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Supersymmetric lattice models: Field theory correspondence, integrability, defects anddegeneracies

Fokkema, T.B.

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Citation for published version (APA):Fokkema, T. B. (2016). Supersymmetric lattice models: Field theory correspondence, integrability, defects anddegeneracies.

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CHAPTER 3

Bethe ansatz solvability andsupersymmetry of the M2 model

on a closed chain

In this chapter a detailed study of the M2

model is presented. We identify asubmanifold in the space of parameters of the model where it is Bethe ansatzsolvable. The relation between this manifold and the existence of additional,so-called dynamic, supersymmetries is discussed. This chapter is based on theresults published in [1].

3.1 IntroductionIntegrability is arguably one of the most powerful analytical tools availablefor the study of d = 2 classical or d = 1 quantum mechanical systems [30].Integrable models are very special and enjoy a high degree of analytical structure.The integrable supersymmetric models provide an interesting opportunity tostudy the interplay between the structure due to integrability and that due tosupersymmetry (see also [31]). This is the topic of this chapter.

We study the staggered version of the M2

model with interactions which aresite-dependent. For a chain of length N these interactions can be parametrisedby 2N parameters: �x and µx, where x = 1, . . . , N labels the sites of the chain.For this model we identify a special two-parameter submanifold in the (�x, µx)-parameter space where the model possesses two types of dynamic supersymmetryin addition to the explicit N = 2 supersymmetry. Furthermore, from completelyindependent considerations we find that the model is solvable via a generalised co-ordinate Bethe ansatz [20] on precisely this same special submanifold in parameterspace.

The special submanifold is given by

�x+2

= �x, µx+2

= µx, µ2

x + µ2

x+1

= 1. (3.1)

The homogeneous model, i.e. the original M2

model with site-independent interac-tions, intersects the special submanifold at the point: �x = 1 and µx = µ = 1/

p2

for all x. These are indeed precisely the parameters for which the original M2

model was found to be integrable [7]. In the original model the constraint onµ follows from spin-reversal symmetry upon mapping the model to the spin�1chain (as seen in sections 2.5 and 2.6). Here we will see that the constraintµ2

x+µ2

x+1

= 1 is also related to a type of spin-reversal symmetry, even though forsite-dependent interactions a mapping to a spin model is not known at present.

17

3 Bethe ansatz solvability and supersymmetry of the M2 model

0

1.00.5

0

0.5

1.00

12

Figure 3.1: We plot the special two-parameter submanifold (3.1), where the modelis integrable and enjoys additional supersymmetry, in a three-parameter subspace forwhich �

x+2 = �x

, µx+2 = µ

x

for all x. We define � = �x

/�x+1 and plot the range

0 < � < 2 and µx

> 0. The red line indicates the parameters for which the original M2

model is recovered: � = 1 and µx

= µx+1 = µ. The red point lies at the intersection of

the red line with the special submanifold: µ = 1/p2. It is the point where the original

model is integrable and maps to the spin-1 chain. The blue point is a generic pointon the special submanifold, and the green point is a generic point that is neither onthe homogeneous line nor on the special submanifold. In figure 3.7 we will comparenumerical data for these three points. Note that the model possesses the explicit N = 2supersymmetry for general �

x

and µx

and thus, in particular, everywhere in the plottedparameter space.

18

3.3 Extension of spin reversal symmetry to the inhomogeneous chain

We investigate how the supersymmetry translates into the structure of theBethe ansatz. We find, in particular, a relation between the action of thesupercharges, the generators of the supersymmetry and exact complete strings ofBethe roots.

In section 3.2 we extend the spin-reversal symmetry discussed in chapter 2to the inhomogeneous M

2

model. In section 3.3 we discuss three dynamicsupersymmetries which exist on the special submanifold. In section 3.4 we studythe model by means of a generalised coordinate Bethe ansatz solution: as weshall see, the requirement that the model be Bethe ansatz solvable leads againto a restriction to the special submanifold. In section 3.5 we study the relationbetween the dynamical supersymmetries and the Bethe ansatz.

3.2 Extension of spin reversal symmetry to theinhomogeneous chain

We have seen above that we can have site dependent amplitudes �[a,b],j in the

model. It is an interesting question what the spin-reversal symmetry (see 2.4.1)will give in the staggered case. We compare the amplitudes for the single hop andpartner swap processes in the M

2

model Hamiltonian see 2.8. As both amplitudesare position-dependent we need a rule to fix their positions on the lattice. Weimpose the following: the single particle which changes its position hops betweenthe same sites in both processes. For the example at hand, we impose that thefollowing amplitudes are equal:

· · ·x

· · · $ · · ·x

· · · · · ·x

· · · $ · · ·x

· · ·

�x�x+1

(1� µ2

x) = �x�x+1

µx�1

µx+1

We see that this implies µx+1

µx�1

= 1� µ2

x. Applying this strategy to the left-and right-split-join processes, that is equating the amplitudes for the processes

· · ·x

· · · $ · · · · · · · · ·j

· · · $ · · ·x

· · ·

�x�x+1

µx+1

= �x�x+1

µx�1

(3.2)

leads to µx�1

= µx+1

and hence we get

µ2

x + µ2

x+1

= 1 (3.3)

which is the second equation of (3.1). It should be emphasized that while itleads to a mod�2 staggering for the µx, the present argument does not impose aconstraint on the �x.

19


3.3 Dynamical supersymmetries of the M2 model

3.3.1 Dynamic supersymmetry Q�

We recall the definition of the supercharge Q+

as discussed in chapter 2. Theaction of the supercharge Q

+,x in the staggered case, is

Q+,x| · · · �

x· · · i = 0,

Q+,x| · · · � •

x� · · · i = (±)�x| · · · � �

x� · · · i,

Q+,x| · · · � •

x• � · · · i = (±)�xµx| · · · � �

x• � · · · i,

Q+,x| · · · � • •

x� · · · i = (±)�xµx�1

| · · · � • �x� · · · i,

where the sign (±) is the usual fermionic string, i.e. minus one to the number offermions located to the left of the site x. The supercharge of the model is definedas the sum of these operators over all sites

Q+

=NX

x=1

Q+,x. (3.4)

The discussion of spin reversal symmetry in the last section leads naturally to thequestion if the “spin-reversal” transformation can be applied to the superchargeQ

+

in order to obtain a second copy of the N = 2 supersymmetry algebra. Letus illustrate this idea with an example. We consider the local action of Q

+

on afilled site, x, with empty neighbouring sites. We map this configuration and itsimage to spin configurations, perform a spin reversal transformation and mapthe result back to fermion configurations:

x

Q+,x

x

fermions

to spins

0

""

spin

reversal

0

##

spins to

fermionsQ�?

The last column suggests that there might be a dynamic supercharge, which wecalled Q�, that inserts 3 fermions and 4 sites. We show in this section that thissupercharge indeed exists, provided that the staggering parameters are periodicwith period two and the model is restricted to subsectors of the Hilbert spacewhich are invariant under translations by four sites.

Definition and properties. As suggested by our example, the dynamic su-percharge inserts four consecutive sites with three particles into the system:Q� : HN,f ! HN+4,f+3

. It is given as a linear superposition of locally acting

20


operators Q�,x which perform the insertion process between sites x and x+ 1:

Q� =

r

N

N + 4

NX

j=�3

Q�,j .

The terms with j = �3,�2, . . . , 0 take care of the insertion process near the“boundary” (near sites N and 1) and need to be present as the four new sites mayhave N + 4 di↵erent locations on a chain with N + 4 sites. Similarly Q� = Q†

� isdefined as

Q� =

r

N

N � 4

N�4

X

j=1

Q�,j .

The application of the spin-reversal rules to the elementary processes induced bythe supercharge Q

+

leads to the following actions on basis vectors:

Q�,x| · · · �x� · · · i = (±)

⇣

ax| · · · �x• • � • � · · · i+ bx| · · · �

x• � • • � · · · i

⌘

,

Q�,x| · · · �x• � · · · i = (±) cx| · · · �

x• • � • • � · · · i, (3.5)

Q�,x| · · · �x• • � · · · i = 0.

The action of Q�,x on configurations which contain a particle on site x is alwayszero. The ± sign corresponds to the fermionic string, it is given by minus one tothe number of particles located to the left of x.

