The fine structure of the finite-size effects for the spectrum of the $OSp(n|2m)$ spin chain

In this paper we investigate the finite-size properties of the spectrum of quantum spin chains with local spins taken to be the fundamental vector representation of the $OSp(n|2m)$ superalgebra.


I. INTRODUCTION
Over the years exactly solvable one-dimensional quantum magnets have been considered as suitable lattice regularization of two-dimensional space-time models of quantum field theory. In principle the respective Bethe ansatz solution offers us a non-perturbative framework to study the properties of the spectrum of the respective spin chain Hamiltonian for large system sizes. In the case of a massless theory it has been showed that the finite size corrections to the spectrum determine the conformal central charge and the anomalous dimensions of the underlying conformal field theory [1]. The study of the finite-size effects of integrable spin chains with generators on some simply laced Lie algebra G suggested that their critical behaviour are governed by the properties of a field theory of Wess-Zumino-Witten type on the same group G. By way of contrast when the underlying invariance of the spin chain is based on supergroups the identification of the respective field theory appears to be more involved [2]. Indeed, it has been observed that the finite size spectrum of the OSp(3|2) superspin chain present the unusual feature of having states with the same conformal dimension as the trivial identity operator [3,4]. Later on similar phenomena have been found to be present in a staggered sl(2|1) spin chain whose degrees of freedom alternate between the fundamental and dual representations [5] as well as in staggered six-vertex model [6].
The degeneracy of many states of the spectrum was found to grow with the size of the chain and this was interpreted as the signature of the existence of non-compact degrees of freedom in the continuum limit [6].
The purpose of this paper is to study the subleading corrections to the finite-size spectrum of a number of spin chains invariant by the OSp(n|2m) super Lie algebra. The results obtained here arXiv:1802.05191v1 [cond-mat.stat-mech] 14 Feb 2018 extend in a substantial way our recent analysis performed for the specific case of the OSp(3|2) superalgebra [4]. In particular, we find a tower of states over the lowest energy with the same leading effective central charge c eff as the size of the chain L → ∞. More precisely, denoting the eigenenergies of such set of states by E k (L) we have, where the integer k ∞ is typically bounded by system size L. The symbol e ∞ denotes the energy density of the ground state in the thermodynamic limit while ξ refers to the velocity of the elementary low-lying excitations. We shall notice that the amplitude β k can be connected to a subset of the possible eigenvalues of the quadratic Casimir operator of the respective underlying OSp(n|2m) superalgebra.
We recall here that the OSp(n|2m) superspin chain realizes a gas of loops on the square lattice in which intersections are allowed [3]. The integer n and m parameterize the fugacity z given to every configuration of closed loops which is z = n − 2m. In the context of the loop model the above peculiar finite-size behaviour was argued to be an indication that for z < 2 the crossing of loops becomes a relevant perturbation driving the system to an unusual critical phase [7]. In particular it was conjectured that the correlations functions in the loop model should be those of the Goldstone phase of the O(z) sigma model. The universal behaviour of the two point correlators has long been computed in [8] and it was found to decrease logarithmically with the distance. More recently this calculation has been extended to two point functions of operators composed by the product of k field components at the same point usually denominated k-leg watermelon correlators [9]. This observable measures the probability of k distinct loop segments connecting two arbitrary lattice points x and y. Here we shall argue that the asymptotic behaviour of such correlation functions of the intersecting loop model can be inferred from the finite-size amplitudes β k in analogy to the known connection among critical exponents and finite-size scaling amplitudes [1]. More precisely we observe that for large distances r = |x − y| this family of correlators can be rewritten as for a suitable choice of the k 0 state.

