deconfined quantum criticality - harvard universityqpt.physics.harvard.edu/talks/osu_seminar.pdf ·...
TRANSCRIPT
-
Deconfined quantum criticality
Science 303, 1490 (2004); Physical Review B 70, 144407 (2004), 71, 144508 and 71, 144509 (2005), cond-mat/0502002
Leon Balents (UCSB) Lorenz Bartosch (Harvard) Anton Burkov (Harvard) Matthew Fisher (UCSB) Subir Sachdev (Harvard)
Krishnendu Sengupta (HRI, India) T. Senthil (MIT and IISc)
Ashvin Vishwanath (Berkeley)
Talk online at http://sachdev.physics.harvard.edu
-
OutlineOutline
I. Magnetic quantum phase transitions in “dimerized” Mott insulators:Landau-Ginzburg-Wilson (LGW) theory
II. Magnetic quantum phase transitions of Mott insulators on the square latticeA. Breakdown of LGW theoryB. Berry phasesC. Spinor formulation and deconfined criticality
-
I. Magnetic quantum phase transitions in “dimerized” Mott insulators:
Landau-Ginzburg-Wilson (LGW) theory:Second-order phase transitions described by
fluctuations of an order parameterassociated with a broken symmetry
-
TlCuCl3
M. Matsumoto, B. Normand, T.M. Rice, and M. Sigrist, cond-mat/0309440.
-
Coupled Dimer AntiferromagnetM. P. Gelfand, R. R. P. Singh, and D. A. Huse, Phys. Rev. B 40, 10801-10809 (1989).N. Katoh and M. Imada, J. Phys. Soc. Jpn. 63, 4529 (1994).J. Tworzydlo, O. Y. Osman, C. N. A. van Duin, J. Zaanen, Phys. Rev. B 59, 115 (1999).M. Matsumoto, C. Yasuda, S. Todo, and H. Takayama, Phys. Rev. B 65, 014407 (2002).