The requirement that Q2

� = 0 can be analysed locally by acting with thesupercharge twice on the simple configurations shown in (3.5). A detailed analysisshows the only non-vanishing terms come from the first configuration. Imposingthat this term vanishes results in the constraint

axcx+3

+ bxcx = 0.

This is a clear analogue of equation (2.8), which constrains the parameters in thedefinition of the supercharge Q

+

. After having imposed this first restriction, wewould like to establish Q� as a symmetry of our model. Hence we look for valuesof the parameters ax, bx, cx such that locally it generates the same Hamiltonianas Q

+

(i.e. the hopping amplitudes and rules for the potential energy are thesame):

H = {Q�, Q�} (3.6)

where Q� = Q†�. The explicit comparison of the two sides is a cumbersome task.

It leads to a system of quadratic di↵erence equations for the parameters ax, bx, cxwhich allow to express them in terms of �x, µx. Out of the many equations, letus just write

a2x+2

= a2x, b2x+2

= b2x, c2x = a2x + b2x+1

, (3.7)

21


which show that once more a staggering with period 2 is obtained. We skip thedetails, and report only the solution to the complete set of di↵erence equationswhich is given by

ax = µx�x+1

, bx = �µx�x, cx = �x+1

. (3.8)

Its insertion into (3.7) implies that for real positive �x, µx the parameters of themodel lie on the line (3.1).

We note that starting from the definition

Q� = SQ+

S, (3.9)

where S denotes the spin-reversal operation, does not give the amplitudes abovein the case that µx 6= µx+1

. One would instead get (by just a simple calculation)the amplitudes

ax = µx+1

�x+1

, bx = �µx+1

�x, cx = �x+1

(3.10)

which we denote by a tilde to distinguish them from the ones above. The di↵erencewith eq. (3.8) is that µx is shifted by one site. The reason for this is exactly theshift that we noted in section 3.2 and the amplitudes a, b, c do therefore not makethe anti-commutator {Q�, Q�} locally equal to the Hamiltonian. In the caseµx = µx+1

the two prescriptions are equal, this is what we will use in chapter 7.

Translation invariance. So far we ignored the boundary terms Q�,x withx = �3, . . . , 0, because all the constraints on the parameters in the definition ofQ� could be derived from local considerations in the bulk. We will now considerthe boundary terms and find that it leads to a restriction on the Hilbert space.That is, we find that the supersymmetry Q� generates the Hamiltonian onlyin a subsector of the Hilbert space. This feature is quite di↵erent from thesupersymmetry generated by Q

+

which exists on the entire Hilbert space.To see this we first define the boundary terms by taking advantage of the

periodicity of the staggering parameters. To this end, we use the translationoperator T . Clearly, the staggering with period 2 implies that, for the localoperators in the bulk,

T 2Q�,xT�2 = Q�,x+2

, x = 1, . . . , N � 2. (3.11)

Note that the translation operator on the left (right) of Q�,x acts on states of achain of length N +4 (N). We now use this equation to define the local operators,Q�,x, for x = �3, . . . , 0 and thus extend (3.11) to x = �3, . . . , 0. This definitionnow also implies that for x = �3, . . . , 0 we have Q�,x+N = TNQ�,xT�N . UsingT�N = 1 when it acts on a chain of length N and similarly TN = T�4 for a chainof length N + 4, we can rewrite this as

T 4Q�,N+x = Q�,x, x = �3, . . . , 0.

22


From this relation we conclude, in particular, that Q� is a well-defined mappingbetween eigenspaces of T 4 only when T 4 ⌘ 1 (one easily checks this by computingT 4Q�T�4, using the relations above and imposing it to be equal to Q�). Sincethe Hamiltonian H = {Q

+

, Q+

} of our model commutes with T 4, we concludethat H = {Q�, Q�} can only hold on subspaces where the translation operatorby four sites acts like the identity. Finally, let us mention that as for the non-dynamic supercharge Q

+

one may show that the construction of Q� only worksfor non-zero twist angles.

Extended supersymmetry algebra. As we have two copies of the N = 2supersymmetry algebra in the case of periodic boundary conditions, it appearsnatural to find two distinct fermion numbers in order to characterise the system.We define the following two operators in terms of the system size N and thenumber of fermions on the lattice f :

FV = N � f, and F = f �N/2.

Using the mapping to the spin chain, FV and F correspond to the length of thespin chain and its magnetisation, respectively. With these two fermion numbers,we obtain a lattice representation of the N = (2, 2) supersymmetry algebra [32](up to a sign convention for FV ) with vanishing central charges

Q2

± = Q2

± = 0, {Q±, Q⌥} = {Q±, Q⌥} = 0,

{Q±, Q±} = H ± P, (3.12)

[FV , Q±] = Q±, [FV , Q±] = �Q±,

[F,Q±] = ⌥Q±, [F, Q±] = ±Q±,

and zero momentumP = 0.

For these commutators and anti-commutators, the action on chains of appropriatelength is implied. Moreover, the algebra exists of course only in translationsubsectors with T 4 = 1, where the presence of the dynamic supersymmetryis guaranteed. The fact that the momentum P is zero is consistent with thisrestriction as we shall see in chapter 4 where the relation between the latticemodel and its field-theory limit is discussed.

3.3.2 Dynamic supersymmetry Q0

The dynamic supercharge defined in the last section increases the length of thechain by four sites. A priori, the periodicity of the staggering parameters doesnot exclude the existence of another length-changing operator which adds onlytwo sites to the system. We will show here that such an operator Q

0

exists indeedin certain subsectors of the Hilbert space. The existence of such a symmetry issomewhat expected. In [33] it was shown that the Fateev-Zamolodchikov chainpossesses a dynamic supersymmetry of the same structure as the one constructedbelow.

23


Supercharge and Hamiltonian. Let us show how the operator Q0

: HN,f !HN+2,f+1

, which inserts two sites and one particle, is constructed. As for thedynamic supercharge Q�, this operator can be written as a sum over localoperators

Q0

=

r

N

N + 2

NX

x=�1

(�1)xQ0,x. (3.13)

Unlike for Q� there is an additional site-dependent string (�1)x: it impliesthat in the homogeneous limit Q

0

inserts a particle with momentum ⇡ into thesystem. We will see later that this picture harmonises well with the Bethe ansatzinterpretation of this supersymmetry. The operator (Q

0

)† = Q0

is only definedon N � 2 sites

Q0

=

r

N

N � 2

N�2

X

x=1

(�1)xQ0,x. (3.14)

The Q0,x are fermionic operators which insert two sites and one particle. Like

for Q�,x their action is non-vanishing only if the site x is empty. In this case, theaction depends on the occupation of the subsequent sites. Let us illustrate thethree di↵erent possible scenarios by the action on basis vectors for x = 1, . . . , N :

Q0,x| · · · �

x� · · · i = (±)

⇣

↵x| · · · �x• � � · · · i+ �x| · · · �

x� • � · · · i

⌘

Q0,x| · · · �

x• � · · · i = (±)

⇣

�x| · · · �x• • � � · · · i+ �x| · · · �

x� • • � · · · i

⌘

Q0,x| · · · �

x• • � · · · i = (±)

⇣

✏x| · · · �x• � • • � · · · i+ ⌘x| · · · �

x• • � • � · · · i

⌘

(3.15)The sign ± represents the fermionic string, it is given by �1 to the number offermions located to the left of x for x > 1, and if x = 1 it is given by +1.

The site-dependent parameters, ↵x,�x, . . . , ⌘x, are constrained by the require-ment Q2

0

= 0. This can be done locally, by acting twice on the three configurationswhich we use in order to define the action of Q

0,x, and imposing that the resultvanishes locally up to boundary terms. An explicit calculation shows then thatthis is possible if and only if all parameters are periodic in x with period 2.Furthermore, it leads to

�x = �↵x+1

, ⌘x = ↵x+1

, ✏x = �↵x (3.16)

and to the relation

↵x�x + ↵x+1

�x+1

= 0, (3.17)

which can be thought of as an analogue of (2.8). The periodicity of the parametersleads to the following relation

T 2Q0,xT

�2 = Q0,x+2

, x = 1, . . . , N � 2.

24

3.4 The coordinate Bethe ansatz for the staggered M2 model

The remaining operators Q0,�1

, Q0,0 which take into account the insertion of a

pair of sites between x = N and x = 1 are defined by extending this relationto x = �1, 0. Using a similar reasoning as for the dynamic supercharge Q�presented above, we find that they can be equivalently expressed as

T 2Q0,x+N = Q

0,x, x = �1, 0.

