Cylinder partition function of the 6-vertex model from algebraic geometry
Zoltan Bajnok, Jesper Lykke Jacobsen, Yunfeng Jiang, Rafael I. Nepomechie, Yang Zhang
CCERN-TH-2020-019, UMTG-303, USTC-ICTS/PCFT-20-04
Cylinder partition function of the 6-vertex model fromalgebraic geometry
Zoltan Bajnok , Jesper Lykke Jacobsen , , , Yunfeng Jiang ,Rafael I. Nepomechie , Yang Zhang , Wigner Research Centre for Physics, Konkoly-Thege Mikl´os u. 29-33, 1121 Budapest, Hungary Institut de Physique Th´eorique, Paris Saclay, CEA, CNRS, 91191 Gif-sur-Yvette, France Laboratoire de Physique de l’ ´Ecole Normale Sup´erieure, ENS, Universit´e PSL,CNRS, Sorbonne Universit´e, Universit´e de Paris, F-75005 Paris, France Sorbonne Universit´e, ´Ecole Normale Sup´erieure, CNRS,Laboratoire de Physique (LPENS), F-75005 Paris, France CERN Theory Department, Geneva, Switzerland Physics Department, P.O. Box 248046, University of Miami, Coral Gables, FL 33124 USA Peng Huanwu Center for Fundamental Theory, Hefei, Anhui 230026, China Interdisciplinary Center for Theoretical Study, University of Science and Technology of China, Hefei,Anhui 230026, China
Abstract
We compute the exact partition function of the isotropic 6-vertex model on a cylinder geom-etry with free boundary conditions, for lattices of intermediate size, using Bethe ansatz andalgebraic geometry. We perform the computations in both the open and closed channels. Wealso consider the partial thermodynamic limits, whereby in the open (closed) channel, theopen (closed) direction is kept small while the other direction becomes large. We computethe zeros of the partition function in the two partial thermodynamic limits, and comparewith the condensation curves. a r X i v : . [ h e p - t h ] J u l ontents Q -system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.3 Algebraic geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.3 Odd N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.4 BAE and Q -system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.5 Algebraic geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 A.1 Polynomial ring and ideal . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50A.2 Gr¨obner basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51A.3 Quotient ring and companion matrix . . . . . . . . . . . . . . . . . . . . . . 52
B More details on AG computation 54C The overlap (4.34) 55D The relation (4.38) 55 Parity of states with paired Bethe roots 57F Exact partition functions 57
F.1 N = 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58F.2 N = 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60F.3 N = 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62F.4 N = 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63F.5 N = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 Computing partition functions of integrable vertex models at intermediate lattice size is ahard problem. For small lattice size, the partition function can be computed simply bybrute force. For large lattice size, where the thermodynamic limit is a good approximation,various methods are available, including the Wiener-Hopf method [1–3], non-linear integralequations [4, 5] and a distribution approach [6]. At intermediate lattice size, brute force isno longer an option, and the thermodynamic approximation is inaccurate. In a previouswork [7], three of the authors developed an efficient method to compute the exact partitionfunction of the 6-vertex model analytically for intermediate lattice size. They considered the6-vertex model at the isotropic point on the torus, i.e. with periodic boundary conditionsin both directions. The method is based on the rational Q -system [8] and computationalalgebraic geometry (AG). The algebro-geometric approach to Bethe ansatz was initiatedin [9], with the general goal of exploring the structure of the solution space of Bethe ansatzequations (BAE) and developing new methods to obtain analytic results in integrable models.The simplest example for such a purpose is the BAE of the SU (2)-invariant Heisenberg XXXspin chain with periodic boundary conditions. It is an interesting question to generalize thesemethods to more sophisticated cases such as higher-rank spin chains, quantum deformationsand non-trivial boundary conditions.In the current work, we take one step forward in this direction and consider the partitionfunction of the 6-vertex model on the cylinder. Namely, we take one direction of the latticeto be periodic and impose free open boundary conditions in the other direction. This set-uphas several new features compared to the torus geometry already considered in [7].First of all, to consider open boundary conditions for the vertex model, we put the modelon a diagonal square lattice where each square is rotated by 45 ◦ , as is shown in figure 2.1.The partition function on such a lattice can be formulated in terms of a diagonal-to-diagonal transfer matrix [10], which does not commute for different values of the spectral parameter.Nevertheless, the R -matrix approach (the so-called Quantum Inverse Scattering Method) canbe applied by reformulating this transfer matrix in terms of an inhomogeneous double-row depend on the spectral parameter . As a result, the BAE depend on a freeparameter ; hence, the Bethe roots are functions of this parameter, instead of pure numbers.In general, this new feature makes it significantly more difficult to solve the BAE. However,in the algebro-geometric approach, there is no extra difficulty, because the computations arepurely algebraic and analytic – there is not much qualitative difference between manipulatingnumbers and algebraic expressions. Therefore, the AG computations can be adapted to caseswith free parameters straightforwardly, which further demonstrates the power of our method.Secondly, in the torus case, the computation of the partition function can be done intwo directions which are equivalent. For the cylinder case, however, the computations ofthe partition function in the two directions or channels are quite different. In the openchannel, we need to diagonalize the transfer matrix corresponding to open spin chains. Thepartition function is given by the sum over traces of powers of the open-channel transfermatrix, similarly to the torus case. In the closed channel, we diagonalize transfer matricescorresponding to closed spin chains, and the open boundaries become non-trivial boundarystates. The partition function is thus given by a matrix element, between boundary states,of powers of the closed-channel transfer matrix.For a given lattice size, the final results should be the same in both channels. Nevertheless,we may consider different limits. In the open (closed) channel, we can take the lattice size inthe open (closed) direction to be finite and let the other direction tend to infinity. This partialthermodynamic limit has been studied in the torus case [7]. In the cylinder case, there aretwo different partial thermodynamic limits (“long narrow straw” and “short wide pancake”),which we study in detail in this paper. In these limits, it follows from the Beraha-Kahane-Weiss theorem [14] that the zeros of the partition function condense on certain curves in thecomplex plane of the spectral parameter.The rest of the paper is structured as follows. In section 2, we give the set-up of thevertex model and its reformulation in terms of a diagonal-to-diagonal transfer matrix. Insections 3 and 4, we discuss the computation of the partition function in the two differentchannels, using Bethe ansatz and algebraic geometry. Section 5 is devoted to some generaldiscussions on the algebro-geometric computations for the BAE/QQ-relation with a freeparameter. In section 6 we present the partition functions which can be written in closedforms for any M or N in the two channels. These include M = 1 in the open channeland N = 1 , , M and N , where wecan perform the computation in both channels and make consistency checks.Some of the results we obtained are too large to be presented in the paper. They canalso be downloaded from the webpage (576 MB in compressed form): http://staff.ustc.edu.cn/~yzhphy/integrability.html We consider the 6-vertex model at the isotropic point on a (2 M + 1) × N medial lattice, forpositive integers M and N . We impose periodic boundary conditions in the vertical direction,and free boundary conditions in the horizontal direction; the geometry under considerationis a cylinder, as is shown in figure 2.1. The partition function on the lattice can be computedFigure 2.1: The 6-vertex model on a cylinder. In the open channel, there are 2 M + 1 sites inthe horizontal direction with free boundary conditions, and 2 N sites in the vertical directionwith periodic boundary conditions.in two different channels. Open channel.
In the open channel, we define the diagonal-to-diagonal transfer matrixt D ( u ) = ˇ R ( u ) ˇ R ( u ) · · · ˇ R M, M +1 ( u ) ˇ R ( u ) ˇ R ( u ) · · · ˇ R M − , M ( u ) , (2.1)shown in figure 2.2; by convention the direction of propagation (the “imaginary time” direc-tion) is upwards in our figures. The subscripts label the spaces being acted upon, and ˇ R is5igure 2.2: Diagonal-to-diagonal transfer matrixrelated to the standard R -matrix of the isotropic 6-vertex model byˇ R jk ( u ) = P jk R jk ( u ) , (2.2)where P is the permutation operator. Written explicitly, the R -matrix is given by R ( u ) = u + iP = a ( u ) 0 0 00 b ( u ) c ( u ) 00 c ( u ) b ( u ) 00 0 0 a ( u ) , (2.3)with the Boltzmann weights a ( u ) = u + i , b ( u ) = u , c ( u ) = i . (2.4)The partition function is given by Z ( u, M, N ) = tr (cid:2) t D ( u ) N (cid:3) . (2.5)For small values of M and N , the results can be directly computed by brute force from thedefinition, for example Z ( u, ,
1) = 2( u + 2 i ) ,Z ( u, ,
2) = 2 (cid:0) u + 8 iu − u − iu + 4 (cid:1) , (2.6) Z ( u, ,
2) = 2 (cid:0) u + 16 iu − u − iu + 268 u + 256 iu − u − iu + 16 (cid:1) . Closed channel.
In the closed channel, the graph is rotated by 90 ◦ degrees, as shown infigure 2.3. The rotated R -matrix which will be denoted by R c takes the following form R c ( u ) = b ( u ) 0 0 00 a ( u ) c ( u ) 00 c ( u ) a ( u ) 00 0 0 b ( u ) . (2.7)6igure 2.3: In the closed channel, there are 2 N sites in the horizontal direction with periodicboundary conditions, and 2 M +1 sites in the vertical direction with free boundary conditions.Notice that this R -matrix does not satisfy the Yang-Baxter equation. In the closed channel,the partition function is no longer given by a trace, since periodic boundary conditions arenot imposed in the vertical direction. Instead, the open boundary conditions give rise to non-trivial boundary states in the closed channel. The partition function in the closed channelis given by Z c ( u, M, N ) = (cid:104) Ψ | U † ˜t D ( u ) M | Ψ (cid:105) , (2.8)where U is the one-site shift operator U = P P · · · P N − , N , (2.9)and ˜t D ( u ) is defined as˜t D ( u ) = ˇ R c12 ( u ) ˇ R c34 ( u ) · · · ˇ R c2 N − , N ( u ) ˇ R c23 ( u ) ˇ R c45 ( u ) · · · ˇ R c2 N − , N − ( u ) ˇ R c2 N, ( u ) , (2.10)where ˇ R c ij ( u ) = P ij R c ij ( u ). The boundary state | Ψ (cid:105) is given by | Ψ (cid:105) = | ψ (cid:105) ⊗ N , | ψ (cid:105) = | ↑ (cid:105) ⊗ | ↓ (cid:105) + | ↓ (cid:105) ⊗ | ↑ (cid:105) , (2.11)where we have used the notation | ↑ (cid:105) ≡ (cid:18) (cid:19) , | ↓ (cid:105) ≡ (cid:18) (cid:19) . (2.12)7he result for the partition function of course does not depend on how we perform thecomputation, so we have Z ( u, M, N ) = Z c ( u, M, N ) . (2.13)To verify the correctness of our various computations (see below), we have explicitly checkedthis identity for small value of M and N .Our goal is to compute analytic expressions of Z ( u, M, N ) explicitly for different in-termediate values of M and N . When both M and N are large, the system can be wellapproximated by the computation in the thermodynamic limit. Here we instead focus onthe interesting intermediate case where we keep one of M , N to be finite (namely, the onethat determines the dimension of the transfer matrix) and the other to be large. For finite M ( ≤
10) and large N (around a few hundred to thousands), we perform the computationin the open channel using (2.5); whereas for finite N and large M , we work in the closedchannel using (2.8). We discuss the computation of the partition function in both channelsfrom the perspective of Bethe ansatz and algebraic geometry. In this section, we discuss the computation of the partition function in the open channelusing Bethe ansatz and algebraic geometry. Using this method, we are able to compute thepartition function for finite M ≤
10 and large N (ranging from a few hundred to thousands). In order to apply the R -matrix machinery, the first step is to re-express the diagonal-to-diagonal transfer matrix t D ( u ) (2.1) in terms of an integrable open-chain transfer matrixwith 2 M + 1 sites and with inhomogeneities { θ j } [11]. Let us definet( u ; { θ j } ) = tr a K + ( u ) T (2 M +1) a ( u ; { θ j } ) K − ( u ) (cid:98) T (2 M +1) a ( u ; { θ j } ) , (3.1)where the monodromy matrices are given by T ( l ) a ( u ; { θ j } ) = R a ( u − θ ) . . . R a l ( u − θ l ) , (cid:98) T ( l ) a ( u ; { θ j } ) = R a l ( u + θ l ) . . . R a ( u + θ ) . (3.2)For our isotropic problem, the K -matrices are simply K + ( u ) = K − ( u ) = I .The eigenvalues Λ( u ; { θ j } ) of the transfer matrix t( u ; { θ j } ) (3.1), which can be obtained8sing algebraic Bethe ansatz [11], are given byΛ( u ; { θ j } ) = 2( u + i )(2 u + i ) (cid:34) M +1 (cid:89) j =1 ( u − θ j + i )( u + θ j + i ) (cid:35) K (cid:89) k =1 ( u − u k − i )( u + u k − i )( u − u k + i )( u + u k + i )+ 2 u (2 u + i ) (cid:34) M +1 (cid:89) j =1 ( u − θ j )( u + θ j ) (cid:35) K (cid:89) k =1 ( u − u k + i )( u + u k + i )( u − u k + i )( u + u k + i ) , (3.3)where the { u k } are solutions of the BAE M +1 (cid:89) j =1 ( u k − θ j + i )( u k + θ j + i )( u k − θ j − i )( u k + θ j − i ) = K (cid:89) j =1; j (cid:54) = k ( u k − u j + i )( u k + u j + i )( u k − u j − i )( u k + u j − i ) . (3.4)The key point (due to Destri and de Vega [12]) is to choose alternating spectral-parameter-dependent inhomogeneities as follows θ j = θ j ( u ) = ( − j u , j = 1 , . . . , M + 1 . (3.5)One can then show [13] that the diagonal-to-diagonal transfer matrix t D ( u ) is given by t D ( u ) = 1 i M +1 ( u + 2 i ) t( u ; { θ j ( u ) } ) , (3.6)noting that half of the R -matrices become proportional to permutation operators, and thespin chain geometry is transformed into the vertex one, see Figure 3.4. Specifying in (3.3)-Figure 3.4: The R -matrices with zero argument act as permutation operators depicted withavoiding lines. This transforms the double-row transfer matrix of the spin-chain geometryinto the diagonal-to-diagonal transfer matrix of the boundary vertex model.(3.4) the inhomogeneities as in (3.5), it follows that the eigenvalues Λ D ( u ) of t D ( u ) are given We note that t D ( u ) does not commute with t D ( v ).