S=1/2 spins on coupled dimers
JλJ
jiij
ij SSJH ⋅= ∑><
10 ≤≤ λ
-
close to 0λ Weakly coupled dimers
-
close to 0λ Weakly coupled dimers
0iS =Paramagnetic ground state
( )↓↑−↑↓=2
1
-
close to 0λ Weakly coupled dimers
( )↓↑−↑↓=2
1
Excitation: S=1 quasipartcle
-
close to 0λ Weakly coupled dimers
( )↓↑−↑↓=2
1
Excitation: S=1 quasipartcle
-
close to 0λ Weakly coupled dimers
( )↓↑−↑↓=2
1
Excitation: S=1 quasipartcle
-
close to 0λ Weakly coupled dimers
( )↓↑−↑↓=2
1
Excitation: S=1 quasipartcle
-
close to 0λ Weakly coupled dimers
( )↓↑−↑↓=2
1
Excitation: S=1 quasipartcle
-
close to 0λ Weakly coupled dimers
( )↓↑−↑↓=2
1
Excitation: S=1 quasipartcle
Energy dispersion away fromantiferromagnetic wavevector
2 2 2 2
2x x y y
p
c p c pε
+= ∆ +
∆spin gap∆ →
-
TlCuCl3
N. Cavadini, G. Heigold, W. Henggeler, A. Furrer, H.-U. Güdel, K. Krämer and H. Mutka, Phys. Rev. B 63 172414 (2001).
S=1 quasi-
particle
-
Coupled Dimer Antiferromagnet
-
close to 1λ Weakly dimerized square lattice
-
close to 1λ Weakly dimerized square lattice
Excitations: 2 spin waves (magnons)
2 2 2 2p x x y yc p c pε = +
Ground state has long-range spin density wave (Néel) order at wavevector K= (π,π)
0 ϕ ≠
spin density wave order parameter: ; 1 on two sublatticesii iSS
ϕ η η= = ±
-
TlCuCl3
J. Phys. Soc. Jpn 72, 1026 (2003)
-
λc = 0.52337(3)M. Matsumoto, C. Yasuda, S. Todo, and H. Takayama,
Phys. Rev. B 65, 014407 (2002)T=0
λ 1 cλ Pressure in TlCuCl3
Quantum paramagnetNéel state
0ϕ ≠ 0ϕ =
The method of bond operators (S. Sachdev and R.N. Bhatt, Phys. Rev. B 41, 9323 (1990)) provides a quantitative description of spin excitations in TlCuCl3 across the quantum phase transition (M. Matsumoto, B. Normand, T.M. Rice, and M. Sigrist,
Phys. Rev. Lett. 89, 077203 (2002))
-
LGW theory for quantum criticality
write down an effective action for the antiferromagnetic order parameter by expanding in powers
of and its spatial and temporal d
Landau-Ginzburg-Wilson theor
erivatives, while preserving
y:
all s
ϕϕ
ymmetries of the microscopic Hamiltonian
( ) ( ) ( )( ) ( )22 22 2 2 212 4!x cud xd cSϕ ττ ϕ ϕ λ λ ϕ ϕ
⎡ ⎤= ∇ + ∂ + − +⎢ ⎥⎣ ⎦∫S. Chakravarty, B.I. Halperin, and D.R. Nelson, Phys. Rev. B 39, 2344 (1989)
-
LGW theory for quantum criticality
write down an effective action for the antiferromagnetic order parameter by expanding in powers
of and its spatial and temporal d
Landau-Ginzburg-Wilson theor
erivatives, while preserving
y:
all s
ϕϕ
ymmetries of the microscopic Hamiltonian
( ) ( ) ( )( ) ( )22 22 2 2 212 4!x cud xd cSϕ ττ ϕ ϕ λ λ ϕ ϕ
⎡ ⎤= ∇ + ∂ + − +⎢ ⎥⎣ ⎦∫S. Chakravarty, B.I. Halperin, and D.R. Nelson, Phys. Rev. B 39, 2344 (1989)
For , oscillations of about 0 constitute the excitation
c
triplonλ λ ϕ ϕ< =
A.V. Chubukov, S. Sachdev, and J.Ye, Phys. Rev. B 49, 11919 (1994)
-
OutlineOutline
I. Magnetic quantum phase transitions in “dimerized” Mott insulators:Landau-Ginzburg-Wilson (LGW) theory
II. Magnetic quantum phase transitions of Mott insulators on the square latticeA. Breakdown of LGW theoryB. Berry phasesC. Spinor formulation and deconfined criticality
-
II. Magnetic quantum phase transitions of Mott insulators on the square lattice:
A. Breakdown of LGW theory
-
Ground state has long-range Néel order
Order parameter 1 on two sublattices
0
i i
i
Sϕ ηη
ϕ
==
≠
±
; spin operator with =1/2ij i j iij
H J S S S S= ⇒∑ iSquare lattice antiferromagnet
-
; spin operator with =1/2ij i j iij
H J S S S S= ⇒∑ iSquare lattice antiferromagnet
Destroy Neel order by perturbations which preserve full square lattice symmetry e.g. second-neighbor or ring exchange.
What is the state with 0 ?ϕ =
-
; spin operator with =1/2ij i j iij
H J S S S S= ⇒∑ iSquare lattice antiferromagnet
Destroy Neel order by perturbations which preserve full square lattice symmetry e.g. second-neighbor or ring exchange.
