SciPost Phys. 10, 100 (2021) ·
published 4 May 2021
We study the nodes of the wavefunction overlap between ground states of a
parameter-dependent Hamiltonian. These nodes are topological, and we can use
them to analyze in a unifying way both equilibrium and dynamical quantum
phase transitions in multi-band systems. We define the Loschmidt index as the
number of nodes in this overlap and discuss the relationship between this index
and the wrapping number of a closed auxiliary hypersurface. This relationship
allows us to compute this index systematically, using an integral representation of
the wrapping number. We comment on the relationship between the Loschmidt
index and other well-established topological numbers. As an example, we classify
the equilibrium and dynamical quantum phase transitions of the XY model by
counting the nodes in the wavefunction overlaps.
SciPost Phys. 10, 020 (2021) ·
published 28 January 2021
We study the Killing vectors of the quantum ground-state manifold of a parameter-dependent Hamiltonian. We find that the manifold may have symmetries that are not visible at the level of the Hamiltonian and that different quantum phases of matter exhibit different symmetries. We propose a Bianchi-based classification of the various ground-state manifolds using the Lie algebra of the Killing vector fields. Moreover, we explain how to exploit these symmetries to find geodesics and explore their behaviour when crossing critical lines. We briefly discuss the relation between geodesics, energy fluctuations and adiabatic preparation protocols. Our primary example is the anisotropic transverse-field Ising model. We also analyze the Ising limit and find analytic solutions to the geodesic equations for both cases.
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