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Register a new userThe open supersymmetric Haldane-Shastry spin chain and its associated motifs
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Added by
Artemio Gonzalez-Lopez
Publication date
2020
fields
Physics
and research's language is
English
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We study the open version of the su$(m|n)$ supersymmetric Haldane-Shastry spin chain associated to the $BC_N$ extended root system. We first evaluate the models partition function by modding out the dynamical degrees of freedom of the su$(m|n)$ supersymmetric spin Sutherland model of $BC_N$ type, whose spectrum we fully determine. We then construct a generalized partition function depending polynomially on two sets of variables, which yields the standard one when evaluated at a suitable point. We show that this generalized partition function can be written in terms of two variants of the classical skew super Schur polynomials, which admit a combinatorial definition in terms of a new type of skew Young tableaux and border strips (or, equivalently, extended motifs). In this way we derive a remarkable description of the spectrum in terms of this new class of extended motifs, reminiscent of the analogous one for the closed Haldane-Shastry chain. We provide several concretes examples of this description, and in particular study in detail the su$(1|1)$ model finding an analytic expression for its Helmholtz free energy in the thermodynamic limit.
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The Haldane-Shastry model is one of the most studied interacting spin systems. The Yangian symmetry makes it exactly solvable, and the model has semionic excitations. We introduce disorder into the Haldane-Shastry model by allowing the spins to sit at random positions on the unit circle and study the properties of the eigenstates. At weak disorder, the spectrum is similar to the spectrum of the clean Haldane-Shastry model. At strong disorder, the long-range interactions in the model do not decay as a simple power law. The eigenstates in the middle of the spectrum follow a volume law, but the coefficient is small, and the entropy is hence much less than for an ergodic system. In addition, the energy level spacing statistics is neither Poissonian nor of the Wigner-Dyson type. The behavior at strong disorder hence serves as an example of a non-ergodic phase, which is not of the many-body localized kind, in a model with long-range interactions and SU(2) symmetry.
Gapped periodic quantum systems exhibit an interesting Localization Dichotomy, which emerges when one looks at the localization of the optimally localized Wannier functions associated to the Bloch bands below the gap. As recently proved, either these Wannier functions are exponentially localized, as it happens whenever the Hamiltonian operator is time-reversal symmetric, or they are delocalized in the sense that the expectation value of $|mathbf{x}|^2$ diverges. Intermediate regimes are forbidden. Following the lesson of our Maestro, to whom this contribution is gratefully dedicated, we find useful to explain this subtle mathematical phenomenon in the simplest possible model, namely the discrete model proposed by Haldane (Phys. Rev. Lett. 61, 2017 (1988)). We include a pedagogical introduction to the model and we explain its Localization Dichotomy by explicit analytical arguments. We then introduce the reader to the more general, model-independent version of the dichotomy proved in (Commun. Math. Phys. 359, 61-100 (2018)), and finally we announce further generalizations to non-periodic models.
Based on the inhomogeneous T-Q relation and the associated Bethe Ansatz equations obtained via the off-diagonal Bethe Ansatz, we construct the Bethe-type eigenstates of the SU(2)-invariant spin-s chain with generic non-diagonal boundaries by employing certain orthogonal basis of the Hilbert space.
An orthogonal basis of the Hilbert space for the quantum spin chain associated with the su(3) algebra is introduced. Such kind of basis could be treated as a nested generalization of separation of variables (SoV) basis for high-rank quantum integrable models. It is found that all the monodromy-matrix elements acting on a basis vector take simple forms. With the help of the basis, we construct eigenstates of the su(3) inhomogeneous spin torus (the trigonometric su(3) spin chain with antiperiodic boundary condition) from its spectrum obtained via the off-diagonal Bethe Ansatz (ODBA). Based on small sites (i.e. N=2) check, it is conjectured that the homogeneous limit of the eigenstates exists, which gives rise to the corresponding eigenstates of the homogenous model.
By combining the algebraic Bethe ansatz and the off-diagonal Bethe ansatz, we investigate the trigonometric SU(3) model with generic open boundaries. The eigenvalues of the transfer matrix are given in terms of an inhomogeneous T-Q relation, and the corresponding eigenstates are expressed in terms of nested Bethe-type eigenstates which have well-defined homogeneous limit. This exact solution provides a basis for further analyzing the thermodynamic properties and correlation functions of the anisotropic models associated with higher rank algebras.
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