UMTG315 Flag Integrable Models and Generalized Graded Algebras

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UMTG–315

Flag Integrable Models and Generalized Graded

Algebras

Marius de Leeuw1, Rafael I. Nepomechie2and Ana L. Retore3,4

Abstract

We introduce new classes of integrable models that exhibit a structure similar to

that of ﬂag vector spaces. We present their Hamiltonians, R-matrices and Bethe-ansatz

solutions. These models have a new type of generalized graded algebra symmetry.

1School of Mathematics & Hamilton Mathematics Institute, Trinity College Dublin, Dublin, Ireland,

m.deleeuw1@gmail.com

2Physics Department, P.O. Box 248046, University of Miami, Coral Gables, FL 33124 USA,

nepomechie@miami.edu

3School of Mathematics & Hamilton Mathematics Institute, Trinity College Dublin, Dublin, Ireland

4Department of Mathematical Sciences, Durham University, Durham DH1 3LE, UK,

ana.retore@durham.ac.uk, (Current)

arXiv:2210.06495v2 [hep-th] 30 May 2023

Contents

1 Introduction 2

2 Derivation of the models 4

2.1 TheHamiltonians................................. 4

2.2 R-matrices..................................... 7

2.2.1 ModelI .................................. 8

2.2.2 ModelII.................................. 8

2.2.3 ModelIII ................................. 10

2.2.4 Properties of R-matrices for models I, II and III . . . . . . . . . . . . 10

2.2.5 ModelIV ................................. 11

3 Generalized graded algebra 11

3.1 Deﬁnition ..................................... 11

3.2 Examples ..................................... 13

3.3 AlgebraforﬂagmodelI ............................. 14

3.4 Generalized graded Yangians . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.5 Symmetries for model II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4 Bethe ansatz for model II 17

4.1 Firstlevelofnesting ............................... 17

4.2 Transfer-matrix eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4.3 Betheequations.................................. 25

5 Bethe ansatz for model I 27

5.1 Firstlevelofnesting ............................... 27

5.2 Transfer-matrix eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

5.3 Betheequations.................................. 31

6 Bethe ansatz for model III 33

6.1 Transfer-matrix eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

6.2 Betheequations.................................. 38

7 Discussion and outlook 39

A Comparison between gl(k0−k1|...|kd−2−kd−1|kd−1)

and gl(m|n−m)41

A.1 R-matrices..................................... 41

A.2 Betheansatz.................................... 43

A.2.1 Remark about gradings . . . . . . . . . . . . . . . . . . . . . . . . . . 45

A.2.2 Fermionic duality transformation . . . . . . . . . . . . . . . . . . . . 45

B Trigonometric solution 47

C Completeness checks 48

1 Introduction

The study of integrable spin chains is by now a mature subject — many inﬁnite families of

such models have already been identiﬁed and solved. Many of these models were derived from

quantum (super) algebras [1–4]. The best known examples are of course Yangians [5, 6] and

quantum aﬃne algebras [7, 8]. In fact, there is even a close relation between the functional

form of the R-matrix and the symmetry algebra [9]. Rational R-matrices typically have a

symmetry of Yangian type, while trigonometric R-matrices typically have a symmetry of a

quantum aﬃne type. Hence, it may come as a surprise that new rational solutions of the

Yang-Baxter equation, and corresponding integrable spin chains, can still be found.

Recently, a more direct approach to classifying solutions of the Yang-Baxter equation has

been put forward which employs the so-called boost operator [10–13]. One of the advantages

of this approach is that it does not rely on symmetry arguments and gives a complete

classiﬁcation. Several new solutions of the Yang-Baxter equation have been found that are

rational, trigonometric and elliptic. The natural follow-up question is then whether there

are quantum algebras that underlie these models. For some of the new models, the algebras

seem closely related to centrally extended algebras [14]. However, in [11] very simple rational

solutions (Models 4 and 6) were found for which the symmetry algebra was still unclear. More

precisely, Models 4 and 6 from [11] have a 4-dimensional Hilbert space at each site, and have

16 ×16 R-matrices that take the form

R∼uI(4,4) −P(4,4) +uI(2,4) −P(2,4),(1.1)

where I(4,4) and P(4,4) are the usual identity and permutation matrix, but I(2,4) and P(2,4) are

the identity and permutation operator restricted to a two-dimensional subspace, see (2.1),

(2.2). These models look like combinations of simple XXX type models. Similar models were

found in work on so-called multiplicity A-models [15] (building on earlier work in [16, 17]),

which were further studied and generalized in [18] and in [19] .

