vortex lattice solutions of the zhk chern-simons equationsdogge/anyon/rajaratnam.pdf · vortex...

Vortex latticesolutions of the

ZHKChern-Simons

equations

KrishanRajaratnamUniversity of

Toronto

Introduction

Abrikosov latticesolutions

Stability ofAbrikosov latticesolutions

Vortex lattice solutions of the ZHK

Chern-Simons equations

Krishan RajaratnamUniversity of Toronto

March 11, 2019

Supervised by Professor I. M. Sigal. Thanks to Li Chen, DmitriChouchkov and Afroditi Talidou for useful discussions and support.

Thanks also to Professor Bruchard for help improving this presentation.

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Introduction

Chern-Simons action

The (abelian) Chern-Simons (CS) action on R3 is [CS71]

SCS(a) = −1

a ∧ da

where a is a 1-form.

It is one of the two gauge theories occurring in odd dimensionalspace-times, the other being the Maxwell action

SMax(a) = −1

‖da‖2 dx

The CS action is gauge invariant on R3, and in general wheneverboundary terms could be neglected.

ZHKChern-Simons

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Introduction

Chern-Simons action

The (abelian) Chern-Simons (CS) action on R3 is [CS71]

SCS(a) = −1

a ∧ da

where a is a 1-form.

It is one of the two gauge theories occurring in odd dimensionalspace-times, the other being the Maxwell action

SMax(a) = −1

‖da‖2 dx

The CS action is gauge invariant on R3, and in general wheneverboundary terms could be neglected.

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Introduction

Motivation - Condensed Matter Physics

I The CS action is known to be related to topological invariantsof three manifolds, however its Euler-Lagrange equations aretrivial. We will couple it with matter fields to get non-trivialEuler-Lagrange equations.

I The CS term occurs specifically in planar physics, and there aregeneral arguments showing that it can be used to attachnon-trivial (fractional) quantum statistics to particles [Wil90].

I We study a theory involving the Chern-Simons term, a constantexternal magnetic field and a double well potential, common inphysics. This theory was first written down by Zhang, Hansonand Kivelson and is called the ZHK model [ZHK89].

I The ZHK model appears in the study of the fractional quantumhall effect in condensed matter physics [ZHK89].

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The ZHK Chern-Simons action

The matter action we study, in the variables (x0, x1, x2) = (t, x1, x2),is

Smat(Ψ, a,Ab) =

(iΨD0Ψ− 1

2|∇a+AbΨ|2 − g

2(|Ψ|2 − 1)2dtdx

where D0Ψ = ∂0Ψ + i(a0 + Ab0)Ψ, ∇a+AbΨ = ∇Ψ + i(a + Ab)Ψ is

the covariant derivative, Ab = b2

(−x2, x1) satisfies curlAb = b > 0and g > 0.

We study the Euler-Lagrange equations of the ZHK action, which is

Smatter (Ψ, a,Ab) + SCS(a)

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The Zhang-Hanson-Kivelson equations

We define A = Ab + a, then the Euler-Lagrange equations of the

above action in terms of A = (A0,A) = (A0,A1,A2) are [ZHK89]

i∂tΨ = −1

2∆AΨ + A0Ψ + g(|Ψ|2 − 1)Ψ

0 = curlA + |Ψ|2 − b (ZHK)

∗ ∂tA = −curl∗A0 + Im(Ψ∇AΨ)

where −∆A = ∇∗A∇A, curlA =∂A2

∂x1− ∂A1

∂x2,

curl∗A0 = (∂A0

∂x2,−∂A0

∂x1) is the adjoint of curl, and ∗ denotes the

Hodge star.

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The Ginzburg-Landau equations

∂tΨ = ∆AΨ− A0Ψ + κ2(1− |Ψ|2)Ψ

∂tA = − curl∗ curlA−∇A0 + Im(Ψ∇AΨ) (GL)

These equations describe superconductors near phase transitions.

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The Ginzburg-Landau equations

∂tΨ = ∆AΨ− A0Ψ + κ2(1− |Ψ|2)Ψ

∂tA = − curl∗ curlA−∇A0 + Im(Ψ∇AΨ) (GL)

These equations describe superconductors near phase transitions.

