Magnetic phases and critical points of insulators and superconductors
• Colloquium article in Reviews of Modern Physics, July 2003, cond-mat/0211005.
• cond-mat/0109419
Talks online:Sachdev
Quantum Phase TransitionsCambridge University Press
What is a quantum phase transition ?
Non-analyticity in ground state properties as a function of some control parameter g
E
g
True level crossing:
Usually a first-order transition
E
g
Avoided level crossing which becomes sharp in the infinite
volume limit:
second-order transition
T Quantum-critical
Why study quantum phase transitions ?
ggc• Theory for a quantum system with strong correlations: describe phases on either side of gc by expanding in deviation from the quantum critical point.
~ zcg g ν∆ −
Important property of ground state at g=gc : temporal and spatial scale invariance;
characteristic energy scale at other values of g:
• Critical point is a novel state of matter without quasiparticle excitations
• Critical excitations control dynamics in the wide quantum-critical region at non-zero temperatures.
OutlineI. Quantum Ising Chain
II. Coupled Dimer AntiferromagnetA. Coherent state path integral B. Quantum field theory near critical point
III. Coupled dimer antiferromagnet in a magnetic fieldBose condensation of “triplons”
IV. Magnetic transitions in superconductorsQuantum phase transition in a background Abrikosov flux lattice
V. Antiferromagnets with an odd number of S=1/2 spins per unit cell. Class A: Compact U(1) gauge theory: collinear spins,
bond order and confined spinons in d=2Class B: Z2 gauge theory: non-collinear spins, RVB,
visons, topological order, and deconfined spinons
VI. Conclusions
Single order parameter.
Multiple order
parameters.
I. Quantum I. Quantum IsingIsing chainchain
I. Quantum Ising Chain
( ) ( )
Degrees of freedom: 1 qubits, "large"
,
1 1 or , 2 2
j j
j jj j j j
j N N=
=→
↓
↓
↑
↑ ↑+ =← − ↓
…
0
Hamiltonian of decoupled qubits: x
jj
H Jg σ= − ∑ 2Jg
j→
j←
1 1
Coupling between qubits: z z
j jj
H J σ σ += − ∑
1 1
Prefers neighboring qubits
are
(not entangle
d)j j j j
either or+ +
↓ ↓↑ ↑
( )( )1 1j j j j+ ++ +← ←→ ←→ →← →
( )0 1 1x z zj j j
jJ gH H H σ σ σ += + = − +∑
Full Hamiltonian
leads to entangled states at g of order unity
Weakly-coupled qubitsGround state:
1 2
G
g
→→→→→→→→→→
→→→→ →→→
=
−←←− →
Lowest excited states:
jj →→→→ +→←= →→→→
Coupling between qubits creates “flipped-spin” quasiparticle states at momentum p
Entire spectrum can be constructed out of multi-quasiparticle states
jipxj
jp e= ∑
( ) ( )
( )
2 1
1
Excitation energy 4 sin2
Excitation gap 2 2
pap J O g
gJ J O g
ε −
−
= ∆ + +
∆ = − + p
∆
aπ
aπ
−
( )pε
( )1g
Dynamic Structure Factor : Cross-section to flip a to a (or vice versa) while transferring energy
and momentum
( , )
p
S p
ω
ω→ ←
ω
( ),S p ω( )( )Z pδ ω ε−
Three quasiparticlecontinuum
Quasiparticle pole
Structure holds to all orders in 1/g
At 0, collisions between quasiparticles broaden pole to a Lorentzian of width 1 where the
21is given by
Bk TB
T
k Te
ϕ ϕ
ϕ
τ τ
τ π−∆
>
=
phase coherence time
S. Sachdev and A.P. Young, Phys. Rev. Lett. 78, 2220 (1997)
~3∆
Weakly-coupled qubits ( )1g
Ground states:
2
G
g
=
−
↑ ↑↑↑↑↑↑↑↑↑↑
↑↑↑↑ −↑↑↓↑↑↑
Lowest excited states: domain walls
jjd ↓↓↑ ↓↑↑ ↓ +↑↑= ↓Coupling between qubits creates new “domain-
wall” quasiparticle states at momentum pjipx
jj
p e d= ∑( ) ( )
( )
2 2
2
Excitation energy 4 sin2
Excitation gap 2 2
pap Jg O g
J gJ O g
ε = ∆ + +
∆ = − +
p
∆
aπ
aπ
−
( )pε
Second state obtained by
and mix only at order N
G
G G g
↓ ↓
↑↓
⇔↑ 0
Ferromagnetic moment0zN G Gσ= ≠
Strongly-coupled qubits ( )1g
Dynamic Structure Factor : Cross-section to flip a to a (or vice versa) while transferring energy
and momentum
( , )
p
S p
ω
ω→ ←
ω
( ),S p ω( ) ( ) ( )22
0 2N pπ δ ω δ
Two domain-wall continuum
Structure holds to all orders in g
At 0, motion of domain walls leads to a finite ,
21and broadens coherent peak to a width 1 where Bk TB
T
k Te
ϕ
ϕϕ
τ
ττ π
−∆
>
=
phase coherence time
S. Sachdev and A.P. Young, Phys. Rev. Lett. 78, 2220 (1997)
~2∆
Strongly-coupled qubits ( )1g
Entangled states at g of order unity
ggc
“Flipped-spin” Quasiparticle
weight Z
( )1/ 4~ cZ g g−
A.V. Chubukov, S. Sachdev, and J.Ye, Phys. Rev. B 49, 11919 (1994)
ggc
Ferromagnetic moment N0
( )1/80 ~ cN g g−
P. Pfeuty Annals of Physics, 57, 79 (1970)
ggc
Excitation energy gap ∆ ~ cg g∆ −
Dynamic Structure Factor : Cross-section to flip a to a (or vice versa) while transferring energy
and momentum
( , )
p
S p
ω
ω→ ←
ω
( ),S p ω
Critical coupling ( )cg g=
c p
( ) 7 /82 2 2~ c pω−
−
No quasiparticles --- dissipative critical continuum
Quasiclassicaldynamics
Quasiclassicaldynamics
1/ 41~z z
j kj k
σ σ−
P. Pfeuty Annals of Physics, 57, 79 (1970)
( )∑ ++−=i
zi
zi
xiI gJH 1σσσ
( ) ( )
( )
0
7 / 4
( ) , 0
1 / ...
