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数理解析研究所講究録

$T$-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

Published Aug 1, 2002• 0 citations
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Abstract

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.)$()\mathrm{I}1\mathrm{B}(^{\backslash }\mathrm{t}1_{1\mathrm{t}^{1}}$

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T-system and thermodynamic Bethe ansatz equations for solvable lattice models associated with superalgebras

Zengo Tsuboi

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.

arXiv:0912.0073•Dec 1, 2009•1 citation

Analytic Bethe ansatz and functional relations related to tensor-like representations of type-II Lie superalgebras B ( r | s ) and D ( r | s )

Zengo Tsuboi

An analytic Bethe ansatz is carried out related to tensor-like representations of the type-II Lie superalgebras B ( r | s ) = osp (2 r +1|2 s ) ( r 0 , s 1 ) and D ( r|s ) = osp (2 r|2s ) ( r 2 , s 1 ). We present eigenvalue formulae of transfer matrices in dressed vacuum forms (DVFs) labelled by Young (super) diagrams. A class of transfer matrix functional relations ( T -system) is discussed. In particular for the B (0| s ) = osp (1|2 s ) ( s 1 ) case, a complete set of functional relations is proposed by using duality among DVFs.

arXiv:0911.5393•Journal of Physics A Mathematical and General•Sep 28, 1999•20 citations

THERMODYNAMIC BETHE ANSATZ EQUATION FROM FUSION HIERARCHY OF osp (1|2) INTEGRABLE SPIN CHAIN

KAZUMITSU SAKAI, ZENGO TSUBOI

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.

arXiv:math-ph/9912014•International Journal of Modern Physics A•Jun 20, 2000•11 citations

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

Zengo Tsuboi

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.

arXiv:cond-mat/0108358•International Journal of Modern Physics A•Jul 10, 2002•12 citations

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

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

•Jan 1, 2001•0 citations

Fusion approach for quantum integrable system associated with the $\mathfrak{gl}(1|1)$ Lie superalgebra

Xiaotian Xu, Wen, Wuxiao, Tao Yang +2 more

In this work we obtain the exact solution of quantum integrable system associated with the Lie superalgebra $\mathfrak{gl}(1|1)$, both for periodic and for generic open boundary conditions. By means of the fusion technique we derive a closed set of operator identities among the fused transfer matrices. These identities allow us to determine the complete energy spectrum and the corresponding Bethe ansatz equations of the model. Our approach furnishes a systematic framework for studying the spectra of quantum integrable models based on Lie superalgebras, in particular when the $U(1)$ symmetry is broken. The derivation of the Bethe states from the exact spectrum is also addressed.

arXiv:2510.27312•Oct 31, 2025•0 citations
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 (^{\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, 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​
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.
An analytic Bethe ansatz is carried out related to tensor-like representations of the type-II Lie superalgebras B ( r | s ) = osp (2 r +1|2 s ) ( r 0 , s 1 ) and D ( r|s ) = osp (2 r|2s ) ( r 2 , s 1 ). We present eigenvalue formulae of transfer matrices in dressed vacuum forms (DVFs) labelled by Young (super) diagrams. A class of transfer matrix functional relations ( T -system) is discussed. In particular for the B (0| s ) = osp (1|2 s ) ( s 1 ) case, a complete set of functional relations is proposed by using duality among DVFs.
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.
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.
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
Fusion approach for quantum integrable system associated with the gl(1∣1)\mathfrak{gl}(1|1)gl(1∣1) Lie superalgebra
In this work we obtain the exact solution of quantum integrable system associated with the Lie superalgebra gl(1∣1)\mathfrak{gl}(1|1)gl(1∣1), both for periodic and for generic open boundary conditions. By means of the fusion technique we derive a closed set of operator identities among the fused transfer matrices. These identities allow us to determine the complete energy spectrum and the corresponding Bethe ansatz equations of the model. Our approach furnishes a systematic framework for studying the spectra of quantum integrable models based on Lie superalgebras, in particular when the U(1)U(1)U(1) symmetry is broken. The derivation of the Bethe states from the exact spectrum is also addressed.