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Gauge equivariance

For any function η : C∞(R2)→ R, and any solution (Ψ(x), A(x))of the ZHK (GL) equations, the state T gauge

η (Ψ(x), A(x)) definedby

T gaugeη (Ψ(x), A(x)) = (e iη(x)Ψ(x), A(x) +∇η(x))

is also a solution of the ZHK (GL) equations.

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Energy and ground state

E (Ψ,A) =

2(|∇AΨ|2 + g(|Ψ|4 − 2 |Ψ|2))dx

The gauge invariant ground state (Ψ,A) = (0, (0,Ab)) wherecurlAb = b, is called the normal state u0.

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Brief History

1957 Abrikosov found vortex lattice solutions of the GLequations.

1989 The Zhang-Hanson-Kivelson equations were firstwritten down to describe the fractional quantum halleffect.

2011 Tzaneteas and Sigal rigorously proved the existence ofAbrikosov lattice solutions of the GL equations.

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Brief History

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Brief History

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Abrikosov lattice states in Superconductivity

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Sketch of a lattice L.

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Abrikosov lattice sate

We want time-independent solutions (Ψ, (A0,A)) of the ZHKequations such that the quantities

ρ = |Ψ|2 J = Im(Ψ∇AΨ)

B = curlA A0

are periodic with respect to a lattice L. Such statesu = (Ψ, (A0,A)) are called Abrikosov lattice states.

These states are more general than requiring Ψ and A to beL-periodic.

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Introduction

Abrikosov lattice sate

We want time-independent solutions (Ψ, (A0,A)) of the ZHKequations such that the quantities

ρ = |Ψ|2 J = Im(Ψ∇AΨ)

B = curlA A0

are periodic with respect to a lattice L. Such statesu = (Ψ, (A0,A)) are called Abrikosov lattice states.

These states are more general than requiring Ψ and A to beL-periodic.

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Abrikosov lattice solutions

At b = b0 := 2g an Abrikosov lattice state bifurcates from thenormal state, as the following theorem states.

Theorem (Existence of a Bifurcation [RS18])For any g > 0 and some b satisfying 0 < |2g − b| � 1

1. There exists an Abrikosov lattice state ub, in a neighbourhoodof the normal branch u0, which solves the ZHK equations.

2. If g < 12, the hexagonal lattice minimizes the average energy

per lattice cell.

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Abrikosov lattice states in Superconductivity

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Abrikosov lattice solutions on Riemann surfaces

It turns out that (Ψ, (A0,A)) is an Abrikosov lattice state iff Ψ liveson a line bundle over the T2 = R2

L , and A is a connection on it. Thislatter view point generalizes to arbitrary Riemann surfaces, and sodoes the bifurcation result.

Theorem (Existence of a Bifurcation [RS18])Let g > 0 and suppose b satisfies 0 < |2g − b| � 1. Then on aRiemann surface of genus h, as long as the first Chern number n ofthe line bundle satisfies 1 ≤ n ≤ h, there exists an Abrikosov latticestate ub, in a neighbourhood of the normal branch u0.

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Orbital stability of Abrikosov lattice solutions

Theorem ([Raj18])For any solution Ψ(t) ∈ C 1(R+,H1(T2)) of the ZHK equations, wecan show that

‖Ψb −Ψ(0)‖ < δ ⇒∥∥e−iγ(t)Ψb −Ψ(t)

∥∥ < ε for all t

and some function γ(t).

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Key ideas behind Bifurcation theorem

I Write the time-independent ZHK equations as

F (b, u) = 0 (3)

I A bifurcation point b0 occurs when dFu(b0, u0) is not invertible.

I The map dFu(b0, u0) is always non-invertible. However, this isfixed by working in the Coloumb gauge.

I The change in invertibility of duF (b, u0) is controlled by thefollowing operator

2∆Ab − g

Its spectrum shall be studied using a Weitzenbock-type identity.