In complete analogy with the case of Q� we conclude from this equation thatthe supercharge Q

0

is a well-defined mapping between translation sectors only ifT 2 ⌘ 1. In these subsectors it generates a supersymmetric Hamiltonian

H = {Q0

, Q0

}. (3.18)

Let us count how many remaining parameters there are at this stage: we are leftwith ↵x, �x, �x which are periodic under x ! x+2, and subject to (3.17). Takinginto account that we are free to rescale them, we find thus three free parameters.Their number is furthermore reduced if we impose that the Hamiltonian Hcoincide with the one for the M

2

model. Quite interestingly, it implies also arestriction to the line of couplings (3.1), and the 2-periodicity for �x, µx. Indeed,the analysis of the potential energies on both sides of (3.18) shows that equalitycan hold only if

�2x + �2x = 4↵2

x+1

. (3.19)

Adjusting the hopping terms one finds that the parameters are related to theoriginal staggering parameters according to

↵x =�x2, �x = ��x+1

2, �x = �x+1

µx+1

,

✏x = ��x2, ⌘x =

�x+1

2, �x = ��x+1

µx.(3.20)

This implies periodicity, and via reinsertion into (3.19) the special line (3.1).Furthermore, we checked explicitly that in the subsectors where T 2 ⌘ 1, thedynamic supercharge Q

0

anti-commutes with the other ones:

{Q0

, Q±} = 0, {Q0

, Q±} = 0.

3.4 The coordinate Bethe ansatz for the stag-gered M2 model

In this section, we show that the Hamiltonian of the M2

model can be diagonalisedby means of the coordinate Bethe ansatz along the two-parameter submanifold inthe space of staggering parameters µ2

x + µ2

x+1

= 1 which coincides precisely withthe submanifold with enhanced supersymmetry identified in the previous section.

The coordinate Bethe ansatz was introduced by Hans Bethe [34]. He foundthe exact eigenstates and energies of the one-dimensional spin- 1

2

Heisenberg

25


model. The technique employed here is a combination of the Bethe ansatz forthe staggered M

1

model found by Nienhuis and Blom [20], the coordinate Betheansatz for higher spin XXX chains [35], and an asymptotic analysis which allowsan easy determination of the integrable manifold in the space of parameters.We start by specifying the basis, and Bethe ansatz form for the wave functionin section 3.4.1. In section 3.4.2 we analyse the one-particle problem: it is notdirectly solvable for general staggering, but will allow to determine some usefulasymptotic expansions of the single-particle wave function. The two-particleproblem is solved in section 3.4.3. We show that it fixes the choice of admissiblestaggering parameters as well as the period of the staggering to the specialsubmanifold. Furthermore, we introduce a useful elliptic parametrisation for it.The many-particle case is considered afterwards, and leads to the Bethe ansatzequations.

3.4.1 Basis vectors and Bethe ansatz form of the wave func-tion

Basis vectors. Our principal goal is to diagonalise the Hamiltonian H, i.e. tosolve the Schrodinger equation

H| i = E| i. (3.21)

To this end we need to choose a suitable basis in the fermion Hilbert space whichsimplifies our problem as much as possible. The most natural choice appearsto be the canonical occupation number basis: a basis vector is labelled by thepositions of the particles in a given configurations:

|x1

, . . . , xf i = | � · · · � •x1

� · · · � •xf

� · · · �i.

This basis is orthonormal which is convenient for many applications. However,the Bethe ansatz is more conveniently formulated when using basis vectors||x

1

, . . . , xf ii which di↵er from the canonical ones by configuration-dependentfactors. Let us denote by x0

1

, x02

, . . . the positions of the first members of pairs inthe configuration x

1

, x2

, . . . , xf . We introduce the non-orthonormal basis

||x1

, x2

, . . . , xf ii =

0

@

Y

j

Cx0j

1

A |x1

, x2

, . . . , xf i (3.22)

with normalisation factors Cx to be determined. This modified basis is similar tothe basis used in the coordinate Bethe ansatz solution for the higher-spin XXXchains studied in [35]. It allows to absorb a trivial part of the wave function intothe basis itself.

Bethe ansatz. The Hamiltonian of our model, twisted or not, commutesobviously with the fermion number operator F , and can therefore be diagonalised

26


separately in each subsector HN,f . We expand its eigenstates in HN,f in themodified basis

| i =X

{x}

(x1

, x2

, . . . , xf )||x1

, x2

, . . . , xf ii. (3.23)

Here, the sum is taken over all positions 1 x1

< x2

< · · · < xf N ofthe particles which respect the exclusion constraint of the M

2

model, and theboundary conditions. For the wave function (x

1

, . . . , xf ) we make the Betheansatz by writing it as a linear combination of products of single-particle wavefunctions '(x; z):

(x1

, . . . , xf ) =X

�2Sf

B�'(x1

; z�(1)) · · ·'(xf ; z�(f)). (3.24)

The sum is over all permutations � of f objects, weighted by certain amplitudesB�. The variables z

1

, . . . , zf are the rapidities of the particles. In order togive meaning to these rapidity variables we need to specify the structure of thesingle-particle wave functions. In fact, we shall assume that the weights of themodel are periodic: �x+p = �x, µx+p = µx. Hence, single particles are describedby Bloch wave functions

'(x; z) = Ax(z)zx, Ax+p(z) = Ax(z). (3.25)

Our main objective here is to show that the model is Bethe ansatz solvable if andonly if p = 2, and the staggering parameters are chosen from the submanifold ofequation (3.1).

Given the basis and Bethe ansatz form of the wave function we proceednow through a series of standard steps for the solution of the Schrodinger equa-tion (3.21). We project this equation on simple basis vectors, and resolve theresulting system of di↵erence equations for the wave functions (x

1

, . . . , xf ). Inthe following sections, we address first the case of f = 1 and 2 particles, mentionbriefly the case of f = 3 and 4 particles, and deduce the result for general fthrough a standard argument.

3.4.2 The one-particle problem

In the subsector of the Hilbert space where no particles are present the diagonali-sation of the Hamiltonian is trivial: the empty state | � � · · · �i is an eigenvector

of H with eigenvalue E =PN

x=1

�2x. It serves as a reference state for the Betheansatz and will allow to build eigenstates with non-zero fermion numbers. Let usstart with a single particle f = 1. We write thus the eigenvalue as

E =NX

x=1

�2x + ✏(z),

27


where ✏(z) denotes the excitation energy for a (pseudo-)particle with rapidity zabove the reference-state level. The corresponding Bethe ansatz wave function isin fact simply given by '(x; z). Using its Bloch-wave structure (3.25) we find thedi↵erence equation

(✏(z) + �2x�1

(1� µ2

x�1

) + �2x+1

(1� µ2

x))Ax(z)

= �x�x+1

(1� µ2

x)Ax+1

(z)z + �x�x�1

(1� µ2

x�1

)Ax�1

(z)z�1

Given the periodicity in x, we conclude that this leads to a system of p homo-geneous linear equations for the quantities A

1

(z), . . . , Ap(z). In order to have anon-trivial solution its coe�cient matrix needs to have zero determinant. Thisleads to a polynomial equation of order p for the excitation energy, and determinesthe dispersion relation, ✏ = ✏(z). Without knowing p it is not very useful towrite down this system and its solution explicitly. Instead, we will analyse itin the formal limit where z ! 0, and z ! 1, in order to obtain an asymptoticexpansion for ✏(z) and the amplitude ratio

fx(z) =�xAx+1

(z)z

�x+1

Ax(z).

Let us start with large rapidity. It is clear from (3.26) that both the excitationenergy, and the amplitude ratio diverge linearly for large z. Writing

✏(z) = ��1

z + �0

+O(z�1), fx(z) = ↵xz(1 + �xz�1 +O(z�2))

we find that the coe�cients ↵x,�x are given by

↵x =��1

�2x+1

(1� µ2

x), �x =

�0

+ �2x+1

(1� µ2

x) + �2x�1

(1� µ2

x�1

)

��1

.

Here the constants ��1

and �0

may in principle be determined from the periodicityAx+p(z) = Ax(z) which leads to f

1

(z)f2

(z) · · · fp(z) = 1, but we will not needtheir explicit form. The limit of small rapidity leads to similar results. We findthat the excitation energy and amplitude ratio have the expansions

✏(z) = ��1

z�1 + �0

+O(z), fx(z) = ⇢xz(1 + ⌘xz +O(z2))

where the coe�cients ⇢x,�x are given by

⇢x =�2x(1� µ2

x)

��1

, ⌘x = ��0 + �2x(1� µ2

x) + �2x+2

(1� µ2

x+1

)

��1

.

