supersymmetric mirror duality and half-ﬁlled … mirror duality and half-ﬁlled landau level...

Supersymmetric Mirror Duality and Half-filled

Landau level

Huajia Wang University of Illinois Urbana Champaign

S. Kachru, M Mulligan, G Torroba and H. Wang Phys.Rev. B92 (2015) 235105

Introduction/Motivation

Planar electrons moving in transverse magnetic field

B

ne

⌫ =ne

B/2⇡

⌫ 2 Integer IQHE: understood via single-particle dynamics:

⌫ 2 Fractions FQHE: emergent collective phenomena

Exhibits a gap, with quantized hall conductivities when

:

E.g. Laughlin states, hierarchical extensions, etc


Composite fermion theory (Jain 89; Halperin, Lee, Read, 92):

electron composite fermion (CF)e

flux4⇡

�B = B � 4⇡neWhen not exact half-filled, CF feels an reduced magnetic field

Jain’s sequence corresponds to IQHE for CF at ⌫ =p

2p+ 1⌫ = p

At exact half-filling, CF is feeling zero effective magnetic field, forms a Fermi surface

Pairing instabilities of CF fermi surface: Moore-Read Pfaffian states

Be↵ = 0

At Half-filling, a metallica state (R. Willettt, et al, 1993):

(at 5/2)⌫ =


Issue: particle/hole (PH) symmetry

approximate symmetry at half-fillings, exact in the limit

PH pairs of Jain states have different CF description

At Half-filling, a metallica state (R. Willettt, et al, 1993):

electron composite fermion (CF)e

flux4⇡ Be↵ = 0

!c � Eint

P

H{LLL ⇠

p+ 1

2p+ 1

p

2p+ 1

⌫

1/2

⌫p+ 1

�p

Composite fermion theory (Jain 89; Halperin, Lee, Read, 92):


New proposal: duality (Son, 14)

B =1

2✏ij@iAjbackground magnetic field

Lel = i el�µ (@µ + iAµ) el

LCF = i cf�µ (@µ + 2iaµ) cf +

1

2⇡✏µ⌫�Aµ@⌫a�

density of composite fermion:

electron density

⇢cf = h cf�0 cfi =

B

4⇡

⇢el = � b

4⇡2⌫el = � 1

2⌫cf

CF dirac fermion, map to itself underP-H symmetry

B.F. coupling (EM coupled to vortices)

no Chern-simons term for emergent gauge field

Variants of this duality has been proved/studied in the context of (3+1D) bulk-(2+1 D) boundary correspondence (C. Wang, T. Senthil; M. Metlitski, A. Vishawanath; M. Metlitski, 2016), but it would also be nice if one can show this duality in a dynamical way that is intrinsically 2+1 D.


There exists a dynamical duality that is similar in nature, called “mirror duality”, which is supersymmetric. The goal of this talk is to introduce this duality, and demonstrate that despite being supersymmetric, the response to external magnetic field is dominated by the “fermionic” part in the low energy dynamics analogous to that of Son’s duality, therefore making it a useful tool to study dirac fermions near half-filling.

Son’s duality (conjectured)

B

el cf

Mirror duality (dynamical)

B

+ �

aµ+ + aµ++

+ �

Introducing Mirror Duality

Mirror Duality: (K. Intriligator and N. Seiberg 96’)

N = 4

N = 4 SUSY free multiplet (Theory A)

U(1)

RG flow

SUSY gauge theory with one flavor of charged multiplets (Theory B)

+ SUSY partners…L = � 1

4g2f2µ⌫ + i �µ (@µ + iaµ)

L = i �µ@µ + SUSY partners…

In anticipation to connect with Son’s duality, we identify the theory with dynamicalgauge field (Theory B) as the composite fermion theory; and the free theory (Theory A) as the one containing physical electrons.