9y Λ D ( u ) = Λ D,K ( u ) = 1 i M +1 ( u + 2 i ) Λ( u ; { θ j ( u ) } ) (3.7)= ( u + i ) M K (cid:89) k =1 ( u − u k − i )( u + u k − i )( u − u k + i )( u + u k + i ) , (3.8)where the { u k } are solutions of the Bethe equations (cid:20) ( u k − u + i )( u k + u + i )( u k − u − i )( u k + u − i ) (cid:21) M +1 = K (cid:89) j =1; j (cid:54) = k ( u k − u j + i )( u k + u j + i )( u k − u j − i )( u k + u j − i ) . (3.9)Here k = 1 , . . . , K and K = 0 , , . . . , M . Note that the BAE (3.9) depend on the spectralparameter u , which is an unusual feature.We observe that t D ( u ) has su (2) symmetry (cid:104) t D ( u ) , (cid:126)S (cid:105) = 0 , (cid:126)S = M +1 (cid:88) j =1 12 (cid:126)σ j . (3.10)The Bethe states are su (2) highest-weight states, with spin s = s z = 12 (2 M + 1) − K . (3.11)For a given value of K , the corresponding eigenvalue therefore has degeneracy2 s + 1 = 2 M + 2 − K . (3.12)We conclude that the partition function (2.5) is given by Z ( u, M, N ) = M (cid:88) K =0 (cid:88) sol( M,K ) (2 M + 2 − K ) Λ D,K ( u ) N , (3.13)where Λ D,K ( u ) is given by (3.8). Here sol( M, K ) stands for physical solutions { u , . . . , u K } of the BAE (3.9) with 2 M + 1 sites and K Bethe roots. The number N ( M, K ) of suchsolutions has been conjectured to be given by [15] N ( M, K ) = (cid:18) M + 1 K (cid:19) − (cid:18) M + 1 K − (cid:19) . (3.14)In order to find the explicit expressions for the partition function (3.13), we need to find10he eigenvalues Λ D,K ( u ). They depend on the values of rapidities which are solutions ofthe BAE (3.9). We encounter two difficulties. Firstly, the solution set of the BAE (3.9)contains some redundancy, since not all solutions are physical; therefore one needs to imposeextra selection rules [15]. Secondly, generally Bethe equations are a complicated system ofalgebraic equations, which cannot be solved analytically. What is worse, our BAE (3.9)depend on a free parameter u , which means that the Bethe roots are functions of u , therebymaking the BAE even harder than usual to solve.In order to overcome these two difficulties, we need new tools, namely the rational Q -system and computational algebraic geometry. These methods have been applied successfullyin computing the torus partition function of the 6-vertex model [7]. The BAE can bereformulated as a set of QQ -relations, with appropriate boundary conditions [8]. The benefitof working with the Q -system is twofold. Firstly, it is much more efficient to solve the rational Q -system than to directly solve the BAE. Secondly, all the solutions of the Q -system arephysical, so there is no need to impose further selection rules [16, 17]. The rational Q -system, which was first developed for isotropic (XXX) spin chains with periodic boundaryconditions [8], was recently generalized to anisotropic (XXZ) spin chains and to spin chainswith certain open boundary conditions [17, 18]. We briefly review the Q -system for openboundary conditions in section 3.2.Turning to the second difficulty, finding all solutions of the BAE (or of the corresponding Q -system) is in general only possible numerically. However, it was realized in [9] that if thegoal is to sum over all the solutions of the BAE/ Q -system for some rational function f ( { u j } )of the Bethe roots, then it can be done without knowing all the solutions explicitly. The ideais based on computational algebraic geometry. The solutions of the BAE/ Q -system form afinite-dimensional linear space called the quotient ring . The dimension of the quotient ringis the number of physical solutions of the BAE/ Q -system. A basis of the quotient ring canbe constructed by standard methods using a Gr¨obner basis. Once a basis for the quotientring is known, one can construct the companion matrix for the function f ( { u j } ), which isa finite-dimensional representation of this function in the quotient ring. Taking the trace ofthe companion matrix gives the sought-after sum. For a more detailed introduction to thesenotions and explicit examples in the context of toroidal boundary conditions, we refer to theoriginal papers [7, 9] and the textbooks [19, 20].The same strategy can be applied to the open boundary conditions. The new featurethat appears in this case is the dependence on a free parameter u . While this creates extradifficulty for numerical computations, it does not cost more effort in the algebro-geometricapproach. The reason is that the constructions of the Gr¨obner basis, the basis for the quotientring and companion matrices are purely algebraic; and it does not make much qualitative11ifference whether we have to manipulate numbers or algebraic expressions. Q -system In this section, we review the rational Q -system for the SU (2)-invariant XXX spin chain withopen boundary conditions [18]. Let us first consider the BAE with generic inhomogeneities { θ j } (3.4) L (cid:89) l =1 ( u j − θ l + i )( u j + θ l + i )( u j − θ l − i )( u j + θ l − i ) = K (cid:89) k (cid:54) = j ( u j − u k + i )( u j + u k + i )( u j − u k − i )( u j + u k − i ) , (3.15)where L = 2 M + 1. For given value of L and K , we consider a two-row Young tableau withnumber of boxes ( L − K, K ). At each vertex of the Young tableau, we associate a Q -functiondenoted by Q a,s . The BAE (3.15) can be obtained from the following QQ -relations v Q a +1 ,s ( v ) Q a,s +1 ( v ) ∝ Q + a +1 ,s +1 ( v ) Q − a,s ( v ) − Q − a +1 ,s +1 ( v ) Q + a,s ( v ) , (3.16)where f ± ( v ) := f ( v ± i ), and the Q -functions Q a,s ( v ) are even polynomials of vQ a,s ( v ) = v M a,s + M a,s − (cid:88) k =0 c ( k ) a,s v k , (3.17)where M a,s is the number of boxes in the Young tableau to the right and top of the vertex( a, s ). The boundary conditions are chosen such that Q ,s = 1, Q ,s>K = 1 and Q , ( v ) = L (cid:89) j =1 ( v − θ j )( v + θ j ) ,Q , ( v ) = Q ( v ) = K (cid:89) k =1 ( v − u k )( v + u k ) . (3.18)Here Q , ( v ) is the usual Baxter Q -function, whose zeros are the Bethe roots. Comparingto the periodic QQ -relations [8], the main differences are an extra factor v that appears onthe left-hand side of (3.16), and the degree of the polynomial of Q a,s which is twice the onefor the periodic case. More details can be found in section 4.2 of [18].For the Bethe equations (3.9), corresponding to the alternating inhomogeneities (3.5), In practice, due to implementations of the algorithm in packages, the efficiencies for manipulating num-bers and algebraic expressions can be different.
12e simply have Q , ( v ) = (cid:2) ( v − u )( v + u ) (cid:3) M +1 . (3.19)To solve the Q -system, we impose the condition that all the Q a,s functions are polynomials.This requirement generates a set of algebraic equations called zero remainder conditions (ZRC) for the coefficients c ( k ) a,s . In principle, one can then solve the ZRC’s and find Q a,s , inparticular the main Q -function Q , . The zeros of Q , are the Bethe roots { u k } , which arefunctions of the parameter u .After finding the Q -functions, the next step is to find the eigenvalues Λ D (3.8), which interms of Q -functions are given simply byΛ D ( u ) = ( u + i ) M Q ( u − i ) Q ( u + i ) . (3.20)Plugging these into (3.13), we finally obtain the partition function. In this subsection, we give the main steps for the algebro-geometric computation of thepartition function:1. Generate the set of zero remainder conditions (ZRC) from the rational Q -system;2. Compute the Gr¨obner basis of the ZRC;3. Construct the quotient ring of the ZRC;4. Compute the companion matrix for the eigenvalues Λ D,K ( u ) (3.8) which will be denotedby T M,K ( u );5. Compute the matrix power of T M,K ( u ) and take the trace Z ( u, M, N ) = M (cid:88) K =0 (2 M + 2 − K ) tr [ T M,K ( u )] N . (3.21)Most steps listed above can be done straightforwardly, adapting the corresponding workingof [7]. The only step that requires some additional work is step 4. The variables of ZRC are c ( k ) a,s which are coefficients of the Q -functions. From these variables, it is easy to construct thecompanion matrix of the Q -function. For fixed M and K , we denote the companion matrix Recall that the argument of the double-row transfer matrix t in (3.6) is u , rather than u . Q M,K . To find the companion matrix of Λ D , which is essentially the companion matrixof Λ (3.20) up to some multiplicative factors, the most direct way is to use homomorphismproperty of the companion matrix and write T M,K ( u ) = ( u + i ) M Q M,K ( u − i ) Q M,K ( u + i ) , (3.22)where T M,K ( u ) is the companion matrix for Λ D ( u ) with fixed M and K . Unfortunately, thismethod involves taking the inverse of the matrix Q M,K ( u + i ) analytically, which can beslow when the dimension of the matrix is large.We find that a much more efficient way is to use the following T Q -relation u T ( u − i ) Q ( u ) = ( u + i ) (cid:2) ( u + i ) − ( z ) (cid:3) L Q ( u − i ) (3.23)+ ( u − i ) (cid:2) ( u − i ) − ( z ) (cid:3) L Q ( u + i ) . In our case, we need to take L = 2 M + 1 and z = u . To solve the T Q relation (3.23), wemake the following ansatz for the two polynomials T ( u ) = t L u L + t L − u L − + · · · + t , (3.24) Q ( u ) = u K + s K − u K − + · · · + s . Notice that Q ( u ) is an even polynomial and only even powers of u appear, which is not thecase for T ( u ). Plugging the ansatz (3.24) into (3.23), we obtain a system of algebraic equa-tions for the coefficients { t , t , · · · , t L , s , · · · , s K − } . In fact, solving these set of algebraicequations is yet another way to find the Bethe roots. For our purpose, we only solve theequation partially , namely we view { s , · · · , s K − } as parameters and solve { t k } in terms of { s j } . This turns out to be much simpler since the equations are linear. We find that t k ( { s j } )are polynomials in the variables { s j } . From ZRC and algebro-geometric computations, wecan find the companion matrix of s j which we denote by s j . Replacing s j by s j and theproducts by matrix multiplication in t k ( { s j } ), we find the companion matrix t k = t k ( { s j } ).Then the companion matrix of the eigenvalues of the transfer matrix is given by T M,K ( u ) = t L u L + t L − u L − + · · · + t . (3.25)More details on the implementation of the algebro-geometric computations are given inappendix B.Using the AG approach, we have computed the partition functions for M up to 6, with N up to 2048. We also calculated some partition functions with higher M and lower N . Theresults for 2 ≤ M, N ≤ Partition function in the closed channel
In this section, we compute the partition function in the closed channel. There are bothsimplifications and complications due to the presence of non-trivial boundary states. Indeed,the presence of boundary states imposes selection rules for the allowed solutions of the BAE.Firstly, it restricts to the states with zero total spin. This implies that the length of the spinchain must be even, which we denote by 2 N ; and the only allowed number of Bethe rootsis K = N . In contrast, for the periodic (torus) case [7], one must consider all the sectors K = 0 , , . . . , N . Moreover, the Bethe roots must form Cooper-type pairs (4.33), which leadsto significant simplification in the computation of the Gr¨obner basis and quotient ring.This simplification comes with a price. Recall that the partition function in the closedchannel takes the form of a matrix element given by (2.8). To evaluate this matrix element,we need the overlaps between the boundary states and the Bethe states. These overlaps area new feature, which is not present in the open channel. They are complicated functions ofthe rapidities, which makes the computation of the companion matrix more difficult. To compute the expression (2.8) for the partition function in the closed channel, the firststep is to rewrite ˜t D (2.10) in terms of integrable closed-chain transfer matrices. To this end,we observe that R c ( u ) (2.7) is related to R ( u ) by R c12 ( u ) = − σ z R (˜ u ) σ z = − σ z R (˜ u ) σ z , (4.1)where ˜ u is the ‘crossing transformed’ spectral parameter defined by˜ u = − u − i . (4.2)The corresponding “checked” R -matrices are therefore related byˇ R c12 ( u ) = − σ z ˇ R (˜ u ) σ z . (4.3)For later convenience, we define˜ V (1) = ˇ R c23 ( u ) ˇ R c45 ( u ) · · · ˇ R c2 N − , N − ( u ) ˇ R c2 N, ( u ) , ˜ V (2) = ˇ R c12 ( u ) ˇ R c34 ( u ) · · · ˇ R c2 N − , N ( u ) , (4.4)15n terms of which ˜t D (2.10) is given by˜t D ( u ) = ˜ V (2) ( u ) ˜ V (1) ( u ) . (4.5)It follows from (4.3) that ˜ V (1) ( u ) = ( − N Ω (2) V (1) (˜ u ) Ω (1) , ˜ V (2) ( u ) = ( − N Ω (1) V (2) (˜ u ) Ω (2) , (4.6)where Ω (1) = σ z σ z · · · σ z N − , Ω (2) = σ z σ z · · · σ z N , (4.7)and the V ( i ) are the same as the corresponding ˜ V ( i ) , but with R ’s instead of R c ’s: V (1) ( u ) = ˇ R ( u ) ˇ R ( u ) . . . ˇ R N − , N − ( u ) ˇ R N, ( u ) ,V (2) ( u ) = ˇ R ( u ) ˇ R ( u ) . . . ˇ R N − , N ( u ) . (4.8)Let us now introduce integrable inhomogeneous closed-chain transfer matrices of length 2 Nτ ( u ; { θ j } ) = tr a T (2 N ) a ( u ; { θ j } ) , (cid:98) τ ( u ; { θ j } ) = tr a (cid:98) T (2 N ) a ( u ; { θ j } ) , (4.9)where the monodromy matrices are defined in (3.2). Using crossing symmetry R (˜ u ) = − σ y R t ( u ) σ y , (4.10)where ‘ t ’ stands for transposition in the first quantum space, one can show that τ (˜ u ; { θ j } ) = (cid:98) τ ( u ; { θ j } ) . (4.11)The transfer matrices (4.9) can be diagonalized by algebraic Bethe ansatz. We define theoperators A , B , C , D as matrix elements of the monodromy matrix (3.2) as usual T (2 N ) a ( u − i ; { θ j } ) = (cid:32) A ( u ) B ( u ) C ( u ) D ( u ) (cid:33) , (4.12)but note the shift in the spectral parameter. We consider the reference state or pseudovacuum | (cid:105) = | ↑(cid:105) ⊗ N . (4.13)16he Bethe states and their duals are constructed by acting with B - and C -operators on thereference state | u (cid:105) = B ( u ) · · · B ( u K ) | (cid:105) , (cid:104) u | = (cid:104) | C ( u ) · · · C ( u K ) . (4.14)These states are eigenstates of the transfer matrix τ ( u ; { θ j } ) (4.9) τ ( u ; { θ j } ) | u (cid:105) = Λ c ( u ; { θ j } ) | u (cid:105) , (4.15)with eigenvalues Λ c ( u ; { θ j } ) given byΛ c ( u ; { θ j } ) = N (cid:89) j =1 ( u − θ j + i ) K (cid:89) k =1 u − u k − i u − u k + i + N (cid:89) j =1 ( u − θ j ) K (cid:89) k =1 u − u k + i u − u k + i , (4.16)provided that the rapidities u = { u , · · · , u K } satisfy the BAE N (cid:89) j =1 u k − θ j + i u k − θ j − i = K (cid:89) j =1; j (cid:54) = k u k − u j + iu k − u j − i . (4.17)In contradistinction with (3.9) these BAE do not depend on the spectral parameter u , as isusually the case. The eigenvalues (cid:98) Λ c ( u ; { θ j } ) of (cid:98) τ ( u ; { θ j } ) are given by (cid:98) Λ c ( u ; { θ j } ) = Λ c (˜ u ; { θ j } ) , (4.18)as follows from (4.11).To make contact with ˜t D , we again choose alternating spectral-parameter-dependentinhomogeneities θ j = θ j ( u ) = ( − j +1 u, j = 1 , . . . , N . (4.19)The V ’s (4.8) can then be related to the closed-chain transfer matrices (4.9) by V (2) ( u ) V (1) ( u ) = ( − N τ ( u ; { θ j ( u ) } ) (cid:98) τ ( u ; { θ j ( u ) } ) , (4.20)which is similar to (3.6), see Figure 4.5. In view of the relations (4.6) between ˜ V ’s and V ’s, It should be kept in mind that the Bethe states depend on the inhomogeneities { θ j } ; in order to lightenthe notation, this dependence is not made explicit. R -matrices with zero argument act as permutation operators depicted withavoiding lines. This transforms the product of periodic transfer matrices of the spin-chaingeometry into the periodic diagonal-to-diagonal transfer matrix of the vertex model.we conclude that ˜t D (4.5) is given by˜t D ( u ) = ˜ V (2) ( u ) ˜ V (1) ( u )= Ω (1) V (2) (˜ u ) V (1) (˜ u ) Ω (1) = ( − N Ω (1) τ ( ˜ u ; { θ j ( ˜ u ) } ) (cid:98) τ ( ˜ u ; { θ j ( ˜ u ) } ) Ω (1) . (4.21)The expression (2.8) for the partition function in the closed channel can therefore be recastas Z c ( u, M, N ) = ( − MN (cid:104) Ψ | U † Ω (1) (cid:2) τ ( ˜ u ; { θ j ( ˜ u ) } ) (cid:98) τ ( ˜ u ; { θ j ( ˜ u ) } ) (cid:3) M Ω (1) | Ψ (cid:105) = ( − ( M +1) N (cid:104) Φ | U † (cid:2) τ ( ˜ u ; { θ j ( ˜ u ) } ) (cid:98) τ ( ˜ u ; { θ j ( ˜ u ) } ) (cid:3) M | Φ (cid:105) , (4.22)where | Φ (cid:105) is the so-called dimer state | Φ (cid:105) = Ω (1) | Ψ (cid:105) = ( − N Ω (2) | Ψ (cid:105) = | φ (cid:105) ⊗ N , | φ (cid:105) = | ↑ (cid:105) ⊗ | ↓ (cid:105) − | ↓ (cid:105) ⊗ | ↑ (cid:105) , (4.23)and we have also used the fact that U † Ω (1) U = Ω (2) . We now insert in (4.22) the completenessrelation in terms of Bethe states (which are SU (2) highest-weight states) and their lower-weight descendants (cid:88) K (cid:88) sol c ( N,K ) N ( u ) | u (cid:105)(cid:104) u | + . . . = , (4.24)where sol c ( N, K ) stands for physical solutions u of the closed-chain BAE (4.17) with 2 N sites and K Bethe roots. Moreover, the normalization factor is given by N ( u ) = (cid:104) u | u (cid:105) , (4.25)18nd the ellipsis denotes the descendant terms. However, these descendant terms do notcontribute to the matrix element (4.22), since the dimer state is annihilated by the spinraising and lowering operators S ± | Φ (cid:105) = 0 , where S ± = S x ± iS y , (cid:126)S = N (cid:88) j =1 12 (cid:126)σ j . (4.26)Moreover, in view of the fact0 = (cid:104) u | S z | Φ (cid:105) = ( N − K ) (cid:104) u | Φ (cid:105) , (4.27)the overlap (cid:104) u | Φ (cid:105) vanishes unless K = N . The matrix element (4.22) therefore reduces to Z c ( u, M, N ) = ( − ( M +1) N (cid:88) sol c ( N,N ) (cid:104) Φ | U † | u (cid:105)(cid:104) u | Φ (cid:105)N ( u ) (cid:104) Λ c ( ˜ u ; { θ j ( ˜ u ) } ) (cid:98) Λ c ( ˜ u ; { θ j ( ˜ u ) } ) (cid:105) M , (4.28)where the sum runs over all physical solutions of the closed-chain BAE (4.17) in the K = N sector. We note that the expressions involving the eigenvalues are given byΛ c ( ˜ u ; { θ j ( ˜ u ) } ) = ( − iu ) N N (cid:89) k =1 − u − u k − i − u − u k , (cid:98) Λ c ( ˜ u ; { θ j ( ˜ u ) } ) = ( − iu ) N N (cid:89) k =1 u − u k + i u − u k , (4.29)as follows from (4.16) and (4.18). The normalization factor (4.25) is given by the Gaudinformula [21, 22] N ( u ) = ( − N N (cid:89) j =1 (cid:104)(cid:0) u j + ˜ u (cid:1) + (cid:105) N (cid:104)(cid:0) u j − ˜ u (cid:1) + (cid:105) N (cid:34) N (cid:89) j,k =1 ; j (cid:54) = k u j − u k − iu j − u k (cid:35) det N ( G jk ) , (4.30)where G jk = δ jk (cid:40) N (cid:104) K ( u j − ˜ u ) + K ( u j + ˜ u ) (cid:105) − N (cid:88) l =1 K ( u j − u l ) (cid:41) + K ( u j − u k ) , (4.31)19nd K a ( u ) = 2 au + a . (4.32)Overlaps similar to (cid:104) u | Φ (cid:105) have been studied extensively, see e.g. [23–30], see also [31, 32].The cases of even and odd N must be analyzed separately. N Let us first consider even values of N . Interestingly, only Bethe states with “paired” Betheroots of the form { u , − u , . . . , u N , − u N } (4.33)have non-zero overlaps [25, 26]. Such Bethe states have even parity, see (E.5) below. Forsuch Bethe states, the overlaps are given by (see Appendix C) (cid:104) u | Φ (cid:105) = (cid:0) ˜ u + i (cid:1) N N (cid:89) j =1 u j (cid:113) u j + (cid:118)(cid:117)(cid:117)(cid:116) det N (cid:0) G + jk (cid:1) det N (cid:0) G − jk (cid:1) (cid:112) N ( u ) , (4.34)where G ± jk = δ jk N (cid:104) K ( u j − ˜ u ) + K ( u j + ˜ u ) (cid:105) − N (cid:88) l =1 K (+)1 ( u j , u l ) + K ( ± )1 ( u j , u k ) , (4.35)with K ( ± ) a ( u, v ) = K a ( u − v ) ± K a ( u + v ) , (4.36)and K a ( u ) is defined in (4.32). We remark that, for these states,det N ( G jk ) = det N (cid:0) G + jk (cid:1) det N (cid:0) G − jk (cid:1) . (4.37)Moreover, we show in Appendix D the relation (cid:104) Φ | U † | u (cid:105) = i N (˜ u + i ) − N Λ c ( ˜ u ; { θ j ( ˜ u ) } ) (cid:104) u | Φ (cid:105) . (4.38)20he matrix element (4.28) therefore reduces to Z c ( u, M, N ) = i N ( − ( M +1) N N (˜ u + i ) N (cid:88) sol( u ,...,u N ) N (cid:89) j =1 u j (cid:0) u j + (cid:1) det N (cid:0) G + jk (cid:1) det N (cid:0) G − jk (cid:1) × Λ c ( ˜ u ; { θ j ( ˜ u ) } ) (cid:104) Λ c ( ˜ u ; { θ j ( ˜ u ) } ) (cid:98) Λ c ( ˜ u ; { θ j ( ˜ u ) } ) (cid:105) M , = 2 − N u N ( M +1) (cid:88) sol( u ,...,u N ) N (cid:89) j =1 u j (cid:0) u j + (cid:1) det N (cid:0) G + jk (cid:1) det N (cid:0) G − jk (cid:1) × N (cid:89) k =1 (cid:18) u − u k + i u − u k (cid:19) (cid:18) u + u k + i u + u k (cid:19) M +1 , (4.39)where we have used (4.29) to pass to the second equality, and the sum is over all physicalsolutions of the BAE (4.17) with paired Bethe roots (4.33), see (4.48) below. N For odd values of N , the only Bethe states with non-zero overlaps have one 0 Bethe root,and all the other Bethe roots form pairs; i.e. , the Bethe roots are of the form { u , − u , . . . , u N − , − u N − , } . (4.40)Such Bethe states have odd parity, see (E.6) below. The overlaps are now given by (cid:104) u | Φ (cid:105) = − (cid:0) ˜ u + i (cid:1) N N − (cid:89) j =1 u j (cid:113) u j + (cid:118)(cid:117)(cid:117)(cid:116) det N +12 ( H jk )det N − (cid:0) G − jk (cid:1) (cid:112) N ( u ) , (4.41)where H is the block matrix H = (cid:32) G + CC t D (cid:33) N +12 × N +12 , (4.42)21nd G ± are now the N − × N − matrices given by G ± jk = δ jk N (cid:104) K ( u j − ˜ u ) + K ( u j + ˜ u ) (cid:105) − K ( u j ) − N − (cid:88) l =1 K (+)1 ( u j , u l ) + K ( ± )1 ( u j , u k ) , j , k = 1 , . . . , N − , (4.43)with K ( ± ) a ( u, v ) and K a ( u ) defined as before, see (4.36), (4.32). Moreover, in (4.42), C is an N − -component column vector, C t is the corresponding row vector, and D is a scalar, whichare given by C j = K ( u j ) , j = 1 , . . . , N − ,D = 2 N K ( ˜ u ) − N − (cid:88) l =1 K ( u l ) . (4.44)We remark that, for these states,det N ( G jk ) = det N +12 ( H jk ) det N − (cid:0) G − jk (cid:1) . (4.45)Moreover, (cid:104) Φ | U † | u (cid:105) = i N (˜ u + i ) − N Λ c ( ˜ u ; { θ j ( ˜ u ) } ) (cid:104) u | Φ (cid:105) . (4.46)The matrix element (4.28) now reduces to Z c ( u, M, N ) = i N ( − ( M +1) N N (˜ u + i ) N (cid:88) sol( u ,...,u N − ) N − (cid:89) j =1 u j (cid:0) u j + (cid:1) det N +12 ( H jk )det N − (cid:0) G − jk (cid:1) × Λ c ( ˜ u ; { θ j ( ˜ u ) } ) (cid:104) Λ c ( ˜ u ; { θ j ( ˜ u ) } ) (cid:98) Λ c ( ˜ u ; { θ j ( ˜ u ) } ) (cid:105) M , = 2 − N u N ( M +1) (cid:88) sol( u ,...,u N − ) N − (cid:89) j =1 u j (cid:0) u j + (cid:1) det N +12 ( H jk )det N − (cid:0) G − jk (cid:1) × (cid:18) u + 2 iu (cid:19) N − (cid:89) k =1 (cid:18) u − u k + i u − u k (cid:19) (cid:18) u + u k + i u + u k (cid:19) M +1 , (4.47)where we have used (4.29) to pass to the second equality, and the sum is over all physicalsolutions of the BAE (4.17) with paired Bethe roots (4.40), see (4.53) below.22 .4 BAE and Q -system We now summarize the BAE and Q -systems in the closed channel. They are special casesof those for the spin chain with periodic boundary condition with length 2 N and magnonnumber N . Even N . For the paired Bethe roots (4.33), the closed-chain Bethe equations (4.17) reduceto open-chain-like Bethe equations (cid:18) u k − ˜ u + i u k − ˜ u − i (cid:19) N (cid:18) u k + ˜ u + i u k + ˜ u − i (cid:19) N = (cid:18) u k + i u k − i (cid:19) N (cid:89) j =1; j (cid:54) = k (cid:18) u k − u j + iu k − u j − i (cid:19) (cid:18) u k + u j + iu k + u j − i (cid:19) , (4.48)where k = 1 , . . . , N . The corresponding QQ -relations are Q a +1 ,s ( v ) Q a,s +1 ( v ) ∝ Q + a +1 ,s +1 ( v ) Q − a,s ( v ) − Q − a +1 ,s +1 ( v ) Q + a,s ( v ) , (4.49)where Q a,s ( v ) are even polynomial functions of v . In particular, the main Q -function is givenby Q , ( v ) = N (cid:89) j =1 ( v − u j )( v + u j ) = N (cid:88) k =0 c (2 k )1 , v k = v N + c ( N − , v N − + · · · + c (0)1 , . (4.50)Moreover, Q , ( v ) = (cid:2) ( v − ˜ u )( v + ˜ u ) (cid:3) N . (4.51)Therefore, to obtain the ZRC for this case, we can simply take the ZRC for the genericperiodic case and add the following constraints c (2 k +1)1 , = 0 , k = 0 , , . . . , N − . (4.52) Odd N . For odd N , the nonzero paired Bethe roots (4.40) satisfy the open-chain-like Betheequations (cid:18) u k − ˜ u + i u k − ˜ u − i (cid:19) N (cid:18) u k + ˜ u + i u k + ˜ u − i (cid:19) N = (cid:18) u k + i u k − i (cid:19) (cid:18) u k + iu k − i (cid:19) N − (cid:89) j =1; j (cid:54) = k (cid:18) u k − u j + iu k − u j − i (cid:19) (cid:18) u k + u j + iu k + u j − i (cid:19) , (4.53)23here k = 1 , . . . , ( N − /
2. The corresponding QQ -relations are again given by (4.49), with Q , ( v ) given by (4.51). The Q -functions are odd polynomials in this case. In particular, themain Q -function takes the form Q , ( v ) = v N − (cid:89) j =1 ( v − u j )( v + u j ) = N − (cid:88) k =0 c (2 k +1)1 , v k +1 = v N + c ( N − , v N − + · · · c (1)1 , v . (4.54)Therefore, to obtain the ZRC in this case, we take the general ZRC for the generic periodiccase and impose the conditions c (2 k )1 , = 0 , k = 0 , , . . . , ( N − / . (4.55)For both even and odd values of N , we conjecture that the number N ( N ) of such physicalsolutions of the BAE (4.48), (4.53) is given simply by N ( N ) = (cid:18) N (cid:98) N/ (cid:99) (cid:19) , (4.56)where (cid:98) x (cid:99) denotes the integer part of x . The first 10 values are given by { , , , , , , , , , } , (4.57)which we checked by explicit computations. The procedure for algebro-geometric computations follows the same steps as in the openchannel. As we mentioned before, the computation of the Gr¨obner basis and quotient ring issimpler. The complication comes from computing the companion matrices. The companionmatrix of the transfer matrices Λ c ( v ; { θ j ( u ) } ) can be constructed similarly from the T Q relation Q ( v ) T ( v − i ) = (cid:2) ( v + i ) − u (cid:3) N Q ( v − i ) + (cid:2) ( v − i ) − u (cid:3) N Q ( v + i ) . (4.58)The most complicated part is the ratio of determinants in (4.39) and (4.47). These arecomplicated functions in terms of rapidities u . As in the open channel, the natural variablesthat enter the AG computation are c ( k ) a,s . Therefore, in order to construct the companionmatrices of the ratio of determinants, we need to first convert it to be functions c ( k ) a,s . Thiscan be done because the ratio of determinants are symmetric rational functions.24 ven N . For even N , after expanding the determinant the result can be written in theform N( u , . . . , u N/ )D( u , . . . , u N/ ) , (4.59)where N( u , . . . , u N/ ) and D( u , . . . , u N/ ) are symmetric polynomials in { u , . . . , u N/ } . Bythe fundamental theorem of symmetric polynomials, they can be written in terms of elemen-tary symmetric polynomials of { u , . . . , u N/ } , which we denote by { s , s , . . . , s N/ − } : s = u u · · · u N , (4.60) · · · s N − = u u + u u + . . . + u N − u N ,s N − = u + u + . . . + u N − . They are related to the coefficients c (2 k )1 , in (4.50) as c (2 k )1 , = ( − N + k s k , k = 0 , , . . . , N − . (4.61) Odd N . For odd N , the result can be written asN( u , . . . , u N − )D( u , . . . , u N − ) (4.62)Similarly, we can do the symmetry reduction and write the result in terms of the elementarysymmetric polynomials s = u u · · · u N − , (4.63) · · · s N − − = u u + u u + . . . + u N − − u N − ,s N − − = u + u + . . . + u N − − . They are related to the coefficients c (2 k +1)1 , in (4.54) as c (2 k +1)1 , = ( − N − + k s k , k = 0 , , . . . , N − − . (4.64)There are two sources of complication worth mentioning. Firstly, computing the determi-nant explicitly and performing the symmetric reduction is straightforward in principle, but25ecomes cumbersome very quickly. It would be desirable to have a simpler form for thesequantities. Secondly, the companion matrix of the quantity 1 / D is the inverse of the com-panion matrix of D. Computing the inverse of a matrix analytically is also straightforward,but it has a negative impact on the efficiency of the computations when the dimension ofthe matrix becomes large. For the eigenvalues of the transfer matrix, we saw in (3.23) thatthe problem of computing inverses can be circumvented by using the
T Q -relations. For theexpression of the overlaps, it is not clear whether we can find better means to compute thecompanion matrix of the ratio N / D so as to avoid taking matrix inverses.Using the algebro-geometric approach in the closed channel, we computed partition func-tions for N up to 7 and M up to 2048. The results for 2 ≤ M, N ≤ In this section, we discuss the Gr¨obner basis of the ZRC in the closed channel in more detail.This will demonstrate further the power of the algebro-geometric approach for algebraicequations, especially for cases with free parameters.The system of algebraic equations we consider depends on a parameter u . This meansthat the coefficients of the equations are no longer pure numbers, but functions of u . Asa result, the solutions also depend on the parameter u . As we vary the parameter u , thesolutions also change. One important question is if there are any special values u where thesolution space changes drastically. To understand this point, let us consider the followingsimple equation for x whose coefficients depend on the free parameter u ( u − x + ux − . (5.1)At generic values of u , this is a quadratic equation with two solutions. However, when u = ±