What is the state with 0 ?ϕ =
-
LGW theory for quantum criticality
write down an effective action for the antiferromagnetic order parameter by expanding in powers
of and its spatial and temporal d
Landau-Ginzburg-Wilson theor
erivatives, while preserving
y:
all s
ϕϕ
ymmetries of the microscopic Hamiltonian
( ) ( )( ) ( )22 22 2 2 212 4!xud xd c rSϕ ττ ϕ ϕ ϕ ϕ
⎡ ⎤= ∇ + ∂ + +⎢ ⎥⎣ ⎦∫
( )
The ground state for 0 has no broken symmetry and a gapped S=1 quasiparticle excitation oscillations of about 0
r
ϕ ϕ
>
=
-
Problem: there is no state with a gapped, stable S=1 quasiparticle and no broken symmetries
-
Problem: there is no state with a gapped, stable S=1 quasiparticle and no broken symmetries
“Liquid” of valence bonds has fractionalized S=1/2 excitations
-
Problem: there is no state with a gapped, stable S=1 quasiparticle and no broken symmetries
“Liquid” of valence bonds has fractionalized S=1/2 excitations
-
Problem: there is no state with a gapped, stable S=1 quasiparticle and no broken symmetries
“Liquid” of valence bonds has fractionalized S=1/2 excitations
-
Problem: there is no state with a gapped, stable S=1 quasiparticle and no broken symmetries
“Liquid” of valence bonds has fractionalized S=1/2 excitations
-
Problem: there is no state with a gapped, stable S=1 quasiparticle and no broken symmetries
“Liquid” of valence bonds has fractionalized S=1/2 excitations
-
Problem: there is no state with a gapped, stable S=1 quasiparticle and no broken symmetries
“Liquid” of valence bonds has fractionalized S=1/2 excitations
-
Another possible state, with 0, is the valence bond solid (VBS)ϕ =
-
Another possible state, with 0, is the valence bond solid (VBS)ϕ =
vbsΨ
( ) ( )arbsc
vtan j ii
i jij
i S S e −Ψ = ∑ i r rvbs vbs
Such a state breaks the symmetry of rotations by / 2 about lattice sites,and has 0, where is the
nπΨ ≠ Ψ VBS order parameter
-
Another possible state, with 0, is the valence bond solid (VBS)ϕ =
( ) ( )arbsc
vtan j ii
i jij
i S S e −Ψ = ∑ i r rvbs vbs
Such a state breaks the symmetry of rotations by / 2 about lattice sites,and has 0, where is the
nπΨ ≠ Ψ VBS order parameter
vbsΨ
-
Another possible state, with 0, is the valence bond solid (VBS)ϕ =
vbsΨ
( ) ( )arbsc
vtan j ii
i jij
i S S e −Ψ = ∑ i r rvbs vbs
Such a state breaks the symmetry of rotations by / 2 about lattice sites,and has 0, where is the
nπΨ ≠ Ψ VBS order parameter
-
Another possible state, with 0, is the valence bond solid (VBS)ϕ =
vbsΨ
( ) ( )arbsc
vtan j ii
i jij
i S S e −Ψ = ∑ i r rvbs vbs
Such a state breaks the symmetry of rotations by / 2 about lattice sites,and has 0, where is the
nπΨ ≠ Ψ VBS order parameter
-
Another possible state, with 0, is the valence bond solid (VBS)ϕ =
vbsΨ
( ) ( )arbsc
vtan j ii
i jij
i S S e −Ψ = ∑ i r rvbs vbs
Such a state breaks the symmetry of rotations by / 2 about lattice sites,and has 0, where is the
nπΨ ≠ Ψ VBS order parameter
-
Another possible state, with 0, is the valence bond solid (VBS)ϕ =
vbsΨ
( ) ( )arbsc
vtan j ii
i jij
i S S e −Ψ = ∑ i r rvbs vbs
