Chiral Phases in Frustrated 2D Antiferromagnets and Fractional Chern Insulators

Eduardo FradkinDepartment of Physics and Institute for Condensed Matter Theory

University of Illinois, Urbana, Illinois, USA Seminar at the Interdisciplinary Theoretical and Mathematical Sciences (iTHEMS)

Program, RIKEN, Japan, December 21, 2017

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Collaborators: Krishna Kumar, Kai Sun, Hitesh Changlani, Bryan Clark, Raman Sohal, Luiz Santos

Kumar, Sun Fradkin, Phys. Rev. B 90, 174409 (2014); 92, 724433 (2015) Sun, Kumar, Fradkin, Phys. Rev. B 92, 115148 (2015) Kumar, Changlani, Clark, Fradkin, Phys. Rev. B 94, 134410 (2016) Sohal, Santos, Fradkin, arXiv:1707.06118

What is a topological phase?

• Fractional Quantum Hall fluids of 2DEG in high magnetic fields

• Deconfined phases of discrete gauge theories

• Topological phases do not break any symmetries

• Natural representation in terms of gauge theories

• Gapped bulk excitations and gapless edge states (not always)

• Fractionalized bulk excitations: vortices of the incompressible fluid and carry fractional charge (charged fluid) and fractional statistics

• Ground state degeneracy kg (k is an integer that depends on the topological phase) on a closed surface with g handles

• Effective field theory: topological QFT (Chern-Simons, BF, discrete gauge theory)

• Entanglement entropy: SvN=const. ℓ- ln 𝒟+… (with 𝒟2=∑k dk2)

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Outline• We will consider two types of fractionalized phases in 2D

• Chiral phases of interacting hard-core bosons on frustrated lattices (Kagome)

• Magnetization plateaus of 2D Heisenberg kagome antiferromagnets

• Chiral phases with spontaneous time-reversal symmetry

• Fermionic chiral theories: Fractional Chern insulators

• Time reversal is broken explicitly

• Free fermion bands have non-zero Chern numbers (Chern insulator)

• Analogs of the FQH states on lattices

• Novel feature: fractionalized translation symmetry

• Key tool: lattice Chern-Simons and BF theory

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Chern-Simons Gauge Theory

• Chern-Simons gauge theory is topological field theory

• Local constraint (“Gauss Law”) relating local density to local flux

• Canonical commutation relations

• The local constraints must be consistent

• Without a coupling to matter fields it has a vanishing Hamiltonian

L =k

4⇡✏µ⌫�Aµ@⌫A� � jµAµ

j0(x) =k

2πB(x); B = ϵij∂iAj

!Ai(x),Aj(y)

"= i

2π

kϵijδ(x − y)

!B(x),B(y)

"= 0

H = ji(x)Ai(x)

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Lattice Chern-Simons Theory• We will use the “Hamiltonian” formulation in which time remains continuous

• Matter fields are defined on the sites and gauge fields are defined on the links

• Gauge fluxes are defined on plaquettes (sites of the dual lattice)

• Local constraints require a one-to-one correspondence between sites and plaquettes

• This condition is satisfied by many lattices (square, kagome lattices, others), but is violated in many others of interest (triangular, honeycomb, etc)

• Compatibility condition: fluxes on all plaquettes must commute with each other

• The naïve construction on the square lattice (Fradkin 1989) violates the compatibility condition

• Corrected by Eliezer and Semenoff (1992), and generalized recently by Kumar, Sun and Fradkin (2014, 2015)

• Commutation relations are somewhat less local than naïvely expected

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Chern-Simons on the Kagome Lattice• Three inequivalent sites, a, b, and

c, in the unit cell of the kagome lattice

• Three plaquettes: one hexagon and two triangles (oppositely oriented)

• We define six link components of the gauge field, A1 through A6, and three fluxes, Ba, Bb, and Bc

• The gauge-invariant action for the discrete CS theory has the form

• The matrix Kij defines the commutation relations

• For each sublattice site, the vector Ji defines the flux attachment, i.e. the relative displacement of the local flux (at the center of a plaquette) to the sublattice site

SCS = S(1)CS + S(2)

CS

S(1)CS =

!dt

"

x,y

A0(x, t)Ji(x − y)Ai(y, t)

S(2)CS = −1

2

!dt

"

x,y

Ai(x, t)Kij(x − y)∂tAj(y, t)

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Flux Attachment and Commutation Relations

• The matrix Kij is defined such that the commutation relations are “local” (only gauge fields on links sharing a site do not commute)

• Si is a shift operator along the lattice directions ei

• It enforces the condition that all fluxes commute with each other, even on plaquettes sharing a common link

