Line of Critical Points in2+1Dimensions: Quantum Critical Loop Gases and Non-Abelian Gauge Theory

Abstract

We (1) construct a one-parameter family of lattice models of interacting spins; (2) obtain their exact ground states; (3) derive a statistical-mechanical analogy which relates their ground states to O(n) loop gases; (4) show that the models are critical for d ≤ √ 2, where d parametrizes the models; (5) note that for the special values d = 2 cos(π/(k + 2)), they are related to doubled level-k SU(2) Chern-Simons theory; (6) conjecture that they are in the universality class of a non-relativistic SU(2) gauge theory; and (7) show that its one-loop β-function vanishes for all values of the coupling constant, implying that it is also on a critical line.Family of Microscopic Lattice Models. Consider the following models of s = 1/2 spins on the links of a honeycomb lattice, inspired by a model of Kitaev [1]:where N (v) is the set of 3 links neighboring vertex v, andif 1, 2, . . . , 6 label the six edges of plaquette p. This Hamiltonian is a sum of projection operators with positive coefficients and, therefore, is positive-definite. The eigenvalues of the first term in (1) are 2, 0, corresponding to whether there is an even or odd number of σ z = −1 spins neighboring this vertex. The zero eigenvalue corresponds to the latter case. When eigenvalue zero is obtained at every vertex, the σ z = 1 links form connected loops. Hence, the zero-energy subspace of the first term is spanned by all configurations of multi-loops. Loops on the honeycomb lattice cannot cross. The term on the second line is a projection operator which annihilates a state |Ψ if the amplitude for all of the spins on a given plaquette to be up is a factor of d times the amplitude for them to all be down, i.e. if the amplitude for a configuration with a small loop encircling a single plaquette is a factor of d times the amplitude for an otherwise identical configuration without the small loop. The other three lines of the Hamiltonian vanish on a state |Ψ if it accords the same value to a configuration if a loop is deformed to enclose an additional plaquette. It is useful to think of these multi-loops as continuous curves, in which case, the ground state is invariant under smooth deformation of a curve and it loses a factor of d if a small, contractible curve is erased. Together, these conditions have been dubbed 'd-isotopy' [2].Exact Ground State Wavefunctions. The terms on the final four lines of (1) do not commute with each other, but they are compatible in the sense that they all annihilate the ground state, which is a superposition of all configurations α of multiloops weighted by a factor of d to the number of loops n α in each configuration:On an L × L torus, the ground state degeneracy is ∼ L 2 because the Hamiltonian does not mix states |α with different winding numbers. The different ground states are given by (3) but with the sum over α restricted to a single topological class. More generally, the ground state on any genus g ≥ 1 surface is infinitely degenerate in the thermodynamic limit. At d = 1, there is an additional condition called the JonesW...

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

“…a weight of d per loop in the SU (2) case, and a weight of d 2 − 1 = Q − 1 per isolated loop in the SO(3) case). Such a lattice model for the SU (2) case was introduced by Freedman, Nayak and Shtengel 30 . We repeat some of these arguments here, and then use the S matrix picture to define an analogous model for the SO(3) case.…”

“…An explicit expression for the H i in the SU (2) case on the honeycomb lattice can be found in Ref. 30. Since we will not need the explicit Hamiltonian, we will not give it here -it's rather ugly, but it does the job.…”

“…(4.1) applies to the O(n) model in its dilute phase, where the energy per unit length is larger than the entropy, so that the loops cover a small part of the lattice. In order to get a purely topological field theory, the weight per unit length of loop must be 1, so that no length scale is set for the loops 30 . Such an O(n) model for n < 2 is in its dense phase, where entropy wins and the loops cover a set of measure 1 of the lattice.…”

“…In both cases we discuss in detail, the Hilbert space is that of a loop gas: in the SU (2) k case, the loops are selfavoiding and mutually-avoiding 30 , while in the SO(3) k case, the loops intersect. The latter are thus more akin to nets than loops 34 .…”

“…A natural description of the configuration space of models in a topological phase is in terms of loops 5,6,7,30,31 . This holds in both Abelian and nonAbelian cases.…”