doubled chern-simons theory of quantum spin liquids and their quantum phase transitions

8/3/2019 Doubled Chern-Simons theory of quantum spin liquids and their quantum phase transitions

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Doubled Chern-Simons theory

of quantum spin liquids and

their quantum phase transitions Talk online: sachdev.physics.harvard.edu
http://sachdev.physics.harvard.edu/http://sachdev.physics.harvard.edu/http://sachdev.physics.harvard.edu/


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Cenke Xu

arXiv:0809.0694

Yang Qi


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Antiferromagnet


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Spin liquid=


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Spin liquid=


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Spin liquid=


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Spin liquid=


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Spin liquid=


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Spin liquid=


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General approach

Look for spin liquids acrosscontinuous (or weakly first-order)

quantum transitions fromantiferromagnetically ordered states


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Ground state has long-range Nel order

Square lattice antiferromagnet

H=

ij

JijSi

Sj

Order parameter is a single vector field = iSii = 1 on two sublattices

= 0 in Neel state.


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Destroy Neel order by perturbations whichpreserve full square lattice symmetry

Square lattice antiferromagnet


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Theory for loss of Neel order

Write the spin operator in terms ofSchwinger bosons (spinons) zi, =, :

Si = zizi

where are Pauli matrices, and the bosons obey the local con-straint

zizi = 2S

Effective theory for spinons must be invariant under the U(1) gaugetransformation

zi eizi


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Perturbation theoryLow energy spinon theory for quantum disordering the Neel stateis the CP1 model

Sz =

d2xd

c2 |(x iAx)z|

2+ |( iA)z|

2+ s |z|

2

+ u

|z|

2

2

+1

4e2(A)

2

here A is an emergent U(1) gauge field (the photon) whichdescribes low-lying spin-singlet excitations.

Phases:

z = 0 Neel (Higgs) state

z = 0 Spin liquid (Coulomb) state


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ssc

Spin liquid with a

photon collective mode

Neel state

z = 0z = 0


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ssc

Neel state

z = 0z = 0

[Unstable to valence bond solid (VBS) order]

Spin liquid with a


N. Read and S. Sachdev,Phys. Rev. Lett. 62, 1694 (1989)


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A spin density wave with

Si (cos(K ri, sin(K ri))

andK

= (,

).

From the square to the triangular lattice


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From the square to the triangular lattice

A spin density wave with

Si (cos(K ri, sin(K ri))

andK

= (

+,

+

).


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Interpretation of non-collinearity

Its physical interpretation becomes clearfrom the allowed coupling to the spinons:

Sz, =

d

2

rd [

zxz + c.c.]

is a spinon pair field



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ssc

Neel state

z = 0z = 0

[Unstable to valence bond solid (VBS) order]

Spin liquid with a



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ssc

z = 0 , = 0non-collinear Neel state

z = 0 , = 0

Z2 spin liquid with a

vison excitation


?


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What is a vison ?

A vison is an Abrikosov vortex in the spinon

pair field .

In the Z2 spin liquid, the vison is S = 0

quasiparticle with a finite energy gap


Wh ?


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Z2 Spin liquid =

What is a vison ?

Wh ?


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=Z2 Spin liquid

What is a vison ?

N. Read and B. Chakraborty,Phys. Rev. B 40, 7133 (1989)N. Read and S. Sachdev,Phys. Rev. Lett. 66, 1773 (1991)

Wh ?


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=Z2 Spin liquid

What is a vison ?


Wh i i ?


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=Z2 Spin liquid

What is a vison ?


Wh i i ?


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=Z2 Spin liquid

What is a vison ?


Wh i i ?


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=Z2 Spin liquid

What is a vison ?


Wh i i ?


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=Z2 Spin liquid

What is a vison ?


