Duality in Condensed Matter Physics and Quantum Field Theory

Eduardo FradkinDepartment 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

�1

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

!2

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 ℤ2 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

!3

Electromagnetic Duality

• EM duality: E ⟺ B

• electric charge e ⟺ magnetic monopole m

• Dirac quantization em=2𝜋

!4

r · E =⇢, r · B = 0

r⇥B � 1c

@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, J𝜇*=𝜀𝜇𝜈 J𝜈, and the dual of a 2nd rank tensor is a scalar, F𝜇𝜈=𝜀𝜇𝜈 𝜃

• In D=3 the dual of a vector is a 2nd rank tensor, J𝜇*=1/2 𝜀𝜇𝜈𝜆 F𝜈𝜆 (and the dual of a 2nd rank tensor is a vector), etc. Duality exchanges the vector potential A𝜇 with a compactified scalar 𝜃

• In D=4 duality exchanges F𝜇𝜈 ⟺ F𝜇𝜈*=1/2 𝜀𝜇𝜈𝜆ρ F𝜆ρ, 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)

!5

Duality the Maxwell field in D=2+1

• Canonical quantization of the Maxwell field in the gauge A0=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

!6

r · E = 0 ) Ei = ✏ij@j✓<latexit sha1_base64="uatnUDCA1vxO+5fOWQXSQGU9Hv0=">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</latexit>

[Ei(x), Aj(y)] = i�ij�(x� y)<latexit sha1_base64="U57ramoO4TXu51dEKlO5I0249zM=">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</latexit>

[Ei(x), B(y)] = i✏ij@j�(x� y)<latexit sha1_base64="7qJ30+ARGjYVOcVHJV2V1EzDBvc=">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</latexit>

[✓(x), B(y)] = i�(x� y)) ✓ ⌘ ⇧<latexit sha1_base64="JW63Lvc5io1O1AAreF8e8+lxhYg=">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</latexit>

H =12

�E2 + B2

�

=12

�(r✓)2 + ⇧2

�<latexit sha1_base64="lv4+YHWu6c+2SmQmaGs3nkt433I=">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</latexit>

r · E = 2⇡n�(x)) �✓� =I

�dxi@i✓ = 2⇡n

<latexit sha1_base64="sj51cBGqP/VdbVJKJ0mZprBnqrI=">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</latexit>

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 ℤ2 gauge theory ⟺ Ising model

• high T loops and 3D surfaces of domains

• 3D: order ⟺ confinement; disorder ⟺ deconfinement

!7

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sha1_base64="HTfC/67isQZy1Vj392LsZMZcovY=">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</latexit><latexit sha1_base64="uBu8Qp+NreH5heyAdS5HrnUxTTk=">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</latexit>

• 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) τ3 and τ1 defined on the dual lattice

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

• Order and disorder• Disordered phase is a kink

condensate

!8

n n + 1n − 1

n n + 1n − 1

H = �X

n

�1(n)� �

X

n

�3(n)�3(n)<latexit sha1_base64="mB6F3fMVV+eUBrGXwpvRjJA7ZUk=">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</latexit>

⌧3(n) =Y

jn

�1(j)

⌧1(n) =�3(n)�3(n + 1)<latexit sha1_base64="SDT5+c2+BfEiR9Aco59PuTxyV6s=">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</latexit>

⌧3(n� 1)⌧3(n) = �1(n)<latexit sha1_base64="lvtScLWp6jiJyt3JSiI4YKHX7qM=">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</latexit>

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

�3(n)2 = �1(n)2 = 1) ⌧3(n)2 = ⌧1(n)2 = 1<latexit sha1_base64="wPWjJYmsO4v5aUTYi7KVnvh9Jxc=">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</latexit>

H = �X

n

⌧3(n)⌧3(n + 1)� �

X

n

⌧1(n)<latexit sha1_base64="F/v3pRwEUuS1sDW+u3ScXG/ABX4=">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</latexit>

What happens in 2+1 dimensions?• The dual of the 2d quantum Ising

model is the 2d ℤ2 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 ℤ2 magnetic charge (“monopole”)

!9

H = �X

r

�1(r)� �

X

r,j=1,2

�3(r)�3(r + ej)<latexit sha1_base64="s8xztKXew06Sv7m6nu1reGVO2zE=">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</latexit>

H = ��

X

r,j

⌧1(r, j)�X

r

⌧3(r, 1)⌧3(r + e1, 2)⌧3(r, 2)⌧3(r1 + e2)<latexit sha1_base64="Odf+TuV7KJhXc8zK9j1lnxyA7gM=">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</latexit>

Gauss Law : ⌧1(r, 1)⌧1(r � e1, 1)⌧1(r, 2)⌧1(r � e2, 2) = 1<latexit sha1_base64="MSOMrsyQs7FOEgz2pKg1BUoBh4w=">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</latexit>

