ˜ φ k ( v − m +2) . . . ⊗ Nk m +1 = n m +1 ˜ φ k m ( v ) , (30) ≤ n < n . . . < n m ≤ N, m = 0 , , . . . (cid:102) M , where (cid:102) M = N − M − . Theorem.
The open
XXZ
Hamiltonian, satisfying PBR (25) with ≤ M < N is block-diagonalized into two comple-mentary invariant subspaces G + M and G − M of dimensions dim G + M = (cid:80) Mk =0 (cid:0) Nk (cid:1) and dim G − M = 2 N − dim G + M , respectively. G + M is spanned by a family of “descending” azimuthal shocks { Φ + ( n , . . . , n m ) } Mm =0 . Subspace G − M is spanned by a familyof “ascending” azimuthal shocks { Φ − ( n , . . . , n m ) } (cid:102) Mm =0 . The eigenvalues of H belonging to G + M are given by the BAE (35),while those belonging to G − M are given by (39).The proof of the “invariance property” of the subspaces G ± M for the theorem is given in the Appendix.For the rest of the theorem it remains to demonstrate that the eigenstates of H belonging to the invariant subspaces G + M and G − M are precisely those given by the BAE (18) for M and BAE (21) for (cid:102) M , respectively, leading further below to the sets ofBAE (35) and (39). Indeed, we observe that precisely the BAE (35) appear as consistency conditions when we construct theBethe eigenstates via a coordinate Bethe Ansatz for M = 1 and M = 2 , and for larger M > . The set of basis states generatedin the alternative modified Algebraic Bethe Ansatz approach [13, 14] is equivalent to that given by the theorem.Below, we demonstrate how this works for M = 1 , see section “Mixtures of phantom and regular roots: “semi-phantom”Bethe states”. Remark 1.
The theorem is valid for an arbitrary Hamiltonian H of type (1) satisfying (25), whether it is Hermitian or not. Forthe applications, we will consider hermitian H , i.e. with boundary parameters satisfying (23) or (24). Remark 2.
If one chooses M outside of the range [0 , N − in (25) or (17), a splitting of the Hilbert space will not occur.However, one can still argue that as a consequence, the whole spectrum will be governed by BAE of type (18) or (21) alone,with the total number of regular Bethe roots larger or equal to N, see elsewhere for details.The proof of the theorem is our main result.To evaluate observables in the phantom Bethe states we need further knowledge about the Bethe amplitudes. In the followingwe perform an exhaustive analysis of phantom Bethe states for the M = 0 and M = 1 cases. Similar results for (cid:102) M = 0 , holdafter a substitution of the boundary parameters.Below we shall explore the consequences of the theorem and construct phantom Bethe states belonging to simple invariantsubspaces, corresponding to a mixtures of regular and phantom Bethe excitations. SPIN HELIX STATES AS “PERFECT” PHANTOM BETHE STATES
By “perfect” phantom Bethe states we mean the phantom Bethe states consisting of exclusively infinite Bethe rapidities. Thiscase has been considered in detail in [7], and it corresponds to the choice M = 0 in the phantom Bethe roots existence criterionin (25).The invariant subspace G + M for M = 0 consists of a single state, the so-called spin helix state (SHS) (cid:104) SHS | = φ ( v ) φ ( v ) · · · φ N ( v ) , (31)which has the energy E given by (6). Another SHS corresponds to G − N − , with M = N − , or equivalently, (cid:102) M = 0 , | SHS (cid:105)(cid:105) = ˜ φ ( v ) ˜ φ ( v ) · · · ˜ φ N ( v ) , (32)which has the energy − E . Despite being factorized states, SHS are rather nontrivial states of chiral nature, characterized byperiodic modulations of the polarization and large O (1) magnetic current in the easy plane regime η = i γ . The magnetic currentsfor (31) and (32) are of opposite signs, reflecting the opposite chiralities, j zSHS = ± γ cosh ( β + ) , (33)with + and − corresponding to (31) and (32) respectively. Spin helix states can be prepared experimentally via coherent [1, 19]and dissipative protocols [20, 21]. Their chiral properties, and in particular their large O (1) current (33) make them very differentfrom typical eigenstates of many-body interacting systems. This fact leads to singular features in the magnetization current’sdependence on various system parameters in the proximity of “phantom Bethe roots” manifolds in the dissipative protocols, see[22, 23].In the following we describe generalizations of the spin helix states, and show that their distinct chiral features persist also inpresence of regular Bethe roots, as long as phantom Bethe roots are present.Before proceeding to the description of Bethe states corresponding to mixtures of phantom and regular Bethe roots, we rewritethe BAE in a momentum representation which will be convenient for the further analysis. XXZ
OPEN SPIN-1/2 CHAIN WITH PHANTOM ROOTS: MOMENTUM REPRESENTATION
Following the traditional coordinate Bethe Ansatz method, we use the single particle quasi-momentum p j related to x j by e ip j = sinh (cid:0) x j + η (cid:1) sinh (cid:0) x j − η (cid:1) . (34)The BAE (18) become e N p j (cid:89) σ = ± a σ − e ip j − a σ e ip j b σ − e ip j − b σ e ip j = (cid:89) σ = ± M (cid:89) k (cid:54) = j − η e ip j + e ip j +i σ p k − η e i σ p k + e ip j +i σ p k , j = 1 , . . . , M. (35)where a ± = sinh( α ± + η )sinh( α ± ) , b ± = cosh( β ± + η )cosh( β ± ) , (36)and the selection rules of Bethe roots are e ip j (cid:54) = e ± ip k , e ip j (cid:54) = ± , e ip j (cid:54) = e ± η . (37)The energy is given by E = 4 M (cid:88) j =1 (cos(p j ) − cosh η ) + E . (38)The second set of BAE (21) and the respective energy in terms of single particle quasi-momenta are given by e N p j (cid:89) σ = ± ˜ a σ − e ip j − ˜ a σ e ip j ˜ b σ − e ip j − ˜ b σ e ip j = (cid:89) σ = ± (cid:102) M (cid:89) k (cid:54) = j − η e ip j + e ip j +i σ p k − η e i σ p k + e ip j +i σ p k , j = 1 , . . . , (cid:102) M , (39) ˜ a ± = sinh( α ± − η )sinh( α ± ) , ˜ b ± = cosh( β ± − η )cosh( β ± ) , (40) E = 4 (cid:102) M (cid:88) j =1 (cos(p j ) − cosh η ) − E . (41)We will show that solutions of (35) and (39) constitute the complete set of eigenstates and eigenvalues in the case with phantomBethe roots present, i.e. under the criterion (25). MIXTURES OF PHANTOM AND REGULAR ROOTS: “SEMI-PHANTOM” BETHE STATES
