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Spin Chains with Dynamical Lattice Supersymmetry

Christian Hagendorf

Published Jan 1, 2013• 35 citations
OpenAlex

Abstract

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 $\\mathcal{N}=(2,2)$ 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

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Open spin chains with dynamic lattice supersymmetry

Christian Hagendorf, Jean Liénardy

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.

arXiv:1612.02951•Journal of Physics A Mathematical and Theoretical•Apr 5, 2017•26 citations

Spinon excitations in the spin-1 XXZ chain and hidden supersymmetry

Chihiro Matsui

We study spinon excitations of the integrable spin-1 (Fateev–Zamolodchikov; FZ) chain and their relation to the hidden supersymmetry. Using the notion of the supercharges earlier introduced to the spin chains, which change the system length by one, we found that they nontrivially act on one of two kinds of the degrees of freedom for the FZ chain. Their actions were obtained to be the same as those of the supercharges defined on the supersymmetric sine-Gordon model, the low-energy effective field theory of the FZ chain. Moreover, we construct the eigenstates which are invariant under the supersymmetric Hamiltonian given as the anti-commutator of the supercharges.

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We investigate a family of lattice models with manifest supersymmetry. The models describe fermions on a 1D lattice, subject to the constraint that no more than k consecutive lattice sites may be occupied. We discuss the special properties arising from the supersymmetry, and present Bethe ansatz solutions of the simplest models. We display the connections of the k = 1 model with the spin- antiferromagnetic XXZ chain at Δ = −1/2, and the k = 2 model with both the su (2|1)-symmetric tJ model in the ferromagnetic regime and the integrable spin-1 XXZ chain at . We argue that these models include critical points described by the superconformal minimal models.

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Lattice SUSY for the DiSSEP at λ2 = 1 (and λ2 = −3)

Desmond A. Johnston

Abstract We investigate whether the dynamical lattice supersymmetry discussed for various Hamiltonians, including one-dimensional quantum spin chains, by (Fendley et al 2003 J. Phys. A 36, 12399424; Yang and Fendley 2004 J. Phys. A 37, 893748; Hagendorf and Fendley 2012 J. Stat. Phys. 146, 112255) and (Hagendorf et al 2013 J. Stat. Phys. 150, 60957; Hagendorf and Liénardy 2017 J. Phys. A 50, 185202; Hagendorf and Liénardy 2018 J. Stat. Mech. 033106) might also exist for the Markov matrices of any one-dimensional exclusion processes, which have the additional constraint of zero column sums by comparison with the spin chains. We find that the DiSSEP (Dissipative Symmetric Simple Exclusion Process), introduced by Crampe et al in (Crampe et al 2016 Phys. Rev. E 94, 032102; Vanicat [arXiv: 1708.02440 ]), provides one such example for suitably chosen parameters. The DiSSEP Markov matrix admits the supersymmetry in these cases because it is conjugate to Ising (λ 2 = 1) and Δ = −1/2 XXZ (λ 2 = −3) spin chain Hamiltonians which also possess the supersymmetry. The consequences for the spectrum of the DiSSEP Markov matrix are discussed. We also note that the length-changing supersymmetry relation for the DiSSEP Markov matrix M L and the supercharge Q L† for L sites, <?CDATA ${M}^{L}{Q}^{L\dagger }={Q}^{L\dagger }{M}^{L-1}$?> M L Q L † = Q L † M L − 1 , is reminiscent of a ‘transfer matrix’ symmetry that has been observed in other exclusion processes and discuss the similarity.

arXiv:1906.07765•Journal of Physics Communications•Oct 1, 2019•0 citations

The Eight-Vertex Model and Lattice Supersymmetry

Christian Hagendorf, Paul Fendley

We show that the XYZ spin chain along the special line of couplings J x J y +J x J z +J y J z =0 possesses a hidden $\\mathcal{N}=(2,2)$ supersymmetry. This lattice supersymmetry is non-local and changes the number of sites. It extends to the full transfer matrix of the corresponding eight-vertex model. In particular, it is shown how to derive the supercharges from Baxter's Bethe ansatz. This analysis leads to new conjectures concerning the ground state for chains of odd length. We also discuss a correspondence between the spectrum of this XYZ chain and that of a manifestly supersymmetric staggered fermion chain