Kij =12

0

BBBBBB@

0 �1 1 �S2 S1 + S�12 �1 + S�1

2

1 0 1� S�11 �S2 � S�1

1 S1 �1�1 S1 � 1 0 1� S2 S1 �1S�1

2 S1 + S�12 S�1

2 � 1 0 S1S�12 S�1

2

�S2 � S�11 �S�1

1 �S�11 �S2S

�11 0 1� S�1

1

1� S2 1 1 �S2 S1 � 1 0

1

CCCCCCA

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Fermions+Chern-Simons Theory• Map hard-core bosons to fermions coupled

to a Chern-Simons gauge field with coupling constant 𝜃=1/(2𝜋)

• Mapping fermions to fermions with 𝜃=1/(4𝜋)

• The number of fermions is the same as the number of bosons/fermions

• This mapping works for 𝜃=1/(2𝜋k) for bosons, where k∈ℤ (positive or negative), and 𝜃=1/(4𝜋k) for fermions

• Fermi field on the sites of the kagome lattice and a Chern-Simons gauge field on the links

• Gauss Law constraint replaces the local fermion density by the flux on the adjoining plaquette (defined on the dual site)

• The action is a bilinear function of the fermions coupled to a gauge field

S = SF (ψ, ψ∗,Aµ) + Sint(Aµ) + θSCS(Aµ)

SF (ψ,ψ∗,Aµ) =!

t

"

x

#ψ∗(x) (iD0 + µ) ψ(x)

− Jxy

2

"

⟨x,x′⟩

$ψ∗(x)eiAj(x)ψ(x′) + h.c

% &

Sint(ψ,ψ∗) = − λJz

!

t

"

⟨x,x′⟩

'12− n(x)

( '12− n(x′)

(

n(x) = θB(x)

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Flux attachment on a Torus• On a closed surface the Chern-Simons coupling must be quantized as n/

(4𝜋), where n∈ℤ

• This is required by invariance under large gauge transformations

• Problem: in our construction 𝜃=1/(2𝜋k)

• To make this consistent one must use a lattice version of a BF term

• The resulting theory now has “statistical” fields on the direct (kagome) lattice and “hydrodynamic” fields on the dual dice lattice

• The action of this theory, CS+BF, now has correctly quantized couplings

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a) b) c)

Spin-1/2 Kagome Quantum Heisenberg Antiferromagnet

• The kagome lattice is highly frustrated

• It has three inequivalent sites (a, b, c) in its unit cell.

• Ising regime (Jz≫Jxy ): mapping to a quantum dimer model on the dual lattice (Cabra et al., 2005)

• This mapping works with and without an external magnetic field

• In this regime the evidence is that it is a ℤ2 topological phase or possibly a gaplesss Dirac theory

a b

c

• Recent DMRG results imply that the ℤ2 topological phase extends to the isotropic point

• The behavior in the XY regime is still unclear

• Magnetization plateaus?10

H =Jxy

4

X

hr,r0i

⇣�1(r)�1(r0) + �2(r)�2(r0)

⌘+

Jz

4

X

hr,r0i

n(r)n(r0) + const.

Chiral Phases in Frustrated Quantum Antiferromagnets

• Chiral spin liquids are topological phases of frustrated quantum antiferromagnets with broken time reversal invariance.

• Bosonic analog of the fractional quantum Hall states of 2D electron fluids.

• Chiral ground states were proposed for the triangular lattice (Laughlin and Kalmeyer, 1988), for the square lattice with J1-J2 interactions (Wen, Wilczek, Zee, 1988), for the triangular and kagome lattices (Yang, Warman, Girvin, 1993), and for the Shastry-Sutherland lattice (Misguich, Jolicoeur, Girvin, 2001).

• Current numerical evidence (DMRG+ exact diagonalization) show a non-collinear ordered state for the triangular lattice, a ℤ2 topological phase for the square lattice with J1-J2 interactions (Jiang, Wang, Balents, 2012), and for the J1-J2 model on the kagome lattice (Zhu, Huse, White, 2011).

• Nature of the ground state is less clear for the “simple” Heisenberg model at the isotropic point and in the XY regime (Sheng et al, 2015).

• Strong evidence for a chiral topological phase for the “chiral Hamiltonian” on the kagome lattice whose Hamiltonian is just the chiral operator for the spins (Bauer et al, 2014).

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Magnetization Plateaus: Mapping to Hard-Core Bosons• Magnetization plateau: gapped state at fixed magnetization M, for a range of magnetic

fields h

• For a spin system the magnetic field breaks time reversal invariance (maps up spins into down spins)

• We will use the standard mapping to hard-core bosons

• A site with an up spin is an empty site, and a site with a down spin is occupied

• 𝜎+(r)≡a+(r) creates a hard-core boson; 𝜎3(r)=1-2n(r); n(r)=a+(r)a(r) is the boson number

• The total magnetization is then M=1-2N, where N is the total number of bosons

• For M=0 the average boson occupancy is 1/2; for M=1/3 is1/3, and 1/6 for M=2/3, etc.