Here we obtain generalizations of the spin-helix state (31) for the case when all but one Bethe root are phantom, i.e. there are N − infinite Bethe roots and one regular finite root. This situation arises at the manifold described by (25) for M = 1 .We shall call the corresponding Bethe states semi-phantom Bethe states or, with some abuse of notations, as phantom Bethestates, meaning that we consider cases where the main contribution to the eigenstate is due to phantom Bethe roots.Here we construct explicit phantom Bethe eigenstates for M = 1 .The basis of the invariant subspace G +1 , according to the theorem for the case M = 1 , is given by linearly independent vectors (cid:104) | , (cid:104) | , . . . , (cid:104) N | of the form (cid:104) n | = e nη φ ( v + ) · · · φ n ( v + ) φ n +1 ( v ) · · · φ N ( v ) , (42)where the prefactor e nη is introduced for convenience and x + = x + 2 . The states (cid:104) N | and (cid:104) | are both SHS of type (31), withthe same chirality, differing by an overall phase shift The generic state of the single particle multiplet (cid:104) n | is a state where thepieces of both SHS are joined together at the link n, n + 1 where an additional phase shift occurs.Any eigenstate of H belonging to the invariant subspace G +1 can be expanded as a linear combination of (cid:104) n | as (cid:104) Ψ | = N (cid:88) n =0 (cid:104) n | f n , (43)with the energy given by E = 4 cos(p ) − η + E . (44)where p satisfies the BAE (35) and E is given by (6). Obviously, (43) predicts the existence of N + 1 linearly independentBethe vectors.It is straightforward to verify that the action of (cid:104) n | H produces a linear combination of (cid:104) n − | , (cid:104) n | , and (cid:104) n + 1 | (see also theAppendix) (cid:104) n | H = 2 (cid:104) n − | + 2 (cid:104) n + 1 | − d (cid:104) n | , (cid:104) | H = − ˜ d (cid:104) | + d − (cid:104) | , (cid:104) N | H = ˜ d (cid:104) N | + d + (cid:104) N − | , (45) d = a + + b + + a − + b − , ˜ d = a + + b + − a − − b − ,d ± = 2 − a ± b ± . (46)Inserting the above equations into the eigenvalue problem (cid:104) Ψ | H = E (cid:104) Ψ | gives the following identities f = (2 cos(p) − a − − b − ) f , (47) (1 − a − b − ) f + f = 2 cos(p) f , (48) f n − + f n +1 = 2 cos(p) f n , ≤ n ≤ N − , (49) (1 − a + b + ) f N + f N − = 2 cos(p) f N − , (50) f N − = (2 cos(p) − a + − b + ) f N . (51)where a ± , b ± are given by (36), and p ≡ p .We use the Ansatz f n = A + e i n p + A − e − i n p , ≤ n ≤ N − ,f = 11 − a − b − ( A + + A − ) ,f N = 11 − a + b + (cid:0) A + e i N p + A − e − i N p (cid:1) , (52)which satisfies Eqs. (48) - (50). To satisfy (47) and (51), we need A + A − = − a − − e ip − a − e ip b − − e ip − b − e ip , (53) A − A + = − e N p a + − e ip − a + e ip b + − e ip − b + e ip . (54)Multiplying the above equations, we retrieve BAE (35) for M = 1 e N p a + − e ip − a + e ip b + − e ip − b + e ip a − − e ip − a − e ip b − − e ip − b − e ip = 1 . (55)which thus serves as a compatibility condition of Eqs. (53), (54). To sum up, the Bethe vectors of the single particle multiplet(invariant subspace G +1 ) are given by (43) with f , . . . , f N being simplified to f n = A + e i n p + A − e − i n p = sin( n p + α ) , n = 1 , . . . , N − , (56) f = 11 − a − b − sin α, f N = 11 − a + b + sin( N p + α ) ,α = 12i ln (cid:18) a − − e ip − a − e ip b − − e ip − b − e ip (cid:19) . (57)and p satisfying BAE (55). Special cases : For the generic case, the structure of the invariant subspace G +1 spanned by (cid:104) n | is a fully connected one;meaning that with a repeated action (cid:104) n | H , (cid:104) n | H , etc. on any basis vector (cid:104) n | one gets the full basis. Consequently, anyeigenvector from the single particle multiplet has nonzero components of all basis vectors, and all phantom Bethe eigenvectorsare given by the Eqs. (54) - (55). This remains true also for higher invariant subspaces G ± M which are generically fully connected.However, in special cases, the invariant subspaces G ± M (as well as higher G ± M ) may have additional internal structure, i.e. containinvariant subspaces of smaller sizes. This case of further partitioning, can already be illustrated on our example of G +1 , whichhas internal structure if one or both coefficients d ± in (46) vanish, as is illustrated below.( i ) d − = 0 . This happens for a − b − = 1 , for which a one-dimensional invariant subspace of G +1 containing just one state, thebasis state (cid:104) | , appears, see (45). Consequently, (cid:104) | itself is a Bethe eigenvector with eigenvalue − ˜ d , while the remaining N phantom Bethe eigenvectors have nonzero components of all basis vectors, given by the BAE which instead of N + 1 solutionsnow has N solutions and α = 0 .( ii ) d + = 0 . This happens for a + b + = 1 with similar structures as in the d − = 0 case. The SHS (cid:104) N | is an eigenstate of H with corresponding eigenvalue ˜ d . The other N eigenstates are given by Eqs. (52) , (55) and (56) with α = − N p .( iii ) d + = d − = 0 . When a ± b ± = 1 , i.e. cosh( α ± + β ± + η ) = 0 (see Eq. (93)), G +1 contains two one-dimensional invariantsubspaces, (cid:104) | and (cid:104) N | , as follows from (45). Consequently (cid:104) N | and (cid:104) | become the eigenvectors of H with eigenvalues ˜ d and − ˜ d , respectively. The remaining N − phantom Bethe states are given by BAE (55) which acquire the remarkably simple form e N p = 1 , (58)yielding N − real solutions p = mπN , m = 1 , . . . , N − , (59)where due to (37), p = 0 , π are not allowed. From Eqs. (53) - (57), we find α = 0 , f n = sin( n p) , n = 1 , . . . , N − ,f = lim a − b − → sin( α )1 − a − b − = sin(p)2 cos(p) − a − − b − ,f N = lim a + b + → sin( N p+ α )1 − a + b + = sin(( N − − a + − b + . (60) PROPERTIES OF PHANTOM BETHE VECTORS FOR M = 1 . Distribution of regular Bethe roots in the single particle multiplet.
