Phantom Bethe excitations and spin helix eigenstates in integrable periodic and open spin chains
PPhantom Bethe excitations and spin helix eigenstates in integrable periodic and open spin chains
Vladislav Popkov, Xin Zhang, and Andreas Kl¨umper Department of Physics, University of Wuppertal, Gaussstraße 20, 42119 Wuppertal, Germany
We demonstrate the existence of special phantom excitations for open and periodically closed integrablesystems at the example of the
XXZ
Heisenberg spin chain. The phantom excitations do not contribute tothe energy of the Bethe state and correspond to special solutions to the Bethe Ansatz equations with infinite“phantom” Bethe roots. The phantom Bethe roots lead to degeneracies between different magnetization sectorsin the periodic case and to the appearance of spin helix states (SHS), i.e. periodically modulated states of chiralnature in both open and closed systems. For the periodic chain, phantom Bethe root (PBR) solutions appearfor anisotropies ∆ = cosh η with exp( η ) being a root of unity, thus restricting the phenomenon to the criticalregion | ∆ | < . For the open chain, PBR solutions appear for any value of anisotropy, both in the critical and inthe non-critical region, provided that the boundary fields satisfy a criterion which we derive in this paper. Thereexist PBR solutions with all Bethe roots being phantom, and PBR solutions that consist of phantom roots aswell as regular (finite) roots. Implications of our results for an experiment are discussed. The spin-1/2
XXZ chain, being an integrable interactingmany-body system, is one of the best studied paradigmaticmodels in quantum statistical mechanics [1]. Despite its longhistory, the
XXZ model remains a source of inspiration andfascinating new progress, such as finding a set of quasi-localconserved quantities [2], calculations of finite temperaturecorrelation functions [3, 4], and making major contributionsto the theory of finite-temperature quantum transport [5].The quasi-particles’ rapidities are usually pairwise distinct[1, 6–8], ground-states are often realized by dense distribu-tions of rapidities which in this sense have similarities withsingle particle momenta of ideal Fermi systems. A properanalysis, however, of the statistics of integrable systems haslead to the concept of “fractional statistics” and the classifi-cation of spinons in the anti-ferromagnetic spin-1/2 Heisen-berg chain as semions [9]. Still, in interacting systems underspecific and quite relevant conditions, exotic excitations arepossible.In the present letter we unveil the existence of novel “phan-tom” excitations based on rapidities that contribute zero en-ergy, but finite momentum. Each of the “phantom” excitationscarries the same “quantum” of momentum. In this mannerthey are Bose-like in contrast to usual quasi-particles whichare Fermi-like, in the sense of carrying strictly different mo-menta. Adding extra phantom excitation does not alter the en-ergy but increases the total momentum and of course changesthe properties of the state.Phantom Bose-like quasi-particles can appear as the onlyexcitations in the system or they may coexist with regularFermi-like excitations. Although some fine-tuning is alwaysrequired, phantom excitations are not rare. For instance, inthe periodic
XXZ spin chain, they appear for commensuratevalues of the anisotropy, while in the open
XXZ model withboundary fields, they appear at any value of anisotropy.Bethe eigenstates of a system with phantom excitations arealways chiral and possess a nonzero magnetic current. If thereare only phantom excitations, the eigenstates are remarkablysimple and have the form of fully factorized states, the spinhelix states (SHS) (5). SHS are notably the simplest non-trivial eigenstates of both the periodic and the open
XXZ spin chains. In addition, SHS can be created by two coher-ent experimental protocols [10, 11] or be generated as a non-equilibrium steady state with the help of boundary dissipation[12, 13].Here we demonstrate the existence of phantom excitationsin integrable models and describe the respective novel solu-tions of Bethe ansatz equations, which we call phantom Betheroots (PBR). We introduce the PBR concept for the exampleof the periodic
XXZ spin chain. Next, we find criteria for thephantom Bethe roots’ existence for both periodic and openboundaries. We construct “phantom Bethe” eigenstates, anddiscuss the simplest experimental setup, where these statescan be realized.
Factorized eigenstates in the spectrum of the
XXZ modelat commensurate values of anisotropy.
