Duality in Condensed Matter Physics and Quantum Field Theory

Duality in Condensed Matter

Physics and Quantum Field Theory

Eduardo Fradkin

Department of Physics and Institute for Condensed Matter Theory University of Illinois, Urbana, Illinois, USA

Lectures at the Quantum Connections Summer School, Lidingö,

Sweden, June 10-22, 2019

Motivation

• Dualities in CM and QFT

• Particle-Vortex duality

• Applications to the Fractional Quantum Hall Effect

• Conjectured dualities, bosonization and fermionization

• Loop models: flux attachment, duality and periodicity

• Periodicity vs Fractional Spin

• Implications for Fractional Quantum Hall fluids

Dualities

• EM duality: E ⟺ B, electric charges ⟺ magnetic monopoles Dirac quantization

• 2D Ising Model: Kramers-Wannier duality, high T ⟺ low T, order ⟺ disorder

• Duality of the 3D ℤ ² gauge theory ⟺ 3D Ising model, order ⟺ confinement

• Particle-Vortex duality: electric charge ⟺ vortex (magnetic charge)

• Mappings between phases of matter, most often between different theories

• Conjectured web of dualities between CFTs in 2+1 dimensions

Electromagnetic Duality

• EM duality: E ⟺ B

• electric charge e ⟺ magnetic monopole m

• Dirac quantization em=2𝜋

r · E =⇢, r · B = 0 r ⇥ B 1

c

@E

@t = j, r ⇥ E + @B

@t = 0

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Electric-magnetic asymmetry of Maxwell’s equations

Duality of Forms

• Geometric duality

• p forms in D dimensions are dual to D-p forms

• In D=2 the dual of a vector is a vector, J 𝜇* =𝜀 𝜇𝜈 J ^𝜈 , and the dual of a 2nd rank tensor is a scalar, F 𝜇𝜈 =𝜀 𝜇𝜈 𝜃

• In D=3 the dual of a vector is a 2nd rank tensor, J 𝜇* =1/2 𝜀 𝜇𝜈𝜆 F ^𝜈𝜆 (and the dual of a 2nd rank tensor is a vector), etc. Duality exchanges the vector potential A 𝜇 with a

compactified scalar 𝜃

• In D=4 duality exchanges F 𝜇𝜈 ⟺ F 𝜇𝜈* =1/2 𝜀 𝜇𝜈𝜆ρ F ^𝜆ρ , E ⟺ B

• Lattice duality: in D=2 the dual of a link is a link, and the dual of a plaquette is a site

• In D=3 the dual of a link is an (oriented) plaquette (and viceversa), and the dual of a 3

volume is a site (and viceversa)

Duality the Maxwell field in D=2+1

• Canonical quantization of the Maxwell field in the gauge A 0 =0

• In 2+1 dimensions there is only one transverse degree of freedom

• It is equivalent (dual!) to a compactified scalar

• The compactified scalar is a Goldstone field

• Charge quantization implies

compactification (periodicity) of the dual scalar field

r · E = 0 ) E ⁱ = ✏ _ij @ _j ✓

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[E _i (x), A _j (y)] = i _ij (x y)

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[E _i (x), B(y)] = i✏ _ij @ _j (x y)

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[✓(x), B(y)] = i (x

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y) ) ✓ ⌘ ⇧

H = 1

2 E ² + B ²

= 1

2 (r✓) ² + ⇧ ²

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r · E = 2⇡n (x) ) ✓ = I

dx _i @ _i ✓ = 2⇡n

Duality in Classical Statistical Mechanics

• 2D Ising Model: Kramers- Wannier (self) duality

• Partition function as a sum over closed domain walls in the low T expansion

• Partition function as a sum over loops of the high T expansion

• high T ⟺ low T

• 2D: order ⟺ disorder

• 3D: Duality of the ℤ ² gauge theory ⟺ Ising model

• high T loops and 3D surfaces of domains

• 3D: order ⟺ confinement;

disorder ⟺ deconfinement

Z _domains [exp( 2/T )] = Z _loops [tanh(1/T )]

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• Closed loops of the high T expansion: Euclidean worldlines of massive neutral scalar particles of the symmetric phase

