| 总结: | 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.
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