Magnetic phases and critical points of insulators and...
Transcript of Magnetic phases and critical points of insulators and...
Magnetic phases and critical points of insulators and superconductors
Colloquium article:Reviews of Modern Physics, 75, 913 (2003).
Talks online:Sachdev
What is a quantum phase transition ?Non-analyticity in ground state properties as a function of some control parameter g
Why study quantum phase transitions ?
T Quantum-critical
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. • 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.
OutlineOutline
A. Coupled dimer antiferromagnetEffect of a magnetic field
B. Magnetic transitions in a superconductorEffect of a magnetic field
C. Spin gap state on the square latticeSpontaneous bond order
(A) InsulatorsCoupled dimer antiferromagnet
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 1λSquare lattice antiferromagnetExperimental realization: 42CuOLa
Ground state has long-rangemagnetic (Neel or spin density wave) 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
( )↓↑−↑↓=2
1
0iS =Paramagnetic ground state
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 ?
λc = 0.52337(3)M. Matsumoto, C. Yasuda, S. Todo, and H. Takayama,
Phys. Rev. B 65, 014407 (2002)
TlCuCl3M. Matsumoto, B. Normand, T.M. Rice, and
M. Sigrist, cond-mat/0309440.
J. Phys. Soc. Jpn72, 1026 (2003)
Field theory for quantum criticality
λ close to λc : use “soft spin” field
αφ 3-component antiferromagnetic order parameter
( ) ( ) ( )( ) ( )22 22 2 2 212 4!b x c
ud xd cα τ α α ατ φ φ λ λ φ φ⎡ ⎤= ∇ + ∂ + − +⎢ ⎥⎣ ⎦∫S
Quantum criticality described by strongly-coupled critical theory with universal dynamic response functions dependent on
Triplon scattering amplitude is determined by kBT alone, and not by the value of microscopic coupling u
S. Sachdev and J. Ye, Phys. Rev. Lett. 69, 2411 (1992).
Bk Tω
( ) ( ), BT T g k Tηχ ω ω=
(A) InsulatorsCoupled dimer antiferromagnet:
effect of a magnetic field.
Effect of a field on paramagnet
Energy of zero
momentum triplon states
∆
0
Bose-Einstein condensation of
Sz=1 triplon
H
TlCuCl3
Ch. Rüegg, N. Cavadini, A. Furrer, H.-U. Güdel, K. Krämer, H. Mutka, A. Wildes, K. Habicht, and P. Vorderwisch, Nature 423, 62 (2003).
TlCuCl3
Ch. Rüegg, N. Cavadini, A. Furrer, H.-U. Güdel, K. Krämer, H. Mutka, A. Wildes, K. Habicht, and P. Vorderwisch, Nature 423, 62 (2003).
“Spin wave (phonon) above critical field
Phase diagram in a magnetic field.
1/λ
Spin singlet state with a spin gap
SDW
1 Tesla = 0.116 meV
HgµBH = ∆
( )( )
( )( ) ( )( )
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= ∆ −
Related theory applies to double layer quantum Hall systems at ν=2
Phase diagram in a magnetic field.
( )( )
( )( ) ( )( )
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 meV
gµBH = ∆[ ]
[ ]2
Elastic scattering intensity
0
I H
HI aJ
=
⎛ ⎞+ ⎜ ⎟⎝ ⎠
~c cH λ λ−
Related theory applies to double layer quantum Hall systems at ν=2
TlCuCl3
M. Matsumoto, B. Normand, T.M. Rice,
and M. Sigrist, cond-mat/0309440.
(B) SuperconductorsMagnetic transitions in a superconductor:
effect of a magnetic field.
Interplay of SDW and SC order in the cuprates
T=0 phases of LSCOky
•
kx
π/a
π/a0
Insulator
δ~0.12-0.140.0550.020SCSC+SDWSDWNéel
(additional commensurability effects near δ=0.125)
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).
Interplay of SDW and SC order in the cuprates
T=0 phases of LSCOky
• •• •
kxπ/a0
Insulatorπ/a
δ~0.12-0.140.0550.020SCSC+SDWSDWNéel
(additional commensurability effects near δ=0.125)
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).
Interplay of SDW and SC order in the cuprates
T=0 phases of LSCOky
••• • Superconductor with Tc,min =10 K
kxπ/a0
π/a
δ~0.12-0.140.0550.020SCSC+SDWSDWNéel
(additional commensurability effects near δ=0.125)
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).