Such a state breaks the symmetry of rotations by / 2 about lattice sites,and has 0, where is the
nπΨ ≠ Ψ VBS order parameter
-
Another possible state, with 0, is the valence bond solid (VBS)ϕ =
vbsΨ
( ) ( )arbsc
vtan j ii
i jij
i S S e −Ψ = ∑ i r rvbs vbs
Such a state breaks the symmetry of rotations by / 2 about lattice sites,and has 0, where is the
nπΨ ≠ Ψ VBS order parameter
-
A )nother possible state, with 0, is the valence bond solid (VBSϕ =
( ) ( )arbsc
vtan j ii
i jij
i S S e −Ψ = ∑ i r rvbs vbs
Such a state breaks the symmetry of rotations by / 2 about lattice sites,and has 0, where is the
nπΨ ≠ Ψ VBS order parameter
vbsΨ
-
A )nother possible state, with 0, is the valence bond solid (VBSϕ =
( ) ( )arbsc
vtan j ii
i jij
i S S e −Ψ = ∑ i r rvbs vbs
Such a state breaks the symmetry of rotations by / 2 about lattice sites,and has 0, where is the
nπΨ ≠ Ψ VBS order parameter
vbsΨ
-
Another possible state, with 0, is the valence bond solid (VBS)ϕ =
vbsΨ
( ) ( )arbsc
vtan j ii
i jij
i S S e −Ψ = ∑ i r rvbs vbs
Such a state breaks the symmetry of rotations by / 2 about lattice sites,and has 0, where is the
nπΨ ≠ Ψ VBS order parameter
-
The VBS state does have a stable S=1 quasiparticle excitation
vbs 0, 0ϕΨ ≠ =
-
The VBS state does have a stable S=1 quasiparticle excitation
vbs 0, 0ϕΨ ≠ =
-
The VBS state does have a stable S=1 quasiparticle excitation
vbs 0, 0ϕΨ ≠ =
-
The VBS state does have a stable S=1 quasiparticle excitation
vbs 0, 0ϕΨ ≠ =
-
The VBS state does have a stable S=1 quasiparticle excitation
vbs 0, 0ϕΨ ≠ =
-
The VBS state does have a stable S=1 quasiparticle excitation
vbs 0, 0ϕΨ ≠ =
-
LGW theory of multiple order parameters
[ ] [ ][ ]
[ ]
vbs vbs int
2 4vbs vbs 1 vbs 1 vbs
2 42 2
2 2int vbs
F F F F
F r u
F r u
F v
ϕ
ϕ
ϕ
ϕ ϕ ϕ
ϕ
= Ψ + +
Ψ = Ψ + Ψ +
= + +
= Ψ +
Distinct symmetries of order parameters permit couplings only between their energy densities
-
LGW theory of multiple order parametersFirst order transitionϕ
vbsΨ
Neel order VBS orderg
g
Coexistence
Neel order
ϕ
VBS order
vbsΨ
g
"disordered"
Neel order
ϕvbsΨ
VBS order
-
LGW theory of multiple order parametersFirst order transitionϕ
vbsΨ
Neel order VBS orderg
g
Coexistence
Neel order
ϕ
VBS order
vbsΨ
g
"disordered"
Neel order
ϕvbsΨ
VBS order
-
OutlineOutline
I. Magnetic quantum phase transitions in “dimerized” Mott insulators:Landau-Ginzburg-Wilson (LGW) theory
II. Magnetic quantum phase transitions of Mott insulators on the square latticeA. Breakdown of LGW theoryB. Berry phasesC. Spinor formulation and deconfined criticality
-
II. Magnetic quantum phase transitions of Mott insulators on the square lattice:
B. Berry phases
-
Quantum theory for destruction of Neel order
Ingredient missing from LGW theory: Ingredient missing from LGW theory: Spin Berry PhasesSpin Berry Phases
AiSAe
-
Quantum theory for destruction of Neel order
Ingredient missing from LGW theory: Ingredient missing from LGW theory: Spin Berry PhasesSpin Berry Phases
AiSAe
-
Quantum theory for destruction of Neel order
-
Quantum theory for destruction of Neel orderDiscretize imaginary time: path integral is over fields on the
