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A NOTE ON THE osp(1|2s) THERMODYNAMIC BETHE ANSATZ EQUATION

Published Jan 1, 2001• 0 citations
OpenAlex

Abstract

A Bethe ansatz equation associated with the Lie superalgebra osp(1|2s) is studied. A thermodynamic Bethe ansatz (TBA) equation is derived by the string hypothesis. The high temperature limit of the entropy density is expressed in terms of the solution of the osp(1|2s) version of the Q-system. In particular for the fundamental representation case, we also derive a TBA equation from the osp(1|2s) version of the T-system and the quantum transfer matrix method. This TBA equation is identical to the one from the string hypothesis. The central charge is expressed by the Rogers dilogarithmic function, and identified to s. 1

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A NOTE ON THE osp(1|2s) THERMODYNAMIC BETHE ANSATZ EQUATION

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A Bethe ansatz equation associated with the Lie superalgebra osp(1|2s) is studied. A thermodynamic Bethe ansatz (TBA) equation is derived by the string hypothesis. The high temperature limit of the entropy density is expressed in terms of the solution of the osp(1|2s) version of the Q-system. In particular for the fundamental representation case, we also derive a TBA equation from the osp(1|2s) version of the T-system and the quantum transfer matrix method. This TBA equation is identical to the one from the string hypothesis. The central charge is expressed by the Rogers dilogarithmic function and identified to s.

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$T$-system and thermodynamic Bethe ansatz equations for solvable lattice models associated with superalgebras (Bilinear Method in the Study of Integrable Systems and Related Topics)

Zengo Tsuboi

An analytic Bethe ansatz is carried out related to the Lie superal- gebra $osp(1|2s)$ .We present an eigenvahie formula of atransfer matrix in dressed vacuum form (DVF) labeled by aYoung (super) diagram.Remarkable duality among DVFs is found.Acomplete set of transfer matrix functional relations ( $T$ -systern)is proposed as areduction of a Hirota-Miwa equation.We also derive athermodynamic Bethe ansatz (TBA) equation from this $T$ -system and the quantum transfer matrix method.This TBA equation is identical to the one from the string hypothesis.1Introduction Solvable lattice models related to Lie superalgebras [1] $\mathrm{h}_{\dot{\mathrm{e}}}\iota \mathrm{v}(^{\backslash }$ rpceived $\mathrm{I}\mathrm{n}\iota 1(j\mathrm{h}$ attentions [2, 3,4,5,6,7,8,9].To construct cigellval\iota \iota e formulae of transfer matrices for such models is an irnportant problem $\mathrm{i}\mathrm{r}\mathrm{l}$ mathernatical pbysics.To achieve this $\mathrm{p}\mathrm{r}()\mathrm{g}\mathrm{l}\cdot \mathrm{a}\mathrm{l}\mathrm{l}\mathrm{l},$ $\mathrm{t}11(^{\mathrm{Y}}$ Betbe $\dot{\epsilon}\iota \mathrm{r}1_{\iota}\mathrm{s}^{1}\mathrm{a}\mathrm{t}_{l}\mathrm{z}$ has been often $\mathrm{t}1.\mathrm{b}^{\tau}\mathrm{t}^{\backslash}\mathrm{r}1$ .Nowadays, there is nmch $1\mathrm{i}\mathrm{t}\mathrm{I}(^{\mathrm{Y}},1^{\cdot}\dot{c}\mathrm{t}\mathrm{f}\mathrm{u}1^{\cdot}\mathrm{t}^{1}$ (see for $(^{\backslash }\mathrm{X}\dot{\subset}\mathrm{l}_{l}\mathrm{I}\mathrm{r}11^{)}1\mathrm{t}^{1},.'[4.1\mathrm{t}],$$11,12,13$ , 14, 15, 16, 17, 18, 19, $2\mathrm{t}\mathrm{I},$ $21,22,23,\underline{9}4$ ] allcl $\mathrm{r}(^{1\mathrm{f}\backslash }$ ( $\Gamma \mathrm{t}^{\backslash }11\mathrm{C}\mathrm{e}\mathrm{s}$ therein.)$()\mathrm{I}1\mathrm{B}(^{\backslash }\mathrm{t}1_{1\mathrm{t}^{1}}$