• In D=2 the closed domain walls represent the Euclidean evolution of kinks (solitons)

• in D=3 the closed domain walls represent the

evolution of closed strings

Duality and the d=1 Quantum Ising Model

• Define a Pauli (Clifford) algebra in terms of (the kink operator) τ 3

and τ 1 defined on the dual lattice

• Maps 𝜆 to 1/𝜆 (strong coupling and weak coupling)

• Order and disorder

• Disordered phase is a kink condensate

n n + 1 n − 1

˜n ˜n + 1

˜n − 1

H = X

n

1 (n) X

n

3 (n) ₃ (n)

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⌧ ₃ (˜n) = Y

j n

1 (j)

⌧

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₁ (˜n) = ₃ (n) ₃ (n + 1)

⌧ ₃ (˜n 1)⌧ ₃ (˜n) = ₁ (n)

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{ ¹ (n), ₃ (k)} = 0 ) {⌧ ¹ (n), ⌧ ₃ (k)} = 0

3 (n) ² = ₁ (n) ² = 1 ) ⌧ ³ (n) ² = ⌧ ₁ (n) ² = 1

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H = X

⌧ ₃ (˜n)⌧ ₃ (˜n + 1) X

⌧ ₁ (˜n)

What happens in 2+1 dimensions?

• The dual of the 2d quantum Ising model is the 2d ℤ 2 gauge theory

• The gauge fields reside on the links of the dual lattice

• Duality maps order to

confinement and disorder to deconfinement

• Ising order parameter maps onto a ℤ 2 magnetic charge

(“monopole”)

H = X

r

1 (r) X

r,j=1,2

3 (r) ₃ (r + e _j )

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H = X

˜ r,j

⌧ ₁ (r, j) X

˜ r

⌧ ₃ (˜r, 1)⌧ ₃ (˜r + e ₁ , 2)⌧ ₃ (˜r, 2)⌧ ₃ (˜r ₁ + e ₂ )

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Gauss Law :

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⌧ ₁ (˜r, 1)⌧ ₁ (˜r e ₁ , 1)⌧ ₁ (˜r, 2)⌧ ₁ (˜r e ₂ , 2) = 1

1 (r) = ⌧ ₃ (˜r, 1)⌧ ₃ (˜r + e ₁ , 2)⌧ ₃ (˜r, 2)⌧ ₃ (˜r + e ₂ , 1)

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⌧ ₁ (˜r, 1) = ₃ (r) ₃ (r + e ₂ )

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Bosonization in 1+1 dimensions as duality

• 1d free fermions at low energies are equivalent to a free massless Dirac field, 𝜓(x)=(𝜓 R (x), 𝜓 L (x))

• Conserved current j 𝜇

• Current algebra

• Equivalent to the algebra of a

canonical massless compactified boson

• Addition of one fermion Q=1, implies that the boson must obey twisted boundary conditions

• Axial current j 𝜇5 =𝜀 𝜇𝜈 j ^𝜈 is not conserved (axial anomaly)

H = i _R ^† @ _{x R} + i _L ^† @ _{x L}

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[j ₀ (x), j ₁ (y)] = i

⇡

0 (x y)

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H = 1

2 ⇧ ² + 1

2 (@ _x ) ²

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Q = Z

j ₀ (x)dx = 1 p ⇡ j ₀ = : _R ^† _R : + : _L ^† _L : j ₁ = : _R ^† _R : : _L ^† _L :

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j ₀ ⌘ 1

p ⇡ @ _x , j ₁ ⌘ 1

p ⇡ ⇧

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j _µ = 1

p ⇡ ✏ _µ⌫ @ ^⌫ , @ _µ j ^µ = 0

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@ ^µ j _µ ⁵ = e

2⇡ ✏ _µ⌫ F ^µ⌫ , @ ² = e

p F ^⇤

Duality in the Classical 3D XY Model

At high T the partition function is a sum over closed particle loops

Z _XY = Y

r

Z 2⇡

0

d✓(r)

2⇡ exp( X

r,µ

cos( _µ ✓(r))

/ X

` _µ (r) 2Z

Y

r

( _µ ` _µ (r)) exp( X

r,µ

` _µ (r) ² 2 )