2 tan16
z z i tj k
k
R
BR
i dt t e
AT i
k T
ωχ ω σ σ
ω
π
∞
=
=− Γ +
Γ =
∑∫
S. Sachdev and J. Ye, Phys. Rev. Lett. 69, 2411 (1992).S. Sachdev and A.P. Young, Phys. Rev. Lett. 78, 2220 (1997).
OutlineI. Quantum Ising Chain
II. Coupled Dimer AntiferromagnetA. Coherent state path integral B. Quantum field theory near critical point
III. Coupled dimer antiferromagnet in a magnetic fieldBose condensation of “triplons”
IV. Magnetic transitions in superconductorsQuantum phase transition in a background Abrikosov flux lattice
V. Antiferromagnets with an odd number of S=1/2 spins per unit cell. Class A: Compact U(1) gauge theory: collinear spins,
bond order and confined spinons in d=2Class B: Z2 gauge theory: non-collinear spins, RVB,
visons, topological order, and deconfined spinons
VI. Conclusions
Single order parameter.
Multiple order
parameters.
II. Coupled II. Coupled DimerDimer AntiferromagnetAntiferromagnet
S=1/2 spins on coupled dimers
jiij
ij SSJH ⋅= ∑><
10 ≤≤ λ
JλJ
II. 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).
close to 1λSquare lattice antiferromagnetExperimental realization: 42CuOLa
Ground state has long-rangemagnetic (Neel) order
( ) 01 0 ≠−= + NS yx iii
Excitations: 2 spin waves (magnons) 2 2 2 2p x x y yc p c pε = +
close to 0λ Weakly coupled dimers
Paramagnetic ground state 0iS =
( )↓↑−↑↓=2
1
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∆ →
close to 0λ Weakly coupled dimers
( )↓↑−↑↓=2
1
S=1/2 spinons are confined by a linear potential into a S=1 triplon
λ 1
λc
Quantum paramagnet
0=S
Neelstate
0S N=
Neel order N0 Spin gap ∆
T=0
δ in cuprates ?
Key ingredient: Spin Berry PhasesKey ingredient: Spin Berry Phases
iSAe
A
Path integral for quantum spin fluctuations
II.A Coherent state path integral
iSAe
A
Path integral for quantum spin fluctuations
Key ingredient: Spin Berry PhasesKey ingredient: Spin Berry Phases
II.A Coherent state path integral
II.A Coherent state path integralSee Chapter 13 of Quantum Phase Transitions, S. Sachdev, Cambridge University Press (1999).
[ ]( )( ) ( ) ( )( )
/
2
Tr
1 exp ( )
H TZ e
iS A d d H Sττ δ τ τ τ τ
−=
= − − − ∫ ∫ ∫
S
N N ND
Path integral for a single spin
( )( ) ( ) 0
Oriented area of triangle on surface of unit sphere bounded
by , , and a fixed reference
A d
dτ τ τ
τ τ τ
=
+N N NAction for lattice antiferromagnet
( ) ( ) ( ), ,j j j jx xτ η τ τ= +N n L
1 identifies sublatticesjη = ±
n and L vary slowly in space and time
( ) ( )
( ) ( )( )
2
2 22 2
, 1 exp ( , )
1 2
1 identifies sublattices
j jj
x
j
Z x iS A x d
d x d cg
τ
τ
τ δ η τ τ
τ
η
= − −
− ∂ + ∇
= ±
∑∫ ∫
∫
Dn n
n n
Integrate out L and take the continuum limit
c
c
c
c
g g
g gλ λ
λ λ
< ⇔>
> ⇔
<
Discretize spacetime into a cubic lattice cubic lattice sites, ,
;;
ax yµ τ
→→
( ),
0
oriented area of spherical triangle
formed by and an arbitrary reference poi
11 exp
t
2
, , n
2a a a a a a
a aa
a
a a
iZ d Ag
A
µ τµ
µ
µ
δ η+
+
= − ⋅ −
→
∑ ∑∏∫ n n n n
n n n
( ) ( )
( ) ( )( )
2
2 22 2
, 1 exp ( , )
1 2
1 identifies sublattices
j jj
x
j
Z x iS A x d
d x d cg
τ
τ
τ δ η τ τ
τ
η
= − −
− ∂ + ∇
= ±
∑∫ ∫
∫
Dn n
n n
Integrate out L and take the continuum limit
Discretize spacetime into a cubic lattice cubic lattice sites, ,
;;
ax yµ τ
→→
c
c
c
c
g g
g gλ λ
λ λ
< ⇔>
> ⇔
<
( ),
11 exp2a a a a
aa
Z dg µ
µ
δ +
= − ⋅
∑∏∫ n n n n
Berry phases can be neglected for coupled dimer antiferromagent(justified later)
Quantum path integral for two-dimensional quantum antiferromagnetPartition function of a classical three-dimensional ferromagnetat a “temperature” g
Quantum transition at λ=λc is related to classical Curie transition at g=gc
⇔
λ close to λc : use “soft spin” field
αφ 3-component antiferromagneticorder parameter
Oscillations of about zero (for λ < λc ) spin-1 collective mode
αφ
T=0 spectrum
ω
Im ( , )pχ ω
( ) ( ) ( )( ) ( )22 22 2 2 212 4!b x c
ud xd cα τ α α ατ φ φ λ λ φ φ = ∇ + ∂ + − + ∫S
2 2
2pc pε = ∆ +
∆
II.B Quantum field theory for critical point
c cλ λ∆ = −
ω
( )Im ,pχ ω
Critical coupling ( )cλ λ=
c p
( ) (2 ) / 22 2 2~ c pη
ω− −
−
No quasiparticles --- dissipative critical continuum
Dynamic spectrum at the critical point
OutlineI. Quantum Ising Chain
II. Coupled Dimer AntiferromagnetA. Coherent state path integral B. Quantum field theory near critical point
III. Coupled dimer antiferromagnet in a magnetic fieldBose condensation of “triplons”
IV. Magnetic transitions in superconductorsQuantum phase transition in a background Abrikosov flux lattice
V. Antiferromagnets with an odd number of S=1/2 spins per unit cell. Class A: Compact U(1) gauge theory: collinear spins,
bond order and confined spinons in d=2Class B: Z2 gauge theory: non-collinear spins, RVB,
visons, topological order, and deconfined spinons
VI. Conclusions
Single order parameter.