II. THE OSp(n|2m) SPIN CHAIN
The vertex model with rational weights which is invariant by the superalgebra OSp(n|2m) was first discovered by Kulish in the context of the graded formulation of the Yang-Baxter equation [10]. The respective R-matrix R ab (λ) with spectral parameter λ can be represented as a linear combination of three basic operators, where R ab (λ) acts on the tensor product V a × V b of two (n + 2m)-dimensional graded vector spaces and I a denotes the identity matrix in one of such spaces. The integers n and 2m stand for the number of bosonic (b) and fermionic (f ) degrees of freedom.
The operator P ab permutes two graded vector spaces and its expression is, where p i = 0 for the n bosonic basis vectors while for the 2m fermionic coordinates we have p i = 1.
The elementary matrices e (a) ij ∈ V a have only one non-vanishing element with value 1 at row i and column j.
The operator E ab plays the role of a typical monoid operator which can formally be represented as, where the non-null matrix elements α ij are always ±1. Their precise distribution within the matrix α depends on the grading sequence we set up for the basis of the vector space. A convenient grading sequence is the basis ordering f 1 · · · f m b 1 · · · b n f m+1 · · · f 2m since it encodes in an explicit way the many U (1) symmetries of the OSp(n|2m) superalgebra. For this choice of grading the structure of the matrix α is [11], where O N ×N and I N ×N are the null and the anti-diagonal N ×N matrices, respectively. The matrix representation for other grading choices can be obtained from Eq.(2.4) by direct permutation of the vector space basis.
In the intersecting loop model realized by this superspin chain the different terms in the R- spectral parameter for which the R is proportional to the graded permutator. Let us denote such transfer matrix by T (λ) on a L × L square lattice with toroidal boundary conditions. It follows that this operator can be written as the supertrace of an auxiliary operator called monodromy matrix [10], where elements of the monodromy matrix T ij are given by an ordered product of R-matrices acting on the same auxiliary space but with distinct quantum space components, As usual considering the logarithmic derivatives of T (λ) around the regular point λ = 0 we obtain the local integrals of motion. The first non-trivial charge turns to be the Hamiltonian whose expression is, where periodic boundary conditions for both bosonic and fermionic degrees of freedom is assumed.
The spectrum of this Hamiltonian can be studied by Bethe ansatz methods and is parametrized by solutions to a set of algebraic Bethe equations. Since these Bethe equations depend on the particular choice of the grading their root configurations are grading dependent. We can however infer on the infinite volume properties of such superspin chain without the need of choosing an specific Bethe ansatz solution [3]. This can be done establishing certain a functional relation for the largest eigenvalue of the transfer matrix usually by means of the matrix inversion method [12,13]. In our case this identity can be derived combining the unitarity property of the R-matrix (2.1) together with its crossing symmetry under translation λ → (2 − n + 2m)/2 − λ of the spectral parameter. Let us denote by [Λ 0 (λ)] L the largest eigenvalue which dominates the partition function of the vertex model per site in the thermodynamic limit. We find that Λ 0 (λ) satisfies the following constraint, Using unitarity Λ 0 (λ)Λ 0 (−λ) = (1 − λ 2 ) we can solve the above functional relation under the assumption of analyticity in the region 0 ≤ λ < |2 − n + 2m|/2. The final result is, where Γ(x) is the Euler's integral of the second kind.
The ground state energy per site e ∞ of the OSp(n|2m) spin chain (2.7) is obtained by taking the logarithmic derivative of Λ 0 (λ) at the spectral point λ = 0. After some simplifications we find, is the Euler psi function.
The same reasoning as above can be used to obtain the dispersion relation for the low-lying excitations, see for instance Ref. [14]. These states correspond to next largest eigenvalues of the transfer matrix and their ratios with the ground state [Λ 0 (λ)] L defines the excitation function γ(λ).
