Realizing non-abelian statistics in quantum loop models Paul Fendley Experimental and theoretical successes have made us take a close look at quantum physics in two spatial dimensions. We have now found (mostly theoretically, but also experimentally) behavior unseen in three dimensions. 1
One potential application of this novel physics is a topological quantum computer. This idea relies on one particularly unusual property possible only in two dimensions: non-abelian statistics. There are several possibilities for realizing non-abelian statistics experimentally: a two-dimensional electron gas in a large magnetic field (the fractional quantum Hall effect); rapidly rotating Bose-Einstein condensates frustrated magnets. 2
Outline: 1. What are non-abelian statistics? 2. What does this have to do with loops? 3. How do we construct models realizing this? my contribution to this story is in collaboration with E. Fradkin 3
The statistics of identical particles comes from the behavior of the two-particle wavefunction ψ(x 1, x 2 ) under the interchange x 1 x 2. In 2+1 dimensions, this can be thought of as braiding. Each particle traces out a worldline in time, and these braid around each other. Braiding is a purely topological property. figure from Skjeltorp et al 4
When braided, particles can change quantum numbers! ψ a (x 1, x 2... ) = B ab ψ b (x 2, x 1... ) When the matrix B is diagonal with entries ±1, particles are bosons and fermions. When B is diagonal with entries e iα, particles are anyons. When B is not diagonal, can get non-abelian statistics. With non-abelian statistics, the probability amplitude changes depending on the order in which the particles are braided. 5
Braiding particles with non-abelian statistics entangles them! This is the fundamental idea behind topological quantum computation. excellent review in Preskill s lecture notes 6
We describe braiding by projecting the world line of the particles onto the plane. Then the braids become overcrossings B = or undercrossings B 1 = In 2+1 dimensions, it matters which direction you braid. 7
The problem: find a quantum Hamiltonian acting on a two-dimensional Hilbert space so that the quasiparticles have non-abelian statistics. The trick: work backwards. First find a ground-state wavefunction which has the right sort of structure. Then engineer a Hamiltonian which has this as its ground state. In other words, attempt to be Laughlin without experimental input! 8
Where to look: The planar projection of the braids suggests that we look at models where the degrees of freedom in the two-dimensional Hilbert space are loops. The model needs to be in a liquid phase, where the ground state is not ordered, but is a sum over many loop configurations. 9
The excitations with non-abelian statistics will be defects in the sea of loops. Cutting a loop raises the energy and creates two quasiparticles. If the defects are deconfined, then there will be no energy cost to move them far from each other. However, they must still be attached by a string. A model like this is said to have topological order, and quasiparticles with fractionalized charge. 10
Constructing Quantum Loop Models Spin models can often be converted to loop models. Consider the quantum eight-vertex model. The degrees of freedom are an Ising spin on the links of a square lattice. The potential is chosen to require that in the ground state, there must be an even number of spins up touching each vertex. 11
Each spin configuration can be converted to a loop configuration simply by relabeling a spin with a link of loop. The constraint that there be an even number of up spins at each vertex means that the loops are closed (except maybe at the boundary). 12
The simplest non-trivial Hamiltonian preserving the 8-vertex condition flips the arrows around one plaquette. The resulting 2+1d quantum model is a liquid with fractionalized excitations. All local order parameters are zero, but it does possess topological order. One way of seeing this is to study the model on different genus surfaces: on genus g, there are 2 g ground states; the flip preserves the number of up arrows mod 2 around a given cycle. Kitaev 13
By varying the Hamiltonian in a particular fashion, one finds a quantum critical line separating the topological phase from a conventionally-ordered phase d 2 Z order 2 confining critical line 2 KT Z gauge 2 defects deconfined spin liquid critical line Kitaev Z order 2 confining 6 vertex KT 2 c 2 Ardonne, Fendley and Fradkin 14
Now, to the non-abelian case. We again study models whose degrees of freedom are loops. Let s again work backwards, and engineer models whose braiding is non-abelian. The first thing to do is understand what possible braidings are mathematically consistent. Luckily, mathematicians have been studying this problem for decades, in the guise of knot theory. 15