Multiple order
parameters.
III. Coupled III. Coupled DimerDimer AntiferromagnetAntiferromagnet in a magnetic fieldin a magnetic field
Pressure,exchange constant,….
( )( )
cos .
sin .
jj
j
K r
K r
=
+
1
2
S N
N
Collinear spins: 0Non-collinear spins: 0
× =× ≠
1 2
1 2
N NN N
0j =S
T=0
Quantum critical point
Both states are insulators
SDW
H
Evolution of phase diagram in a magnetic field
Effect of a field on paramagnet
Energy of zero
momentum triplon states
H
∆
0
Bose-Einstein condensation of
Sz=1 triplon
H
1/λ
Spin singlet state with a spin gap
SDW
1 Tesla = 0.116 meVRelated theory applies to double layer quantum Hall systems at ν=2
III. Phase diagram in a magnetic field.
gµBH = ∆
H
1/λ
Spin singlet state with a spin gap
SDW
1 Tesla = 0.116 meVRelated theory applies to double layer quantum Hall systems at ν=2
III. Phase diagram in a magnetic field.
gµBH = ∆
( )( )
( )( ) ( )( )
2 *
2 2 2 2 2
2
Zeeman term leads to a uniform precession of spins
Take oriented along the direction.
.
, ~ , while for ,
Then
For
c x y c x y
c x c c c
i H i H
H
H H
H zτ α τ α ασρ σ ρ τ α αβγ β γφ φ ε φ φ ε φ
λ λ φ φ λ λ φ φ
λ λ φ λ λ λ λ
∂ ⇒ ∂ − ∂ −
− + ⇒ − − +
> − + < ~ cλ λ= ∆ −
( )( )
( )( ) ( )( )
2 *
2 2 2 2 2
2
Zeeman term leads to a uniform precession of spins
Take oriented along the direction.
.
, ~ , while for ,
Then
For
c x y c x y
c x c c c
i H i H
H
H H
H zτ α τ α ασρ σ ρ τ α αβγ β γφ φ ε φ φ ε φ
λ λ φ φ λ λ φ φ
λ λ φ λ λ λ λ
∂ ⇒ ∂ − ∂ −
− + ⇒ − − +
> − + < ~ cλ λ= ∆ −
H
1/λ
Spin singlet state with a spin gap
SDW
1 Tesla = 0.116 meVRelated theory applies to double layer quantum Hall systems at ν=2
III. Phase diagram in a magnetic field.
gµBH = ∆[ ]
[ ]2
Elastic scattering intensity
0
I H
HI aJ
=
+
~c cH λ λ−
III. Phase diagram in a magnetic field.
M
H∆
III. Phase diagram in a magnetic field.
M
H∆
At very large H, magnetization
saturates
1
III. Phase diagram in a magnetic field.
M
H∆
1
1/2
Respulsive interactions between triplons can lead to
magnetization plateau at any rational fraction
ij zi zji j
J S S<∑
III. Phase diagram in a magnetic field.
M
H∆
1
1/2
Quantum transitions in and out of plateau are
Bose-Einstein condensations of “extra/missing”
triplons
OutlineI. Quantum Ising Chain
II. Coupled Dimer AntiferromagnetA. Coherent state path integral B. Quantum field theory near critical point
III. Coupled dimer antiferromagnet in a magnetic fieldBose condensation of “triplons”
IV. Magnetic transitions in superconductorsQuantum phase transition in a background Abrikosov flux lattice
V. Antiferromagnets with an odd number of S=1/2 spins per unit cell. Class A: Compact U(1) gauge theory: collinear spins,
bond order and confined spinons in d=2Class B: Z2 gauge theory: non-collinear spins, RVB,
visons, topological order, and deconfined spinons
VI. Conclusions
Single order parameter.
Multiple order
parameters.
IV. Magnetic transitions in superconductorsIV. Magnetic transitions in superconductors
Pressure,carrier concentration,….
0j =S
T=0
Quantum critical point
We have so far considered the case where both states are insulators
SDW
( ) ( ) .
cos . sin .
= Re
j
j jj
i K r
K r K r
e
i
= +
≡ −
1 2
1 2
S N N
N N
Φ
Φ
Collinear spins: 0Non-collinear spins: 0
× =× ≠
1 2
1 2
N NN N
Pressure,carrier concentration,….
( ) ( ) .
cos . sin .
= Re
j
j jj
i K r
K r K r
e
i
= +
≡ −
1 2
1 2
S N N
N N
Φ
Φ
Collinear spins: 0Non-collinear spins: 0
× =× ≠
1 2
1 2
N NN N
0j =S
T=0
Quantum critical point
Now both sides have a “background” superconducting (SC) order
SC+SDW SC
ky
•
kx
π/a
π/a0
Insulator
δ~0.12-0.140.055SC
0.020
J. M. Tranquada et al., Phys. Rev. B 54, 7489 (1996). G. Aeppli, T.E. Mason, S.M. Hayden, H.A. Mook, J. Kulda, Science 278, 1432 (1997).
S. Wakimoto, G. Shirane et al., Phys. Rev. B 60, R769 (1999). Y.S. Lee, R. J. Birgeneau, M. A. Kastner et al., Phys. Rev. B 60, 3643 (1999)
S. Wakimoto, R.J. Birgeneau, Y.S. Lee, and G. Shirane, Phys. Rev. B 63, 172501 (2001).
(additional commensurability effects near δ=0.125)
T=0 phases of LSCOInterplay of SDW and SC order in the cuprates
SC+SDWSDWNéel
• •• •
ky
kx
π/a
π/a0
Insulator
δ~0.12-0.140.055SC
0.020
J. M. Tranquada et al., Phys. Rev. B 54, 7489 (1996). G. Aeppli, T.E. Mason, S.M. Hayden, H.A. Mook, J. Kulda, Science 278, 1432 (1997).
