Breakdown of the Landau-Ginzburg-Wilson paradigm at quantum ...
Transcript of Breakdown of the Landau-Ginzburg-Wilson paradigm at quantum ...
Breakdown of the Landau-Ginzburg-Wilson paradigm at quantum phase transitions
Science 303, 1490 (2004); cond-mat/0312617cond-mat/0401041
Leon Balents (UCSB) Matthew Fisher (UCSB)
S. Sachdev (Yale) T. Senthil (MIT)
Ashvin Vishwanath (MIT)
OutlineOutlineA. Magnetic quantum phase transitions in “dimerized”
Mott insulatorsLandau-Ginzburg-Wilson (LGW) theory
B. Mott insulators with spin S=1/2 per unit cellBerry phases, bond order, and the breakdown of the LGW paradigm
A. 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 triplon
close to 0λ Weakly coupled dimers
( )↓↑−↑↓=2
1
Excitation: S=1 triplon
close to 0λ Weakly coupled dimers
( )↓↑−↑↓=2
1
Excitation: S=1 triplon
close to 0λ Weakly coupled dimers
( )↓↑−↑↓=2
1
Excitation: S=1 triplon
close to 0λ Weakly coupled dimers
( )↓↑−↑↓=2
1
Excitation: S=1 triplon
close to 0λ Weakly coupled dimers
( )↓↑−↑↓=2
1
Excitation: S=1 triplon(exciton, spin collective mode)
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).
“triplon”
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)
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 c
ud 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 c
ud 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)
Key reason for validity of LGW theory
There is a simple high temperature disordered state with =0 and e
Clas
xpon
sical statistical
entially decaying
mechani
correla
:
t
s
ns
c
ioϕ
There is a " " non-degenerate ground state with =0Quantum mechani
and an energy gap to all excitations
: s
c quantum disorderedϕ
B. Mott insulators with spin S=1/2 per unit cell:
Berry phases, bond order, and the breakdown of the LGW paradigm
Mott insulator with two S=1/2 spins per unit cell
Mott insulator with one S=1/2 spin per unit cell
Mott insulator with one S=1/2 spin per unit cell
Ground state has Neel order with 0ϕ ≠
Mott insulator with one S=1/2 spin per unit cell
Destroy Neel order by perturbations which preserve full square lattice symmetry e.g. second-neighbor or ring exchange.The strength of this perturbation is measured by a coupling g.
Small ground state has Neel order with 0
Large paramagnetic ground state with 0
g
g
ϕ
ϕ
⇒ ≠
⇒ =
Mott insulator with one S=1/2 spin per unit cell
Destroy Neel order by perturbations which preserve full square lattice symmetry e.g. second-neighbor or ring exchange.The strength of this perturbation is measured by a coupling g.
Small ground state has Neel order with 0
Large paramagnetic ground state with 0
g
g
ϕ
ϕ
⇒ ≠
⇒ =
Mott insulator with one S=1/2 spin per unit cell
Possible large paramagnetic ground state with 0g ϕ =
Mott insulator with one S=1/2 spin per unit cell
bondΨ
Possible large paramagnetic ground state with 0g ϕ =
( ) ( )arond
cb
tan j iii j
iji S S e −Ψ = ∑ i r r
bond bond
Such a state breaks the symmetry of rotations by / 2 about lattice sites,and has 0, where is the
nπΨ ≠ Ψ bond order parameter
Mott insulator with one S=1/2 spin per unit cell
( ) ( )arond
cb
tan j iii j
iji S S e −Ψ = ∑ i r r
bond bond
Such a state breaks the symmetry of rotations by / 2 about lattice sites,and has 0, where is the
nπΨ ≠ Ψ bond order parameter
bondΨ
Possible large paramagnetic ground state with 0g ϕ =
Mott insulator with one S=1/2 spin per unit cell
( ) ( )arond
cb
tan j iii j
iji S S e −Ψ = ∑ i r r
bond bond