1, the leading term vanishes and the equation become linear. The number of solutionsbecomes one. Therefore at these ‘singular’ points, the structure of the solution space changesdrastically.A similar phenomena occurs in the BAE of the Heisenberg spin chain. Consider for amoment the more general XXZ spin chain and take the anisotropy parameter ( alias quantumgroup deformation parameter) q as the free parameter of the BAE. It is well-known that thesolution space is very different between generic q and q being a root of unity. The traditionalway to see this is by studying representation theory of the U q ( sl (2)) symmetry of the spinchain [33]. A more straightforward way to see this fact is by the algebro-geometric approach.We can compute the Gr¨obner basis of the corresponding BAE/ Q -system and analyze thecoefficients as functions of q . We shall discuss this problem in more detail in a future26ublication.Related to the current work, we consider the ZRC in the closed channel for the XXX spinchain. Here the free parameter is the inhomogeneity u . We want to know whether there arespecial singular points of u where the structure of solution space changes drastically. Recallthat from elementary algebraic geometry, the number of solutions equals the linear dimensionof the quotient ring. Furthermore, the quotient ring dimension is completely determined bythe leading terms of the Gr¨obner basis. Therefore, to this end, we compute the Gr¨obnerbasis explicitly. For N = 3, the ideal can be written as (cid:104) g , g , g (cid:105) where the elements of theGr¨obner basis g i are given by g = 192 s + (192 u − s − (192 u + 288 u − s − u + 48 u + 60 u + 9 , (5.2) g = s ,g = s . Here we have chosen the ordering s ≺ s ≺ s . (5.3)We see from (5.2) that the leading terms are independent of u . This implies that the dimen-sion of the quotient ring C [ s , s , s ] / (cid:104) g , g , g (cid:105) , or equivalently the number of solutions, isindependent of the value of u . Of course, the explicit solutions of the BAE will depend onthe value of u , but there will always be 3 solutions to the ZRC for N = 3 at any value of u .Similarly, we can write down a slightly more non-trivial example for N = 4. The ideal isgiven by (cid:104) g , · · · , g (cid:105) where the Gr¨obner basis elements g i are given by g = + ( − − u − u ) s + (3 + 48 u + 288 u + 768 u + 768 u ) s (5.4)+ ( − u + 2304 u − u ) s s + ( − − u − u − u − u ) s + ( − − u − u + 3328 u − u ) s + ( − − u − u − u − u ) ,g = s − s s + (768 + 3072 u ) s + ( − − u − u − u − u ) s + ( − − u − u + 1024 u ) s + ( − − u − u − u − u ) ,g = s + ( − u ) s s + 1280 s + ( − − u − u ) s + ( − − u − u − u − u ) ,g = + ( −
48 + 64 u ) s − s s + ( − − u − u ) s + (16 − u ) s + (3 + 12 u − u − u ) g = s g = s . u .For all the values of N which we compute, this is true. It would be nice to prove this forgeneral N .Therefore from the algebro-geometric computation, we conclude that for any value of u ,there exist solutions with definite parity (parity even/odd for even/odd N ). For fixed N ,the number of solutions is the same for any value of u , which has been given in (4.56).We end this section by the following comment. The conclusion that there always ex-ist solutions with definite parity for any u is far from obvious from the ZRC or originalBAE. It is also not easy to see this from numerical computations. On the contrary, it is astraightforward observation from the algebro-geometric computation. This shows again thatalgebro-geometric approach is a powerful tool to analyze the solution space of BAE. In this section, we discuss the analytical results which can be written in closed forms forarbitrary N and M in the open and closed channel respectively. We first discuss the open channel. The partition function takes the same form as the toruscase, which is written as the trace of the N -th power of the transfer matrix. If the eigenvaluesof the transfer matrix can be found analytically, we can write down the partition functionfor any N . Here by analytical we mean more precisely expressible in terms of radicals . Inthe algebro-geometric approach, we first compute the companion matrix of the eigenvalue ofthe transfer matrix. The dimension of the companion matrix equals the number of physicalsolutions of the open channel BAE/Q-system. The eigenvalues of the companion matrixgive the eigenvalues of the transfer matrices evaluated at each solution. From Galois theory,if the dimension of the companion matrix is less than 5, the eigenvalues can be expressedin terms of radicals. Therefore for the values of M where all the companion matrices havedimension less than 5, we can obtain the analytical expression for any N . This requirementis only met by M = 1. Already for M = 2, we need to consider the sectors K = 0 , , K = 1 and 2 the dimensions of the companion matrices are 4 and 6 respectively.For larger M , the dimensions of the companion matrices are even larger. We give the closedform expression for M = 1 and any N in what follows.28 he M = 1 case. We need to consider K = 0 ,
1. For K = 0, the eigenvalue of the transfermatrix is given by Λ D , ( u ) = ( u + i ) . (6.1)For K = 1, the solution of BAE takes the form { u , − u } . The companion matrix is 2-dimensional. The two eigenvalues of the transfer matrix in this sector are given by λ ( u ) = − u − − iu √ u + 4 , (6.2) λ ( u ) = − u − iu √ u + 4 . The closed-form expression of the partition function, taking into account the su (2) multi-plicities (3.12), is then Z ( u, , N ) = 4( u + i ) N + 2 (cid:0) λ ( u ) N + λ ( u ) N (cid:1) . (6.3)Let us make one comment on the comparison with the torus case. The closed-form re-sults have been found up to M = 6 in the torus case [7]. There we also used the fact thatcertain companion matrices can be further decomposed into smaller blocks, which impliesthe existence of non-trivial primary decompositions over the field Q . Physically, this primarydecomposition is related to decomposing the solutions of BAE/Q-system according to the to-tal momentum. In the cylinder case, however, the total momentum is automatically zero forall allowed solutions, due to the presence of the boundary. Therefore, further decompositionaccording to the total momentum is not possible in the current case. The situation is more interesting in the closed channel. The expression for the partitionfunction is qualitatively different from the torus case, since we have a new ingredient: thenon-trivial overlap between Bethe states and the boundary state. To find the analyticalexpressions, we first compute the companion matrices, both for the transfer matrix and theoverlaps. For the values of N where the dimensions of the companion matrices are lessthan 5, we can express the final result in terms of radicals for any M . This is satisfied by N = 1 , ,
3. The dimensions of the companion matrices are 1 , , Note that in the torus case, M denoted the length of the spin chain [7], while here, in the cylinder case,the length of the spin chain is given by 2 M + 1. he N = 1 case. This case is somewhat trivial, but we give it here for completeness.There is only one allowed solution to the BAE, which is { } . The eigenvalue of the transfermatrix is given by Λ c = − i ( u + 2 i ) . (6.4)The contribution from the overlaps only comes from det H jk which is given bydet H jk = 8 u ( u + 2 i ) . (6.5)The partition function is thus given by (4.47). Z ( u, M,
1) = i ( − M u ( − i ( u + 2 i )) M +1 u ( u + 2 i ) = 2( u + 2 i ) M . (6.6) The N = 2 case. This is the simplest non-trivial case where N is even. The Bethe rootstake the form { u , − u } . There are two such solutions, which can be found straightforwardlyby directly solving Bethe equations. Let us denote the companion matrices of Λ c (cid:0) ˜ u , { ˜ u } (cid:1) by T ( u ) and companion matrix of the following factor1 u ( u + ) det G + jk det G − jk (6.7)by F ( u ). The partition function is given by (4.39) Z ( u, M,
2) = − u tr (cid:2) T M +12 ( u ) · F ( u ) (cid:3) . (6.8)Let us denote the eigenvalues of T ( u ) and F ( u ) by λ T ,i ( u ) and λ F ,i , i = 1 , λ T , ( u ) = 1 − iu − (cid:112) u + 2 i )( u + i ) u , (6.9) λ T , ( u ) = 1 − iu + (cid:112) u + 2 i )( u + i ) u and λ F , ( u ) = 32 u ( u + 2 i ) + 16( u + 2 iu − (cid:112) u + 2 i )( u + i ) uu ( u + 2 i ) ( u + 4 iu − u − iu + 1) , (6.10) λ F , ( u ) = 32 u ( u + 2 i ) − u + 2 iu − (cid:112) u + 2 i )( u + i ) uu ( u + 2 i ) ( u + 4 iu − u − iu + 1) . N = 2 and any M is givenby Z ( u, M,
2) = − u (cid:0) [ λ T , ( u )] M +1 λ F , ( u ) + [ λ T , ( u )] M +1 λ F , ( u ) (cid:1) . (6.11)One may check that this agrees in particular with (2.6) for M = 2. We see here thatthe eigenvalues take rather complicated forms in terms of radicals whose arguments arepolynomials of u . Nevertheless, the final result is a polynomial, as it should be. The N = 3 case. This is the simplest non-trivial case where N is odd. The results arebulky, therefore it is more convenient to write them in terms of smaller building blocks. Tothis end, we recall the solution for cubic polynomial equations. Let us consider the followinggeneric cubic equation ax + bx + cx + d = 0 , (6.12)where a (cid:54) = 0. We define∆ = b − ac, ∆ = 2 b − abc + 27 a d (6.13)and C = (cid:32) ∆ + (cid:112) ∆ − (cid:33) / . (6.14)Then the three solutions of the cubic equation (6.12) are given by x k = − a (cid:18) b + ξ k C + ∆ ξ k C (cid:19) , k = 1 , , ξ = ( − i √ / N = 3, the solutions of the Bethe equations take the form { u , − u , } . There arethree physical solutions. Let us denote the companion matrix of Λ c (cid:0) ˜ u , { ˜ u } (cid:1) by T ( u ) andthe companion matrix of the following factor1 u ( u + ) det H jk det G − jk (6.16)31y F ( u ). The partition function is given by Z ( u, M,
3) = − i ( − M u tr (cid:2) T M +13 ( u ) · F ( u ) (cid:3) , (6.17)where the trace is over the 3-dimensional quotient ring. One can check explicitly that T ( u )and F ( u ) commute with each other and can thus be diagonalized simultaneously. Let usdenote their eigenvalues by λ T ,i ( u ) and λ F ,i ( u ), with i = 1 , ,
3. The characteristic equationsof T ( u ) and F ( u ) take cubic forms x + b T x + c T x + d T = 0 , x + b F x + c F x + d F = 0 , (6.18)where the coefficients are rational functions of u . The characteristic equations can be solvedby radicals using (6.15). The relevant quantities are given as follows. For the eigenvalues of T ( u ), we have a T = 1 , b T = 2 u + 3 iu − T0 = − iu + 31 u + 54 iu − u − iu + 4 , (6.20)∆ T1 = 27 iu − u − iu + 1096 u + 1035 iu − u − iu + 300 u + 72 iu − . The eigenvalues { λ T , , λ T , , λ T , } are given by λ T ,i ( u ) = − (cid:18) b T + ξ i C T + ∆ T0 ξ i C T (cid:19) , i = 1 , , C T is defined as in (6.14). For the eigenvalues of F , we have a F = 1 , b F = − u (2 i + u ) (6.22)and ∆ F0 = 16384 u (2 i + u ) P ( u ) P ( u ) , ∆ F1 = − u (2 i + u ) P ( u ) P ( u ) , (6.23)32here P ( u ) = 27 u + 270 iu − u − iu + 3345 u + 2934 iu − u − iu − u − iu + 64 ,P ( u ) = 27 u + 270 iu − u − iu + 10680 u + 16704 iu (6.24) − u − iu − u − iu + 1024 ,P ( u ) = 27 u + 270 iu − u − iu + 15360 u + 29664 iu − u − iu + 11520 u − iu + 4096 . The eigenvalues are given by λ F ,i ( u ) = − (cid:18) b F + ξ i − C F + ∆ F0 ξ i − C F (cid:19) , i = 1 , , Z ( u, M,
3) = − i ( − M u (cid:88) i =1 ( λ T ,i ) M +1 λ F ,i ( u ) . (6.26) The study of partition function zeros is a well-known tool to access the phase diagram ofmodels in statistical physics. The seminal works by Lee and Yang [34] and by Fisher [35]studied the zeros of the Ising model partition function, respectively with a complex magneticfield (at the critical temperature) and at a complex temperature (in zero magnetic field).But more generally, any statistical model depending on one (or more) parameters can bestudied in the complex plane of the corresponding variable(s). In particular, the chromaticpolynomial with Q ∈ C colors has been used as a test bed to develop a range of numerical,analytical and algebraic tools for computing partition function zeros and analyzing theirbehavior as the (partial) thermodynamic limit is approached [36–42]. Further informationabout the physical relevance of studying partition function zeros can be found in [43] andthe extensive list of references in [36].In the case at hand, we are interested in zeros of the partition function Z ( u, M, N ) of thesix-vertex model, in the complex plane of the spectral parameter, u ∈ C . As explained in Notice that the powers of ξ in (6.25) are slightly different from (6.21). The reason for this conventionis to make sure that λ T i and λ F i correspond to the same eigenvector. Working directly with characteristicequations, it is not immediately clear which eigenvalues correspond to the same eigenvector. We establishthe correspondence by making numerical checks. We choose u to be some purely imaginary numbers suchthat the arguments in the radicals are real and positive. Z ( u, M, N ) closeto the partial thermodynamic limits N (cid:29) M (open channel) or M (cid:29) N (closed channel),and more precisely for aspect ratios ρ := N/M of the order ∼ and ∼ − , respectively. An important result for analyzing these cases is the Beraha-Kahane-Weiss (BKW) theorem[14]. When applied to partition functions of the form (3.13) for the open channel, respectively(4.39) or (4.47) for the closed channel, it states that the partition function zeros in thepartial thermodynamic limits ( ρ → ∞ or ρ →
