Topological order and conformal quantum critical points

Abstract

We discuss a certain class of two-dimensional quantum systems which exhibit conventional order and topological order, as well as two-dimensional quantum critical points separating these phases. All of the ground-state equal-time correlators of these theories are equal to correlation functions of a local two-dimensional classical model. The critical points therefore exhibit a time-independent form of conformal invariance. These theories characterize the universality classes of two-dimensional quantum dimer models and of quantum generalizations of the eight-vertex model, as well as Z 2 and non-abelian gauge theories. The conformal quantum critical points are relatives of the Lifshitz points of three-dimensional anisotropic classical systems such as smectic liquid crystals. In particular, the ground-state wave functional of these quantum Lifshitz points is just the statistical (Gibbs) weight of the ordinary 2D free boson, the 2D Gaussian model. The full phase diagram for the quantum eight-vertex model exhibits quantum critical lines with continuously-varying critical exponents separating phases with long-range order from a Z 2 deconfined topologically-ordered liquid phase. We show how similar ideas also apply to a well-known field theory with non-Abelian symmetry, the strong-coupling limit of 2 + 1-dimensional Yang-Mills gauge theory with a Chern-Simons term. The ground state of this theory is relevant for recent theories of topological quantum computation.

Lifshitz scale anomalies

“…The d = 2, z = 2 case is of particular importance, as certain condensed matter systems have been shown to exhibit a Lifshitz scaling symmetry with this value of the dynamical critical exponent [34]. The parity even sector for this case has been extensively studied in the literature (see e.g.…”

Realizing non-Abelian statistics in time-reversal-invariant systems

“…The resulting effective Chern-Simons theory is doubled to restore the time-reversal symmetry. To give the theory a gap while keeping the topological theory as its ground state, one can include the electric-field part of the Maxwell term d 2 x E · E in the Hamiltonian 27,28,29 . Hence in these topological phases, the ground-state wave function is a superposition of configurations of Wilson loops in two-dimensional space, while the world-lines of the excitations correspond to Polyakov loops in the 2+1-dimensional field theory 7 .…”

“…In a 2+1-dimensional picture, this means it is a good idea to look for a system where the lowenergy degrees of freedom are loops in the plane, a quantum loop gas. In a number of cases it has been argued that quantum loop gases turn into gauge theories with Chern-Simons theories in the continuum 6,7,29,34 . The excitations can be non-Abelian in a topological phase, where the ground state contains a superposition of Wilson loops (loops in the spatial plane).…”

“…Precisely, each basis state in the Hilbert space of the quantum theory is a loop configuration in two dimensions. Quantum dimer models and their generalizations 29,33 can also be viewed as quantum loop gases.…”

Quantum Field Theory of Many-Body Systems

“…Much of this progress has occurred in three areas of research: (1) the study of topological phases in condensed matter systems such as FQH systems [Wen and Niu (1990); Blok and Wen (1990); Read (1990); Fröhlich and Kerler (1991)], quantum dimer models [Rokhsar and Kivelson (1988); Read and Chakraborty (1989); Moessner and Sondhi (2001); Ardonne et al (2004)], quantum spin models [Kalmeyer and Laughlin (1987) ;Wen et al (1989); Wen (1990); Read and Sachdev (1991); Wen (1991a); Senthil and Fisher (2000); Wen (2002b); Sachdev and Park (2002) ;Balents et al (2002)], or even superconducting states [Wen (1991b); Hansson et al (2004)], (2) the study of lattice gauge theory [Wegner (1971); Banks et al (1977); Kogut and Susskind (1975); Kogut (1979)], and (3) the study of quantum computing by anyons [Kitaev (2003); Ioffe et al (2002); Freedman et al (2002)]. The phenomenon of string condensation is important in all of these fields, though the string picture is often de-emphasized.…”