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The quantum mechanics of vortices in superfluids near a Mott transition
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)
Talk online: Google Sachdev
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The quantum order of superfluids: why all superfluids are not the same
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)
Talk online: Google Sachdev
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Recent experiments on the cuprate superconductors show:
• Proximity to insulating ground states with density wave order at carrier density =1/8
• Vortex/anti-vortex fluctuations for a wide temperature range in the normal state
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T. Hanaguri, C. Lupien, Y. Kohsaka, D.-H. Lee, M. Azuma, M. Takano, H. Takagi, and J. C. Davis, Nature 430, 1001 (2004).
The cuprate superconductor Ca2-xNaxCuO2Cl2
Multiple order parameters: superfluidity and density wave.Phases: Superconductors, Mott insulators, and/or supersolids
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Distinct experimental charcteristics of underdoped cuprates at T > Tc
Measurements of Nernst effect are well explained by a model of a liquid of vortices and anti-vortices
N. P. Ong, Y. Wang, S. Ono, Y. Ando, and S. Uchida, Annalen der Physik 13, 9 (2004).
Y. Wang, S. Ono, Y. Onose, G. Gu, Y. Ando, Y. Tokura, S. Uchida, and N. P. Ong, Science 299, 86 (2003).
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STM measurements observe “density” modulations with a period of ≈ 4 lattice spacings
LDOS of Bi2Sr2CaCu2O8+ at 100 K. M. Vershinin, S. Misra, S. Ono, Y. Abe, Y. Ando, and A. Yazdani, Science, 303, 1995
(2004).
Distinct experimental charcteristics of underdoped cuprates at T > Tc
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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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Recent experiments on the cuprate superconductors show:
• Proximity to insulating ground states with density wave order at carrier density =1/8
• 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 filling fraction fQuantum mechanics of vortices in a superfluid proximate to a commensurate Mott insulator
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 filling fraction f
Quantum mechanics of vortices in a superfluid proximate to a
commensurate Mott insulator
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Bosons at density 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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Approaching the transition from the insulator (f=1)
Excitations of the insulator:
†Particles ~
Holes ~
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Approaching the transition from the superfluid (f=1)Excitations of the superfluid: (A) Superflow (“spin waves”)
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Approaching the transition from the superfluid (f=1)Excitations of the superfluid: (B) Vortices
vortex
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Approaching the transition from the superfluid (f=1)Excitations of the superfluid: (B) Vortices
vortex
E
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Approaching the transition from the superfluid (f=1)Excitations of the superfluid: Superflow and vortices
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Dual theories of the superfluid-insulator transition (f=1)
C. Dasgupta and B.I. Halperin, Phys. Rev. Lett. 47, 1556 (1981);
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A vortex in the vortex field is the original boson
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A vortex in the vortex field is the original boson
vortexboson
2ie
The wavefunction of a vortex acquires a phase of 2 each time the vortex encircles a boson
Current of
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Bosons at density f(equivalent to S=1/2 AFMs)
Weak interactions: superfluidity
Strong interactions: Candidate insulating states
1
2( + )
.All insulating phases have density-wave order with 0ie Q rQ Q
Q
r
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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Predictions of LGW theory
(Supersolid)
Coexistence
1 2r r
First order transition
1 2r rSuperconductor
Superconductor Charge-ordered insulator
Q
Q
Charge-ordered insulator
1 2r rSuperconductor
Q( topologically ordered)
" "
0, 0sc
Disordered
Q
Charge-ordered
insulator
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Predictions of LGW theory
(Supersolid)
Coexistence
1 2r r
First order transition
1 2r rSuperconductor
Superconductor Charge-ordered insulator
Q
Q
Charge-ordered insulator
1 2r rSuperconductor
Q( topologically ordered)
" "
0, 0sc
Disordered
Q
Charge-ordered
insulator
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vortexboson
2ie
The wavefunction of a vortex acquires a phase of 2 each time the vortex encircles a boson
Boson-vortex duality
Strength of “magnetic” field on vortex field = density of bosons = f flux quanta per plaquette
C. Dasgupta and B.I. Halperin, Phys. Rev. Lett. 47, 1556 (1981); D.R. Nelson, Phys. Rev. Lett. 60, 1973 (1988); M.P.A. Fisher and D.-H. Lee, Phys. Rev. B 39, 2756 (1989);
Current of
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Quantum mechanics of the vortex “particle” is invariant under the square lattice space group:
Boson-vortex duality
, : 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
Strength of “magnetic” field on vortex field = density of bosons = f flux quanta per plaquette
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Boson-vortex duality
At density = / ( , 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 density = / ( , 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”
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
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Boson-vortex duality
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)
At density = / ( , 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 density = / ( , 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”
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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
Boson-vortex duality
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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
Boson-vortex duality
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Field theory with projective symmetry
Degrees of freedom:
complex vortex fields
1 non-compact U(1) gauge field
q
A
Degrees of freedom:
complex vortex fields
1 non-compact U(1) gauge field
q
A
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Fluctuation-induced, weak, first order transitionsc
1 2r r
Q
Superconductor
0, 0mn
Charge-ordered insulator
0, 0mn
Field theory with projective symmetry
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sc
1 2r r
Q
Superconductor
0, 0mn
Charge-ordered insulator
0, 0mn
sc
1 2r r
Q
Superconductor
0, 0mn
Charge-ordered insulator
0, 0mn
Supersolid
0, 0mn
Field theory with projective symmetryFluctuation-induced,
weak, first order transition
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sc
1 2r r
Q
Superconductor
0, 0mn
Charge-ordered insulator
0, 0mn
sc
1 2r r
Q
Superconductor
0, 0mn
Charge-ordered insulator
0, 0mn
Supersolid
0, 0mn
Second order transitionsc
1 2r r
Q
Superconductor
0, 0mn
Charge-ordered insulator
0, 0mn
Field theory with projective symmetryFluctuation-induced,
weak, first order transition
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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
Each pinned vortex in the superfluid has a halo of density wave order over a length scale ≈ the zero-point quantum motion of the
vortex. This scale diverges upon approaching the insulator
* 2
1
2Density operators at wavevectors ,
q
i mnf i mfmn
mn
n
pm n
q
e e
Q Q
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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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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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Conclusions
I. Superfluids near commensurate insulators with “boson” density p/q have q species of vortices. The projective transformations of these vortices under the lattice space group defines a “quantum order” which distinguishes superfluids from each other. (Note: only the density of the insulator, and not the superfluid, is exactly p/q).
II. Vortices carry the quantum numbers of both superconductivity and the square lattice space group (in a projective representation).
III. Vortices carry halo of density wave order, and pinning of vortices/anti-vortices leads to a unified theory of STM modulations in zero and finite magnetic fields.
IV. Field theory of vortices with projective symmetries describes superfluids with precursor fluctuations of density wave order and its transitions to supersolids and insulators.
Conclusions
I. Superfluids near commensurate insulators with “boson” density p/q have q species of vortices. The projective transformations of these vortices under the lattice space group defines a “quantum order” which distinguishes superfluids from each other. (Note: only the density of the insulator, and not the superfluid, is exactly p/q).
II. Vortices carry the quantum numbers of both superconductivity and the square lattice space group (in a projective representation).
III. Vortices carry halo of density wave order, and pinning of vortices/anti-vortices leads to a unified theory of STM modulations in zero and finite magnetic fields.
IV. Field theory of vortices with projective symmetries describes superfluids with precursor fluctuations of density wave order and its transitions to supersolids and insulators.