The quantum phase transition between a superfluid and an insulator: applications to trapped ultracold atoms and the cuprate superconductors. The quantum phase transition between a superfluid and an insulator: applications to trapped ultracold atoms and the cuprate superconductors.

Leon Balents (UCSB) Lorenz Bartosch (Harvard) Anton Burkov (Harvard) Predrag Nikolic (Harvard) Subir Sachdev (Harvard) Krishnendu Sengupta (HRI, India)

Talk online at http://sachdev.physics.harvard.edu OutlineOutline I. Bose-Einstein condensation and superfluidity.

II. The superfluid-insulator quantum phase transition.

III. The cuprate superconductors, and their proximity to a superfluid-insulator transition.

IV. Landau-Ginzburg-Wilson theory of the superfluid- insulator transition.

V. Beyond the LGW paradigm: continuous quantum transitions with multiple order parameters.

VI. Experimental tests in the cuprates. I. Bose-Einstein condensation and superfluidity Superfluidity/superconductivity occur in: • liquid 4He • metals Hg, Al, Pb, Nb,

Nb3Sn….. • liquid 3He • neutron stars

• cuprates La2-xSrxCuO4, YBa2Cu3O6+y…. •M3C60 • ultracold trapped atoms

•MgB2 The Bose-Einstein condensate:

A macroscopic number of bosons occupy the lowest energy quantum state

Such a condensate also forms in systems of fermions, where the bosons are Cooper pairs of fermions:

ky Pair wavefunction in cuprates: 22 Ψ =−()kkxy() ↑↓−↓↑ k x S = 0 Velocity distribution function of ultracold 87Rb atoms

M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman and E. A. Cornell, Science 269, 198 (1995) Superflow: The wavefunction of the condensate iθ r Ψ→Ψe ( ) Superfluid velocity v = ∇θ s m (for non-Galilean invariant superfluids, the co-efficient of ∇θ is modified) Excitations of the superfluid: Vortices Observation of quantized vortices in rotating 4He

E.J. Yarmchuk, M.J.V. Gordon, and R.E. Packard, Observation of Stationary Vortex Arrays in Rotating Superfluid Helium, Phys. Rev. Lett. 43, 214 (1979). Observation of quantized vortices in rotating ultracold Na

J. R. Abo-Shaeer, C. Raman, J. M. Vogels, and W. Ketterle, Observation of Vortex Lattices in Bose-Einstein Condensates, Science 292, 476 (2001). Quantized fluxoids in YBa2Cu3O6+y

J. C. Wynn, D. A. Bonn, B.W. Gardner, Yu-Ju Lin, Ruixing Liang, W. N. Hardy, J. R. Kirtley, and K. A. Moler, Phys. Rev. Lett. 87, 197002 (2001). OutlineOutline I. Bose-Einstein condensation and superfluidity.

II. The superfluid-insulator quantum phase transition.

III. The cuprate superconductors, and their proximity to a superfluid-insulator transition.

IV. Landau-Ginzburg-Wilson theory of the superfluid- insulator transition.

V. Beyond the LGW paradigm: continuous quantum transitions with multiple order parameters.

VI. Experimental tests in the cuprates. II. The superfluid-insulator quantum phase transition Velocity distribution function of ultracold 87Rb atoms

M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman and E. A. Cornell, Science 269, 198 (1995) Apply a periodic potential (standing laser beams) to trapped ultracold bosons (87Rb)

M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Nature 415, 39 (2002). 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). Superfluid-insulator quantum phase transition at T=0

V0=0Er V0=3Er V0=7Er V0=10Er

V0=13Er V0=14Er V0=16Er V0=20Er Bosons at filling fraction f = 1 Weak interactions: superfluidity

Strong interactions: Mott insulator which preserves all lattice symmetries

M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Nature 415, 39 (2002). Bosons at filling fraction f = 1

