Supersymmetric t-J models with long-range interactions: partition function and spectrum

Abstract

We study the spectrum of the long-range supersymmetric su(m) t-J model of Kuramoto and Yokoyama in the presence of an external magnetic field and a charge chemical potential. To this end, we first establish the precise equivalence of a large class of models of this type to a family of su(1|m) spin chains with long-range exchange interactions and a suitable chemical potential term. We exploit this equivalence to compute in closed form the partition function of the long-range t-J model, which we then relate to that of an inhomogeneous vertex model with simple interactions. From the structure of this partition function we are able to deduce an exact formula for the restricted partition function of the long-range t-J model in subspaces with well-defined magnon content in terms of its analogue for the equivalent vertex model. This yields a complete analytical description of the spectrum in the latter subspaces, including the precise degeneracy of each level, by means of the supersymmetric version of Haldane's motifs and their related skew Young tableaux. As an application, we determine the structure of the motifs associated with the ground state of the spin 1/2 model in the thermodynamic limit in terms of the magnetic field strength and the charge chemical potential. This leads to a complete characterization of the distinct ground state phases, determined by their spin content, in terms of the magnetic field strength and the charge chemical potential.Keywords: integrable spin chains and vertex models, solvable lattice models arXiv:1811.10297v2 [cond-mat.str-el]

Appearance of branched motifs in the spectra of BC type Polychronakos spin chains

“…The above mentioned equivalence between the eigenvalues of Yangian invariant spin chains with long-range interactions and energy functions of onedimensional vertex models with only local interactions can be extended even in the presence of chemical potentials [19,20]. As a result, by using transfer matrices associated with those vertex models, one can calculate various thermodynamic quantities of this type of spin chains even in the presence of chemical potentials and investigate the critical properties of those systems [19,20,41,42]. Thus the expressions of the generalized partition functions in terms of Schur polynomials lead to a powerful method for classifying the degenerate multiplets of the corresponding spectra and for studying various thermodynamic properties of the related spin chains.…”

“…(4.6) as well as (4.7), we obtain an extended boson-fermion duality relation between two deformed super Schur polynomials as N´r`1 y conjugate to xk 1 , k 2 , ..., k r y . Expanding both sides of the above equation by using (4.8) and redefining the variable N´l as l, we obtainComparing the powers of q in both sides of the above equation, we find that restricted super Schur polynomials satisfy an extended boson-fermion duality relation of the form Next, we consider the reversed border strip x k rev y " xk r , k r´1 , ..., k 1 y corresponding to the border strip x ky " xk 1 , k 2 , ..., k r y, and the reverse conjugate border strip x k N´r , ..., k 1 y corresponding to x k Following the procedure outlined in the Appendix of Ref [41],. it can be easily shown that any q-deformed super Schur Polynomial, defined through a determinant relation of the form (4.4), would remain invariant under the reversal of the corresponding border strip: S pn 2 ,n 1 |m 2 ,m 1 q x k 1 y px,x; y,ȳ; qq " S Expanding both sides of the above equation by using (4.8), we find that the restricted super Schur polynomials also remain invariant under the above mentioned reversal, i.e.,…”

Supersymmetric t-J models with long-range interactions: thermodynamics and criticality

“…As shown in our previous paper [10], the Hamiltonian of the supersymmetric su(m) KY model can be written as ‡…”

“…As shown in Ref. [10], the KY Hamiltonian (2.1) can be mapped to (a suitable modification of) the su(1|m) Haldane-Shastry spin chain Hamiltonian by identifying the holes of the KY model with the bosons of the HS spin chain. Indeed, the Hilbert space of the latter chain isĤ = ⊗ N i=1Ĥ i , whereĤ i is the linear span of the one-particle…”

“…The starting point in our method is the explicit formula for the partition function of the KY model with an arbitrary (finite) number of sites N obtained in Ref. [10], which can be recast into that of a related inhomogeneous vertex model. This key observation makes it feasible to apply the transfer matrix method in Refs.…”