arXiv subjects

Shi-shyr Roan

Publications and source records attributed to Shi-shyr Roan.

At least 19 recordsLinked to original sources

arXiv2013

Fermat Surface and Group Theory in Symmetry of Rapidity Family in Chiral Potts Model

The present paper discusses various mathematical aspects about the rapidity symmetry in chiral Potts model (CPM) in the context of algebraic geometry and group theory . We re-analyze the symmetry group of a rapidity curve in $N$-state CPM, explore the universal group structure for all $N$, and further enlarge it to modular symmetries of the complete rapidity family in CPM. As will be shown in the article that all rapidity curves in $N$-state CPM constitute a Fermat hypersurface in $\PZ^3$ of degree 2N as the natural generalization of the Fermat K3 elliptic surface $(N=2)$, we conduct a thorough algebraic geometry study about the rapidity fibration of Fermat surface and its reduced hyperelliptic fibration via techniques in algebraic surface theory. Symmetries of rapidity family in CPM and hyperelliptic family in $τ^{(2)}$-model are exhibited through the geometrical representation of the universal structural group in mathematics.

math.AG↗

arXiv2012

Quantum Group Theory in $τ^{(2)}$-model, Duality of $τ^{(2)}$-model and XXZ-model with Cyclic ${\bf U_q(sl_2)}$-representation for ${\bf q^n =1}$, and Chiral Potts Model

We identify the quantum group ${\Large\textsl{U}}_\textsl{w}(sl_2)$ in the $L$-operator of $τ^{(2)}$-model for a generic $\textsl{w}$ as a subalgebra of $U_{\sf q} (sl_2)$ with $\textsl{w} = {\sf q}^{-2}$. In the roots of unity case, ${\sf q}=q, \textsl{w} = ω$ with $q^{\bf n} = ω^N = 1$, the eigenvalues and eigenvectors of XXZ-model with the $U_q (sl_2)$-cyclic representation are determined by the $τ^{(2)}$-model with the induced ${\Large\textsl{U}}_ω(sl_2)$-cyclic representation, which is decomposed as a finite sum of $τ^{(2)}$-models in non-superintegrable inhomogeneous $N$-state chiral Potts model. Through the theory of chiral Potts model, the $Q$-operator of XXZ-model can be identified with the related chiral Potts transfer matrices, with special features appeared in the ${\bf n}=2N$, e.g. $N$ even, case. We also establish the duality of $τ^{(2)}$-models related to cyclic representations of $U_q (sl_2)$, analogous to the $τ^{(2)}$-duality in chiral Potts model; and identify the model dual to the XXZ model with $U_q (sl_2)$-cyclic representation.

math-ph↗

arXiv2011

Eigenvectors of an Arbitrary Onsager Sector in Superintegrable $τ^{(2)}$-model and Chiral Potts Model

We study the eigenvector problem in homogeneous superintegrable $N$-state chiral Potts model (CPM) by the symmetry principal. Using duality symmetry and (spin-)inversion in CPM, together with Onsager-algebra symmetry and $sl_2$-loop-algebra symmetry of the superintegrable $τ^{(2)}$-model, we construct the complete $k'$-dependent CPM-eigenvectors in the local spin basis for an arbitrary Onsager sector. In this paper, we present the complete classification of quantum numbers of superintegrable $τ^{(2)}$-model. Accordingly, there are four types of sectors. The relationships among Onsager sectors under duality and inversion, together with their Bethe roots and CPM-eigenvectors, are explicitly found. Using algebraic-Bethe-ansatz techniques and duality of CPM, we construct the Bethe states and the Fabricius-McCoy currents of the superintegrable $τ^{(2)}$-model through its equivalent spin-$\frac{N-1}{2}$-XXZ chain. The $τ^{(2)}$-eigenvectors in a sector are derived from the Bethe state and the $sl_2$-product structure determined by the Fabricius-McCoy current of the sector. From those $τ^{(2)}$-eigenvectors, the $k'$-dependence of CPM state vectors in local-spin-basis form is obtained by the Onsager-algebra symmetry of the superintegrable chiral Potts quantum chain.

