Finite-size effects in the spectrum of the $OSp(3|2)$ superspin chain

The low energy spectrum of a spin chain with $OSp(3|2)$ supergroup symmetry is studied based on the Bethe ansatz solution of the related vertex model. This model is a lattice realization of intersecting loops in two dimensions with loop fugacity $z=1$ which provides a framework to study the critical properties of the unusual low temperature Goldstone phase of the $O(N)$ sigma model for $N=1$ in the context of an integrable model. Our finite-size analysis provides strong evidence for the existence of continua of scaling dimensions, the lowest of them starting at the ground state. Based on our data we conjecture that the so-called watermelon correlation functions decay logarithmically with exponents related to the quadratic Casimir operator of $OSp(3|2)$. The presence of a continuous spectrum is not affected by a change to the boundary conditions although the density of states in the continua appears to be modified.


I. INTRODUCTION
This paper is concerned with the study of the finite-size properties of a solvable two-dimensional vertex model based on the five-dimensional representation of the OSp(3|2) superalgebra. This system has a close relation with a particular Lorentz lattice gas used to model the diffusion of particles through randomly placed obstacles on the square lattice [1]. In this cellular automata the particle moves along the bonds of the lattice and is scattered according to scattering rules fixed a priori once it reaches a given node. Here the scatterers are constituted of mirrors tilted right and left, i.e. by ± π 4 , with respect to the lattice [2,3]. When the particle collides with a mirror it will turn right or left, depending on the orientation of the latter. In the absence of a mirror at a node the particle passes the node on a straight path. The corresponding scattering rules are depicted in Figure 1. Amplitudes w 1 , w 2 and w 3 represent the fraction of right and left mirrors and node vacancies on the lattice, respectively. The kinetic properties of this lattice Lorentz gas have been investigated by numerical simulations where an anomalous diffusive behavior was observed [4]. In the case of partially occupied lattice by mirrors (w 3 = 0) the fractal dimension of large trajectories was argued to be d f = 2 with the presence of logarithmic corrections [5,6]. supersymmetric vertex model which is the subject of this paper. The integrability of this model provides a framework for the study of its critical behaviour and -exploiting the equivalences listed above -of the peculiar properties observed in the Lorentz lattice gas. In fact, the finite-size analysis of the lowest excitation of the OSp(3|2) superspin chain found the respective critical exponent to be very small [1]. This was taken as an indication for the presence of a zero conformal dimension on the spectrum implying the superdiffusive behaviour (d f = 2) predicted for the Lorentz gas.
In the context of the loop model, it has been argued that this behaviour signals the existence of an unusual critical phase of intersecting loops: in two dimensions the crossing of loops, w 3 = 0, is a relevant perturbation to the low temperature dense loops phase [7]. There has been a series of attempts towards the identification of the other characteristic feature of this Goldstone phase starting from integrable lattice models, i.e. a finite density of vanishing critical exponents. The existence of a continuous spectrum of conformal weights has been established in several staggered superspin chains [8][9][10], including a model based on the four-dimensional representations of U q [sl(2|1)] alternating with their duals which, in the self-dual case, is isomorphic a deformation of the OSp(2|2) chain relevant to the loop model with fugacity N = 0 [11]. Further details of the spectral properties of these models have been uncovered when it was realized that hidden within the zero charge sector of this superspin chain there exists a staggered six-vertex model which already displays a continuous low energy spectrum. For the latter strong evidence has been accumulated that the effective theory describing the low energy excitations of the model is the SL(2, R)/U (1) sigma model at a level related to the anisotropy [12][13][14][15]. We note, however, that the focus of these studies has been on the anisotropic deformations of the vertex models: while it has been established that the isotropic N = 0 models are on the boundary of the critical region, the question of the critical properties at the isotropic point itself has not been addressed.
In this paper we want to return to the case of N = 1 as described by the OSp(3|2) supersymmetric vertex model. As argued above, the critical properties of this model are those of the proposed Goldstone phase of the O(1) sigma model and can be studied based on its solutions by means of the algebraic Bethe ansatz. By means of an extensive finite-size study of the model we accumulate ample evidence for the existence of continua of critical exponents with lower edges at scaling dimensions X = 0, 1, 2, 4, 6,. . . . With the exception of the the ground state of the superspin chain all of the states considered show strong logarithmic corrections to scaling governed by the flow of the model to weak coupling in the Goldstone phase. This is complemented by the observation that many excitations which have energies ∝ 1/L for small system sizes but disappear from the low energy spectrum as the system size is increased. Both of these features of the spectrum require very large system sizes to be studied for a reliable identification of the low energy effective theory: here we consider lattices with up to 4096 sites.
Finally, since recent numerical studies of the N = 1 intersecting loop model have emphasized the importance of boundary conditions for the long distance behaviour of correlation functions [16,17] we consider both periodic boundary conditions to the superspin chain and twisted ones depending on the fermion number. Comparing the results we find that the amplitudes of the subleading (logarithmic in the system size) finite-size corrections do depend on the choice of boundary conditions.

