The quantum phases of matter and gauge-gravity dualityqpt.physics.harvard.edu/talks/fortaleza13.pdf · Fortaleza, Brazil, April 10, 2013 Subir Sachdev Scientific American 308, 44

The quantum phases of matter and

gauge-gravity duality

HARVARD

Frontiers in Condensed Matter Science,Fortaleza, Brazil, April 10, 2013

Subir Sachdev

Scientific American 308, 44 (January 2013)

Wednesday, April 10, 13

Sommerfeld-Bloch theory of metals, insulators, and superconductors:many-electron quantum states are adiabatically

connected to independent electron states




Band insulators

E

MetalMetal

carryinga current

InsulatorSuperconductor

kAn even number of electrons per unit cell


E

MetalMetal

carryinga current


k

E

MetalMetal

carryinga current


k



Metals

An odd number of electrons per unit cell


E

MetalMetal

carryinga current


k



Superconductors


E

MetalMetal

carryinga current


k

E

MetalMetal

carryinga current


k

Boltzmann-Landau theory of dynamics of metals:

Long-lived quasiparticles (and quasiholes) have weak interactions which can be described by a Boltzmann equation

Metals



E

MetalMetal

carryinga current


k

E

MetalMetal

carryinga current


k

Boltzmann-Landau theory of dynamics of metals:

Long-lived quasiparticles (and quasiholes) have weak interactions which can be described by a Boltzmann equation

Metals



Modern phases of quantum matterNot adiabatically connected

to independent electron states:


Modern phases of quantum matterNot adiabatically connected

to independent electron states:many-particle

quantum entanglement,and no quasiparticles


_

Quantum Entanglement: quantum superposition with more than one particleHydrogen molecule:


_

Quantum Entanglement: quantum superposition with more than one particle


_



_


Einstein-Podolsky-Rosen “paradox”: Non-local correlations between observations arbitrarily far apart


Ishida, Nakai, and HosonoarXiv:0906.2045v1

Iron pnictides: a new class of high temperature superconductors


TSDW Tc

T0

2.0

0

α"

1.0 SDW

Superconductivity

BaFe2(As1-xPx)2

AF

Resistivity⇠ ⇢0 +AT↵

S. Kasahara, T. Shibauchi, K. Hashimoto, K. Ikada, S. Tonegawa, R. Okazaki, H. Shishido, H. Ikeda, H. Takeya, K. Hirata, T. Terashima, and Y. Matsuda,

Physical Review B 81, 184519 (2010)Wednesday, April 10, 13

TSDW Tc

T0

2.0

0

α"

1.0 SDW

Superconductivity

BaFe2(As1-xPx)2

AF



Physical Review B 81, 184519 (2010)

Short-range entanglement in state with Neel (AF) order


TSDW Tc

T0

2.0

0

α"

1.0 SDW

Superconductivity

BaFe2(As1-xPx)2

AF




SuperconductorBose condensate of pairs of electrons

Short-range entanglementWednesday, April 10, 13

TSDW Tc

T0

2.0

0

α"

1.0 SDW

Superconductivity

BaFe2(As1-xPx)2

AF




Ordinary metal(Fermi liquid)


TSDW Tc

T0

2.0

0

α"

1.0 SDW

Superconductivity

BaFe2(As1-xPx)2

AF



Physical Review B 81, 184519 (2010)Wednesday, April 10, 13

TSDW Tc

T0

2.0

0

α"

1.0 SDW

Superconductivity

BaFe2(As1-xPx)2

AF




StrangeMetal


TSDW Tc

T0

2.0

0

α"

1.0 SDW

Superconductivity

BaFe2(As1-xPx)2

AF




StrangeMetal

no quasiparticles,Landau-Boltzmann theory

does not apply


1. Superfluid-insulator transition of ultracold atoms in optical lattices:

Quantum criticality and conformal field theories

2. Gauge-gravity duality

3. Black-hole horizons and quasi-normal modes

4. Strange metals: What lies beyond the horizon ?

Outline







Outline


M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Nature 415, 39 (2002).

