Some Phase Transitions Between Topological...

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


Toric Toric Code (Z2 gauge)

enforces enforces

flips all + ↔ - around a plaquette

violation is electric defect e

closed loops of

enforces


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


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


(i,j neighbors)

enforces


Toric

Exact mapping to Transverse Field Ising Model

Toric

Commutes with electric term. Never creates e defects. Can work in reduced Hilbert Space

Toric


Creates/annihilates/hops m bosons (i,j neighbors)

enforces


Toric


Toric

Toric

on edge creates m on two adjacent plaquettes Operate with


Creates/annihilates/hops m bosons (i,j neighbors)

enforces


Toric


Toric


String tension for loops Toric


enforces


Toric



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


Hilbert space: Edge labels:

Levin and Wen model of

Equivalent to bilayer

superconductor

L

R



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



creates/annihilates/hops b‘s

b is equivalent to a flux through a plaquette

Creates only this one type of defect




on edge creates b on two adjacent plaquettes

String tension for σ loops!

Edges labeled With 1 or only

⇒Toric Code


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



superconductor

L

R



In superconductor - language

Turns on interlayer s-pairing



superconductor

L

R



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

Some Phase Transitions Between Topological...

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