�1(r) = ⌧3(r, 1)⌧3(r + e1, 2)⌧3(r, 2)⌧3(r + e2, 1)<latexit sha1_base64="SFLgbWJiQHPADY0qqqEFerwcbtQ=">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</latexit>

⌧1(r, 1) = �3(r)�3(r + e2)<latexit sha1_base64="UBLp+w3UOjHQZt14oACAgB3U188=">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</latexit>

Bosonization in 1+1 dimensions as duality• 1d free fermions at low energies are

equivalent to a free massless Dirac field, 𝜓(x)=(𝜓R(x), 𝜓L(x))

• Conserved current j𝜇

• 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 j𝜇5=𝜀𝜇𝜈 j𝜈 is not conserved (axial anomaly)

!10

H = �i †R@x R + i †

L@x L<latexit sha1_base64="bOuSQbq0BYJSQBkgKFumSsReJhs=">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</latexit>

[j0(x), j1(y)] = � i

⇡�0(x� y)

<latexit sha1_base64="/MlVccF1mjoi0v0yEpqaF45YEgU=">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</latexit>

H =12⇧2 +

12(@x�)2

<latexit sha1_base64="h1bIl3MlaLQTXLcnBrKciHCt+LE=">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</latexit>

Q =Z

j0(x)dx =1p⇡

��<latexit sha1_base64="GeeOcu6QGQlNCQrz68Z0L6Ab2Ms=">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</latexit>

j0 = : †R R : + : †

L L :

j1 = : †R R : � : †

L L :<latexit sha1_base64="VwVMhkPh+7PbI4pZxLq9P0VpHoI=">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</latexit>

j0 ⌘1p⇡

@x�, j1 ⌘ �1p⇡

⇧<latexit sha1_base64="Be/mwHhmssyHxVLEnbyzVAXDA9w=">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</latexit>

jµ =1p⇡

✏µ⌫@⌫�, @µjµ = 0<latexit sha1_base64="3njB6WAiLqRkMb9O8IaGB7PWBnY=">AAACXnicdVDLSgMxFE3HV62vqhvBTbAILqTM1PdCKbhxWcGqaGrJZDKdaGYm5iGUYf7N33Dnyp3oJ5hpK1jRC4Gbc07OzT2+4Exp130pOROTU9Mz5dnK3PzC4lJ1eeVSpUYS2iYpT+W1jxXlLKFtzTSn10JSHPucXvkPpwV/9USlYmlyofuCdmLcS1jICNYW6lZv7rsoNscolJhkXp4h9Sh1hgTLc4ioUIxbVWYlECUmRwJLzTC/sxeIRMS2IXo0OIDfRGEG7+8KS7dbrbl1d1DwR7Pnekf7HvRGSA2MqtWtvqMgJSamiSYcK3XruUJ3ssKZcJpXkFFUYPKAezQbbJ7DTQsFMEylPYmGA3RMh2Ol+rFvlTHWkfrNFeCfnB+Pzbs1OjzsZCwRRtOEDAeHhkOdwiJWGDBJieZ922Aimf0xJBG2oWob/pizsqtFNMgrNp/vEOD/zWWj7u3UG+e7tebJKKkyWAcbYAt44AA0wRlogTYg4Bm8gQ/wWXp1pp0FZ2kodUqjN6tgrJy1L6l7uhw=</latexit>

@µj5µ =

e

2⇡✏µ⌫Fµ⌫ , @2� =

ep⇡

F ⇤

<latexit sha1_base64="fHqo1kQSAcMD9pLDqf9TKG3q6U8=">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</latexit>

Duality in the Classical 3D XY Model

!11

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

ZXY =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�)

<latexit sha1_base64="YnsXtjKC+8D3d8/GTopNl5uQEAw=">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</latexit>

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

ZXY /X

`µ(r)2Z

Y

r

�(�µ`µ(r)) exp(�X

r,µ

`µ(r)2

2�)

=X

sµ(r)2Z

Y

r

�(�µsµ) exp(� 12�

X

r,µ

(✏µ⌫��⌫s�)2)

=X

mµ(r)

Y

r,µ

d�µ(r) exp(� 12�

X

r,µ

(✏µ⌫��⌫��)2 + i2⇡X

r,µ

mµ(r)�µ(r))

=X

mµ(r)

exp(�2⇡2�X

mµ(r)Gµ⌫(r � r0)m⌫(r0))<latexit sha1_base64="bGLc7b5FqaoOzq4XA2djKarnvhs=">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</latexit>

where Gµ⌫(r � r0) = h�µ(r)�⌫(r0)i<latexit sha1_base64="gpwXo+f4A/eRqsEpIC3P4YtzX1k=">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</latexit>