Ultracold 87Rbatoms - bosons

Superfluid-insulator transition


g

T

gc

0

InsulatorSuperfluid

Quantumcritical

TKT

��c

h i 6= 0 h i = 0


g

T

gc

0

InsulatorSuperfluid

Quantumcritical

TKT

��c

h i 6= 0 h i = 0

! a complex field representing the

Bose-Einstein condensate of the superfluid


g

T

gc

0

InsulatorSuperfluid

Quantumcritical

TKT

��c

h i 6= 0 h i = 0

S =

Zd2rdt

⇥|@t |2 � c2|rr |2 � V ( )

⇤

V ( ) = (�� c)| |2 + u�| |2

�2


g

T

gc

0

InsulatorSuperfluid

Quantumcritical

TKT

��c

h i 6= 0 h i = 0

system with a recently developed scheme based on single-atom-resolved detection24. It is the high sensitivity of this method thatallowed us to reduce the modulation amplitude by almost an orderof magnitude compared with earlier experiments20,21 and to stay wellwithin the linear response regime (Supplementary Information).

The results for selected lattice depths V0 are shown in Fig. 2b. Weobserve a gapped response with an asymmetric overall shape that willbe analysed in the following paragraphs. Notably, the maximumobserved temperature after modulation is well below the ‘melting’temperature for a Mott insulator in the atomic limit25, Tmelt < 0.2U/kB

(kB, Boltzmann’s constant), demonstrating that our experiments probethe quantum gas in the degenerate regime. To obtain numerical valuesfor the onset of spectral response, we fitted each spectrum with an errorfunction centred at a frequency n0 (Fig. 2b, black lines). With japproaching jc, the shift of the gap to lower frequencies is alreadyvisible in the raw data (Fig. 2b) and becomes even more apparent forthe fitted gap n0 as a function of j/jc (Fig. 2a, filled circles). The n0 valuesare in quantitative agreement with a prediction for the Higgs gap nSF atcommensurate filling (solid line):

hnSF=U~ 3!!!2p

{4" #

1zj=jc! "$ %1=2

j=jc{1! "1=2

Here h denotes Planck’s constant. This value is based on an analysis ofvariations around a mean-field state7,16 (throughout the manuscript,we have rescaled jc in the theoretical calculations to match the valuejc<0:06 obtained from quantum Monte Carlo simulations26).

The sharpness of the spectral onset can be quantified by the width ofthe fitted error function, which is shown as vertical dashed lines inFig. 2a. Approaching the critical point, the spectral onset becomessharper, and the width normalized to the centre frequency n0 remainsconstant (Supplementary Fig. 3). The constancy of this ratio indicatesthat the width of the spectral onset scales with the distance to thecritical point in the same way as the gap frequency.

We observe similar gapped responses in the Mott insulating regime(Supplementary Information and Fig. 5a), with the gap closing con-tinuously when approaching the critical point (Fig. 2a, open circles).We interpret this as a result of combined particle and hole excitationswith a frequency given by the Mott excitation gap that closes at thetransition point16. The fitted gaps are consistent with the Mott gap

hnMI=U~ 1z 12!!!2p

{17" #

j=jc$ %1=2

1{j=jc! "1=2

where nMI is the Mott gap as predicted by mean-field theory16 (Fig. 2a,dashed line).

The observed softening of the onset of spectral response in thesuperfluid regime has led to an identification of the experimentalsignal with a response from collective excitations of Higgs type. Togain further insight into the full in-trap response, we calculated theeigenspectrum of the system in a Gutzwiller approach16,22 (Methodsand Supplementary Information). The result is a series of discreteeigenfrequencies (Fig. 3a), and the corresponding eigenmodes showin-trap superfluid density distributions, which are reminiscent of thevibrational modes of a drum (Fig. 3b). The frequency of the lowest-lying amplitude-like eigenmode n0,G closely follows the long-wave-length prediction for homogeneous commensurate filling nSF over awide range of couplings j/jc until the response rounds off in the vicinityof the critical point due to the finite size of the system (Fig. 3c). Fittingthe low-frequency edge of the experimental data can be interpreted asextracting the frequency of this mode, which explains the goodquantitative agreement with the prediction for the homogeneous com-mensurate filling in Fig. 2a. Modes at different frequencies from thelowest-lying amplitude-like mode broaden the spectrum only abovethe onset of spectral response.