sites of a cubic lattice of points a
-
Quantum theory for destruction of Neel orderDiscretize imaginary time: path integral is over fields on the
sites of a cubic lattice of points aa a
a
Recall = (0,0,1) in classical Neel state; 1 on two
square subl
2attices ;
a aSϕ η ϕη
→→ ±=
a+µϕ
0ϕ
aϕ
( ), ,x yµ τ=
-
Quantum theory for destruction of Neel orderDiscretize imaginary time: path integral is over fields on the
sites of a cubic lattice of points aa a
a
Recall = (0,0,1) in classical Neel state; 1 on two
square subl
2attices ;
a aSϕ η ϕη
→→ ±=
a+µϕ
0ϕ
aϕ
a a+ 0
oriented area of spherical triangle
formed by and an arbitrary reference , poi, nta hA alfµ
µϕ ϕ ϕ
→
2 aA µ
-
Quantum theory for destruction of Neel orderDiscretize imaginary time: path integral is over fields on the
sites of a cubic lattice of points aa a
a
Recall = (0,0,1) in classical Neel state; 1 on two
square subl
2attices ;
a aSϕ η ϕη
→→ ±=
a+µϕ
0ϕ
aϕ
a a+ 0
oriented area of spherical triangle
formed by and an arbitrary reference , poi, nta hA alfµ
µϕ ϕ ϕ
→
-
Quantum theory for destruction of Neel orderDiscretize imaginary time: path integral is over fields on the
sites of a cubic lattice of points aa a
a
Recall = (0,0,1) in classical Neel state; 1 on two
square subl
2attices ;
a aSϕ η ϕη
→→ ±=
a+µϕ
0ϕ
aϕ
a a+ 0
oriented area of spherical triangle
formed by and an arbitrary reference , poi, nta hA alfµ
µϕ ϕ ϕ
→
aγ
-
a µγ +
Quantum theory for destruction of Neel orderDiscretize imaginary time: path integral is over fields on the
sites of a cubic lattice of points aa a
a
Recall = (0,0,1) in classical Neel state; 1 on two
square subl
2attices ;
a aSϕ η ϕη
→→ ±=
a+µϕ
0ϕ
aϕ
a a+ 0
oriented area of spherical triangle
formed by and an arbitrary reference , poi, nta hA alfµ
µϕ ϕ ϕ
→
aγ
-
a µγ +
Quantum theory for destruction of Neel orderDiscretize imaginary time: path integral is over fields on the
sites of a cubic lattice of points aa a
a
Recall = (0,0,1) in classical Neel state; 1 on two
square subl
2attices ;
a aSϕ η ϕη
→→ ±=
a+µϕ
0ϕ
aϕ
a a+ 0
oriented area of spherical triangle
formed by and an arbitrary reference , poi, nta hA alfµ
µϕ ϕ ϕ
→
aγ
Change in choice of is like a “gauge transformation”
2 2a a a aA Aµ µ µγ γ+→ − +
0ϕ
-
a µγ +
Quantum theory for destruction of Neel orderDiscretize imaginary time: path integral is over fields on the
sites of a cubic lattice of points aa a
a
Recall = (0,0,1) in classical Neel state; 1 on two
square subl
2attices ;
a aSϕ η ϕη
→→ ±=
a+µϕ
0ϕ
aϕ
a a+ 0
oriented area of spherical triangle
formed by and an arbitrary reference , poi, nta hA alfµ
µϕ ϕ ϕ
→
aγ
Change in choice of is like a “gauge transformation”
2 2a a a aA Aµ µ µγ γ+→ − +
0ϕ
The area of the triangle is uncertain modulo 4π, and the action has to be invariant under 2a aA Aµ µ π→ +
-
Quantum theory for destruction of Neel order
Ingredient missing from LGW theory: Ingredient missing from LGW theory: Spin Berry PhasesSpin Berry Phases
exp a aa
i A τη⎛ ⎞⎜ ⎟⎝ ⎠
∑Sum of Berry phases of all spins on the square lattice.
