On Relations between One-Dimensional Quantum and Two-Dimensional Classical Spin Systems

Hutchinson, Joanna; Keating, Jon P; Mezzadri, Francesco

doi:10.1155/2015/652026

Cited by 5 publications

(5 citation statements)

References 26 publications

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“…III we compute the ground state correlators σ , and n i=1 σ z i g , which exhibit quasi-long-range order behavior when the quantum system is gapless, decreasing as a power law of the distance n, with an exponent dependent upon the symmetry class. In a companion paper to this [20], we then use the mapping between one-dimensional quantum spin chains and two-dimensional classical spin models to extend our results to classical systems.…”

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confidence: 99%

“…In particular, in Section 2 we compute the critical exponents s, ν and z, which are related to the energy gap, correlation length and dynamic exponent respectively, and in Section 3 we compute the ground state correlators σ x i σ x i+n g , σ y i σ y i+n g and n i=1 σ z i g , which exhibit quasilong-range order behaviour when the quantum system is gapless, decreasing as a power law of the distance n, with an exponent dependent upon the symmetry class. In a companion paper to this [14], we then use the mapping between 1-D quantum spin chains and 2-D classical spin models to extend our results to classical systems.…”

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confidence: 99%

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Random matrix theory and critical phenomena in quantum spin chains

2015

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We compute critical properties of a general class of quantum spin chains which are quadratic in the Fermi operators and can be solved exactly under certain symmetry constraints related to the classical compact groups U(N ), O(N ), and Sp(2N ). In particular we calculate critical exponents s, ν, and z, corresponding to the energy gap, correlation length, and dynamic exponent, respectively. We also compute the ground state correlators σ , and n i=1 σ z i g , all of which display quasi-long-range order with a critical exponent dependent upon system parameters. Our approach establishes universality of the exponents for the class of systems in question.

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confidence: 99%

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confidence: 99%

Random matrix theory and critical phenomena in quantum spin chains

2015

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“…• In Section 6, we saw that the spectrum of the transfer matrix has a free-fermion form, suggesting that there could exist a free-fermion Hamiltonian H eff (in general, non-hermitian) such that E I = e −H eff . This is a natural question, with connections to quantum-classical mappings and imaginary time evolution under our class of Hamiltonians [83,88]. Relatedly, in Section 6.5, we showed how our results can be used to diagonalise the transfer matrix in a simple case.…”

Section: Discussionmentioning

confidence: 76%

Exact correlations in topological quantum chains

Jones¹,

Verresen²

2021

Preprint

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Despite free-fermion systems being dubbed exactly solvable, they generically do not admit closed expressions for nonlocal quantities such as topological string correlations or measures of entanglement. We derive closed expressions for such nonlocal quantities for a dense subclass of certain classes of topological fermionic wires (classes BDI and AIII). Our results also apply to spin chains known as generalised cluster models. While there is a bijection between general models in these fermionic classes and Laurent polynomials, restricting to polynomials with degenerate zeros leads to a plethora of exact results. In particular, (1) we derive closed expressions for the string correlation functions-the order parameters for the topological phases in these classes; (2) we obtain an exact formula for the characteristic polynomial of the correlation matrix, giving insight into the entanglement of the ground state; (3) the latter implies that the ground state can be described by a matrix product state (MPS) with a finite bond dimension in the thermodynamic limit-an independent and explicit construction for the BDI class is given in a concurrent work (Jones, Bibo, Jobst, Pollmann, Smith, Verresen, arXiv:2105.12143); (4) for BDI models with even integer topological invariant, all non-zero eigenvalues of the transfer matrix are identified as products of zeros and inverse zeros of the aforementioned polynomial. We show how general models in these classes can be obtained by taking limits of the models studied in this work, which can be used to apply limits of our results to the general case. To the best of our knowledge, these results constitute the first application of Day's formula and Gorodetsky's formula from the theory of Toeplitz determinants to the context of many-body quantum physics.

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“…In Section 9, we saw that the spectrum of the transfer matrix has a free-fermion form, suggesting that there could exist a free-fermion Hamiltonian H eff (in general, non-hermitian) such that E I = e −H eff . This is a natural question, with connections to quantum-classical mappings and imaginary time evolution under our class of Hamiltonians [37,95]. Relatedly, in Section 9.5, we showed how our results can be used to diagonalise the transfer matrix in a simple case.…”

Section: Discussionmentioning

confidence: 77%

Exact Correlations in Topological Quantum Chains

Jones,

Verresen

2023

SIGMA

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Although free-fermion systems are considered exactly solvable, they generically do not admit closed expressions for nonlocal quantities such as topological string correlations or entanglement measures. We derive closed expressions for such quantities for a dense subclass of certain classes of topological fermionic wires (classes BDI and AIII). Our results also apply to spin chains called generalised cluster models. While there is a bijection between general models in these classes and Laurent polynomials, restricting to polynomials with degenerate zeros leads to a plethora of exact results: (1) we derive closed expressions for the string correlation functions - the order parameters for the topological phases in these classes; (2) we obtain an exact formula for the characteristic polynomial of the correlation matrix, giving insight into ground state entanglement; (3) the latter implies that the ground state can be described by a matrix product state (MPS) with a finite bond dimension in the thermodynamic limit - an independent and explicit construction for the BDI class is given in a concurrent work [Phys. Rev. Res. 3 (2021), 033265, 26 pages, arXiv:2105.12143]; (4) for BDI models with even integer topological invariant, all non-zero eigenvalues of the transfer matrix are identified as products of zeros and inverse zeros of the aforementioned polynomial. General models in these classes can be obtained by taking limits of the models we analyse, giving a further application of our results. To the best of our knowledge, these results constitute the first application of Day's formula and Gorodetsky's formula for Toeplitz determinants to many-body quantum physics.

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On Relations between One-Dimensional Quantum and Two-Dimensional Classical Spin Systems

Cited by 5 publications

References 26 publications

Random matrix theory and critical phenomena in quantum spin chains

Random matrix theory and critical phenomena in quantum spin chains

Exact correlations in topological quantum chains

Exact Correlations in Topological Quantum Chains

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