-connections on circle bundles over space-time - PowerPoint Presentation

Slide1

-connections on circle bundles over space-time

Distortion of gauge fields and order parametersMichael FreedmanRoman LutchynSeptember, 2014

1Slide2

OutlineI will discuss the possibility of slightly generalizing -principal bundles familiar in

ElectromagnetismSuperfluidsSuperconductors

 2Slide3

What we propose is looking at things “halfway” between​

-principal bundle with

-connection and​-principal bundle with

-connection.

1

3

1. Going this far is problematic: the

definite Killing directions yield, in quantum theories, excitations of

negative energy

.

 Slide4

Namely​-principal bundle with local

gauge symmetry of the Lagrangian but

with an -connection (i.e., not “left invariant”)

Or an

associated

bundle to

built from the quotient action:

This bundle has an

connection.

There is no

canonical

-symmetry. But imposing one leads back to case 1.

4

repelling

attractingSlide5

Energy PenaltyPossible to measure distortion of the Mobius structure on

and charge some cost:

,“

nematic distortion energy” or “Beltrami

energy”

,

where

are the

Killing direction of

.

Idea:

Consider models which tolerate a little (elliptical) distortion (at a price). Study the limit:

distortion .

This

new flexibility has some surprising consequences

.At the end of this talk I’ll consider and sl(2,R) G-L theory: 5Slide6

6Preview:One may write a Ginzburg

-Landau Lagrangian in the

context

Beltrami energy

​

is (local)

-gauge invariant:

,

 Slide7

LiteratureWitten: – gravity

(Nucl. Phys. B311 (1988), 46–78, 0712.0155, 1001.2933)

Haldane: anisotropic model for FQHE (1201.1983, 1202.5586) Effective mass tensor

compared to Coulomb

​

(kinetic energy of free electrons in

crystal with

-field)

Coulomb

interaction energy (inside lattice)

7Slide8

Mathematical Starting PointA surprising flexibility of circle bundles over surfaces with flat

connection

, real, ,

Lie algebra, real,

,

via

polarization

,

8Slide9

Mathematical Starting Point9

commutes with

commutes with

elliptic

parabolic

hyperbolicSlide10

FactFor

a closed surface of genus

has a flat

connection (acting

projectively

between fibers

).

10

Proof

Unwrap

to

,

. G

eodesic flow

canonically identifies all

unit tangent

circles

to

with

the circle at infinity

. This integrates

the connection .

 Slide11

FactThis gives a (actually, many) irreps

.

(Such geometric reps and their Galois conjugates are the chief source of examples.)“Chern

number”:

for

Although

,

defines a

flat

-bundle with structure group

:

,

11Slide12

FactFor ,

cannot extend as

rep over any bounding -manifold ,

but by

Thurston’s

orbifold

theorem

extensions over

For

,

“

tripus

”

12

 Slide13

FactThis can be used to make pairs of -monopoles if one

allows

not to act near a point.Recall: In EM,

over a spatial surface

,

.

If topology is standard

and

(

no

monopoles

).

13Slide14

FactBut expanding the nominal fiber from to

and letting the

connection (potential) of EM take values near a point creates a

-

monopole (charge

).

14

time

Tripus

flux appears from projection of

to

.

This projection creates curvature.

fibers

No flux,

flat

 Slide15

Chern-Weil TheoryWhat is going on? How can you have a characteristic class without curvature?char. class func

. (curvature) for rational classes has an exceptions when structure group is noncompact:

Exception for Euler Class, group non-compact.For

connections

,

where

However, for

, there is

no

such

formula

even

though

and

have equivalent bundle theories

.

15

“

Pfaffian

”Slide16

Chern-Weil TheoryFact: There exist

-bundles with which admit flat

connections.Milnor (1958) proved a sharp threshold for surfaces

:

Given

,

a

flat

linear

connection, and

Wood

(1970) showed

a flat

projective

connection.

16

 Slide17

Infinitesimal Milnor

For genus g

,

has an

-flat bundle

with

and

holonomies

,

,

radian rotation

(

Commutating boots yield a rotation

.)

17Slide18

Infinitesimal MilnorTo second order:

But this is not

exact.

However

, topology implies an exact solution.

is

to

.

Since this is

non-

contractable

,

perturbations remain

surjective

.

This defines

a representation

near boost on meridian and longitude and pure rotation around puncture. Band summing copies

“infinitesimal Milnor

”

18Slide19

A principal bundle with flat

satisfies

const.

Proof

:

Use flat

to trivialize

over top cell

. Comparing this

trivialization with “round”

structure on each

fiber gives a map

.

,

where

is the

th

component

of

,

,

. But

.