It would be well justified to question the use of these expansions at thispoint. The idea is the following. For the two-particle problem we will derivethe S-matrix of the model as a complicated combination of the amplitude ratiosfx(zj), j = 1, 2. The expression carries an x-dependence which should howeverbe spurious. The formal limit where one of the rapidities tends to zero orinfinity allows to understand the (pole) structure of the S-matrix, and determineconditions on the staggering parameters which yield a position-independentS-matrix. The expressions derived above will be instrumental in this procedure.

28


3.4.3 The two-particle problem and the S-matrix

Next, we consider the case of two particles f = 2. As long as we project theSchrodinger equation on configurations where the two particles are far apart(and far from the boundaries), the Bethe ansatz wave function (3.24) solves theresulting di↵erence equation with the eigenvalue

E =NX

x=1

�2x + ✏(z1

) + ✏(z2

). (3.26)

If, however, the particles are next-to-nearest or nearest neighbours, we have totake into account that the action of the Hamiltonian induces pair formation, pairsplitting, and pair hopping.

Next-to-nearest neighbours

We start with the projection of the Schrodinger equation on a configuration wheretwo particles are next-to-nearest neighbours · · · � • � • � · · · , i.e. on some basisvector |x, x+ 2i. For this case we find the rather long equation

E (x, x+ 2) = �x�x�1

(1� µ2

x�1

) (x� 1, x+ 2)

+ �x+2

�x+3

(1� µ2

x+2

) (x, x+ 3)

+ �x�x+1

µx+1

Cx+1

(x+ 1, x+ 2)

+ �x+1

�x+2

µxCx (x, x+ 1)

+

NX

y=1

�2y � �2x�1

(1� µ2

x�1

)� �2x+1

� �2x+3

(1� µ2

x+2

)

!

(x, x+ 2).

If the Bethe ansatz holds together with the form of the eigenvalue as writtenin (3.26) then each of the arguments of the wave function can formally be treatedas for isolated particles. This leads to a second eigenvalue equation. Equatingthe two expressions we find the di↵erence equation

�x�x+1

(µx+1

Cx+1

+ µ2

x � 1) (x+ 1, x+ 2)

+ �x+1

�x+2

(µxCx + µ2

x+1

� 1) (x+ 1, x+ 2)

+ �2x+1

(1� µ2

x � µ2

x+1

) (x, x+ 2) = 0.

Given this equation there are two ways to proceed. We could impose the equa-tion as a constraint on the wave function, leaving the normalisation factors Cx

undetermined, but fixing the structure of the S-matrix. However, one can showthat this leads to a contradiction for other particle arrangements. Therefore, wewill instead require that the coe�cients multiplying the wave functions in thisequation be identically zero, so that the equation does not lead to any constraintsfor the S-matrix [35]. In the present case, the only non-trivial solution is

Cx = µx, and µ2

x + µ2

x+1

= 1, (3.27)

29


for arbitrary x. We conclude that our requirement implies the 2-periodicity ofthe staggering parameters µx while it does not fix the �x. As we shall see, thelatter will be constrained by the nearest-neighbour problem.

Nearest neighbours

Next, we project the Schrodinger equation for two particles onto a configurationwith a single pair · · ·��••�� · · · , i.e. on the basis vector |x, x+1i. We follow thesame procedure as in the last section and find the following di↵erence equationfor the wave function

�x�1

(�x+1

(x� 1, x)� �x (x� 1, x+ 1) + �x�1

(x, x+ 1)) + �x�x+1

(x, x)

+ �x+2

(�x+2

(x, x+ 1)� �x+1

(x, x+ 2) + �x (x+ 1, x+ 2))

+ �x�x+1

(x+ 1, x+ 1) = 0. (3.28)

Unlike in the case of next-to-nearest neighbours this equation cannot vanishidentically for non-trivial choices of the staggering parameters. Hence it willlead to constraints on the parameters in the Bethe ansatz wave function. Indeed,using (3.24) we find that the two amplitudes B

12

and B21

are related by

B12

Px(z, w) +B21

Px(w, z) = 0, (3.29)

where Px(z, w) denotes the complicated expression

Px(z, w) = �2x�1

✓

1� fx(w)

fx�1

(z)+ fx(w)

◆

+ �2x + �2x+1

fx(z)fx(w)

+ �2x+2

(fx(w)fx+1

(w)(fx(z)� 1) + fx(w)) .

Let us suppose that z and w are such that both Px(z, w) and Px(w, z) arenon-vanishing for all x. In this case we find the S-matrix of the model

S(z, w) =B

12

B21

= �Px(w, z)

Px(z, w).

The fact that this expression is rather implicit, since we have not yet found ageneral expression for fx(z), is perhaps less dramatic than its x-dependence. In-deed, the Bethe ansatz assumes that the amplitudes B� are position-independent.For generic choices of the staggering parameters the formula for S(z, w) leads,however, to an x-dependent expression. It follows that the only possible choicefor the model to be Bethe ansatz solvable is to impose the site-independencefor arbitrary z, w. One may try to do this by working directly with the givenexpression, but the resulting equations are quite involved. Hence, we choose totake advantage of our asymptotic expansions for fx(z) derived in section 3.4.2,and analyse the limit z ! 1. We find

S(z, w) = z�(w) +O(1), with �(w) =��1

(1� fx(w))

µ2

xfx(w)(�2

x+1

+ �2x+2

fx+1

(w)).

30


Like S(z, w) the function �(w) has to be position-independent. Finding whichstaggering parameters lead to this independence is still a delicate task, and hencewe analyse again only the leading terms of the expansion of �(w) as w ! 0:

�(w) =��1

µ2

xµ2

x+1

�2x�2

x+1

�

��1

w�1 � �0

+O(w)�

.

Every term in this expansion needs to be independent of x. Using the 2-periodicityfor µx, we see that this can at leading order only be true if the product �2x�

2

x+1

isindependent of x. This condition implies trivially that for real positive staggeringparameters we have

�x+2

= �x.

Hence, we find that the expression for the S-matrix found above is independent ofthe position x only if the staggering parameters are 2-periodic, and satisfy (3.1),which confirms that the latter is a necessary condition for Bethe ansatz solvability.

Reduced equation. If all staggering parameters have period two, then theequation for the wave function can be simplified. Indeed, notice that for �x+2

= �xthe first four terms equal the last four terms of the left-hand side of (3.28) up toa shift x ! x + 1. We may formalise this by introducing the shift operator Tdefined through T f(x) = f(x+ 1). Then we find

(1 + T ) [�x�1

(�x+1

(x� 1, x)� �x (x� 1, x+ 1)

+�x�1

(x, x+ 1)) + �x�x+1

(x, x)] = 0

Hence the expression within brackets lies in the kernel of (1 + T ), i.e. it is of theform (�1)x ⇥ const. In fact, one can show that this constant needs to be zero (weomit this tedious and not very illuminating discussion here). Using this result wefind the reduced equation

�x+1

( (x� 1, x) + (x, x+ 1)) + �x( (x, x)� (x� 1, x+ 1)) = 0. (3.30)

This relation has two advantages. First of all, it allows to write a somewhatsimpler version of (3.29). We find that

B12

Rx(z, w) +B21

Rx(w, z) = 0 (3.31a)

where Rx(z, w) is given by the expression

Rx(z, w) = 1 +�2x+1

�2x

✓

1� fx(w)

fx�1

(z)+ fx(w)

◆

. (3.31b)

This leads to a simplified expression for the S-matrix, and will be used below toderive a closed expression for it in terms of Jacobi theta functions. Second, (3.30)proves to be quite useful in order to show that processes involving three and fourparticles are indeed coherent.

31


Elliptic parametrisation

In the previous section, we saw that it is necessary to restrict the parameters ofthe model to the special submanifold. The aim of this and the following sectionis to show that this restriction is also su�cient for the model to be Bethe ansatzsolvable. To this end, it is convenient to re-examine the one-particle problem,and determine an explicit parametrisation for the rapidities and the excitationenergy, which will lead to an explicit and simple form for the S-matrix.