As expected, there are N + 1 physical solutions of theBAE (35) for one quasi-momentum p ≡ p ( M = 1 ). These solutions for p and α are denoted by p j and α j with j = 1 , ..., N +1 .We refer to the sets { p j } N +1 j =1 and { α j } N +1 j =1 of solutions as the single particle multiplet.Note that for a hermitian Hamiltonian H it follows from (44) that p can be either real or purely imaginary. In the following weshall concentrate on the easy plane case | ∆ | < which is physically more interesting since it produces eigenstates with multiplewindings of the magnetization vector along the chain for large systems, see Fig. 3.It can be shown (see Appendix) that in the single particle multiplet { p j } N +1 j =1 there are , or purely imaginary p j solutionsdepending on the system size N and on the values a ± , b ± ; the remaining N + 1 , N or N − solutions are real.It can be argued that in the thermodynamic limit N (cid:29) the points τ j = e i p j for real p j densely populate the upper unitsemicircle. Let us order the real p m in order of increasing energy E m . One finds min j p j = | O (1 /N ) | and max j p j = π − | O (1 /N ) | . Thus, the energies of the real members of the single particle multiplet { p j } N +1 j =1 densely populate the interval ofenergies E j ∈ ( E − − , E − , see Fig. 1. Amplitudes of phantom Bethe vectors in the single particle multiplet: Standing waves structure of the coefficients f n . It can be shown that the distribution of the shock amplitudes f n near the lower part of the energy spectrum (inside the singleparticle multiplet) obeys Re[ f n ] ≈ ( − n sin( mπN ) , m = 1 , , , . . . valid for m/N (cid:28) . Likewise, the shock amplitudes f n in the upper part of the energy spectrum (inside the single particle multiplet apart from the states with exponentially decayingamplitudes f n ) obey Re[ f n ] ≈ sin( mπN ) , m = 1 , , , . . . where m/N (cid:28) respectively. The functions { f n } thus have the formof discrete standing waves, with knots near the “edges” n = 0 , N , see Fig. 2. On the contrary, imaginary solutions p j correspondto exponentially decaying amplitudes f n , see Fig. 2. Chirality of phantom Bethe vectors in the single particle multiplet: high current and modulations in the density profile
Before calculating explicit expressions for the magnetization current j z by using the explicit form of the Bethe function (43),let us make a rough estimate. The basis of the invariant subspace (42) consists of N + 1 basis vectors, {(cid:104) n |} . The two states (cid:104) | , (cid:104) N | are pure SHS. The expectation value of the spin current operator (cid:104) n | j zl | n (cid:105) for n = 0 , N does not depend on the site andis given by (33), or, in terms of boundary parameters, by j (0) z = 2 sin γ cosh ( β + ) . (61)The remaining basis states (cid:104) n | , i.e. (cid:104) | , (cid:104) | . . . (cid:104) N − | have a kink in the phase at the link n, n + 1 , where the qubit phasedifference flips sign. The expectation value of the spin operator (cid:104) n | j zl | n (cid:105) with j zl = 2( σ xl σ yl +1 − σ yl σ xl +1 ) depends on l : we find (cid:104) n | j zl | n (cid:105) = j (0) z for all links, except for the link l, l + 1 with kink: (cid:104) l | j zl | l (cid:105) = − j (0) z . The local current (cid:104) n | j zl | n (cid:105) averaged overall links is given by N − N − (cid:88) l =1 (cid:104) n | j zl | n (cid:105) = j (0) z (cid:18) − N − (cid:19) , n = 1 , . . . N − , (62) N − N − (cid:88) l =1 (cid:104) n | j zl | n (cid:105) = j (0) z , n = 0 , N. (63)Consequently, the local current averaged over all links and all the basis states of the G +1 single particle multiplet is (cid:104) j zlocal (cid:105) G +1 = 1( N + 1)( N − N − (cid:88) l =1 N (cid:88) n =0 (cid:104) n | j zl | n (cid:105) = j (0) z (cid:18) − N + 1 (cid:19) . (64)Analogously to (64), for phantom Bethe states with two regular Bethe roots M = 2 we obtain (cid:104) j zlocal (cid:105) G +2 = 1( N −
1) dim G +2 N − (cid:88) l =1 dim G +2 (cid:88) n =1 (cid:104) n | j zl | n (cid:105) = j (0) z (cid:18) − N + 1 + N (cid:19) = j (0) z (cid:18) − N + 1 (cid:19) + O (cid:18) N (cid:19) , (65)and so on.The quantity (cid:104) j zlocal (cid:105) G +1 from (64) can be regarded as a rough estimate for a typical current of the single particle multiplet;it cannot be precise since we made the equal amplitude assumption f n ≡ and in addition ignored the non-orthogonality ofthe basis states (cid:104) n | m (cid:105) (cid:54) = δ nm . However, it renders our idea that typical currents in the single particle multiplet can differ fromthe SHS current j (0) z at most by O (1 /N ) corrections, which are strictly negative, hence decreasing its amplitude. Moreover, thecalculations performed for special boundary parameters, confirm the estimate (64) even quantitatively, see Eq.(71).Using similar arguments for the magnetization profile, we conjecture that typical transversal magnetization components mustbe quasi-periodic. In fact this conjecture is confirmed by numerical simulations, see Fig. 3.Summarizing, remarkable qualitative chirality features of the SHS are conserved if a regular excitation is added ( M = 1 ).Quantitatively, they get only slightly distorted, with degree of distortion that can be quantified.Thus, all states of the single particle multiplet have distinct chiral features: quasi-periodicity of a magnetization profile andlarge magnetization current. This is especially evident for imaginary p j solutions, if the system admits any: in fact, in thiscase the major contribution to the Bethe state is given by either the pure chiral SHS (cid:104) | or the SHS (cid:104) N | , while the other statescontribute with exponentially decaying amplitudes f n .Let us illustrate the calculation of a physical observable for phantom Bethe states belonging to the single particle multipletat the example of the special case ( iii ) in the previous section. For a hermititian Hamiltonian in the easy plane regime (see Eq.(23)) we can satisfy the constraints a ± b ± = 1 with the following choice of the boundary parameters, without losing generality: β + = β − = 0 , α ± = − i γ + i π mod π i , θ − − θ + = i( N − γ mod 2 π i . (66)0In this special case, there are two SHS: (cid:104) | and (cid:104) N | and the current in these two states can be calculated exactly as j z = (cid:104) | j zl | (cid:105)(cid:104) | (cid:105) = (cid:104) N | j zl | N (cid:105)(cid:104) N | N (cid:105) = 2 sin γ. (67)For the remaining N − states, we know the Bethe roots p m = πm/N, m = 1 , . . . , N − from (58). After some tediouscalculations, we get the norm of the phantom Bethe vectors (cid:104) Ψ | Ψ (cid:105) = 2 N (cid:88) ≤ n ,n ≤ N f n f n ∆ | n − n | = 2 N − N (1 − ∆ )(1 + ∆ −