N = 4 hypermultiplet Q : = {(u+, +), (u�, �)}

Introducing Mirror DualityWhat are the “ ( ) SUSY partners” ? N = 4

Theory BN = 4 vector multiplet V : = {aµ, ~�,�ia}aµ

L = � 1

4g2f2µ⌫ + i �µ (@µ + iaµ)

SUSY extension

LB(V,Q) = L(V) + L(Q,V)

L(Q,V) = |Dua|2 + i i /D i � |�ij |2|ua|2 � �ij i j

+p2�i�iau

a i + h.c.�� g2

2(uaub)2

L(V) = 1

g2

✓�1

4f2µ⌫ +

1

2

⇣@~�

⌘2+ i�/@�

◆Theory B

SUSY: doubling of degrees of freedoms boson fermion

+ SUSY partners + SUSY couplings compatible with i aµ�µ

Introducing Mirror DualityWhat are the “ ( ) SUSY partners” ? N = 4

Theory A el N = 4 hypermultiplet: {(v+, +), (v�, �)}

L = el�µ@µ el

SUSY extension

LA =X

±

�|@v±|2 + i ± /@ ±� Theory A

An important component of Son’s duality is the way external electromagnetic sources are coupled across the duality. To connect it with mirror duality, we need to introduce the physical electromagnetic for the mirror pair. U(1)em

SUSY: doubling of degrees of freedoms boson fermion

There are various global symmetries on both sides of the duality. The power of mirror duality allows us to identify these global symmetries across.


(Theory B) topological symmetry : J top

µ =1

4⇡✏µ⌫⇢f

⌫⇢

The topological symmetry acts in Theory B by shifting the “dual photon”:

fµ⌫ = ✏µ⌫⇢@⇢�defined by , therefore only the (non-perturbative) dynamical vortices

� ! � + c

e2⇡ig �“ ” are charged under the topological symmetry.

(Theory A) phase rotation :

U(1)J

U(1)A (v±, ±) ! e±i✓(v±, ±)

In particular, the charges are carried by vortices in one theory, but by fundamental particles in the other. This demonstrates an important aspect of mirror duality: particle/vortex duality, augmented by super-partners.

Identified by mirror duality

Introducing Mirror DualityWe therefore identify the physical U(1)em = U(1)J = U(1)A

Coupling to external source :Aµ

Theory BLB(V,Q, Aµ) = L(V) + L(Q,V) + 1

4⇡✏µ⌫⇢fµ⌫A⇢

LA(Aµ) =X

±{|(@µ ± iAµ)v±|2 + ±�

µ (i@µ ⌥Aµ) ±} Theory A

In conclusion, with the appropriate identification of , mirror duality becomes a SUSY variation of Son’s conjectured duality.

U(1)em

Reminder (Son’s conjecture):

Lel = i el�µ (@µ + iAµ) el LCF = i cf�

µ (@µ + 2iaµ) cf +1

2⇡✏µ⌫�Aµ@⌫a�

A very quick glimpse of why mirror duality works (S. Sachdev, X. Yin 2008):

H(�) = 1 +g2

4⇡|~�|~r⇥ ~!(�) = ~rH(�)


Start from theory B (gauge theory), one can integrate out the charged matter to obtain the effective action for the gauge fields (vector multiplet). Due to the largenumber of SUSY present, the effective action is one-loop exact, and is described explicitly by a non-linear sigma model on the coulomb branch:

low energy regime corresponds to the limit , in which we can find an explicit change of variables that brings the effective action into a free theory form

Le↵ =

1

g2

H(�)

⇣@µ~�

⌘2+H�1

(�)

✓@µ� +

1

2⇡~!(�) · @µ~�

◆2!

+ fermionic part

g ! 1

~� = |~�|(cos ✓, sin ✓ cos', sin ✓ sin')

vi =

✓v+v⇤�

◆=

s|~�|2⇡

e2⇡i�/g2

✓cos ✓/2

ei' sin ✓/2

◆

a =

✓ +

⇤�

◆=

1p2|~�|

�aivi ,

Theory A!


Change of variables:

Le↵ =

1

g2

H(�)

⇣@µ~�

⌘2+H�1

(�)

✓@µ� +

1

2⇡~!(�) · @µ~�

◆2!

+ fermionic part

Le↵ =X

i

|@µvi|2 +X

a

a�µ@µ a +O(|~�|/g2)

change of variable

g ! 1

A very quick glimpse of why mirror duality works (S. Sachdev, X. Yin 2008):

~� = |~�|(cos ✓, sin ✓ cos', sin ✓ sin')

vi =

✓v+v⇤�

◆=

s|~�|2⇡

e2⇡i�/g2

✓cos ✓/2

ei' sin ✓/2

◆

a =

✓ +

⇤�

◆=

1p2|~�|

�aivi ,

Theory A!


