cond-mat/0408329, cond-mat/0409470, and to appear
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Classifying two-dimensional superfluids: why there is more to cuprate superconductivity than the condensation of charge -2 e Cooper pairs. cond-mat/0408329, cond-mat/0409470, and to appear. - PowerPoint PPT PresentationTRANSCRIPT
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Classifying two-dimensional superfluids: why there is more to cuprate superconductivity than the
condensation of charge -2e Cooper pairs
cond-mat/0408329, cond-mat/0409470, and to appear
Leon Balents (UCSB) Lorenz Bartosch (Yale) Anton Burkov (UCSB)
Predrag Nikolic (Yale) Subir Sachdev (Yale)
Krishnendu Sengupta (Toronto)
MIT Chez PierreDecember 13, 2004
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Experiments on the cuprate superconductors show:
• Tendency to produce “density” wave order near wavevectors (2/a)(1/4,0) and (2/a)(0,1/4).
• Proximity to a Mott insulator at hole density =1/8 with long-range “density” wave order at wavevectors (2/a)(1/4,0) and (2/a)(0,1/4).
• Vortex/anti-vortex fluctuations for a wide temperature range in the normal state
• STM studies of Ca2-xNaxCuO2Cl2 at low T, T. Hanaguri, C. Lupien, Y. Kohsaka, D.-H. Lee, M. Azuma, M. Takano, H. Takagi, and J. C. Davis, Nature 430, 1001 (2004).• Measurements of the Nernst effect, Y. Wang, S. Ono, Y. Onose, G. Gu, Y. Ando, Y. Tokura, S. Uchida, and N. P. Ong, Science 299, 86 (2003).• STM studies of Bi2Sr2CaCu2O8+ above Tc, M. Vershinin, S. Misra, S. Ono, Y. Abe, Y. Ando, and A. Yazdani, Science, 303, 1995 (2004).
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Experiments on the cuprate superconductors show:
• Tendency to produce “density” wave order near wavevectors (2/a)(1/4,0) and (2/a)(0,1/4).
• Proximity to a Mott insulator at hole density =1/8 with long-range “density” wave order at wavevectors (2/a)(1/4,0) and (2/a)(0,1/4).
• Vortex/anti-vortex fluctuations for a wide temperature range in the normal state
Needed: A quantum theory of transitions between superfluid/supersolid/insulating phases at fractional filling, and a deeper
understanding of the role of vortices
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OutlineOutline
A. Superfluid-insulator transitions of bosons on the square lattice at fractional fillingQuantum mechanics of vortices in a superfluid proximate to a commensurate Mott insulator at filling f
B. Extension to electronic models for the cuprate superconductorsDual vortex theories of the doped
(1) Quantum dimer model(2)“Staggered flux” spin liquid
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A. Superfluid-insulator transitions of bosons on the square lattice at fractional filling
Quantum mechanics of vortices in a superfluid proximate to a commensurate
Mott insulator at filling f
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Bosons at filling fraction f
M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Nature 415, 39 (2002).
Weak interactions: superfluidity
Strong interactions: Mott insulator which preserves all lattice
symmetries
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Vortices proliferate as the superfluid approaches the insulator.
In two dimensions, we can view the vortices as point particle excitations of the superfluid. What is
the quantum mechanics of these “particles” ?
Excitations of the superfluid: Vortices
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In ordinary fluids, vortices experience the Magnus Force
FM
mass density of fluid velocity circulationMF
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In a Galilean-invariant superfluid at 0, the Magnus force
on a quantized vortex with vo
rticity ( integer) is
where is the number densi
o
ˆ
y
t
M
m m
dF mh z
d
T
t
rv
f bosons,
is local superfluid velocity, and
is the position of the vortex.
v
r
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( ) ( )
In the presence of a lattice, we must distinguish two
physically distinct situations, and write
with
1 A stationary vortex in a moving superfluid
E BM M MF F F
( )
( )
( )
where
(2) A moving vortex in a stationary superfluid
ˆ
ˆ where
The expression for is basically correct (with
-
.