2∆ cos p m ) , ∆ = cos γ. (68)The explicit expression of the current is j z ( m ) = (cid:104) Ψ | j zl | Ψ (cid:105)(cid:104) Ψ | Ψ (cid:105) = 2 sin γ (cid:80) ≤ n ,n ≤ N f n f n ∆ | n − n | + 2 f − f ∆ − (cid:80) ≤ n ,n ≤ N f n f n ∆ | n − n | = 2 sin γ (cid:18) − − cos p m ) N (1 + ∆ −
2∆ cos p m ) (cid:19) , p m = πmN , m = 1 , . . . , N − , (69)where we emphasize the dependence of the current on the particular state by using the state’s ordinal number as argument in j z ( m ) . The expression (69) can also be applied to the other two SHS letting m = 0 , N . Suppose < γ < π . The respectiveSHS current j z (0) = j z ( N ) is maximal and the minimal value is among j z ( m ) and j z ( m +1) with mπN ≤ γ ≤ ( m +1) πN . In thecase N (cid:29) m j z ( m ) = j (0) z (cid:18) − N (cid:19) + o (cid:18) N (cid:19) . (70)We can calculate the average current of the single particle multiplet, (cid:104) j z (cid:105) G +1 = 1 N + 1 N (cid:88) m =0 j z ( m ) = j (0) z (cid:18) − N + 1 (cid:19) + O (cid:18) N (cid:19) , (71)in qualitative accordance with our naive estimate (64). The fact that corrections to the current are strictly negative originatesfrom the influence of the kinks in the states as explained in the paragraph following (61). The links with kinks sustain localcurrent of opposite sign, reducing the current amplitude. The invariant subspace for M = 0 contains one state (SHS) with nokinks and the respective SHS current j (0) z is maximal. The N + 1 -dimensional invariant subspace for M = 1 consists of stateswith or kink and the average current reduces by the fraction N +1 , see (71), (64). For M = 2 , the invariant subspace consistsof states with , or kinks, leading to a further decrease of the average current, as predicted by Eq.(65).We conclude that the inclusion of further regular Bethe roots (in case of larger M ) makes the chiral properties of the phantomBethe states less pronounced. However the average multiplet current can decrease significantly, only if typical multiplet basisstates contain sizable proportions of kinks, meaning M/N = O (1) . The accurate analysis of the quantity (71) for arbitrary M requires further investigation and is out of the present scope.A high average current is not the only chiral feature of phantom Bethe states. Another typical feature is the large periodicmodulation of the magnetization profile. We find that inclusions of regular Bethe roots distort the perfectly periodic spin helixstructure. The degree of distortion naturally depends on the number M of regular Bethe roots involved. If M/N (cid:28) , modula-tions of the magnetization profile are clearly visible for all members of the multiplet. We show typical magnetization profiles inFig. 3 for M = 1 .Our analytic results are fully confirmed by numerical simulations, done for large system size N . In Fig. 2 we show typicalamplitudes of phantom Bethe vectors for M = 1 . In Fig. 3 we show typical magnetization profiles. DISCUSSION
We have analyzed the integrable
XXZ
Heisenberg spin chain with open boundary conditions and have described a noveltype of solutions to the Bethe Ansatz equations containing phantom (infinite) Bethe roots, as well as regular (finite) Bethe roots.These solutions appear under condition (17) which leads to a complete decoupling of the Bethe equations for infinite and finiteBethe roots. Phantom Bethe roots do not contribute to the energy of the system, which in case of spin chains with periodicboundary condition leads to degeneracies of the energies [7] which we refer to as phantom excitations.1
20 40 60 80 100 m - - - - - E m Figure 1: Energy levels E m = E −
4∆ + 4 cos( p m ) plotted versus m of a single particle multiplet for M = 1 . Dotted lines show strict lowerand upper bounds E − − and E −
4∆ + 4 , E given by (6), for energies E m of real p m solutions. Two separate points representing thelargest energies correspond to two imaginary p m solutions. Parameters are: N = 100 , ∆ = cos( π/ . ≈ . , α + = 0 . , α − = 0 . , β + = − β − = 1 . . The structure of selected phantom Bethe states belonging to the single particle multiplet is shown in Fig. 2. Condition (17) has appeared in [9, 13, 16, 17] as a technical condition for the applicability of a variant of the Algebraic BetheAnsatz, based on special properties of Sklyanin’s K -matrices. In the present manuscript, we have unveiled its meaning as acondition for the splitting of the Hilbert space into two invariant chiral subspaces G ± M . The integer parameter ≤ M < N entering (17) determines the dimensions of the invariant subspaces, dim G + M = (cid:80) Mk =0 (cid:0) Nk (cid:1) and dim G − M = 2 N − dim G + M .The meaning of the key parameter M , and its dual (cid:102) M = N − M − , parametrizing the hyperplane (17) is twofold. On onehand, M is the maximal number of kinks in the chiral basis vectors (29) of G + M . Analogously, (cid:102) M is the maximal number ofkinks in the chiral basis vectors of G − M . On the other hand, the fulfillment of (17) entails a reduction of the number of regularBethe roots from N in the general “inhomogeneous” BAE to M or (cid:102) M in the reduced sets of BAE. The solutions to the two BAEsets give all Bethe eigenstates, which necessarily possess chiral properties since the nature of the invariant subspaces is chiral.Conversely, we understand condition (17) as a criterion for the occurrence of phantom Bethe roots [7] among the N initialBethe roots: the phantom Bethe roots decouple from the general “inhomogeneous” BAE, leading to the two (dual) homogeneousBAE sets for M or (cid:102) M regular Bethe roots. The BAE for the phantom Bethe roots are not trivially satisfied just by these rootslying at infinity. The equations require a specific arrangement of the roots in a string that is unrelated to the usual TBA strings.The appearance of invariant subspaces and the splitting of the set of eigenvectors into blocks is sowewhat similar to theoccurrence of invariant subspaces for the periodic XXZ spin chain with fixed values of magnetization. There are howevercrucial differences: in the periodic case there are N + 1 blocks, with magnetizations − N/ , − N/ , ..., N/ and BAE witha number of Bethe roots specific for each block, ranging from 0 to N/ for even N and to ( N − / for odd N .In contrast, in the open XXZ model fulfilling (17) with ≤ M < N all eigenstates split into just two blocks, with basisvectors that are chiral, which leads to two sets of BAE with a total number of M roots or (cid:102) M roots. The latter fact leads to highlyunusual properties of the respective eigenstates such as high magnetization currents and quasiperiodic magnetization profiles.The main result of this paper is the proof that in the open integrable XXZ spin chain, the occurrence of phantom Bethe rootsleads to the splitting of the Hilbert space into two chiral invariant subspaces. Based on the splitting, we are able to constructexplicit Bethe vectors, the eigenstates of the open
XXZ model with fine-tuned non-diagonal boundary fields, and investigatetheir properties to the same degree of detail as for the periodic spin chain with U (1) symmetry for a given magnetization sector.The phantom Bethe eigenstates in open systems are very unusual and carry distinct chiral properties. This is due to the underlyingchiral nature of the basis states, constituting the respective invariant subspaces. Our results can be used for the generation ofstable spin helix states in experimental setups allowing to realize a paradigmatic XXZ model with tunable anisotropy [1] asargued in [7].It would be interesting to extend our results to other integrable systems with phantom Bethe roots e.g. to the spin-1 Fateev-Zamolodchikov model with open boundary conditions, see [7].
ACKNOWLEDGMENTS
Financial support from the Deutsche Forschungsgemeinschaft through DFG project KL 645/20-1, is gratefully acknowledged.Xin Zhang thanks the Alexander von Humboldt Foundation for financial support. Xin Zhang would like to thank Yupeng Wang2
20 40 60 80 100 n - - - Re [ f n ]
20 40 60 80 100 n Re [ f n ] ,Im [ f n ]
20 40 60 80 100 n - - - Re [ f n ]
20 40 60 80 100 n - - - Re [ f n ] ,Im [ f n ]
20 40 60 80 100 n - - - Re [ f n ]
20 40 60 80 100 n - - - Re [ f n ] ,Im [ f n ] n - Re [ f n ]
90 92 94 96 98 100 n - - Re [ f n ] Figure 2: Amplitudes of the Bethe vectors f n in the expansion (cid:104) Ψ | = (cid:80) Nn =0 (cid:104) n | f n for selected states in the single particle multiplet Fig. 1,obtained by numerical solutions of BAE (55). Top three panels in the left column:
Phantom Bethe states with the three lowest energies E j of the single particle multiplet, (the energy increases from top to bottom). The thin lines connecting the points are guides for the eye. Weshow only Re[ f n ] in the graphs. Im[ f n ] looks similar, with a shift by a quarter of a period as in the right panel. Top three panels in the rightcolumn:
Phantom Bethe states with the three highest energies E j of the real p j solutions (the energy decreases from top to bottom). Large andsmall points correspond to Re[ f n ] and Im[ f n ] respectively. Bottom row:
Two phantom Bethe states corresponding to the two highest energylevels in the single particle multiplet, corresponding to imaginary p j solutions. The largest (the second largest) energy of the single particlemultiplet corresponds to the right (left) Panel in the bottom row. k - - 〈σ kX 〉 , 〈σ kY 〉 Figure 3: Transversal components of the magnetization profiles (cid:104) σ xk (cid:105) , (cid:104) σ yk (cid:105) (blue and yellow curves, respectively), for a typical phantomBethe state from G +1 (one regular Bethe root), versus site number k . Parameters: N = 29 , η = i γ = i π , ∆ = cos π ≈ . , α ± = − η, β ± = 0 , θ − = i π, θ + = i( π − Φ) , and Φ = ( N − γ . The energy of the chosen state is E m = − . and the correspondingcurrent is j z = 1 . , close to the asymptotic SHS current j SHS ≈ . . The dotted line shows the y -component of an “ideal chiral” SHSprofile for the same value of the anisotropy and is given for comparison. and Junpeng Cao for discussions. [1] N. Jepsen, et al. Spin transport in a tunable Heisenberg model realized with ultracold atoms.
Nature , 403 (2020).[2] M. Gaudin.
The Bethe Wavefunction (Cambridge University Press, 2014).[3] T. Prosen, E. Ilievski. Families of quasilocal conservation laws and quantum spin transport.
Phys. Rev. Lett. , 057203 (2013).[4] F. G¨ohmann, A. Kl¨umper, A. Seel. Integral representations for correlation functions of the XXZ chain at finite temperature.
J. Phys. A:Math. Gen. , 7625 (2004).[5] H. E. Boos, F. G¨ohmann, A. Kl¨umper, J. Suzuki. Factorization of multiple integrals representing the density matrix of a finite segment ofthe Heisenberg spin chain. J. Stat. Mech.
P04001 (2006).[6] B. Bertini, et al.
Finite-temperature transport in one-dimensional quantum lattice models. arXiv:2003.03334.[7] V. Popkov, X. Zhang, A. Kl¨umper. Phantom bethe excitations and spin helix eigenstates in integrable periodic and open spin chains;(letter) arxiv:???[8] Y. Wang, W.-L. Yang, J. Cao, K. Shi.
Off-Diagonal Bethe Ansatz for Exactly Solvable Models (Springer, 2015).[9] J. Cao, W.-L. Yang, K. Shi, Y. Wang. Off-diagonal Bethe ansatz solutions of the anisotropic spin-1/2 chains with arbitrary boundaryfields.
Nucl. Phys. B , 152 (2013).[10] G. Niccoli. Non-diagonal open spin-1/2 XXZ quantum chains by separation of variables: complete spectrum and matrix elements ofsome quasi-local operators.
J. Stat. Mech.
P10025 (2012).[11] S. Faldella, N. Kitanine, G. Niccoli. The complete spectrum and scalar products for the open spin-1/2 XXZ quantum chains with non-diagonal boundary terms.
