Post on 23-Mar-2020
Magnetic phases and critical points of insulators and superconductors
Colloquium article:Reviews of Modern Physics, 75, 913 (2003).
Reviews:http://onsager.physics.yale.edu/qafm.pdf
cond-mat/0203363
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.
OutlineOutlineA. “Dimerized” Mott insulators with a spin gap
Tuning quantum transitions by applied pressure
B. Spin gap state on the square latticeSpontaneous bond order
C. Tuning quantum transitions by a magnetic field1. Mott insulators2. Cuprate superconductors
(A) “Dimerized” Mott Insulators with a spin gapTuning quantum transitions by applied pressure
TlCuCl3
M. Matsumoto, B. Normand, T.M. Rice, and M. Sigrist, cond-mat/0309440.
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
( )↓↑−↑↓=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
TlCuCl3
N. Cavadini, G. Heigold, W. Henggeler, A. Furrer, H.-U. Güdel, K. Krämer and H. Mutka, Phys. Rev. B 63 172414 (2001).
“triplon” or spin exciton
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 = +
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
Neel state
0S N=
δ in cuprates ?Pressure in TlCuCl3
Quantum paramagnet0=S
PHCC – a two-dimensional antiferromagnet
b
c
PHCC = C4H12N2Cu2Cl6
a
cCu
ClC
N
M. B. Stone, I. A. Zaliznyak, D. H. Reich, and C. Broholm, Phys. Rev. B 64, 144405 (2001).
PHCC – a two-dimensional antiferromagnet
ω(m
eV)
Dispersion to “chains”Dispersion to “chains”
Not chains but planesNot chains but planes
⊥ M. B. Stone, I. A. Zaliznyak, D. H. Reich, and C. Broholm, Phys. Rev. B 64, 144405 (2001).
PHCC – a two-dimensional antiferromagnet
ω(m
eV)
Dispersion to “chains”Dispersion to “chains”
Not chains but planesNot chains but planes
⊥
ω(m
eV)
0
1 0
1
h
M. B. Stone, I. A. Zaliznyak, D. H. Reich, and C. Broholm, Phys. Rev. B 64, 144405 (2001).
Triplon dispersion
S. Sachdev and R.N. Bhatt, Phys. Rev. B 41, 9323 (1990).
Quantitative theory of experiments and simulations: method of bond operators
Operators algebra for all states on a single dimer
( )† 102
s s≡ ≡ ↑↓ − ↓↑
( )
( )
( )
†
†
†
102
02
102
x x
y y
z z
t t
it t
t t
≡ ≡ ↑↑ − ↓↓
≡ ≡ ↑↑ + ↓↓
≡ ≡ ↑↓ + ↓↑
† †
†
†
1
, 1
,
s s t t
s s
t t
α α
α β αβδ
+ =
⎡ ⎤ =⎣ ⎦⎡ ⎤ =⎣ ⎦
Canonical Bose operators with a hard
core constraint
( )
( )
† † †1
† † †1
1212
S s t t s i t t
S s t t s i t t
α α α αβγ β γ
α α α αβγ β γ
ε
ε
= + −
= − − −
Spin operators on both sites can be expressed in terms of bond operators
Quantitative theory of experiments and simulations: method of bond operators
S. Sachdev and R.N. Bhatt, Phys. Rev. B 41, 9323 (1990). A. V. Chubukov and Th. Jolicoeur, Phys. Rev. B 44, 12050 (1991).
†
Hamiltonian for coupled dimers
Solve c 1 ,
o
nstraint by s t tα α= −
( ) ( ) ( )
( ) ( ) ( )
† † †
2 2
Triplon
2
d
ispe
rs
io
n
:
t k k k k k kk
B kH A k t t t t t t
k A k B k
α α α α α α
ε
− −
⎛ ⎞= + + +⎜ ⎟
⎝ ⎠
= −
∑
( ) ( )222 2 2
Transition to magnetically ordered state occurs when 0 and the bosons condense, lea
