Exact correlations in topological quantum chains

Abstract

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.

References in corpus (8)

• Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells
• Classification of topological insulators and superconductors in three spatial dimensions
• Three dimensional topological invariants for time reversal invariant Hamiltonians and the three dimensional quantum spin Hall effect
• String order and symmetries in quantum spin lattices
• Matrix Product States: Symmetries and Two-Body Hamiltonians
• Entanglement entropy in quantum spin chains with finite range interaction
• Distinct trivial phases protected by a point-group symmetry in quantum spin chains
• Disorder lines, modulation, and partition function zeros in free fermion models

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