0, respectively) will condense on a set ofcurves in the complex u -plane that we shall refer to as condensation curves . In particular,the condensation set cannot comprise isolated points, or areas. By standard theorems ofcomplex analysis, each closed region delimited by these curves constitutes a thermodynamicphase (in the partial thermodynamic limit).To be more precise, let Λ i ( u ) denote the eigenvalues of the relevant transfer matrix (forthe open or closed channel, respectively) that effectively contributes to Z ( u, M, N ). For agiven u , we order these eigenvalues by norm, so that | Λ ( u ) | ≥ | Λ ( u ) | ≥ · · · , and we saythat an eigenvalue Λ i ( u ) is dominant at u if there does not exist any other eigenvalue havinga strictly greater norm. Under a mild non-degeneracy assumption (which is satisfied for theexpressions of interest here), the BKW theorem [14] states that the condensation curves aregiven by the loci where there are (at least) two dominant eigenvalues, | Λ ( u ) | = | Λ ( u ) | .It is intuitively clear that this defines curves, since the relative phase φ ( u ) ∈ R defined byΛ ( u ) = e iφ ( u ) Λ ( u ) is allowed to vary along the curve. Moreover, a closer analysis [36]shows that the condensation curves may have bifurcation points (usually called T-points) orhigher-order crossings when more than two eigenvalues are dominant. They may also haveend-points under certain conditions; see [36] for more details.A numerical technique for tracing out the condensation curves has been outlined inour previous paper on the toroidal geometry [7]. It builds on an efficient method for thenumerically exact diagonalization of the relevant transfer matrix, and on a direct-searchmethod that allows us to trace out the condensation curves. We refer the reader to [7]for more details, and focus instead on a technical point that is important (especially in theclosed channel) for correctly computing the condensation curves for the cylindrical boundaryconditions studied in this paper.One might of course choose to obtain the eigenvalues by solving the BAE, either analyti-cally or numerically. However, the Bethe ansatz does not provide a general principle to orderthe eigenvalues by norm. It is of course well known that in many, if not most, Bethe-ansatzsolvable models, for “physical” values of the parameters the dominant eigenvalue and its low-lying excitations are characterized by particularly nice and symmetric arrangements of the34ethe roots, and hence one can easily single out those eigenvalues. However, we here wish toexamine our model for all complex values of the parameter u , and it is quite possible—and infact true, as we shall see—that there will be a complicated pattern of crossings (in norm) ofeigenvalues throughout the complex u -plain. To apply the BAE one would therefore have tomake sure to obtain all the physical eigenvalues and compare their norm for each value of u .By contrast, the numerical scheme (Arnoldi’s method) that we use for the direct numericaldiagonalization of the transfer matrix is particularly well suited for computing only the firstfew eigenvalues (in norm), so we shall rely on it here. We shall later compare the computedcondensation curves with the zeros of partition functions obtained using Bethe ansatz andalgebraic geometry.The reader will have noticed that above we have twice referred to the diagonalization ofa “relevant” transfer matrix. By this we mean a transfer matrix whose spectrum containsonly the eigenvalues that provide non-zero contributions to Z ( u, M, N ), after taking accountof the boundary conditions via the trace (2.5) in the open channel, or the sandwich betweenboundary states (2.8) in the closed channel. These contributing eigenvalues correspond tothe physical solutions in (3.13) for the open channel, or in (4.39) and (4.47) for the closedchannel. A “relevant” transfer matrix is thus not only a linear operator that can buildup the partition function Z ( u, M, N ), but it must also have the correct dimension, namely (cid:80) K N ( M, K ) given by (3.14) in the open channel, or N ( N ) given by (4.56) in the closedchannel. Ensuring this is an issue of representation theory. We begin by discussing it in theopen channel, which is easier. The defining ingredient of the transfer matrix is the ˇ R -matrix. Using (2.2)–(2.3), it readsˇ R ( u ) = a ( u ) 0 0 00 c ( u ) b ( u ) 00 b ( u ) c ( u ) 00 0 0 a ( u ) , (7.1)with a ( u ) = u + i , b ( u ) = u and c ( u ) = i . The most immediate transfer matrix approach isto let the diagonal-to-diagonal transfer matrix t D ( u ) given by (2.1) act in the 6-vertex modelrepresentation, that is, on the space {| ↑(cid:105) , | ↓(cid:105)} ⊗ M +1 of dimension 2 M +1 .If we constrain to a fixed magnon number K , the dimension reduces to (cid:0) M +1 K (cid:1) . This islarger than N ( M, K ) given by (3.14), because we have not restricted to su (2) highest-weightstates. Therefore, each eigenvalue would appear with a multiplicity given by (3.12). Sinceeach eigenvalue actually does contribute to Z ( u, M, N ), dealing with this naive representation35rovides a feasible route to computing the condensation curves (and this was actually theapproach used in [7]). However, the appearance of multiplicities is cumbersome and impedesthe efficiency of the computations. To overcome this problem, notice that in the more general XXZ model with quantum-groupdeformation parameter q , the integrable ˇ R -matrix may be taken asˇ R i,i +1 ( u ) = αI + βE i , (7.2)for certain coefficients α , β depending on u and q . Here I denotes the identity operator and E i is a generator of the Temperley-Lieb (TL) algebra. The defining relations of this algebra,acting on L = 2 M + 1 sites, are E i E i = δE i ,E i E i ± E i = E i , (7.3) E i E j = E j E i for | i − j | > , where i, j = 1 , , . . . , L − δ := q + q − . A representation of E i , writtenin the same 6-vertex model representation as (7.1), reads E i = q − q
00 0 0 0 . (7.4)By taking tensor products, one may check that this satisfies the relations (7.3). We canmatch with (7.2) by taking α = u + i , β = u , q = − , δ = − . (7.5)The trick is now that there exists another representation of the TL algebra having exactlythe required dimension N ( M, K ). The basis states of this representation are link patterns on L sites with d := L − K defects. A link pattern consists of a pairwise matching of L − d = 2 K points (usually depicted as K arcs) and d defect points, subject to the constraint of planarity:two arcs cannot cross, and an arc is not allowed to straddle a defect point. We show here36wo possible link patterns for L = 5 and d = 1 (hence M = 2 and K = 2):and (7.6)The TL generator E i acts on sites i and i + 1 by first contracting them, then adding anew arc between i and i + 1. This can be visualized by placing the graphical representation E i = on top of the link pattern. If a loop is formed in the contraction, it is removedand replaced by the weight δ . If a contraction involves an arc and a defect point, the defectpoint moves to the other extremity of the arc. If a contraction involves two distinct arcs,the opposite ends of those arcs become paired by an arc. For instance, the action of E onthe two link patterns in (7.6) produces and δ × (7.7)Recall from (3.11) that the spin s associated with the K -magnon sector in the chain of L = 2 M + 1 sites reads s = L − K . The generators E i can decrease s by contracting a pairof defects and replacing them by an arc. It is however possible to define a representation ofthe TL algebra in which s is fixed, by defining the action of E i to be zero whenever there is apair of defects at sites i and i + 1. In the literature on the TL algebra, these representationsin terms of link patterns with a conserved number of defects are known as standard modules and denoted W s . Meanwhile, in TL representation theory, the partition function in the openchannel is no longer written in terms of a trace as in (2.5). Instead it is written as a so-calledMarkov trace Z ( u, M, N ) = Mtr (cid:2) t D ( u ) N (cid:3) , (7.8)which can be interpreted diagramatically as the stacking of N rows of diagrams, followedby a gluing operation in which the top and the bottom of the system are identified andeach resulting loop replaced by the corresponding weight δ . It is a remarkable fact that thisMarkov trace can be computed as a linear combination of ordinary matrix traces over thestandard modules, as follows: Z ( u, M, N ) = M +1 / (cid:88) s =1 / , / ,... (1 + 2 s ) q tr W s (cid:2) t D ( u ) N (cid:3) . (7.9)We have here defined the q -deformed numbers( n ) q = q n − q − n q − q − = U n − (cid:18) δ (cid:19) , (7.10)37here U p ( x ) denotes the p -th order Chebyshev polynomial of the second kind. This result(7.9) can be proved by using the quantum group symmetry U q ( su (2)) enjoyed by the spinchain in the open channel [44], or alternatively by purely combinatorial means [45].The factors (1 + 2 s ) q appearing in (7.9) account for the multiplicities in the problem.In the limit q → − q -deformed numbersbecome ( n ) q = n for n odd, and ( n ) q = − n for n even. The latter minus sign can beeliminated at the price of an overall sign change of the partition function (7.9), since onlyeven n = 1 + 2 s occur in the problem. This corresponds to q (cid:55)→ − q , so that the quantumgroup symmetry U q ( su (2)) becomes just ordinary su (2) in the limit. The multiplicities thenbecome (1 + 2 s ) q = 1 + 2 s = 2 M + 2 − K , in agreement with (3.11).In conclusion, we see that not only do the link pattern representations of the TL alge-bra lead to the correct dimensions N ( M, K ), but they also account for the correct su (2)multiplicities 2 M + 2 − K of eigenvalues in the XXX spin chain. We have computed the condensation curves by applying the numerical methods of [7] tothe transfer matrix t D ( u ) given by (2.1). The latter is taken to act on the representationgiven by the union of link patterns on L = 2 M + 1 sites with K ∈ { , , . . . , M } arcs and d = L − K defects.The results for the condensation curves with M = 2 , , , u ≤
0, and they are invariant under changing thesign of Re u . Therefore it is enough to consider them in the fourth quadrant: Re u ≥ u ≤
0. The condensation curves display several noteworthy features:1. Outside the curves and in the enclosed regions delimited by blue curves, the dominanteigenvalue belongs to the K = M magnon sector (i.e., d = 1 defect in the TL represen-tation). For the largest size M = 5 there are also enclosed regions delimited by greencurves: in this case the dominant eigenvalue belongs to the K = M − d = 3).2. The whole real axis forms part of the curve. In fact, when u ∈ R , all the eigenvalues areequimodular and have norm ( u + 1) M . Above the real axis (Im u >
0) the dominanteigenvalue is the unique eigenvalue in the K = 0 sector.3. There is a segment of the imaginary axis, Re u = 0 and Im u ≤ u c ( M ) which alsobelongs to the condensation curve. Along this segment, the two dominant eigenvaluescome from the K = M sector. For the end-point u c ( M ) we find the following results: M u c ( M ) -1.091487 -1.065097 -1.050552 -1.041328 -1.034954 -1.03028538 -6-5-4-3-2-1 0 0 1 2 3 4 5L=9T-points -7-6-5-4-3-2-1 0 0 1 2 3 4 5L=11T-points Figure 7.6: Condensation curves for partition function zeros on a (2 M + 1) × N cylinder,in the limit N → ∞ (aspect ratio ρ → ∞ , open channel). The panels show, in readingdirection, the cases M = 2 , , , u c ( M ) → − M → ∞ , (7.11)with a finite-size correction proportional to 1 /M . We also note that at this asymptoticend-point, u = − i , for all finite M there is a unique dominant eigenvalue which belongsto the K = M sector and has norm 1, while all other eigenvalues have norm 0.5. The remainder of the condensation curve forms a single connected component with noend-points. It however has a number of T-points that grows fast with M . Notice thatwe have taken great care to determine all of these T-points, some of which are veryclose and thus hard to distinguish in the figures. To help the reader identifying them,they have been marked by small crosses.39. For the leftmost point u (cid:63) of this connected component (i.e., the point with the smallestimaginary part) we find the following results: M u (cid:63) ( M ) 0.496489 0.338134 0.258384 0.209593 0.176490 0.152498Im u (cid:63) ( M ) -1.307913 -1.196739 -1.146652 -1.117510 -1.098264 -1.0845397. It seems compelling from these data thatRe u (cid:63) ( M ) → M → ∞ , Im u (cid:63) ( M ) → − M → ∞ , (7.12)both with finite-size corrections proportional to 1 /M . We conclude that the leftmostpoint of the connected component converges to the same value as the end-point, namely u = − i . This kind of “pinching” is characteristic of a phase transition [34, 35]; notehowever that the limit u → − i of the XXX model is singular and does not present acritical point in the usual sense.We now compare the condensation curves with the partition function zeros. The partitionfunctions Z ( u, M, N ) were first computed from the algebro-geometric approach, for M =2 , , , N = 1024, which corresponds to a very large aspect ratio ρ ∼ . Thezeros of Z ( u, M, N ) were then computed by the program MPSolve [46] [47], which is amultiprecision implementation of the Ehrlich-Aberth method [48, 49], an iterative approachto finding all zeros of a polynomial simultaneously.The resulting zeros are shown in Figure 7.7, as red points superposed on the condensationcurves of Figure 7.6. The agreement is in general very good, although some portions of thecondensation curves are very sparsely populated with zeros; in those cases the zeros are stillat a discernible distance from the curves, in spite of the large aspect ratio. We have verifiedthat the agreement improves upon increasing ρ . One may notice that some of the roots (in particular for M = 5, in the region 0 < Re u <
1) stray offthe curves in a seemingly erratic fashion. We believe that this is an artifact of
MPSolve when applied topolynomials of very high degree. -6-5-4-3-2-1 0 0 1 2 3 4 57 x 1024L=7 -6-5-4-3-2-1 0 0 1 2 3 4 59 x 1024L=9T-points -7-6-5-4-3-2-1 0 0 1 2 3 4 511 x 1024L=11T-points Figure 7.7: Comparison between the partition function zeros on a (2 M + 1) × N cylinder,with N = 1024, and the corresponding condensation curves in the N → ∞ limit (openchannel). The panels show, in reading direction, the cases M = 2 , , , In the closed channel the ˇ R -matrix can be inferred from (2.7) and (2.2). It readsˇ R c ( u ) = b ( u ) 0 0 00 c ( u ) a ( u ) 00 a ( u ) c ( u ) 00 0 0 b ( u ) , (7.13)still with a ( u ) = u + i , b ( u ) = u and c ( u ) = i . However, as we shall soon see, it is convenientto apply a diagonal gauge transformation D = diag(1 , −
1) in the left in-space and the rightout-space of ˇ R c ; that is, ˇ R c12 (cid:55)→ D ˇ R c12 D . This has the effect of changing the sign of c ( u )while leaving the partition function unchanged: the gauge matrices square to the identity atthe intersections between ˇ R -matrices when taking powers of the transfer matrix ˜t D ( u ) given41y (2.10). To complete the transformation, the first and last row of gauge transformationshave to be absorbed into a redefinition of the boundary states (cid:104) Ψ | and | Ψ (cid:105) appearing in(2.8). As in the open channel, we can rewrite the ˇ R -matrix in terms of TL generators (7.4):ˇ R c i,i +1 ( u ) = αI + βE i . (7.14)To match (7.13), with c ( u ) = − i after the gauge transformation, we must now set α = u , β = u + i , q = − , δ = − . (7.15)In the closed channel, the TL algebra is defined on L = 2 N sites. The goal is now to finda representation having the same dimension N ( N ), see (4.56), as the number of physicalsolutions appearing in the closed-channel expressions of the partition function, (4.39) and(4.47). This issue is more complicated than in the open channel.As a first step, we let the TL generators act on the basis of link patterns, as before. Sincethe boundary states restrict to zero total spin, the only allowed number of Bethe roots is K = N (see section 4). This implies that the link patterns are free of defects ( d = 0). Thetransfer matrix ˜t D ( u ) is then given by (2.10) with (7.14), where the TL generators E i act onthe link patterns as described in section 7.2.1. To reproduce the partition function (2.8) wealso need to interpret the boundary state (2.11) within the TL representation. The naturalobject is the quantum-group singlet of two neighboring sites | ψ (cid:105) TL = q / | ↑(cid:105) ⊗ | ↓(cid:105) + q − / | ↓(cid:105) ⊗ | ↑(cid:105) , (7.16)which is represented in terms of link patterns as a short arc joining the neighboring sites. Weshould however remember at this stage the gauge transformation that allowed us to switchthe sign of c ( u ). To compensate this, we need to insert a minus sign for a down-spin in thesecond tensorand, to obtain | ˜ ψ (cid:105) TL = − q / | ↑(cid:105) ⊗ | ↓(cid:105) + q − / | ↓(cid:105) ⊗ | ↑(cid:105) , (7.17)With q = −