Staggering parameters. We need a suitable parametrisation of the parame-ters �x and µx. It turns out that a convenient choice is to write them in termsof Jacobi theta functions #j(u) = #j(u, q), j = 1, . . . , 4 where q is the so-calledelliptic nome. We follow the conventions of Whittaker and Watson [36]. In fact,it is su�cient to define

#1

(u, q) = �i1X

j=�1(�1)jq(j+1/2)2e(2j+1)iu.

The other theta functions are obtained by shifting the argument by ⇡/2, ⇡⌧/2and ⇡/2 + ⇡⌧/2 where ⌧ is related to the elliptic nome by q = ei⇡⌧ . Forinstance, #

4

(u) = iq1/4e�iu#1

(u� ⇡⌧/2). These functions can be thought of asgeneralisations of the trigonometric or the exponential functions. They satisfy ahost of identities, in particular various addition theorems which are at the heartof the simplifications in the following.

The staggering parameters are given as functions of two real parameters: tand the elliptic nome 0 q < 1. In terms of theta functions they read

µ2

x =

✓

#1

(✓)

#1

(2✓)

◆

2 #4

(t+ 2x✓)2

#4

(t+ (2x� 1)✓)#4

(t+ (2x+ 1)✓), ✓ =

⇡

4,

and

�2x = 2

✓

#1

(✓)

#1

(2✓)

◆

2 #4

(t+ (2x� 1)✓)2

#4

(t+ 2(x� 1)✓)#4

(t+ 2x✓), ✓ =

⇡

4.

One may check that this choice is compatible with both periodicity, and theequation µ2

x + µ2

x+1

= 1. The second equation implies that �2x + �2x+1

= 2, anormalisation which we are free to choose. The latter is designed to recover�x = 1 and µx = 1/

p2 in the trigonometric limit where the elliptic module q

tends to zero, and the Hamiltonian becomes translation invariant. For non-zeroq, we note that the transformation t ! t+ 2✓ is equivalent to a shift x ! x+ 1,and thus a translation of the system by one site. The elliptic parametrisationgiven here can be justified and derived in a systematic analysis of the Mk modelsfor all k = 1, 2, 3, . . . [14]. In figure 3.2 we plot the parameters as a function of twith 0 t 2✓ for q = 0, 0.05, . . . , 0.5. It is clear that this parametrisation mapsout the special submanifold.

32


0

5

10

15

20

l

0.0

0.5

1.0mx+1

0.0

0.5

1.0

mx

Figure 3.2: We plot the parameters µx

, µx+1 and � = �

x

/�x+1 as a function of t with

0 t 2✓ for q = 0, 0.05, . . . , 0.5. The black dot corresponds to q = 0, the outer redline corresponds to q = 0.5. Finally, we also show the special submanifold given by (3.1),where the model is integrable and enjoys additional supersymmetry. The constant qlines lie on the special submanifold.

Excitation energy, rapidity and one-particle wave function. Now let ususe this parametrisation in order to find convenient expressions for the excitationenergy ✏(z). As we know, it is determined by the homogeneous linear p ⇥ psystem (3.26). The period p = 2 of the staggering parameters implies that theperiodic part of the single-particle Bloch wave function satisfies Ax+2

(z) = Ax(z).We may use this in the recursion relation (3.26) which becomes of first order (asopposed to second order for arbitrary p > 2):

(✏(z) + �2x+1

)Ax(z) = �x�x+1

�

µ2

x+1

z + µ2

xz�1

�

Ax+1

(z)

Applying the periodicity property after shifting x ! x+ 1, we obtain that ✏(z)solves the second-order polynomial equation

✏(z)(✏(z) + 2) = ⇤2(z � z�1)2, ⇤ = 2

✓

#1

(✓)

#1

(2✓)

◆

4

.

Notice that this equation is independent of the parameter t. As we shall see thespectrum of the Hamiltonian is t-independent as a consequence. Solving this

33


equation for ✏(z) leads to roots of quartic polynomials in z which are neither elegantnor useful. Instead, we will uniformise this equation through the introduction oftheta-function parametrisations of the rapidities. One checks that the choice

z(u) = �#1(u+ ✓)

#1

(u� ✓)(3.32)

leads to

✏(u) ⌘ ✏(z(u)) = �2

✓

#1

(✓)

#1

(2✓)

◆

2 #1

(u)2

#1

(u� ✓)#1

(u+ ✓). (3.33)

The main tools in all these calculations are the addition theorems for the Jacobitheta functions mentioned above. Using these two relations, one may determinethe functions Ax(z) as a function of the parameter u from the recursion relationgiven above up to an overall factor. We fix the latter by the requirementAx(z = 1) = �x. This gives

Ax(u) ⌘ Ax(z(u)) =�x#4(t� u+ (2x� 1)✓)

#4

(t+ (2x� 1)✓).

The S-matrix. We are now in the position to derive an explicit expression forthe S-matrix in terms of Jacobi theta functions. To this end, we use (3.31), andexpress all the amplitude ratios fx(z) in terms of the expressions given in the lastparagraph. We find that

Rx(z(u), z(v)) =

� #1

(2✓)2#4

(t+ 2x✓)#4

(t+ 2(x�1)✓)#4

(t� (u+v) + (2x�1)✓)

#1

(✓)#4

(t+ (2x�1)✓)#4

(t� u+ (2x�1)✓)#4

(t� v + (2x�1)✓)r(u, v)

where r(u, v) has the simple form

r(u, v) =#1

(u� v + ✓)

#1

(u+ ✓)#1

(v � ✓). (3.34)

Notice that the x-dependent part is symmetric in u, v, and that r(u, v) does notdepend on the parameter t. This implies that we are left with the rather simpleequation B

12

r(u, v) +B21

r(v, u) = 0, and hence find the S-matrix

S(u, v) ⌘ S(z(u), z(v)) =z(u)

z(v)

#1

(u� v � ✓)

#1

(u� v + ✓), ✓ =

⇡

4.

This expression does not have the di↵erence property, i.e. it does not dependon u, v only through the di↵erence u � v. The reason for this is the exclusionconstraint of the model, which leads to the prefactor z(u)/z(v). Otherwise, theexpression is similar to the one for the eight-vertex model. The result of Blom andNienhuis for the M

1

model is similar [20], one needs to choose ✓ = ⇡/3 instead of✓ = ⇡/4.

34


3.4.4 Many particles: boundary conditions and the Betheequations

Consistency. Considering the one- and two-particle problems is not su�cientto conclude that the model is Bethe ansatz solvable. The next level of di�cultycomes from testing the Bethe ansatz for local three- and four-particle interactions,i.e. for projections on configurations · · · � • • � • � · · · and · · · � • � • • � · · · inthe sector with f = 3 particles, and · · · � • • � • • � · · · in the sector with f = 4particles.

The key to show consistency for these situations is the reduced di↵erenceequation (3.30), which we derived for the two-particle wave function. In fact, thenature of the Bethe ansatz implies that it holds in fact for any f in the followingway

�x+1

( (. . . , x� 1,x, . . . ) + (. . . , x, x+ 1, . . . ))

+ �x( (. . . , x, x, . . . )� (. . . , x� 1, x+ 1, . . . )) = 0. (3.35)

Let us start with f = 3 particles. If we consider (3.35) with an additionalparticle on the site x+ 1 or x+ 2, and re-apply it to the existing two particles,we obtain quite straightforwardly the following equations for the three-particlewave function:

(x� 1, x+ 1, x+ 1) + (x, x+ 1, x+ 2) =

(x� 1, x, x+ 1) + (x, x, x+ 2) = 0.(3.36)

These equations are su�cient to prove that the three-particle problem is consistent.Indeed, when projecting the Schrodinger equation onto the basis vector |x, x+2, x + 3i, and comparing it as for the one- and two-particle problems, to thecase when all particles are treated as if they were free, one obtains the followingconstraint on the wave function:

�x+1

( (x,x+ 2, x+ 4)� (x, x+ 2, x+ 2))

� �x( (x, x+ 3, x+ 4) + (x, x+ 2, x+ 3))

= �x( (x+ 1, x+ 1, x+ 3) + (x+ 1, x+ 2, x+ 3)).

The left-hand side vanishes as a consequence of (3.35) whereas the vanishingof the right-hand side is due to (3.36). The projection on the basis vector|x, x+ 1, x+ 3i leads to the same type of relation which holds identically. Thisexhausts all possible three-particle interactions, and shows that they can bereduced to two-particle processes via (3.35).