• The Hamiltonian is real (time-reversal for spins maps to particle-hole symmetry for bosons)

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H =Jxy

2

X

hr,r0i

a†(r)a(r0) + h.c. + Jz

X

hr,r0i

n(r)n(r0) + const.

Mapping hard-core bosons to fermions

• The simplest way to investigate phases with broken-time reversal invariance in 2D is to use flux attachment

• In the case of interest here time reversal (in the bosonic sense) is a mirror symmetry for the spins which has to be broken spontaneously

• This is a mapping from hard-core bosons to fermions by attaching an odd number of flux quanta to each boson.

• Flux attachment is a constraint relating the local particle density to the local flux

• Equivalent system: fermions (with the same total particle number) coupled to a lattice abelian Chern-Simons gauge field with a suitable Chern-Simons “level” (Fradkin, 1989)

• Similarity with the Laughlin states and other fractional quantum Hall states (Jain, 1989; López and Fradkin, 1991)

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• The naïvely discretized Chern-Simons theory was applied to frustrated lattices early on (Yang, Warman, Girvin, 1993; Misguich, Jolicoeur, Girvin, 2001) who found chiral spin liquid states.

• The inconsistent commutation relations do not show up in mean-field theory

• However, the problem with the commutation relations appear already at the leading level in quantum fluctuations and the mean field results were questionable

• For the S=1/2 Heisenberg antiferromagnet on the square lattice the correctly discretized CS theory was used by López, Rojo, Fradkin (1995). After some subtle computations it was found to describe a Néel state (as it should!)

• Here we will use it to study the magnetization plateaus of the S=1/2 anisotropic Heisenberg antiferromagnet on the kagome lattice.

• For the most part we will discuss the case of the XY model (Jz=0)

Flux attachment and antiferromagnets

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Semiclassical Theory: Integrating-out the Fermions

• The action is a quadratic in the fermions and we will integrate them out• The effective action is a function S[A𝜇] of the CS gauge fields• We will seek a classical solution with uniform flux• We have a Hofstadter problem of Landau levels for the kagome lattice• Seek solutions with a gap: an integer number of Landau levels are filled• Compute the Chern number C of the occupied Landau sub-bands to

determine the chirality of the state

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Seff(Aµ) = −itr ln[iD0 + µ − h(A)] + Sint(Aµ) + θSCS(Aµ)

h(A) =J

2

!

<x;x′>

"eiAj(x,t)|x, ⟩⟨x′, t| + h.c

#

Magnetization plateaus for the XY model

• Consider the XY model and set Jz=0

• Seek solutions with the same flux in the three plaquettes of the unit cell: Ba=Bb=Bc=𝜙=2𝜋 p/q (with p, q ∈ℤ)

• Since 𝜃=1/(2𝜋) and the total flux in the unit cell is 3𝜙, the average density of all three sites of the unit cell is ⟨n⟩=p/q

• Gapped states for ⟨n⟩=1/3, 2/9, 1/6, corresponding to M=1/3, 5/9, 2/3

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Hofstadter spectrum for the XY model• Bottom solid line: top-most occupied

single-particle state • Vertical jumps at densities1/3, 2/9, 1/6• The Chern numbers of the occupied

bands are C=+1, +2, +1

Chiral magnetization plateaus are similar to FQH states for bosons!

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Seff =SeffCS + Seff

M + . . .

SeffCS =

!d3x

12

(θ + θF ) ϵµνλAµ∂νAλ

SeffM =

!d3x

"12ϵE2 − 1

2χB2

#

θF =C

2π

θ + θF =12π

(1 + C)

spin Hall conductance σxy =θeff

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θeff=

1θ

+1θF

For the 1/3 and 2/3 plateaus σxy =12

12π

For the 5/9 plateau σxy =23

12π

Bosonic Laughlin state

Bosonic Jain state

The excitations are “vortices” with fractional boson charge (spin!) 1/2 and 1/3, and fractional statistics 𝜋/2 and 𝜋/3, and chiral edge states

Exact Diagonalization Study• Checked by exact diagonalization the

existence of a magnetization plateau at M=2/3 and found with good accuracy that it is a Laughlin state for bosons with 𝜎xy=1/2 (1/2𝜋)

• We considered clusters of up to 48 sites on a torus and found it gapped

• To control finite size effects we added a chiral operator for every triangle and studied the changes of the low energy spectrum

• Computed the entanglement spectrum on cylindrical cuts of the torus and used it to compute the modular S matrix of the topological state and found it to be described by a U(1)2 Chern-Simons theory

