Quantum phase transitions of correlated electrons and atoms
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Quantum phase transitions of correlated electrons and atoms
Physical Review B 71, 144508 and 144509 (2005),cond-mat/0502002
Leon Balents (UCSB) Lorenz Bartosch (Yale) Anton Burkov (UCSB)
Subir Sachdev (Yale) Krishnendu Sengupta (Toronto)
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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.
~z
cg g
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.
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OutlineOutlineI. The Quantum Ising chain
II. The superfluid-Mott insulator quantum phase transition
III. The cuprate superconductorsSuperfluids proximate to finite doping Mott insulators
with VBS order ?
IV. Vortices in the superfluid
V. Vortices in superfluids near the superfluid-insulator quantum phase transition The “quantum order” of the superconducting state: evidence for vortex flavors
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I. Quantum Ising Chain
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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:
xj
j
H Jg 2Jg
j
j
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1 1
Coupling between qubits:
z zj j
j
H J
1 1
Prefers neighboring qubits
are
(not entangle
d)
j j j jeither or
1 1j j j j
0 1 1x z zj j j
j
J gH H H Full Hamiltonian
leads to entangled states at g of order unity
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LiHoF4
Experimental realization
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Weakly-coupled qubits Ground 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
j
p e
2 1
1
Excitation energy 4 sin2
Excitation gap 2 2
pap J O g
gJ J O g
p
a
a
p
1g
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Dynamic Structure Factor :
Cross-section to flip a to a (or vice versa)
while transferring energy
and momentum
( , )
p
S p
,S p Z p
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
phase coherence time
S. Sachdev and A.P. Young, Phys. Rev. Lett. 78, 2220 (1997)
~3
Weakly-coupled qubits 1g
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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
p
a
a
p
Second state obtained by
and mix only at order N
G
G G g
0
Ferromagnetic moment
0zN G G
Strongly-coupled qubits 1g
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Dynamic Structure Factor :
Cross-section to flip a to a (or vice versa)
while transferring energy
and momentum
( , )
p
S p
,S p 22
0 2N p
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
phase coherence time
S. Sachdev and A.P. Young, Phys. Rev. Lett. 78, 2220 (1997)
~2
Strongly-coupled qubits 1g
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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
1/8
0 ~ cN g g
P. Pfeuty Annals of Physics, 57, 79 (1970)
ggc
Excitation energy gap ~ cg g
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Dynamic Structure Factor :
Cross-section to flip a to a (or vice versa)
while transferring energy
and momentum
( , )
p
S p
,S p
Critical coupling cg g
c p
7 /82 2 2~ c p
No quasiparticles --- dissipative critical continuum
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Quasiclassical dynamics
Quasiclassical dynamics
1/ 4
1~z z
j kj k
P. Pfeuty Annals of Physics, 57, 79 (1970)
i
zi
zi
xiI gJH 1
0
7 / 4
( ) , 0
1 / ...
2 tan16
z z i tj k
k
R
BR
idt t e
A
T i
k T
S. Sachdev and J. Ye, Phys. Rev. Lett. 69, 2411 (1992).S. Sachdev and A.P. Young, Phys. Rev. Lett. 78, 2220 (1997).
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II. The superfluid-Mott insulator quantum phase transition
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M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman and E. A. Cornell, Science 269, 198 (1995)
Bose condensationVelocity distribution function of ultracold 87Rb atoms
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Apply a periodic potential (standing laser beams) to trapped ultracold bosons (87Rb)
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Momentum distribution function of bosons
Bragg reflections of condensate at reciprocal lattice vectors
M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Nature 415, 39 (2002).