ding to 0
x x x y y yc k
t t
K c k K
α α
= ∆ + −
∆ →
−
≠
+
Quantitative theory of experiments and simulations: method of bond operators
S. Sachdev and R.N. Bhatt, Phys. Rev. B 41, 9323 (1990).
Field theory for quantum criticality
αϕ 3-component antiferromagnetic order parameter
( ) ( ) ( )( ) ( )22 22 2 2 212 4!x c
ud xd cSϕ α τ α α ατ ϕ ϕ λ λ ϕ ϕ⎡ ⎤= ∇ + ∂ + − +⎢ ⎥⎣ ⎦∫
( )For oscillations of about 0 lead to the following structure in the dynamic structure factor ,
c
S pα αλ λ ϕ ϕ
ω< =
( )2 2
; 2 c
c pp cε λ λ= ∆ + ∆ = −∆
ω
( ),S p ω
( )( )Z pδ ω ε−
Three triplon continuum
Triplon pole
~3∆Structure holds to all orders in u
Field theory for quantum criticality
αϕ 3-component antiferromagnetic order parameter
( ) ( ) ( )( ) ( )22 22 2 2 212 4!x c
ud xd cSϕ α τ α α ατ ϕ ϕ λ λ ϕ ϕ⎡ ⎤= ∇ + ∂ + − +⎢ ⎥⎣ ⎦∫
( )0For oscillations of about 0 lead to the
following dynamic structure factor ,c z
zz
Nlongitudinal S p
αλ λ ϕ ϕω
> = ≠
ω
( ),zzS p ω
Two spin-wave continuum
Structure holds to all orders in u~3∆
( ) ( ) ( )220 2N pπ δ ω δ
Entangled states at λ of order λc
1/λλc
( )~ cZ ηνλ λ−Triplonquasiparticle
weight Z A.V. Chubukov, S. Sachdev, and J.Ye, Phys. Rev. B 49, 11919 (1994)
Antiferromagneticmoment N0
( )0 ~ cN βλ λ−
1/λλc
Triplon energy gap ∆ ( )~ c
νλ λ∆ −
1/λλc
Field theory for quantum criticalityDynamic spectrum at the critical point
ω
Critical coupling ( )cλ λ=
c p
( ) (2 ) / 22 2 2~ c pη
ω− −
−( ),S p ω
No quasiparticles --- dissipative critical continuum
Field theory for quantum criticality
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=
(B) 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 ?
Can such a state with bond order be the ground state of a system with full square lattice symmetry ?
Need additional exchange interactions with full square lattice symmetry to move out of Neel state into
paramagnet e.g. a second neighbor exchange J2. This defines a dimensionless coupling g = J2 / J
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
0n
a µ+nan
aA µ
a
Neel order parameter; 1 on two square sublattices ;
oriented area of spherical triangle
formed by and an ar
~
, ,
a a a
a
a a
S
A µ
µ
ηη
+
→→ ±
→
n
n n 0bitrary reference poi tn n
Discretize imaginary time: path integral is over fields on the sites of a
cubic lattice of points a
Collinear spins and compact U(1) gauge theory
( ),
11 exp2
2a a a a a a
a aa
iZ d Ag µ τ
µ
δ η+
⎛ ⎞= − ⋅ −⎜ ⎟
⎝ ⎠∑ ∑∏∫ n n n n
Partition function on square lattice
Modulus of weights in partition function: those of a classical ferromagnet at “temperature” g
0Small ground state has Neel order with 0
Large paramagnetic ground state with 0 Berry phases lead to large cancellations between different time histories need an effective action for
a
a
g N
g
⇒ = ≠
⇒ =
→
n
n
at large aA gµ
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⎛ ⎞⎛ ⎞= − ∆ − ∆ −⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
±
∑ ∑∏∫
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).
N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989).
N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989).
For S=1/2 and 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
There is a definite average height of the interfaceGround state has bond order.
⇒⇒
N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989).
0
1/2
1/4
3/4
0
1/2
1/4
3/4
0 1/4 0 1/4
Smooth interface with average height 3/8
W. Zheng and S. Sachdev, Phys. Rev. B 40, 2704 (1989)
1/41 1/41
3/4 1/2 3/4 1/2
1/41/4 11
Smooth interface with average height 5/8
W. Zheng and S. Sachdev, Phys. Rev. B 40, 2704 (1989)
5/41 5/41
3/4 1/2 3/4 1/2
5/45/4 11
Smooth interface with average height 7/8
W. Zheng and S. Sachdev, Phys. Rev. B 40, 2704 (1989)
1/40 1/40
1/2-1/4 -1/4 1/2
1/41/4 00
Smooth interface with average height 1/8
W. Zheng and S. Sachdev, Phys. Rev. B 40, 2704 (1989)
1/4
“Disordered-flat” interface with average height 1/2
1/23/4 1/2
1/40 1↔0 1↔
3/4
1/41/4 0 1↔0 1↔
W. Zheng and S. Sachdev, Phys. Rev. B 40, 2704 (1989)
“Disordered-flat” interface with average height 3/4
1/ 4 5 / 4
↔ 1/ 4 5 / 4
↔1 1
3/4 1/2 3/4 1/2
1 1 1/ 4 5 / 4
↔1/ 4 5 / 4
↔
W. Zheng and S. Sachdev, Phys. Rev. B 40, 2704 (1989)
0 01/4
-1/4 -1/4
W. Zheng and S. Sachdev, Phys. Rev. B 40, 2704 (1989)
1/ 2-1/ 2
↔
0 0
“Disordered-flat” interface with average height 0
1/4
1/4
1/ 2-1/ 2
↔
1/4
1/4
W. Zheng and S. Sachdev, Phys. Rev. B 40, 2704 (1989)
3 / 4 -1/ 4
↔
0 0
0 0
“Disordered-flat” interface with average height 1/4
1/4
1/4
3/ 4-1/ 4
↔
1/4
1/2 1/2
Two possible bond-ordered paramagnets for S=1/2
Distinct lines represent different values of on linksi jS Si
There is a broken lattice symmetry, and the ground state is at least four-fold degenerate.