•Jan 1, 2012•17 citations

Non-local spacetime supersymmetry on the lattice

Xiao Yang, Paul Fendley

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.

arXiv:cond-mat/0404682•Journal of Physics A Mathematical and General•Sep 9, 2004•43 citations
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
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.
We study spinon excitations of the integrable spin-1 (Fateev–Zamolodchikov; FZ) chain and their relation to the hidden supersymmetry. Using the notion of the supercharges earlier introduced to the spin chains, which change the system length by one, we found that they nontrivially act on one of two kinds of the degrees of freedom for the FZ chain. Their actions were obtained to be the same as those of the supercharges defined on the supersymmetric sine-Gordon model, the low-energy effective field theory of the FZ chain. Moreover, we construct the eigenstates which are invariant under the supersymmetric Hamiltonian given as the anti-commutator of the supercharges.
We investigate a family of lattice models with manifest supersymmetry. The models describe fermions on a 1D lattice, subject to the constraint that no more than k consecutive lattice sites may be occupied. We discuss the special properties arising from the supersymmetry, and present Bethe ansatz solutions of the simplest models. We display the connections of the k = 1 model with the spin- antiferromagnetic XXZ chain at Δ = −1/2, and the k = 2 model with both the su (2|1)-symmetric tJ model in the ferromagnetic regime and the integrable spin-1 XXZ chain at . We argue that these models include critical points described by the superconformal minimal models.
Abstract We investigate whether the dynamical lattice supersymmetry discussed for various Hamiltonians, including one-dimensional quantum spin chains, by (Fendley et al 2003 J. Phys. A 36, 12399424; Yang and Fendley 2004 J. Phys. A 37, 893748; Hagendorf and Fendley 2012 J. Stat. Phys. 146, 112255) and (Hagendorf et al 2013 J. Stat. Phys. 150, 60957; Hagendorf and Liénardy 2017 J. Phys. A 50, 185202; Hagendorf and Liénardy 2018 J. Stat. Mech. 033106) might also exist for the Markov matrices of any one-dimensional exclusion processes, which have the additional constraint of zero column sums by comparison with the spin chains. We find that the DiSSEP (Dissipative Symmetric Simple Exclusion Process), introduced by Crampe et al in (Crampe et al 2016 Phys. Rev. E 94, 032102; Vanicat [arXiv: 1708.02440 ]), provides one such example for suitably chosen parameters. The DiSSEP Markov matrix admits the supersymmetry in these cases because it is conjugate to Ising (λ 2 = 1) and Δ = −1/2 XXZ (λ 2 = −3) spin chain Hamiltonians which also possess the supersymmetry. The consequences for the spectrum of the DiSSEP Markov matrix are discussed. We also note that the length-changing supersymmetry relation for the DiSSEP Markov matrix M L and the supercharge Q L† for L sites, <?CDATA MLQL†=QL†ML−1{M}^{L}{Q}^{L\dagger }={Q}^{L\dagger }{M}^{L-1}MLQL†=QL†ML−1?> M L Q L † = Q L † M L − 1 , is reminiscent of a ‘transfer matrix’ symmetry that has been observed in other exclusion processes and discuss the similarity.
We show that the XYZ spin chain along the special line of couplings J x J y +J x J z +J y J z =0 possesses a hidden mathcalN=(2,2)\\mathcal{N}=(2,2)mathcalN=(2,2) supersymmetry. This lattice supersymmetry is non-local and changes the number of sites. It extends to the full transfer matrix of the corresponding eight-vertex model. In particular, it is shown how to derive the supercharges from Baxter's Bethe ansatz. This analysis leads to new conjectures concerning the ground state for chains of odd length. We also discuss a correspondence between the spectrum of this XYZ chain and that of a manifestly supersymmetric staggered fermion chain
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.