The case to be checked for f = 4 particles is the projection of the Schrodingerequation on the state |x, x+ 1, x+ 3, x+ 4i. The comparison to the case of freeparticles leads to the consistency constraint

µ2

x+1

⇣

�x( (x� 1, x+ 1, x+ 3, x+ 4)� (x, x, x+ 3, x+ 4))

35


� �x+1

( (x� 1, x, x+ 3, x+ 4) + (x, x+ 1, x+ 3, x+ 4))⌘

+ µ2

x

⇣

�x( (x, x+ 1, x+ 3, x+ 5)� (x, x+ 1, x+ 4, x+ 4))

� �x+1

( (x, x+ 1, x+ 4, x+ 5) + (x, x+ 1, x+ 3, x+ 4))⌘

= �x⇣

µ2

x+1

( (x+ 1, x+ 1, x+ 3, x+ 4) + (x, x+ 1, x+ 2, x+ 4)

+ µ2

x( (x, x+ 2, x+ 3, x+ 4) + (x, x+ 1, x+ 3, x+ 3)⌘

.

The left-hand side of this lengthy equation vanishes by applying (3.35) to thesecond two variables, whereas the right-hand side gives zero by applicationof (3.36) to the first and last three variables. Hence, also the four-particleproblem is consistent. This exhausts all cases which need to be checked: theconsistency for configurations with higher particle numbers can be reduced tolinear superpositions the one-, two-, three- and four-particle situations. Weconclude that the Bethe ansatz works consistently for the staggered M

2

modelalong the special line in parameter space (3.1).

Translation symmetry. We consider from now on an arbitrary number offermions f > 1 on the chain. In order to discuss the Bethe ansatz equations weneed to specify the boundary conditions of our model. We will consider the modelwith a twist. Because of the 2-periodicity of the staggering, the Hamiltoniancommutes with the square of the twisted translation operator T 0, [H, (T 0)2] = 0,as explained in section 2.3. Hence we impose the solutions of the Schrodingerequation to be eigenvectors of (T 0)2:

(T 0)2| i = t2| i.

The equation is written in a suggestive form: for the homogeneous model t issimply the eigenvalue of T 0, which becomes a proper symmetry of the Hamiltonianin this case. In the general, staggered case, we use (3.23) and project the resultingequation on a configuration |x

1

, . . . , xf i. Assuming that the first m = 0, 1, 2particles are located on the first 2 lattice sites, the projection leads to

(�1)m(f�1)ei�(m�2f/N) (xm+1

� 2, . . . , xf � 2, x1

+N � 2, . . . , xm +N � 2)

= t2 (x1

, . . . , xf ).

In the casem = 0, we use the Bethe ansatz (3.24) and conclude that the eigenvaluet2 is given by

t2 = e�2i�f/N

fY

j=1

z�2

j . (3.37)

If we require this result to be compatible with the other choices m = 1, 2 then weobtain a common constraint on the transformation behaviour of the amplitudes

36


B� under a cyclic shift of the permutation �. Let ⇡ be the cyclic shift, i.e.⇡(1) = 2,⇡(2) = 3, . . .⇡(f � 1) = f,⇡(f) = 1, then we have

B� = (�1)f�1ei�zN�(1)B�·⇡. (3.38)

Bethe equations. The cyclic shift property allows to derive the Bethe equa-tions for the model. For f particles the amplitudes B� are required to solve thesystem of equations

B···�(i),�(i+1),···r(u�(i), u�(i+1)

) +B···�(i+1),�(i),···r(u�(i+1)

, u�(i)) = 0, � 2 Sf .(3.39)

where r(u, v) is the function defined in (3.34). The system can be solved by

B� = C�1 sgn�Y

1m<nf

b�(m)�(n).

Here C is an arbitrary normalisation factor, and the numbers bmn are solutionsto the equations

bmnr(um, un) = bnmr(un, um).

If all the r(um, un) are finite and non-zero for allm,n then this system of equationshas the simple solution bmn = r(un, um). Using the cyclic shift property, weobtain in this case the Bethe equations of the model:

z(uk)N�f = e�i�

fY

j=1

#1

(uj � ✓)

#1

(uj + ✓)

#1

(uk � uj � ✓)

#1

(uk � uj + ✓)(3.40)

A solution of this equation leads to an eigenstate of the Hamiltonian whose energyis given by

E = N +fX

j=1

✏(uj) (3.41)

where ✏(u) is the elliptic form of the excitation energy found in (3.33). Inparticular, as the Bethe equations do not depend on the parameter t whichparametrises the constants µx,�x, nor does the excitation energy itself, weconclude that the energy is t-independent. It follows that the spectrum does notchange as one moves along the constant q lines on the special submanifold inparameter space plotted in figure 3.2. The eigenstate itself can be reconstructedfrom the amplitudes B�. With an appropriate choice of normalisation, we findthat they are given by

B� = sgn�fY

n=1

z(u�(n))�n

Y

1m<nf

#1

(u�(n) � u�(m)

+ ✓).

There are however cases, where r(um, un) vanishes or becomes infinite forcertain pairs m,n. This happens for so-called exact strings or bound states,

37


which need to be treated separately. We will show in the next section, that thesesomewhat exceptional cases are actually quite relevant in order to understandthe supersymmetry from the point of view of the Bethe ansatz.

3.5 Supersymmetry and the Bethe ansatzIn this section we analyse the relation between the di↵erent symmetries of themodel and the Bethe ansatz equations. We show that the action of the operatorsQ

+

and Q0

can be derived rather straightforwardly from the Bethe equations.The dynamic supersymmetry generated by Q� is however more subtle. As weshall see it is related to the existence of so-called exact strings of Bethe roots,which are present in the model essentially because the parameter ✓ is a rationalmultiple of ⇡.

3.5.1 Non-dynamic supersymmetry Q+

Let us start with the supersymmetry that was originally used to define the model.We claim that the action of Q

+

is equivalent to adding to a set of Bethe rootsu1

, . . . , uf that solve (3.40) an additional root uf+1

= 0, i.e. rapidity zf+1

= 1,without changing the number of sites. This is readily verified by comparing theBethe equations at f and f + 1 particles, which confirms our statement providedthat the twist angle is � = 0. The relation between the wave functions with fand f + 1 particles can be evaluated explicitly for zf+1

= 1:

(x1

, . . . , xf+1

) = const.⇥f+1

X

k=1

(�1)k�1�xk (x1

, . . . , xk�1

, xk+1

, . . . , xf+1

)

The string (�1)k�1 is a clear sign of a fermionic operator. Promoting thisrelation between wave functions to a relation between the corresponding statesby multiplying each side with ||x

1

, . . . , xf+1

ii, followed by a summation over allallowed particle arrangements, leads straightforwardly to the definition of Q

+

(up to a constant).

3.5.2 Dynamic supersymmetry Q0

The neutral dynamic supersymmetry can also be understood through a simpleaddition of a Bethe root to a given solution u

1

, . . . , uf of (3.40). The new memberhas uf+1

= ⇡/2, and corresponds therefore to a particle with rapidity zf+1

= �1.In addition to this particle insertion, one needs to increase the length of the chainby two.

From the Bethe ansatz form of the wave function we obtain a relation betweenthe wave functions for f + 1 particles with zf+1

= �1, and f particles witharbitrary rapidities:

(x1

, . . . , xf+1

) = const.⇥f+1

X

k=1

(�1)xk+k�1�xk+1

(x1

, . . . , xk�1

, xk+1

�2, . . . , xf+1

�2)

38

3.5 Supersymmetry and the Bethe ansatz

It is not di�cult to translate this equation into a relation between the correspond-ing eigenvectors of the Hamiltonian at N and N + 2 sites. The correspondingoperator is fermionic and carries “momentum” ⇡ as can be seen from the string(�1)xk+k�1. Working out its amplitudes leads precisely to the dynamic super-charge Q

0

discussed in section 3.3.2.