EM
BM
EM
F m h z
dF m h z
dt
F
E E v
rB B
s
( )
),
while that for is correct. The latter is modified
by the periodic potential of the lattice close to a
Mott insulator.................
BM notF
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( )BMF can be re-interpreted as a Lorentz force on
a vortex “particle” due to a “magnetic” field B=h
So we need to consider the quantum mechanics of a particle moving in a “magnetic” field B and a periodic
lattice potential --- the Hofstadter problem.
At filling fraction f=1, the B field is such that there is exactly one flux quantum per unit cell. Such a B field is
“invisible”, and the vortex “particle” moves in conventional Bloch waves ( ) 0.B
MF
2 2
0 0
0
At densities close to the Mott insulator density the effective field is
where and is the lattice spaci1 , n g
.
ˆMI
MI
MI
B
f aa a
B h z
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Bosons at filling fraction f(equivalent to S=1/2 AFMs)
Weak interactions: superfluidity
0
C. Lannert, M.P.A. Fisher, and T. Senthil, Phys. Rev. B 63, 134510 (2001) S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002)
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Bosons at filling fraction f(equivalent to S=1/2 AFMs)
Weak interactions: superfluidity
0
C. Lannert, M.P.A. Fisher, and T. Senthil, Phys. Rev. B 63, 134510 (2001) S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002)
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Bosons at filling fraction f(equivalent to S=1/2 AFMs)
Weak interactions: superfluidity
0
C. Lannert, M.P.A. Fisher, and T. Senthil, Phys. Rev. B 63, 134510 (2001) S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002)
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Bosons at filling fraction f(equivalent to S=1/2 AFMs)
Weak interactions: superfluidity
0
C. Lannert, M.P.A. Fisher, and T. Senthil, Phys. Rev. B 63, 134510 (2001) S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002)
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Bosons at filling fraction f(equivalent to S=1/2 AFMs)
Strong interactions: insulator
0
C. Lannert, M.P.A. Fisher, and T. Senthil, Phys. Rev. B 63, 134510 (2001) S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002)
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Bosons at filling fraction f(equivalent to S=1/2 AFMs)
Strong interactions: insulator 0
C. Lannert, M.P.A. Fisher, and T. Senthil, Phys. Rev. B 63, 134510 (2001) S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002)
1
2( + )
.All insulating phases have "density" wave order with 0ie Q rQ Q
Q
r
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Quantum mechanics of the vortex “particle” in a periodic potential with f flux quanta per unit cell
Vortices in a superfluid near a Mott insulator at filling f
, : Translations by a lattice spacing in the , directions
: Rotation by 90 degrees.
x yT T x y
R
2
1 1 1 4
Magnetic space group:
;
; ; 1
ifx y y x
y x x y
T T e T T
R T R T R T R T R
2
1 1 1 4
Magnetic space group:
;
; ; 1
ifx y y x
y x x y
T T e T T
R T R T R T R T R
Space group symmetries of Hamiltonian:
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At filling = / ( , relatively
prime integers) there are species
of vortices, (with =1 ),
associated with gauge-equivalent
regions of the Brillouin zone
f p q p q
q
q
q
At filling = / ( , relatively
prime integers) there are species
of vortices, (with =1 ),
associated with gauge-equivalent
regions of the Brillouin zone
f p q p q
q
q
q
Hofstadter spectrum of the quantum vortex “particle” with field operator
2
1 1 1 4
Magnetic space group:
;
; ; 1
ifx y y x
y x x y
T T e T T
R T R T R T R T R
2
1 1 1 4
Magnetic space group:
;
; ; 1
ifx y y x
y x x y
T T e T T
R T R T R T R T R
Vortices in a superfluid near a Mott insulator at filling f=p/q
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At filling = / ( , relatively
prime integers) there are species
of vortices, (with =1 ),