□

There is a converse

19

“Beltrami energy”

 Slide20

Application of hyperbolic geometry in condensed matter physicsQuantum Hall effect: Haldane et al., 2011–12, Maciejko et al., 2013)Superfluids and superconductors: Freedman and Lutchyn, 2014(this talk ☺

)Properties of two dimensional models on a space with negative curvature are very different!

No long range interaction between vortices in XY model: Callan and Wilczek, Nucl. Phys. B340 (1990), 366–386In contrast, the last bit of this talk is about curvature the target space

Connections may boost as well as rotateExotic quantization condition

20Slide21

Synthetic Gauge Fields in Cold AtomsThe cold atoms community is now proficient at simulating

and

gauge fields.We suspect that a similar technique would permit simulation of gauge field-gravity in the lab. Specifically, a

rotationally symmetric, pure boost

might be

imposed on a ring of cold atoms

.

21Slide22

Cold AtomsRecall the Poincaré disk model:

22

Metric:

Stereographic projection:

On the next slide we see that the boost connection

is precisely the angular component of the

Levi-

Civita

connection

.

also

has the interpretation of the tangents to the circle

of radius

in the hyperbolic plane

.

 Slide23

Cold AtomsHyperbolic geometry arises as

when has the bi-invariant Killing metric.

23

loop of all ellipses of eccentricity

 Slide24

Cold AtomsAn order parameter

coupled to

(not

!)

will have energy

All zero-energy solutions:

(

)

The quantization condition

is integral

.

24Slide25

The uninitiated would have an experimental surpriseA pure boost integrates to purely rotational holonomy

Hyperbolic quantization conditionsEnergy vs.

, not:

but

25Slide26

superconductivity?

Let us assume EM is pure . Is there a role for

in an effective theory of superconductivity

?

The simplest

opportunity is

a

spin polarized

,

two-dimensional

,

superconductor.

Consider the GL

Lagrangian density:

26Slide27

Understanding the Order Parameter  

First, in what complex line should a spin-polarized

(fixed -vector) -wave superconducting order parameter

take its values?It is a section of

, where

τ

is the tangent bundle to the superconducting

space-time,

and

is the

bundle of electromagnetism,

EM.. The

comes from

: , , , etc… .GL-Hamiltonian has symmetry

-wave:

ground state symmetry

,

since

and if

implements

and

implements

spatial

,

then,

27Slide28

Since the tangent bundle to a sample is only an abstraction coarsely connected to the experimental reality, it does not seem essential to postulate

.

A small amount of “slop” in metric transport is modeled at lowest order by elliptic distortion ().

28

Thus

lies in

.

Considering the effect of

,

-shifts, one calculates that

is a section of

.

 Slide29

One may write a Ginzburg-Landau Lagrangian in the

context

Notation:

​

is,

as in EM, a

-connection.

is an enhancement, an

-connection. Using orthogonal basis:

,

project

:

29

-part, boost

-part rotation

Beltrami energySlide30

All bundles have structure group , but

possess an connection,

.Dynamic variables: ,

​

is (local)

-gauge invariant:

,

Under a local

-gauge transformation

,

is invariant.

Since conjugation

by

is an

isometry

of

and the

contribution is

parallel

to

and thus projected

out.

30Slide31

In such a Lagrangian there are new ways to trade energy around, specifically between

,

,

and the Beltrami term

.

We expect

some

modification to

Meissner

physics

,

and

vortex geometry.

Also new Josephson equations, if the Beltrami energy

were coupled to

(as is charge density).

 31Slide32

U(1) Meissner effect – geometry of vortices

picture

winding

Energy balance

32

SC

SC

h/2e

h/2e

Large

Small

|^2

Small

Large

|^2

 Slide33

Vortices

Small SC stiffness Type II, for usual

connection

Moderate

SC stiffness and small

Beltrami coefficient

“genus transition”

,

In this transition, a

planar

2DEG becomes high genusEither by spontaneously adding handles, or

In a bilayer system via “

genons

”*High genus makes independent of

, enabling both

and

to be

reduced

at the expense of

increasing

the energy of nematic distortion,

“Infinitesimal Milnor” allows “designer connections”

33

*[Barkeshli, Jian, Qi] arXiv:1208.4834Slide34

genus transition

Since a polarized chiral -wave 2DEG supports non-abelian Ising excitations, genus

ground state degeneracy (for a fixed order parameter configuration).This degeneracy could have observable consequences, e.g., entropy as it affects specific heat.

34

2DE6

vortex core

 Slide35

Summary connections on

-principal bundles allow surprising flexibilities and may be useful in model building:High energy (We discussed the

-sector of the standard model only)Low energy, superfluids, superconductors, and possibly QHENon-compact forms of other Lie algebras can be considered in a similar vein.

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