We consider the
XXZ spin-1/2 Hamiltonian for periodic and open boundary condi-tions. For the periodically closed chain we have H XXZ = N (cid:88) n =1 h n,n +1 (∆) , (1) h n,n +1 (∆) = J (cid:2) σ xn σ xn +1 + σ yn σ yn +1 + ∆ (cid:0) σ zn σ zn +1 − I (cid:1)(cid:3) , (2)with boundary conditions (cid:126)σ N +1 ≡ (cid:126)σ . For the open chain wehave H XXZ = N − (cid:88) n =1 h n,n +1 (∆) + (cid:126)h (cid:126)σ + (cid:126)h N (cid:126)σ N , (3)where we allow for boundary fields (cid:126)h and (cid:126)h N on the firstand on the last sites. In both cases a shift − J ∆ in thenearest-neighbour interaction (1) is added for convenience.Both models (1), (3) possess the remarkable property of hav-ing commuting families of parameter-dependent transfer ma-trices, rendering the models integrable via the Bethe Ansatzmethod [6–8]. We parametrize the anisotropy ∆ of the ex-change interaction as ∆ = cos γ or ∆ = cosh η .We want to construct simple, factorized eigenstates of theHamiltonians and introduce to this end for each site the (un- a r X i v : . [ c ond - m a t . s t a t - m ec h ] F e b normalized) qubit state | y (cid:105) = (cid:18) e y (cid:19) . (4)The qubit state (4) with y = f + i F corresponds to afully polarized spin / pointing into the direction (cid:126)n =(sin θ cos F, sin θ sin F, cos θ ) with tan θ = e f . A site-factorized state, the so-called spin-helix state (SHS) [12, 14] | SHS ( y , ϕ ) (cid:105) = | y (cid:105) | y + i ϕ (cid:105) . . . | y + i( N − ϕ (cid:105) N , (5)subscripts indicating the site number, with fixed polar an-gle and uniformly increasing azimuthal angles becomes aneigenstate of the XXZ
Hamiltonian if: (i) the increase ϕ ofthe azimuthal angle is identical to ± γ , the parameter of theanisotropy ∆ = cos γ , and (ii) the boundary conditions canbe accounted for.Indeed, the bulk interaction of the XXZ
Hamiltonian ap-plied to any SHS state (5) results in 0 due to the followingeasily checkable “divergence” relation h (∆) | y (cid:105) ⊗ | y + i γ (cid:105) == | y (cid:105) ⊗ ( κσ z | y + i γ (cid:105) ) − ( κσ z | y (cid:105) ) ⊗ | y + i γ (cid:105) , (6) κ = i sin γ, valid for any γ . For the periodic model (1), the periodic clo-sure requirement is γN = 2 πm with m an integer. SHSeigenstates in the periodic case can exist only for anisotropy | ∆ | ≤ .The model with open boundary condition will be treated indetail further below. Here we would like to note already thatcondition (ii) on the boundary can be satisfied not only in thecase | ∆ | ≤ , but also for | ∆ | > . For | ∆ | > we may usethe expression (5) with the replacement ϕ = i η which resultsin a spin-helix state with fixed azimuthal angle and uniformlyincreasing polar angles, see (32). The eigenstate condition foran SHS is fulfilled, if the boundary interactions h l = (cid:126)h (cid:126)σ and h r = (cid:126)h N (cid:126)σ N satisfy h l | y (cid:105) = κ σ z | y (cid:105) + λ − | y (cid:105) , (7) h r | y N − (cid:105) = − κ σ z | y N − (cid:105) + λ + | y N − (cid:105) , (8)where y N − = y + i( N − γ . The corresponding (total)energy eigenvalue is E = λ − + λ + where λ ± appear in (7)and (8). Note that in the open spin chain case a condition onthe anisotropy ∆ like in the periodic closure case is absentand γ in (5) can be real or imaginary. The relations (7) and (8)impose a joint condition on the anisotropy and all boundaryparameters.The factorized SHS state is perhaps the simplest eigenstateof both the periodic and open XXZ spin chains. Only thefully polarized ferromagnetic state is simpler and included inthe limits
Re[ y ] → ±∞ . Yet the state (5) is quite nontrivial,and describes a “frozen” spin precession along the z -axis withperiod π/ϕ , see Fig. 1. Due to the chiral nature, the SHS carries a remarkably high magnetization current, of the orderof O (1) , i.e. not decaying in the thermodynamic limit: (cid:104) j z (cid:105) SHS = (cid:104) σ + n σ − n +1 − h.c. ) (cid:105) SHS = ± γ cosh (Re[ y ]) , where the sign ± corresponds to the choice ϕ = ± γ in (5).The very existence of an eigenstate (5) for the periodicspin chain, characterized by periodic modulations in the mag-netization profile seems to contradict well known facts: in-deed, due to U (1) symmetry, XXZ eigenvectors split inblocks with well defined values of the global magnetization S z = (cid:80) n (cid:104) σ zn (cid:105) . For any eigenvector belonging to a block withfixed S z the expectation values (cid:104) σ + n (cid:105) = (cid:104) σ − n (cid:105) = 0 vanish, andso do the expectations (cid:104) σ xn (cid:105) = (cid:104) σ yn (cid:105) = 0 .This paradox is resolved by the energetical degeneracy ofeigenstates with different values of the total magnetization S z . We will show this explicitly by use of the Bethe ansatz.We will show the existence of virtual phantom Bethe rootswhich render Bethe states from different blocks (actually,from blocks with all possible S z values!) energetically de-generate, and allow for superpositions of states from differentblocks yielding periodic magnetization profiles. In fact, thestate (5) is not an eigenstate of operator ˆ S z and therefore itcannot belong to a single block.For simplicity, we set in the following the overall couplingstrength of the bulk J = 1 . Phantom Bethe roots at commensurate anisotropies in peri-odic
XXZ chains.
The eigenstates and eigenvalues are givenin terms of rapidities µ j ( j = 1 , , . . . n ) whose total number n may take any value out of , , . . . N . For any solution ofthe Bethe Ansatz equations (BAE) sinh N ( µ j − i γ/ N ( µ j + i γ/
2) = n (cid:89) k (cid:54) = j sinh( µ j − µ k − i γ )sinh( µ j − µ k + i γ ) , (9)there is an eigenstate with energy eigenvalue E µ ,...,µ n = − n (cid:88) j =1 γ cosh(2 µ j ) − cos γ , (10)and Bethe eigenvector Ψ µ ,...µ n = B ( µ ) . . . B ( µ n ) | (cid:105) . (11)Here | (cid:105) = | ↑↑ . . . ↑(cid:105) is the reference state of fully polarizedspins and B ( µ ) is the creation operator of magnons [7], seealso Appendix A. Definition.