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At low T it can be written a sum over vortex loops

Z _XY / X

` _µ (r) 2Z

Y

r

( _µ ` _µ (r)) exp( X

r,µ

` _µ (r) ² 2 )

= X

s _µ ( ˜ r) 2Z

Y

˜ r

( _µ s _µ ) exp( 1 2

X

˜ r,µ

(✏ _{µ⌫ ⌫} s ) ² )

= X

m _µ (r)

Y

˜ r,µ

d _µ (˜r) exp( 1 2

X

˜ r,µ

(✏ _{µ⌫ ⌫} ) ² + i2⇡ X

˜ r,µ

m _µ (r) _µ (r))

= X

m _µ (r)

exp( 2⇡ ² X

m _µ (˜r)G _µ⌫ (˜r ˜r ⁰ )m _⌫ (˜r ⁰ ))

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where G _µ⌫ (˜r ˜r ⁰ ) = h ^µ (˜r) _⌫ (˜r ⁰ )i

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Particle-Vortex Duality

• Theories with a global U(1) symmetry, e.g. the 3D XY model

• High T expansion loop gas: worldlines of charged particles with short-range interactions

• Low T expansion: closed vortex loops with Biot-Savart long-range interactions

• Particle-Vortex duality: electric charge ⟺ vortex (magnetic charge)

• The situation reverses for a XY model is coupled to a fluctuating Maxwell field: Particle loops have long range Coulomb interactions, and vortex loops have short range

interactions (Higgs mechanism)

• The two models are dual to each other!

• In field theory language

Z( , e) ' Z

✓ e ²

4⇡ , 1 2

◆

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|(@ ^µ + iA _µ ) | ² + m ² | | ² + | | ⁴ $ |(@ ^µ + ia _µ ) ˜| ² m ² | ˜| ² + |˜| ⁴ + 1

2⇡ ✏ _µ⌫ b _µ @ _⌫ A

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j _µ $ 1

2⇡ ✏ _µ⌫ @ _⌫ a

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The Fractional Quantum Hall Effect

• Two-dimensional system of N e electrons in Landau levels created by a large external uniform magnetic field with N 𝛷 fluxes

• Filling fraction: 𝜈= N e /N 𝛷

• Quantized Hall conductivity 𝜎 xy =p/(2np±1) e ² /h (Jain fractions) (p, n ∈ ℤ)

• Laughlin states: 𝜈=1/m (m ∈ ℤ) (m odd for fermions, even for bosons)

• Laughlin wavefunction:

• Statistical transmutation of charge-flux composites (Wilczek)

• Composite bosons: m fluxes attached to bosons (Zhang, Hansson and Kivelson; Read)

• Composite fermions: (m-1) fluxes attached to fermions (Jain)

• Field theory: Chern-Simons gauge field encodes flux attachment

(z ₁ , . . . , z _N ) =

Y N i<j=1

(z _i z _j ) ^m exp( 1 4` ² ₀

X N i=1

|z ⁱ | ² )

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L = m

4⇡ ✏ _µ⌫ a ^µ @ ^⌫ a j _µ a ^µ ) j ⁰ = m

2⇡ ✏ _ij @ _i a _j and [a _i (x), a _j (y)] = i 2⇡

m ✏ _ij (x y)

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Composite Boson Picture of the FQHE

• Landau-Ginzburg theory (Zhang, Hansson and Kivelson): Non-Relativistic abelian-Higgs model with a Chern-Simons term: composite bosons

coupled to m fluxes S ^B = Z

d ³ z

⇢

⇤ (z)[iD ₀ + µ] (z) + ~ ²

2M | D (z) | ² + 1

4⇡m ✏ _µ⌫ a ^µ @ ^⌫ a 1

2 Z

d ³ z Z

d ³ z ⁰ (| (z)| ² ⇢ ₀ )V (|z z ⁰ |)(| (z ⁰ )| ² ⇢ ₀ )

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• FQH plateau: composite bosons condense

(x) = p

⇢(x)e ^i!(x)

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1

2⇡m ✏ _ij @ _i a _j + | (x)| ² = 0, Z

d ³ x | (x)| ² = ⇢ ₀ L ² T

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D _µ = @ _µ + i(A _µ + a _µ )