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

!12

Z(�, e) ' Z

✓e2

4⇡,

12�

◆

<latexit sha1_base64="uPHF/biTGwwu0Q147UArxngrvBk=">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</latexit>

|(@µ + iAµ)�|2 + m2|�|2 + �|�|4 $ |(@µ + iaµ)�|2 �m2|�|2 + �|�|4 +12⇡

✏µ⌫�bµ@⌫A�<latexit sha1_base64="VxlVhHCg2QTPbstQDBZJmFKvvAQ=">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</latexit>

jµ $12⇡

✏µ⌫�@⌫a�<latexit sha1_base64="Gk6naDmxMMr5eHr9KJVwtHoqtyM=">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</latexit>

The Fractional Quantum Hall Effect• Two-dimensional system of Ne electrons in Landau levels created by a large external uniform

magnetic field with N𝛷 fluxes

• Filling fraction: 𝜈= Ne/N𝛷

• Quantized Hall conductivity 𝜎xy=p/(2np±1) e2/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

!13

(z1, . . . , zN ) =NY

i<j=1

(zi � zj)m exp(� 14`20

NX

i=1

|zi|2)<latexit sha1_base64="5aYM8F1B+OI1mvOGD+s8fKFBrtA=">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</latexit>

L =m

4⇡✏µ⌫�aµ@⌫a� � jµaµ ) j0 =

m

2⇡✏ij@iaj and [ai(x), aj(y)] = i

2⇡

m✏ij�(x� y)

<latexit sha1_base64="/Wg1zF4Ns4t6hWFfc+ujRbScGmw=">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</latexit>

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

!14

SB =Z

d3z

⇢�⇤(z)[iD0 + µ]�(z) +

~2

2M|D�(z)|2 +

14⇡m

✏µ⌫�aµ@⌫a�

�

� 12

Zd3z

Zd3z0 (|�(z)|2 � ⇢0)V (|z � z0|)(|�(z0)|2 � ⇢0)

<latexit sha1_base64="n/m/13Txfls1TRuY72O/88wpUg8=">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</latexit>

• FQH plateau: composite bosons condense

�(x) =p

⇢(x)ei!(x)<latexit sha1_base64="UKB1B/Yg7M9m0zXSqUdU/o8dg90=">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</latexit>

12⇡m

✏ij@iaj + |�(x)|2 = 0,

Zd3x|�(x)|2 = ⇢0L

2T<latexit sha1_base64="ntKpGcnqwPBev2U2MUzXlVxpwOM=">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</latexit>

Dµ = @µ + i(Aµ + aµ)<latexit sha1_base64="hFHtoAWVwYby4Hur6l4T+s4iujU=">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</latexit>

haii + Ai = 0<latexit sha1_base64="6iIVdMe8z5vcJWxqMpHSYGrmRps=">AAACHXicdVDLSgMxFM3UV62vUVciSLAIglBmKr4WSsWNywr2AZ0yZDJpG5rJDElGKKX4Hbpx5Va/wp24FT/CfzAzreCIHggczjnJvTlexKhUlvVh5KamZ2bn8vOFhcWl5RVzda0uw1hgUsMhC0XTQ5IwyklNUcVIMxIEBR4jDa9/kfiNGyIkDfm1GkSkHaAupx2KkdKSa244DPEuIxC5FDoi5XvnLj21XLNolawU8Ac5sOyTQxvaE6VY2bpLcF91zU/HD3EcEK4wQ1K2bCtS7SESimJGRgUnliRCuI+6ZJguPoI7WvJhJxT6cAVTNZNDgZSDwNPJAKme/O0l4p+eF2TmtWLVOW4PKY9iRTgeD+7EDKoQJq1AnwqCFRtogrCgemOIe0ggrHR3mZel/lqP+KOC7ue7BPg/qZdL9n6pfKWLOgNj5MEm2Aa7wAZHoAIuQRXUAAa34BE8gWfjwXgxXo23cTRnTO6sgwyM9y9iU6WE</latexit>

|�|2 = ⇢0 =⌫

2⇡`20<latexit sha1_base64="prdtSGbONXe0Q/n12NIz1rs8Oj0=">AAACLXicdVDLSgMxFM34rFVr1aUIwSLopmRGfIGVghuXFWwVOu2QSTNOaOZBkhHKdL7BT3HlxoV+hQtB3LryH0wfghU9EDicc5Kbe9yYM6kQejWmpmdm5+ZzC/nFpeXCSnF1rSGjRBBaJxGPxLWLJeUspHXFFKfXsaA4cDm9crtnA//qlgrJovBS9WLaCvBNyDxGsNKSU9zt27HP+m2rYgs/clDF9gQmqR0mWWrZMYM25dxBbStziiVURkPAH2QfmccHJjTHSql68ig3ywVUc4qfdiciSUBDRTiWsmmiWLVSLBQjnGZ5O5E0xqSLb2g6XCSD21rqQC8S+oQKDtWJHA6k7AWuTgZY+fK3NxD/9NxgYl4zUd5RK2VhnCgaktFgL+FQRXDQEuwwQYniPU0wEUz/GBIf62aU7nLiZalX82kny+t+vkuA/5OGVTb3ytaFLuoUjJADG2AL7AATHIIqOAc1UAcE3IEH8ASejXvjxXgz3kfRKWN8Zx1MwPj4ApKZquw=</latexit>

