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

TL;DR

This paper characterizes the network of exactly solvable matrix product state models within the 1D BDI class, revealing their structure, connectivity, and implications for topological phases and quantum computation.

Contribution

It provides a complete characterization of the MPS skeleton in the BDI class, linking Hamiltonian parameters to MPS solvability and exploring the phase connectivity.

Findings

MPS-solvability corresponds to a polynomial being a perfect square.

Any two topological phases in this class can be connected via MPS-solvable models.

MPS-solvable models are dense and can approximate the Kitaev chain.

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.