Considering that Eq.(2.8) applies also for the excitations such function is expected to satisfies the constraint γ(λ)γ(λ + 2−n+2m 2 ) = 1. This means that γ(λ) has the real period |2 − n + 2m| and consequently it can be expressed in terms of product of trigonometric functions. We can now follow the reasoning discussed in [14] and conclude that the dispersion relation e(p) for the lowlying excitations with momenta p is, e(p) = 2π |2 − n + 2m| sin(p) (2.11) and therefore the speed of sound is ξ = 2π |2−n+2m| . We would like to note that for the results so far it has implicitly been assumed that n − 2m = 2.
For n − 2m = 2 we can not derive the Hamiltonian from the R-matrix (2.1) since there is no point λ 0 such that R ab (λ 0 ) ∼ P ab . These are the cases in which the Killing form of the OSp(n|2m) superalgebra is degenerated. One way to circumvent this problem is to scale the spectral parameter λ → λ(2 − n + 2m)/2 and afterwards take the limit n − 2m → 2 in Eq.(2.1) to obtain, which is the R-matrix of the so-called Temperley-Lieb model with E 2 ab = 2E ab [13]. We note that in this case the respective loop model realization does not permit configurations involving intersecting paths since the identity operator is not present in the R-matrix (2.12). We further recall that for n = 2 and m = 0 the vertex model corresponds to the isotropic six-vertex model. The expression of the respective anti-ferromagnetic Hamiltonian is, The inversion method can also provide us with exact results for the vertex model with weights based on the R-matrix (2.12). It turns out that the respective partition function per site is, while the ground state energy and dispersion relation associated to the Hamiltonian (2.13) are, From the above results we conclude that the bulk behaviour depends only on the loop model fugacity z = n − 2m. In next sections we shall present evidences that this feature still remains valid for the central charge and for the compact part of the critical exponents of the underlying conformal field theory.

III. SMALL SIZE RESULTS
In order to gain some insight on the spectrum properties of the OSp(n|2m) superspin chain we have numerically diagonalize the respective Hamiltonians for lattice sizes L ≤ 8. We have limited our analysis to Hamiltonians with maximum number of seven states per site n + 2m = 7. We find that the ground state is generically degenerated for spin chains with n − 2m ≤ 0 while when n − 2m ≥ 1 the ground state is always a singlet for L even. In Table I we present the ground state degeneracies for the OSp(n|2m) spin chains studied in this paper for even and odd lattice sizes.
We have noted that for a fixed fugacity n − 2m the eigenspectrum are basically the same apart degeneracies up to the size L = 4 for distinct values of n and m. Considerable number of new OSp(3|4) 23 7 OSp(2|2) 8 4 OSp(3|2) 1 5 OSp(2|4) 16 32 OSp(5|2) 1 7 eigenvalues start to emerge for L = 6 but they occur at the higher energy part of the spectrum.
These findings suggests that for large enough L the spectrum should satisfies the following sequence of inclusions, such that the ground state and the low-lying excitations for a given fugacity n − 2m is described by the superspin chain with the lowest possible values of the integers n and m. This feature is present in spin chains with different supergroup symmetries, e.g. for gl(m|n) where a spectral embedding of models with given m − n has been observed [15].
This above observation can be used in order to predict the value for the effective central charge.
For n − 2m ≥ 2 the sequence can be started with the orthogonal invariance O(n − 2m) and the respective conformal field theory should be that of the Wess-Zumino-Witten model on this group see for instance [16,17]. The partition function is expected to be dominated by n − 2m Ising degrees of freedom and therefore the central charge is, On the other hand when n − 2m < 2 the orthogonal invariance is somehow broken and the partition function is effectively dominated by n − 2m − 1 bosonic degrees of freedom with effective central charge [3], At this point we remark that in the context of the intersecting loop model these two regimes are distinguished by the behaviour of the respective Boltzmann weights. We note that for n − 2m < 2 the three weights in Eq.(2.1) can be chosen positive and consequently they can be interpreted as probabilities. However when n − 2m > 2 one of the weights is always negative and the probability interpretation is lost. Therefore it is not a surprise that the continuum limit of these regimes are described by two different conformal field theories.