associated with gauge-equivalent
regions of the Brillouin zone
f p q p q
q
q
q
At filling = / ( , relatively
prime integers) there are species
of vortices, (with =1 ),
associated with gauge-equivalent
regions of the Brillouin zone
f p q p q
q
q
q
Hofstadter spectrum of the quantum vortex “particle” with field operator
21
2
1
The vortices form a representation of the space group
: ; :
1 :
i fx y
qi mf
mm
q projective
T T e
R eq
21
2
1
The vortices form a representation of the space group
: ; :
1 :
i fx y
qi mf
mm
q projective
T T e
R eq
See also X.-G. Wen, Phys. Rev. B 65, 165113 (2002)
Vortices in a superfluid near a Mott insulator at filling f=p/q
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The vortices characterize
superconducting and density wave orders
q both The vortices characterize
superconducting and density wave orders
q both
Superconductor insulator : 0 0
Vortices in a superfluid near a Mott insulator at filling f=p/q
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The vortices characterize
superconducting and density wave orders
q both The vortices characterize
superconducting and density wave orders
q both
ˆ
* 2
1
Density wave order:
Status of space group symmetry determined by
2density operators at wavevectors ,
: ;
:
qi mnf i mf
mn
i xx
n
y
mn
pm n
e
T
e
q
e T
Q
QQ Q
Q
ˆ
:
i ye
R R
QQ Q
Q Q
Vortices in a superfluid near a Mott insulator at filling f=p/q
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The vortices characterize
superconducting and density wave orders
q both The vortices characterize
superconducting and density wave orders
q both
*
* 2
Vorticity modulations:
In the presence of an applied magnetic field, there
are also modulations in the vorticity at the same
2wavevectors ,
mn
i mnf i mfnmn n
pm n
V e e
q
Q
1
q
Vortices in a superfluid near a Mott insulator at filling f=p/q
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Field theory with projective symmetry
( ) ( )
Degrees of freedom:
complex vortex fields
1 non-compact U(1) gauge field which mediates and E BM M
q
A F F
( ) ( )
Degrees of freedom:
complex vortex fields
1 non-compact U(1) gauge field which mediates and E BM M
q
A F F
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Field theory with projective symmetry
( ) ( )
Degrees of freedom:
complex vortex fields
1 non-compact U(1) gauge field which mediates and E BM M
q
A F F
( ) ( )
Degrees of freedom:
complex vortex fields
1 non-compact U(1) gauge field which mediates and E BM M
q
A F F
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Spatial structure of insulators for q=2 (f=1/2)
Field theory with projective symmetry
1
2( + )
.All insulating phases have density-wave order with 0ie Q rQ Q
Q
r
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Spatial structure of insulators for q=4 (f=1/4 or 3/4)
unit cells;
, , ,
all integers
a b
q q aba b q
unit cells;
, , ,
all integers
a b
q q aba b q
Field theory with projective symmetry
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Field theory with projective symmetry
Any pinned vortex must chose an orientation in flavor space. This necessarily leads to modulations in the local density of states over the spatial region where the vortex
executes its quantum zero point motion.
* 2
1
2Density operators at wavevectors ,
q
i mnf i mfmn
mn
n
pm n
q
e e
Q Q
MI
In the cuprates, assuming boson density=density of Cooper pairs we have
, and (both models in part B yield this value of ). So
modulation must have period
7 /16 16
1 with ,6 / 16 / , and /16
q q
a b a b ab
all integers.
Pinned vortices in the superfluid
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100Å
b7 pA
0 pA
Vortex-induced LDOS of Bi2Sr2CaCu2O8+ integrated from 1meV to 12meV at 4K
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).
Vortices have halos with LDOS modulations at a period ≈ 4 lattice spacings
Prediction of VBS order near vortices: K. Park and S. Sachdev, Phys.
Rev. B 64, 184510 (2001).