-
Quantum theory for destruction of Neel order
,
1a a
agµ
µ
ϕ ϕ +⎛ ⎞
⋅⎜ ⎟⎝ ⎠
∑( )2,
11 expa a a a a aa aa
Z d i Ag µ τµ
ϕ δ ϕ ϕ ϕ η+⎛ ⎞
= − ⋅ +⎜ ⎟⎝ ⎠
∑ ∑∏∫
Partition function on cubic lattice
LGW theory: weights in partition function are those of a classical ferromagnet at a “temperature” g
Small ground state has Neel order with 0
Large paramagnetic ground state with 0
g
g
ϕ
ϕ
⇒ ≠
⇒ =
-
Quantum theory for destruction of Neel order
( )2,
11 expa a a a a aa aa
Z d i Ag µ τµ
ϕ δ ϕ ϕ ϕ η+⎛ ⎞
= − ⋅ +⎜ ⎟⎝ ⎠
∑ ∑∏∫
Partition function on cubic lattice
Modulus of weights in partition function: those of a classical ferromagnet at a “temperature” g
Berry phases lead to large cancellations between different time histories need an effective action for at large aA gµ→
S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002)
Small ground state has Neel order with 0
Large paramagnetic ground state with 0
g
g
ϕ
ϕ
⇒ ≠
⇒ =
-
OutlineOutline
I. Magnetic quantum phase transitions in “dimerized” Mott insulators:Landau-Ginzburg-Wilson (LGW) theory
II. Magnetic quantum phase transitions of Mott insulators on the square latticeA. Breakdown of LGW theoryB. Berry phasesC. Spinor formulation and deconfined criticality
-
II. Magnetic quantum phase transitions of Mott insulators on the square lattice:
C. Spinor formulation and deconfinedcriticality
-
Quantum theory for destruction of Neel order
( )2,
11 expa a a a a aa aa
Z d i Ag µ τµ
ϕ δ ϕ ϕ ϕ η+⎛ ⎞
= − ⋅ +⎜ ⎟⎝ ⎠
∑ ∑∏∫
Partition function on cubic lattice
*
Rewrite partition function in terms of spino
rs
,
with , and
a
a a a
z
z z
α
α αβ β
α
ϕ σ=↑ ↓
=
S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002)
-
Quantum theory for destruction of Neel order
( )2,
11 expa a a a a aa aa
Z d i Ag µ τµ
ϕ δ ϕ ϕ ϕ η+⎛ ⎞
= − ⋅ +⎜ ⎟⎝ ⎠
∑ ∑∏∫
Partition function on cubic lattice
*
Rewrite partition function in terms of spino
rs
,
with , and
a
a a a
z
z z
α
α αβ β
α
ϕ σ=↑ ↓
=*
,
Remarkable identity fromspherical trigonometry
Arg a aaz z Aα µµ α+⎡ ⎤⎢ ⎥⎣ ⎦ =
S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002)
-
Quantum theory for destruction of Neel order
( )2,
11 expa a a a a aa aa
Z d i Ag µ τµ
ϕ δ ϕ ϕ ϕ η+⎛ ⎞
= − ⋅ +⎜ ⎟⎝ ⎠
∑ ∑∏∫
Partition function on cubic lattice
S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002)
( )2
*,
,
1
1 exp a
a a aa
iAa a a a
a a
Z dz dA z
z e z i Ag
µ
α µ α
α µ α τµ
δ
η+
≈ −
⎛ ⎞× +⎜ ⎟
⎝ ⎠
∏∫
∑ ∑
Partition function expressed as a gauge theory of spinordegrees of freedom
-
Large g effective action for the Aaµ after integrating zαµ
( )22 2
,
withThis is compact QED in 3 spacetime dimensions with
static charges 1 on
1exp c
two sublat
tices.
o2
~
sa a a a aaa
Z dA A A i Ae
e g
µ µ ν ν µ τµ
η⎛ ⎞= ∆ − ∆ −⎜ ⎟⎝ ⎠
±
∑ ∑∏∫
This theory can be reliably analyzed by a duality mapping.