Change of variables:

Le↵ =

1

g2

H(�)

⇣@µ~�

⌘2+H�1

(�)

✓@µ� +

1

2⇡~!(�) · @µ~�

◆2!

+ fermionic part

Le↵ =X

i

|@µvi|2 +X

a

a�µ@µ a +O(|~�|/g2)

change of variable

g ! 1

Theory A Theory B

v± =vortices

{~�}

± =vortices�

e2⇡i�/g2

e2⇡i�/g2 {~�}

A very quick glimpse of why mirror duality works

Summary of Mirror duality

So far, we have introduced the mirror duality, and shown that by identifying the topological on the gauge theory side as the physical , mirror duality becomes a SUSY version of Son’s conjecture.

U(1)J U(1)em

Mirror duality can be verified by a change of variables that is explicitly particle/vortex duality, dressed by SUSY-partners.

Comparison:

CF theory U(1)em U(1)gauge

cf 0 2

2⇡ vortex 1 0

el theory U(1)em U(1)gauge

el 1 0

theory A U(1)em U(1)gauge

+, v+ 1 0

�, v� -1 0

theory B U(1)em U(1)gauge

+, u+ 0 1

�, u� 0 -1

2⇡ vortex 1 0

~�, � 0 0

Son’s conjecture Mirror dualityv± = {2⇡ vortex, ~�}

± = {2⇡ vortex, ~�, �}

Although we have established the formal similarity between Son’s conjecture and mirror duality, due to the complicated interplay among the SUSY partners, the low energy dynamics upon deforming with external EM source may be completely different from Son’s result (i.e. Half-filling —> CF fermi-surface).

Breaking SUSY — duality at half-filling

The goal of the this section is to argue that by deforming mirror duality with external magnetic field, the bosonic SUSY partners decouple at low energies on both sides, the effective descriptions are therefore driven by the fermionic responses. We derive the low energy theories on both sides, by identifying them we therefore provide a dynamical realization of the composite fermi-surface(s) picture for half-filled Dirac fermions.

Concern: does the deformation, which breaks SUSY, invalidate the duality, which relies on SUSY?

The SUSY duality has an intrinsic scale set by the dimensionful bare gaugecoupling . By turning on deformations of scale , SUSY-breaking corrections are controlled by .

gO(M/g2)


M

Theory A = Free theory + O(E/g2)



g MO(M/g2)


Theory A = Free theory + O(E/g2)E/g2 ! 0



g MO(M/g2)



SUSY-breaking sources

Theory A = Free theory +O(M/g2)

⇠ M

In particular, here we shall require that the background B field be weak:

pB ⌧ g2



g MO(M/g2)



SUSY-breaking sources

Theory A = Free theory +O(M/g2)

⇠ M

Theory A: (free) charged Dirac fermions + relativistic bosons in constant background magnetic field B

spectrum: relativistic Landau levels

Efermion

n = ±p2nB

Eboson

n = ±p(2n+ 1)B

Project down to low energies: , scalars are gapped out.

Theory A: two copies of LLL Dirac fermions at neutrality

2⇥ Half-filled Landau Levels

E ⌧pB ⌧ g2


0

pB

�pB

�p2B

p2B

(v+, v�) ( +, �)

}E

Theory B:

Project down to low energies: , the kinetic term for the vector multiplet is suppressed by a factor of .

g ! 1


1/g2

LB = L(V) + L(Q,V)� 1

2⇡a0B

L(V)

+p2�i�iau

a i + h.c.�� g2

2

�uaub

�2 � 1

4⇡a0B

Solving the system:

Charged scalars : quartic potential

hu+i = hu�i = 0

a0 Eqn of Motion:

Total fermi surface , shared between ±

h †+ +i � h †

� �i = � 1

2⇡B

1

2⇡B

Theory B:


LB = L(V) + L(Q,V)� 1

2⇡a0B

LB = |Dua|2 + i i�µ@µ i � |~�|2|ua|2 � �ij

i j

How do the SUSY field content respond to the source ?� 1

2⇡a0B

(u+, u�) V (u) =g2

2(uaub)2

g ! 1 V (u) as , turns into a hard constraint:

How is the total fermi surface distributed between ? ±

! h †+ +i = �h †

� �i


Theory B: h †+ +i � h †

� �i = � 1

2⇡B

There are other global symmetries that we can identify across the duality:

SU(2) rotation of the doublet( +, ⇤�)Theory B:

SU(2) rotation of the doubletTheory A: (v+, v⇤�)

Mirror duality (v+, v⇤�) massive

due to laudau level.