We shall call a Bethe root µ p satisfying (9), a phantom Bethe root, if it does not give a contribution to therespective energy eigenvalue (10) i.e. if
Re[ µ p ] = ±∞ . (12)The next Lemma affirms that such phantom Bethe roots doexist: Lemma 1:
For anisotropy γ = 2 πm/N with integer m there exist the following “phantom” solutions of the BAE(9) for any given n = 1 , , . . . Nµ p = ±∞ + i π pn , p = 1 , . . . n. (13)These distributions remind of the string solutions to the Betheansatz equations. Note however that (13) holds for any finitesystem size N with a total number n of roots appearing in theequidistant distribution with separation π/n . In general nγ isnot a multiple of π . Proof.
Assume µ j = ± µ ∞ + i πj/n , where µ ∞ has a largereal part which we let to ∞ when evaluating the LHS of theBethe ansatz equations. As γ = 2 πm/N the LHS of (9) be-comes LHS → e ∓ i γN = 1 . On the RHS the term µ ∞ dropsout leaving finite differences µ j − µ k = i π ( j − k ) /n . Denoting ω = e i π/n , we have RHS j = n (cid:89) k =1 k (cid:54) = j ω j − k e − i γ − ω − ( j − k ) e i γ ω j − k e i γ − ω − ( j − k ) e − i γ = n − (cid:89) k =1 ω k e − i γ − ω − k e i γ ω k e i γ − ω − k e − i γ = n − (cid:89) k =1 ω k e − i γ − ω − k e i γ − ω − k e i γ + ω k e − i γ = 1 . In the passage to the second line we first used that the set of ω j − k with k = 1 , ..., n (and (cid:54) = j ) is identical to the set of ω k with k = 1 , ..., n − as we have ω n = − (in fact occasionalminus signs may enter which however drop out in the ratioabove). Subsequently, in the denominator we substitute theindex k by n − k and use ω n − k = − ω − k . Phantom Bethe vectors
The Bethe vectors corresponding tothe PBR solution (13), under the conditions of Lemma 1, canbe explicitly constructed using (11). The two signs ± in (13)correspond to different Bethe vectors which upon normaliza-tion read |± , n (cid:105) = 1 n ! (cid:113)(cid:0) Nn (cid:1) N − (cid:88) k ,...,k n =0 e ± i γ ( k + ... + k n ) σ − k . . . σ − k n | (cid:105) ,n = 0 , , . . . , N. (14)Each multiplication by the B ( µ ) -operator in (11) adds a quasi-particle with momentum p ( µ ) where e i p ( µ ) = sinh( µ +i γ )sinh( µ − i γ ) .Within the standard picture [7, 8] quasi-particles obey a“Fermi rule”: all p ( µ j ) are usually different. This propertyis violated for phantom Bethe roots µ p for which all p ( µ p ) areexactly the same: either p ( µ p ) = + γ or p ( µ p ) = − γ depend-ing on the sign of singularity in (13). Repeated action of B in(11) generates “phantom” Bethe states (14) with “quantized”momenta ± nγ and zero energy for all magnetization sectors n , yielding the degeneracy of the eigenvalue E = 0 betweendifferent sectors. The dimension of the degenerate subspace,is deg = 2( N − N since the states | + , n (cid:105) , |− , n (cid:105) for n = 1 , , . . . N − are linearly independent and for n = 0 , N the states | + , n (cid:105) , |− , n (cid:105) coincide. The degeneracy between sectors with different magnetization leads to eigenstates withperiodic modulations in the density profile. Indeed, the SHS(5) with positive chirality and ϕ = + γ = 2 πm/N (cid:54) = π is alinear combination of phantom Bethe states | + , n (cid:105) , and SHS(5) with opposite chirality ϕ = − γ is a linear combination of |− , n (cid:105)| SHS ( y , ± πm/N ) (cid:105) = (cid:18) Nn (cid:19) / N (cid:88) n =0 e y n |± , n (cid:105) , (15)see Appendix E for the proof. A set of SHS (15) with different y form an alternative basis in the degenerate subspace with E = 0 . Finally, note that the states (14) are chiral, which isevidenced by nonzero expectation values of the magnetizationcurrent, see Appendix B, (cid:104)± , n | j z |± , n (cid:105) = ± (cid:0) N − n − (cid:1)(cid:0) Nn (cid:1) sin γ . (16) Mixtures of regular and phantom excitations for the peri-odic
XXZ model.
Now as we have shown the existenceof phantom Bethe roots in the
XXZ model at anisotropies ∆ = cos γ , γ = 2 πm/N , we shall show that phantom Betheroots can appear also alongside with usual finite Bethe roots,however for other special values of the anisotropy.Let us assume that within a sector of n flipped spins, thereexists a BAE solution with n phantom Bethe roots µ , ..., µ n and the remaining r = n − n Bethe roots are regular. Wedenote the regular roots as x , . . . , x r where x j = µ n + j . Letus consider separately the BAE (9) subsets for phantom µ p and for regular x j . For the phantom BAE subset, substituting(13) we obtain e i γ ( N − r ) = 1 , (17)since each factor of the RHS containing a mixed pair µ p , x j contributes a term exp(2i γ ) . The product over the remainingfactors of the RHS involving just phantom roots µ p results in +1 by precisely the reasoning that led to Lemma 1 . Thecriterion (17) fixes the anisotropy parameter to γ ( N − r ) = 2 πm, (18)while the BAE subset for regular roots simplifies to sinh N ( x k − i γ/ N ( x k +i γ/
2) = e ± γn r (cid:89) l (cid:54) = kl =1 sinh( x k − x l − i γ )sinh( x k − x l +i γ ) ,k = 1 , . . . , r, (19)which has the structure of the BAE of a twisted XXZ chain,because of the presence of a constant phase factor. The signs ± match the ± sign in (13). The number of solutions of (19)in the sector with n spins down and fixed sign in the exponentis (cid:0) Nn − n (cid:1) . Phantom excitations in the open
XXZ chain.