N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989).
( ),
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
g0
Bond order in a frustrated S=1/2 XY magnet
A. W. Sandvik, S. Daul, R. R. P. Singh, and D. J. Scalapino, Phys. Rev. Lett. 89, 247201 (2002)
First large scale numerical study of the destruction of Neel order in a S=1/2antiferromagnet with full square lattice symmetry
( ) ( )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=
( ),
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
g0
g0S. 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
Nature of quantum critical point
Use a sequence of simpler models which can be analyzed byduality mappings
A. Non-compact QED with scalar matterB. Compact QED with scalar matterC. N=1: Compact QED with scalar matter and Berry phasesD. theoryE. Easy plane case for N=2
N → ∞
Nature of quantum critical point
Use a sequence of simpler models which can be analyzed byduality mappings
A. Non-compact QED with scalar matterB. Compact QED with scalar matterC. N=1: Compact QED with scalar matter and Berry phasesD. theoryE. Easy plane case for N=2
N → ∞
A. N=1, non-compact U(1), no Berry phases
C. Dasgupta and B.I. Halperin, Phys. Rev. Lett. 47, 1556 (1981).
Nature of quantum critical point
Use a sequence of simpler models which can be analyzed byduality mappings
A. Non-compact QED with scalar matterB. Compact QED with scalar matterC. N=1: Compact QED with scalar matter and Berry phasesD. theoryE. Easy plane case for N=2
N → ∞
Nature of quantum critical point
Use a sequence of simpler models which can be analyzed byduality mappings
A. Non-compact QED with scalar matterB. Compact QED with scalar matterC. N=1: Compact QED with scalar matter and Berry phasesD. theoryE. Easy plane case for N=2
N → ∞
B. N=1, compact U(1), no Berry phases
Nature of quantum critical point
Use a sequence of simpler models which can be analyzed by duality mappings
A. Non-compact QED with scalar matterB. Compact QED with scalar matterC. N=1: Compact QED with scalar matter and Berry phasesD. theoryE. Easy plane case for N=2
N → ∞
Nature of quantum critical point
Use a sequence of simpler models which can be analyzed by duality mappings
A. Non-compact QED with scalar matterB. Compact QED with scalar matterC. N=1: Compact QED with scalar matter and Berry phasesD. theoryE. Easy plane case for N=2
N → ∞
C. N=1, compact U(1), Berry phases
C. N=1, compact U(1), Berry phases
C. N=1, compact U(1), Berry phases
Nature of quantum critical point
Use a sequence of simpler models which can be analyzed byduality mappings
A. Non-compact QED with scalar matterB. Compact QED with scalar matterC. N=1: Compact QED with scalar matter and Berry phasesD. theoryE. Easy plane case for N=2
N → ∞
Identical critical theories!
Nature of quantum critical point
Use a sequence of simpler models which can be analyzed byduality mappings
A. Non-compact QED with scalar matterB. Compact QED with scalar matterC. N=1: Compact QED with scalar matter and Berry phasesD. theoryE. Easy plane case for N=2
N → ∞
Identical critical theories!