• Computed the spin Hall conductivity by averaging over twisted boundary conditions on the torus (the Niu-Thouless-Wu formula)

• Verified that the chiral state remains the ground state up to a critical value of Jz~Jxy

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S =1√2

!1 11 −1

", U = ei 2π

24 1

!1 00 i

"

Chern Insulators on the Kagome Lattice(Green, Santos, Chamon 2010)

• Spinless free fermions on a kagome lattice with 𝜙±=±𝜋/2 (for simplicity)

• Time reversal is broken explicitly even if the total flux per unit cell is zero

• Three bands with Chern numbers C=-1, 0, +1

• Analog of the Haldane honeycomb lattice model

• We will fractionally fill the lowest band (with C=1) with density nL and seek a fractionalized Chern insulator state using flux attachment

• We will use Chern-Simons+BF theory to represent flux attachment on a torus

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Results from Mean Field Theory and Semiclassical Corrections

• Similarly to the FQH states in the 2DEG (Lopez and Fradkin, 1991) we seek a gapped state with uniform flux

• At the mean field level the three sublattices generally have different occupancies and a broken point group symmetry

• More complex Hofstadter spectrum with effective bands with Chern numbers C

• Diophantine equation: nL=-3𝜙C/(2𝜋)+r (with r∈ℤ) (𝜙: average flux per unit cell)

• One loop quantum fluctuations: families of fractional Chern insulators (FCI)

• The Hall conductance of the states is 𝜎xy= C/(2kC + 1), and nL= r 𝜎xy/C

• Jain states with r=C and nL= 𝜎xy

• “non-Jain” states with r≠C and nL≠𝜎xy

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Fractional Chern Insulator States

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C=1nL=1/3

C=2nL=2/5

C=3nL=3/7 C=1

nL=2/3

C=-3nL=3/5C=-4

nL=4/7

C=3nL=5/7

• Vertical lines: gapped states• C: effective Chern number• nL: band filling fraction• Jain states: 1/3 (C=1), 2/5

(C=2), 3/7 (C=3), 3/5 (C=-3), 4/7 (C=-4)

• Non-Jain states: e.g. nL=2/3 (with C=+1 instead of C=-1,

which does not exist) and 𝜎xy=1/3; nL=5/7 (with C=+3)

and 𝜎xy=3/7

Topological Field Theory

As expected, in all cases the effective field theory of the hydrodynamic gauge fields is a Chern-Simons theory with the standard form (Wen and Zee)

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L =KIJ

4⇡✏µ⌫�aI

µ@⌫aJ� �

qI

2⇡✏µ⌫�Aµ@⌫aI

� + lIjµqpaI

µ

KIJ =

0

@�2k 1 01 C 00 0 1

1

A , qI =

0

@100

1

A , aµI =

0

@Bµ

Aµ

Cµ

1

A .

�xy = �qT K�1q =C

2kC + 1

Ql = �lT K�1q, ✓ll0 = �2⇡lT K�1l0

|detK| = |2kC + 1|

Hall conductance:

Quasiparticle charge and statistics:

Vacuum degeneracy of a torus:

Modular Properties

The modular S and T matrices of these states are

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Sab =1p

|detK|e�2⇡ilTa K�1lb , Taa = e�⇡ilTa K�1la

D =p|detK| = S�1

00Total quantum dimension:

These quantities determine the universal entanglement properties of these states (Levin and Wen, Kitaev and Preskill)

Translation Symmetry Fractionalization• We also found several “anomalous” states that have the same quantum

order (Hall conductance, charge, statistics, vacuum degeneracy, etc.) but different filling fraction.

• Example: nL=1/7, 2/7, 3/7, 5/7

• How are such states classified?

• Contrary to the continuum 2DEGs, FCIs have lattice translation symmetries

• The anyons of these topological states transform projectively under lattice translations which are fractionalized

• Translations by Ti a of an anyon a along the direction ei satisfy

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(T a2 )�1(T a

1 )�1T a2 T a

1 = ei✓a,b

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• We have m states on a torus of size N1 and N2 unit cells and det K=m

• The set of symmetry fractionalization classes is the 2nd cohomology group H2(G,A)

• Here A=ℤm and G=ℤ2 × U(1) (lattice translations and charge conservation)

• This results in the ground states on a torus having different lattice

momentum k

Translation Symmetry Fractionalization

Conclusions and Outlook

• Strongly coupled 2D lattice models can exhibit a host of non-trivial gapped topological phases

• Chiral phases of frustrated antiferromagnets

• These are examples of bosonic FCIs

• Fermionic FCIs exhibit an array of features not found in the continuum 2DEGs

• All the sequences discussed here have compressible (gapless) states

• How are these related to the recent conjectures on fermionic dualities?

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