Shor–Movassagh chain leads to unusual integrable model

Tong, Bin; Salberger, Olof; Hao, Kun; Korepin, V. E.

doi:10.1088/1751-8121/ac1f3f

Cited by 6 publications

(7 citation statements)

References 30 publications

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“…This work was then generalized by Movassagh and Shor to any integer spin s ≥ 1 in [MS16] and called s−colored Motzkin quantum spin chain, where each color represents a spin value s. In particular, it was proved that for any number of colors s > 1 the model exhibits tremendous amount of entanglement in its ground state which was not previously believed possible for ground states of physical quantum spin chains. Later, Korepin et al found that the s−colored Motzkin chain is an unusual integrable model [TSHK21]. The frustration-freeness along with the combinatorial nature of the ground state structure allows one to map these quantum models to classical Markov processes in classical probability theory for analyzing the entanglement entropy of the ground state and the gap above the ground state.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

The spin-one Motzkin chain is gapped for any area weight $t<1$

Andrei¹,

Lemm²,

Movassagh³

2022

Preprint

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We prove a conjecture by Zhang, Ahmadain, and Klich that the spin-1 Motzkin chain is gapped for any area weight t < 1. Existence of a finite spectral gap is necessary for the Motzkin Hamiltonian to belong to the Haldane phase, which has been argued to potentially be the case in recent work of Barbiero, Dell'Anna, Trombettoni, and Korepin. Our proof rests on the combinatorial structure of the ground space and the analytical verification of a finite-size criterion. Contents 1 Introduction and main results 2 The model and the proof of the main result 3 Characterization of open-chain ground states 4 The finite-size criterion 5 Analytical verification of the finite-size criterion: overview 6 Low-imbalance approximations 7 High-imbalance approximations 8 Implementing the approximations 9 Normalizations and the x r factors 10 Completing the low-imbalance proof A Analysis of the ratios of normalization factors B Auxiliary Results

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Section: Introduction and Main Resultsmentioning

confidence: 99%

The spin-one Motzkin chain is gapped for any area weight $t<1$

Andrei¹,

Lemm²,

Movassagh³

2022

Preprint

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“…We studied the free Motzkin chain with periodic boundary conditions. Based on intrinsic properties of the R-matrix, we derive the functional relations (27) of the transfer matrix. These relations, together with the asymptotic behavior (30), allow us to construct a generalized T − Q relation (31) for the eigenvalues of the transfer matrix and the associated Bethe equations (33).…”

Section: Discussionmentioning

confidence: 99%

“…Let ã and d represent the coefficients before a(λ) and d(λ) terms, respectively. Based on the identities (27) and the asymptotic behavior ( 24) and ( 30), these coefficients…”

Section: Structure Of Operator ô and Its Eigenvaluesmentioning

confidence: 99%

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Can a spin chain relate combinatorics to number theory?

Hao¹,

Salberger²,

Korepin³

2022

Preprint

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The Motzkin spin chain is a spin-1 frustration-free model introduced by Shor & Movassagh. The ground state is constructed by mapping of random walks on upper half of the square lattice to spin configurations. It has unusually large entanglement entropy [quantum fluctuations]. We simplify the model by removing one of the local equivalence moves of the Motzkin paths. The system becomes integrable [similar to the XXX spin chain]. We call it free Motzkin chain. From the point of view of quantum integrability, the model is special since its R-matrix does not have crossing-unitarity. We solve the periodic free Motzkin chain by generalized functional Bethe Ansatz method. We construct a T − Q relation with an additional parameter to formulate the energy spectrum. This new parameter is related to the roots of unity and can be described by the Möbius function of the number theory. We observe further patterns of the number theory.

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“…Both the RE scheme and the Lax pair formulation can demonstrate the integrability of the model. There is one special case where only the Lax pair formulation works for the model without crossing unitarity [15], while the RE scheme does not. When solving the RE, one has to take into account the spectral parameter and the crossing parameter.…”

Section: Discussionmentioning

confidence: 99%

Lax pair formulation for the open boundary Osp(1∣2) spin chain

Zhang¹,

Hao²

2022

Commun. Theor. Phys.

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Based on the Lax pair formulation, we study the integrable conditions of the $Osp(1|2)$ spin chain with open boundaries. We consider both the non-graded and graded versions of the $Osp(1|2)$ chain. The Lax pair operators $M_\pm$ for the boundaries can be induced by the Lax operator $M_j$ for the bulk of the system. They correspond to the reflection equations and the Yang-Baxter equation, respectively. We further calculate the boundary $K$-matrices for both the non-graded and graded versions of the model with open boundaries. The double row monodromy matrix and transfer matrix of the spin chain have also been constructed. The $K$-matrices obtained from the Lax pair formulation are consistent with those from Sklyanin's reflection equations. This construction is another way to prove the quantum integrability of the $Osp(1|2)$ chain. We find that the Lax pair formulation has advantages in dealing with the boundary terms of the supersymmetric model.

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Shor–Movassagh chain leads to unusual integrable model

Cited by 6 publications

References 30 publications

The spin-one Motzkin chain is gapped for any area weight $t<1$

The spin-one Motzkin chain is gapped for any area weight $t<1$

Can a spin chain relate combinatorics to number theory?

Lax pair formulation for the open boundary Osp(1∣2) spin chain

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