S. Wakimoto, G. Shirane et al., Phys. Rev. B 60, R769 (1999). Y.S. Lee, R. J. Birgeneau, M. A. Kastner et al., Phys. Rev. B 60, 3643 (1999)
S. Wakimoto, R.J. Birgeneau, Y.S. Lee, and G. Shirane, Phys. Rev. B 63, 172501 (2001).
(additional commensurability effects near δ=0.125)
T=0 phases of LSCO
SC+SDWSDWNéel
Interplay of SDW and SC order in the cuprates
•••Superconductor with Tc,min =10 K•
ky
kx
π/a
π/a0
δ~0.12-0.140.055SC
0.020
J. M. Tranquada et al., Phys. Rev. B 54, 7489 (1996). G. Aeppli, T.E. Mason, S.M. Hayden, H.A. Mook, J. Kulda, Science 278, 1432 (1997).
S. Wakimoto, G. Shirane et al., Phys. Rev. B 60, R769 (1999). Y.S. Lee, R. J. Birgeneau, M. A. Kastner et al., Phys. Rev. B 60, 3643 (1999)
S. Wakimoto, R.J. Birgeneau, Y.S. Lee, and G. Shirane, Phys. Rev. B 63, 172501 (2001).
(additional commensurability effects near δ=0.125)
T=0 phases of LSCO
SC+SDWSDWNéel
Interplay of SDW and SC order in the cuprates
Collinear magnetic (spin density wave) order
( ) ( )cos . sin .j jj K r K r= +1 2S N NCollinear spins
( ), 0K π π= =2; N
( )3 4, 0K π π= =2; N
( )
( )3 4,
2 1
K π π=
= −2 1
;
N N
•••Superconductor with Tc,min =10 K•
ky
kx
π/a
π/a0
δ~0.12-0.140.055SC
0.020
T=0 phases of LSCO
SC+SDWSDWNéel
Use simplest assumption of a direct second-order quantum phase transition between SC and SC+SDW phases
Interplay of SDW and SC order in the cuprates
If does not exactly connect two nodal points, critical theory is as in an insulator
K
Otherwise, new theory of coupled excitons and nodal quasiparticles
L. Balents, M.P.A. Fisher, C. Nayak, Int. J. Mod. Phys. B 12, 1033 (1998).
Magnetic transition in a d-wave superconductor
Coupling to the S=1/2 Bogoliubov quasiparticles of the d-wave superconductor
Trilinear “Yukawa” coupling
is prohibited unless ordering wavevector is fine-tuned.
2d rd ατ Φ ΨΨ∫
22 † is allowed
Scaling dimension of (1/ - 2) 0 irrelev t.an
d rd αα
κ τ
κ ν
Φ Ψ Ψ
= < ⇒
∑∫
( )2 22 2rd rd c Vα τ α ατ = ∇ Φ + ∂ Φ + Φ ∫S
Similar terms present in action for SDW ordering in the insulator
Magnetic transition in a d-wave superconductor
Neutron scattering measurements of dynamic spin correlations of the superconductor (SC) in a magnetic field
B. Lake, G. Aeppli, K. N. Clausen, D. F. McMorrow, K. Lefmann, N. E. Hussey, N. Mangkorntong, M. Nohara, H. Takagi, T. E. Mason,
and A. Schröder, Science 291, 1759 (2001).
2- 4Neutron scattering off La Sr CuO ( 0.163, ) δ δ δ = SC phase
Peaks at (0.5,0.5) (0.125,0)and (0.5,0.5) (0,0.125)
dynamic SDW of period 8
±±
⇒
red dat lo otsw temperatures in =0 ( ) and =7.5T blue d )s( otH H
Neutron scattering measurements of dynamic spin correlations of the superconductor (SC) in a magnetic field
B. Lake, G. Aeppli, K. N. Clausen, D. F. McMorrow, K. Lefmann, N. E. Hussey, N. Mangkorntong, M. Nohara, H. Takagi, T. E. Mason,
and A. Schröder, Science 291, 1759 (2001).
2- 4Neutron scattering off La Sr CuO ( 0.163, ) δ δ δ = SC phase
Peaks at (0.5,0.5) (0.125,0)and (0.5,0.5) (0,0.125)
dynamic SDW of period 8
±±
⇒
red dat lo otsw temperatures in =0 ( ) and =7.5T blue d )s( otH H
D. P. Arovas, A. J. Berlinsky, C. Kallin, and S.-C. Zhang, Phys. Rev. Lett. 79, 2871 (1997) proposed static magnetism localized within vortex cores, but signal was much larger
than anticipated.
Dominant effect of magnetic field: Abrikosov flux lattice
2 2
2
Spatially averaged superflow kinetic energy3 ln c
sc
HHvH H
∼ ∼
1sv
r∼
r
( )1/ 2 22 2 2 22 2 21 2
0 2 2
T
b rg gd r d c sα τ α α α ατ = ∇ Φ + ∂ Φ + Φ + Φ + Φ ∫ ∫S
2 22
2c d rd ατ ψ = Φ ∫S v
( )4
222
2GL rF d r iAψ
ψ ψ
= − + + ∇ −
∫
( ) ( )( )
( )
,
ln 0
GL b cFZ r D r e
Z rr
ψ τ
δ ψδψ
− − −= Φ
=
∫ S S
(extreme Type II superconductivity)Effect of magnetic field on SDW+SC to SC transition
Quantum theory for dynamic and critical spin fluctuations
Static Ginzburg-Landau theory for non-critical superconductivity
1 2N iNα α αΦ = −
Triplon wavefunction in bare potential V0(x)
Energy
x0
Spin gap ∆
Vortex cores
( ) ( ) 20
Bare triplon potential
V s ψ= +r rv
( ) ( ) ( ) 20
Wavefunction of lowest energy triplon
after including triplon interactions: V V g
α
α
Φ
= + Φr r r
E. Demler, S. Sachdev, and Y
. Zhang, . , 067202 (2001).A.J. Bray and
repulsive interactions between excitons imply that triplons must be extended as 0.