1, this is proportional to | ψ (cid:105) of (2.11).On the other hand it is easy to check from (7.4) that the TL generator E i is nothing butthe (unnormalized) projector onto the quantum-group singlet. Therefore, just as E i = ,the initial boundary state | Ψ (cid:105) can be represented graphically by the defect-free link pattern42n which sites 2 j − j are connected by an arc, for each j = 1 , , . . . , N . Similarly, thefinal boundary state (cid:104) Ψ | is interpreted as the TL contraction of the corresponding pairs ofsites. With these identifications, we have explicitly verified for small N and M that the TLformalism produces the correct partition functions, such as (2.6).With the spin-zero constraint imposed, the TL dimension is thus equal to the number ofdefect-free link patterns on L = 2 N sites. This is easily shown to be given by the Catalannumbers Cat( N ) = 1 N + 1 (cid:18) NN (cid:19) , (7.18)for which the first 10 values are given by { , , , , , , , , , } . (7.19)Although this is smaller than the dimension of the 6-vertex-model representation constrainedto the S z = 0 sector, viz. (cid:0) NN (cid:1) , it is not as small as (4.56), so further work is needed.The transfer matrix and the boundary states are also symmetric under cyclic shifts (inunits of two lattice spacings) of the L = 2 N sites. This symmetry can be used to furtherreduce the dimension of the transfer matrix. Indeed, after acting with ˜t D ( u ) we project eachlink pattern obtained onto a suitably chosen image under the cyclic group Z N . In this wayeach orbit under Z N is mapped onto a unique representative link pattern. The dimension ofthe corresponding rotation invariant transfer matrix then reduces to [50]dim Z N ( N ) = 1 N (cid:88) m | N ϕ ( N/m ) (cid:18) mm (cid:19) − Cat( N ) , (7.20)where the sum is over the divisors of N , and ϕ ( x ) denotes the Euler totient function. Thefirst 10 values are given by { , , , , , , , , , } . (7.21)But one can go a bit further, since the transfer matrix and boundary states are alsoinvariant under reflections. This gives rise to a symmetry under the dihedral group D N . Thedimension of the rotation-and-reflection invariant transfer matrix then becomes [50]dim D N ( N ) = 12 N (cid:88) m | N ϕ ( N/m ) (cid:18) mm (cid:19) − Cat( N ) + (cid:18) N (cid:98) N/ (cid:99) (cid:19) , (7.22)43f which the first 10 values are { , , , , , , , , , } . (7.23)The process of imposing more and more symmetries and reducing the dimension of therelevant transfer matrices might be realized at the ZRC level by imposing more and moreconstraints on the Q -functions. Consider the ZRC of a closed spin chain with length L = 2 N and N magnons. The corresponding Bethe states are in the S z = 0 sector. In order to restrictto the parity symmetric solutions, we need to impose the condition Q ( u ) = Q ( − u ) for any u . This leads to further constraints to the ZRC and reduces the number of allowed solutionsdown to N ( N ). Since Q ( u ) is a polynomial of order N , the constraint Q ( u ) = Q ( − u ) canalso be imposed by Q ( x k ) = Q ( − x k ) at N different values. Now the main observation is thatat certain values of x k , the constraints have a clear physical meaning. For example, taking x = i/
2, the constraint Q ( i/
2) = Q ( − i/
2) is equivalent to Q ( − i/ Q (+ i/
2) = N (cid:89) k =1 u k + i u k − i = 1 , (7.24)which restricts to the solutions with zero total momentum. It is therefore an interestingquestion to see whether the dihedral symmetry can be realized in this way. If so, at whichfurther value(s) of x k would we need to impose Q ( x k ) = Q ( − x k ) ?We do not presently know if and how one can identify a TL representation whose dimen-sion equals N ( N ) = (cid:0) N (cid:98) N/ (cid:99) (cid:1) given by (4.56). It certainly appears remarkable at this stagethat 2 dim D N ( N ) − dim Z N ( N ) = N ( N ) , (7.25)as already noticed in [50]. It is also worth pointing out that N ( N ) can be interpreted as thenumber of defect-free link patterns on 2 N sites which are symmetric around the mid-point.We leave the further investigation of this question for future work. We have computed the condensation curves in the closed channel, using the TL link-patternrepresentations identified above, namely using: 1) spin-zero (i.e., defect free) link patterns,2) spin-zero link patterns with cyclic symmetry Z N , and 3) spin-zero link patterns withdihedral symmetry.Figure 7.8 shows the results using only the spin-zero constraint. In the regions enclosed bycurves of blue color there is a unique dominant eigenvalue, whereas in the regions enclosedby green (resp. yellow) color the dominant eigenvalue has multiplicity two (resp. three).44 -5-4-3-2-1 0 0 1 2 3 4 5L=8T-pointsEnd-points -5-4-3-2-1 0 0 1 2 3 4 5L=10T-pointsEnd-points -5-4-3-2-1 0 0 1 2 3 4 5L=12T-pointsEnd-points Figure 7.8: Condensation curves for partition function zeros on a (2 M + 1) × N cylinder,in the limit M → ∞ (closed channel), for a system exhibiting only spin-zero symmetry. Thepanels show, in reading direction, the cases N = 3 , , , N ( N ) is expected to be multiplicity-free, so the corresponding condensation curve should be free of green and yellow branches.Nevertheless, the curves in Figure 7.8 are expected to correctly produce the condensationcurves of partition function zeros for any system described by the transfer matrix ˜t D ( u ) andwith boundary states that impose only the spin-zero symmetry, while breaking any othersymmetry (e.g., by imposing spatially inhomogeneous weights).Next we show in Figure 7.9 the results using both the spin-zero and the cyclic symmetry Z N . When compared to Figure 7.8 it can be seen that many branches of the curves areunchanged. However all of the yellow and some of the green curves have now disappeared,reflecting the fact that the eigenvalues which were formerly dominant inside the regionsenclosed by green and yellow colors have now been eliminated from the spectrum, sincethey do not correspond to Z N symmetric eigenstates. The curves in Figure 7.9 should givethe correct condensation curves for systems having the spin-zero and cyclic symmetries.45 -5-4-3-2-1 0 0 1 2 3 4 5L=10T-pointsEnd-points -5-4-3-2-1 0 0 1 2 3 4 5L=12T-pointsEnd-points -5-4-3-2-1 0 0 1 2 3 4 5L=14T-pointsEnd-points Figure 7.9: Condensation curves for partition function zeros on a (2 M + 1) × N cylinder, inthe limit M → ∞ (closed channel), for a system exhibiting spin-zero and cyclic symmetry Z N . The panels show, in reading direction, the cases N = 4 , , , N = 4 , D N .For N = 6 the condensation curve is identical to the one found with cyclic symmetry,meaning that none of the four eliminated eigenvalues (when going from dim Z N (6) = 28 todim D N (6) = 24) was dominant anywhere in the complex u -plane. It should provide thecorrect result for the full “Cooper-pair” symmetry (4.33) if the elimination of four moreeigenvalues (going from dim D N (6) = 24 to N (6) = 20) turned out to be equally innocuous.The N = 7 curve with dihedral symmetry has only branches corresponding to multiplicity-free eigenvalues, so it may also apply to the full paired symmetry, although a greater amountof eigenvalues are redundant in this case.All the curves contain an end-point u e close to the origin for which we have found thefollowing results: 46 -5-4-3-2-1 0 0 1 2 3 4 5L=14T-pointsEnd-points Figure 7.10: Condensation curves for partition function zeros on a (2 M +1) × N cylinder, inthe limit M → ∞ (closed channel), for a system exhibiting spin-zero and dihedral symmetry D N . The panels show the cases N = 6 , N u e ( N ) 0.234690 0.186435 0.154775 0.132364 0.115649 0.102696Im u e ( N ) -0.057271 -0.045012 -0.037154 -0.031665 -0.027604 -0.024475It seems compelling from these data that u e ( N ) → N → ∞ , (7.26)with finite-size correction in both the real and imaginary parts proportional to 1 /N .To finish this section, we now compare the condensation curves with the actual partitionfunction zeros. This is done for N = 4 , , , M = 2048, except for N = 7 where we have only M = 1024; this ensuresan aspect ratio ρ < − in all cases. The agreement with the condensation curves appearsexcellent, with the possible exception of the bubble-shaped region with − < Im u < − N = 7 case. We have computed the exact partition functions for the 6-vertex model on intermediate sizelattices with periodic boundary condition in one direction and free open boundary condi-tions in the other. This work is a natural continuation of a previous work [7] by three of theauthors on the partition function of the 6-vertex model where periodic boundary conditionswere imposed in both directions. The presence of free open boundary conditions brings new47 -5-4-3-2-1 0 0 1 2 3 4 55 x 2048L=10T-pointsEnd-points -5-4-3-2-1 0 0 1 2 3 4 56 x 2048L=12T-pointsEnd-points -5-4-3-2-1 0 0 1 2 3 4 57 x 1024L=14T-pointsEnd-points
Figure 7.11: Comparison between the partition function zeros on a (2 M + 1) × N cylinder,with M = 2048, and the corresponding condensation curves in the M → ∞ limit (closedchannel). The panels show, in reading direction, the cases N = 4 , , , Q -systems for openspin chains [18] and the exact formulae for overlaps between integrable boundary states andBethe states [23–27]. We have also developed powerful algorithms to perform the algebraicgeometry computations, such as the construction of Gr¨obner bases and companion matrices,in the presence of a free parameter .Equipped with these new developments, we obtained the following exact results for thecylinder partition function Z ( u, M, N ). • Open channel.
In the open channel, for M = 1, we obtained a closed-form expression(6.3) valid for any N . For M = 2 , , , ,
6, we have computed the partition functionfor fixed N , both for small values and large values. For small values, we computed N = 2 , , , ,
6. These results are given in appendix F. For large values, we computed48 = 128 , , , , N are notsuitable to be put in the paper, so we have uploaded them as ancillary data files. • Closed channel.
In the closed channel, for N = 1 , ,
3, we obtained closed-form ex-pressions valid for any M . The results are given in (6.6), (6.8) and (6.26), respectively.For N = 4 , , , M , both for small andlarge values. For small values of M , we computed M = 2 , , , , N = 4 , , M , we computed M = 128 , , , , M = 4 , , M = 128 , , , M = 7. The results for large values of M have been used to obtain the zeros of the partition function in the closed channeland are uploaded as ancillary files.We studied the partial thermodynamic limit of the partition function in both channels usingthe exact results. In particular, we computed the zeros of the partition functions in theselimits and found that they condense on certain curves. The condensation curves in the partialthermodynamic limit can be found by a numerical approach based on the BKW theorem.This numerical approach has been applied in the torus case and was further developed inthe current context by taking into account the new features, especially in the closed channel.Comparing the distribution of the zeros obtained from the exact partition function and thecondensation curve obtained from the numerical approach, we found nice agreement andwere able to shed light on several interesting features. The condensation curves in boththe open and closed channels were found to involve very intricate features with multiplebifurcation points and enclosed regions. We believe that the further study of these curvesmight be of independent interest.There are many other questions which deserve further investigation.One of the most interesting directions is to compute the partition function for the q -deformed case. For generic values of q (namely, when q is not a root of unity), Q -systems forboth closed and open chains have been formulated in a recent work [18]. This should providea good starting point for developing an algebro-geometric approach, since QQ -relations aremore efficient than Bethe equations and give only physical solutions. It would presumablybe easier to first study the torus case where the relevant Bethe equations are those of theperiodic XXZ spin chain. After that, one could move to the more complicated cylinder case.We have focused here on the cylinder geometry with free boundary conditions. The caseof fixed boundary conditions (with two arbitrary boundary parameters) may now also be inreach, using the new Q -system [51].From the perspective of algebro-geometric computations, it will be desirable to sharpenthe computational power of our method. For instance, we will try to apply the modern im-49lements of Faug`ere’s F4 algorithm [53], which is in general more efficient than Buchberger’salgorithm. Acknowledgements
ZB and RN are grateful for the hospitality extended to them at the University of Miamiand the Wigner Research Center, respectively. ZB was supported in part by the NKFIHgrant K116505. RN was supported in part by a Cooper fellowship. YZ thanks Janko Boehmfor help on applied algebraic geometry. YJ and YZ acknowledge support from the NSF ofChina through Grant No. 11947301. JLJ acknowledges support from the European ResearchCouncil through the advanced grant NuQFT.
A Basic notions of computational algebraic geometry
In this appendix, we give a brief introduction to some basic notions of computational alge-braic geometry which are used in the main text.
A.1 Polynomial ring and ideal
Polynomial ring
Let us start with the notion of polynomial ring which is denoted by A K [ z , . . . , z n ] or A K for short. It is the set of all polynomials in n variables z , z , . . . , z n whose coefficients are in the field K . In our case, the field is often taken to be the set ofcomplex numbers C or rational numbers Q . Ideal
An ideal I of A K is a subset of A K such that1. f + f ∈ I , if f ∈ I and f ∈ I ,2. gf ∈ I , for f ∈ I and g ∈ A K .Importantly, any ideal I of the polynomial ring A K is finitely generated . This means, for anyideal I , there exists a finite number of polynomials f i ∈ I such that any polynomial F ∈ I can be written as F = k (cid:88) i =1 f i g i , g i ∈ A K . (A.1)We can write I = (cid:104) f , f , . . . , f k (cid:105) . Here the polynomials { f k } are called a basis of the ideal.50 .2 Gr¨obner basis As mentioned before, an ideal is generated by a set of basis { f , . . . , f k } . The choice of thebasis is not unique. Namely, the same ideal can be generated by several different choices ofbases I = (cid:104) f , . . . , f k (cid:105) = . . . = (cid:104) g , . . . , g s (cid:105) . (A.2)Notice that in general k does not have to be the same as s . For many cases, a convenientbasis is needed. For solving polynomial equations and the polynomial reduction problem, itis most convenient to work with a Gr¨obner basis .We can introduce the Gr¨obner basis by considering polynomial reduction . A polynomialreduction of a given polynomial F over a set of polynomials { f , . . . , f k } is given by F = (cid:88) g i f i + r, g i ∈ A K (A.3)where r is a polynomial that cannot be reduced further by any of the f i . The polynomial r iscalled the remainder of the polynomial reduction. One important fact is that the polynomialreduction is not unique for a generic basis { f , . . . , f k } . As a simple example, let us take f = y − f = xy −