The energyof integrable open spin-1/2
XXZ
Hamiltonian with boundaryfields (3) is given by E µ ,...,µ N = N (cid:88) j =1 η cosh(2 µ j ) − cosh η + E , (20) E = − sinh η (coth α − + coth α + + tanh β − + tanh β + ) , (21)where the boundary fields are parametrized as (cid:126)h = sinh η sinh α − cosh β − (cosh θ − , i sinh θ − , cosh α − sinh β − ) , (22) (cid:126)h N = sinh η sinh α + cosh β + (cosh θ + , i sinh θ + , − cosh α + sinh β + ) . (23)and the Bethe roots µ j satisfy BAE of a somewhat bulky form,see Appendix F [15–17]. After some algebra, see Appendix E,we find that if the following condition on bulk and boundaryparameters ± ( θ + − θ − ) ≡ (2 M − N + 1) η + α − + β − + α + + β + mod 2 π i , (24)is satisfied with some integer M = 0 , , . . . , N − , each setof N Bethe roots contains n phantom Bethe roots of type (13), µ p = ∞ + i π pn , j = 1 , , . . . n, (25)where n takes one of two values n + = N − M and n − = M + 1 [17]. The remaining N − n Bethe roots x j (= µ n + j ) are regular and satisfy reduced BAE G ± ( x p − η ) sinh N ( x p + η ) G ± ( − x p − η ) sinh N ( x p − η ) = N − n ± (cid:89) q =1 q (cid:54) = p sinh( x p − x q + η )sinh( x p − x q − η ) ×× sinh( x p + x q + η )sinh( x p + x q − η ) , p = 1 , . . . , N − n ± , (26) G ± ( u ) = (cid:89) κ = ± sinh( u ∓ α κ ) cosh( u ∓ β κ ) , (27)while the total eigenvalue (20) has contributions from the reg-ular Bethe roots only. We like to note that (20),(21) holdliterally for case n = n + . For n = n − the + E contri-bution in (20) is to be replaced by − E , see Appendix E.. We find that the BAE (26) for n = N − M describes dim G + M = (cid:80) Mm =0 (cid:0) Nm (cid:1) Bethe states, while the remaining N − dim G + M eigenstates are contained in the other, comple-mentary BAE set for n = M + 1 [17]. Unlike in the periodicsetup, where some Bethe eigenstates contain PBR modes, andother eigenstates are fully regular, in open systems, satisfy-ing criterion (24), all N eigenstates include phantom Betheroots. Remarkably, exactly the same condition (24) appearsin [18–21] as a condition for the application of the Algebraic Bethe Ansatz, based on special properties of Sklyanin’s K -matrices. The BAE set (26) coincides with that found by analternative method [18, 19, 21].Now we focus on the simplest Bethe states, correspondingto all Bethe roots being phantom, n + = N , the respectiveenergy given by E in (21). We demonstrate that such “phan-tom” Bethe states are spin-helix states (5) with appropriatelychosen parameters. The phantom Bethe states for mixtures ofphantom and regular Bethe roots can be also obtained explic-itly. They retain qualitative chiral features of the spin-helixstates and are discussed in [17]. Phantom Bethe states: open
XXZ chain. Easy planeregime ∆ < . It is straightforward to verify that the state | SHS ( y , γ ) (cid:105) from (5), with Re[ y ] = β − and the phase Im[ y ] = π + i α − + i θ − (note that α − , θ − are imaginaryand β − = − β + are real to ensure hermiticity of H ), is aneigenstate of H . Indeed, one can check that Eqs. (7), (8) aresatisfied with λ ± = − sinh η (coth α ± − tanh β ± ) , so thatthis SHS is an eigenvector of (3) with eigenvalue λ − + λ + ,which coincides with the phantom Bethe vector eigenvalue E , given by (21). The magnetization profile of this SHS isgiven in Fig. 1, left panel. Unlike for the periodic chain, herethe eigenvalue E is generically non-degenerate. Simplest experimental setup.
Here we give an exampleof a possible experimental setup for the realization of a stableSHS. Consider the following Hamiltonian H = J (cid:32) N − (cid:88) n =1 h n,n +1 (∆) + σ x + cos Φ σ xN + sin Φ σ yN (cid:33) , (28)realizable for instance with the technique [10], where bound-ary fields have the same amplitude, lie in the xy plane, anddiffer by an azimuthal angle Φ . Using (6) one can verify thatfor each of the following values of the anisotropy: ∆ m = cos γ m , γ m = Φ + 2 πmN + 1 , m = 0 , , . . . N, (29)the SHS (5) with y = i γ m , ϕ = γ m becomes an eigenvectorof (28). The respective wave vector of the spin helix structureis given by γ m , see Fig. 1.Conversely, for fixed anisotropy ∆ = cos γ there exist twofine-tuned positions of the mismatch angle Φ ± = ± ( N + 1) γ ,for which two SHS with opposite chiralities (wave vectors ± γ ) become stable, i.e. become eigenstates of the Hamilto-nian (28). Remark.