Phys. Rev. Lett
Strongly relevant∆ →
87 M.A. Moore, . C , L7 65 (1982).
J.A. Hertz, A. Fleishman, and P.W. Anderson, . , 942 (1979).J. Phys
Phys. Rev. Lett15
43
Energy
x0
Spin gap ∆
Vortex cores
( ) ( ) 20
Bare triplon potential
V s ψ= +r rv
2 2
2
Spatially averaged superflow kinetic energy3 ln c
sc
H HvH H
∝
1sv
r∝
r
Phase diagram of SC and SDW order in a magnetic field
( ) 2eff
2
The suppression of SC order appears to the SDW order as a effective "doping" :3 ln c
c
HHH CH H
δ
δ δ = −
uniform
E. Demler, S. Sachdev, and Ying Zhang, Phys. Rev. Lett. 87, 067202 (2001).
E. Demler, S. Sachdev, and Ying Zhang, Phys. Rev. Lett. 87, 067202 (2001).
( )
( )( )
eff
( )~ln 1/
c
c
c
H
H
δ δ
δ δδ δ
= ⇒
−−
Phase diagram of SC and SDW order in a magnetic field
[ ] [ ]
[ ]
eff
2
2
Elastic scattering intensity, 0,
3 0, ln c
c
I H I
HHI aH H
δ δ
δ
≈
≈ +
Structure of long-range SDW order in SC+SDW phase
Magnetic order
parameter
( ) ( ) ( ) ( )3 2
Dynamic structure factor
, 2
reciprocal lattice vectors of vortex lattice. measures deviation from SDW ordering wavevector
S fω π δ ω δ= − +
→
∑ GG
k k G
G k K
s – sc = -0.3
20 ln(1/ )f H Hδ ∝
E. Demler, S. Sachdev, and Ying Zhang, Phys. Rev. Lett. 87, 067202 (2001).
•••Superconductor with Tc,min =10 K•
ky
kx
π/a
π/a0
δ~0.12-0.140.055SC
0.020
T=0 phases of LSCO
SC+SDWSDWNéel
H
Follow intensity of elastic Bragg spots in a magnetic field
Use simplest assumption of a direct second-order quantum phase transition between SC and SC+SDW phases
Interplay of SDW and SC order in the cuprates
2- 4Neutron scattering of La Sr CuO at =0.1x x x
B. Lake, H. M. Rønnow, N. B. Christensen, G. Aeppli, K. Lefmann, D. F. McMorrow, P. Vorderwisch, P. Smeibidl, N. Mangkorntong, T. Sasagawa, M. Nohara, H. Takagi, T. E. Mason, Nature, 415, 299 (2002).
2
2
Solid line - fit ( ) nto : l c
c
HHI H aH H
=
See also S. Katano, M. Sato, K. Yamada, T. Suzuki, and T. Fukase, Phys. Rev. B 62, R14677 (2000).
E. Demler, S. Sachdev, and Ying Zhang, Phys. Rev. Lett. 87, 067202 (2001).
Neutron scattering observation of SDW order enhanced by
superflow.
( )
( )( )
eff
( )~ln 1/
c
c
c
H
H
δ δ
δ δδ δ
= ⇒
−−
Phase diagram of a superconductor in a magnetic field
Prediction: SDW fluctuations enhanced by superflow and bond order pinned by vortex cores (no
spins in vortices). Should be observable in STM
K. Park and S. Sachdev Physical Review B 64, 184510 (2001); Y. Zhang, E. Demler and S. Sachdev, Physical Review B 66, 094501 (2002).
( ) ( ) 2
2
1 triplon energy30 ln c
c
SHHH b
H Hε ε
=
= −
100Å
b7 pA
0 pA
Vortex-induced LDOS of Bi2Sr2CaCu2O8+δ integrated from 1meV to 12meV
J. Hoffman E. W. Hudson, K. M. Lang, V. Madhavan, S. H. Pan, H. Eisaki, S. Uchida, and J. C. Davis, Science 295, 466 (2002).
Our interpretation: LDOS modulations are
signals of bond order of period 4 revealed in
vortex halo
See also: S. A. Kivelson, E. Fradkin, V. Oganesyan, I. P. Bindloss, J. M. Tranquada, A. Kapitulnik, and C. Howald, cond-mat/0210683.
Fourier Transform of Vortex-Induced LDOS map
J. Hoffman et al. Science, 295, 466 (2002).
K-space locations of vortex induced LDOS
Distances in k –space have units of 2π/a0a0=3.83 Å is Cu-Cu distance
K-space locations of Bi and Cu atoms
C. Howald, H. Eisaki, N. Kaneko, and A. Kapitulnik, Phys. Rev. B 67, 014533 (2003).
Spectral properties of the STM signal are sensitive to the microstructure of the charge order
Measured energy dependence of the Fourier component of the density of states which modulates with a period of 4 lattice spacings M. Vojta, Phys. Rev. B 66, 104505 (2002);
D. Podolsky, E. Demler, K. Damle, and B.I. Halperin, Phys. Rev. B in press, cond-mat/0204011
Theoretical modeling shows that this spectrum is best obtained by a modulation of bond variables, such as the exchange, kinetic or pairing energies.
OutlineI. Quantum Ising Chain
II. Coupled Dimer AntiferromagnetA. Coherent state path integral B. Quantum field theory near critical point
III. Coupled dimer antiferromagnet in a magnetic fieldBose condensation of “triplons”
IV. Magnetic transitions in superconductorsQuantum phase transition in a background Abrikosov flux lattice
V. Antiferromagnets with an odd number of S=1/2 spins per unit cell. Class A: Compact U(1) gauge theory: collinear spins,
bond order and confined spinons in d=2Class B: Z2 gauge theory: non-collinear spins, RVB,
visons, topological order, and deconfined spinons
VI. Conclusions
Single order parameter.
Multiple order
parameters.