Ψ ≠ 0

Weak interactions: superfluidity Bosons at filling fraction f = 1

Ψ ≠ 0

Weak interactions: superfluidity Bosons at filling fraction f = 1

Ψ ≠ 0

Weak interactions: superfluidity Bosons at filling fraction f = 1

Ψ ≠ 0

Weak interactions: superfluidity Bosons at filling fraction f = 1

Ψ = 0

Strong interactions: insulator Bosons at filling fraction f = 1/2

Ψ ≠ 0

Weak interactions: superfluidity Bosons at filling fraction f = 1/2

Ψ ≠ 0

Weak interactions: superfluidity Bosons at filling fraction f = 1/2

Ψ ≠ 0

Weak interactions: superfluidity Bosons at filling fraction f = 1/2

Ψ ≠ 0

Weak interactions: superfluidity Bosons at filling fraction f = 1/2

Ψ ≠ 0

Weak interactions: superfluidity Bosons at filling fraction f = 1/2

Ψ = 0

Strong interactions: insulator Bosons at filling fraction f = 1/2

Ψ = 0

Strong interactions: insulator Bosons at filling fraction f = 1/2

Ψ = 0

Strong interactions: insulator

Insulator has “density wave” order Bosons on the square lattice at filling fraction f=1/2

?

Insulator Charge density Superfluid wave (CDW) order Interactions between bosons Bosons on the square lattice at filling fraction f=1/2

?

Insulator Charge density Superfluid wave (CDW) order Interactions between bosons Bosons on the square lattice at filling fraction f=1/2 1 = ( + ) 2

?

Insulator Valence bond Superfluid solid (VBS) order Interactions between bosons

N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989). Bosons on the square lattice at filling fraction f=1/2 1 = ( + ) 2

?

Insulator Valence bond Superfluid solid (VBS) order Interactions between bosons

N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989). Bosons on the square lattice at filling fraction f=1/2 1 = ( + ) 2

?

Insulator Valence bond Superfluid solid (VBS) order Interactions between bosons

N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989). Bosons on the square lattice at filling fraction f=1/2 1 = ( + ) 2

?

Insulator Valence bond Superfluid solid (VBS) order Interactions between bosons

N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989). Bosons on the square lattice at filling fraction f=1/2

?

Insulator Valence bond Superfluid solid (VBS) order Interactions between bosons

N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989). Bosons on the square lattice at filling fraction f=1/2

?

Insulator Valence bond Superfluid solid (VBS) order Interactions between bosons

N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989). Bosons on the square lattice at filling fraction f=1/2

?

Insulator Valence bond Superfluid solid (VBS) order Interactions between bosons

N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989). Bosons on the square lattice at filling fraction f=1/2

?

Insulator Valence bond Superfluid solid (VBS) order Interactions between bosons

N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989). Bosons on the square lattice at filling fraction f=1/2

?

Insulator Valence bond Superfluid solid (VBS) order Interactions between bosons

N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989). OutlineOutline I. Bose-Einstein condensation and superfluidity.

II. The superfluid-insulator quantum phase transition.

III. The cuprate superconductors, and their proximity to a superfluid-insulator transition.

IV. Landau-Ginzburg-Wilson theory of the superfluid- insulator transition.

V. Beyond the LGW paradigm: continuous quantum transitions with multiple order parameters.

VI. Experimental tests in the cuprates. III. The cuprate superconductors and their proximity to a superfluid-insulator transition La2CuO4 La

O

Cu La2CuO4

Mott insulator: square lattice antiferromagnet

∑ ij ⋅= SSJH ji ij>< La2-δSrδCuO4

Superfluid: condensate of paired holes

S = 0 The cuprate superconductor Ca2-xNaxCuO2Cl2

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

Evidence that holes can form an insulating state with period ≈ 4 modulation in the density

T. Hanaguri, C. Lupien, Y. Kohsaka, D.-H. Lee, M. Azuma, M. Takano, H. Takagi, and J. C. Davis, Nature 430, 1001 (2004). Sr24-xCaxCu24O41

Nature 431, 1078 (2004); cond-mat/0604101

Resonant X-ray scattering evidence that the modulated state has one hole pair per unit cell. Sr24-xCaxCu24O41

Nature 431, 1078 (2004); cond-mat/0604101

Similar to the superfluid-insulator transition of bosons at fractional filling OutlineOutline I. Bose-Einstein condensation and superfluidity.