⌫ =1m

, `20 =1B

<latexit sha1_base64="bXc2rxZRAhuZsO9FQd1ByjOjRqA=">AAACKXicdVDLSgMxFM3UV62vqks3wVJwIWWmvhdK0Y3LCvYBnTpkMpk2NPMgyQhlmIVrf8B/cOW2foW4Ubeuu/APTB+iFT0QOJxzkpt77JBRIXX9VUtNTc/MzqXnMwuLS8sr2dW1qggijkkFByzgdRsJwqhPKpJKRuohJ8izGanZnbOBX7smXNDAv5TdkDQ91PKpSzGSSrKyedOPjk2XIxwbSewl29AkjFn6VfFbPU2sbE4v6EPAH2RPN472DWiMlVxpt3dz9/HcL1vZvukEOPKILzFDQjQMPZTNGHFJMSNJxowECRHuoBaJh0skMK8kB7oBV8eXcKhO5JAnRNezVdJDsi1+ewPxT8/2JuY1IukeNmPqh5EkPh4NdiMGZQAHDUGHcoIl6yqCMKfqxxC3kepCqh4nXhZqtTZxkozq56sE+D+pFgvGTqF4oYo6ASOkwQbYBFvAAAegBM5BGVQABrfgAfTAo3avPWkv2tsomtLGd9bBBLT3TyLxq+8=</latexit>

Fluctuations and the FQHE

!15

• Effective action for quantum fluctuations: a𝜇=⟨a𝜇⟩+𝛿a𝜇; probe field 𝛿Ai

Le↵ =

2(@0! � �a0 � e�A0)

2 � ⇢s

2(r! � �a� e�A)2 +

12⇡m

✏µ⌫��aµ@⌫�a�

<latexit sha1_base64="0kPhe8GQBbZz35cZ6xIGZa+HMRc=">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</latexit>

Le↵[�Aµ] =e2

4⇡m✏µ⌫��Aµ@⌫�A� + . . .) �xy =

e2

2⇡m=

1m

✓e2

h

◆

<latexit sha1_base64="WgXQ8swb23U9MPTIu5qTYny14Tw=">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</latexit>

• Integrating out the fluctuations 𝛿a𝜇

• Vortices: lim|x|!1

�(x) =p

⇢0 ei'(x), �a0 = 0, '(x) = tan�1(y/x)

lim|x|!1

�ai = ± @i' = ±✏ijxj

|x|2

)I

��a · dx = ± 2⇡

<latexit sha1_base64="B7y5QdKMIfwyWAV1qukGXNgP4zw=">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</latexit>

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

Q =e

2⇡m

Z

⌃d2x ✏ij@i�aj =

e

2⇡m

I

@⌃dx · �a = ± e

m<latexit sha1_base64="4+60RhEiiDZSVnf0Sx+9VssoRKs=">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</latexit>

Vortex Partition Function

• Using the identity

!16

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

• jv𝜇 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 b𝜇 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!)

• In FQH insulator one obtains a similar expression for the fermions (with m=1)

Le↵ = �m

4⇡✏µ⌫�bµ@⌫b� +

e

2⇡✏µ⌫��Aµ@⌫b� + jv

µbµ

<latexit sha1_base64="tJ8QwnwFuMK/qhUKeolTnw3aq/8=">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</latexit>

Se↵ [jvµ] =

⇡

m

Zd3x

Zd3y jv

µ(x)✏µ⌫�hx| 1@2

|x0i @y� jv

⌫ (y)<latexit sha1_base64="t3a7oaZuqUoBs2PMgZgjyLyNFbI=">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</latexit>

L =m

4⇡✏µ⌫�bµ@⌫b� +

12⇡

✏µ⌫�bµ@⌫a� ⌘ � 14⇡m

✏µ⌫�aµ@⌫a�

<latexit sha1_base64="gGBk8QJpz1ega3JDLAwp30aRIGw=">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</latexit>

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)

!17

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 𝜎xy=𝜈 e2/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)

• This theory predicts that the FQH fluid becomes compressible for 𝜈=1/2n (as p ↦∞)!