V.V. AntiferromagnetsAntiferromagnets with an odd number with an odd number of S=1/2 spins per unit cellof S=1/2 spins per unit cell
Class AClass A
V. Order in Mott insulatorsMagnetic order ( ) ( )cos . sin . j jj K r K r= +1 2S N N
Class A. Collinear spins
( ), 0K π π= =2; N
( )3 4, 0K π π= =2; N
( )
( )3 4,
2 1
K π π=
= −2 1
;
N N
( ) ( )cos . sin . j jj K r K r= +1 2S N NV. Order in Mott insulatorsMagnetic order
Class A. Collinear spins
Order specified by a single vector N.
Quantum fluctuations leading to loss of
magnetic order should produce a paramagnetic state with a vector (S=1) quasiparticle excitation.
Key property
Class A: Collinear spins and compact U(1) gauge theory
Key ingredient: Spin Berry PhasesKey ingredient: Spin Berry Phases
iSAe
A
Write down path integral for quantum spin fluctuations
Class A: Collinear spins and compact U(1) gauge theory
iSAe
A
Write down path integral for quantum spin fluctuations
Key ingredient: Spin Berry PhasesKey ingredient: Spin Berry Phases
Class A: Collinear spins and compact U(1) gauge theory
S=1/2 square lattice antiferromagnet with non-nearest neighbor exchange
ij i ji j
H J S S<
= ⋅∑Include Berry phases after discretizing coherent state path
integral on a cubic lattice in spacetime
( ),
a 1 on two square sublattices ;
Neel order parameter; oriented area of spheri
11
cal trian
exp2
~g
l
2a a a a a a
a aa
a a a
a
iZ d Ag
SA
µ τµ
µ
δ η
η
η
+
→ ±
= − ⋅ −
→
→
∑ ∑∏∫ n n n n
n
0, e
formed by and an arbitrary reference point ,a a µ+n n n
Spin-wave theory about Neel state receives minor modifications from Berry phases.
Berry phases are crucial in determining structure of "qua
nt
g
g
→
→
Small
Largeum-disordered" phase with
0
a
a aA µ
=
Integrate out to obtain effective action forn
n
( ),
11 exp2
2a a a a a a
a aa
iZ d Ag µ τ
µ
δ η+
= − ⋅ −
∑ ∑∏∫ n n n n
a µ+n
0n
an
aA µ
a µ+n
0n
an
aA µ
aγ a µγ +
Change in choice of n0 is like a “gauge transformation”
a a a aA Aµ µ µγ γ+→ − +
(γa is the oriented area of the spherical triangle formed by na and the two choices for n0 ).
aA µ
The area of the triangle is uncertain modulo 4π, and the action is invariant under4a aA Aµ µ π→ +
These principles strongly constrain the effective action for Aaµ which provides description of the large g phase
0′n
( )2,
2 2withThis is compact QED in
1 1exp
+1 dimensions with static charges 1 on two sublattic
cos2 22
e
~
.
s
a a a a aaa
d
iZ dA A A
e g
Aeµ µ ν ν µ τ
µ
η
±
= ∆ − ∆ −
∑ ∑∏∫
Simplest large g effective action for the Aaµ
This theory can be reliably analyzed by a duality mapping.
d=2: The gauge theory is always in a confiningconfining phase and there is bond order in the ground state.
d=3: A deconfined phase with a gapless “photon” is possible.
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).
For large e2 , low energy height configurations are in exact one-to-one correspondence with dimer coverings of the square lattice
2+1 dimensional height model is the path integral of the Quantum Dimer Model⇒
There is no roughening transition for three dimensional interfaces, which are smooth for all couplings
There is a definite average height of the interfaceGround state has bond order.
⇒⇒
V. Order in Mott insulatorsParamagnetic states 0j =S
Class A. Bond order and spin excitons in d=2
( )↓↑−↑↓=2
1
N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989).
S=1/2 spinons are confinedby a linear potential into a
S=1 spin triplon
Spontaneous bond-order leads to vector S=1 spin excitations
A. W. Sandvik, S. Daul, R. R. P. Singh, and D. J. Scalapino, Phys. Rev. Lett. 89, 247201 (2002)
Bond order in a frustrated S=1/2 XY magnet
( ) ( )2 x x y yi j i j i j k l i j k l
ij ijklH J S S S S K S S S S S S S S+ − + − − + − +
⊂
= + − +∑ ∑g=
First large scale numerical study of the destruction of Neel order in a S=1/2antiferromagnet with full square lattice symmetry
OutlineI. Quantum Ising Chain
II. Coupled Dimer AntiferromagnetA. Coherent state path integral B. Quantum field theory near critical point
III. Coupled dimer antiferromagnet in a magnetic fieldBose condensation of “triplons”
IV. Magnetic transitions in superconductorsQuantum phase transition in a background Abrikosov flux lattice
V. Antiferromagnets with an odd number of S=1/2 spins per unit cell. Class A: Compact U(1) gauge theory: collinear spins,
bond order and confined spinons in d=2Class B: Z2 gauge theory: non-collinear spins, RVB,
visons, topological order, and deconfined spinons
VI. Conclusions
Single order parameter.
Multiple order
parameters.
V. V. AntiferromagnetsAntiferromagnets with an odd number with an odd number of of SS=1/2 spins per unit cell =1/2 spins per unit cell
Class BClass B
V.B Order in Mott insulatorsMagnetic order
Class B. Noncollinear spins
( )3 4,K π π=(B.I. Shraiman and E.D. Siggia, Phys. Rev. Lett. 61, 467 (1988))
( ) ( )cos . sin . j jj K r K r= +1 2S N N
2 , 0= =22 1 1 2N N N .N
( )4 / 3, 4 3K π π=
V.B Order in Mott insulatorsMagnetic order
Class B. Noncollinear spins
( )
( )
2 2
2 2
Solve constraints by expressing in terms of two complex numbers ,
2
Order in ground state specified by a spinor ,
z z
z z
i i z z
z z
z z
↑ ↓
↓ ↑
↓ ↑
↑ ↓
↑ ↓
−
+ = +
1,2
1 2
N
N N
(modulo an overall sign).
