Automorphic Equivalence within Gapped Phases of Quantum Lattice Systems

Abstract: Gapped ground states of quantum spin systems have been referred to in the physics literature as being `in the same phase' if there exists a family of Hamiltonians H(s), with finite range interactions depending continuously on $s \in [0,1]$, such that for each $s$, H(s) has a non-vanishing gap above its ground state and with the two initial states being the ground states of H(0) and H(1), respectively. In this work, we give precise conditions under which any two gapped ground states of a given quantum spin system that 'belong to the same phase' are automorphically equivalent and show that this equivalence can be implemented as a flow generated by an $s$-dependent interaction which decays faster than any power law (in fact, almost exponentially). The flow is constructed using Hastings' 'quasi-adiabatic evolution' technique, of which we give a proof extended to infinite-dimensional Hilbert spaces. In addition, we derive a general result about the locality properties of the effect of perturbations of the dynamics for quantum systems with a quasi-local structure and prove that the flow, which we call the {\em spectral flow}, connecting the gapped ground states in the same phase, satisfies a Lieb-Robinson bound. As a result, we obtain that, in the thermodynamic limit, the spectral flow converges to a co-cycle of automorphisms of the algebra of quasi-local observables of the infinite spin system. This proves that the ground state phase structure is preserved along the curve of models $H(s), 0\leq s\leq 1$.

Automorphic Equivalence Within Gapped Phases of Quantum Lattice Systems

The paper "Automorphic Equivalence Within Gapped Phases of Quantum Lattice Systems" by Sven Bachmann, Spyridon Michalakis, Bruno Nachtergaele, and Robert Sims provides a rigorous analysis of the automorphic equivalence of ground states in the context of gapped quantum spin systems. The focus is on understanding how distinct ground states, considered to be in the same quantum phase, can be transformed into one another via automorphisms when connected by a family of Hamiltonians with a persistent spectral gap.

Summary of Results

The authors propose a precise definition of when two gapped ground states are considered to be in the same phase. They define a phase connection through a continuous family of gapped Hamiltonians $H(s)$, with $s$ ranging from 0 to 1, such that the initial and terminal states of this family are the ground states of $H(0)$ and $H(1)$ respectively. The principal result is a demonstration that these ground states are automorphically equivalent, employing Hastings' quasi-adiabatic evolution technique. This equivalence is extended to infinite-dimensional Hilbert spaces.

The paper proves that in the thermodynamic limit, the spectral flow converges to a cocycle of automorphisms on the algebra of quasi-local observables in an infinite spin system. This result ensures that the phase structure is preserved along the path $H(s)$.

Technical Contributions

1. Quasi-Adiabatic Continuation: The authors extend the quasi-adiabatic evolution technique to infinite-dimensional Hilbert spaces, providing a detailed account that leverages Lieb-Robinson bounds to control locality properties of dynamics in quantum systems.
2. Spectral Projection and Automorphism Flow: By showing that the spectral projection associated with isolated spectral parts of a Hamiltonian can be represented as unitary evolution, the authors facilitate the automorphic match between quantum phases.
3. Thermodynamic Limit Existence: The study establishes the thermodynamic limit for the spectral flow, demonstrating that the finite-volume equivalence extends naturally to the infinite volume case, thereby reinforcing the applicability of their results in practical systems.

Implications and Future Directions

The theoretical implications of this work are significant for the understanding of quantum phase transitions, particularly in relation to the concept of topological order and stability against perturbations. This framework provides powerful insights into the classification of quantum phases based on their automorphic properties.

From a practical standpoint, this research may impact quantum information science, particularly in the context of designing robust quantum computation architectures that leverage the stability properties of gapped phases. By establishing automorphic equivalence via quasi-local interactions, the paper suggests pathways for exploring computational phases resistant to local perturbations.

Future research could potentially explore the necessary conditions under which different simpices of ground states belong to distinct phases, as the current work outlines sufficient conditions. Moreover, applications to models beyond discrete spin systems, incorporating continuum models or models with more complex symmetries, could further extrapolate the reach of these findings.

In conclusion, this comprehensive investigation into the automorphic equivalence within gapped quantum phases offers a robust mathematical framework that aligns with physical intuition about phase continuity and stability in quantum systems. It sets a foundational ground for both theoretical explorations and practical implementations in quantum technologies.