In next section we shall begin our study of the finite-size effects for large L for the superspin chains in Table I by using convenient grading choice for the Bethe ansatz solution. We will investigate two specific sequences of models with the same fugacity and argue that the potential extra eigenvalues does not lead to new conformal dimensions.

IV. FINITE SIZE EFFECTS
In this section we will investigate the finite size properties of the super spin chains with the help of their Bethe ansatz solution. As mentioned above it is a common feature of integrable spin models based on super Lie algebras that the Bethe equations for the rapidities parametrizing their spectrum depend on the choice of grading. In a first step we have to choose the formulation which is most convenient for the numerical solution of the respective Bethe ansatz equations for large system sizes. By now it is well know that for rational vertex models there exists a direct connection between the form of the Bethe ansatz equations with the specific Dynkin diagram representation of the underlying superalgebra. In Figure 2 we exhibit the diagrams with the respective grading ordering for the orthosympletic superalgebras suitable for each super spin chain studied in this paper. The explicit form of the Bethe equations and the basic root distributions is presented in the next subsections.
Based on the numerical solution of the Bethe equations we can analyze the finite size scaling of the spectrum. For a conformally invariant theory the finite size gaps are expected to scale as [18,19] where X k are the scaling dimensions of the corresponding operator in the continuum limit and the effective central charge c eff governs the finite size scaling of the ground state energy E 0 (L) of the lattice model. Similarly, from the momentum of the states the conformal spin of the corresponding operator can be determined, s(k; L) = (L/2π)(P k (L) − P 0 ).
As we shall see below the spectrum of scaling dimensions of the OSp(n|2m) models is highly degenerate in the thermodynamic limit. In a finite system this degeneracy is lifted by subleading corrections to scaling which can be studied in conformal perturbation theory [20,21]. With re- For the OSp(2|4) model it turns out to be most convenient to use the grading Bethe equations for the grading f f bbf f (4.4) The roots of (4.3) corresponding to the ground state and many of the low-lying excitations are found to be real with finite densities N 1 /L → 1, N ± /L → 1 2 in the thermodynamic limit. This fact allows to study their finite size scaling analytically based on linear integral equations [22][23][24]. In the present case we find that the Bethe ansatz integral equations have a singular kernel, similar as in the staggered sl(2|1) superspin chains and the staggered six-vertex model where this has been found to lead to a continuous spectrum of scaling dimensions [5,6,25,26].  with finite scaling dimension X). Taking these constraints into account we find that the conformal weights in the low energy effective theory are non-negative integers.
The ground state of the chain for even L appears in the sector with (n 1 , n 2 , n 3 ) = (1, 2, 0) (and also (1, 0, 0)) and m 1 = m 2 = 0. Both from (4.5) and the extrapolation of data obtained by numerically solving (4.3) we find Among these is the lowest energy state of the odd length super spin chains for k = ±1 -are also described by real Bethe roots. As expected from (4.5) they exhibit the same leading finite size scaling as the ground state but different subleading corrections, see Figure 3. From our numerical data based on the solution of the Bethe equations we find that the corrections to the scaling dimension of these states vanish as 1/ log L, as expected from perturbative renormalization group analysis of the low temperature Goldstone phase of the loop models with n−2m < 2 [7]. Analyzing the subleading corrections in detail we find Similar groups of excitations parameterized by real Bethe roots appear in the sectors (n 1 , n 2 , n 3 ) = (2, 3, k) and (2, 4, k) with n 2 + k ∼ L mod 2, see Figure 3. The finite size analysis shows that the corresponding primaries have a scaling dimension X (2,3,k) = 1 and X (2,4,k) = 2, their conformal spins are s = 1 and s = 0, 2, respectively. Again, the subleading corrections to finite size scaling are found to vanish as 1/ log L with k-dependent amplitudes.
