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-CONNECTIONS ON CIRCLE BUNDLES OVER SPACE-TIME

DISTORTION OF GAUGE FIELDS AND ORDER PARAMETERS

Michael Freedman

Roman Lutchyn

September, 2014

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Outline

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

1. Electromagnetism

2. Superfluids

3. Superconductors

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What we propose is looking at things “halfway” between• -principal bundle with -connection and• -principal bundle with -connection.1

1. Going this far is problematic: the definite Killing directions yield, in quantum theories, excitations of negative energy.

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Namely

1. -principal bundle with local gauge symmetry of the Lagrangian but with an -connection (i.e., not “left invariant”)

2. 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.

repelling attracting

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Energy Penalty• Possible 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:

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Preview:

•One may write a Ginzburg-Landau Lagrangian in the context

ℒ𝐺𝐿𝑠𝑙 ( 2 ,ℝ )=−

14‖𝐹 𝐴‖

2+𝜌‖𝐷~

𝐴𝜙‖2+¿

Beltrami energy

is (local) -gauge invariant:,

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Literature• Witten: – 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

1. (kinetic energy of free electrons in crystal with -field)

2. Coulomb interaction energy (inside lattice)

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Mathematical Starting Point

A surprising flexibility of circle bundles over surfaces with flat connection• , real, , • Lie algebra, real, , • via polarization ,

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Mathematical Starting Point

𝑆𝐿 (2 ,ℝ )

𝑆𝑈 (1 ,1)

Con(1 −𝑖1 𝑖 )

commutes with

commutes with

{𝑎 𝑏𝑏 𝑎

∣|𝑎|2−|𝑏|2

=1)→( 𝑎|𝑎| ,𝑏𝑎 )∈𝑆1× �̊�2

⊂

⊂

¿

𝑆𝐿(2 ,ℂ)

𝑃𝑆𝐿 (2 ,ℝ ) : idellipticparabolic

hyperbolic

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Fact

For a closed surface of genus has a flat connection (acting projectively between fibers).

ProofUnwrap to , . Geodesic flow canonically identifies all unit tangent circles to with the circle at infinity . This integrates the connection .

𝛾∈Γ

ℍ2

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Fact• This 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 :

,

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Fact• For , cannot extend as rep over any bounding -manifold , but

by Thurston’s orbifold theorem extensions over

• For ,

“tripus”

↩ ↩

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Fact• This 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).

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Fact

But expanding the nominal fiber from to and letting the connection (potential) of EM take values near a point creates a -monopole (charge).

time

⋯

Tripus𝑢(1)

flux appears from projection of to . This projection creates curvature.

fibers

𝑢(1)

No flux, flat

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Chern-Weil Theory• What 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.

“Pfaffian”

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Chern-Weil Theory• Fact: 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.

→

Σ𝑔

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𝑚1

𝑙1 𝑙2 𝑙3𝑙𝑔

𝑚2 𝑚3 𝑚𝑔

Infinitesimal Milnor

For genus g , has an -flat bundle with and holonomies , ,

radian rotation

(Commutating boots yield a rotation.)

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Infinitesimal Milnor

To 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”

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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 . □

h (𝐶 )ℍ2

There is a converse

“Beltrami energy”

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Application of hyperbolic geometry in condensed matter physics• Quantum 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), 366–386

• In contrast, the last bit of this talk is about curvature the target space

• Connections may boost as well as rotate• Exotic quantization condition

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Synthetic Gauge Fields in Cold Atoms• The 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.

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Cold Atoms• Recall the Poincaré disk model:

𝑟Γ

ℍ2

Metric: Stereographic projection:

𝑟=tanh(𝜏2 )

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 .

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Cold Atoms

Hyperbolic geometry arises as when has the bi-invariant Killing metric.

ℍ2=𝑆𝐿(2 ,ℝ )/𝑈 (1)={ellipses of area=1 modulo translation}

𝑑𝑠ℍ2=4

(1 −𝑟 2)2 𝑑 𝑠𝐸2

dis tℍ2(|1 00 1|,|𝑛 0

01𝑛|)=log𝑛2

loop of all ellipses of eccentricity

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Cold Atoms• An order parameter coupled to (not !) will have energy

• All zero-energy solutions:

()• The quantization condition

is integral.

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The uninitiated would have an experimental surprise1. A pure boost integrates to purely rotational holonomy

2. Hyperbolic quantization conditions

3. Energy vs. , not:

but

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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:

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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,

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• 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 ().

𝐻

Thus lies in .

𝑆𝑂 (2)

𝑈 (1)

Considering the effect of , -shifts, one calculates that is a section of .

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One may write a Ginzburg-Landau Lagrangian in the context

1. Notation: is, as in EM, a -connection. is an enhancement, an -connection. Using orthogonal basis: ,

project: 𝑐𝑏𝑎-part, boost

-part rotation

ℒ𝐺𝐿𝑠𝑙 ( 2 ,ℝ )=−

14‖𝐹 𝐴‖

2+𝜌‖𝐷~

𝐴𝜙‖2+¿

Beltrami energy

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2. All bundles have structure group , but possess an connection, .

3. Dynamic variables: ,

4. 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.

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• 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).

U(1) Meissner effect – geometry of vortices

picture

winding

Energy balance

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∬𝐹SC SC

h/2e h/2e

Large Small |^2

Small Large |^2

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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 genus• Either 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”

*[Barkeshli, Jian, Qi] arXiv:1208.4834

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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.

2DE6vortex core

𝜆London

�⃗�𝜆 topological

�⃗�

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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 QHE

• Non-compact forms of other Lie algebras can be considered in a similar vein.