II. The superfluid-insulator quantum phase transition.

III. The cuprate superconductors, and their proximity to a superfluid-insulator transition.

IV. Landau-Ginzburg-Wilson theory of the superfluid- insulator transition.

V. Beyond the LGW paradigm: continuous quantum transitions with multiple order parameters.

VI. Experimental tests in the cuprates. IV. Landau-Ginzburg-Wilson theory of the superfluid-insulator transition Bosons on the square lattice at filling fraction f=1/2 1 = ( + ) 2

?

Superfluid Insulator Valence bond Ψ≠0 solid (VBS) order Interactions between bosons

N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989). Insulating phases of bosons at filling fraction f = 1/2

1 = ( + ) 2

Charge density Valence bond Valence bond wave (CDW) order solid (VBS) order solid (VBS) order

iQr. Can define a common CDW/VBS order using a generalized "density" ρρ(r)= ∑ Qe Q All insulators have Ψ= 0 and ρQ ≠ 0 for certain Q

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) Landau-Ginzburg-Wilson approach to multiple order parameters:

⎡⎤ FF=Ψ+sc [ ] Fcharge ⎣⎦ρQ + Fint 24 Frusc []Ψ=11 Ψ + Ψ + 24 ⎡⎤ Frucharge ⎣⎦ρρQQ=+2 2 ρ Q + 2 2 Fvint =ΨρQ +

Distinct symmetries of order parameters permit couplings only between their energy densities Predictions of LGW theory First order Ψ transition ρQ Superconductor Charge-ordered insulator

rr12− Predictions of LGW theory First order Ψ transition ρQ Superconductor Charge-ordered insulator

rr12− Coexistence (Supersolid) Ψ ρQ Superconductor Charge-ordered insulator

rr12− Predictions of LGW theory First order Ψ transition ρQ Superconductor Charge-ordered insulator

rr12− Coexistence (Supersolid) Ψ ρQ Superconductor Charge-ordered insulator

rr12−

""Disordered ρ Ψ Q (≠ topologically ordered) Charge-ordered Superconductor Ψ ==0,ρ 0 sc Q insulator rr12− Predictions of LGW theory First order Ψ transition ρQ Superconductor Charge-ordered insulator

rr12− Coexistence (Supersolid) Ψ ρQ Superconductor Charge-ordered insulator

rr12−

""Disordered ρ Ψ Q (≠ topologically ordered) Charge-ordered Superconductor Ψ ==0,ρ 0 sc Q insulator rr12− OutlineOutline I. Bose-Einstein condensation and superfluidity.

II. The superfluid-insulator quantum phase transition.

III. The cuprate superconductors, and their proximity to a superfluid-insulator transition.

IV. Landau-Ginzburg-Wilson theory of the superfluid- insulator transition.

V. Beyond the LGW paradigm: continuous quantum transitions with multiple order parameters.

VI. Experimental tests in the cuprates. V. Beyond the LGW paradigm: continuous transitions with multiple order parameters Excitations of the superfluid: Vortices and anti-vortices

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” ? In ordinary fluids, vortices experience the Magnus Force

FM

FM = (mass density of air)ii( velocity of ball) ( circulation)

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)

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)

Bosons on the square lattice at filling fraction f=p/q Bosons on the square lattice at filling fraction f=p/q Bosons on the square lattice at filling fraction f=p/q A Landau-forbidden continuous transitions

Ψ ρQ Superfluid Charge-ordered insulator

rr12−

Vortices in the superfluid have associated quantum numbers which determine the local “charge order”, and their proliferation in the superfluid can lead to a continuous transition to a charge-ordered insulator Bosons on the square lattice at filling fraction f=1/2 1 = ( + ) 2

?

Superfluid Insulator Valence bond Ψ≠0 solid (VBS) order Interactions between bosons

N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989). Bosons on the square lattice at filling fraction f=1/2 1 = ( + ) 2

Superfluid Insulator Valence bond Ψ≠0 solid (VBS) order

“Aharanov-Bohm” or “Berry” phases lead to surprising kinematic duality relations between seemingly distinct orders. These phase factors allow for continuous quantum phase transitions in situations where such transitions are forbidden by Landau-Ginzburg-Wilson theory. OutlineOutline I. Bose-Einstein condensation and superfluidity.

