Post on 05-Jul-2018
Some Phase Transitions Between Topological Phases
Steven H. Simon w/ Fiona J. Burnell and Joost Slingerland (NUI Maynooth)
In Celebration of Mike Freedman’s Birthday
Topologically Nontrivial Birthday Cake
The Landau Paradigm:
Phases, and Phase Transitions Described by Local Order Parameters; Local Broken Symmetries
Ex: Boson Condensation
Normal Phase Condensed Phase: Broken U(1)
Forcing Condensate Order
Boson Number Uncertainty in Condensed Phases
Ex.
Transitions between Topological Phases?
Order is Nonlocal
Our Objective • Realize these transitions in an (almost) solvable lattice model. • Examine the critical theory. • Relevance to “experiments” ?
• Can something condense to make a transition between two topological phases?
Algebraic structure of topological condensation transitions:
…
First Example: Phase Transition in Perturbed Toric Code
Prior Work By…
Toric
Ground state = equal superposition of all loop configurations
Toric Code (Z2 gauge)
closed loops of
enforces enforces
flips all + ↔ - around a plaquette
enforces
Ground state = equal superposition of all loop configurations
Toric Toric Code (Z2 gauge)
enforces enforces
flips all + ↔ - around a plaquette
violation is electric defect e
closed loops of
enforces
Ground state = equal superposition of all loop configurations
Toric Code (Z2 gauge) Toric
violation is electric defect e
closed loops of
enforces
flips all + ↔ - around a plaquette violation is magnetic defect m
enforces
m is a Z2 boson : Can we condense it? (trivial self braiding and m x m = 1)
Toric
Want:
but no local operator creates single m. Toric
Toric Code (Z2 gauge) Toric
enforces
flips all + ↔ - around a plaquette violation is magnetic defect m
Instead Creates/annihilates/hops m bosons
Commutes with electric term. Never creates e defects Can work in reduced Hilbert Space
Toric
Toric
m is a Z2 boson : Can we condense it? (trivial self braiding and m x m = 1)
Want:
but no local operator creates single m.
Toric
Toric Code (Z2 gauge)
(i,j neighbors)
enforces
flips all + ↔ - around a plaquette violation is magnetic defect m
Toric
Exact mapping to Transverse Field Ising Model
Toric
Commutes with electric term. Never creates e defects. Can work in reduced Hilbert Space
Toric
Toric Code (Z2 gauge)
Creates/annihilates/hops m bosons (i,j neighbors)
enforces
flips all + ↔ - around a plaquette violation is magnetic defect m
Toric
Exact mapping to Transverse Field Ising Model
Toric
Toric
on edge creates m on two adjacent plaquettes Operate with
Toric Code (Z2 gauge)
Creates/annihilates/hops m bosons (i,j neighbors)
enforces
flips all + ↔ - around a plaquette violation is magnetic defect m
Toric
Exact mapping to Transverse Field Ising Model
Toric
on edge creates m on two adjacent plaquettes Operate with
String tension for loops Toric
Toric Code (Z2 gauge)
enforces
flips all + ↔ - around a plaquette violation is magnetic defect m
Toric
Exact mapping to Transverse Field Ising Model
on edge creates m on two adjacent plaquettes Operate with
String tension for loops Toric
Frustrated
Pure Toric Code Deconfined loops = Toric Code Phase
Confined loops = Topologically trivial
Paramagnetic Ferromagnetic
No Loops
spins in x direction = (1 + m) maximum uncertainty of # of bosons;
Transitions between two topologically nontrivial phases: Much the same physics applies! (with some twists)
1. Start with Levin-Wen Lattice Model
Provides description of Uncondensed Phase
2. Find a ZN “plaquette boson” (no vertex defect; trivial self braiding ; b x b .. x b = 1 )
3. Add condensation term to HLevin-Wen
4. Exact mapping to “transverse field” ZN spin model ⇒ confinement transition for certain “loops”
N times
Provides “solvable” realizations of (some) transitions proposed by
Ground state = weighted superposition of string net configurations
Levin and Wen model of IsingR x IsingL
Hilbert space: Edge labels:
Spectrum: both vertex and plaquette defects = IsingR x IsingL Particles = ( )R x ( )L
enforces
flips edge variables around a plaquette violation is “magnetic” defect
Enforces “ ” where allowed vertices are
(note: σ forms closed loops)
condense this boson b
Levin and Wen model of IsingR x IsingL
Hilbert space: Edge labels:
Levin and Wen model of
Equivalent to bilayer
superconductor
L
R
Levin and Wen model of IsingR x IsingL
condense this boson b
Levin and Wen model of Levin and Wen model of IsingR x IsingL
condense this boson
b is equivalent to a flux through a plaquette
condense this boson b
Levin and Wen model of Levin and Wen model of IsingR x IsingL
creates/annihilates/hops b‘s
b is equivalent to a flux through a plaquette
Creates only this one type of defect
Exact mapping to Transverse Field Ising Model
Levin and Wen model of Levin and Wen model of IsingR x IsingL
creates/annihilates/hops b‘s
b is equivalent to a flux through a plaquette
Creates only this one type of defect
Exact mapping to Transverse Field Ising Model
Levin and Wen model of Levin and Wen model of IsingR x IsingL
Exact mapping to Transverse Field Ising Model
on edge creates b on two adjacent plaquettes
String tension for σ loops!
Edges labeled With 1 or only
⇒Toric Code
Exact mapping to Transverse Field Ising Model
String tension for σ loops!
Frustrated
Pure IsingR x IsingL
Deconfined σ loops = IsingR x IsingL Phase Confined σ loops
Paramagnetic Ferromagnetic
No σ loops
Levin and Wen model of
Equivalent to bilayer
superconductor
L
R
Levin and Wen model of IsingR x IsingL
condense this boson b
In superconductor - language
Turns on interlayer s-pairing
Levin and Wen model of
Equivalent to bilayer
superconductor
L
R
Levin and Wen model of IsingR x IsingL
condense this boson b
In superconductor - language
Turns on interlayer s-pairing
Excitations:
Fermion
Boson
Boson
General Structure:
• Start with (chiral) Chern-Simons theory (or any MTC)
• Choose a ZN simple current
• Double CS theory (Levin Wen Lattice Model)
• Add term to condense
• Maps to ZN spin model
• Condensed phase is Drinfeld Double of the subcategory of particles from the original CS theory which braid trivially with .
Some Phase Transitions Between Topological Phases
Steven H. Simon w/ Fiona J. Burnell and Joost Slingerland (NUI Maynooth)
In Celebration of Mike Freedman’s Birthday
Topologically Nontrivial Birthday Cake