!18

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 mg

• 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 𝜔𝜇 through the topological spin s=m/2 of the vortices

• New “universal” numbers: the Hall viscosity 𝜂H=sρ0/2=mρ0/4, the shift (Wen-Zee term), and a gravitational Chern-Simons term (edge thermal conductivity) (c=1)

!19

L = ⇢0�A0 +m

2⇢0!0 �

m

4⇡✏µ⌫�bµ@⌫b� � 1

2⇡✏µ⌫��Aµ@⌫b�

� m

212⇡

✏µ⌫�!µ@⌫b� � 148⇡

✏µ⌫�!µ@⌫!�

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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 Bc

• In mean field theory it is a Fermi liquid

• Successful to explain several experiments

• Predicts quantum oscillations as a function of B-Bc (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)

!20

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

!21

L = (i/@ � /a) +12⇡✏µ⌫�Aµ@⌫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)

22

Z[Aex] =RD

⇥ ,

⇤exp

�iSF [ , , Aex]

�

To compute current correlators

hjµ1(x1)jµ2(x2) · · · i =1i

�

�Aexµ1

(x1)1i

�

�Aexµ2

(x2)· · · lnZ[Aex]

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

23

Z[Aex + a] = Z[Aex].

fµ⌫ [a] = 0

Z[Aex] =ZD[a]pureZ[Aex + a]

hjµ1(x1)jµ2(x2) · · · i = h✏µ1⌫1�1···@⌫1b�1···(x1)✏µ2⌫2�2···@⌫2b�2···(x2) · · · i

jµ(x) ⌘ ✏µ⌫�⇢···@⌫b�⇢···(x) , @µjµ = 0

Z[Aex] =ZD[a, b]Z[a]⇥ exp

⇣� i

2

ZdDx bµ⌫···✏

µ⌫···↵� (f↵� [a]� f↵� [Aex])⌘

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

24

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

25

✓ R

L

◆! ei�3✓

✓ R

L

◆⇢(x)! ⇢(x + a)) ✓ = kF a

(x) = eikF x R(x) + e�ikF x L(x)

A uniform displacement of the charge profile is equivalent to a chiral transformation

26

Topological invariant(Goldstone & Wilczek) ⌫ =

✓(+1)� ✓(�1)2⇡

Z[Aex] =ZD[a, b] exp

i

ZdDxL

!

L = �b✏µ⌫@µ(a⌫ �Aex⌫ ) +

✓

2⇡✏µ⌫@µa⌫ + · · ·

Charge Fractionalization•At half-filling this system has a Peierls instability•Spontaneous breaking of translation invariance: CDW on the bonds with wave vector Q=2kF=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

27

L = �bµ✏µ⌫�@⌫(a� �Aex� ) +

Ch

4⇡✏µ⌫�aµ@⌫a�.

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

• The first term is the BF Lagrangian

• The hydrodynamic field b𝜇 couples to flux tubes

• The statistical gauge field a𝜇 couples to quasiparticle worldlines

• Quantized Hall conductivity �xy = Ch e2

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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

28

L = �bµ⌫✏µ⌫�⇢@�(a⇢ �Aex⇢ ) +

✓

8⇡2✏µ⌫�⇢@µa⌫@�a⇢ �

14⇡2g2

@µa⌫@µa⌫ + · · ·

If time-reversal (particle-hole) is imposed, the topological class is ℤ2

with 𝜃=𝜈𝜋 (mod 2𝜋)

• The bulk gapped (massive) fermionic excitations are represented by their worldlines jμ which are minimally coupled to the gauge field aμ

• Flux tubes of aμ are coupled minimally to the curl of bμν

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

• (Qi, Hughes, Zhang, 2009)

Web of Dualities in 2+1 DimensionsSeiberg, Senthil, Wang and Witten (2016)

• Recently conjectured dualities between fixed points (relativistic CFTs)

• A new look at particle-vortex duality of theory with a U(1) symmetry (Peskin; Thomas and Stone; Dasgupta and Halperin) (on the r.h.s. there is also a Maxwell term)

!29

|D(A)�|2 �m2|�|2 � �|�|4 ! |D(a)'|2 + m2|'|2 � �|'|4 +12⇡

✏µ⌫�aµ@⌫A�

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• m2<0: l.h.s. in the broken symmetry phase with a Goldstone boson and quantized vortices with long range interactions; r.h.s. in the unbroken phase with a transverse photon and charged particle with long range interactions: compactified scalar ⟺ Maxwell field.

• m2>0: l.h.s. in the unbroken phase with particle loops with short range interactions; r.h.s. in the Higgs phase with massive photons and vortices with short range interactions

• 𝜙⟷𝜑 bound to a monopole of a𝜇 (end of a vortex), and j𝜇 ⟷ 1/2𝜋 𝜀𝜇𝜈𝜆 ∂𝜈 a𝜆• Maps the Wilson-Fisher fixed point (l.h.s.) to the gauged Wilson-Fisher fixed point

(r.h.s)

Bosonization Duality(Conjectured by Seiberg, Senthil, Wang and Witten)

• Maps two different fixed point theories

!30

• The 1/2 quantized Chern-Simons term can be regarded as due to have “fermion doublers” (lattice models) or as shorthand for the 𝜂 invariant in a time-reversal invariant regularization

• Conjecture proven for SU(N)k gauge fields in the ’t Hooft limit: N, k ↦∞, with N/k fixed (Minwalla et al; Aharony et al)