This spinor can become a =1/2 spinon in paramagnetic state.S
( ) ( )cos . sin . j jj K r K r= +1 2S N N
A. V. Chubukov, S. Sachdev, and T. Senthil Phys. Rev. Lett. 72, 2089 (1994)
2 , 0= =22 1 1 2N N N .N
3 2
2
Order parameter space: Physical observables are invariant under the gauge transformation a a
S ZZ z z→ ±
V.B Order in Mott insulatorsMagnetic order
Class B. Noncollinear spins
( ) ( )cos . sin . j jj K r K r= +1 2S N N
S3
(A) North pole
(B) South pole x
y(A)
(B)
Vortices associated with π1(S3/Z2)=Z2 (visons)
Such vortices (visons) can also be defined in the phase in which spins are “quantum disordered”. A Z2 spin liquid with deconfined spinons must
have visons supressedN. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773 (1991)
Model effective action and phase diagram
2
h.c.
gau ge fie d l
ij i j ijij
ij
J z z K
Z
α α σ
σ
σ ∗= − +
→
−∑ ∑ ∏S (Derivation using Schwinger bosons on a quantum antiferromagnet: S. Sachdev and
N. Read, Int. J. Mod. Phys. B 5, 219 (1991)).
P. E. Lammert, D. S. Rokhsar, and J. Toner, Phys. Rev. Lett. 70, 1650 (1993) ; Phys. Rev. E 52, 1778 (1995). (For nematic liquid crystals)
Magnetically ordered
Confined spinonsFree spinons andtopological order
First order transition
P. Fazekas and P.W. Anderson, Phil Mag 30, 23 (1974).G. Misguich and C. Lhuillier, Eur. Phys. J. B 26, 167 (2002).R. Moessner and S.L. Sondhi, Phys. Rev. Lett. 86, 1881 (2001).
A topologically ordered state in which vortices associated with π1(S3/Z2)=Z2 [“visons”] are gapped out. This is an RVB state with
deconfined S=1/2 spinons za
Recent experimental realization: Cs2CuCl4R. Coldea, D.A. Tennant, A.M. Tsvelik, and Z. Tylczynski, Phys. Rev. Lett. 86, 1335 (2001).
N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773 (1991). X. G. Wen, Phys. Rev. B 44, 2664 (1991). A.V. Chubukov, T. Senthil and S. S., Phys. Rev. Lett.72, 2089 (1994). T. Senthil and M.P.A. Fisher, Phys. Rev. B 62, 7850 (2000).
V.B Order in Mott insulatorsParamagnetic states 0j =S
Class B. Topological order and deconfined spinons
V.B Order in Mott insulatorsParamagnetic states 0j =S
Class B. Topological order and deconfined spinons
Number of valence bonds cutting line is conserved
modulo 2. Changing sign of each such bond does not
modify state. This is equivalent to a Z2 gauge transformation
with on sites to the right of dashed line.
Number of valence bonds cutting line is conserved
modulo 2. Changing sign of each such bond does not
modify state. This is equivalent to a Z2 gauge transformation
with on sites to the right of dashed line.
Direct description of topological order with valence bonds
a az z→ −
D. Rokhsar and S.A. Kivelson, Phys. Rev. Lett. 61, 2376 (1988); N. Read and B. Chakraborty, Phys. Rev. B 40, 7133 (1989).
V.B Order in Mott insulatorsParamagnetic states 0j =S
Class B. Topological order and deconfined spinons
Number of valence bonds cutting line is conserved
modulo 2. Changing sign of each such bond does not
modify state. This is equivalent to a Z2 gauge transformation
with on sites to the right of dashed line.
Number of valence bonds cutting line is conserved
modulo 2. Changing sign of each such bond does not
modify state. This is equivalent to a Z2 gauge transformation
with on sites to the right of dashed line.
Direct description of topological order with valence bonds
a az z→ −
D. Rokhsar and S.A. Kivelson, Phys. Rev. Lett. 61, 2376 (1988); N. Read and B. Chakraborty, Phys. Rev. B 40, 7133 (1989).
V.B Order in Mott insulatorsParamagnetic states 0j =S
Class B. Topological order and deconfined spinons
Terminating the line creates a plaquette with Z2 flux at the X
--- a vison.
Terminating the line creates a plaquette with Z2 flux at the X
--- a vison.
Direct description of topological order with valence bonds
X
D. Rokhsar and S.A. Kivelson, Phys. Rev. Lett. 61, 2376 (1988); N. Read and B. Chakraborty, Phys. Rev. B 40, 7133 (1989).
Effect of flux-piercing on a topologically ordered quantum paramagnetN. E. Bonesteel, Phys. Rev. B 40, 8954 (1989). G. Misguich, C. Lhuillier, M. Mambrini, and P. Sindzingre, Eur. Phys. J. B 26, 167 (2002).
DD
a DΨ = ∑D =
1 2 3Lx-1Lx-2 Lx
Ly
Φ
Effect of flux-piercing on a topologically ordered quantum paramagnetN. E. Bonesteel, Phys. Rev. B 40, 8954 (1989). G. Misguich, C. Lhuillier, M. Mambrini, and P. Sindzingre, Eur. Phys. J. B 26, 167 (2002).
DD
a DΨ = ∑D =
1 2 3Lx-1Lx-2 Lx
( )Number of bonds cutting dashed line
After flux insertion
1 ;
D
D
⇒
−
vison
Equivalent to inserting a inside hole of the torus.
This leads to a ground state degeneracy.
vison
Ly
VI. ConclusionsI. Quantum Ising Chain
II. Coupled Dimer AntiferromagnetA. Coherent state path integral B. Quantum field theory near critical point
III. Coupled dimer antiferromagnet in a magnetic fieldBose condensation of “triplons”
IV. Magnetic transitions in superconductorsQuantum phase transition in a background Abrikosov flux lattice
V. Antiferromagnets with an odd number of S=1/2 spins per unit cell. Class A: Compact U(1) gauge theory: collinear spins,
bond order and confined spinons in d=2Class B: Z2 gauge theory: non-collinear spins, RVB,
visons, topological order, and deconfined spinons
VI. Cuprates are best understood as doped class A Mott insulators.
Single order parameter.
