deconfined quantum criticality - harvard university
TRANSCRIPT
Deconfined quantum criticality
Leon Balents (UCSB) Lorenz Bartosch (Frankfurt)
Anton Burkov (Harvard) Matthew Fisher (UCSB) Subir Sachdev (Harvard)
Krishnendu Sengupta (HRI, India) T. Senthil (MIT)
Ashvin Vishwanath (Berkeley)
Talks 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.
S=1/2 spins on coupled dimers
jiij
ij SSJH ⋅= ∑><
10 ≤≤ λ
JλJ
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).
Weakly coupled dimersclose to 0λ
close to 0λ Weakly coupled dimers
Paramagnetic ground state 0iS =
( )↓↑−↑↓=2
1
( )↓↑−↑↓=2
1
close to 0λ Weakly coupled dimers
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
Energy dispersion away fromantiferromagnetic wavevector
2 2 2 2
2x x y y
p
c p c pε
+= Δ +
Δspin gapΔ →
Excitation: S=1 quasipartcle
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 i
SS
ϕ η η= = ±
TlCuCl3
J. Phys. Soc. Jpn 72, 1026 (2003)
λ 1 cλ
Néel state
T=0
Pressure in TlCuCl3
Quantum paramagnet
λc = 0.52337(3)M. Matsumoto, C. Yasuda, S. Todo, and H. Takayama,
Phys. Rev. B 65, 014407 (2002)
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))
0ϕ ≠ 0ϕ =
( ) ( ) ( )( ) ( )22 22 2 2 212 4!x c
ud xd cSϕ ττ ϕ ϕ λ λ ϕ 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
S. Chakravarty, B.I. Halperin, and D.R. Nelson, Phys. Rev. B 39, 2344 (1989)
( ) ( ) ( )( ) ( )22 22 2 2 212 4!x c
ud xd cSϕ ττ ϕ ϕ λ λ ϕ 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
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)
S. Chakravarty, B.I. Halperin, and D.R. Nelson, Phys. Rev. B 39, 2344 (1989)
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= ⇒∑ i
Square lattice antiferromagnet
Destroy Neel order by perturbations which preserve full square lattice symmetry e.g. second-neighbor or ring exchange.
; spin operator with =1/2ij i j iij
H J S S S S= ⇒∑ i
Square lattice antiferromagnet
What is the state with 0 ?ϕ =
Destroy Neel order by perturbations which preserve full square lattice symmetry e.g. second-neighbor or ring exchange.
; spin operator with =1/2ij i j iij
H J S S S S= ⇒∑ i
Square lattice antiferromagnet
What is the state with 0 ?ϕ =
( ) ( )( ) ( )22 22 2 2 212 4!x
ud xd c rSϕ ττ ϕ ϕ ϕ 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
( )
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
Large scale Quantum Monte Carlo studies
A.W. Sandvik, cond-mat/0611343
A.W. Sandvik, S. Daul, R. R. P. Singh, and D. J. Scalapino, Phys. Rev. Lett.89, 247201 (2002); A.W. Sandvik and R.G. Melko, cond-mat/0604451.
Easy-plane model
Spin stiffness
Easy-plane model
Valence bond solid (VBS) order in expectation values of plaquette and exchange terms
N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989)