Global phase diagram


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sNeel state

[Unstable to valence bond solid VBS order]

Spin liquid with a



z = 0 , = 0


vison excitation

S. Sachdev and N. Read,Int. J. Mod. Phys B5, 219 (1991), cond-mat/0402109

z = 0 , = 0 z = 0 , = 0

s


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A Simple Toy Model (A. Kitaev, 1997)

Spins S

living on the links of a square lattice:

Hence, Fp's and Ai's form a set of conserved quantities.


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Spins S




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Spins S




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Spins S




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Spins S




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Spins S




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Spins S




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Spins S




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Spins S




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Spins S




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Spins S




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Properties of the Ground State

Ground state: all Ai=1, Fp=1 Pictorial representation: color each link with an up-spin. Ai=1 : closed loops.

Fp=1 : every plaquette is an equal-amplitude superposition of

inverse images.

The GS wavefunction takes the same value on configurations

connected by these operations. It does not depend on the geometryof the configurations, only on their topology.


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Properties of Excitations

Electric particle, orAi = 1 endpoint of a line

Magnetic particle, orvortex: Fp= 1 a flip of this plaquette

changes the sign of a given term in the superposition. Charges and vortices interact via topological Aharonov-Bohm

interactions.


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Electric particle, orAi = 1 endpoint of a line (a spinon) Magnetic particle, orvortex: Fp= 1 a flip of this plaquette

changes the sign of a given term in the superposition. Charges and vortices interact via topological Aharonov-Bohm

interactions.


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changes the sign of a given term in the superposition (a vison).

Charges and vortices interact via topological Aharonov-Bohminteractions.


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changes the sign of a given term in the superposition (a vison).

Charges and vortices interact via topological Aharonov-Bohminteractions.

Spinons and visons are mutual semions



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sNeel state


Spin liquid with a



z = 0 , = 0


vison excitation

S. Sachdev and N. Read,Int. J. Mod. Phys B5, 219 (1991), cond-mat/0402109

z = 0 , = 0 z = 0 , = 0

s

Mutual Chern Simons Theory


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Mutual Chern-Simons Theory

Express theory in terms of the physical excitations: the spinons,

z, and the visons. After accounting for Berry phase effects, thevisons can be described by a complex field v, which transformsnon-trivially under the square lattice space group operations.

The spinons and visons have mutual semionic statistics, and thisleads to the continuum theory:

S =

d2xd

c2 |(x iAx)z|

2+ |( iA)z|

2+ s |z|

2+ . . .

+ c2 |(x iBx)v|2 + |( iB)v|2 + s |v|2 + . . .+

i

BA

Cenke Xu and S. Sachdev, to appear



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S = d2xd c2 |(x iAx)z|2 + |( iA)z|2 + s |z|2 + . . .+

c2 |(x iBx)v|

2+ |( iB)v|

2+

s |v|

2+ . . .

+i

BAThis theory fully accounts for all the phases, including

their global topological properties and their broken sym-

metries. It also completely describe the quantum phasetransitions between them.




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p g

sNeel state


Spin liquid with a


non-collinear Neel state


vison excitation

s

z = 0 , v = 0

z = 0 , v = 0

z = 0 , v = 0

z = 0 , v = 0




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S=

d2xd c

2 |(x iAx)z|2+ |( iA)z|

2+ s |z|

2+ . . .

+ c2 |(x iBx)v|2 + |( iB)v|2 + s |v|2 + . . .+ iBA

Low energy states on a torus:

Z2 spin liquid has a 4-fold degeneracy.

Non-collinear Neel state has low-lying tower of states de-scribed by a broken symmetry with order parameter S3/Z2.

Photon spin liquid has a low-lying tower of states describedby a broken symmetry with order parameter S1/Z2. This isthe VBS order v2.

Neel state has a low-lying tower of states described by a bro-ken symmetry with order parameter S3S1/(U(1)U(1)) S2. This is the usual vector Neel order.

doubled chern-simons theory of quantum spin liquids and their quantum phase transitions

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