Even if the desired boundary fields (28) cannot befaithfully realized in experiment, long-lived SHS can still berealized. The initial state should have the form of a spin-helixprepared in the xy -plane, with helix period λ SHS matchingthe z -anisotropy ∆ = cos γ via λ SHS = 2 πγ . (30)Indeed a state (5) | SHS ± (cid:105) ≡ | SHS (i F , ± γ ) (cid:105) with period(30) will remain invariant in the bulk and change initially onlyat the boundaries, since N − (cid:88) n =1 h n,n +1 (cos γ ) | SHS ± (cid:105) = ∓ i sin γ ( σ z − σ zN ) | SHS ± (cid:105) , (31)as follows from (6). The ends of the spin chain will thus playthe role of defects, and the state in the bulk will be alteredonly by propagation of the information from the boundaries.Thus the state can be fully destroyed only after times of or-der t = N a/v char , where v char is the sound velocity, N isthe number of spins in the chain and a is the lattice constant.For example, in experiment [10], the process of the expan-sion of the defect in the bulk can be monitored. On the otherhand, if the SHS period does not match the anisotropy via(30), then the initial SHS will be destroyed after times of order t = a/v char . So, on one hand, the effect is robust (w.r.t. thephase of the helix and chain length N ), and on the other hand,it is sensitive w.r.t. the matching condition for the anisotropy ∆ . This sensitivity can be used as a benchmark for calibrat-ing the anisotropy, or the wave-length of the produced SHS,or both. Phantom Bethe states: Easy axis ∆ = cosh η > . Choose ϕ = γ = i η in Eqs. (5-8), and consider the factorizedstate | SHS polar ( y , η ) (cid:105) = N − (cid:79) k =0 | i F + f − kη (cid:105) , (32) f = α − + β − , F = π + i θ − . We find that the Eqs. (7), (8) are satisfied with κ → − sinh η and λ ± = − sinh η (coth α ± + tanh β ± ) . Consequently, state(32) is an eigenstate of H with eigenvalue λ + + λ − = E .Thus, the state (32) is the phantom Bethe vector. It describesspins on the lattice with fixed azimuthal angle and changingpolar angle along the chain, as visualized in Fig. 1, lowerpanel. Unlike the “azimuthal” spin helix state (5), the “po-lar” SHS Eq. (32) carries no spin current, (cid:104) j z (cid:105) SHS polar = 0 . Discussion
We have described a novel type of excitationsin integrable systems, namely phantom Bethe excitations, andthe respective phantom Bethe roots (PBR) corresponding tounbounded rapidities. The existence criterion for these statesis formulated and depends on the boundary conditions of thesystem. Under this criterion a certain subset of Bethe roots islocated at infinity with relative positions at equidistant points.This resembles a perfect TBA string, but is of entirely differ-ent nature. The occurrence of the phantom roots and their dis-tribution is intrinsically related to the underlying chiral natureof the state.For models with U (1) symmetry the PBR are responsiblefor degeneracies between sectors with different total magne-tization, and lead to factorized spin helix eigenstates at cer-tain root of unity anisotropies. Remarkably, also for the open XXZ model the PBR related eigenstates have the form of
10 20 30 40 50 n - - 〈σ n x,y,z 〉 SHS
10 20 30 40 50 n - - 〈σ n x,y,z 〉 SHS polar
Figure 1. Components of local magnetization (cid:104) σ xn (cid:105) , (cid:104) σ yn (cid:105) , (cid:104) σ zn (cid:105) forSHS/phantom Bethe states versus site number n , for the easy plane(upper panel) and the easy axis case (lower panel), indicated withblack, red and blue points respectively. Upper panel:
SHS (5), thephantom Bethe eigenstate of (3) or (1) for ∆ < . Parameters: y =i γ + 1 / √ , ϕ = γ = 2 π/ . Curves connecting points serve asa guide for the eye. Lower panel: “Polar” SHS (32), the phantomBethe eigenstate of (3) for ∆ > . Parameters: y = i π/ , η =2 π/ , f = ( N/ η . spin helix states with winding polarization vector, in the easyplane regime, and the “polar angle”-version of the latter, in theeasy axis regime. Our results can be used for the generationof stable spin helix states in experimental setups where a DXXZ model can be realized [10].While our discussion was restricted to the
XXZ model, thePBR existence can be shown in other integrable models, e.g.in the spin-1 Fateev-Zamolodchikov model, see Appendix Dwith periodic boundary conditions, and arbitrary spin s gener-alizations of the abovesee Appendix E. Acknowledgments.