• Dirac fermion operator ⟷ magnetic monopole of the gauge field a𝜇 bound to the complex scalar field 𝜙

• Dirac current:

i /D(A) � 1

8⇡AdA ! |D(a)�|2 � |�|4 + 1

4⇡ada+

1

2⇡adA

jµ $ 12⇡ ✏µ⌫�@⌫a�

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Fermionic Particle-Vortex Duality• Conjectured fermionic particle-vortex duality (“QED3”) (Son, Metlitski-

Vishwanath):

!31

• l.h.s: time reversal invariant free massless Dirac fermion

• r.h.s. charge conjugation invariant gauged Dirac fermion (QED3) (Maxwell terms committed)

• time reversal 𝒯 ⟷ charge conjugation 𝐶 (PH); B ⟷𝜇

• l.h.s. is 𝒯 invariant and the r.h.s. is 𝐶 invariant

• fermion masses will have opposite signs of the duality

• The 1/2 integer Chern-Simons terms can be viewed as coming from regularization (“fermion doublers”)

• Another interpretation is that this theory is at the boundary of a 3+1 dimensional topological insulator which has 𝜃=𝜋

• Duality maps a fermion to a fermion bound to a monopole

i /D(A) +18⇡✏µ⌫�Aµ@⌫A� !

i� /D(a)�+12⇡✏µ⌫�aµ@⌫b� � 2

4⇡✏µ⌫�bµ@⌫b� +

12⇡✏µ⌫�aµ@⌫A� � 1

8⇡✏µ⌫�Aµ@⌫A�

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A way to “derive” the dualities with loop models

• “Derive” this web of dualities using quantum loop models near criticality, but still in the gapped phases.

• These models are related to modular invariant models we originally introduced with Kivelson

• Modular invariance cannot be kept close to the CFT.

• “Fractional spin” breaks modular invariance, and gives rise to Dirac fermions, leading to loop model based “proofs” of the CFT duality web.

!32

Quantum Loop Models and Duality(EF and Kivelson 1996)

• Non-intersecting linked loops [J𝜇] in 3D Euclidean space-time (with no spin) with exact particle-hole symmetry

• flux attachment with fractional statistics 𝜃, long ranged interactions with coupling g, and short-range repulsion (to avoid crossings)

• The imaginary part of the action is given in terms of the loops linking number

!33

L = |Dµ�|2 �m2|�|2 � �|�|4 � 14g2

fµ⌫1p�@2

fµ⌫ +k

4⇡✏µ⌫�aµ@⌫a�

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Field theory picture: 2+1 D complex scalar field coupled to 3+1 D Maxwell field with a 𝜃 term

Z[g, ✓] =X

{Jµ}2Z�(�µJ

µ)e�S[Jµ]

S[Jµ] =g2

2

X

x,y

Jµ(x)Gµ⌫(x� y)J⌫(y) + i✓X

x,y

Jµ(x)Kµ⌫(x� y)J⌫(y)

Gµ⌫(p) =1pp2

✓�µ⌫ � pµp⌫

p2

◆, Kµ⌫(p) = i✏µ⌫�

p�p2

long ranged interactions linking number =𝜃𝛷[J]

Superconducting order parameter field at the boundary of a 3D topological insulator

Self-Duality and Modular Invariance

• The partition functions of loop models regularized without self-linking (fractional spin) have the symmetries

• 𝑆: duality: Z[τ]=Z[-1/τ], and 𝒯: Periodicity: Z[τ]=Z[τ+1]

• 𝑆 and 𝒯 generate the modular group PSL(2,ℤ)

• The partition function is self dual at the fixed points of the modular group

• Two types of PSL(2,ℤ) fixed points: “bosonic” and “fermionic”

• FK showed that the finite modular fixed points are quantum critical points with 𝜎xx ≠ 0 and 𝜎xy = 0

• The predicted conductivities are different in the FK loop models and the relativistic web of dualities

⌧ =✓

⇡+ i

g2

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Modular parameter:

!34

The Role of Fractional Spin

• The linking number of two separate loops 𝑙1 and 𝑙2 is

• Witten: point-split the loops into ribbons so that the writhe is a frame-dependent topological invariant W[𝑙] = SL[𝑙] = integer. Only consistent deep in the topological phase, not as the critical point is approached.

• Polyakov: no-point splitting and W[𝑙] = SL[𝑙] - T[𝑙] (writhe = self-linking - twist)

T[𝑙] is a Berry phase (fractional spin) and e is the tangent vector to the loop. The twist T[𝑙] is not quantized and depends on the metric.