3.5.3 Dynamic supersymmetry Q�

The Bethe equations for the staggered M2

model resemble those of the eight-vertexmodel at so-called root-of-unity points, i.e. points where ✓ is a rational multipleof ⇡. It is known that at such points so-called exact strings appear in finite-sizesystems, i.e. configurations of Bethe roots which are arranged in the pattern

uj = u+ (j � 1)✓, ✓ = ⇡/4, (3.42)

with j = 1, 2, 3, 4. The aim of this section is to discuss a relation between theseexact strings and the dynamic supercharge Q�. To this end, we derive the wavefunction for a single exact string and then relate it to the action of Q� and Q

+

in the limit where the so-called string centre u tends to zero.Let us first discuss a few properties of an exact string of Bethe roots. From

the elliptic parameterisation of the Bethe roots given in (3.32), we infer that itstotal rapidity is given by

4

Y

j=1

z(uj) = 1,

and hence does not carry any net momentum. Moreover, the sum of the singleparticle excitation energies for its members yields

4

X

j=1

✏(uj) = �4,

irrespectively of the value u for the string centre. Comparing this with theexpression of the total energy in (3.41), we conclude that adding an exact stringto a configuration of Bethe roots decreases the energy by four. Therefore, if weadd simultaneously four sites to the system, the total energy remains unchanged.This observation hints at a dynamic symmetry relating the Hamiltonians forchains of length N and N + 4.

Here, we investigate the simplest case of a single exact string in order toestablish a relation with dynamic supersymmetry. To this end, we compute thewave function by following Baxter’s calculation for the six-vertex model [37].Concretely, for four particles we have to solve the equations, (3.39),

B···�(i),�(i+1),···r(u�(i), u�(i+1)

) +B···�(i+1),�(i),···r(u�(i+1)

, u�(i)) = 0.

These give the relations between the di↵erent amplitudes of the wave function, B�,where � 2 S

4

in the present case. For the configuration of Bethe roots that form

39


the exact string (3.42) it can easily be seen that the function r(ui, uj) definedin (3.34) vanishes whenever j = i+ 1 for i = 1, 2, 3, or i = 4, j = 1. Conversely,r(uj , ui) is nonzero for these cases. If we normalise the amplitudes such thatB

1234

6= 0 then our equations imply that all other amplitudes are finite, and thatB

2134

= 0 because r(u2

, u1

) = 0, B1324

= 0 because r(u3

, u2

) = 0 etc. We findthat all B� are zero except those for which the permutation � is an integer powerof the cyclic shift ⇡ = (1234). The remaining amplitudes are related because ofthe translation symmetry (3.38). For zero twist angle, we find the simple relation

Bj+1,...,4,1,...,j = (�1)jzNj+1

. . . zN4

B1234

.

We obtain the wave function for a single exact string by plugging into theBethe wave function, (3.24), the expression for the non-vanishing B�. Writingout the rapidities in their elliptic parametrisation, we find

(x1

, x2

, x3

, x4

) = C3

X

j=0

(�1)x1+x2

4

Y

k=1

Axk#1(u+ (k + j � 2)✓)xk+2�xk�3,

where

Axk = �xk

#4

(t� u1

+ (2xk � (k + j))✓)

#4

(t+ (2xk � 1)✓)

and C is some normalisation constant.We now take the limit where the centre of the exact string u tends to zero

and establish the relation with the supercharge Q�. To this end, observe thatfor u = 0 the wave function is only nonzero for a special set of configurations.In this case, the products in the exact-string wave function contain a factor#1

((k+ j � 2)✓) which is zero when k = 2� j, 6� j, so for every j there is a casefor which this becomes zero. The whole wave function vanishes therefore unlessalso xk+2

�xk � 3 = 0 simultaneously. This means that the fermion configurationcontains a pair of particles located at xk and xk+2

, such that xk+2

�xk = 3. Now,recall that because the particles are ordered there needs to be one particle on thetwo sites in between them. This fixes the relative positions of three particles. Theremaining one is itinerant: it can be anywhere as long as the exclusion constraintsof the model hold. For simplicity, we consider only the configurations for whichthe first site is occupied. The non-vanishing values of the Bethe wave functionsfor a chain of N sites are given by:

•1

� •3

•4

� · · · •x· · · �

N

(1, 3, 4, x) = C(#1

(✓)#1

(2✓))N�6�21

�0

�x#4

(t)2

#4

(t+ ✓)2

•1

•2

� •4

� · · · •x· · · � �

N

(1, 2, 4, x) = �C(#1

(✓)#1

(2✓))N�6�20

�1

�x#4

(t)2

#4

(t+ 3✓)2

40

3.5 Supersymmetry and the Bethe ansatz

•1

� •3

•4

� •6

� · · · �N

(1, 3, 4, 6) = C(#1

(✓)#1

(2✓))N�6#4

(t)2�20

�21

✓

1

#4

(t+ 3✓)2+

1

#4

(t+ ✓)2

◆

•1

•2

� •4

•5

� · · · �N

(1, 2, 4, 5) = �C(#1

(✓)#1

(2✓))N�6�21

�22

✓

#4

(t)2

#4

(t+ 3✓)2+#4

(t+ 2✓)2

#4

(t+ 3✓)2

◆

.

All other non-vanishing amplitudes can be recovered from these either frominvariance under translation by two sites, or by shifting t ! t + 2✓, whichamounts to a translation by one site. We now normalise the amplitudes by setting

C = � 1

(#1

(✓)#1

(2✓))N�6

#4

(t+ ✓)2

#4

(t)21

�21

µ0

µ1

.

The Bethe wave function gives the amplitudes of the non-orthonormal basis wedefined in (3.22). To find the amplitudes of the configurations in the basis inwhich we defined Q� (3.5) we have to include an extra normalisation factorµx for any pair starting a site x. When comparing with the definitions of thenon-dynamic and dynamic supercharges in section 3.3 and 3.3 we observe that weobtain after some algebra precisely the amplitudes of Q�Q+

(= �Q+

Q�) actingon the empty chain:

µ3

(1, 3, 4, x) = �µ0

�0

�x, µ1

(1, 2, 4, x) = µ0

�1

�x, for 5 < x < N,

µ3

(1, 3, 4, 6) = �2µ0

, µ1

µ4

(1, 2, 4, 5) = �21

.(3.43)

Remember that we showed that Q+

acts on Bethe states by adding a Betheroot u = 0 to a configuration. We conclude that Q� adds a set of three Betheroots which is centred around the Bethe root u = 0, in a pattern of an exact stringwith exactly the central root missing. To be more precise, the above argumentonly shows this for Q� acting on an empty chain, but we expect the action tobe the same when starting from a general configuration of Bethe roots. Anotherpossible way to prove this statement might be to study the insertion of the threeBethe roots u = ⇡/4,⇡/2, 3⇡/4 to a given solution of the Bethe equations, andincrease the number of sites by four. However, it appears that this has to be doneby employing a suitable limiting procedure which involves breaking translationsymmetry, and is therefore technically very challenging.

3.5.4 Multiplet structure

The eigenstates of the Hamiltonian organise in representations of its symmetryalgebra. From our discussion in section 3.3, we conclude that the amount ofsupersymmetry of the model on the special submanifold actually depends onthe translation subsector of the Hilbert space we consider. Here we provide abrief analysis of the resulting multiplet structure in the case of periodic boundaryconditions.

41


First of all, by definition the model always contains a copy of the N = 2 super-symmetry algebra with non-dynamic supercharges Q

+

, Q+

. As the Hamiltonianis given by their anticommutator, and thus a positive operator, its eigenvaluesare either zero or positive. It is well-known that the zero-energy eigenstatescorrespond to supersymmetry singlets, i.e. eigenstates which are annihilated byboth supercharges. Conversely, all positive eigenvalues are doubly-degenerate, andthe corresponding eigenvalues come as doublets (| i, Q

+

| i) where Q+

| i = 0.Second, in translation sectors where T 4 ⌘ 1, the multiplets have a richer

structure because of the presence of a second N = 2 algebra with dynamicsupercharges Q�, Q�. Since [T 2, H] = 0, we need to distinguish two caseshere. (i) If T 2 ⌘ �1, which can be the case only if the number of sites is amultiple of four, we find that states with positive energy organise in quartets(| i, Q

+

| i, Q�| i, Q+

Q�| i) where | i is annihilated by both Q+

and Q�. So-called short (or BPS) multiplets are not present, because the central charges ofthe algebra generated by the supercharges are all zero. (ii) If on the other handT 2 ⌘ 1, then a third dynamic copy of the supersymmetry algebra generated byQ

0

, Q0

is present. In this case, the non-zero energy states organise in octets,which are generated from a cyclic state, that is, a state annihilated by all theadjoint supercharges Q

+

, Q�, Q0

.

