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arXiv:1612.02951 · Journal of Physics A Mathematical and Theoretical

Open spin chains with dynamic lattice supersymmetry

Christian Hagendorf, Jean Liénardy

Published Apr 5, 2017• 26 citations
arXivHTMLPDF

Abstract

Abstract The quantum spin 1/2 XXZ chain with anisotropy parameter Δ = − 1 / 2 possesses a dynamic supersymmetry on the lattice. This supersymmetry and a generalisation to higher spin are investigated in the case of open spin chains. A family of non-diagonal boundary interactions that are compatible with the lattice supersymmetry and depend on several parameters is constructed. The cohomology of the corresponding supercharges is explicitly computed as a function of the parameters and the length of the chain. This cohomology is shown to be non-trivial for certain specific values of the parameters that lead to diagonal boundary interactions. For spin 1/2, they were previously studied by Fendley and Yang. The non-trivial cohomology implies that the spin-chain ground states are supersymmetry singlets. Special scalar products involving an arbitrary number of these supersymmetry singlets for chains of different lengths are exactly computed. In the case of spin 1/2, the scalar products are used to determine the logarithmic bipartite fidelity. Its scaling limit is shown to match the predictions of conformal field theory.

DOI
10.1088/1751-8121/aa67ff

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Abstract The quantum spin 1/2 XXZ chain with anisotropy parameter Δ = − 1 / 2 possesses a dynamic supersymmetry on the lattice. This supersymmetry and a generalisation to higher spin are investigated in the case of open spin chains. A family of non-diagonal boundary interactions that are compatible with the lattice supersymmetry and depend on several parameters is constructed. The cohomology of the corresponding supercharges is explicitly computed as a function of the parameters and the length of the chain. This cohomology is shown to be non-trivial for certain specific values of the parameters that lead to diagonal boundary interactions. For spin 1/2, they were previously studied by Fendley and Yang. The non-trivial cohomology implies that the spin-chain ground states are supersymmetry singlets. Special scalar products involving an arbitrary number of these supersymmetry singlets for chains of different lengths are exactly computed. In the case of spin 1/2, the scalar products are used to determine the logarithmic bipartite fidelity. Its scaling limit is shown to match the predictions of conformal field theory.
Spin chains with exact supersymmetry on finite one-dimensional lattices are considered. The supercharges are nilpotent operators on the lattice of dynamical nature: they change the number of sites. A local criterion for the nilpotency on periodic lattices is formulated. Any of its solutions leads to a supersymmetric spin chain. It is shown that a class of special solutions at arbitrary spin gives the lattice equivalents of the mathcalN=(2,2)\\mathcal{N}=(2,2)mathcalN=(2,2) superconformal minimal models. The case of spin one is investigated in detail: in particular, it is shown that the Fateev-Zamolodchikov chain and its off-critical extension possess a lattice supersymmetry for all its coupling constants. Its supersymmetry singlets are thoroughly analysed, and a relation between their components and the weighted enumeration of alternating sign matrices is conjectured
We show that several well-known one-dimensional quantum systems possess a hidden non-local supersymmetry. The simplest example is the open XXZ spin chain with Δ = −1/2. We use the supersymmetry to place lower bounds on the ground-state energy with various boundary conditions. For an odd number of sites in the periodic chain, and with a particular boundary magnetic field in the open chain, we can derive the ground-state energy exactly. The supersymmetry thus explains why it is possible to solve the Bethe equations for the ground state in these cases. We also show that a similar spacetime supersymmetry holds for the t–J model at its integrable ferromagnetic point, where the spacetime supersymmetry and the Hamiltonian it yields coexist with a global u (1|2) graded Lie algebra symmetry. Possible generalizations to other algebras are discussed.
Abstract We reconsider the open XXZ chain of Yang and Fendley. This model possesses a lattice supersymmetry which changes the length of the chain by one site. We perform an algebraic Bethe ansatz analysis of the model and derive the commutation relations of the lattice SUSY operators with the four elements of the open-chain monodromy matrix. Hence we give the action of the SUSY operator on off-shell and on-shell Bethe states. We show that this action generally takes one on-shell Bethe eigenstate to another. The exception is that a zero-energy vacuum state will be a SUSY singlet. The SUSY pairings of Bethe roots we obtain are analogous to those found previously for closed chains by Fendley and Hagendorf by analysing the Bethe equations.
We show that several well-known one-dimensional quantum systems possess a hidden nonlocal supersymmetry. The simplest example is the open XXZ spin chain with ∆ = −1 2. We use the supersymmetry to place lower bounds on the ground state energy with various boundary conditions. For an odd number of sites in the periodic chain, and with a particular boundary magnetic field in the open chain, we can derive the ground-state energy exactly. The supersymmetry thus explains why it is possible to solve the Bethe equations for the ground state in these cases. We also show that a similar space-time supersymmetry holds for the t-J model at its integrable ferromagnetic point, where the space-time supersymmetry and the Hamiltonian it yields coexist with a global u(1|2) graded Lie algebra symmetry. Possible generalizations to other algebras are discussed. 1
We study a model of strongly correlated fermions in one dimension with extended N = 2 supersymmetry. The model is related to the spin S = 1/2 XXZ Heisenberg chain at anisotropy Delta = -1/2 with a real magnetic field on the boundary. We exploit the combinatorial properties of the ground state to determine its exact wave function on finite lattices with up to 30 sites. We compute several correlation functions of the fermionic and spin fields. We discuss the continuum limit by constructing lattice observables with well defined finite-size scaling behavior. For the fermionic model with periodic boundary conditions we give the emptiness formation probability in closed form.
Abstract The open XXZ spin chain with the anisotropy parameter Δ = − 1 2 and diagonal boundary magnetic fields that depend on a parameter x is studied. For real x > 0, the exact finite-size ground-state eigenvalue of the spin-chain Hamiltonian is explicitly computed. In a suitable normalisation, the ground-state components are characterised as polynomials in x with integer coefficients. Linear sum rules and special components of this eigenvector are explicitly computed in terms of determinant formulas. These results follow from the construction of a contour-integral solution to the boundary quantum Knizhnik–Zamolodchikov equations associated with the R -matrix and diagonal K -matrices of the six-vertex model. A relation between this solution and a weighted enumeration of totally-symmetric alternating sign matrices is conjectured.