J. Stat. Mech.
P01011 (2014).[12] N. Kitanine, J. M. Maillet, G. Niccoli. Open spin chains with generic integrable boundaries: Baxter equation and Bethe ansatz complete-ness from separation of variables.
J. Stat. Mech.
P05015 (2014).[13] J. Cao, H.-Q. Lin, K.-J. Shi, Y. Wang. Exact solution of XXZ spin chain with unparallel boundary fields.
Nucl. Phys. B , 487 (2003).[14] S. Belliard, R. Pimenta. Modified algebraic Bethe ansatz for XXZ chain on the segment – II – general cases.
Nucl. Phys. B , 527(2015).[15] X. Zhang, et al.
Bethe states of the XXZ spin-1/2 chain with arbitrary boundary fields.
Nucl. Phys. B , 70 (2015).[16] R. I. Nepomechie, F. Ravanini. Completeness of the Bethe Ansatz solution of the open XXZ chain with nondiagonal boundary terms.
J.Phys. A: Math. Gen. , 11391 (2003).[17] R. I. Nepomechie. Bethe ansatz solution of the open XXZ chain with nondiagonal boundary terms. J. Phys. A: Math. Gen. , 433(2003).[18] R. I. Nepomechie. Functional relations and Bethe Ansatz for the XXZ chain. J. Stat. Phys. , 1363 (2003).[19] S. Hild, et al.
Far-from-equilibrium spin transport in heisenberg quantum magnets.
Phys. Rev. Lett. , 147205 (2014).[20] V. Popkov, C. Presilla. Obtaining pure steady states in nonequilibrium quantum systems with strong dissipative couplings.
Phys. Rev. A , 022111 (2016).[21] V. Popkov, G. M. Sch¨utz. Solution of the Lindblad equation for spin helix states. Phys. Rev. E , 042128 (2017).[22] V. Popkov, T. Prosen, L. Zadnik. Exact Nonequilibrium Steady State of Open XXZ/XYZ Spin-1/2 Chain with Dirichlet BoundaryConditions. Phys. Rev. Lett. , 160403 (2020).[23] V. Popkov, T. Prosen, L. Zadnik. Inhomogeneous matrix product ansatz and exact steady states of boundary-driven spin chains at largedissipation.
Phys. Rev. E , 042122 (2020). Proof of Completeness
Here we prove that the basis of states spanning G + M and G − M is complete, i.e. H = G + M ⊕ G − M . (72)where H is the full Hilbert space. First we prove, that the vectors (cid:104) g + ,k | ∈ G + M , k = 1 , , . . . , dim G + M are all independentfor η (cid:54) = 0 , i π (for the cases η = 0 , i π , a jump φ n ( x ) → φ n ( x + 2) ≡ φ n ( x ) is trivial). We restrict our proof to the most“unfavourable”, extreme case of η = i π/ , β − = 0 under the hermiticity condition (23). For this special case, the property φ n ( x ) φ † n ( x + 2) = 0 renders all states (cid:104) g + ,k | ∈ G + M being pairwise orthonormal, (cid:104) g + ,k | g + ,l (cid:105) = 2 N δ kl (cid:104) g + ,k | g + ,k (cid:105) . Thus, evenin the most “unfavourable” setting all (cid:104) g + ,k | are independent, and the same is valid for the basis vectors | g − ,k (cid:105)(cid:105) spanning G − M .In the next step, we return to the general setup and show that any two vectors g − ∈ G − M and g + ∈ G + M are orthogonal. Definethe function y ( n, j n , k n ) as φ n ( v + 2 j n ) ˜ φ n ( v − k n ) = 1 − e y ( n,j n ,k n ) , y ( n, j n , k n ) = N − n − j n − k n . (73)When y ( n, j n , k n ) = 0 , the local vectors φ n ( v + 2 j n ) and ˜ φ n ( v − k n ) are orthogonal. From the definition of (29) and (30),any basis vector belonging to G + M is a tensor product of φ n ( v + 2 j n ) with ≤ j N ≤ j N − . . . ≤ j ≤ M and any basis vectorbelonging to G − M is a tensor product of ˜ φ n ( v − k n ) with ≤ k N ≤ k N − . . . ≤ k ≤ (cid:102) M . It is easy to find y ( n + 1 , j n +1 , k n +1 ) − y ( n, j n , k n ) = 0 , ± ,y ( N, j N , k N ) ≤ , y (1 , j , k ) ≥ . (74)so that y ( n, j n , k n ) = 0 holds at least for one point n ( ≤ n ≤ N ). It shows that any pair of vectors g − ∈ G − M and g + ∈ G + M is orthogonal.The dimension of G + M is equal to the total number of different tuples ( n , . . . , n k ) with ≤ n < n < . . . < n k ≤ N overall k = 0 , . . . M , which is given by dim G + M = M (cid:88) m =0 (cid:18) Nm (cid:19) = (cid:18) N (cid:19) + (cid:18) N (cid:19) + . . . (cid:18) NM (cid:19) . (75)Analogously, the dimension of G − M is dim G − M = (cid:102) M (cid:88) m =0 (cid:18) Nm (cid:19) = N (cid:88) n = M +1 (cid:18) Nn (cid:19) . (76)The sum of dimensions dim G + M + dim G − M = (cid:80) Nn =0 (cid:0) Nn (cid:1) = 2 N is identical to the dimension of the total Hilbert space H .Hence the basis vectors in (29) and (30) span G + M and G − M respectively, hence (72) is proved. The invariance property of G ± M Here we prove the theorem for arbitrary Hamiltonian H of type (1), satisfying (25), whether being hermitian or not. It is easyto prove that φ n ( x ) φ n +1 ( x ) h n,n +1 = sinh η φ n ( x ) σ zn φ n +1 ( x ) − sinh η φ n ( x ) φ n +1 ( x ) σ zn +1 , (77) φ n ( x + ) φ n +1 ( x ) h n,n +1 = sinh η φ n ( x + ) φ n +1 ( x ) σ zn +1 − sinh η φ n ( x + ) σ zn φ n +1 ( x ) , (78) φ ( x ) h = − sinh η cosh( ρ − +( v +2 