II. THE INTEGRABLE LATTICE MODEL
The statistical configurations of the Lorentz lattice gas or intersecting loop model mentioned in the introduction have a one-to-one correspondence with the generators of the braid-monoid algebra. This fact has been elaborated previously in the work [1] but for sake of completeness we have summarized this equivalence in Appendix A. This algebra can be used to built solvable models and it turns out that integrability is assured when the weights are parameterized as where λ is a free spectral parameter and w 0 is an arbitrary normalization. This scale can be chosen to interpret the weights as probabilities but in our context we set it to unity. Note that all three weights are positive for 0 ≤ λ ≤ 1 2 . Furthermore, it has been shown that this integrable manifold can be realized in terms of a standard local vertex model. Its bond states are constituted of three bosonic and two fermionic degrees of freedom realized in terms of the five-dimensional representation of the OSp(3|2) superalgebra, see Appendix B. The possible configurations on a vertex with their respective Boltzmann weights are encoded in the R-matrix ab are the 5 × 5 Weyl matrices acting either on the auxiliary space for k = 0 or on the quantum space associated to the sites of a chain of length L for k = 1, · · · , L. The symbol p a denotes the Grassmann parities distinguishing the bosonic (p a = 0) and fermionic (p a = 1) degrees of freedom. The 5 × 5-matrix α ab is the basic ingredient to built an explicit representation for the monoid operator and its expression depends much on the grading order basis, see for instance [18]. Here we will consider two specific Grassmann orderings in which the two U (1) charges of the OSp(3|2) algebra commuting with the operator R 0j (λ) are organized in a way which is suitable to perform the Bethe ansatz analysis. This turns out to be the f bbbf and bf bf b basis ordering and the corresponding forms for the matrix α are For the vertex model on the square lattice with L × L vertices and periodic boundary conditions for both bosonic and fermionic configurations in the horizontal direction we now construct the vertex model row-to-row transfer matrix. This operator is given as the supertrace over the auxiliary space of an ordered product of L matrices R 0j (λ), where periodic boundary conditions for bosonic and fermionic degrees of freedom are assumed.
In terms of the transfer matrix the partition function of the vertex model is given by a supertrace of the L th power of T (λ), now taken on the 5 L -dimensional quantum space. For system sizes up to 4 × 4 we find by direct computation in agreement with the triviality of the partition sum (1.1) of the loop model with z = 1. Here p k 1 , · · · , p k L are the Grassmann parities of the degrees of freedom composing a given k-state of the Hilbert space. The partition function of any statistical model is dominated by the largest eigenvalue of the transfer matrix. In the present case the contribution of a single vertex to the partition function is the sum of the three weights w 1 +w 2 +w 3 . Therefore, as long as the Boltzmann weights are all non-negative (i.e. in the regime 0 ≤ λ ≤ 1/2), we conclude that the largest eigenvalue of the transfer matrix Λ max (λ) for the lattice with linear dimension L is (2.7) The fact that there are no subleading (in L) corrections is a consequence of the grading of the states together with the properties of the OSp(3|2) representations appearing in the Hilbert space, We note that as an immediate consequence of the expression (2.7) for the largest eigenvalue of the transfer matrix the ground state energy of the superspin chain (2.5) is Eq. (2.6) implies that the partition function can be normalized to Z = 1 by rescaling of the local Boltzmann weights. Note that this does not necessarily mean that the low-lying excitations in the spectrum of the transfer matrix are trivial. In general, we can only infer that the critical properties are governed by a conformal field theory (CFT) with central charge c = 0. Since the Hamiltonian (2.5) is a non-Hermitian operator the continuum limit is not expected to be described by a unitary c = 0 conformal field theory.
In order to study the finite-size properties of the low-lying spectrum of this quantum spin chain we turn to its Bethe ansatz solution.