!35

T [`] =12⇡

Z L

0ds

Z 1

0due · @se⇥ @ue

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�[J = `1 + `2] = 2⇥ (Linking number of `1 with `2) +W [`1] +W [`2]

“Writhe.” Associated with self linking. Not necessarily a topological invariant (Hansson, Karlhede, et al)

Fractional Spin: Periodicity Lost, 3D Bosonization Regained

!36

L[J]: length of loop, 𝛷[J]: linking number (including the spin factor)

• T[𝑙] is not quantized. Means Duality 𝑆 remains a symmetry, but periodicity 𝒯 is lost

• Polyakov: fractional spin leads to the (IR) duality between a complex massive scalar with CS at k = 1 and a massive Dirac spinor (with a parity anomaly)

• Loop model representation

Zfermion = det[i/@ �M ] =ZDJ �(@µJµ) e�|m|L[J]�isign(M)⇡�[J]

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For general statistical angle 𝜃 we have the loop model

Z =ZDJ �(@µJµ) e�|m|L[J]+i✓�[J]

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!37

• Can we use this to “derive” the web of dualities? Yes!

• First step: We introduce background fields to the boson side of Polyakov’s duality in the unbroken phase

Exact rewriting as loop model coupled to gauge fields

Fractional Spin: Periodicity Lost, 3D Bosonization Regained

LB = |D[a]�|2 �m20|�|2 � |�|4 + 1

4⇡ ada + 12⇡ adA

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S[J, a,A] =Z

d3x

J(a�A) +

14⇡

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Integrating-out a results in a term involving the linking number and the spin factor

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with M<0

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Loop model representation

Sfermion[J,A;M < 0] =Z

d3x

✓JA� 1

8⇡AdA

◆� ⇡�[J ]

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The bosonization identity in the phase with broken time reversal, M>0, is obtained by a particle vortex duality in the bosonic theory

Loop Models: Tools for Deriving Dualities

!39

1. Start with a proposed duality and write down boson loop models for each theory using Polyakov’s duality.

2. Use path integral manipulations to equate the two loop model partition functions.

3. Match both sides of the critical point using bosonic particle-vortex duality. Relates superfluid of particles to insulator of vortices.

4. The dualities are IR identities

5. In the bosonic theories the short-distance repulsion between loops become the 𝜙4 coupling, which in the massless limit flow in the IR into the WF fixed point

Example: Fermion particle-vortex duality

!40

• Case for opposite mass signs (QH phase) follows from the same logic

• Current mapping also natural upon integrating out b:

Use loop models to derive the duality between free Dirac fermion and QED3 with (quantized) Chern-Simons terms

Zd3xJµA

µ + ⇡�[J ] �⇡�[J ] +

Zd3x

Jµa

µ � 1

2⇡adb+

2

4⇡bdb� 1

2⇡bdA

�

Integrate out a, b

�µ ! 1

4⇡✏µ⌫�@

⌫a�

i /D[A] � 18⇡ AdA$ i /D[a] + 1

8⇡ ada� 12⇡ adb + 2

4⇡ bdb� 12⇡ bdA

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ZF[A;M < 0] = ZQED3[A;M 0 > 0], ZF[A;M > 0] = ZQED3

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Compressible “FQH” states and Duality

• The Jain sequences of FQH states 𝜈(p, n)=p/(2np±1) converge to 𝜈=1/(2n) where the FQH gap vanishes ↦ Halperin, Lee, Read theory of a composite Fermi liquid

• This theory had great successes. It also has problems: in the simplest case, n=1, 𝜈↦ 1/2 and PH symmetry is expected (for large B).

• HLR is not compatible with PH (DH Lee)

• The “Fermi liquid” is a “Non-Fermi liquid”

• Son proposed a relativistic version of HLR which satisfies PH

• At finite 𝜇 (Fermi surface!) this is still a “non-Fermi liquid”

• What about the 𝜈=1/2n compressible states where PH should not hold?

!41

𝜈=1/2n Compressible States

• Compressible states with 𝜈=1/2n are predicted by the Jain sequences

• They are seen in experiment

• PH does not hold for general n

• Reflection symmetry of the I-V curves at plateau transitions

• Interpreted as evidence of particle-vortex duality (Shahar, Shimshoni, Sondhi)

• For 𝜈=1/2 PH symmetry relates ρxx to 𝜎xx and 𝜈⟷1-𝜈

!42

Symmetries at 1/2n Compressible States

• The same reflection symmetry is seen at 𝜈=1/4, locus of 𝜈=1/3 ↦ 0 transition (where 𝜈↛ 1-𝜈), with ρxy=-3e2/h

• This is not PH symmetry!