D. , compact U(1), Berry phasesN → ∞
E. Easy plane case for N=2
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
OutlineOutlineA. “Dimerized” Mott insulators with a spin gap
Tuning quantum transitions by applied pressure
B. Spin gap state on the square latticeSpontaneous bond order
C. Tuning quantum transitions by a magnetic field1. Mott insulators2. Cuprate superconductors
(C) Tuning quantum transitions by a magnetic field
1. Mott insulators
λc = 0.52337(3)M. Matsumoto, C. Yasuda, S. Todo, and H. Takayama,
Phys. Rev. B 65, 014407 (2002)T=0
λ 1 cλ
Neel state
0S N=
δ in cuprates ?Pressure in TlCuCl3
Quantum paramagnet0=S
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
H
1/λ
Spin singlet state with a spin gap
Canted magnetic order
gµBH = ∆
Phase diagram in a magnetic field
H
1/λ
Spin singlet state with a spin gap
Canted magnetic order gµBH = ∆[ ]
[ ]2
Elastic scattering intensity
0
I H
HI aJ
=
⎛ ⎞+ ⎜ ⎟⎝ ⎠
~c cH λ λ−
1 Tesla = 0.116 meV
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.
Canted magnetic order
Spin gap paramagnet
Phase diagram in a strong magnetic field.
Magnetization =density of triplons
H∆
Spin gap
Canted magnetic order
Phase diagram in a strong magnetic field.
1
∆
Magnetization =density of triplons
H
Spin gapAt very large H, magnetization
saturatesCanted magnetic order
Phase diagram in a strong magnetic field.
M
∆
1
1/2
Magnetization =density of triplons
H
Spin gap ij zi zji j
J S S<∑
Respulsive interactions between triplons can lead to
magnetization plateau at any rational fraction
Canted magnetic order
Phase diagram in a strong magnetic field.
∆
1
M
1/2
Quantum transitions in and out of plateau are
Bose-Einstein condensations of “extra/missing”
triplons
Magnetization =density of triplons
H
Partial magnetization plateau observed in SrCu2(BO3)2 and NH4CuCl3
Spin gap
Canted magnetic order
(C) Tuning quantum transitions by a magnetic field
2. Cuprate superconductors
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
.Re ; order parameter is complex vector ji K rj e⎡ ⎤= ⎣ ⎦S Φ Φ
( ), 0K π π θ= =;
( )3 4, 0K π π θ= =;
( )3 4, / 8K π π θ π= =;
Collinear spins ie θ⇒ = nΦ
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~0.12-0.140.055SC
0.020SC+SDWSDWNéel
Use simplest assumption of a direct second-order quantum phase transition between SC and SC+SDW phases
If does not exactly connect two nodal points, critical theory is as in an insulator
K
Magnetic transition in a d-wave superconductor
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
( )2 22 2rd rd c Vα τ α ατ ⎡ ⎤= ∇ Φ + ∂ Φ + Φ⎣ ⎦∫S
Similar terms present in action for SDW ordering in the insulator
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
= < ⇒
∑∫
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 second-order quantum phase transition between SC and SC+SDW phases
Recall, in an insulator intensity would increase ~ H2
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 fluctuations
( )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) suggested nucleation of static magnetism (with ∆=0) within vortex scores in a first-order transition. However,
given the small size of the vortex cores, the magnetism must become dynamic as in a spin gap state.
S. Sachdev, Phys. Rev. B 45, 389 (1992); N. Nagaosa and P. A. Lee, Phys. Rev. B 45, 966 (1992)
( ) ( ) ( ) 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
TlCuCl3
M. Matsumoto, B. Normand, T.M. Rice,
and M. Sigrist, cond-mat/0309440.
Canted magnetic order
Spin gap paramagnet
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
= ⇒
−−
( ) ( ) 2
2
1 triplon energy30 ln c
c
SHHH b
H Hε ε
=
⎛ ⎞= − ⎜ ⎟⎝ ⎠
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).
Peaks at (0.5,0.5) (0.125,0)and (0.5,0.5) (0,0.125)
dynamic SDW of period 8
±±
⇒
2- 4Neutron scattering off La Sr CuO ( 0.163, ) δ δ δ = SC phasered 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
Collinear magnetic (spin density wave) order
.Re ; order parameter is complex vector ji K rj e⎡ ⎤= ⎣ ⎦S Φ Φ
( )Collinear spins , and there is modulation
in the parameter at
wavevector 2x
i
j j j a
e
Qbond order r
K
θ
+
⇒ =
≡ i
n
S S
Φ
( )3 4, 0K π π θ= =;
( )3 4, / 8K π π θ π= =;
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
=
⎛ ⎞= − ⎜ ⎟⎝ ⎠
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).
ConclusionsI. Introduction to magnetic quantum criticality in coupled
dimer antiferromagnet.
II. Berry phases and bond order in square lattice antiferromagnets.
III. 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.
IV. 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. Berry phases and bond order in square lattice antiferromagnets.
III. 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.
IV. Spontaneous bond order in spin gap state on the square lattice: possible connection to modulations observed in vortex halo.