math-ph↗

arXiv2011

Toric Representation and Positive Cone of Picard Group and Deformation Space in Mirror Symmetry of Calabi-Yau Hypersurfaces in Toric Varieties

We derive the combinatorial representations of Picard group and deformation space of anti-canonical hypersurfaces of a toric variety using techniques in toric geometry. The mirror cohomology correspondence in the context of mirror symmetry is established for a pair of Calabi-Yau (CY) ${\sf n}$-spaces in toric varieties defined by reflexive polytopes for an arbitrary dimension ${\sf n}$. We further identify the Kahler cone of the toric variety and degeneration cone of CY hypersurfaces, by which the Kahler cone and degeneration cone for a mirror CY pair are interchangeable under mirror symmetry. In particular, different degeneration cones of a CY 3-fold are corresponding to flops of its mirror 3-fold.

math.AG↗

arXiv2011

Picard Groups of Hypersurfaces in Toric Varieties

We study the structure of rational Picard groups of hypersurfaces of toric varieties. By using the fan structure associated to the ambient toric variety, an explicit basis of the Picard group is described by certain combinatorial data. We shall also discuss the application to Calabi-Yau spaces.

math.AG↗

arXiv2010

Duality and Symmetry in Chiral Potts Model

We discover an Ising-type duality in the general $N$-state chiral Potts model, which is the Kramers-Wannier duality of planar Ising model when N=2. This duality relates the spectrum and eigenvectors of one chiral Potts model at a low temperature (of small $k'$) to those of another chiral Potts model at a high temperature (of $k'^{-1}$). The $τ^{(2)}$-model and chiral Potts model on the dual lattice are established alongside the dual chiral Potts models. With the aid of this duality relation, we exact a precise relationship between the Onsager-algebra symmetry of a homogeneous superintegrable chiral Potts model and the $sl_2$-loop-algebra symmetry of its associated spin-$\frac{N-1}{2}$ XXZ chain through the identification of their eigenstates.

cond-mat.stat-mech↗

arXiv2009

On $τ^{(2)}$-model in Chiral Potts Model and Cyclic Representation of Quantum Group $U_q(sl_2)$

We identify the precise relationship between the five-parameter $τ^{(2)}$-family in the $N$-state chiral Potts model and XXZ chains with $U_q (sl_2)$-cyclic representation. By studying the Yang-Baxter relation of the six-vertex model, we discover an one-parameter family of $L$-operators in terms of the quantum group $U_q (sl_2)$. When $N$ is odd, the $N$-state $τ^{(2)}$-model can be regarded as the XXZ chain of $U_{\sf q} (sl_2)$ cyclic representations with ${\sf q}^N=1$. The symmetry algebra of the $τ^{(2)}$-model is described by the quantum affine algebra $U_{\sf q} (\hat{sl}_2)$ via the canonical representation. In general for an arbitrary $N$, we show that the XXZ chain with a $U_q (sl_2)$-cyclic representation for $q^{2N}=1$ is equivalent to two copies of the same $N$-state $τ^{(2)}$-model.

cond-mat.stat-mech↗

arXiv2008

Bethe Equation of $τ^{(2)}$-model and Eigenvalues of Finite-size Transfer Matrix of Chiral Potts Model with Alternating Rapidities

We establish the Bethe equation of the $τ^{(2)}$-model in the $N$-state chiral Potts model (including the degenerate selfdual cases) with alternating vertical rapidities. The eigenvalues of a finite-size transfer matrix of the chiral Potts model are computed by use of functional relations. The significance of the "alternating superintegrable" case of the chiral Potts model is discussed, and the degeneracy of $τ^{(2)}$-model found as in the homogeneous superintegrable chiral Potts model.

cond-mat.stat-mech↗

arXiv2008

On the Equivalent Theory of the Generalized τ^{(2)}-model and the Chiral Potts Model with two Alternating Vertical Rapidities