数理解析研究所講究録•Aug 1, 2002•0 citations
A Bethe ansatz equation associated with the Lie superalgebra osp(1|2s) is studied. A thermodynamic Bethe ansatz (TBA) equation is derived by the string hypothesis. The high temperature limit of the entropy density is expressed in terms of the solution of the osp(1|2s) version of the Q-system. In particular for the fundamental representation case, we also derive a TBA equation from the osp(1|2s) version of the T-system and the quantum transfer matrix method. This TBA equation is identical to the one from the string hypothesis. The central charge is expressed by the Rogers dilogarithmic function, and identified to s. 1
A Bethe ansatz equation associated with the Lie superalgebra osp(1|2s) is studied. A thermodynamic Bethe ansatz (TBA) equation is derived by the string hypothesis. The high temperature limit of the entropy density is expressed in terms of the solution of the osp(1|2s) version of the Q-system. In particular for the fundamental representation case, we also derive a TBA equation from the osp(1|2s) version of the T-system and the quantum transfer matrix method. This TBA equation is identical to the one from the string hypothesis. The central charge is expressed by the Rogers dilogarithmic function and identified to s.
The thermodynamic Bethe ansatz (TBA) and the excited state TBA equations for an integrable spin chain related to the Lie superalgebra osp (1|2) are proposed by the quantum transfer matrix (QTM) method. We introduce the fusion hierarchy of the QTM and derive the functional relations among them (T-system) and their certain combinations (Y-system). Their analytical property leads to the nonlinear integral equations which describe the free energy and the correlation length at any finite temperatures. With regard to the free energy, they coincide with the TBA equation based on the string hypothesis.
An analytic Bethe ansatz is carried out related to the Lie superalgebra osp(1|2s). We present an eigenvalue formula of a transfer matrix in dressed vacuum form (DVF) labeled by a Young (super) diagram. Remarkable duality among DVFs is found. A complete set of transfer matrix functional relations (T-system) is proposed as a reduction of a Hirota-Miwa equation. We also derive a thermodynamic Bethe ansatz (TBA) equation from this T-system and the quantum transfer matrix method. This TBA equation is identical to the one from the string hypothesis.
This note is a supplement to our previous papers: Mod. Phys. Lett. A14 (1999) 2427 (math-ph/9911010); Int. J. Mod. Phys. A15 (2000) 2329 (math-ph/9912014). The thermodynamic Bethe ansatz (TBA) equation for an integrable spin chain related to the Lie superalgebra osp(1|2) is analyzed. The central charge determined by low temperature asymptotics of the specific heat can be expressed by the Rogers dilogarithmic function, and identified to be 1. Solving the TBA equation numerically, we evaluate the several thermodynamic quantities. The excited state TBA equation is also discussed.
TTT-system and thermodynamic Bethe ansatz equations for solvable lattice models associated with superalgebras (Bilinear Method in the Study of Integrable Systems and Related Topics)
An analytic Bethe ansatz is carried out related to the Lie superal- gebra osp(1∣2s)osp(1|2s)osp(1∣2s) .We present an eigenvahie formula of atransfer matrix in dressed vacuum form (DVF) labeled by aYoung (super) diagram.Remarkable duality among DVFs is found.Acomplete set of transfer matrix functional relations ( TTT -systern)is proposed as areduction of a Hirota-Miwa equation.We also derive athermodynamic Bethe ansatz (TBA) equation from this TTT -system and the quantum transfer matrix method.This TBA equation is identical to the one from the string hypothesis.1Introduction Solvable lattice models related to Lie superalgebras [1] he˙ιv(\\mathrm{h}_{\dot{\mathrm{e}}}\iota \mathrm{v}(^{\backslash }he˙​ιv(\ rpceived Inι1(jh\mathrm{I}\mathrm{n}\iota 1(j\mathrm{h}Inι1(jh attentions [2, 3,4,5,6,7,8,9].To construct cigellval\iota \iota e formulae of transfer matrices for such models is an irnportant problem irl\mathrm{i}\mathrm{r}\mathrm{l}irl mathernatical pbysics.To achieve this pr()gl⋅alll,\mathrm{p}\mathrm{r}()\mathrm{g}\mathrm{l}\cdot \mathrm{a}\mathrm{l}\mathrm{l}\mathrm{l},pr()gl⋅alll, t11(Y\mathrm{t}11(^{\mathrm{Y}}t11(Y Betbe ϵ˙ιr1ιs1atlz\dot{\epsilon}\iota \mathrm{r}1_{\iota}\mathrm{s}^{1}\mathrm{a}\mathrm{t}_{l}\mathrm{z}ϵ˙ιr1ι​s1atl​z has been often t1.bτt\r1\mathrm{t}1.\mathrm{b}^{\tau}\mathrm{t}^{\backslash}\mathrm{r}1t1.bτt\r1 .Nowadays, there is nmch 1itI(Y,1⋅c˙tfu1⋅t11\mathrm{i}\mathrm{t}\mathrm{I}(^{\mathrm{Y}},1^{\cdot}\dot{c}\mathrm{t}\mathrm{f}\mathrm{u}1^{\cdot}\mathrm{t}^{1}1itI(Y,1⋅c˙tfu1⋅t1 (see for (\X⊂˙llIr11)1t1,.′[4.1t],(^{\backslash }\mathrm{X}\dot{\subset}\mathrm{l}_{l}\mathrm{I}\mathrm{r}11^{)}1\mathrm{t}^{1},.'[4.1\mathrm{t}],(\X⊂˙​ll​Ir11)1t1,.′[4.1t],11,12,1311,12,1311,12,13 , 14, 15, 16, 17, 18, 19, 2tI,2\mathrm{t}\mathrm{I},2tI, 21,22,23,9‾421,22,23,\underline{9}421,22,23,9​4 ] allcl r(1f\\mathrm{r}(^{1\mathrm{f}\backslash }r(1f\ ( Γt\11Ces\Gamma \mathrm{t}^{\backslash }11\mathrm{C}\mathrm{e}\mathrm{s}Γt\11Ces therein.)()I1B(\t11t1()\mathrm{I}1\mathrm{B}(^{\backslash }\mathrm{t}1_{1\mathrm{t}^{1}}()I1B(\t11t1​