II. The superfluid-insulator quantum phase transition.

III. The cuprate superconductors, and their proximity to a superfluid-insulator transition.

IV. Landau-Ginzburg-Wilson theory of the superfluid- insulator transition.

V. Beyond the LGW paradigm: continuous quantum transitions with multiple order parameters.

VI. Experimental tests in the cuprates. VI. Experimental tests in the cuprates STM around vortices induced by a magnetic field in the superconducting state J. E.3.0 Hoffman, E. W. Hudson, K. M. Lang, V. Madhavan, S. H. Pan, H. Eisaki, S. Uchida, and J. C. Davis, Science 295, 466 (2002).

2.5 Local density of states (LDOS) Regular QPSR 2.0 Vortex 1Å spatial resolution image of integrated 1.5 LDOS of

Bi2Sr2CaCu2O8+δ 1.0 ( 1meV to 12 meV) at B=5 Tesla.

Differential Conductance (nS) 0.5

0.0 -120 -80 -40 0 40 80 120 Sample Bias (mV) S.H. Pan et al. Phys. Rev. Lett. 85, 1536 (2000). Vortex-induced LDOS of Bi2Sr2CaCu2O8+δ integrated from 1meV to 12meV at 4K

7 pA Vortices have b halos with LDOS modulations at a period ≈ 4 lattice 0 pA spacings

100Å

J. Hoffman E. W. Hudson, K. M. Lang, Prediction of periodic LDOS V. Madhavan, S. H. Pan, H. Eisaki, S. Uchida, modulations near vortices: and J. C. Davis, Science 295, 466 (2002). K. Park and S. Sachdev, Phys. Rev. B 64, 184510 (2001).

Influence of the quantum oscillating vortex on the LDOS

ω /ωv 2 2 mv α ==F 1 ωv Influence of the quantum oscillating vortex on the LDOS

No zero bias peak.

ω /ωv 2 2 mv α ==F 1 ωv Influence of the quantum oscillating vortex on the LDOS

Resonant feature near the vortex oscillation frequency

ω /ωv 2 2 mv α ==F 1 ωv 3.0 Influence of the quantum oscillating vortex on the LDOS

2.5 Regular QPSR 2.0 Vortex

1.5

1.0

Differential Conductance (nS) 0.5

0.0 -120 -80 -40 0 40 80 120 Sample Bias (mV)

ω /ωv I. Maggio-Aprile et al. Phys. Rev. Lett. 75, 2754 (1995) 2 S.H. Pan et al. Phys. Rev. Lett. 85, 1536 (2000). 2 mv α ==F 1 ωv ConclusionsConclusions •• QQuantumuantum zero zero point point motion motion of of th thee vortex vortex provides provides a a natural natural explanationexplanation for for LDOS LDOS modulations modulations observed observed in in STM STM experiments. experiments. •• SSizeize of of modulation modulation halo halo allows allows estimate estimate of of the the inertial inertial mass mass of of a a vortexvortex •• DDirectirect detection detection of of vortex vortex zero-p zero-pointoint motion motion may may be be possible possible in in inelasticinelastic neutron neutron or or light-scattering light-scattering experiments experiments •• TThehe quantum quantum zero-point zero-point motion motion of of the the vortices vortices influences influences the the spectrumspectrum of of the the electronic electronic quasiparti quasiparticles,cles, in in a a manner manner consistent consistent with with LDOSLDOS spectrum spectrum •• ““Aharanov-Bohm”Aharanov-Bohm”oorr “Berry” “Berry”pphaseshases lead lead to to surprising surprising kinematic kinematic dualityduality relations relations between between seemin seeminglygly distinct distinct orders. orders. These These phase phase factorsfactors allow allow for for continuous continuous quantum quantum phase phase transitions transitions in in situations situations wherewhere such such trans transitionsitions are are forb forbiddenidden by by Landau-Ginzburg-Wilson Landau-Ginzburg-Wilson theory.theory.