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The eigenstates of an integrable Hamiltonian, within the Algebraic Bethe Ansatz approach, are given by Ψ µ ,...µ n = B ( µ ) . . . B ( µ n ) | (cid:105) . (S1)Here | (cid:105) = | ↑↑ . . . ↑(cid:105) is the reference state of fully polarized spins and B ( µ ) is the creation operator of magnons. This operatoris found as a special element of the monodromy matrix [7] T ( µ ) = L N ( x ) L N − ( µ ) . . . L ( µ ) , (S2)where the L k are × matrices with elements acting nontrivially in the qubit quantum space of site k . Explicitly we have ( L k ) ( µ ) = a ( µ ) e k, + d ( µ ) e k, , ( L k ) ( µ ) = d ( µ ) e k, + a ( µ ) e k, , ( L k ) = σ + k , ( L k ) = σ − k , (S3)with ( e ij ) i (cid:48) j (cid:48) = δ i,i (cid:48) δ j,j (cid:48) , σ ± k = ( σ xk ± i σ yk ) and a ( µ ) = i sinh ( µ − i γ/ γ , d ( µ ) = i sinh ( µ + i γ/ γ . (S4)The 2-dimensional index space of the matrices T and L k is usually referred to as the auxiliary space. The monodromy matrixhas four operator valued elements T ( µ ) = (cid:18) A ( µ ) B ( µ ) C ( µ ) D ( µ ) (cid:19) , (S5)which act in the physical Hilbert space (cid:0) C (cid:1) ⊗ N of N qubits.The monodromy matrix T ( µ ) satisfies the Yang-Baxter equation which implies a set of quadratic relations for the operators A ( µ ) , B ( µ ) , C ( µ ) , D ( µ ) . Here we note the additional important properties A ( µ ) | (cid:105) = a ( µ ) N | (cid:105) , D ( µ ) | (cid:105) = d ( µ ) N | (cid:105) ,C ( µ ) | (cid:105) = 0 , B ( µ ) | (cid:105) (cid:54) = 0 . (S6)The trace of the monodromy matrix yields the transfer matrix T ( µ ) = A ( µ ) + D ( µ ) which is a generating family of the integralsof motion among which we have the momentum operator and the Hamiltonian itself. Due to the properties (S6) the referencestate | (cid:105) is a simple factorized eigenstate to T ( µ ) and of course to the Hamiltonian. Appendix B. Proof of Eq.(16)
We start with the definition j zn = 2( σ xn σ yn +1 − σ yn σ xn +1 ) = 4i( σ + n σ − n +1 − σ − n σ + n +1 ) . For a sector with p spins down we have (cid:104)− , p | σ + n σ − n +1 |− , p (cid:105) = N − (cid:88) k ,...,k p =0 k (cid:48) ,...,k (cid:48) p =0 e i γ ( k (cid:48) + ... + k (cid:48) p − k − ... − k p ) ( p !) (cid:0) Np (cid:1) (cid:104) k (cid:48) . . . k (cid:48) p | σ + n σ − n +1 | k . . . k p (cid:105) = p N − (cid:88) k ,...,k p =0 , (cid:54) = nk (cid:48) ,...,k (cid:48) p =0 e i γ ( k (cid:48) + ... + k (cid:48) p − n − k − ... − k p ) ( p !) (cid:0) Np (cid:1) (cid:10) k (cid:48) . . . k (cid:48) p | n + 1 , k . . . k p (cid:11) = p e i γ ( p !) (cid:0) Np (cid:1) ( p − N − (cid:88) k ,...,k p =0 (cid:54) = n,n +1 (cid:104) k . . . k p | k , . . . k p (cid:105) = p e i γ ( p !) (cid:0) Np (cid:1) (( p − (cid:18) N − p − (cid:19) = (cid:0) N − p − (cid:1)(cid:0) Np (cid:1) e i γ . Analogously, we obtain (cid:104)− , p | σ − n σ + n +1 |− , p (cid:105) = (cid:0) N − p − (cid:1)(cid:0) Np (cid:1) e − i γ Finally, (cid:104)− , p | j z |− , p (cid:105) = 4i (cid:0) N − p − (cid:1)(cid:0) Np (cid:1) (cid:0) e i γ − e − i γ (cid:1) = − (cid:0) N − p − (cid:1)(cid:0) Np (cid:1) sin γ, (S7)which is Eq. (16) for the minus sign case. Repeating the calculations for (cid:104) + , p | j z | + , p (cid:105) , we obtain (16) for the plus sign. Herewe have used that the states |± , p (cid:105) are normalized. This is shown by a calculation simpler than the above presented calculation,but using similar steps. Appendix C. Proof of Eq.(15)
Let us directly check Eq. (15), by comparing projections of the LHS and RHS of (15) on the full basis. Let us denote by | k k . . . k n (cid:105) = σ − k . . . σ − k n | (cid:105) where | (cid:105) = | ↑↑ . . . ↑(cid:105) is the reference state with all spins up. It is straightforward to check (cid:104) k k . . . k n | SHS ( y , γ ) (cid:105) = e y n e i γ ( k + ... + k n ) . On the other hand, using the rhs of (15) yields the same expression, N (cid:88) m =0 a m (cid:104) k k . . . k n | + , m (cid:105) = e y n n ! N − (cid:88) p p ...p n =0 e i γ ( p + ... + p n ) (cid:104) k k . . . k n | p p . . . p n (cid:105) = e y n (cid:88) p p ...p n e i γ ( p + ... + p n ) δ p ,k . . . δ p n ,k n = e y n e i γ ( k + ... + k n ) . The vectors | k k . . . k n (cid:105) for all possible sets of k j and all n form a full basis, thus (15) is proved. Appendix D. Fateev-Zamolodchikov model