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ha

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ⁱ i + A ⁱ = 0

| | ² = ⇢ ₀ = ⌫ 2⇡` ² ₀

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⌫ = 1

m , ` ² ₀ = 1 B

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Fluctuations and the FQHE

• Effective action for quantum fluctuations: a 𝜇 =⟨a 𝜇 ⟩+𝛿a 𝜇 ; probe field 𝛿A i

L ^e↵ = 

2 (@ ₀ ! a ₀ e A ₀ ) ² ⇢ _s

2 (r! a e A) ² + 1

2⇡m ✏ _µ⌫ a ^µ @ ^⌫ a

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Le↵[ A ^µ ] = e ²

4⇡m ✏ _µ⌫ A ^µ @ ^⌫ A + . . . ) ^xy = e ²

2⇡m = 1 m

✓ e ² h

◆

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• Integrating out the fluctuations 𝛿a 𝜇

• ^{Vortices: lim}

|x|!1 (x) =p⇢ ₀ e ^i'(x) , a ₀ = 0, '(x) = tan ¹ (y/x)

|x|!1 lim a _i = ± @ ⁱ ' = ±✏ ^ij x _j

|x| ² )

I

a · dx = ± 2⇡

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• Vortices have finite energy (not logarithmic!) and fractional charge:

Q = e 2⇡m

Z

⌃

d ² x ✏ _ij @ _i a _j = e 2⇡m

I

@⌃

dx · a = ± e

m

Vortex Partition Function

• Using the identity

• Effective topological field theory in terms of the hydrodynamic field b 𝜇 (Wen)

• ^{j v𝜇} is a current that represents slowly-varying vortex worldlines

• We recover the correct Hall conductivity and the fractional charge of the vortex

• Integrating out the field b 𝜇 we obtain the partition function of the worldlines of the vortices whose action is i times the Hopf invariant (or linking number)

• Fractional statistics!

• The vortex excitations of a FQH state are represented by a the worldlines of vortices with this effective action (a loop model!)

L ^e↵ = m

4⇡ ✏ _µ⌫ b ^µ @ ^⌫ b + e

2⇡ ✏ _µ⌫ A ^µ @ ^⌫ b + j _µ ^v b ^µ

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S _e↵ [j _µ ^v ] = ⇡ m

Z

d ³ x Z

d ³ y j _µ ^v (x)✏ ^µ⌫ hx| 1

@ ² |x ⁰ i @ ^y j _⌫ ^v (y)

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L = m

4⇡ ✏ _µ⌫ b ^µ @ ^⌫ b + 1

2⇡ ✏ _µ⌫ b ^µ @ ^⌫ a ⌘ 1

4⇡m ✏ _µ⌫ a ^µ @ ^⌫ a

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Duality in the FQHE

• Both vortices and fermions are described by a model of loops that close in imaginary time

• Both sides of the plateau transition are described by worldlines representing massive particles

• The transition between FQH states can then be thought as the condensation of some anyons with the two phases being related by duality

• Other states can be thought of being obtained by “addition of Landau levels”

• Duality and Landau level addition do not commute as operations

• ^SL(2, ℤ) symmetry: Universal phase diagram for the FQH states based on particle-vortex duality (Kivelson, Lee and Zhang) with “super-universal”

transitions (superconductor-insulator transition)

• Suggests that there is self-duality at the plateau transitions (I ⟷ V)

(Shimshoni, Sondhi and Shahar)

Composite Fermion Perspective

• We can also use flux attachment to map fermions to composite fermions by attaching an even number of fluxes

• Non-relativistic composite fermions at finite density coupled to a Chern-Simons gauge field with prefactor 1/2𝜋(m-1) (López and Fradkin)

• For the Jain electron filling fractions 𝜈 ± =p/(2np±1), the composite fermions fill p Landau levels of a reduced effective magnetic field with a gap ~1/(2np±1)

• These are the fractions seen in experiment!

• Upon the computation of quantum fluctuations at the quadratic level one obtains a FQHE with 𝜎 xy =𝜈 e ² /h.