By the Baxter's $Q_{72}$-operator method, we demonstrate the equivalent theory between the generalized $τ^{(2)}$-model (other than two special cases with a pseudovacuum state) and the $N$-state chiral Potts model with two alternating vertical rapidities, where the degenerate models are included. As a consequence, the theory of the XXZ chain model associated to cyclic representations (with the parameter $ς$) of $U_{\sf q}(sl_2)$ with ${\sf q}^N=1$ for odd $N$ is identified with either (for $ς^N=1$) the chiral Potts model with two superintegrable vertical rapidities, or (for $ς^N \neq 1$) the degenerate model for the selfdual solution of the star-triangle relation. In all these identifications, the transfer matrices $T, \hat{T}$ of the chiral Potts model (including the degenerate ones) serve as the $Q_R, Q_L$-operators of the corresponding $τ^{(2)}$-model, so that the functional relations hold as in the solvable $N$-state chiral Potts model.

cond-mat.stat-mech↗

arXiv2007

The Transfer Matrix of Superintegrable Chiral Potts Model as the Q-operator of Root-of-unity XXZ Chain with Cyclic Representation of $U_q(sl_2)$

We demonstrate that the transfer matrix of the inhomogeneous $N$-state chiral Potts model with two vertical superintegrable rapidities serves as the $Q$-operator of XXZ chain model for a cyclic representation of $U_{\sf q}(sl_2)$ with $N$th root-of-unity ${\sf q}$ and representation-parameter for odd $N$. The symmetry problem of XXZ chain with a general cyclic $U_{\sf q}(sl_2)$-representation is mapped onto the problem of studying $Q$-operator of some special one-parameter family of generalized $τ^{(2)}$-models. In particular, the spin-$\frac{N-1}{2}$ XXZ chain model with ${\sf q}^N=1$ and the homogeneous $N$-state chiral Potts model at a specific superintegrable point are unified as one physical theory. By Baxter's method developed for producing $Q_{72}$-operator of the root-of-unity eight-vertex model, we construct the $Q_R, Q_L$- and $Q$-operators of a superintegrable $τ^{(2)}$-model, then identify them with transfer matrices of the $N$-state chiral Potts model for a positive integer $N$. We thus obtain a new method of producing the superintegrable $N$-state chiral Potts transfer matrix from the $τ^{(2)}$-model by constructing its $Q$-operator.

cond-mat.stat-mech↗

arXiv2007

The Q-operator and Functional Relations of the Eight-vertex Model at Root-of-unity $η= \frac{2m K}{N}$ for odd N

Following Baxter's method of producing Q_{72}-operator, we construct the Q-operator of the root-of-unity eight-vertex model for the crossing parameter $η= \frac{2m K}{N}$ with odd $N$ where Q_{72} does not exist. We use this new Q-operator to study the functional relations in the Fabricius-McCoy comparison between the root-of-unity eight-vertex model and the superintegrable N-state chiral Potts model. By the compatibility of the constructed Q-operator with the structure of Baxter's eight-vertex (solid-on-solid) SOS model, we verify the set of functional relations of the root-of-unity eight-vertex model using the explicit form of the Q-operator and fusion weights of SOS model.

cond-mat.stat-mech↗

arXiv2007

Structure of Certain Chebyshev-type Polynomials in Onsager's Algebra Representation

In this report, we present a systematic account of mathematical structures of certain special polynomials arisen from the energy study of the superintegrable $N$-state chiral Potts model with a finite number of sizes. The polynomials of low-lying sectors are represented in two different forms, one of which is directly related to the energy description of superintegrable chiral Potts $\ZZ_N$-spin chain via the representation theory of Onsager's algebra. Both two types of polynomials satisfy some $(N+1)$-term recurrence relations, and $N$th order differential equations; polynomials of one kind reveal certain Chebyshev-like properties. Here we provide a rigorous mathematical argument for cases $N=2, 3$, and further raise some mathematical conjectures on those special polynomials for a general $N$.