III. THE BETHE ANSATZ
The diagonalization of the transfer matrix can be carried out within the algebraic Bethe ansatz framework. The essential tools have already been discussed before [18] and here we shall present only the main results. It turns out to be convenient to re-scale the spectral parameter by the imaginary unit to bring the resulting equations into a canonical form for performing numerical analysis. Denoting by Λ(λ) the eigenvalues of the transfer matrix (2.4) we find that they can be parametrized in terms of two sets of rapidities {λ for all states except the OSp(3|2) singlet (0; 0) for which n 1 = n 2 = 0 (note that only OSp(3|2) irreps with integer p appear in the Hilbert space of the superspin chain).
The corresponding transfer matrix eigenvalue is given by the following expression: The eigenspectrum of the Hamiltonian can be derived from this expression giving The sets of variables {λ (a) j } are constrained by the Bethe ansatz equations which for the f bbbf grading are given by In this grading the Bethe ansatz uses a highest weight state of the (8L − 4)-dimensional representation (0; L/2) as reference state. States in the sector (p; q) with (3.1) are now parameterized by (L − n 2 − 1) rapidities λ (1) j and (L − n 1 − n 2 ) rapidities λ (2) j ). 1 In terms of these parameters the corresponding transfer matrix eigenvalue is The expression for energies of the Hamiltonian in this grading differs from the previous one by an overall minus sign: The Bethe equations for the rapidities λ  1 We use the same notation for the Bethe ansatz rapidities for both gradings. Therefore, whenever specific root configurations are discussed, they need to be seen in the context of the underlying grading. • (0; 0): the singlet ground state is again parametrized by two roots on each level, which have to satisfy (3.7) in this grading. It is straightforward to find the solution to be λ 1,2 = 0 (degenerate roots) giving the ground state energy E = −6.
Note that the degeneration of Bethe roots in the 'flat' ground state is a feature which has also been observed in other (super-)spin chains [8,9,19].

IV. GROUND STATE AND LOWEST EXCITATIONS
As discussed above the ground state energy of the superspin chain is exactly given by (2.8).
For L = 2 the corresponding root configurations in the Bethe ansätze (3.4) and (3.7) have been obtained in the previous section where they were found to be singular in the sense that Bethe roots may degenerate. For odd chain lengths L the situation turns out to be easier: here we find that the root configurations describing the (0; 1 2 ) ground state are non-degenerate. In the grading f bbbf it is given by collections of (L − 1)/2 pairs of complex conjugate rapidities on each level a = 1, 2. In the thermodynamic limit, L → ∞, the deviations from these 'strings' become small. This allows to compute the ground energy density and the Fermi velocity v F of the gapless low lying excitations within the root density approach [20] giving lim L→∞ E 0 (L)/L = −3 and v F = 2π, see also Ref. 1.
As a consequence of conformal invariance the leading terms in the finite-size scaling of energy levels are predicted to be [21,22] Here c is the central charge of the effective low energy theory. As a consequence of (2.   1|2n). In the present case with N = 1, and using the system size L as a long distance cutoff, the single coupling constant of the resulting sigma model on this supersphere is found to be within a perturbative RG approach [23,24] (note that log L 0 has to be negative for g σ ≥ 0 as expected on physical grounds). With this as an input we extrapolate the finite-size data for the scaling dimensions assuming a rational dependence on 1/ log L. Our results for the lowest (0; q) states are displayed in Figure 3. From these data we conclude that all dimensions X (0;q) with finite q ≥ 1 vanish in the thermodynamic limit, showing subleading scaling corrections proportional to 1/ log L. For q > 2 the fact that X (0;q) → 0 can be obscured by the latter for quite large L rendering a finite-size analysis based on small system sizes impossible.
To determine the amplitude of these subleading terms we note that this class of (p = 0; q)-states can be extended to include the ground states of the superspin chain, i.e. the singlet (0; 0) for L even and the quintet (0, 1 2 ) for L odd. Since there are no finite-size corrections to the energies of the ground states (with q = 0 and q = 1 2 for L even and odd, respectively) we conjecture that the amplitudes of the logarithms are related to the quadratic Casimir (B1) of OSp(3|2) as Comparing this conjecture with our numerical data we find very good agreement, see Figure 3(b).
The lattice sizes L considered here do not allow for a realiable estimate of the non-universal scale log L 0 though. In the context of the loop model the amplitudes (4.5) determine the long distance asymptotics of the 'watermelon' correlation functions G k (r) measuring the probability of k loop segments connecting two points at distance r: these correlators -two-point functions of the socalled k-leg operators -vanish with a power of 1/ log r, i.e.
G k (r) ∼ 1/ (log r) α k . (B1) of OSp(3|2). Identifying the (0; q) primary with the k = 2q-leg operator this agrees with RG calculations and numerical results for G 2 (r) and G 4 (r) [17]. As will be seen below, however, the asymptotic behaviour of the watermelon correlators is likely to depend on the boundary conditions.
We further note that the reference state in the second Bethe ansatz is a highest weight state in j . In this situation the second set of the Bethe equations (3.7) is automatically satisfied while the first level ones become, after taking their logarithm, Here ψ a (x) = 2 arctan(x/a) and the numbers Q j define the many possible branches of the logarithm being given by the expression, Within this approach the eigenenergies corresponding to this state can be obtained using the following expression (4.9) Comparing the numerical solution of these 'string' equations with the those of (3.7) we conclude that the finite-size energies in the sectors (0; q) with q > 2 are reproduced by Eqs. (4.7)-(4.9), see Figure 3(a). In this formulation the root density approach can be applied to compute the finitesize energies. In this approach we find again that for L → ∞ the conformal dimensions are zero, independent of q. This observation provides an additional analytical support to the existence of a continuum of zero conformal weights starting at zero.