SU(2) invariant model
Strong evidence for a continuous “deconfined”quantum critical point
T. Senthil, A. Vishwanath, L. Balents, S. Sachdev and M.P.A. Fisher, Science 303, 1490 (2004).
A.W. Sandvik, cond-mat/0611343
( ) ( )arbs
cv
tan j iii j
iji S S e −Ψ = ∑ i r r
vbs 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
Characterization of VBS state with 0ϕ =
( ) ( )arbs
cv
tan j iii j
iji S S e −Ψ = ∑ i r r
vbs 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
Characterization of VBS state with 0ϕ =
( ) ( )arbs
cv
tan j iii j
iji S S e −Ψ = ∑ i r r
vbs 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
Characterization of VBS state with 0ϕ =
( ) ( )arbs
cv
tan j iii j
iji S S e −Ψ = ∑ i r r
vbs 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
Characterization of VBS state with 0ϕ =
( ) ( )arbs
cv
tan j iii j
iji S S e −Ψ = ∑ i r r
vbs 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
Characterization of VBS state with 0ϕ =
( ) ( )arbs
cv
tan j iii j
iji S S e −Ψ = ∑ i r r
vbs 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
Characterization of VBS state with 0ϕ =
( ) ( )arbs
cv
tan j iii j
iji S S e −Ψ = ∑ i r r
vbs 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
Characterization of VBS state with 0ϕ =
( ) ( )arbs
cv
tan j iii j
iji S S e −Ψ = ∑ i r r
vbs 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
Characterization of VBS state with 0ϕ =
( ) ( )arbs
cv
tan j iii j
iji S S e −Ψ = ∑ i r r
vbs 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
Characterization of VBS state with 0ϕ =
( ) ( )arbs
cv
tan j iii j
iji S S e −Ψ = ∑ i r r
vbs 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
Characterization of VBS state with 0ϕ =
SU(2) invariant model
Probability distribution of VBS order Ψ at quantum critical point
Emergent circular symmetry is a consequence of a gapless photon excition
T. Senthil, A. Vishwanath, L. Balents, S. Sachdev and M.P.A. Fisher, Science 303, 1490 (2004).
A.W. Sandvik, cond-mat/0611343
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 parameters
g
First order transitionϕ
vbsΨ
g
Coexistence
Neel order VBS order
g
"disordered"
Neel order
ϕ
VBS order
vbsΨ
Neel order
ϕvbsΨ
VBS order
LGW theory of multiple order parameters
g
First order transitionϕ
vbsΨ
g
Coexistence
Neel order VBS order
g
"disordered"
Neel order
ϕ
VBS order
vbsΨ
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
iSAe
A
iSAe
A
Quantum theory for destruction of Neel order
Ingredient missing from LGW theory: Ingredient missing from LGW theory: Spin Berry PhasesSpin Berry Phases
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.
,
1a a
ag μμ
ϕ ϕ +
⎛ ⎞⋅⎜ ⎟
⎝ ⎠∑( )2
,
11 expa a a a a aa aa
Z d i Ag μ τ
μ
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 τ
μ
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→
Quantum theory for destruction of Neel order
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
( )2
,
11 expa a a a a aa aa
Z d i Ag μ τ
μ
ϕ δ ϕ ϕ ϕ η+
⎛ ⎞= − ⋅ +⎜ ⎟
⎝ ⎠∑ ∑∏∫
Partition function on cubic lattice
Quantum theory for destruction of Neel order
S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002)
*
Rewrite partition function in terms of spino
rs
,
with , and
a
a a a
z
z z
α
α αβ β
α
ϕ σ=↑ ↓
=
( )2
,
11 expa a a a a aa aa
Z d i Ag μ τ
μ
ϕ δ ϕ ϕ ϕ η+
⎛ ⎞= − ⋅ +⎜ ⎟
⎝ ⎠∑ ∑∏∫
Partition function on cubic lattice
Quantum theory for destruction of Neel order
S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002)
*
Rewrite partition function in terms of spino
rs
,
with , and
a
a a a
z
z z
α
α αβ β
α
ϕ σ=↑ ↓
=*
,
Identity fromspherical trigonometry
Arg a aaz z Aα μμ α+⎡ ⎤⎢ ⎥⎣ ⎦
=
( )2
,
11 expa a a a a aa aa
Z d i Ag μ τ
μ
ϕ δ ϕ ϕ ϕ η+
⎛ ⎞= − ⋅ +⎜ ⎟
⎝ ⎠∑ ∑∏∫
Partition function on cubic lattice
Quantum theory for destruction of Neel order
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
( )2
2 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⎛ ⎞= Δ − Δ −⎜ ⎟⎝ ⎠
±
∑ ∑∏∫
Large g effective action for the Aaμ after integrating zαμ
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).
g0
Neel order 0ϕ ≠
vbs
VBS order0
Not present in LGW theory of orderϕ
Ψ ≠
or
( )2 *,
,
11 exp aiAa a a a a a a
a aa
Z dz dA z z e z i Ag
μα μ α α μ α τ
μ
δ η+
⎛ ⎞≈ − +⎜ ⎟
⎝ ⎠∑ ∑∏∫
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
g0
Neel order 0ϕ ≠
?vbs
VBS order0
Not present in LGW theory of orderϕ
Ψ ≠
or
( )2 *,
,
11 exp aiAa a a a a a a
a aa
Z dz dA z z e z i Ag
μα μ α α μ α τ
μ
δ η+
⎛ ⎞≈ − +⎜ ⎟
⎝ ⎠∑ ∑∏∫
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.
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
→→
T. Senthil, A. Vishwanath, L. Balents, S. Sachdev and M.P.A. Fisher, Science 303, 1490 (2004).
Monopole fugacity
; Arovas-Auerbach state
g
Phase diagram of S=1/2 square lattice antiferromagnet
T. Senthil, A. Vishwanath, L. Balents, S. Sachdev and M.P.A. Fisher, Science 303, 1490 (2004).
*
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=
=