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Superfluid-insulator quantum phase transition at T=0
V0=0Er V0=7ErV0=10Er
V0=13Er V0=14Er V0=16Er V0=20Er
V0=3Er
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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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Bosons at filling fraction f
Weak interactions: superfluidity
0
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Bosons at filling fraction f
Weak interactions: superfluidity
0
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Bosons at filling fraction f
Weak interactions: superfluidity
0
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Bosons at filling fraction f
Weak interactions: superfluidity
0
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Strong interactions: insulator
Bosons at filling fraction f
0
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Bosons at filling fraction f
Weak interactions: superfluidity
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)
0
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Weak interactions: superfluidity
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)
Bosons at filling fraction f
0
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Weak interactions: superfluidity
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)
Bosons at filling fraction f
0
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Weak interactions: superfluidity
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)
Bosons at filling fraction f
0
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Weak interactions: superfluidity
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)
Bosons at filling fraction f
0
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Strong interactions: insulator
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)
Bosons at filling fraction f
0
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Strong interactions: insulator
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)
Bosons at filling fraction f
0
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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( + )
.Can define a common CDW/VBS order using a generalized "density" ie Q rQ
Q
r
Insulating phases of bosons at filling fraction f
Charge density Charge density wave (CDW) orderwave (CDW) order
Valence bond solid Valence bond solid (VBS) order(VBS) order
Valence bond solid Valence bond solid (VBS) order(VBS) order
All insulators have 0 and 0 for certain Q Q
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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( + )
.Can define a common CDW/VBS order using a generalized "density" ie Q rQ
Q
r
Insulating phases of bosons at filling fraction f
Charge density Charge density wave (CDW) orderwave (CDW) order
Valence bond solid Valence bond solid (VBS) order(VBS) order
Valence bond solid Valence bond solid (VBS) order(VBS) order
All insulators have 0 and 0 for certain Q Q
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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( + )
.Can define a common CDW/VBS order using a generalized "density" ie Q rQ
Q
r
Insulating phases of bosons at filling fraction f
Charge density Charge density wave (CDW) orderwave (CDW) order
Valence bond solid Valence bond solid (VBS) order(VBS) order
Valence bond solid Valence bond solid (VBS) order(VBS) order
All insulators have 0 and 0 for certain Q Q
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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( + )
.Can define a common CDW/VBS order using a generalized "density" ie Q rQ
Q
r
Charge density Charge density wave (CDW) orderwave (CDW) order
Valence bond solid Valence bond solid (VBS) order(VBS) order
Valence bond solid Valence bond solid (VBS) order(VBS) order
All insulators have 0 and 0 for certain Q Q
Insulating phases of bosons at filling fraction f
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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( + )
.Can define a common CDW/VBS order using a generalized "density" ie Q rQ
Q
r
Insulating phases of bosons at filling fraction f
Charge density Charge density wave (CDW) orderwave (CDW) order
Valence bond solid Valence bond solid (VBS) order(VBS) order
Valence bond solid Valence bond solid (VBS) order(VBS) order
All insulators have 0 and 0 for certain Q Q
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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)
.Can define a common CDW/VBS order using a generalized "density" ie Q rQ
Q
r
Insulating phases of bosons at filling fraction f
Charge density Charge density wave (CDW) orderwave (CDW) order
Valence bond solid Valence bond solid (VBS) order(VBS) order
Valence bond solid Valence bond solid (VBS) order(VBS) order
All insulators have 0 and 0 for certain Q Q
1
2( + )
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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( + )
.Can define a common CDW/VBS order using a generalized "density" ie Q rQ
Q
r
Insulating phases of bosons at filling fraction f
Charge density Charge density wave (CDW) orderwave (CDW) order
Valence bond solid Valence bond solid (VBS) order(VBS) order
Valence bond solid Valence bond solid (VBS) order(VBS) order
All insulators have 0 and 0 for certain Q Q
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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( + )
.Can define a common CDW/VBS order using a generalized "density" ie Q rQ
Q
r
Charge density Charge density wave (CDW) orderwave (CDW) order
Valence bond solid Valence bond solid (VBS) order(VBS) order
Valence bond solid Valence bond solid (VBS) order(VBS) order
All insulators have 0 and 0 for certain Q Q
Insulating phases of bosons at filling fraction f
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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( + )
.Can define a common CDW/VBS order using a generalized "density" ie Q rQ
Q
r
Charge density Charge density wave (CDW) orderwave (CDW) order
Valence bond solid Valence bond solid (VBS) order(VBS) order
Valence bond solid Valence bond solid (VBS) order(VBS) order
All insulators have 0 and 0 for certain Q Q
Insulating phases of bosons at filling fraction f
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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( + )
.Can define a common CDW/VBS order using a generalized "density" ie Q rQ
Q
r
Insulating phases of bosons at filling fraction f
Charge density Charge density wave (CDW) orderwave (CDW) order
Valence bond solid Valence bond solid (VBS) order(VBS) order
Valence bond solid Valence bond solid (VBS) order(VBS) order
All insulators have 0 and 0 for certain Q Q
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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( + )
.Can define a common CDW/VBS order using a generalized "density" ie Q rQ
Q
r
Insulating phases of bosons at filling fraction f
Charge density Charge density wave (CDW) orderwave (CDW) order
Valence bond solid Valence bond solid (VBS) order(VBS) order
Valence bond solid Valence bond solid (VBS) order(VBS) order
All insulators have 0 and 0 for certain Q Q
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Superfluid-insulator transition of bosons at generic filling fraction f
The transition is characterized by multiple distinct order parameters (boson condensate, VBS/CDW order)
Traditional (Landau-Ginzburg-Wilson) view:Such a transition is first order, and there are no precursor fluctuations of the order of the insulator in the superfluid.
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Superfluid-insulator transition of bosons at generic filling fraction f
The transition is characterized by multiple distinct order parameters (boson condensate, VBS/CDW order)
Traditional (Landau-Ginzburg-Wilson) view:Such a transition is first order, and there are no precursor fluctuations of the order of the insulator in the superfluid.
Recent theories: Quantum interference effects can render such transitions second order, and the superfluid does contain precursor VBS/CDW fluctuations.
N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989).T. Senthil, A. Vishwanath, L. Balents, S. Sachdev and M.P.A. Fisher, Science 303, 1490 (2004).