One famous example comes from the Temperley-Lieb algebra of the Potts model in statistical mechanics, or equivalently the Jones polynomial of knot theory. Every time we see a braid, we disentangle it into the linear combination = q where q is some root of 1. In other words, the braid matrix takes two incoming particles to a linear combination of the state where the two don t meet, and the state where they annihilate and then are created again. 16
To make this precise, define the matrices e i acting on the two particle state of the ith and the (i + 1)th particles, so that I e We then have B i = I qe i B 1 i = I q 1 e i In mathematical language, we ve turned a problem involving topology into an algebraic one. 17
Braids defined this way are consistent if the e i satisfy the Temperley-Lieb algebra e 2 i = de i e i e i±1 e i = e i = d = where d = q + q 1. Note that closed loops are weighted by d. To ensure finite-dimensional representations of this algebra, need for k integer. d = 2 cos[π/(k + 2)] i.e. q = e iπ/(k+2) 18
Doing this algebra tells us what kind of a quantum loop model will have the braiding properties. We want a ground-state wavefunction consisting of a sum over loops where each loop is weighted by the number d = 2 cos(π/(k + 2)). Freedman; Freedman, Nayak and Shtengel This is related to (doubled) SU(2) k Chern-Simons theory, which is a topological field theory. Freedman, Nayak, Shtengel, Walker and Wang 19
However, there s a catch: we re doing quantum mechanics. We must weight configurations by ψ 2 when computing ground-state correlators! Thus effectively, each loop is weighted by d 2, not d. This squaring sounds innocuous, but 2d stat mech aficianados will recall that self- and mutually-avoiding loops with weight n = d 2 are not critical for n > 2. To get an interesting gas of loops thus requires that d 2 The only non-abelian possibility is k = 4, where d = 2. But this won t make a universal quantum computer (the braid group is not dense in the unitary group)! 20
There is an idea on how to construct such more general models, so that we can escape this problem Fendley and Fradkin Think again of the 2+1d world lines projected down to 1+1d. In 1+1d, particles can t go around each other. 1+1d braiding is related to the scattering matrix S! It s well-known from knot theory that if S(θ) obeys the Yang-Baxter equation, then a representation of the braid group is found by Akutsu, Deguchi and Wadati B = lim θ S(θ), B 1 = lim θ S( θ) where θ is the rapidity difference of the two particles. 21
start: scattering matrix in 1+1 dim classical field theory in 2+0 dim finish: classical loop gas in 2+0 dim a 2+1 dim quantum loop gas with non abelian statistics described by this S matrix! quantum loop gas in 2+1 dim 22
Applying this procedure, we found a model which not only avoids this complication, but has the simplest possible possible non-abelian braiding. It possesses only one type of quasiparticle. When one fuses two of these quasiparticles together, we get the linear combination φ φ 1 + φ These are dubbed Fibonacci anyons, because the dimension of the Hilbert space of N such quasiparticles is the N th Fibonacci number, which grows as ( 1 + ) N 5 2 The braiding of these particles is related to SO(3) 3 Chern-Simons theory. 23
The difference between SO(3) 3 and SU(2) 3 is that in SO(3) we project out all half-integer spins. In the loop language, we fuse together the spin-1/2 particles from the SU(2) k to get spin-1 particles in SO(3) k by the projector I e/d: = 1 d In other words, we glue two worldlines together, and then project out the identity. 24
Using this idea, we showed that the loops for SO(3) k are related to the low-temperature expansion of the Q-state Potts model with ( ) π Q = d 2 = 4 cos 2 k + 2 The fact that φ φ = 1 + φ leads to trivalent vertices in the loop gas. 25
Using the correspondence with the Potts model, we showed that the weight of a loop configuration is given by the chromatic polynomial χ Q of the (dual) graph (a topological quantity!) When Q is an integer, the chromatic polynomial χ Q is the number of Q-colorings. 1 2 3 3 3 2 3 3 2 2 For Q not an integer, χ Q is defined recursively. 26
But remember that we re in a quantum theory: we must square the weights! The danger is that this squaring will wreck the loop gas: the Potts model has no critical point for Q > 4. Squaring a chromatic polynomial is in general not so easy, but some heuristic arguments led us to conjecure that the k = 3 Fibonacci case will escape doom. Luckily... 27
When Q = (3 + 5)/2 = τ + 1 (the Fibonacci case), the chromatic polynomial obeys Tutte s golden identity: ( χτ+1 ) 2 = 1 5 τ 3 N tri/2 χ τ+2 where N tri is the number of trivalent vertices. Because τ + 2 = (5 + 5)/2 is less than 4, The 1969 proof by Tutte of our 2005 conjecture proves that our Fibonacci model has a quantum critical point! 28
The upshot: We have some beautiful wavefunctions. But only ugly Hamiltonians. There s much more work to be done! these transparencies at http://rockpile.phys.virginia.edu 29