Skeleton of Matrix-Product-State-Solvable Models Connecting Topological Phases of Matter

Abstract

Models whose ground states can be written as an exact matrix product state (MPS) provide valuable insights into phases of matter. While MPS-solvable models are typically studied as isolated points in a phase diagram, they can belong to a connected network of MPS-solvable models, which we call the MPS skeleton. As a case study where we can completely unearth this skeleton, we focus on the one-dimensional BDI class -- non-interacting spinless fermions with time-reversal symmetry. This class, labelled by a topological winding number, contains the Kitaev chain and is Jordan-Wigner-dual to various symmetry-breaking and symmetry-protected topological (SPT) spin chains. We show that one can read off from the Hamiltonian whether its ground state is an MPS: defining a polynomial whose coefficients are the Hamiltonian parameters, MPS-solvability corresponds to this polynomial being a perfect square. We provide an explicit construction of the ground state MPS, its bond dimension growing exponentially with the range of the Hamiltonian. This complete characterization of the MPS skeleton in parameter space has three significant consequences: (i) any two topologically distinct phases in this class admit a path of MPS-solvable models between them, including the phase transition which obeys an area law for its entanglement entropy; (ii) we illustrate that the subset of MPS-solvable models is dense in this class by constructing a sequence of MPS-solvable models which converge to the Kitaev chain (equivalently, the quantum Ising chain in a transverse field); (iii) a subset of these MPS states can be particularly efficiently processed on a noisy intermediate-scale quantum computer.

21 pages + appendix

References in corpus (12)

• The density-matrix renormalization group in the age of matrix product states
• One-Dimensional Symmetry Protected Topological Phases and their Transitions
• Rise and fall of hidden string order of lattice bosons
• Valence bond solids for SU(n) spin chains: exact models, spinon confinement, and the Haldane gap
• Fermionic Matrix Product States and One-Dimensional Topological Phases
• Class of exactly solvable SO(n) symmetric spin chains with matrix product ground states
• Z_3 symmetry-protected topological phases in the SU(3) AKLT model
• Valence bond solid states with symplectic symmetry
• Finite and Infinite Matrix Product States for Gutzwiller Projected Mean-Field Wavefunctions
• Density matrix renormalization group boosted by Gutzwiller projected wave functions
• Quantum criticality in the SO(5) bilinear-biquadratic Heisenberg chain
• Novel families of $\mathrm{SU}(N)$ AKLT states with arbitrary self-conjugate edge states

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