Exact correlations in topological quantum chains
arXiv:2105.13359 · doi:10.3842/SIGMA.2023.098
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 (45)
- Topological Insulators
- Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells
- Topological Order and the Quantum Spin Hall Effect
- Unpaired Majorana fermions in quantum wires
- Topological Insulators in Three Dimensions
- Classification of topological insulators and superconductors in three spatial dimensions
- Entanglement in quantum critical phenomena
- Novel Symmetry Classes in Mesoscopic Normal-Superconducting Hybrid Structures
- Topological invariants of time-reversal-invariant band structures
- Topological insulators and superconductors: ten-fold way and dimensional hierarchy
- Periodic table for topological insulators and superconductors
- Entanglement spectrum of a topological phase in one dimension
- Calculation of reduced density matrices from correlation functions
- Three dimensional topological invariants for time reversal invariant Hamiltonians and the three dimensional quantum spin Hall effect
- Classification of Gapped Symmetric Phases in 1D Spin Systems
- Topological phases of fermions in one dimension
- Matrix Product States and Projected Entangled Pair States: Concepts, Symmetries, and Theorems
- Classifying quantum phases using Matrix Product States and PEPS
- Symmetry Protected Topological phases of Quantum Matter
- Quantum Spin Chain, Toeplitz Determinants and Fisher-Hartwig Conjecture
- Detection of Symmetry Protected Topological Phases in 1D
- Density-Matrix Spectra of Solvable Fermionic Systems
- Entanglement in XY Spin Chain
- String order and symmetries in quantum spin lattices
- Fermionic Projected Entangled Pair States
- Griffiths effects and quantum critical points in dirty superconductors without spin-rotation invariance: One-dimensional examples
- Quantum phase transitions in matrix product systems
- Topology and edge modes in quantum critical chains
- On the entanglement entropy for a XY spin chain
- Random Matrix Theory and Entanglement in Quantum Spin Chains
- Crossing a topological phase transition with a quantum computer
- Correlation Function of the Spin-1/2 XXX Antiferromagnet
- Matrix Product States: Symmetries and Two-Body Hamiltonians
- Emptiness Formation Probability for the One-Dimensional Isotropic XY Model
- Asymptotics of Toeplitz Determinants and the Emptiness Formation Probability for the XY Spin Chain
- Quantum states on Harmonic lattices
- Ellipses of Constant Entropy in the XY Spin Chain
- Entanglement entropy in quantum spin chains with finite range interaction
- Distinct trivial phases protected by a point-group symmetry in quantum spin chains
- Entanglement Spectrum for the XY Model in One Dimension
- Detecting Majorana fermions in quasi-one-dimensional topological phases using nonlocal order parameters
- Truncating an exact Matrix Product State for the XY model: transfer matrix and its renormalisation
- Disorder lines, modulation, and partition function zeros in free fermion models
- Modified Szego-Widom Asymptotics for Block Toeplitz Matrices with Zero Modes
- Gaussian Matrix Product States