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

Weston, Robert; Yang, Junye

doi:10.1088/1742-5468/aa9f42

Cited by 5 publications

(5 citation statements)

References 22 publications

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“…We now use the lemmas 3.5 and 3.6 to compute a commutation relation between the transfer matrix and the supercharge. This generalises a relation established by Weston and Yang [30] for the six-vertex model, corresponding to d = 0, y = 0.…”

Section: Local Relations We Define Two Operatorssupporting

confidence: 88%

On the transfer matrix of the supersymmetric eight-vertex model. II. Open boundary conditions

Hagendorf

Liénardy

2020

J. Stat. Mech.

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The transfer matrix of the square-lattice eight-vertex model on a strip with L 1 vertical lines and open boundary conditions is investigated. It is shown that for vertex weights a, b, c, d that obey the relation (a 2 + ab)(b 2 + ab) = (c 2 + ab)(d 2 + ab) and appropriately chosen K-matrices K ± this transfer matrix possesses the remarkably simple, non-degenerate eigenvalue ΛL = (a + b) 2L tr(K + K − ). For positive vertex weights, ΛL is shown to be the largest transfer-matrix eigenvalue. The corresponding eigenspace is equal to the space of the ground states of the Hamiltonian of a related XYZ spin chain. An essential ingredient in the proofs is the supersymmetry of this Hamiltonian.

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Section: Local Relations We Define Two Operatorssupporting

confidence: 88%

On the transfer matrix of the supersymmetric eight-vertex model. II. Open boundary conditions

Hagendorf

Liénardy

2020

J. Stat. Mech.

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“…It would also be interesting to investigate supersymmetric properties of the open chain (see for example [56,57]), and to understand how to formulate an ensemble of states in the presence of boundary roots. Namely how to determine for example the steady state after a quantum quench [58] (Generalised Gibbs Ensemble) close to the edge of the chain.…”

Section: Resultsmentioning

confidence: 99%

Open XXZ chain and boundary modes at zero temperature

2019

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We study the open XXZ spin chain in the anti-ferromagnetic regime and for generic longitudinal magnetic fields at the two boundaries. We discuss the ground state via the Bethe ansatz and we show that, for a chain of even length L and in a regime where both boundary magnetic fields are equal and bounded by a critical field, the spectrum is gapped and the ground state is doubly degenerate up to exponentially small corrections in L. We connect this degeneracy to the presence of a boundary root, namely an excitation localized at one of the two boundaries. We compute the local magnetization at the left edge of the chain and we show that, due to the existence of a boundary root, this depends also on the value of the field at the opposite edge, even in the half-infinite chain limit. Moreover we give an exact expression for the large time limit of the spin autocorrelation at the boundary, which we explicitly compute in terms of the form factor between the two quasi-degenerate ground states. This, as we show, turns out to be equal to the contribution of the boundary root to the local magnetization. We finally discuss the case of chains of odd length.

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“…gives the following associativity condition on the local supercharge q for open chains [5,9] q(q ⊗ I) = q(I ⊗ q) (4)…”

Section: Lattice Susymentioning

confidence: 99%

“…1),−8(1),−4(6). Both M 2 and M 3 possess a zero eigenvalue singlet and the eigenvalues −4 for L=2 have partners in the L=3 spectrum.…”

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confidence: 99%

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

Johnston

2019

J. Phys. Commun.

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We investigate whether the dynamical lattice supersymmetry discussed for various Hamiltonians, including one-dimensional quantum spin chains, by (Fendley et al 2003 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,, is reminiscent of a 'transfer matrix' symmetry that has been observed in other exclusion processes and discuss the similarity.

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

Cited by 5 publications

References 22 publications

On the transfer matrix of the supersymmetric eight-vertex model. II. Open boundary conditions

On the transfer matrix of the supersymmetric eight-vertex model. II. Open boundary conditions

Open XXZ chain and boundary modes at zero temperature

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

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

Cited by 5 publications

References 22 publications

On the transfer matrix of the supersymmetric eight-vertex model. II. Open boundary conditions

On the transfer matrix of the supersymmetric eight-vertex model. II. Open boundary conditions

Open XXZ chain and boundary modes at zero temperature

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

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Product

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