Multiple order
parameters.
Competing order parameters in the cuprate superconductors
1. Pairing order of BCS theory (SC)
(Bose-Einstein) condensation of d-wave Cooper pairs
Orders (possibly fluctuating) associated with Orders (possibly fluctuating) associated with proximate Mott insulator in class Aproximate Mott insulator in class A
2. Collinear magnetic order (CM)
3. Bond/charge/stripe order (B)
(couples strongly to half-breathing phonons)
S. Sachdev and N. Read, Int. J. Mod. Phys. B 5, 219 (1991). M. Vojta and S. Sachdev, Phys. Rev. Lett. 83, 3916 (1999); M. Vojta, Y. Zhang, and S. Sachdev, Phys. Rev. B 62, 6721 (2000); M. Vojta, Phys. Rev. B 66, 104505 (2002).
Evidence cuprates are in class A
Evidence cuprates are in class A
• Neutron scattering shows collinear magnetic order co-existing with superconductivity
J. M. Tranquada et al., Phys. Rev. B 54, 7489 (1996). Y.S. Lee, R. J. Birgeneau, M. A. Kastner et al., Phys. Rev. B 60, 3643 (1999). S. Wakimoto, R.J. Birgeneau, Y.S. Lee, and G. Shirane, Phys. Rev. B 63, 172501 (2001).
Evidence cuprates are in class A
• Neutron scattering shows collinear magnetic order co-existing with superconductivity
• Proximity of Z2 Mott insulators requires stable hc/e vortices, visongap, and Senthil flux memory effect
S. Sachdev, Physical Review B 45, 389 (1992)N. Nagaosa and P.A. Lee, Physical Review B 45, 966 (1992)T. Senthil and M. P. A. Fisher, Phys. Rev. Lett. 86, 292 (2001).D. A. Bonn, J. C. Wynn, B. W. Gardner, Y.-J. Lin, R. Liang, W. N. Hardy, J. R. Kirtley, and K. A. Moler, Nature 414, 887 (2001).J. C. Wynn, D. A. Bonn, B. W. Gardner, Y.-J. Lin, R. Liang, W. N. Hardy, J. R. Kirtley, and K. A. Moler, Phys. Rev. Lett. 87, 197002 (2001).
Evidence cuprates are in class A
• Neutron scattering shows collinear magnetic order co-existing with superconductivity
• Proximity of Z2 Mott insulators requires stable hc/e vortices, visongap, and Senthil flux memory effect
• Non-magnetic impurities in underdoped cuprates acquire a S=1/2 moment
Effect of static non-magnetic impurities (Zn or Li)
Zn
Zn
Spinon confinement implies that free S=1/2 moments form near each impurity
Zn
Zn
impurity( 1)( 0)3 B
S STk T
χ +→ =
J. Bobroff, H. Alloul, W.A. MacFarlane, P. Mendels, N. Blanchard, G. Collin, and J.-F. Marucco, Phys. Rev. Lett. 86, 4116 (2001).
Inverse local susceptibilty
in YBCO
7Li NMR below Tc
impurityMeasured with 1/ 2 in underdoped sample.
This behavior does not emerge out of BCS the
( 1)(
ory
0)3
.B
S STk T
Sχ =+
→ =
A.M Finkelstein, V.E. Kataev, E.F. Kukovitskii, G.B. Teitel’baum, Physica C 168, 370 (1990).
Spatially resolved NMR of Zn/Li impurities in the superconducting state
Evidence cuprates are in class A
• Neutron scattering shows collinear magnetic order co-existing with superconductivity
• Proximity of Z2 Mott insulators requires stable hc/e vortices, visongap, and Senthil flux memory effect
• Non-magnetic impurities in underdoped cuprates acquire a S=1/2 moment
Evidence cuprates are in class A
• Neutron scattering shows collinear magnetic order co-existing with superconductivity
• Proximity of Z2 Mott insulators requires stable hc/e vortices, visongap, and Senthil flux memory effect
• Non-magnetic impurities in underdoped cuprates acquire a S=1/2 moment
• Tests of phase diagram in a magnetic field
E. Demler, S. Sachdev, and Ying Zhang, Phys. Rev. Lett. 87, 067202 (2001).
Neutron scattering observation of SDW order enhanced by
superflow.
( )
( )( )
eff
( )~ln 1/
c
c
c
H
H
δ δ
δ δδ δ
= ⇒
−−
Phase diagram of a superconductor in a magnetic field
E. Demler, S. Sachdev, and Ying Zhang, Phys. Rev. Lett. 87, 067202 (2001).
Neutron scattering observation of SDW order enhanced by
superflow.
Phase diagram of a superconductor in a magnetic field
Possible STM observation of predicted bond order in halo
around vortices
K. Park and S. Sachdev Physical Review B 64, 184510 (2001); Y. Zhang, E. Demler and S. Sachdev, Physical Review B 66, 094501 (2002).
( )
( )( )
eff
( )~ln 1/
c
c
c
H
H
δ δ
δ δδ δ
= ⇒
−−
Doping a paramagnetic bond-ordered Mott insulatorsystematic Sp(N) theory of translational symmetry breaking, while
preserving spin rotation invariance.
S. Sachdev and N. Read, Int. J. Mod. Phys. B 5, 219 (1991).
d-wave superconductor
Superconductor with co-existing
bond-order
Mott insulator with bond-order
T=0
VI. Doping Class A
A phase diagram
•Pairing order of BCS theory (SC)
•Collinear magnetic order (CM)
•Bond order (B)
S. Sachdev and N. Read, Int. J. Mod. Phys. B 5, 219 (1991). M. Vojta and S. Sachdev, Phys. Rev. Lett. 83, 3916 (1999); M. Vojta, Y. Zhang, and S. Sachdev, Phys. Rev. B 62, 6721 (2000); M. Vojta, Phys. Rev. B 66, 104505 (2002).
Microscopic theory for the interplay of bond (B) and d-wave
superconducting (SC) order
Vertical axis is any microscopic parameter which suppresses
CM order