N N+2 N+4 N+6

f + 4

f + 2

f

Q�

Q0

Q0

Q+

Q+

Q+

Q+

Figure 3.3: Structure of a multiplet and its quantum numbers at positive energy: allstates are generated through the action of the supercharges on the cyclic state. Thedashed line corresponds to states with the same f as the cyclic state of the multiplet.

Multiplet structure in numerical spectra. The multiplet structure can beobserved directly in spectra obtained from exact diagonalisation of the Hamilto-nian. In this section we present the numerical data for three di↵erent choices ofthe parameters �x and µx, corresponding to the homogeneous point on the specialsubmanifold (red dot in figure 3.1), a di↵erent point on the special submanifold(blue dot in figure 3.1), and finally, for comparison a point away from the specialsubmanifold (green dot in figure 3.1).

In figure 3.4, we plot numerical data obtained by exact diagonalisation ofthe Hamiltonian. In particular, we plot the energy levels of chains of evenlengths N = 2, 4, . . . , 12 for the translation sector t2 = 1. The parameters in the

42

3.6 Conclusion

Hamiltonian are taken at the homogeneous point, i.e. �x = 1, µx = 1/p2 8 x.

To reveal the multiplet structure we split up the spectrum of a single chainlength by fermion number. The symmetry generated by Q

+

is very obviousin this plot, but also the full multiplets, octets in this case, can be detectedupon closer examination. For one of the multiplets, with the cyclic state atN = 2, f = 1, E = 2, we indicate the action of the supercharges. This can becompared directly with figure 3.3. The multiplets that start at length N = 8 orlarger are incomplete, because we truncate the spectrum at length N = 12.

To show that the multiplet structure survives when we move away from thehomogeneous point, while remaining on the manifold with dynamical supersym-metry, we show the same plot for �x = 4/5�x+1

= 1, µx = 1/2 and µx+1

=p3/2

for all even x for the translation sectors t2 = 1 and t2 = �1 (see figures 3.5and 3.6). The supermultiplets in the translation sector t2 = �1 consist of atmost 4 elements, since Q

0

is absent in this sector. Finally, in figure 3.7 we plotthe spectra for a choice of parameters away from the manifold with dynamicalsupersymmetry. The parameters are staggered with period 2, since this is requiredin order for T 2 to commute with the Hamiltonian. It is clear that the doubletstructure generated by Q

+

is preserved, while the multiplets corresponding tothe dynamical supersymmetry are absent.

Finally, we point out that the supersymmetries still do not account for all thedegeneracies in the spectrum (see for instance the degeneracies of the levels withE = 4 in figure 3.4). This suggests the possible presence of further symmetries.

3.6 ConclusionIn this chapter we provided a detailed analysis of the one-dimensional M

2

modelof strongly-interacting fermions and pairs. In particular, we determined a sub-manifold in the space of parameters for which the model presents two hiddendynamic supersymmetries. We showed that this symmetry enhancement is presentprecisely for the choice of parameters for which the model is also diagonalisableby the Bethe ansatz. This allowed to understand its various symmetries in termsof the Bethe equations: in particular, we pointed out a relation between theexistence of a dynamic supersymmetry and the existence of exact strings.

Furthermore, there are a few aspects beyond the M2

model. One of the keyfeatures to detect the Bethe ansatz solvability was an asymptotic analysis forsmall and large rapidities within the one- and two-particle problem of the Betheansatz. This strategy has potential to be applied to other systems, and help findintegrable points when only a Hamiltonian (without additional structures suchas transfer matrices etc.) is given. In fact, the reasoning presented here can beextended to all Mk models with k = 1, 2, 3, . . . [14].

43


1

1

1

1

1

1

1

1 1

1 1

1 1

1 1

1 1

1

1

1

1

3

1

1

1

3

1

1

1

51

1

1

1

15

1

1

1

1

1

2

1

1

1

5

1

1

2

1

1

2121

3

2121

41

1

1

2

1122251122211422

1

1

1

1

1

1

6

1

1

3113

2

23

6

2

231

1

121232

3212122322221323

22213

221

1

1

1

11

2116

1

2

3

21

1

1

32111216243

81212622431822

12

1

1

3

1

1

11

42114

41

24

7

42241

1

1

5

1

1

2

1

1

Q! Q! Q! Q!

Q0 Q0Q"

2 4 6 8 10 12Length

0

2

4

6

8

10

12

000000 111111 22222 3333 4444 555 66 78

Energy

Fermion number

Figure 3.4: The plot shows the energy levels of chains of even lengths N = 2, 4, . . . , 12for the translation sector t2 = 1. The parameters in the Hamiltonian are taken at thehomogeneous point, i.e. �

x

= 1, µx

= 1/p2 8 x. To reveal the multiplet structure we

split up the spectrum of a single chain length by fermion number. At the bottom of theplot we indicate a range containing the levels of a chain of a certain length. At the topof the plot we indicate the fermion number, the grey dashed and drawn lines serve asa guide to the eye, where the drawn lines correspond to fermion number f = 0. Thevertical axis labels the energy. We plot each level as a box, where the numerical valueof the energy corresponds to the centre of the box. The numbers inside the boxes givethe degeneracy of the level and the colours of the boxes are one-to-one with the fermionnumbers.

44

3.6 Conclusion

1

1

1

1

1

1

1

1 1

1 1

1 1

1 1

1 1

1

1

1

1

1

2

1

1

1

3

1

1

1

1

4

1

1

1

1

1

5

11

1

1

1

1

1

1

1

11

4

11

2

1

1

2

112321

1213

1

1

1

2

1211224122112124

211

1

1

1

1

1

5

1

1

1

12

321122621

22

1

1

1221

32212122312232221022213232

21

1

1

1

11

1

111

5

1111

3

1

1

1

1

11122

12411212242162422112224262112

2

1

1

1

2

1

1

11

1

1312

321213127221

223

1

1

11

4

11

2

1

1

2 4 6 8 10 12Length

0

2

4

6

8

10

000000 111111 22222 3333 4444 555 66 7 8

Energy

Fermion number

Figure 3.5: The plot shows the energy levels of chains of even lengths N = 2, 4, . . . , 12for the translation sector t2 = 1. The parameters in the Hamiltonian are taken at somepoint on the manifold with dynamical supersymmetry, specifically �

x

= 4/5�x+1 = 1,

µx

= 1/2 and µx+1 =

p3/2 for all even x. See figure 3.4 for details on how to read the

plot.

45


1

1

2

1

1

2

1

1

1

1

2

11

2

1

1

1

1

1

1

11

2

2

2

1

1

1

1

1

11

2111222

12

2

1

1

1

11

111

111142111

2

2

2

222

1

1

1

1

2

11

1

11

11

21

1112

22112212112222122112

2

1

1

111

111

111111412111124121121221114

1

1

1

11

2111222

12

2

Q! Q!

Q"

4 8 12 16Length

0

1

2

3

4

0000 1111 2222 333 444 555 66 77 88 910

Energy

Fermion number

Figure 3.6: The plot shows the energy levels of chains of even lengths N = 4, 8, . . . , 16for the translation sector t2 = �1. The parameters in the Hamiltonian are taken at somepoint on the manifold with dynamical supersymmetry, specifically �

x

= 4/5�x+1 = 1,

µx

= 1/2 and µx+1 =

p3/2 for all even x. The spectrum is cut-o↵ at E = 4.1. See

figure 3.4 for details on how to read the plot.

46

3.6 Conclusion

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

111

1

1

12

1

1

1

1

131

1

1

1

1

141

1

1

1

1

1

1

1

1

1

1

1

21

1111

1

1

1

111111111111111121

1

1

1

11111111111111111111111111131

111

1

1

11

11

11

2

11

11111111111111111111111111111

1

111111111111111111111111111111111111111111111111111111111111111111

111

1

1

1111

11111111111111111

1

1

11111111111111111111111111111111111111111111111111111111111111111111111111

1

1

1

11

1

11111111111111111111111111111111111111111111

11

11111111111

1

2 4 6 8 10 12Length

0

2

4

6

8

10000000 111111 22222 3333 4444 555 66 7 8

Energy

Fermion number

Figure 3.7: The plot shows the energy levels of chains of even lengths N = 2, 4, . . . , 12for the translation sector t2 = 1. The parameters in the Hamiltonian are staggered withperiod 2, but do not lie on the manifold with dynamical supersymmetry, specifically�x

= 3/4�x+1 = 1, µ

x

= 1/4 and µx+1 = 1/2 for all even x. See figure 3.4 for details on

how to read the plot.

47

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