• For 𝜈=1/2n the symmetry is between the Jain states at 𝜈=p/(2np+1) and 𝜈’=(1+1)/(2n(1+p)-1), both converging to 1/2n

• For reflection symmetry to hold the HLR composite fermions must have 𝜎xy=-e2/2h

• Flux attachment breaks PH and reflection explicitly

• Same problems in Son’s theory which needs to be modified to treat 𝜈 and 𝜈’ equitably

!43

Reflection symmetry at 𝜈=1/2n

• A is the external gauge field of strength B, and a is the Chern-Simons field (flux attachment); b=𝜀ij∂iaj

!44

electron filling: ⌫ = 2⇡B h �L⌫=1/2n

�A0i = 1

2n

�1 + b⇤

B

�<latexit sha1_base64="XzxWkHNXedOGGvE7zsFo9kI2l38=">AAACgXicdVFNa9wwEJWdtkk2bbNNjr2ILilJCxvJJV+Ewm5y6aGHBLJJINoaWZZ3RWTZSHJhMYL+yRx67rn/obKdHLa0A0LDe29mNE9JKYWxCP0MwpVnz1+srq33Nl6+er3Zf7N1bYpKMz5hhSz0bUINl0LxiRVW8ttSc5onkt8k9+cNf/OdayMKdWUXJZ/mdKZEJhi1Hor7jqjqM8k0ZXVESuHqMweJpGomOexgknJpKSQ5tXNGZf3VxXVThPcj5dwTPY6RI7qte2yHXe0FvhnP7C7+2IFJ/KEZQbSYze1eL+4P0DA6QCfHCKIhasMnBwifHGKIH5HBaN+93/w2friI+79JWrAq58oySY25w6i005pqK5jkrkcqw0vK7umM1605Du54KIVZof1RFrboko7mxizyxCubFc3fXAP+k0vypXl3lc2Op7VQZWW5Yt3grJLQFrBxHqZCc2blwieUaeFfDNmceles/5+lzsavNuepa/x5MgH+P7mOhvjTMLrEg9H4B2hjDbwF78AuwOAIjMAXcAEmgIFfwXqwFWyHK+FeiMKok4ZBd4NtsBTh6R9+BsZn</latexit>

b⇤ = 0) ⌫ = 12n

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Composite fermion 𝜓 FS set by a0: ⇢ = 12⇡

�12 �

12n

�b⇤ � 1

2⇡B2n

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⌫ = 2⇡ ⇢ b⇤ = 12 + ⌫

1�2n⌫<latexit sha1_base64="fTJ9bl3pCyuEmAiTd2hwyuLdioI=">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</latexit>

⌫ = p + 12 ) ⌫ = p

2np+1<latexit sha1_base64="0yJ+19TnSyRL6fm8qhB37fdV3Es=">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</latexit>

⌫ ! �⌫ , ⌫ = p2np+1 !

1+p2n(1+p)�1

<latexit sha1_base64="3mfvTvunhEh0pMzaF33UYXKRoFQ=">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</latexit>

L1/2n = i /Da � 14⇡

�12 �

12n

�ada + 1

2⇡12nadA + 1

2n14⇡ AdA

<latexit sha1_base64="5cCBn/+9AePt1iuM+8QRO7yPg5k=">AAACsHicdZHNbtQwEMed8NUuX0s5csBiValVxTYJFLYH0G7hwIFDkdh2pWYVTRxn19RxIttBWll+Gc698TQ8BC/Aqc7uQkkFI1ka/ebvGfs/acWZ0kHww/Nv3Lx1+87GZufuvfsPHnYfbZ2ospaEjknJSzlJQVHOBB1rpjmdVJJCkXJ6mp6/a+qnX6lUrBSf9aKi0wJmguWMgHYo6V7EBeg5AW4+2sSE+5Gwb5iJU5A4rhSzOFYc1Jxm5r1NYMmex7kEYkJrXsZVo+A01zt/YGSvBK5bLNlsrncxZLB3xVcX/5K5+gjvtci1MaNslHR7QT86CA4HAQ76wTJcchCEh69CHK5Jb7j9bffp91/iOOn+jLOS1AUVmrhvqLMwqPTUgNSMcGo7ca1oBeQcZtQsvbR426EM56V0R2i8pC0dFEotitQpG+fU9VoD/1lLi9a8s1rng6lhoqo1FWQ1OK851iVuFoUzJinRfOESIJK5F2MyB2eJdutsdV5vyHacP79NwP9PTqJ++KIffQp7w7doFRvoCXqGdlCIXqMh+oCO0RgRb8sbeCPvyI/8iZ/4sJL63vrOY9QK/8sl6JzYhQ==</latexit>

Reflection symmetry and boson self-duality

• First line: fermion-boson duality

• Second line: boson-vortex duality

• relates 𝜈𝜙 to - 1/𝜈𝜑

• 𝜈=1/2n ⟺ 𝜈𝜙 = - 𝜈𝜑=1

• Reflection related filling fractions 𝜈𝜙 (𝜈) = - 𝜈𝜑(𝜈’)

• Reflection symmetry is boson-vortex exchange

• Reflection symmetry at 𝜈=1/2n ⟺ boson self-duality!!45

L1/2n $ |Dg�A�|2 � |�|4 +14⇡

12n� 1

gdg

$ |Dh'|2 � |'|4 � 2n� 14⇡

hdh +12⇡

hdA<latexit sha1_base64="kP9k4FXvGzUVUsd8j6VtdLjXrjg=">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</latexit>

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!47