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III. The cuprate superconductors
Superfluids proximate to finite doping Mott insulators with VBS order ?
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Cu
O
LaLa2CuO4
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Mott insulator: square lattice antiferromagnet
jiij
ij SSJH
La2CuO4
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Superfluid: condensate of paired holes
La2-SrCuO4
0S
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Many experiments on the cuprate superconductors show:
• Tendency to produce modulations in spin singlet observables at 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 charge modulations at wavevectors (2/a)(1/4,0) and (2/a)(0,1/4).
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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
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Many experiments on the cuprate superconductors show:
• Tendency to produce modulations in spin singlet observables at 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 charge modulations at wavevectors (2/a)(1/4,0) and (2/a)(0,1/4).
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Many experiments on the cuprate superconductors show:
• Tendency to produce modulations in spin singlet observables at 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 charge modulations at wavevectors (2/a)(1/4,0) and (2/a)(0,1/4).
Superfluids proximate to finite doping Mott insulators with VBS order ?
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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).
Experiments on the cuprate superconductors also show strong vortex fluctuations above Tc
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Main claims:
• There are precursor fluctuations of VBS order in the superfluid.
• There fluctuations are intimately tied to the quantum theory of vortices in the superfluid
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IV. Vortices in the superfluid
Magnus forces, duality, and point vortices as dual “electric” charges
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Central question: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 air velocity of ball circulationMF
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Dual picture:The vortex is a quantum particle with dual “electric”
charge n, moving in a dual “magnetic” field of strength = h×(number density of Bose particles)
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V. Vortices in superfluids near the superfluid-insulator quantum phase transition
The “quantum order” of the superconducting state:
evidence for vortex flavors
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A4
A3
A1
A2 A1+A2+A3+A4= 2f
where f is the boson filling fraction.
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Bosons at filling fraction f
• At f=1, the “magnetic” flux per unit cell is 2, and the vortex does not pick up any phase from the boson density.
• The effective dual “magnetic” field acting on the vortex is zero, and the corresponding component of the Magnus force vanishes.
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Quantum mechanics of the vortex “particle” in a periodic potential with f flux quanta per unit cell
, : 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 Hofstadter Hamiltonian:
Bosons at rational filling fraction f=p/q
The low energy vortex states must form a representation of this algebra
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At filling = / , there are species
of vortices, (with =1 ),
associated with degenerate minima in
the vortex spectrum. These vortices realize
the smallest, -dimensional, representation of
the
f p q q
q
q
q
magnetic algebra.
At filling = / , there are species
of vortices, (with =1 ),
associated with degenerate minima in
the vortex spectrum. These vortices realize
the smallest, -dimensional, representation of
the
f p q q
q
q
q
magnetic algebra.
Hofstadter spectrum of the quantum vortex “particle” with field operator
Vortices in a superfluid near a Mott insulator at filling f=p/q
21
2
1
: ; :
1 :
i fx y
qi mf
mm
T T e
R eq
21
2
1
: ; :
1 :
i fx y
qi mf
mm
T T e
R eq
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The vortices characterize
superconducting and VBS/CDW orders
q both The vortices characterize
superconducting and VBS/CDW orders
q both
ˆ
* 2
1
VBS order:
Status of space group symmetry determined by
2density operators at wavevectors ,
: ;
:
qi mnf i mf
m
mn
n n
i x ix y
pm
e
nq
T e T e
e
Q
Q QQ Q Q Q
Q
ˆ
:
y
R R Q Q
Vortices in a superfluid near a Mott insulator at filling f=p/q
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Vortices in a superfluid near a Mott insulator at filling f=p/q
The excitations of the superfluid are described by the
quantum mechanics of flavors of low energy vortices
moving in zero dual "magnetic" field.
The orientation of the vortex in flavor space imp
q
lies a
particular configuration of VBS order in its vicinity.
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Spatial structure of insulators for q=2 (f=1/2)
Mott insulators obtained by “condensing” vortices
1
2( + )
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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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Vortices in a superfluid near a Mott insulator at filling f=p/q
The excitations of the superfluid are described by the
quantum mechanics of flavors of low energy vortices
moving in zero dual "magnetic" field.
The orientation of the vortex in flavor space imp
q
lies a
particular configuration of VBS order in its vicinity.
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Vortices in a superfluid near a Mott insulator at filling f=p/q
The excitations of the superfluid are described by the
quantum mechanics of flavors of low energy vortices
moving in zero dual "magnetic" field.
The orientation of the vortex in flavor space imp
q
lies a
particular configuration of VBS order in its vicinity.
Any pinned vortex must pick an orientation in flavor
space: this induces a halo of VBS order in its vicinity
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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 magnetoplasmon frequency
Future experiments can directly d
10
1 THz = 4 meV
etect vortex zero point motion
by
v e
Preliminar
m m
y
looking for resonant absorption at this frequency.
Vortex oscillations can also modify the electronic density of states.
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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