Lieb-Schultz-Mattis theorem for quasi-topological systems

Abstract

In this paper we address the question of the existence of a spectral gap in a class of local Hamiltonians. These Hamiltonians have the following properties: their ground states are known exactly; all equal-time correlation functions of local operators are short-ranged; and correlation functions of certain non-local operators are critical. A variational argument shows gaplessness with $\omega \propto k^2$ at critical points defined by the absence of certain terms in the Hamiltonian, which is remarkable because equal-time correlation functions of local operators remain short-ranged. We call such critical points, in which spatial and temporal scaling are radically different, quasi-topological. When these terms are present in the Hamiltonian, the models are in gapped topological phases which are of special interest in the context of topological quantum computation.

v2: The version published in Phys. Rev B. A new section has been added; a gap in the earlier version of the proof has been eliminated

References in corpus (19)

• Fault-tolerant quantum computation by anyons
• An RVB phase in the triangular lattice quantum dimer model
• Lieb-Schultz-Mattis in Higher Dimensions
• Z_2 Gauge Theory of Electron Fractionalization in Strongly Correlated Systems
• Quantum Glassiness
• Topological Order and Conformal Quantum Critical Points
• Quantum dimer model on the kagome lattice: solvable dimer liquid and Ising gauge theory
• A Class of $P,T$-Invariant Topological Phases of Interacting Electrons
• Relaxation time for a dimer covering with height representation
• From quantum mechanics to classical statistical physics: generalized Rokhsar-Kivelson Hamiltonians and the "Stochastic Matrix Form" decomposition
• From classical to quantum dynamics at Rokhsar-Kivelson points
• Sufficient Conditions for Topological Order in Insulators
• An extended Hubbard model with ring exchange: a route to a non-Abelian topological phase
• A Line of Critical Points in 2+1 Dimensions: Quantum Critical Loop Gases and Non-Abelian Gauge Theory
• Dual Vortex Theory of Strongly Interacting Electrons: Non-Fermi Liquid to the (Hard) Core
• Quantum three-coloring dimer model and the disruptive effect of quantum glassiness on its line of critical points
• Slow dynamics and aging in a non-randomly frustrated spin system
• Entropy-vanishing transition and glassy dynamics in frustrated spins
• Topological jamming of extended structures and the glass transition

Cited by in corpus (10)

• Topological order from quantum loops and nets
• Topological Entanglement in Abelian and non-Abelian Excitation Eigenstates
• Linear independence of nearest neighbor valence bond states on the kagome lattice and construction of SU(2)-invariant spin-1/2-Hamiltonian with a Sutherland-Rokhsar-Kivelson quantum liquid ground state
• Dzyaloshinskii-Moriya interactions in valence bond systems
• Detecting Topological Order with Ribbon Operators
• Josephson Coupled Moore-Read States
• Fibonacci topological order from quantum nets
• Quantum criticality of loops with topologically constrained dynamics
• Self-dual criticality in three-dimensional ${\mathbb{Z}}_2$ gauge theory with matter
• Topological Phases: An Expedition off Lattice