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F (b, u) = 0 (3)

2∆Ab − g

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Introduction

F (b, u) = 0 (3)

2∆Ab − g

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F (b, u) = 0 (3)

2∆Ab − g

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Weitzenbock-type identityFirst we define

∂Ab= ∂ − iAb

c ∂∗Ab= ∂ − i Ab

1 − iAb2)

Then the Weitzenbock identity states that

2∆Ab = 2∂Ab∂∗Ab

+curlAb

Since ∂Ab∂∗Ab≥ 0 and curlAb =

∂Ab1

∂x2− ∂Ab

∂x1= b, we have

2∆Ab − g ≥ b

2− g �

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Introduction

∂Ab= ∂ − iAb

c ∂∗Ab= ∂ − i Ab

1 − iAb2)

+curlAb

∂Ab1

∂x2− ∂Ab

∂x1= b, we have

2∆Ab − g ≥ b

2− g �

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∂Ab= ∂ − iAb

c ∂∗Ab= ∂ − i Ab

1 − iAb2)

+curlAb

∂Ab1

∂x2− ∂Ab

∂x1= b, we have

2∆Ab − g ≥ b

2− g �

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Hamiltonian form of ZHK Equations

The ZHK equations can be written in Hamiltonian form using theenergy functional

E (Ψ,A) =

2(|∇AΨ|2 +g(|Ψ|2−1)2)+A0(− curlA+ |Ψ|2 +b)dx

Then letting J(Ψ,A) = (iΨ, ∗A), the ZHK equations become

)= ∇Ψ,AE (Ψ,A0,A)

0 = ∇A0E (Ψ,A0,A)

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Stability analysis

If we let H = HessE (ub), then to leading order in b, we have

−∆Ab0 − b0 0 00 0 − curl0 − curl∗ 0

The operator

(0 − curl

− curl∗ 0

has an infinite number of negative eigenvalues, which makes stabilityimpossible.

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Stability analysis

If we let H = HessE (ub), then to leading order in b, we have

−∆Ab0 − b0 0 00 0 − curl0 − curl∗ 0

The operator

(0 − curl

− curl∗ 0

has an infinite number of negative eigenvalues, which makes stabilityimpossible.

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Modified equations

However we can use the second and third ZHK equations to solvefor A and A0 as a function of Ψ. Substituting these into the energyfunctional, we obtain

E (Ψ) =

2(∣∣∇A(Ψ)Ψ

∣∣2 + g(|Ψ|2 − 1)2)dx

whose Euler-Lagrange equation is

i∂tΨ = −1

2∆Ab+A(|Ψ|)Ψ + A0(|Ψ|)Ψ + g(|Ψ|2 − 1)Ψ

hereafter called the non-local ZHK equations.

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The Hessian of the non-local energy to leading order in thebifurcation parameter is

−∆Ab0 − b0

which is non-negative.

For the new equation, we can prove orbital stability.

Theorem ([Raj18])For any solution Ψ(t) ∈ C 1(R+,H1(T2)) of the non-local ZHKequations, we can show that

‖Ψb −Ψ(0)‖ < δ ⇒∥∥e−iγ(t)Ψb −Ψ(t)

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The Hessian of the non-local energy to leading order in thebifurcation parameter is

−∆Ab0 − b0

which is non-negative.

For the new equation, we can prove orbital stability.

Theorem ([Raj18])For any solution Ψ(t) ∈ C 1(R+,H1(T2)) of the non-local ZHKequations, we can show that

‖Ψb −Ψ(0)‖ < δ ⇒∥∥e−iγ(t)Ψb −Ψ(t)

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Introduction

Thank you for listening!

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K. Rajaratnam. “On stability of Abrikosov latticesolutions of the ZHK Chern-Simons equations”. 2018.K. Rajaratnam and I. M. Sigal. “Abrikosov latticesolutions of the ZHK Chern-Simons equations”. 2018.F Wilczek. Fractional Statistics and AnyonSuperconductivity. International journal of modernphysics. World Scientific, 1990, p. 447. isbn:9789810200497. url:https://books.google.ca/books?id=MHf9sBNPszkC.S C Zhang, T H Hansson, and S Kivelson.“Effective-Field-Theory Model for the FractionalQuantum Hall Effect”. In: Phys. Rev. Lett. 62.1 (Jan.1989), pp. 82–85. doi: 10.1103/PhysRevLett.62.82.url: https://link.aps.org/doi/10.1103/PhysRevLett.62.82.

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