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
Interplay of SDW and SC order in the cuprates
T=0 phases of LSCO
•••Superconductor with Tc,min =10 K•
ky
kx
π/a
π/a0
δ~0.12-0.140.055SC
0.020SC+SDWSDWNéel
H
Follow intensity of elastic Bragg spots in a magnetic field
Use simplest assumption of a direct seco een nd-order quantum phase transition betwSC and SC+SDW phases
Dominant effect of magnetic field: Abrikosov flux lattice
1sv
r∼
r
2 2
2
Spatially averaged superflow kinetic energy3 ln c
sc
HHvH H
∼ ∼
Effect of magnetic field on SDW+SC to SC transition (extreme Type II superconductivity)
Quantum theory for dynamic and critical spin fluctuations1 2N iNα α αΦ = −
( )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
( ) ( )( )
( )
,
ln 0
GL b cFZ r D r e
Z rr
ψ τ
δ ψδψ
− − −= Φ⎡ ⎤⎣ ⎦
⎡ ⎤⎣ ⎦ =
∫ S S
2 22
2c d rd ατ ψ⎡ ⎤= Φ⎢ ⎥⎣ ⎦∫S v
( )4
222
2GL rF d r iAψ
ψ ψ⎡ ⎤
= − + + ∇ −⎢ ⎥⎢ ⎥⎣ ⎦
∫
Static Ginzburg-Landau theory for non-critical superconductivity
Triplon wavefunction in bare potential V0(x)
Energy
x0
Spin gap ∆
Vortex cores
( ) ( ) 20
Bare triplon potential
V s ψ= +r rv
D. P. Arovas, A. J. Berlinsky, C. Kallin, and S.-C. Zhang, Phys. Rev. Lett. 79, 2871 (1997) proposed static magnetism
(with ∆=0) localized within vortex cores
( ) ( ) ( ) 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
Phase diagram of SC and SDW order in a magnetic field
2 2
2
Spatially averaged superflow kinetic energy3 ln c
sc
H HvH H
∝
1sv
r∝
r
( ) 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).
Phase diagram of SC and SDW order in a magnetic field
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
= ⇒
−−
[ ] [ ]
[ ]
eff
2
2
Elastic scattering intensity, 0,
3 0, ln c
c
I H I
HHI aH H
δ δ
δ
≈
⎛ ⎞≈ + ⎜ ⎟⎝ ⎠
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).
Neutron scattering measurements of static spin correlations of the superconductor+spin-density-wave (SC+CM) in a magnetic field
( )( )
2
2
2
Solid line --- fit to :
is the only fitting parameterBest fit value - = 2.4 with
3.01 l
= 6
n
0 T
0
c
c
c
I H HHH
a
aI H
a H
⎛ ⎞= + ⎜ ⎟⎝ ⎠
H (Tesla)
2 4
B. Khaykovich, Y. S. Lee, S. Wakimoto, K. J. Thomas, M. A. Kastner, and R.J. Birge
Elastic neutron scatt
neau, B , 014528 (2002)
ering off La C O
.
u y
Phys. Rev.
+
66
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.
( )
( )( )
eff
( )~ln 1/
c
c
c
H
H
δ δ
δ δδ δ
= ⇒
−−
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
=
⎛ ⎞= − ⎜ ⎟⎝ ⎠
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
STM around vortices induced by a magnetic field in the superconducting stateJ. E. Hoffman, E. W. Hudson, K. M. Lang, V. Madhavan, S. H. Pan,
H. Eisaki, S. Uchida, and J. C. Davis, Science 295, 466 (2002).
-120 -80 -40 0 40 80 1200.0
0.5
1.0
1.5
2.0
2.5
3.0
Regular QPSR Vortex
Diff
eren
tial C
ondu
ctan
ce (n
S)
Sample Bias (mV)
Local density of states
1Å spatial resolution image of integrated
LDOS of Bi2Sr2CaCu2O8+δ
( 1meV to 12 meV) at B=5 Tesla.
S.H. Pan et al. Phys. Rev. Lett. 85, 1536 (2000).
Vortex-induced LDOS of Bi2Sr2CaCu2O8+δ integrated from 1meV to 12meV
100Å
b7 pA
0 pA
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.
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).
(C) Spin gap state on the square lattice:Spontaneous bond order
Paramagnetic ground state of coupled ladder model
Can such a state with bond order be the ground state of a system with full square lattice symmetry ?
Collinear spins and compact U(1) gauge theory
Write down path integral for quantum spin fluctuations
Key ingredient: Spin Berry PhasesKey ingredient: Spin Berry Phases
AiSAe
Collinear spins and compact U(1) gauge theory
Write down path integral for quantum spin fluctuations
Key ingredient: Spin Berry PhasesKey ingredient: Spin Berry Phases
AiSAe
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
0n
a µ+nan
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 ).
0′n
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
Simplest large g effective action for the Aaµ
( ),
2 2
2
withThis is compact QED in
1 1e
+1 dimensions with static char
xp co
ges 1 on two sublattice
s2
~
s.
22
a a a a aaa
d
iZ dA A A Ae
e g
µ µ ν ν µ τµ
η⎛ ⎞⎛ ⎞= − ∆ − ∆ −⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
±
∑ ∑∏∫
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).
Phase diagram of S=1/2 square lattice antiferromagnet
g
or
Spontaneous bond order, confined spinons, and “triplon” excitations
Neel order
Critical theory is not expressed in terms of order parameter of either phase, but instead contains spinons interacting the a non-compact U(1) gauge force
T. Senthil, A. Vishwanath, L. Balents, S. Sachdev and M.P.A. Fisher, submitted to Science
ConclusionsI. Introduction to magnetic quantum criticality in coupled
dimer antiferromagnet.
II. Theory of quantum phase transitions provides semi-quantitative predictions for neutron scattering measurements of spin-density-wave order in superconductors; theory also proposes a connection to STM experiments.
III. Spontaneous bond order in spin gap state on the square lattice: possible connection to modulations observed in vortex halo.
ConclusionsI. Introduction to magnetic quantum criticality in coupled
dimer antiferromagnet.
II. Theory of quantum phase transitions provides semi-quantitative predictions for neutron scattering measurements of spin-density-wave order in superconductors; theory also proposes a connection to STM experiments.
III. Spontaneous bond order in spin gap state on the square lattice: possible connection to modulations observed in vortex halo.