Inspired by this, we consider here a generalization of these types of models where we take

the R-matrix to be a linear combination of the identity, permutation and trace operators,

see (2.1)-(2.3), that are restricted to subspaces Vkd−1⊂. . . ⊂Vk1⊂Vk0see Figure 1. We

Vk0

Vk1

Vk2

Vk3

ℙ(k0,n),𝕀(k0,n),𝕂(k0,n)

ℙ(k1,n),𝕀(k1,n),𝕂(k1,n)

ℙ(k2,n),𝕀(k2,n),𝕂(k2,n)

Figure 1: We consider a ﬂag vector space with operators P,I,Kacting on the tensor products

of various subspaces. In particular, I(ki,n)is the characteristic function of that subspace, i.e.

it is the identity for vectors in the subspace and 0 for the complement. The other operators

are similarly deﬁned in the case of the permutation and the trace operator.

recall that, in linear algebra, a ﬂag refers to such an increasing sequence of subspaces of a

vector space, and hence we name these solutions ﬂag integrable models.

By using the boost operator method, we ﬁnd three non-trivial inﬁnite families of inte-

grable spin chains that have such a ﬂag structure. We refer to these as models I, II and III.

These models are characterized by a set ⃗

kof ddecreasing positive integers

⃗

k={k0, k1, . . . , kd−1}, n =k0> k1> . . . > kd−1≥1,(1.2)

where nis the dimension of the Hilbert space at each site. A subset of model II can be

related to a subset of the model in [15]. Despite the simplicity of their Hamiltonians and

R-matrices, these models have nontrivial spectra, symmetries and degeneracies. We ﬁnd a

fourth model, model IV, whose spectrum is purely combinatorial. For given values of nand

d, the number of possible models are n−1

d−1for model I and II, and n−3

d−2for models III and

IV, respectively, as we will see below.

We will show that our models exhibit a type of generalized graded Lie algebra symmetry,

which we will denote by gl(k0−k1|. . . |kd−2−kd−1|kd−1). When the ﬂag has only two stripes

i.e. d= 2, then we return to the usual Lie superalgebra gl(n−k|k). We furthermore show

that Model I admits a Yangian extension of this algebra and is uniquely ﬁxed by it.

We will also work out the nested algebraic Bethe ansatz for models I, II and III. Sur-

prisingly, many of the transfer-matrix eigenvalues are described by inﬁnite, singular and/or

continuous Bethe roots.

2 Derivation of the models

In this section we derive the form of the ﬂag models. Motivated by our work on Hubbard-

type models and the Maassarani-Matthieu models, we will consider Hamiltonians built out

of restrictions of the identity, permutation and trace operators.

2.1 The Hamiltonians

We begin by studying the direct generalization of Models 4 and 6 from [11]. We will see

that these models have R-matrices that are rational and of diﬀerence form, and are similar

to XXX-type models.

Notation Let us ﬁrst deﬁne the restricted operators that we will use to construct our

integrable models. We denote

P(m,n)=

i,j=1

ei,j ⊗ej,i ,(2.1)

I(m,n)=

i,j=1

ei,i ⊗ej,j ,(2.2)

K(m,n)=

i,j=1

ei,j ⊗ei,j ,(2.3)

where ei,j is an n×nmatrix such that (ei,j )α,β =δi,αδj,β , and 1 ≤m≤n. For m=n, the

operator P(m,n)becomes the usual permutation operator for a Hilbert space of dimension n,

and similarly, I(m,n)reduces to the identity matrix.

Hamiltonian Inspired by the simple form of Models 4 and 6 from [11], we consider a

similar nested structure where we combine general Hamiltonians that are built out of the

building blocks of SO(n) spin chains. Consider a set ⃗

kof decreasing positive integers

⃗

k={k0, k1, . . . , kd−1}, n =k0> k1> . . . > kd−1≥1,(2.4)

where n=k0is the dimension of the Hilbert space at each site. We take our Hamiltonian

to be of the form

H⃗

d−1

i=0 aiI(ki,n)+biP(ki,n)+ciK(ki,n).(2.5)

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摘要：

UMTG–315FlagIntegrableModelsandGeneralizedGradedAlgebrasMariusdeLeeuw1,RafaelI.Nepomechie2andAnaL.Retore3,4AbstractWeintroducenewclassesofintegrablemodelsthatexhibitastructuresimilartothatofflagvectorspaces.WepresenttheirHamiltonians,R-matricesandBethe-ansatzsolutions.Thesemodelshaveanewtypeofgenera...

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