Among the remaining low energy levels in the spectrum of small systems found by exact diagonalization we have identified (see Figure 3) • a descendent of the ground state with X = 1, s = 1 in the (n 1 , n 2 , n 3 ) = (1, 2, 0) sector described by a Bethe root configuration containing a single 2-string of complex conjugate Bethe roots λ (1) 0± λ 0 ± i/2 with real λ 0 in addition to the real ones.
• two states in the sectors (2, 4, 0) and (2, 4, 2) disappear from the low energy spectrum as the system size is increased. Such behaviour is expected for levels violating the constraint For the OSp(1|2) model the Bethe equations are [10,27]  (4.10) The operator content of the effective theory describing this superspin chain at low energies is known from Ref. [27], the primary fields have scaling dimensions for states with superspin J = n and vorticity m subject to the constraint (n + m) ∈ Z + 1 2 . * Hence, for the triplet ground state (n, m) = ( 1 2 , 0) we find the central charge c eff = −2. The finite size scaling and finite temperature properties of the OSp(1|2) superspin has recently been studied based on a formulation of the Bethe ansatz in terms of nonlinear integral equations [28]. As a consequence of the boundary conditions used in that work the effective central charge obtained from the low temperature behaviour differs from the one appearing in the finite size scaling behaviour of the ground state. We note that the ground state is degenerate (up to subleading corrections to scaling) with the lowest singlet, (n, m) = (0, 1 2 ). As discussed above, the spectrum of the OSp(1|2) superspin chain is a subset of that of the OSp(3|4) model. Therefore we discuss the finite size scaling in the context of the latter based on the Bethe ansatz for grading bf f bf f b and real centers λ  ((1, 1, k) (4.14) A second tower of primaries with spin s = 1 levels extrapolating to X (3|4) = 1 is found in the corresponding Bethe root configurations one of the N 1 = L − 2 − k roots on the first level is real.
The lowest of these excitations, k = 0, is also present as a triplet in the spectrum of the OSp(1|2).
In addition there are descendents of the (1, 1, k) primaries with scaling dimension X = 1.
The next excitations, both of the OSp(1|2) and the OSp(3|4) chain, for which we have determined the Bethe root configurations correspond to fields with scaling dimension X = 2. Their conformal spin is s = 0 or 2.
We study the spectrum of the OSp(2|2) superspin chain using the Bethe equations in the grading f bbf : Solutions to these equations parametrize states with energy (4.16) The ground state and low energy excitations of the OSp(2|2) superspin chain are described by real roots of (4.15) with densities N k /L → 1/2 in the thermodynamic limit, see [29]. for levels with different n 2 is lifted for finite system sizes, see Figure 5. Analyzing the subleading corrections to scaling of the (n 1 , n 2 ) = (1, k) states, k = 0, 1, 2, . . . ∼ L − 1 mod 2, we find a tower of levels: A tower of spin s = 1 excitations extrapolating to X = 1 is found in the sectors (n 1 , n 2 ) = (2, k) with k = 0, 1, 2, . . . ∼ L mod 2. A state with spin s = 2 in the sector (2, 4) disappears from the low energy spectrum as the system size is increased since the restriction m 2 = 0 is violated. The finite size spectrum of the OSp(3|2) superspin chain has been studied extensively using its solution by means of the algebraic Bethe ansatz in Ref. [4]. The ground state displays no finite size corrections from it has been concluded that c eff = 0. The excitations considered in that work can be grouped into towers extrapolating to integer scaling dimensions X = 0, 1, 2, . . . or disappear from the low energy spectrum in the thermodynamic limit. The degeneracies in the spectrum of scaling dimensions is lifted for finite system sizes: with the exception of the ground state the finite size gaps show strong logarithmic corrections to scaling. For the levels in the X = 0 tower these corrections have been found to scale as E. n − 2m = +3 As mentioned above the continuum limit of the OSp(n|2m) superspin chain for n − 2m > 2 is expected to be different from that for the cases discussed so far. Here first insights into the finite size Its primaries can be written as composite operators built from an Gaussian representing the Kac-Moody algebra with topological charge k = 2 and an Ising field [32]. The lowest scaling dimensions appearing in the lattice model of length L are   Unlike in the OSp(n|2m) models with n − 2m < 2 discussed above the ground state of the superspin chain remains a unique singlet indicating the absence of a symmetry breaking transition into a low temperature phase of the loop models in this regime [7].