math-ph↗

arXiv2007

On $Q$-operators of XXZ Spin Chain of Higher Spin

We provide two methods of producing the $Q$-operator of XXZ spin chain of higher spin, one for $N$th root-of-unity $q$ with odd $N$ and another for a general $q$, as the generalization of those known in the six-vertex model. In the root-of-unity case, we discuss the functional relations involving the constructed $Q$-operator for the symmetry study of the theory. The $Q$-operator of XXZ chain of higher spin for a generic $q$ is constructed by extending Baxter's argument in spin-1/2 case for the six-vertex $Q$-operator.

cond-mat.stat-mech↗

arXiv2007

Bethe Ansatz and Symmetry in Superintegrable Chiral Potts Model and Root-of-unity Six-vertex Model

We examine the Onsager algebra symmetry of $τ^{(j)}$-matrices in the superintegrable chiral Potts model. The comparison of Onsager algebra symmetry of the chiral Potts model with the $sl_2$-loop algebra symmetry of six-vertex model at roots of unity is made from the aspect of functional relations using the $Q$-operator and fusion matrices. The discussion of Bethe ansatz for both models is conducted in a uniform manner through the evaluation parameters of their symmetry algebras.

cond-mat.stat-mech↗

arXiv2007

Fusion Operators in the Generalized $τ^{(2)}$-model and Root-of-unity Symmetry of the XXZ Spin Chain of Higher Spin

We construct the fusion operators in the generalized $τ^{(2)}$-model using the fused $L$-operators, and verify the fusion relations with the truncation identity. The algebraic Bethe ansatz discussion is conducted on two special classes of $τ^{(2)}$ which include the superintegrable chiral Potts model. We then perform the parallel discussion on the XXZ spin chain at roots of unity, and demonstrate that the $sl_2$-loop-algebra symmetry exists for the root-of-unity XXZ spin chain with a higher spin, where the evaluation parameters for the symmetry algebra are identified by the explicit Fabricius-McCoy current for the Bethe states. Parallels are also drawn to the comparison with the superintegrable chiral Potts model.

cond-mat.stat-mech↗

arXiv2006

The Q-operator for Root-of-Unity Symmetry in Six Vertex Model

We construct the explicit $Q$-operator incorporated with the $sl_2$-loop-algebra symmetry of the six-vertex model at roots of unity. The functional relations involving the $Q$-operator, the six-vertex transfer matrix and fusion matrices are derived from the Bethe equation, parallel to the Onsager-algebra-symmetry discussion in the superintegrable $N$-state chiral Potts model. We show that the whole set of functional equations is valid for the $Q$-operator. Direct calculations in certain cases are also given here for clearer illustration about the nature of the $Q$-operator in the symmetry study of root-of-unity six-vertex model from the functional-relation aspect.

cond-mat.stat-mech↗

arXiv2005

The Onsager Algebra Symmetry of $τ^{(j)}$-matrices in the Superintegrable Chiral Potts Model

We demonstrate that the $τ^{(j)}$-matrices in the superintegrable chiral Potts model possess the Onsager algebra symmetry for their degenerate eigenvalues. The Fabricius-McCoy comparison of functional relations of the eight-vertex model for roots of unity and the superintegrable chiral Potts model has been carefully analyzed by identifying equivalent terms in the corresponding equations, by which we extract the conjectured relation of $Q$-operators and all fusion matrices in the eight-vertex model corresponding to the $T\hat{T}$-relation in the chiral Potts model.

cond-mat.stat-mech↗

arXiv2005

Chiral Potts Rapidity Curve Descended from Six-vertex Model and Symmetry Group of Rapidities

In this paper, we present a systematical account of the descending procedure from six-vertex model to the $N$-state chiral Potts model through fusion relations of $τ^{(j)}$-operators, following the works of Bazhanov-Stroganov and Baxter-Bazhanov-Perk. A careful analysis of the descending process leads to appearance of the high genus curve as rapidities' constraint for the chiral Potts models. Full symmetries of the rapidity curve are identified, so is its symmetry group structure. By normalized transfer matrices of the chiral Potts model, the $τ^{(2)}T$ relation can be reduced to functional equations over a hyperelliptic curves associated to rapidities, by which the degeneracy of $τ^{(2)}$-eigenvalues is revealed in the case of superintegrable chiral Potts model.

cond-mat.stat-mech↗