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arXiv:1507.08966 · Physical review. B./Physical review. B

Formalism for the solution of quadratic Hamiltonians with large cosine terms

Sriram Ganeshan, Michael Levin

Published Feb 8, 2016• 19 citations
arXivHTMLPDF

Abstract

We consider quantum Hamiltonians of the form $H={H}_{0}\ensuremath{-}U{\ensuremath{\sum}}_{j}cos({C}_{j})$, where ${H}_{0}$ is a quadratic function of position and momentum variables ${{x}_{1},{p}_{1},{x}_{2},{p}_{2},\ensuremath{\cdots}}$ and the ${C}_{j}$'s are linear in these variables. We allow ${H}_{0}$ and ${C}_{j}$ to be completely general with only two restrictions: we require that (1) the ${C}_{j}$'s are linearly independent and (2) $[{C}_{j},{C}_{k}]$ is an integer multiple of $2\ensuremath{\pi}i$ for all $j,k$ so that the different cosine terms commute with one another. Our main result is a recipe for solving these Hamiltonians and obtaining their exact low-energy spectrum in the limit $U\ensuremath{\rightarrow}\ensuremath{\infty}$. This recipe involves constructing creation and annihilation operators and is similar in spirit to the procedure for diagonalizing quadratic Hamiltonians. In addition to our exact solution in the infinite $U$ limit, we also discuss how to analyze these systems when $U$ is large but finite. Our results are relevant to a number of different physical systems, but one of the most natural applications is to understanding the effects of electron scattering on quantum Hall edge modes. To demonstrate this application, we use our formalism to solve a toy model for a fractional quantum spin Hall edge with different types of impurities.

DOI
10.1103/physrevb.93.075118

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In this thesis we investigate topological phases of matter that have a global, unbroken symmetry group -- also known as symmetry enriched topological (SET) phases. We address three questions about these phases: (1) how can we build exactly solvable models that realize them? (2) how can we determine if their edge theories can be gapped without breaking the symmetry? and (3) how do we understand the phenomenon of decoupled charge and neutral modes which occurs in certain fractional quantum Hall states? ,More specifically, we address the first question by constructing exactly solvable models for a wide class of symmetry enriched topological (SET) phases, which we call symmetry-enriched string nets. The construction applies to 2D bosonic SET phases with finite unitary onsite symmetry group G, and we conjecture that our models realize every phase in this class that can be described by a commuting projector Hamiltonian. As an example, we present a model for a phase with the same anyon excitations as the toric code and with a Z2 symmetry which exchanges the 'e' and 'm' type anyons. We further illustrate our construction with a number of additional examples. ,For the second question, we focus on the edge theories of 2D SET phases with Z2 symmetry. The central problem we seek to solve is to determine which edge theories can be gapped without breaking the symmetry. Previous attempts to answer this question in special cases relied on constructing perturbations of a particular type to gap the edge. This method proves the edge can be gapped when the appropriate perturbations can be found, but is inconclusive if they cannot be found. We build on this previous work by deriving a necessary and sufficient algebraic condition for when the edge can be gapped. Our results apply to Z2 symmetry protected topological phases as well as Abelian Z2 SET phases,Finally, in the fourth chapter, we describe solvable models that capture how impurity scattering in certain fractional quantum Hall edges can give rise to a neutral mode --- i.e. an edge mode that does not carry electric charge. These models consist of two counter-propagating chiral Luttinger liquids together with a collection of discrete impurity scatterers. Our main result is an exact solution of these models in the limit of infinitely strong impurity scattering. From this solution, we explicitly derive the existence of a neutral mode and we determine all of its microscopic properties including its velocity. We also study the stability of the neutral mode and show that it survives at finite but sufficiently strong scattering. Our results are applicable to a family of Abelian fractional quantum Hall states of which the nu = 2/3 state is the most prominent example.