• Composite fermions become anyons with fractional statistics 𝜋/(2np±1), and charge e/

(2np±1)

• The hydrodynamic (topological) field theory has the same general form (Wen)

•

Topology and Geometry

• Both theories lead to a unified description of the FQH states as topological fluids

• Hall conductivity and the quantum numbers of the vortices

• On a closed surface of genus g (g=0 for a sphere, 1 for a torus, 2 for a pretzel, etc) the fluid has a topological degeneracy of m ^g

• For of non-abelian FQH states this leads to the concept of a topological qubit

• In addition, the fluid can also sense the geometry (i.e. the curvature) of the

surface through the coupling to the spin connection 𝜔 𝜇 through the topological spin s=m/2 of the vortices

• New “universal” numbers: the Hall viscosity 𝜂 H =sρ 0 /2=mρ 0 /4, the shift (Wen-Zee term), and a gravitational Chern-Simons term (edge thermal conductivity) (c=1)

L = ⇢ ⁰ A ₀ + m

2 ⇢ ₀ ! ₀ m

4⇡ ✏ _µ⌫ b ^µ @ ^⌫ b 1

2⇡ ✏ _µ⌫ A ^µ @ ^⌫ b m

2 1

2⇡ ✏ _µ⌫ ! ^µ @ ^⌫ b 1

48⇡ ✏ _µ⌫ ! ^µ @ ^⌫ !

Compressible States and the (Non) Fermi Liquid

• Non-relativistic composite fermions at fixed density with a Fermi surface (Halperin, Lee, and Read) as p ↦∞ (the FQH gap collapses)

• At fixed electron density the compressible state is reach at a field B c

• In mean field theory it is a Fermi liquid

• Successful to explain several experiments

• Predicts quantum oscillations as a function of B-B c (seen in experiment)

• Compressible states seen at 𝜈=1/2, 1/4, 3/4

• Pairing of composite fermions in the p+ip channel leads to the Moore-Read

non-abelian FQH state (formally at 𝜈=1/2 but works for 𝜈=5/2)

Problems at 𝜈=1/2

• In the high magnetic field limit, at 𝜈=1/2 we expect to see particle-hole symmetry

• The HLR Fermi liquid is manifestly not particle-hole symmetric (Kivelson and DH Lee)

• It also has a large amount of Landau level mixing (largest in the compressible states!)

• The theory also has dynamical Chern-Simons gauge fields non-Fermi liquid!

• The Jain fractions predict that all compressible states are limits of two converging sequences with 𝜈 ± =p/(2np±1): “mirror symmetry”?

• DT Son proposed to to describe the compressible states in terms of relativistic spinor (Dirac) field 𝜓, which is particle-hole symmetric

L = ¯ (i / @ / a) + 1

2⇡ ✏ _µ⌫ A ^µ @ ^⌫ a

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• Relativistic flux attachement

• FQH states: Dirac mass and a chemical potential

• PH symmetric paired state (Jackiw-Rossi ↦ Read-Green p+ip)

• One of the motivations of the web of dualities

Functional Bosonization

Early approach to bosonization of the fermion path integral deep in a massive phase (EF & F. Schaposnik; C. Burgess and F. Quevedo)

Z[A ^ex ] = R

D ⇥ ¯ , ⇤

exp iS _F [ ¯ , , A ^ex ]

To compute current correlators

hj ^{µ 1} (x ₁ )j ^{µ 2} (x ₂ ) · · · i = 1

i A ^ex _{µ 1} (x ₁ ) 1

i A ^ex _{µ 2} (x ₂ ) · · · ln Z[A ^ex ]

Use gauge invariance of the fermion path integral: shift A ^ex to A ^ex + a, where a is a gauge transformation:

Z[A ^ex + a] = Z[A ^ex ].

f _µ⌫ [a] = 0

Z[A ^ex ] = Z

D[a] ^pure Z[A ^ex + a]

hj ^{µ 1} (x ₁ )j ^{µ 2} (x ₂ ) · · · i = h✏ ^{µ 1 ⌫ 1 1 ···} @ _{⌫ 1} b _{1 ···} (x ₁ )✏ ^{µ 2 ⌫ 2 2 ···} @ _{⌫ 2} b _{2 ···} (x ₂ ) · · · i j ^µ (x) ⌘ ✏ ^{µ⌫ ⇢ ···} @ _⌫ b _{⇢ ···} (x) , @ _µ j ^µ = 0

