© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

arXiv:1906.07765 · Journal of Physics Communications

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

Desmond A. Johnston

Published Oct 1, 2019• 0 citations

Abstract

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.

DOI: 10.1088/2399-6528/ab4ecb

Related papers

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

Spin Chains with Dynamical Lattice Supersymmetry

Christian Hagendorf

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

•Jan 1, 2013•35 citations

Lattice supersymmetry in the open XXZ model: an algebraic Bethe Ansatz analysis

Robert Weston, Junye Yang

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.

arXiv:1709.00442•Journal of Statistical Mechanics Theory and Experiment•Dec 1, 2017•10 citations

Non-local space-time supersymmetry on the lattice

Xiao Yang, Paul Fendley

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

•Nov 15, 2012•28 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

Lattice fermion models with supersymmetry

Paul Fendley, Bernard Nienhuis, Kareljan Schoutens

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.

arXiv:cond-mat/0307338•Journal of Physics A Mathematical and General•Dec 1, 2003•114 citations

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.

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

\\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

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 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.

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.