1, and consider F ( x, y ) = x y + xy + y . The polynomial reductionof F ( x, y ) over f and f can be performed in two different ways F ( x, y ) = ( x + 1) f + x f + (2 x + 1) , (A.4) F ( x, y ) = f + ( x + y ) f + ( x + y + 1) . As we can see, the remainders are r = 2 x + 1 and r = x + y + 1 respectively. Thereforefor a generic basis, the remainder of the polynomial is not well-defined. A basis of an ideal { g , . . . , g s } is said to be a Gr¨obner basis if the polynomial reduction is well-defined in thesense that the remainder is unique. Now we move to the more formal definition of theGr¨obner basis. Monomial ordering
To define a Gr¨obner basis, we first need to define monomial ordersin the polynomial ring. A monomial order ≺ is specified by the following two rules • If u ≺ v then for any monomial w , we have uw ≺ vw . • If u is non-constant monomial, then 1 ≺ u .Some commonly used orders are lex (Lexicographic), deglex (DegreeLexicographic) and degrevlex (DegreeReversedLexicographic). 51 eading term Once a monomial order ≺ is specified, for any polynomial f , we can definethe leading term uniquely. The leading term, which is denoted by LT( f ), is defined as thehighest monomial of f with respect to the monomial order ≺ . Gr¨obner basis
A Gr¨obner basis G ( I ) of an ideal I with respect to the monomial order ≺ is a basis of the ideal { g , . . . , g s } such that for any f ∈ I , there exists a g i ∈ G ( I ) suchthat LT( f ) is divisible by LT( g i ). A Gr¨obner basis for a given ideal with a monomial ordercan by computed by standard algorithms such as Buchberger algorithm [52] or the F4/F5algorithms [53]. For an ideal I , given a monomial order ≺ , the so-called minimal reducedGr¨obner basis is unique. A.3 Quotient ring and companion matrix
Quotient ring
Given an ideal I , we can define the quotient ring A K /I by the equivalencerelation: f ∼ g if and only if f − g ∈ I .Polynomial reduction and Gr¨obner basis provide a canonical representation of the ele-ments in the quotient ring A K /I . Two polynomials F and F belong to the same elementin the quotient ring A K /I if and only if their remainders of the polynomial reduction are thesame. In particular, f ∈ I if and only if its remainder of the polynomial reduction is zero.This gives a very efficient method to determine whether a polynomial f is in the ideal I ornot. Dimension of quotient ring
To our purpose, the dimension of the quotient ring is im-portant. Consider a system of polynomial equations of n variables f ( z , . . . , z n ) = · · · = f k ( z , . . . , z n ) = 0 , (A.5)we can define the ideal and the quotient ring as I = (cid:104) f , . . . , f k (cid:105) , Q I = C [ z , . . . , z n ] / (cid:104) f , . . . , f k (cid:105) . (A.6)One crucial result is that the linear dimension of the quotient ring dim K Q I equals the numberof the solutions of the system (A.5). Therefore, if the number of solutions of the polynomialequations (A.5) is finite, Q I is a finite dimensional linear space. Let G ( I ) be the Gr¨obnerbasis of the ideal I . The linear space is spanned by monomials which are not divisible byany elements in LT[ G ( I )]. Companion matrix
Another important notion for our applications is the companionmatrix . The main idea is that we can represent any polynomial f ∈ A K as a matrix in the52uotient ring which is a finite dimensional linear space. More precisely, let ( m , . . . , m N ) bethe monomial basis of A K /I , which can be constructed by the Gr¨obner basis G ( I ). Givenany polynomial, we can define an N × N matrix as follows1. Multiply f with one of the basis monomials m i , perform the polynomial reduction withrespect to the Gr¨obner basis G ( I ) and find the remainder r i . It is clear that r i sits inthe quotient ring A K /I .2. Since r i is in the quotient ring, we can expand it in terms of the basis ( m , . . . , m N ),namely r i = c ij m j . In terms of formulas, we can write[ f × m i ] G ( I ) = (cid:88) j c ij m j (A.7)where [ F ] G ( I ) means the remainder of the polynomial reduction of F with respect tothe Gr¨obner basis G ( I ).3. The companion matrix of f is defined by( M f ) ij = c ij . (A.8) Properties of companion matrix
Let us denote the companion matrix of the polyno-mials f and g by M f and M g . It is clear that M f = M g if and only if [ f ] = [ g ] in A K /I .Furthermore, we have the following properties M f + g = M f + M g , M fg = M f M g = M g M f . (A.9)If M g is an invertible matrix, we can actually define the companion matrix of the rationalfunction f /g by M f/g = M f M − g . (A.10)The companion matrix is a powerful tool for computing the sum over solutions of the poly-nomial system (A.5). As we mentioned before, the dimension of Q I equals the number ofsolutions of (A.5). Let us denote the N solutions to be ( (cid:126)ξ , . . . , (cid:126)ξ N ). Then we have thefollowing important result N (cid:88) i =1 f ( (cid:126)ξ i ) = Tr M f . (A.11)53 More details on AG computation
In this section, we summarize the algorithm of our algebra-geometry based partition functioncomputation for this paper. • We first compute the Gr¨obner basis and quotient ring linear basis of the TQ relationequations (3.23) and QQ relation equations (4.51). Note that the TQ and QQ relationscontain the free parameter u . It is possible to compute the corresponding Gr¨obner basisanalytically in u via sophisticated computational algebraic-geometry algorithms, like“slimgb” in the software Singular [54].However, we find that it is more efficient to set u to some integral value, and computethe Groebner basis. The computation is done with the standard Gr¨obner basis com-mand “std” in Singular . In this approach, we maximize the power of parallelizationsince the Groebner basis running time for different values of u is quite uniform. • Then we compute the power of companion matrices ( T M,K ) N . Although from theGr¨obner basis it is straightforward to evaluate ( T M,K ) N , T M,K is usually a dense ma-trix and the matrix product is a heavy computation. Instead, we postpone the matrixcomputations to the end, and evaluate the polynomial power F N first. Here F isthe corresponding polynomial of T M,K . After each polynomial multiplication step, wedivide the polynomial by the ideal’s Gr¨obner basis to save RAM usage by trimminghigh-degree terms. To speed up the computation, we apply the binary strategy, i.e., F N = F N/ F N/ . After F N is calculated, a standard polynomial division computa-tion provides the companion matrix power ( T M,K ) N . So the partition function for aparticular u value is obtained. • In previous steps, u is set as an integral number. To get the analytic partition functionin u , we have to repeat the computation, and then interpolate in u . We know thatfor both the closed and open channel partition functions, the maximum degree in u is2 M N . Hence, we compute the partition function with 2
M N + 1 integer values. Theseresults are then interpolated to the analytic partition function in u . The interpolationis carried out with the Newton polynomial method.The whole computation is powered by our codes in Singular . The parallelization isimplemented in the Gr¨obner basis and companion matrix power steps, for different integervalues of u ’s. The interpolation step is not parallelized, although it is also straightforwardto do so in the future.We remark that through the computations, the coefficient field is chosen to be the rationalnumber field Q . We observe that the resulting analytic partition function contains large-integer coefficients, so finite-field techniques may not speed up the computation.54 The overlap (4.34)
The overlap (4.34) can be deduced from results in the paper by Pozsgay and R´akos [27](based on [23, 24]), to which we refer here by PR. Their R -matrix is given by PR (2.6),which has the same form as ours (2.3), except with a ( u ) = sinh( u + η ) , b ( u ) = sinh( u ) , c ( u ) = sinh( η ) . (C.1)Moreover, they work with the “quantum monodromy matrix” given by PR (2.33) T QT M ( u ) = R N, ( u − η + ω ) R N − , ( u − ω ) . . . R , ( u − η + ω ) R , ( u − ω )= (cid:32) A ( u ) B ( u ) C ( u ) D ( u ) (cid:33) . (C.2)By choosing ω = η − u , (C.3)scaling the variables as u (cid:55)→ (cid:15) u , η (cid:55)→ i(cid:15) , (C.4)and keeping the leading order in (cid:15) , our shifted monodromy matrix (cid:98) T (2 N ) a ( u − i ; { θ j ( u ) } ) (3.2)with alternating inhomogeneities (4.19) can be obtained.In order to relate the generic boundary states PR (2.36)–(2.38) to the dimer state, theboundary parameters in PR (2.15) can be chosen as follows α → ∞ , β = 0 , θ = 0 , (C.5)so that the K-matrices are proportional to the identity matrix. In this limit, the overlap PR(3.4) together with PR (3.3) gives our overlap (4.34). D The relation (4.38)
We show here that the relation (4.38) follows from two simpler lemmata.
Lemma 1: (cid:104) Φ | U † = (cid:18) i v − i (cid:19) N (cid:104) Φ | τ ( v ; { θ j ( v ) } ) . (D.1)55his lemma follows from the observation τ ( v ; { θ j ( v ) } ) = i N R (2 v ) R (2 v ) . . . R N − , N (2 v ) P N − , N − . . . P , P , , (D.2)together with (cid:104) Φ | R (2 v ) R (2 v ) . . . R N − , N (2 v ) = (2 v − i ) N (cid:104) Φ | , (D.3)and (cid:104) Φ | P N − , N − . . . P , P , = ( − N (cid:104) Φ | U † . (D.4)Taking the scalar product of (D.1) with transfer-matrix eigenvectors | u (cid:105) (which are con-structed using B-operators with alternating inhomogeneities { θ j ( v ) } ) and setting v = ˜ u , weobtain (cid:104) Φ | U † | u (cid:105) = (cid:18) i ˜ u − i (cid:19) N (cid:104) Φ | τ ( ˜ u ; { θ j ( ˜ u ) } ) | u (cid:105) = (cid:18) i ˜ u − i (cid:19) N Λ c ( ˜ u ; { θ j ( ˜ u ) } ) (cid:104) Φ | u (cid:105) . (D.5) Lemma 2:
The following relation is valid off shell (cid:104) Φ | u (cid:105) = (cid:18) v − i v + i (cid:19) N (cid:104) u | Φ (cid:105) . (D.6)See e.g. (3.3) in [27], with (C.3) and (C.4).For our case, with v = ˜ u , we have (cid:104) Φ | u (cid:105) = (cid:18) ˜ u − i ˜ u + i (cid:19) N (cid:104) u | Φ (cid:105) . (D.7)Inserting (D.7) in the RHS of (D.5), we obtain (cid:104) Φ | U † | u (cid:105) = (cid:18) i ˜ u + i (cid:19) N Λ c ( ˜ u ; { θ j ( ˜ u ) } ) (cid:104) u | Φ (cid:105) , (D.8)which coincides with (4.38). 56 Parity of states with paired Bethe roots
Following [55, 56], the parity operator Π in the closed channel (length 2 N ) is defined byΠ X n Π − = X N +1 − n , (E.1)where X n is any operator at site n ∈ { , , . . . , N } , and is given byΠ = P , N P , N − . . . P N,N +1 , (E.2)hence Π = Π − = Π † . The parity operator has a simple and beautiful action on the B -operator, namely Π B ( u ) Π = − B ( − u ) , (E.3)while the reference state (4.13) remains invariant under parityΠ | (cid:105) = | (cid:105) . (E.4)It follows from (E.3) and (E.4) that Bethe states (4.14) corresponding to the paired Betheroots (4.33) (even N ) are eigenstates of parity with eigenvalue +1Π | u , − u , . . . , u N , − u N (cid:105) = | u , − u , . . . , u N , − u N (cid:105) . (E.5)Similarly, Bethe states corresponding to the paired Bethe roots (4.40) (odd N ) are eigenstatesof parity with eigenvalue − | u , − u , . . . , u N − , − u N − , (cid:105) = −| u , − u , . . . , u N − , − u N − , (cid:105) . (E.6)The dimer state | Φ (cid:105) (4.23) is an eigenstate of parity with eigenvalue ( − N Π | Φ (cid:105) = ( − N | Φ (cid:105) , (E.7)which is consistent with the fact that the overlaps (cid:104) Φ | u (cid:105) are nonzero only for Bethe stateswith paired Bethe roots (4.33), (4.40). F Exact partition functions
In this appendix, we list all the exact partition functions for 2 ≤ M, N ≤ M N , the partition function can be computed in both channels. As a consistency check,computations in the two channels give the same result, as it should be. F.1 N = 6 Z , = 2 u + 288 iu − u − iu + 8317584 u + 127125504 iu (F.1) − u − iu + 138130609500 u + 1041934800608 iu − u − iu + 217832067312960 u + 1049571874084608 iu − u − iu + 68788394401561470 u + 235662122430397008 iu − u − iu + 6098355792769470156 u + 15707717731534712832 iu − u − iu + 182272478003908656432 u + 364591770627856882608 iu − u − iu + 2047050659153089585764 u + 3243098453976454684320 iu − u − iu + 9226096556900912648334 u + 11712585761982875838624 iu − u − iu + 17319816398521272862676 u + 17719150885149621701664 iu − u − iu + 13745313212235461309790 u + 11338458117413798884752 iu − u − iu + 4587036228059022522828 u + 3037167694581592157600 iu − u − iu + 626974446577452175698 u + 329820882193355585184 iu − u − iu + 33342588484553082888 u + 13690734769961746560 iu − u − iu + 633003570392541120 u + 197095658448168960 iu − u − iu + 3728675863226880 u + 839585772859392 iu − u − iu + 5386833271296 u + 806617128960 iu − u − iu + 1206835200 u + 98304000 iu − u − iu + 8192 , = 2 u + 240 iu − u − iu + 4106064 u + 50944128 iu (F.2) − u − iu + 28034725284 u + 167372075712 iu − u − iu + 17320477694784 u + 65956540414464 iu − u − iu + 2108994608155482 u + 5652786955676832 iu − u − iu + 68261467147861284 u + 135034284911783616 iu − u − iu + 686116309138082856 u + 1029410431077285360 iu − u − iu + 2340240191331772812 u + 2702070900176216160 iu − u − iu + 2827960377154344960 u + 2525039773174991216 iu − u − iu + 1218539894552498448 u + 838213811707493088 iu − u − iu + 181939121823624930 u + 95158509487089840 iu − u − iu + 8777163928275864 u + 3401716413897600 iu − u − iu + 120092906906976 u + 32925815343360 iu − u − iu + 368278769664 u + 65577553920 iu − u − iu + 159490560 u + 15134720 iu − u − iu + 2048 Z , = 8 u + 336 iu − u − iu + 1484064 u + 13860000 iu (F.3) − u − iu + 3461257458 u + 15858754400 iu − u − iu + 716128871152 u + 2039662782720 iu − u − iu + 26092620286092 u + 50742484822368 iu − u − iu + 222439906309668 u + 308118607243360 iu − u − iu + 503911508965434 u + 505852778886192 iu − u − iu + 316601908242108 u + 230384778687264 iu − u − iu + 53851823641602 u + 27924428347584 iu − u − iu + 2242521937456 u + 795411629952 iu − u − iu + 18332792928 u + 4082245632 iu − u − iu + 18380160 u + 2105344 iu − u − iu + 512 , = 2 u + 144 iu − u − iu + 371556 u + 2500704 iu (F.4) − u − iu + 222050034 u + 710461408 iu − u − iu + 10235477796 u + 19420445664 iu − u − iu + 67208955660 u + 