The gauge theory is in a confining phase, and there is VBS order in the ground state. (Proliferation of monopoles in
the presence of Berry phases).
This theory can be reliably analyzed by a duality mapping.
The gauge theory is in a confiningconfining phase, and there is VBS order in the ground state. (Proliferation of monopoles in
the presence of Berry phases).
N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989).S. Sachdev and R. Jalabert, Mod. Phys. Lett. B 4, 1043 (1990).
K. Park and S. Sachdev, Phys. Rev. B 65, 220405 (2002).
-
( )2 * ,,
11 exp aiAa a a a a a aa aa
Z dz dA z z e z i Ag
µα µ α α µ α τ
µ
δ η+⎛ ⎞
≈ − +⎜ ⎟⎝ ⎠
∑ ∑∏∫
vbs
VBS order0
Not present in LGW theory of orderϕ
Ψ ≠
or
Neel order 0ϕ ≠
g0
-
Ordering by quantum fluctuations
-
Ordering by quantum fluctuations
-
Ordering by quantum fluctuations
-
Ordering by quantum fluctuations
-
Ordering by quantum fluctuations
-
Ordering by quantum fluctuations
-
Ordering by quantum fluctuations
-
Ordering by quantum fluctuations
-
Ordering by quantum fluctuations
-
( )2 * ,,
11 exp aiAa a a a a a aa aa
Z dz dA z z e z i Ag
µα µ α α µ α τ
µ
δ η+⎛ ⎞
≈ − +⎜ ⎟⎝ ⎠
∑ ∑∏∫
g0
Neel order 0ϕ ≠
?vbs
VBS order0
Not present in LGW theory of orderϕ
Ψ ≠
or
-
Second-order critical point described by emergent fractionalized degrees of freedom (Aµ and zα );
Order parameters (ϕ and Ψvbs ) are “composites” and of secondary importance
Second-order critical point described by emergent fractionalized degrees of freedom (Aµ and zα );
Order parameters (ϕ and Ψvbs ) are “composites” and of secondary importance
→
S. Sachdev and R. Jalabert, Mod. Phys. Lett. B 4, 1043 (1990); G. Murthy and S. Sachdev, Nuclear Physics B 344, 557 (1990); C. Lannert, M.P.A. Fisher, and T. Senthil, Phys. Rev. B 63, 134510 (2001); S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002); O. Motrunich and A. Vishwanath, Phys. Rev. B 70, 075104 (2004)
Theory of a second-order quantum phase transition between Neel and VBS phases
*
At the quantum critical point: +2 periodicity can be ignored
(Monopoles interfere destructively and are dangerously irrelevant). =1/2 , with ~ , are globally
pro
A A
S spinons z z z
µ µ
α α αβ β
π
ϕ σ
• →
•
pagating degrees of freedom.
T. Senthil, A. Vishwanath, L. Balents, S. Sachdev and M.P.A. Fisher, Science 303, 1490 (2004).
-
Phase diagram of S=1/2 square lattice antiferromagnet
g*
Neel order
~ 0z zα αβ βϕ σ ≠
(associated with condensation of monopoles in ),Aµ
or
vbsVBS order 0Ψ ≠
1/ 2 spinons confined, 1 triplon excitations
S zS
α=
=
T. Senthil, A. Vishwanath, L. Balents, S. Sachdev and M.P.A. Fisher, Science 303, 1490 (2004).
-
Conclusions
• New quantum phases induced by Berry phases: VBS order in the antiferromagnet
• Critical resonating-valence-bond states describes the quantum phase transition from the Neel to the VBS
• Emergent gauge fields are essential for a full description of the low energy physics.
Conclusions
• New quantum phases induced by Berry phases: VBS order in the antiferromagnet
• Critical resonating-valence-bond states describes the quantum phase transition from the Neel to the VBS
• Emergent gauge fields are essential for a full description of the low energy physics.