The Fateev-Zamolodchikov model [22–24] is a spin- quantum chain with Hamiltonian H F Z = N (cid:88) n =1 h F Zn,n +1 ( η ) , N + 1 ≡ , (S8) sinh(2 η ) h F Zn,n +1 ( η ) = (cid:126)S n (cid:126)S n +1 − ( (cid:126)S n (cid:126)S n +1 ) +2 sinh η (cid:0) S zn S zn +1 − ( S zn S zn +1 ) +( S zn ) +( S zn +1 ) − I (cid:1) − η (cid:0) ( S xn S xn +1 + S yn S yn +1 ) S zn S zn +1 + h.c. (cid:1) , where S αn are spin- generators of SU(2), acting at site n . This Hamiltonian possesses the following analog of a SHS (5): | SHS ( r, F , ϕ ) (cid:105) = N − (cid:79) k =0 || F + kϕ, r, ϕ (cid:105)(cid:105) , (S9) || F, r, ϕ (cid:105)(cid:105) = e − F r r ϕ e F , where r is an arbitrary parameter (an analog of a polar angle), and F is an arbitrary initial phase. We establish the followingdivergence condition h F Z (i γ ) ( || F , r, γ (cid:105)(cid:105) ⊗ || F + γ, r, γ (cid:105)(cid:105) ) = || F , r, γ (cid:105)(cid:105) ⊗ ( V || F + γ, r (cid:105)(cid:105) ) − ( V || F , r, γ (cid:105)(cid:105) ) ⊗ || F + γ, r, γ (cid:105)(cid:105) ,V = i S z sin 2 γ . It follows, taking into account also h F Z ( − x ) = − h F Z ( x ) , that the state (S9) | SHS ( r, F , ± γ ) (cid:105) is an eigenstate to eigenvalue E = 0 of the Fateev-Zamolodchikov quantum spin chain H F Z = (cid:80) Nn =1 h F Zn,n +1 (i γ ) H F Z | SHS ( r, F , ± γ ) (cid:105) = 0 , m = 0 , , . . . N − , (S10)provided the periodic boundary conditions are satisfied γN = 2 πm .Like in the spin- / XXZ chain, the eigenstates | SHS ( r, F , ± γ ) (cid:105) are not the eigenstates of the z -component S z = (cid:80) n S zn of the total magnetization operator, while the Hamiltonian H F Z commutes with S z . Consequently, we propose that the spin-1helix eigenstates (S9) originate from the degeneracy of the energy eigenvalue in blocks with different values of magnetization S z . The corresponding multiplet of degenerate eigenstates is generated by phantom Bethe roots in the Fateev-Zamolodchikovmodel.Indeed, our conjecture can be verified by looking at the BAE for the closed FZ chain H F Z = (cid:80) Nn =1 h F Zn,n +1 (i γ ) , see [25–27] sinh N ( µ j − i γ )sinh N ( µ j + i γ ) = n (cid:89) k (cid:54) = j sinh( µ j − µ k − i γ )sinh( µ j − µ k + i γ ) , (S11) j = 1 , , . . . n , where the number n of flipped spins may take any value n = 0 , , . . . , N . The corresponding eigenvalue is E µ ,...,µ n = n (cid:88) j =1
2i sin(2 γ )cosh(2 µ j ) − cos γ . (S12)As above, we prove that the ansatz for the phantom Bethe roots (13) µ p = ±∞ + i πp/n solves the BAE (S11) for N γ = πm with integer m . The phantom BAE solution corresponds to the eigenvalue E = 0 as follows from (S12). The degeneracy of theeigenvalue E = 0 in blocks with different n gives rise to the existence of spin-1 helix eigenstates (S9), which are not eigenstatesof S z .Finally, | SHS ( r, F , ϕ ) (cid:105) with ϕ = ± γ will be an eigenstate of the open Fateev-Zamolodchikov chain with Hamiltonian H F Z ( γ ) + h l + h r and boundary fields h l , h r acting on spins , N respectively, provided that h l and h r satisfy analogs of (7),(8) h l || , r, ϕ (cid:105)(cid:105) = V || , r, ϕ (cid:105)(cid:105) + Λ l || , r, ϕ (cid:105)(cid:105) h r || ( N − γ, r, ϕ (cid:105)(cid:105) = − V || ( N − ϕ, r, ϕ (cid:105)(cid:105) + Λ r || ( N − ϕ, r, ϕ (cid:105)(cid:105) , with the energy eigenvalue E = Λ l + Λ r . Appendix E. Phantom Bethe roots in integrable models with arbitrary spin s Our PBR results easily extend to arbitrary spin s , with the respective BAE of the form [28] sinh N ( µ j − i sγ )sinh N ( µ j + i sγ ) = n (cid:89) k (cid:54) = j sinh( µ j − µ k − i γ )sinh( µ j − µ k + i γ ) , (S13) j = 1 , , . . . n ; n = 0 , , . . . sN Also for (S13) we have PBR solutions (13) for any n , for periodic boundary conditions sγN = 2 πm . In particular, for the spin s = 1 case the (S13) describe Bethe roots for the Fateev-Zamolodchikov model [22]. Other PBR examples can be found e.g. inother integrable systems with U (1) symmetry where analogs of SHS have been predicted [14]. Appendix F. Derivation of Eq. (24)
We consider an open spin-1/2
XXZ