Higher energy excitations for even length superspin chains have been found extrapolating to scaling dimensions X = 1, 3/8 + 1, 2, and 3/8 + 2, see Figure 6. The first of these is in the sector with (n 1 , n 2 , n 3 ) = (1, 1, 0) and its energy is that of the zero-spin field with scaling dimensions X = 1 in the O(3) model. Its root configuration differs from the one for the lowest tower by one root λ (1) = 0 on the first level. The energy is that of the zero-spin field with scaling dimensions We have investigated the corrections to finite size scaling of mostly spin zero levels in the OSp(5|2) chain of even length up to some energy cutoff which including the first states extrapolating to X = 3/8 + 2. Among these we find no evidence for the existence of towers of dimensions exept those starting at the descendents of the field with X = 3/8. This resembles the presence of both a continuous and a discrete part in the conformal spectrum of the sl(2|1) superspin chain with alternating quark and antiquark representations and its deformation [5,25,34,35].

V. DISCUSSION
In this paper we have studied the fine structure appearing in the finite size spectrum of the OSp(n|2m) superspin chains. We find that the ground states of these models have a finite degeneracy for n − 2m < 2. For large finite system size L there exists a tower of scaling dimensions extrapolating to that of the identity operator, X = 0, and forming a continuum in the thermodynamic limit, L → ∞. These observations are reminiscent of the appearence of a continuous component in the spectrum of scaling dimensions in staggered (super) spin chains [5,6,25,26]. There are some differences to our present findings though: the fine structure in the finite size spectrum of the staggered models has been argued to be a consequence of a non-compact degree of freedom in the low energy theory and the subleading gaps vanish quadratically with the inverse of log L. This has to be contrasted to the linear dependence predicted from conformal field theory for the WZNW models with a marginal perturbation and observed in the towers of excitations of the OSp(n|2m) models. Similarly, the corrections to scaling of the ground state of the staggered models due to a marginal perturbation by a continuum of excitations differ from (4.23) [36].
From the analysis of our data we have formulated conjectures for the amplitudes of the subleading (logarithmic) corrections to the lowest tower of excitations. For the OSp(n|2m) models considered above these amplitudes are found to be quadratic functions of the single quantum number labelling the different levels in the lowest tower. This suggests that they should be connected where n = 2 (1) for n even (odd). The first term in this expression, 2ν 1 (ν 1 + n − 2m − 2), is present in the Casimir for each of the algebras with given n − 2m for n > 1. We note that, the amplitudes of the subleading logarithmic corrections measured relative to the first level with positive scaling dimension (i.e. after subtracting the smallest non-negative amplitude β (n|2m) ) appearing in the superspin chains with n − 2m < 2 for even L the amplitudes can be directly related to the corresponding Casimir eigenvalue: Being corrections to the scaling dimensions of primary field these amplitudes are expected to determine logarithmic corrections to correlation functions. Again, the scaling dimensions should be measured starting from the smallest non-negative one for the given model. For the OSp(n|2m) models with n − 2m < 2 this predicts two-point functions of these fields to be For the correlation functions to decay at large distances k has to be restricted to k > 2 − n + 2m.
Eq. (5.3) agrees with the exponents for k-leg watermelon correlators proposed for the Goldstone phase of intersecting loop models with fugacity n − 2m < 2 and studied numerically using Monte Carlo simulations for n − 2m = 1 [8,9].