81794924496 iu − u − iu + 78198825096 u + 62348094624 iu − u − iu + 16442440386 u + 8405591184 iu − u − iu + 534186864 u + 162405504 iu − u − iu + 1700640 u + 249600 iu − u − iu + 128 Z , = 2 u + 96 iu − u − iu + 67908 u + 267360 iu (F.5) − u − iu + 4286358 u + 7635536 iu − u − iu + 16767884 u + 15970080 iu − u − iu + 5311314 u + 2637792 iu − u − iu + 103512 u + 22016 iu − u − iu + 32 F.2 N = 5 Z , = 2 u + 240 iu − u − iu + 4185570 u + 53344848 iu (F.6) − u − iu + 32597238180 u + 201557883000 iu − u − iu + 22813265943960 u + 89030269264680 iu − u − iu + 3032155944987750 u + 8282078769490080 iu − u − iu + 105537643218985020 u + 212434648891515000 iu − u − iu + 1136284932112092120 u + 1734011186676432768 iu − u − iu + 4148085867425363430 u + 4871950812684428040 iu − u − iu + 5370865405192114860 u + 4880969524493532440 iu − u − iu + 2487244973239722900 u + 1743350380048456200 iu − u − iu + 401316693215927490 u + 214260526736194800 iu − u − iu + 21095469547685670 u + 8367142598539080 iu − u − iu + 318402566218770 u + 89734056144240 iu − u − iu + 1103026376160 u + 203688568320 iu − u − iu + 561016320 u + 55685120 iu − u − iu + 8192 , = 2 u + 200 iu − u − iu + 1816720 u + 18696280 iu (F.7) − u − iu + 5980862700 u + 29622561920 iu − u − iu + 1680469080490 u + 5162580577360 iu − u − iu + 83189622229650 u + 175077243069600 iu − u − iu + 981612780573350 u + 1481512519939600 iu − u − iu + 3180146358121680 u + 3515139558674328 iu − u − iu + 2998653880187200 u + 2438862766979800 iu − u − iu + 820389106281650 u + 486066456229800 iu − u − iu + 61002271510150 u + 25593377506000 iu − u − iu + 1057993826450 u + 296565427400 iu − u − iu + 3151663680 u + 524266240 iu − u − iu + 780800 u + 51200 iu − Z , = 2 u + 160 iu − u − iu + 728000 u + 5674392 iu (F.8) − u − iu + 752228880 u + 2769883720 iu − u − iu + 63775686230 u + 143577146200 iu − u − iu + 864503010900 u + 1283718138360 iu − u − iu + 2348251006860 u + 2372768013000 iu − u − iu + 1381238400930 u + 950393607264 iu − u − iu + 168978930950 u + 76875351000 iu − u − iu + 3584938290 u + 996761760 iu − u − iu + 8454080 u + 1185280 iu − u − iu + 512 Z , = 2 u + 120 iu − u − iu + 182970 u + 1000824 iu (F.9) − u − iu + 45931380 u + 116188080 iu − u − iu + 785325960 u + 1134545880 iu − u − iu + 1599383250 u + 1395092760 iu − u − iu + 428924070 u + 221637600 iu − u − iu + 12523890 u + 3435384 iu − u − iu + 19680 u + 1920 iu − , = 32 u + 320 iu − u − iu + 28560 u + 87696 iu (F.10) − u − iu + 747090 u + 1031120 iu − u − iu + 880230 u + 572520 iu − u − iu + 46290 u + 12240 iu − u − iu + 32 F.3 N = 4 Z , = 2 u + 192 iu − u − iu + 1695768 u + 17380704 iu (F.11) − u − iu + 5456197416 u + 26727665408 iu − u − iu + 1440779616064 u + 4325994981888 iu − u − iu + 64075191370944 u + 130502828494848 iu − u − iu + 654563752915968 u + 947664700456960 iu − u − iu + 1769626440158208 u + 1857530670661632 iu − u − iu + 1333448855969280 u + 1017287720361984 iu − u − iu + 275828260637184 u + 150674393284608 iu − u − iu + 14290811226112 u + 5382179758080 iu − u − iu + 151954421760 u + 36599955456 iu − u − iu + 217178112 u + 27590656 iu − u − iu + 8192 Z , = 2 u + 160 iu − u − iu + 789416 u + 6439584 iu (F.12) − u − iu + 988466328 u + 3799026688 iu − u − iu + 97056390560 u + 225028607488 iu − u − iu + 1472970252384 u + 2247880822272 iu − u − iu + 4461712923648 u + 4633176305664 iu − u − iu + 2934531772416 u + 2079427166208 iu − u − iu + 405738771968 u + 190745063424 iu − u − iu + 9914835456 u + 2874777600 iu − u − iu + 28375040 u + 4218880 iu − u − iu + 2048 , = 2 u + 128 iu − u − iu + 275920 u + 1721920 iu (F.13) − u − iu + 113839344 u + 324510720 iu − u − iu + 3195196320 u + 5276663040 iu − u − iu + 11469719520 u + 11721735168 iu − u − iu + 6162514560 u + 3908175360 iu − u − iu + 458877312 u + 169820160 iu − u − iu + 3218944 u + 581632 iu − u − iu + 512 Z , = 2 u + 96 iu − u − iu + 76656 u + 321984 iu (F.14) − u − iu + 6375696 u + 12024320 iu − u − iu + 30396896 u + 30257664 iu − u − iu + 11701920 u + 6151680 iu − u − iu + 290688 u + 68096 iu − u − iu + 128 Z , = 2 u + 64 iu − u − iu + 9928 u + 24864 iu (F.15) − u − iu + 85176 u + 80128 iu − u − iu + 16864 u + 5888 iu − u − iu + 32 F.4 N = 3 Z , = 2 u + 144 iu − u − iu + 566172 u + 4421808 iu (F.16) − u − iu + 585564318 u + 2106246952 iu − u − iu + 41667849936 u + 87178071096 iu − u − iu + 400109679918 u + 534347671248 iu − u − iu + 677171433726 u + 594340399176 iu − u − iu + 211359220146 u + 119998562832 iu − u − iu + 10758322656 u + 3726162432 iu − u − iu + 62295552 u + 10997760 iu − u − iu + 8192 , = 2 u + 120 iu − u − iu + 254340 u + 1570464 iu (F.17) − u − iu + 99104634 u + 274915920 iu − u − iu + 2379520968 u + 3712304088 iu − u − iu + 6573306816 u + 6196463352 iu − u − iu + 2424284226 u + 1362333816 iu − u − iu + 102169152 u + 31401216 iu − u − iu + 268800 u + 30720 iu − Z , = 8 u + 168 iu − u − iu + 91506 u + 418992 iu (F.18) − u − iu + 10394316 u + 20622512 iu − u − iu + 60175370 u + 62991480 iu − u − iu + 28353042 u + 15688800 iu − u − iu + 896448 u + 227840 iu − u − iu + 512 Z , = 2 u + 72 iu − u − iu + 22914 u + 71568 iu − u (F.19) − iu + 488124 u + 605480 iu − u − iu + 341826 u + 184392 iu − u − iu + 6624 u + 1152 iu − Z , = 2 u + 48 iu − u (F.20) − iu + 2934 u + 4968 iu − u − iu + 4338 u + 2224 iu − u − iu + 32 F.5 N = 2 Z , = 2 u + 96 iu − u − iu + 131208 u + 717696 iu (F.21) − u − iu + 28851648 u + 66013184 iu − u − iu + 275340800 u + 318861312 iu − u − iu + 193300992 u + 119365632 iu − u − iu + 9934848 u + 2883584 iu − u − iu + 8192 , = 2 u + 80 iu − u − iu + 57240 u + 245376 iu (F.22) − u − iu + 4300128 u + 7203584 iu − u − iu + 10543104 u + 8245248 iu − u − iu + 1245696 u + 430080 iu − u − iu + 2048 Z , = 2 u + 64 iu − u − iu + 20080 u + 62592 iu (F.23) − u − iu + 361824 u + 397312 iu − u − iu + 133504 u + 57344 iu − u − iu + 512 Z , = 2 u + 48 iu − u − iu (F.24)+ 4848 u + 9600 iu − u − iu + 11040 u + 6400 iu − u − iu + 128 References [1] H. J. de Vega and F. Woynarovich, “Method for calculating finite-size corrections in BetheAnsatz systems: Heisenberg chain and six-vertex model,”
Nucl. Phys. B (1985) 439.[2] F. Woynarovich and H.-P. Eckle, “Finite-size corrections and numerical calculations for longspin 1/2 Heisenberg chains in the critical region,”
J. Phys.
A20 (1987) L97.[3] C. J. Hamer, G. R. W. Quispel, and M. T. Batchelor, “Conformal anomaly and surfaceenergy for Potts and Ashkin-teller quantum chains,”
J. Phys.
A20 (1987) 5677.[4] P. A. Pearce and A. Kl¨umper, “Finite-size corrections and scaling dimensions of solvablelattice models: An analytic method,”
Phys. Rev. Lett. (1991) 974.[5] C. Destri and H. J. de Vega, “Unified approach to thermodynamic Bethe Ansatz and finitesize corrections for lattice models and field theories,” Nucl. Phys. B (1994) 413.[6] E. Granet, J. L. Jacobsen, and H. Saleur, “A distribution approach to finite-size correctionsin Bethe Ansatz solvable models,”
Nucl. Phys. B (2018) 96–117, arXiv:1801.05676 .[7] J. L. Jacobsen, Y. Jiang, and Y. Zhang, “Torus partition function of the six-vertex modelfrom algebraic geometry,”
JHEP (2019) 152, arXiv:1812.00447 [hep-th] .[8] C. Marboe and D. Volin, “Fast analytic solver of rational Bethe equations,” J. Phys.
A50 no. 20, (2017) 204002, arXiv:1608.06504 [math-ph] .
9] Y. Jiang and Y. Zhang, “Algebraic geometry and Bethe ansatz. Part I. The quotient ring forBAE,”
JHEP (2018) 087, arXiv:1710.04693 [hep-th] .[10] A. L. Owczarek and R. J. Baxter, “Surface free energy of the critical six-vertex model withfree boundaries,” J. Phys.
A22 (1989) 1141.[11] E. K. Sklyanin, “Boundary Conditions for Integrable Quantum Systems,”
J. Phys.
A21 (1988) 2375.[12] C. Destri and H. J. de Vega, “Bethe ansatz and quantum groups: the light cone latticeapproach. 1. Six vertex and SOS models,”
Nucl. Phys.
B374 (1992) 692–719.[13] C. M. Yung and M. T. Batchelor, “Integrable vertex and loop models on the square latticewith open boundaries via reflection matrices,”
Nucl. Phys.
B435 (1995) 430–462, arXiv:hep-th/9410042 [hep-th] .[14] S. Beraha, J. Kahane, and N. J. Weiss, “Limits of zeroes of recursively defined polynomials,”
Proc. Natl. Acad. Sci. (1975) 4209–4209.[15] A. M. Gainutdinov, W. Hao, R. I. Nepomechie, and A. J. Sommese, “Counting solutions ofthe Bethe equations of the quantum group invariant open XXZ chain at roots of unity,” J.Phys.
A48 no. 49, (2015) 494003, arXiv:1505.02104 [math-ph] .[16] R. I. Nepomechie and C. Wang, “Algebraic Bethe ansatz for singular solutions,”
J. Phys.
A46 (2013) 325002, arXiv:1304.7978 [hep-th] .[17] E. Granet and J. L. Jacobsen, “On zero-remainder conditions in the Bethe ansatz,”
JHEP (2020) 178, arXiv:1910.07797 [hep-th] .[18] Z. Bajnok, E. Granet, J. L. Jacobsen, and R. I. Nepomechie, “On Generalized Q -systems,” JHEP (2020) 177, arXiv:1910.07805 [hep-th] .[19] D. Cox, J. Little, and D. O’Shea, Using Algebraic Geometry . Springer, 1998.[20] D. Cox, J. Little, and D. O’Shea,
Ideals, Varieties, and Algorithms . Springer, 2007.[21] M. Gaudin, B. M. McCoy, and T. T. Wu, “Normalization sum for the Bethe’s hypothesiswave functions of the Heisenberg-Ising chain,”
Phys. Rev.
D23 no. 2, (1981) 417.[22] V. E. Korepin, “Calculation of norms of Bethe wave functions,”
Commun. Math. Phys. (1982) 391–418.[23] B. Pozsgay, “Overlaps between eigenstates of the XXZ spin-1/2 chain and a class of simpleproduct states,” J. Stat. Mech. no. 6, (2014) P06011, arXiv:1309.4593[cond-mat.stat-mech] .
24] M. Brockmann, J. De Nardis, B. Wouters, and J.-S. Caux, “A Gaudin-like determinant foroverlaps of Neel and XXZ Bethe states ,”
J. Phys. A no. 14, (2014) 145003, arXiv:1401.2877 [cond-mat.stat-mech] .[25] M. Brockmann, J. De Nardis, B. Wouters, and J.-S. Caux, “Neel-XXZ state overlaps: oddparticle numbers and Lieb-Liniger scaling limit ,” J. Phys. A (2014) 345003,, arXiv:1403.7469 [cond-mat.stat-mech] .[26] L. Piroli, B. Pozsgay, and E. Vernier, “What is an integrable quench?,” Nucl. Phys.
B925 (2017) 362–402, arXiv:1709.04796 [cond-mat.stat-mech] .[27] B. Pozsgay and O. Rakos, “Exact boundary free energy of the open XXZ chain witharbitrary boundary conditions,”
J. Stat. Mech. no. 11, (2018) 113102, arXiv:1804.09992 [cond-mat.stat-mech] .[28] M. de Leeuw, C. Kristjansen, and K. Zarembo, “One-point Functions in Defect CFT andIntegrability,”
JHEP (2015) 098, arXiv:1506.06958 [hep-th] .[29] I. Buhl-Mortensen, M. de Leeuw, C. Kristjansen, and K. Zarembo, “One-point Functions inAdS/dCFT from Matrix Product States,” JHEP (2016) 052, arXiv:1512.02532[hep-th] .[30] M. de Leeuw, C. Kristjansen, and S. Mori, “AdS/dCFT one-point functions of the SU(3)sector,” Phys. Lett.
B763 (2016) 197–202, arXiv:1607.03123 [hep-th] .[31] O. Tsuchiya, “Determinant formula for the six-vertex model with reflecting end,”
J. Math.Phys. (1998) 5946–5951, arXiv:solv-int/9804010 [nlin.SI] .[32] K. K. Kozlowski and B. Pozsgay, “Surface free energy of the open XXZ spin-1/2 chain,” J.Stat. Mech. (2012) P05021, arXiv:1201.5884 [nlin.SI] .[33] V. Pasquier and H. Saleur, “Common Structures Between Finite Systems and ConformalField Theories Through Quantum Groups,”
Nucl. Phys.
B330 (1990) 523–556.[34] C. N. Yang and T. D. Lee, “Statistical theory of equations of state and phase transitions. I.Theory of condensation,”
Phys. Rev. (Aug, 1952) 404–409. https://link.aps.org/doi/10.1103/PhysRev.87.404 .[35] M. Fisher, “The nature of critical points,” in Lecture notes in theoretical physics , W. Brittin,ed., vol. 7c, pp. 1–159. University of Colorado Press, Boulder, 1965.[36] J. Salas and A. D. Sokal, “Transfer matrices and partition-function zeros forantiferromagnetic Potts models. I. General theory and square-lattice chromatic polynomial,”
J. Stat. Phys. (2001) 609–699, arXiv:cond-mat/0004330 .
37] J. L. Jacobsen and J. Salas, “Transfer matrices and partition-function zeros forantiferromagnetic Potts models. II. Extended results for square-lattice chromaticpolynomial,”
J. Stat. Phys. (2001) 701–723, arXiv:cond-mat/0011456 .[38] J. L. Jacobsen, J. Salas, and A. D. Sokal, “Transfer matrices and partition-function zeros forantiferromagnetic Potts models. III. Triangular-lattice chromatic polynomial,”
J. Stat. Phys. (2003) 921–1017, arXiv:cond-mat/0204587 .[39] J. L. Jacobsen and J. Salas, “Transfer matrices and partition-function zeros forantiferromagnetic Potts models. IV. Chromatic polynomial with cyclic boundary conditions,”
J. Stat. Phys. (2006) 705–760, arXiv:cond-mat/0407444 .[40] J. L. Jacobsen and J. Salas, “Phase diagram of the chromatic polynomial on a torus,”
Nucl.Phys. B (2007) 238–296, arXiv:cond-mat/0703228 .[41] J. L. Jacobsen and J. Salas, “A generalized Beraha conjecture for non-planar graphs,”
Nucl.Phys. B (2013) 678–718, arXiv:1303.5210 .[42] J. L. Jacobsen, J. Salas, and C. R. Scullard, “Phase diagram of the triangular-lattice Pottsantiferromagnet,”
J. Phys. A: Math. Theor. (2017) 345002, arXiv:1702.02006 .[43] C. Itzykson and J.-M. D. Drouffe, Statistical Field Theory: Volume 1, from Brownian Motionto Renormalization and Lattice Gauge Theory . Cambridge University Press, 1991.[44] H. Saleur and M. Bauer, “On Some Relations Between Local Height Probabilities andConformal Invariance,”
Nucl. Phys.
B320 (1989) 591–624.[45] J.-F. Richard and J. L. Jacobsen, “Character decomposition of Potts model partitionfunctions, I: Cyclic geometry,”
Nucl. Phys.
B750 (2006) 250–264, arXiv:math-ph/0605016[math-ph] .[46] D. A. Bini and G. Fiorentino, “Design, analysis, and implementation of a multiprecisionpolynomial rootfinder,”
Numer. Algorithms no. 2-3, (2000) 127–173. https://doi.org/10.1023/A:1019199917103 .[47] D. A. Bini and L. Robol, “Solving secular and polynomial equations: A multiprecisionalgorithm,” J. Comp. Appl. Math. (2015) 276–292.[48] L. W. Ehrlich, “A modified Newton method for polynomials,”
Comm. of the ACM (1967)107–108.[49] O. Aberth, “Iteration methods for finding all zeros of a polynomial simultaneously,” Math.Comp. (1973) 339–344.[50] S.-C. Chang, J. L. Jacobsen, J. Salas, and R. Shrock, “Exact Potts model partition functionsfor strips of the triangular lattice,” J. Stat. Phys. (2004) 763.
51] R. I. Nepomechie, “Q-systems with boundary parameters,” arXiv:1912.12702 [hep-th] .[52] B. Buchberger,
A theoretical basis for the reduction of polynomials to canonical forms , SIGSAM Bull. (Aug., 1976) 19–29.[53] J.-C. Faug`ere, “A new efficient algorithm for computing Gr¨obner bases (F4),” J. Pure Appl.Algebra (1999) 61–88.[54] W. Decker, G.-M. Greuel, G. Pfister, and H. Sch¨onemann, “
Singular , 2019.[55] A. Doikou and R. I. Nepomechie, “Discrete symmetries and S matrix of the XXZ chain,”
J.Phys.
A31 (1998) L621–L628, arXiv:hep-th/9808012 [hep-th] .[56] A. Doikou and R. I. Nepomechie, “Parity and charge conjugation symmetries and S matrix ofthe XXZ chain,” in
Statistical Physics on the Eve of the Twenty-First Century , M. Batchelorand L. Wille, eds., pp. 391–411. World Scientific, 1999. arXiv:hep-th/9810034 [hep-th] ..