From String Nets to Nonabelions

@article{Fidkowski2006FromSN,
  title={From String Nets to Nonabelions},
  author={Lukasz M. Fidkowski and Michael H. Freedman and C. Nayak and Kevin Walker and Zhenghan Wang},
  journal={Communications in Mathematical Physics},
  year={2006},
  volume={287},
  pages={805-827}
}
We discuss Hilbert spaces spanned by the set of string nets, i.e. trivalent graphs, on a lattice. We suggest some routes by which such a Hilbert space could be the low-energy subspace of a model of quantum spins on a lattice with short-ranged interactions. We then explain conditions which a Hamiltonian acting on this string net Hilbert space must satisfy in order for the system to be in the DFib (Doubled Fibonacci) topological phase, that is, be described at low energy by an SO(3)3 × SO(3)3… Expand
Topological phases: An expedition off lattice
Abstract Motivated by the goal to give the simplest possible microscopic foundation for a broad class of topological phases, we study quantum mechanical lattice models where the topology of theExpand
Topological order from quantum loops and nets
Abstract I define models of quantum loops and nets that have ground states with topological order. These make possible excited states comprised of deconfined anyons with non-abelian braiding. WithExpand
Characterization of topological phase transitions from a non-Abelian topological state and its Galois conjugate through condensation and confinement order parameters
  • Wen-Tao Xu, N. Schuch
  • Physics
  • Physical Review B
  • 2021
Topological phases exhibit unconventional order that cannot be detected by any local order parameter. In the framework of Projected Entangled Pair States (PEPS), topological order is characterized byExpand
A pr 2 00 8 Topological order from quantum loops and nets
I define models of quantum loops and nets which have ground states with topological order. These make possible excited states comprised of deconfined anyons with non-abelian braiding. With theExpand
Galois Conjugates of Topological Phases
Galois conjugation relates unitary conformal field theories and topological quantum field theories (TQFTs) to their nonunitary counterparts. Here we investigate Galois conjugates of quantum doubleExpand
A Nonperturbative Proposal for Nonabelian Tensor Gauge Theory and Dynamical Quantum Yang-Baxter Maps
We propose a nonperturbative approach to nonabelian two-form gauge theory. We formulate the theory on a lattice in terms of plaquette as fundamental dynamical variable, and assign U(N) Chan-PatonExpand
Hamiltonian models for topological phases of matter in three spatial dimensions
We present commuting projector Hamiltonian realizations of a large class of (3+1)D topological models based on mathematical objects called unitary G-crossed braided fusion categories. ThisExpand
Tensor Network Approach to Phase Transitions of a Non-Abelian Topological Phase.
TLDR
A generic quantum-net wave function with two tuning parameters dual with each other is proposed, and the norm of the wave function can be exactly mapped into a partition function of the two-coupled ϕ^{2}-state Potts models, where ϕ=(sqrt[5]+1)/2 is the golden ratio. Expand
Tutte chromatic identities from the Temperley-Lieb algebra
This paper introduces a conceptual framework, in the context of quantum topology and the algebras underlying it, for analyzing relations obeyed by the chromatic polynomial . Q/ of planar graphs.Expand
Non-Abelian Anyons and Topological Quantum Computation
Topological quantum computation has emerged as one of the most exciting approaches to constructing a fault-tolerant quantum computer. The proposal relies on the existence of topological states ofExpand
...
1
2
3
...

References

SHOWING 1-10 OF 38 REFERENCES
Realizing non-Abelian statistics in time-reversal-invariant systems
We construct a series of $(2+1)$-dimensional models whose quasiparticles obey non-Abelian statistics. The adiabatic transport of quasiparticles is described by using a correspondence between theExpand
String-net condensation: A physical mechanism for topological phases
We show that quantum systems of extended objects naturally give rise to a large class of exotic phases---namely topological phases. These phases occur when extended objects, called ``string-nets,''Expand
Extended hubbard model with ring exchange: a route to a non-Abelian topological phase.
TLDR
An extended Hubbard model on a 2D kagome lattice with an additional ring exchange term is proposed and it is shown how to arrive at an exactly soluble point whose ground state is the "d-isotopy" transition point into a stable phase with a certain type of non-Abelian topological order. Expand
Fractionalization in an easy-axis Kagome antiferromagnet
We study an antiferromagnetic spin-$1/2$ model with up to third nearest-neighbor couplings on the Kagome lattice in the easy-axis limit, and show that its low-energy dynamics are governed by aExpand
Line of critical points in 2+1 dimensions: quantum critical loop gases and non-Abelian gauge theory.
TLDR
A one-parameter family of lattice models of interacting spins is constructed, obtained their exact ground states, and its one-loop beta function vanishes for all values of the coupling constant, implying that it is also on a critical line. Expand
Nonabelions in the fractional quantum Hall effect
Applications of conformal field theory to the theory of fractional quantum Hall systems are discussed. In particular, Laughlin's wave function and its cousins are interpreted as conformal blocks inExpand
Microscopic models of two-dimensional magnets with fractionalized excitations
We demonstrate that spin-charge separation can occur in two dimensions and note its confluence with superconductivity, topology, gauge theory, and fault-tolerant quantum computation. We construct aExpand
Common structures between finite systems and conformal field theories through quantum groups
Abstract We discuss in this paper algebraic structures that are common to finite integrable lattice systems and conformal field theories. The concept of quantum group plays a major role in our study,Expand
Temperley-Lieb Recoupling Theory and Invariants of 3-Manifolds
I used to keep the entire spin networks literature in a small folder on my shelf. The recent explosion of interest in the subject has made this impossible; more is probably now written every weekExpand
The antiferromagnetic Potts model in two dimensions: Berker-Kadanoff phase, antiferromagnetic transition, and the role of Beraha numbers
Abstract Using methods of integrable systems and conformal field theory, we study the Q -state Potts model on the square lattice with e K real. We discover a surprisingly rich phase diagram thatExpand
...
1
2
3
4
...