V. OTHER EXCITATIONS
As for the sectors (0; q) in the previous section we have identified the Bethe configurations corresponding to the lowest energy states in the sectors (p; q) with p = 0. In Figure 4 we present the corresponding scaling dimensions for p = 1. Again, the energies show strong logarithmic corrections to scaling which are dealt with in the extrapolation by assuming a rational dependence In Table I we have collected the discrete parts of the conformal weights of primary operators in the CFT as identified from our numerical studies. Based on these data we conjecture that the (1, 0), (0, 1) spectrum of conformal weights is given by h k = 1 2 k(k + 1), k = 0, 1, 2, . . . and the lowest levels in the (p; q) sector of the spectrum of the superspin chain correspond to operators with conformal This behaviour is not limited to excitations in the (0; 1) sector but we have found such levels in many other sectors, too. Let us note that the removal of low energy states present in small systems in the scaling limit has also been observed in one phase of the U q [sl(2|1)] staggered superspin chain [11]. It is a direct consequence of the vanishing of a coupling constant, such as in (4.4) for the OSp(3|2) model, and therefore expected to be a generic property of realizations for field theories with a continuous spectrum of critical exponents as lattice models with a compact quantum space. reproducing the numerical value for the energy, we cannot confirm this expectation.

VI. BOUNDARY CONDITIONS
Here we would like to point out that peculiar finite-size behavior reported so far is also present if we had modified the boundary conditions such that the five possible states on the bonds of the vertex model are considered to be bosonic degrees of freedom. In this situation the vertex model transfer matrix is given, instead of (2.4), as the standard trace over the auxiliary space of the following product of operators, bb plays the role of a graded identity matrix. We recall here that this type of transfer matrix has been considered before in the case a vertex model based on the OSp(1|2) superalgebra [25].
The corresponding energies of the Hamiltonian with these boundary conditions are given by Eq. (3.3), as before.
It is evident from this construction that the finite-size spectrum will be different from that of the superspin chain only for sectors where the number of fermions n 1 is even. To see the effect on the scaling dimensions we have to take into account that the Bethe state of highest weight in the even fermion sector of the (p; q)-multiplet is parametrized by rapidities in (6.2) for p even rather than (3.1) which still holds for p odd.
The ground state of the spin chain with twist is the unique singlet (0; 0) as for the superspin chain before. The corresponding root configuration consists of L/2 pairs of rapidities (4.1). Unlike the situation in the superspin chain the ground state of the twisted model has a strong finite-size dependence on L, see Figure 9(a): the additional phase in the Bethe equations (6.2) leads to a central charge c = 3 as noted before in Refs. 1 and 7. In addition there are subleading corrections to scaling which turn out to much stronger than the 1/[log L] 3 behavior usually observed in isotropic spin chains. We stress that this is even in contrast to the finite-size behavior found for other spin chains invariant by superalgebra such as OSp(1|2) and OSp(2|2).
In addition we have solved Eqs. (6.2) for some low energy excitations (p; q) e with even fermion number n 1 . Their root configuration contains n 2 real roots λ (1) in addition to pairs as in the ground state. Our results show that the low energy spectrum of the spin chain with anti-periodic boundary conditions for the fermionic states shows a similar behaviour similar to that of the superspin chain, see Figure 9 (p; q) multiplet of the superspin chain. Here the scaling dimensions, however, have to be computed from (4.2) relative to the new ground state which leads to a shift X o = X (p;q) +c/12 as a consequence of the different effective central charge. As in the superspin chain the subleading corrections to the scaling dimensions vanish as 1/ log L. The amplitudes of these terms display a q-dependence which is clearly different from (4.5), even for the X o as a consequence of the logarithmic corrections to the central charge. Our data do not allow to quantify these amplitudes though: (much) larger system sizes would be needed for an estimate which is beyond the methods used in this work. It has been observed [1] that for integer z this algebra has a realization in terms of a finite dimensional representation of the superalgebra OSp(m|2n) provided that

ACKNOWLEDGMENTS
In this formulation the braid operator becomes the graded permutation between m bosonic and 2n fermionic degrees of freedom, where p a is the Grassmann parity of the a-th degree of freedom assuming values p a = 0 for bosons and p a = 1 for fermions.