Hamiltonian with boundary fields (3), which are parametrized through parameters α ± , β ± , θ ± as (cid:126)h = sinh η sinh α − cosh β − (cosh θ − , i sinh θ − , cosh α − sinh β − ) , (S14) (cid:126)h N = sinh η sinh α + cosh β + (cosh θ + , i sinh θ + , − cosh α + sinh β + ) . (S15)The system (3) is integrable [15]. Parametrizing ∆ = cosh η , all eigenvalues of the Hamiltonian (3) are classified by differentsets of N Bethe roots µ j , as E µ ,...,µ N = N (cid:88) j =1 η cosh(2 µ j ) − cosh η − sinh η (coth α − + coth α + + tanh β − + tanh β + ) , (S16)where the roots satisfy BAE of the form (see e.g. [15], p. 149, [16]) G ( − µ j − η ) Q ( − µ j − η )sinh(2 µ j + η ) sinh(2 µ j ) sinh N ( µ j + η ) + 4 G ( µ j − η ) Q ( µ j − η )sinh(2 µ j − η ) sinh(2 µ j ) sinh N ( µ j − η ) = 2 c, (S17) Q ( u ) = N (cid:89) k =1 sinh( u − µ k − η u + µ k − η ,G ( u ) = (cid:89) k = ± sinh( u − α k ) cosh( u − β k ) ,c = cosh [( N + 1) η + α − + β − + α + + β + ] − cosh( θ − − θ + ) , (S18) j = 1 , . . . N . Note that unlike in the periodic chain, here each Bethe vector is characterized by a set of N Bethe roots µ j .Let us assume that, out of N Bethe roots, N − M roots are phantom, µ p = ∞ + γ p , p = 1 , , . . . N − M , (S19)where γ j are some finite imaginary constants to be defined later. The more precise formulation of (S19) is µ j = µ ∞ + γ j with µ ∞ → ∞ . The remaining M Bethe roots x , . . . , x M where x k = µ N − M + k are supposed to be finite. Inserting (S19) into(S17), for ≤ j ≤ N − M we obtain e W N − M (cid:89) k =1 e γ k − γ j sinh( γ j − γ k + η ) + e − W N − M (cid:89) k =1 e γ k − γ j sinh( γ j − γ k − η ) = 2 c, (S20) W = η ( M + 1) + α − + β − + α + + β + . Let us use the ansatz γ k = i πk/ ( N − M ) , (S21)and denote ω = e i π/ ( N − M ) , so that ω N − M = − and e γ k = ω k . Then we can rewrite a part of (S20) as N − M (cid:89) k =1 e γ k − γ j (2 sinh( γ j − γ k + η )) = e η ( N − M ) N − M (cid:89) n =1 (cid:0) − ω n − e − η (cid:1) = e η ( N − M ) (cid:16) − e − η ( N − M ) (cid:17) = 2 sinh( η ( N − M )) , (S22)where we used the identity N − M (cid:89) n =1 (cid:0) − ω n − z (cid:1) = 1 − z N − M , (S23)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 right hand side of (S23) reduces to the term in brackets of line (S22). Analogously, we obtain N − M (cid:89) k =1 e γ k − γ j sinh( γ j − γ k − η ) = − η ( N − M )) . (S24)The LHS of (S20) can thus be rewritten as W sinh(( N − M ) η ) = 2 cosh( η ( N − M ) + W ) − η ( N − M ) − W ) = 2 c. (S25)Recalling the definition of W in (S20) and c (S18) we note that c = 2 cosh( η ( N − M ) + W ) − θ − − θ + ) . In order tosatisfy (S25) we must require cosh( θ − − θ + ) = cosh( η ( N − M ) − W ) , or ± ( θ + − θ − ) = (2 M − N + 1) η + α − + β − + α + + β + mod 2 π i . (S26)Therefore, under condition (24), N − M out of N Bethe roots in (S17) can be chosen phantom.To obtain the BAE for the M remaining finite roots x p = µ N − M + p , we substitute (25) into (S17) and take j > N − M . TheLHS of (S17) contains phantom terms, so that the finite constant c on the RHS of (S17) can be neglected. The leading ordergives the reduced BAE G ( x p − η ) sinh N ( x p + η ) G ( − x p − η ) sinh N ( x p − η ) = M (cid:89) q =1 q (cid:54) = p sinh( x p − x q + η )sinh( x p − x q − η ) sinh( x p + x q + η )sinh( x p + x q − η ) , p = 1 , . . . , M, (S27)i.e. (26) for N − n + = M , and the respective eigenvalues have contributions from the M finite Bethe roots only, E x ,...,x M = M (cid:88) p =1 η cosh(2 x p ) − cosh η − sinh η (coth α − + coth α + + tanh β − + tanh β + ) . (S28)In addition, we convinced ourselves that valid finite Bethe root solutions x p corresponding to normalizable eigenstates satisfythe selection rule sinh(2 x p ) (cid:89) q (cid:54) = p sinh( x p − x q ) sinh( x p + x q ) (cid:54) = 0 . Complementary BAE set for n − = M + 1 phantom and r − = N − n − regular Bethe roots. Apart from (S27) onecan obtain another set of “phantom” roots eigenvalues for the same Hamiltonian H satisfying (24). This other set will have r − regular Bethe roots and N − r − phantom Bethe roots, with r − = N − M − . To see this, note that exactly the same Hamiltonian H is produced by an alternative parametrization with α ± , β ± , θ ± replaced by α ± → − α ± ,β ± → − β ± , (S29) θ ± → i π + θ ± , which leave the boundary fields (cid:126)h ,(cid:126)h N invariant. Now, note that (24) will be mapped onto itself under substitutions (S29) and M → r − = N − M − . This entails a possibility of choosing ( N − r − ) BAE roots of the form µ p = ∞ + i πN − r − p, (S30)in the original general BAE set (S17), where substitutions (S29 ) are made.As a consequence, by substitutions (S29) and M → r − in (26) and (21), we obtain another set of BAE for r − regular (finite)roots and the respective eigenvalues. These are BAE (26) for r = r − = N − n − . Note also that the “bare” eigenvalue E0