SU(2) is not spontaneously broken

SU(2) rotation symmetry should be respected by the two fermi-surfaces in Theory B:


Theory B: h †+ +i = �h †

� �i = � 1

4⇡B

Therefore, we deduced that the response to external magnetic field is the formation of two equal Fermi-surfaces of the Dirac fermions , with Fermi momenta . In theory A, we projected out the scalars by going down to energies , i.e. the corresponding regime in theory B is the Fermi-liquid regime.

B

kF ⇠pB (v+, v�)

E ⌧pB ⇠ kF

( +, ⇤�)

2 ⇥

( +, ⇤�)

+{aµ, ~�,�}+ {u+, u�} in the UV

E ⌧pB ⇠ kF

?

What survives in the low energy sector that couple to the Fermi-surfaces?


Theory B: low energy effective theory near Fermi surfaces: E ⌧ kF

2 ⇥( +,

⇤�)

+{aµ, ~�,�}+ {u+, u�} in the UV

E ⌧pB ⇠ kF

?

��ij i j +p2�i�iau

a i + h.c.�� g2

2

�uaub

�2LB = |Dua|2 + i �µ@µ

j � |~�|2|~u|2



2 ⇥( +,

⇤�)

+{aµ, ~�,�}+ {u+, u�} in the UV

E ⌧pB ⇠ kF

?


a i + h.c.�� g2

2

�uaub

�2LB = |Dua|2 + i �µ@µ

j � |~�|2|~u|2

m2� ⇠ m2

u ⇠ g2kFPositive quantum correction to scalar masses:

{~�, u±} : without fermi-surfaces, quantum corrections to scalar masses cancel out between fermion loops and boson loops by supersymmetry; with the fermi surfaces, there is a deficit of (negative) fermion loop contribution due to suppression of filled states .

( +, ⇤�)

|~k| < kF

�M2�,u = = 0 �M2

�,u = > 0



2 ⇥( +,

⇤�)

+{aµ, ~�,�}+ {u+, u�} in the UV

E ⌧pB ⇠ kF

?


a i + h.c.�� g2

2

�uaub

�2LB = |Dua|2 + i �µ@µ

j � |~�|2|~u|2

�ia : S� = ip2

Z

p,k�ia(�k)ua(�p) i(p+ k) + h.c.



2 ⇥( +,

⇤�)

+{aµ, ~�,�}+ {u+, u�} in the UV

E ⌧pB ⇠ kF


a i + h.c.�� g2

2

�uaub

�2LB = |Dua|2 + i �µ@µ

j � |~�|2|~u|2

�ia : S� = ip2

Z

p,k�ia(�k)ua(�p) i(p+ k) + h.c.

, integrate out

⇠ 1

g2kF��

Gaugino does not get massive, butit decouples from the Fermi surfaces, and becomes a “spectator” in the low energy sector

m2u ⇠ g2kF

{aµ,�}

In conclusion, for :E ⌧pB ⌧ g2

2⇥ Half-filled Landau Levels under Theory A:

Theory B: composite fermi surface2⇥

Mirror duality

+

B


decoupled gaugino+ �

kF ⇠pB

B

+ �

+ �

�

aµemergent gauge field

aµ

Conclusions/future directions

To summarize, we have shown that:

Mirror duality, with the right identification of , is a supersymmetric variant of Son’s duality

Despite the SUSY spectrum, the response of the mirror pair to externalmagnetic field is dominated by the fermionic part, and we derived a double-copied version of the composite fermi surface — half-filled LL duality.

Effect of coulomb interactions: scaling; no BCS instabilities; beta function predicts fixed points for the BCS couplings,

z = 2

U(1)em

Conclusions/future directions

Some future directions:

Study systematically the IR phase of Theory B (other orders, NFL, etc).

Turn on additional deformations to obtain different limits of the duality.

Study the implications other pairs of mirror symmetric theories.

Thank you!

Turn on electric charges to study the mirror dual of Jain’s sequence states

supersymmetric mirror duality and half-ﬁlled … mirror duality and half-ﬁlled landau level...

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