Post on 06-Apr-2018
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/8/3/2019 Doubled Chern-Simons theory of quantum spin liquids and their quantum phase transitions
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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
photon collective mode
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
N. Read and S. Sachdev,Phys. Rev. Lett. 66, 1773 (1991)
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ssc
Neel state
z = 0z = 0
[Unstable to valence bond solid (VBS) order]
Spin liquid with a
photon collective mode
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ssc
z = 0 , = 0non-collinear Neel state
z = 0 , = 0
Z2 spin liquid with a
vison excitation
N. Read and S. Sachdev,Phys. Rev. Lett. 66, 1773 (1991)
?
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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
N. Read and S. Sachdev,Phys. Rev. Lett. 66, 1773 (1991)
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 ?
N. Read and B. Chakraborty,Phys. Rev. B 40, 7133 (1989)N. Read and S. Sachdev,Phys. Rev. Lett. 66, 1773 (1991)
Wh i i ?
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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 i i ?
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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 i i ?
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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 i i ?
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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)
Global phase diagram
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Global phase diagram
sNeel state
[Unstable to valence bond solid VBS order]
Spin liquid with a
photon collective mode
z = 0 , = 0non-collinear Neel state
z = 0 , = 0
Z2 spin liquid with a
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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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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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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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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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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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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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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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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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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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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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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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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Properties of Excitations
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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Properties of Excitations
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 (a vison).
Charges and vortices interact via topological Aharonov-Bohminteractions.
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Properties of Excitations
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 (a vison).
Charges and vortices interact via topological Aharonov-Bohminteractions.
Spinons and visons are mutual semions
Global phase diagram
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Global phase diagram
sNeel state
[Unstable to valence bond solid VBS order]
Spin liquid with a
photon collective mode
z = 0 , = 0non-collinear Neel state
z = 0 , = 0
Z2 spin liquid with a
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
Mutual Chern Simons Theory
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Mutual Chern-Simons 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
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.
Cenke Xu and S. Sachdev, to appear
Global phase diagram
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p g
sNeel state
[Unstable to valence bond solid VBS order]
Spin liquid with a
photon collective mode
non-collinear Neel state
Z2 spin liquid with a
vison excitation
s
z = 0 , v = 0
z = 0 , v = 0
z = 0 , v = 0
z = 0 , v = 0
Cenke Xu and S. Sachdev, to appear
Mutual Chern Simons Theory
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Mutual Chern-Simons Theory
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.