Z[A ^ex ] = Z

D[a, b]Z[a] ⇥ exp ⇣ i 2

Z

d ^D x b _{µ⌫ ···} ✏ ^{µ⌫ ···↵} (f _↵ [a] f _↵ [A ^ex ]) ⌘

The form of the partition function Z[a] depends on the dimension (and regularization)

• This procedure is meaningful only if the effective action of the gauge field is local

• This works in 1+1 dimensions for massless relativistic fermions

• For D>1+1 it works only as en effective action for low energy degrees of freedom if the theory is massive

• For general dimension Z[a] can be computed only in the massive theory.

• The effective action is an expansion in 1/mass

• This approach does not work in a theory at (or even close to) a fixed point

• This leads to a hydrodynamic description of the massive phase

• For systems with a Fermi surface one obtains the Landau

theory of the Fermi liquid

Example: Polyacetylene

• Fermions in d=1 with a spontaneously broken translation symmetry: broken chiral symmetry (Class AIII)

• It is a half-filled system of spin 1/2 fermions (the π electrons of the carbon atoms) coupled to an optical phonon vibration of the (CH) n chain

• As usual in d=1 we can decompose the electron field into its right and left moving components

✓

R L

◆

! ei ³ ✓ ✓

R L

◆

⇢(x) ! ⇢(x + a) ) ✓ = k ^F a (x) = e ^{ik F x} _R (x) + e ^{ik F x} _L (x)

A uniform displacement of the charge profile is equivalent to a

chiral transformation

Topological invariant

(Goldstone & Wilczek) ⌫ = ✓(+ 1) ✓( 1) 2⇡

Z[A ^ex ] = Z

D[a, b] exp i Z

d ^D x L

!

L = b✏ ^µ⌫ @ _µ (a _⌫ A ^ex _⌫ ) + ✓

2⇡ ✏ ^µ⌫ @ _µ a _⌫ + · · ·

Charge Fractionalization

• At half-filling this system has a Peierls instability

• Spontaneous breaking of translation invariance: CDW on the bonds with wave vector Q=2k F =π

• Gap in the spectrum of fermions

• Effective theory: Dirac (Weyl) fermions with a dynamically generated mass

• This system has soliton excitations which correspond to winding of 𝜃

Charge conjugation (particle-hole) 𝜃=n𝜋 (mod 2𝜋)

D=2+1 Chern Insulator (Class A or D)

• Free fermions with broken time reversal invariance: integer quantum Hall states and the quantum anomalous Hall state

• These states are characterized by a topological invariant, the Chern number Ch ∈ ℤ

• The low energy effective theory is

L = b ^µ ✏ ^µ⌫ @ _⌫ (a A ^ex ) + Ch

4⇡ ✏ ^µ⌫ a _µ @ _⌫ a .

• where we neglected terms in higher derivatives, e.g. a Maxwell term

• The first term is the BF Lagrangian

• The hydrodynamic field b 𝜇 couples to flux tubes

• The statistical gauge field a 𝜇 couples to quasiparticle worldlines

• Quantized Hall conductivity _xy = Ch ^e _h ²

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3D Topological Insulator (Class AIII and DIII)

• Example: massive relativistic fermions with a conserved U(1) charge.

• This system has a topological invariant: the winding number

L = b ^µ⌫ ✏ ^{µ⌫ ⇢} @ (a _⇢ A ^ex _⇢ ) + ✓

8⇡ ² ✏ ^{µ⌫ ⇢} @ _µ a _⌫ @ a _⇢ 1

4⇡ ² g ² @ _µ a _⌫ @ ^µ a ^⌫ + · · ·

If time-reversal (particle-hole) is imposed, the topological class is ℤ ² with 𝜃=𝜈𝜋 (mod 2𝜋)

• The bulk gapped (massive) fermionic excitations are represented by their worldlines j μ which are minimally coupled to the gauge field a ^μ

• Flux tubes of a μ are coupled minimally to the curl of b μν

• The effective action for the external gauge field has an axion term

• (Qi, Hughes, Zhang, 2009)