Such a state breaks the symmetry of rotations by / 2 about lattice sites,and has 0, where is the
nπΨ ≠ Ψ bond order parameter
bondΨ
Possible large paramagnetic ground state with 0g ϕ =
Mott insulator with one S=1/2 spin per unit cell
bondΨ
Possible large paramagnetic ground state with 0g ϕ =
( ) ( )arond
cb
tan j iii j
iji S S e −Ψ = ∑ i r r
bond bond
Such a state breaks the symmetry of rotations by / 2 about lattice sites,and has 0, where is the
nπΨ ≠ Ψ bond order parameter
Mott insulator with one S=1/2 spin per unit cell
bondΨ
Possible large paramagnetic ground state with 0g ϕ =
bond bond
Such a state breaks the symmetry of rotations by / 2 about lattice sites,and has 0, where is the
nπΨ ≠ Ψ bond order parameter
( ) ( )arond
cb
tan j iii j
iji S S e −Ψ = ∑ i r r
Mott insulator with one S=1/2 spin per unit cell
bond bond
Another state breaking the symmetry of rotations by / 2 about lattice sites,which also has 0, where is the
nπΨ ≠ Ψ bond order parameter
( ) ( )arond
cb
tan j iii j
iji S S e −Ψ = ∑ i r r
bondΨ
Possible large paramagnetic ground state with 0g ϕ =
Mott insulator with one S=1/2 spin per unit cell
bond bond
Another state breaking the symmetry of rotations by / 2 about lattice sites,which also has 0, where is the
nπΨ ≠ Ψ bond order parameter
( ) ( )arond
cb
tan j iii j
iji S S e −Ψ = ∑ i r r
bondΨ
Possible large paramagnetic ground state with 0g ϕ =
Mott insulator with one S=1/2 spin per unit cell
bond bond
Another state breaking the symmetry of rotations by / 2 about lattice sites,which also has 0, where is the
nπΨ ≠ Ψ bond order parameter
( ) ( )arond
cb
tan j iii j
iji S S e −Ψ = ∑ i r r
bondΨ
Possible large paramagnetic ground state with 0g ϕ =
Mott insulator with one S=1/2 spin per unit cell
bond bond
Another state breaking the symmetry of rotations by / 2 about lattice sites,which also has 0, where is the
nπΨ ≠ Ψ bond order parameter
( ) ( )arond
cb
tan j iii j
iji S S e −Ψ = ∑ i r r
bondΨ
Possible large paramagnetic ground state with 0g ϕ =
Mott insulator with one S=1/2 spin per unit cell
bond bond
Another state breaking the symmetry of rotations by / 2 about lattice sites,which also has 0, where is the
nπΨ ≠ Ψ bond order parameter
( ) ( )arond
cb
tan j iii j
iji S S e −Ψ = ∑ i r r
bondΨ
Possible large paramagnetic ground state with 0g ϕ =
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
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→
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
⇒ ≠
⇒ =
Simplest large g effective action for the Aaµ
( )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⎛ ⎞= ∆ − ∆ +⎜ ⎟⎝ ⎠
±
∑ ∑∏∫
N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989).S. Sachdev and R. Jalabert, Mod. Phys. Lett. B 4, 1043 (1990).
S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002)
N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989).
For large e2 , low energy height configurations are in exact one-to-one correspondence with nearest-neighbor valence bond pairings of the sites square lattice
There is no roughening transition for three dimensional interfaces, which are smooth for all couplings
⇒⇒
N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989).
D.S. Fisher and J.D. Weeks, Phys. Rev. Lett. 50, 1077 (1983)There is a definite average height of the interfaceGround state has bond order.
.
( )2
,
11 expa a a a a aa aa
Z d i Ag µ τ
µ
ϕ δ ϕ ϕ ϕ η+
⎛ ⎞= − ⋅ +⎜ ⎟
⎝ ⎠∑ ∑∏∫
g0
Neel order 0ϕ ≠
?bond
Bond order0
Not present in LGW theory of orderϕ
Ψ ≠
or