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Measuring the inertial mass of a vortex
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Measuring the inertial mass of a vortex
p
estimates for the BSCCO experiment:
Inertial vortex mass
Vortex m
10
1 THz = 4 meagnetoplasmon frequency
Large uncertainty due to uncertainty in valu f
V
e o
v e
rms
Preliminar
m
y
m
u
Note: With nodal fermionic quasiparticles, mv is expected to be dependent on the magnetic field i.e.
vortex densityG. E. Volovik, JETP Lett. 65, 217 (1997);
N. B. Kopnin, Phys. Rev. B 57, 11775 (1998).
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B. Extension to electronic models for the cuprate superconductors
Dual vortex theories of the doped (1) Quantum dimer model(2)“Staggered flux” spin liquid
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g = parameter controlling strength of quantum fluctuations in a semiclassical theory of the destruction of Neel order
Neel order
La2CuO4
(B.1) Phase diagram of doped antiferromagnets
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g
Neel order
La2CuO4
VBS order
or
N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989).
(B.1) Phase diagram of doped antiferromagnets
T. Senthil, A. Vishwanath, L. Balents, S. Sachdev and M.P.A. Fisher, Science 303, 1490 (2004).
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g
Neel order
La2CuO4
VBS order
orDual vortex theory of doped dimer model for interplay between VBS order and d-wave superconductivity
Dual vortex theory of doped dimer model for interplay between VBS order and d-wave superconductivity
Hole density
(B.1) Phase diagram of doped antiferromagnets
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(B.1) Doped quantum dimer model(B.1) Doped quantum dimer model
E. Fradkin and S. A. Kivelson, Mod. Phys. Lett. B 4, 225 (1990).
dqdH J
t
Density of holes =
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(B.1) Duality mapping of doped quantum dimer model shows:
Vortices in the superconducting state obey the magnetic translation algebra
2
1 with
2
ifx y y x
MI
T T e T T
pf
q
2
1 with
2
ifx y y x
MI
T T e T T
pf
q
where is the density of holes in the proximate
Mott insulator (for 1/ 8, 7 /16 )16MI
MI qf
Most results of Part A on bosons can be applied unchanged with q as determined above
Note: density of Cooper pairsf Note: density of Cooper pairsf
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g
Neel order
La2CuO4
VBS order
Hole density
(B.1) Phase diagram of doped antiferromagnets
132
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g
Neel order
La2CuO4
Hole density
(B.1) Phase diagram of doped antiferromagnets
116
VBS order
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g
Neel order
La2CuO4
Hole density
(B.1) Phase diagram of doped antiferromagnets
18
VBS order
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g
Neel order
La2CuO4
Hole density
(B.1) Phase diagram of doped antiferromagnets
VBS order
d-wave superconductivity above a critical
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(B.2) Dual vortex theory of doped “staggered flux” spin liquid
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(B.2) Dual vortex theory of doped “staggered flux” spin liquid
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(B.2) Dual vortex theory of doped “staggered flux” spin liquid
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(B.2) Dual vortex theory of doped “staggered flux” spin liquid
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Superfluids near Mott insulators
• Vortices with flux h/(2e) come in multiple (usually q) “flavors”
• The lattice space group acts in a projective representation on the vortex flavor space.
• These flavor quantum numbers provide a distinction between superfluids: they constitute a “quantum order”
• Any pinned vortex must chose an orientation in flavor space. This necessarily leads to modulations in the local density of states over the spatial region where the vortex executes its quantum zero point motion.
Superfluids near Mott insulators
• Vortices with flux h/(2e) come in multiple (usually q) “flavors”
• The lattice space group acts in a projective representation on the vortex flavor space.
• These flavor quantum numbers provide a distinction between superfluids: they constitute a “quantum order”
• Any pinned vortex must chose an orientation in flavor space. This necessarily leads to modulations in the local density of states over the spatial region where the vortex executes its quantum zero point motion.
The Mott insulator has average Cooper pair density, f = p/q per site, while the density of the superfluid is close (but need
not be identical) to this value