M − x ) η )sinh( α − )cosh( β − ) φ ( x )+ sinh η sinh( α − )cosh( β − ) (cosh( α − )sinh( β − ) − sinh( ρ − +( v +2 M − x ) η )) φ ( x ) σ z , (79) φ N ( x ) h N = − sinh η cosh( ρ + +( x − v ) η )sinh( α + )cosh( β + ) φ N ( x )+ sinh η sinh( α + )cosh( β + ) (sinh( ρ + +( x − v ) η ) − cosh( α + )sinh( β + )) φ N ( x ) σ zN . (80)5Here we used the notation x + = x + 2 , x ++ = x + 4 , ρ ± = α ± + β ± . (81)From (77) - (80), it is obvious that Φ + ( n , . . . , n k ) H = C ( n , . . . , n k )Φ + ( n , . . . , n k ) + N (cid:88) n =1 C n ( n , . . . , n k )Φ + ( n , . . . , n k ) σ zn , (82)where C n ( n , . . . , n k ) , n = 0 , , . . . , N , are some constants. Using Eqs. (77) - (80) repeatedly and checking the detailedcoefficients, we can prove that the coefficient C n ( n , . . . , n k ) is zero when Φ + ( n , . . . , n k ) has some special structure which isrealized under any of the following conditions Φ + ( n , . . . , n M ) = φ ( v +2 M ) φ ( v +2 M ) · · · , i.e. ≤ n < n . . . < n M ≤ N, C ( n , . . . , n M ) = 0 , Φ + ( n , . . . , n k ) = · · · φ N − ( v ) φ N ( v ) , i.e. ≤ n < n . . . < n k ≤ N − , C N ( n , . . . , n k ) = 0 , Φ + ( n , . . . , n k ) = · · · φ n − ( x ) φ n ( x ) φ n +1 ( x ) · · · , i.e. n m ≤ n − , n m +1 ≥ n +1 , C n ( n , . . . , n k ) = 0 , Φ + ( n , . . . , n k ) = · · · φ n − ( x ++ ) φ n ( x + ) φ n +1 ( x ) · · · , i.e. n m = n − , n m +1 = n, C n ( n , . . . , n k ) = 0 . (83)We have two useful identities φ n ( x ) σ zn = − cosh η sinh η φ n ( x ) + e η sinh η φ n ( x + ) ,φ n ( x + ) σ zn = cosh η sinh η φ n ( x + ) − e − η sinh η φ n ( x ) . (84)Then, we obtain φ n ( x + ) φ n +1 ( x ) σ zn = cosh η sinh η φ n ( x + ) φ n +1 ( x ) − e − η sinh η φ n ( x ) φ n +1 ( x ) ,φ n ( x ) φ n +1 ( x ) σ zn = − cosh η sinh η φ n ( x ) φ n +1 ( x ) + e η sinh η φ n ( x + ) φ n +1 ( x ) ,φ n − ( x + ) φ n ( x ) σ zn = − cosh η sinh η φ n − ( x + ) φ n ( x ) + e η sinh η φ n − ( x + ) φ n ( x + ) ,φ n − ( x + ) φ n ( x + ) σ zn = cosh η sinh η φ n − ( x + ) φ n ( x + ) − e − η sinh η φ n − ( x + ) φ n ( x ) . (85)Let us extend the definition of Φ + ( n , . . . , n k ) as Φ + (0 , n , . . . , n k ) ≡ Φ + ( n , . . . , n k ) , ≤ n < . . . < n k ≤ N. (86)Using Eqs. (82), (85) and the notation (86), we can prove that Φ + ( n , . . . , n k ) H for ( n k − , n k ) (cid:54) = ( N − , N ) is a linearcombination of Φ + ( n , . . . , n k ) and Φ + ( n , . . . , n m ± , . . . , n k ) , m = 1 , . . . , k. Here the actions Φ + ( n = 0 , n . . . , n k ) H → Φ + ( n = 1 , n . . . , n k ) and Φ + ( n = 1 , n . . . , n k ) H → Φ + ( n = 0 , n . . . , n k ) represent the generation and annihilation of a kink at the left boundary respectively. Due to Eqs. (83) and (85), the followingunwanted structures will not appear Φ + ( . . . , n j , n j +1 = n j , . . . ) , Φ + ( n , . . . , n M +1 ) , ≤ n < n < . . . < n M +1 ≤ N. (87)In words, the Hamiltonian H acting on Φ + ( n , . . . , n k ) with ( n k − , n k ) (cid:54) = ( N − , N ) can not degenerate any state beyond ourbasis vectors { Φ + ( . . . ) } . Some nontrivial situations arise when H acts on Φ + ( n , . . . , n k − = N − , n k = N )Φ + ( . . . , N − , N ) H = · · · + F ( . . . , N − , N ) Φ + ( . . . , N, N ) , (88)where · · · on the RHS of (88) denotes a linear combination of some vectors belonging to Φ + ( . . . ) in (29) and F ( . . . , N − , N ) is a constant. These “unwanted” vectors { Φ + ( . . . , N, N ) } are defined by replacing φ N − ( v ++ ) φ N ( v + ) in Φ + ( . . . , N − , N ) with φ N − ( v ++ ) φ N ( v ++ ) . In fact Φ + ( . . . , N, N ) is not an extra independent vector. Following the method in (73) and (74)we can prove Φ + ( . . . , N, N ) is orthogonal to all the basis vectors in G − M and can be expanded in G + M basis. Thus the invarianceproperty of the subspace G + M is proved. Analogously, we prove the invariance property of the set G − M .6 The proof of Eqs. (47)-(51)
Using Eqs. (77) - (80) and (84), we find (cid:104) | H = (cid:18) sinh η cosh( ρ − )sinh( α − )cosh( β − ) − sinh η cosh( ρ + )sinh( α + )cosh( β + ) (cid:19) (cid:104) |− η cosh( ρ − + η )sinh( α − )cosh( β − ) (cid:104) | , (89) (cid:104) N | H = (cid:18) sinh η cosh( ρ + )sinh( α + )cosh( β + ) − sinh η cosh( ρ − )sinh( α − )cosh( β − ) (cid:19) (cid:104) N |− η cosh( ρ + + η )sinh( α + )cosh( β + ) (cid:104) N − | , (90) (cid:104) n | H = − (cid:18) sinh η cosh( ρ − )sinh( α − )cosh( β − ) + sinh η cosh( ρ + )sinh( α + )cosh( β + ) + 4cosh η (cid:19) (cid:104) n | +2 (cid:104) n − | + 2 (cid:104) n +1 | , n = 2 , . . . , N. (91)where ρ ± = α ± + β ± . We have two useful identities sinh η cosh( ρ ± )sinh( α ± ) cosh( β ± ) = a ± + b ± − η, (92) sinh η cosh( ρ ± + η )sinh( α ± ) cosh( β ± ) = a ± b ± − . (93)where a ± , b ± are given by (36). Using Eqs. (92), (93), we can rewrite Eqs. (89) - (91) in terms of a ± and b ± and then easilyobtain Eqs. (47) - (51). The solutions of BAE (35) with M = 1 Recall the BAE for the M = 1 case e N p (cid:89) σ = ± a σ − e ip − a σ e ip b σ − e ip − b σ e ip = 1 . (94)When M = 1 , we can treat e ip as an unknown parameter which satisfies the unary equation (94) with degree N +4 . It is easy toprove that e ip = ± are always two trivial solutions. Replacing e ip with e − ip , the BAE still holds which implies that e ip and e − ip are equivalent (see Eq. (38)). So we can summarize that the BAE (94) has N +1 independent valid solutions. If the Hamiltonian H is hermitian, p can be a real or purely imaginary number. Here we only consider the easy plane regime case with (23). Imaginary solutions
Introduce the following auxiliary function Y ( x ) = x N (cid:89) σ = ± ( x − a σ ) ( x − b σ ) − (cid:89) σ = ± (1 − a σ x ) (1 − b σ x ) . (95)The zero points of Y ( x ) correspond to the solution of BAE (94). Suppose < a − < < a + . We find that Y (1) = 0 , Y (0) = − , Y (cid:0) a − (cid:1) = a − N − (cid:89) σ = ± (1 − a + a σ )(1 − a + b σ ) . (96)When a + a − > , we find Y (cid:0) a − (cid:1) > and equation Y ( x ) = 0 has a solution in the interval (cid:0) , a − (cid:1) . When a + a − < , weconsider the derivative of Y ( x ) at the point x = 1 Y (cid:48) (1) = (cid:32) (cid:89) σ = ± (1 − a σ ) (1 − b σ ) (cid:33) (cid:32) N − (cid:88) σ = ± a σ + 1 a σ − − (cid:88) σ = ± b σ + 1 b σ − (cid:33) . (97)If Y (cid:48) (1) < , i.e. N > (cid:32) (cid:88) σ = ± a σ + 1 a σ − (cid:88) σ = ± b σ + 1 b σ − (cid:33) , (98)7the function Y ( x ) has a zero in the interval (cid:0) a − , (cid:1) .Then suppose that < a − < a + . We can prove that Y (0) < , Y (cid:0) a − (cid:1) > , Y (cid:0) a − − (cid:1) > , Y (1) = 0 . (99)Obviously, there exists a real solution in the interval (cid:0) , a − (cid:1) . If Y (cid:48) (1) > , i.e. the inequality (98) holds, there will be anotherreal solution in the interval (cid:0) a − − , (cid:1) . When N (cid:29) the purely imaginary solutions of (94) are very simple. We find e N p → (cid:40) , − < e ip < , ∞ , e ip < − or e ip > . So there should be a purely imaginary solution at the point e ip = a − σ + O (cid:0) N (cid:1) in case of a σ > or a σ < − . We can use asimilar method to analyze the distribution of real solutions in other cases. Real solutions
Suppose that p is a real number p = (cid:15) with (cid:15) being a small positive real number, then we can prove a σ − e ip − a σ e ip = a σ − (1+i (cid:15) + · · · )1 − a σ (1+i (cid:15) + · · · ) = − a σ + 1 a σ − (cid:15) + · · · , (100)with (cid:12)(cid:12)(cid:12) i (cid:15) − a σ (cid:12)(cid:12)(cid:12) (cid:28) . If (cid:12)(cid:12)(cid:12)(cid:12) i (cid:15) − a σ (cid:12)(cid:12)(cid:12)(cid:12) (cid:28) , (cid:12)(cid:12)(cid:12)(cid:12) i (cid:15) − b σ (cid:12)(cid:12)(cid:12)(cid:12) (cid:28) , σ = ± , p = (cid:15) = mπN + O (cid:0) N (cid:1) is a solution of BAE (94) provided that < mN (cid:28) .Now we suppose that p is a real number p = π − (cid:15) with (cid:15) being a small positive real number, then we find a σ − e ip − a σ e ip = a σ + (1 − i (cid:15) + · · · )1 + a σ (1 − i (cid:15) + · · · ) = 1 + i a σ − a σ + 1 (cid:15) + · · · , (101)with (cid:12)(cid:12)(cid:12) i (cid:15) a σ (cid:12)(cid:12)(cid:12) (cid:28) . If (cid:12)(cid:12)(cid:12)(cid:12) i (cid:15) a σ (cid:12)(cid:12)(cid:12)(cid:12) (cid:28) , (cid:12)(cid:12)(cid:12)(cid:12) i (cid:15) b σ (cid:12)(cid:12)(cid:12)(cid:12) (cid:28) , σ = ± , p = π − (cid:15) = π − πmN + O (cid:0) N (cid:1) is a solution of BAE (94) provided that < mN (cid:28) .Let p = − i η = γ , the LHS of (94) thus becomes e Nη − η − α + − α − . Obviously, p = γ is not a solution in the general case.However, in the thermodynamic limit N → ∞ , we can always find an integer m to ensure η = i πm + α + + α − N − O (cid:18) N (cid:19) , N → ∞ , m → ∞ . (102)So p = γ + O (cid:0) N (cid:1) becomes a solution of (94) in the thermodynamic limit. M = 1 current: General case Consider a hermitian Hamiltonian in the easy plane regime (23) with α ± , θ ± and η = i γ being purely imaginary and β + = − β − being real. The norm of the eigenvector (cid:104) Ψ | is (cid:104) Ψ | Ψ (cid:105) = (cid:0) e − β + (cid:1) N (cid:88) n ≤ m (cid:16) f n f ∗ m ˜ b m − n + + f ∗ n f m b m − n + (cid:17) − (cid:88) n f n f ∗ n , (103)where f ∗ n is the complex conjugate of f n and b + ( x ) and ˜ b + ( x ) are defined in (36) and (40) respectively. Then, the current canbe obtained by j z = (cid:104) Ψ | j zl | Ψ (cid:105)(cid:104) Ψ | Ψ (cid:105) = 2 sin γ cosh ( β + ) (cid:104) Ψ | Ψ (cid:105) − N (cid:88) n =0 f n f ∗ n + (cid:88) n ≤ n