An eigenmode analysis, however, does not yield any informationabout the finite spectral width of the modes, which stems from theinteraction between amplitude and phase excitations. We will considerthe question of the spectral width by analysing the low-, intermediate-and high-frequency parts of the response separately. We begin byexamining the low-frequency part of the response, which is expectedto be governed by a process coupling a virtually excited amplitudemode to a pair of phase modes with opposite momenta. As a result,the response of a strongly interacting, two-dimensional superfluid is

a1

2

V

Re( )Im( )

Higgs modeNambu–

Goldstonemode

j/jc 1

0 100 200 300 4000

5

10

15

20

Time (ms)

Lat

tice

dept

h (E

r)

Lattice loading Modulation Hold time Ramp to atomic limit

Temperaturemeasurement

V0

Ttot = 200 ms

A = 0.03V0

Tmod = 20WW

b

3

j/jc * 1

j/jc , 1

ΨΨ

Figure 1 | Illustration of the Higgs mode and experimental sequence.a, Classical energy density V as a function of the order parameter Y. Within theordered (superfluid) phase, Nambu–Goldstone and Higgs modes arise fromphase and amplitude modulations (blue and red arrows in panel 1). As thecoupling j 5 J/U (see main text) approaches the critical value jc, the energydensity transforms into a function with a minimum at Y 5 0 (panels 2 and 3).Simultaneously, the curvature in the radial direction decreases, leading to acharacteristic reduction of the excitation frequency for the Higgs mode. In thedisordered (Mott insulating) phase, two gapped modes exist, respectivelycorresponding to particle and hole excitations in our case (red and blue arrow inpanel 3). b, The Higgs mode can be excited with a periodic modulation of thecoupling j, which amounts to a ‘shaking’ of the classical energy densitypotential. In the experimental sequence, this is realized by a modulation of theoptical lattice potential (see main text for details). t 5 1/nmod; Er, lattice recoilenergy.

LETTER RESEARCH

2 6 J U L Y 2 0 1 2 | V O L 4 8 7 | N A T U R E | 4 5 5

Macmillan Publishers Limited. All rights reserved©2012




hnSF=U~ 3!!!2p

{4" #

1zj=jc! "$ %1=2

j=jc{1! "1=2




hnMI=U~ 1z 12!!!2p

{17" #

j=jc$ %1=2

1{j=jc! "1=2




a1

2

V

Re( )Im( )

Higgs modeNambu–

Goldstonemode

j/jc 1

0 100 200 300 4000

5

10

15

20

Time (ms)

Lat

tice

dept

h (E

r)



V0

Ttot = 200 ms

A = 0.03V0

Tmod = 20WW

b

3

j/jc * 1

j/jc , 1

ΨΨ


LETTER RESEARCH

2 6 J U L Y 2 0 1 2 | V O L 4 8 7 | N A T U R E | 4 5 5





hnSF=U~ 3!!!2p

{4" #

1zj=jc! "$ %1=2

j=jc{1! "1=2




hnMI=U~ 1z 12!!!2p

{17" #

j=jc$ %1=2

1{j=jc! "1=2




a1

2

V

Re( )Im( )

Higgs modeNambu–

Goldstonemode

j/jc 1

0 100 200 300 4000

5

10

15

20

Time (ms)

Lat

tice

dept

h (E

r)



V0

Ttot = 200 ms

A = 0.03V0

Tmod = 20WW

b

3

j/jc * 1

j/jc , 1

ΨΨ


LETTER RESEARCH

2 6 J U L Y 2 0 1 2 | V O L 4 8 7 | N A T U R E | 4 5 5





hnSF=U~ 3!!!2p

{4" #

1zj=jc! "$ %1=2

j=jc{1! "1=2




hnMI=U~ 1z 12!!!2p

{17" #

j=jc$ %1=2

1{j=jc! "1=2




a1

2

V

Re( )Im( )

Higgs modeNambu–

Goldstonemode

j/jc 1

0 100 200 300 4000

5

10

15

20

Time (ms)

Lat

tice

dept

h (E

r)



V0

Ttot = 200 ms

A = 0.03V0

Tmod = 20WW

b

3

j/jc * 1

j/jc , 1

ΨΨ


LETTER RESEARCH

2 6 J U L Y 2 0 1 2 | V O L 4 8 7 | N A T U R E | 4 5 5


Particles and holes correspond

to the 2 normal modes in the

oscillation of about = 0.

S =

Zd2rdt

⇥|@t |2 � c2|rr |2 � V ( )

⇤

V ( ) = (�� c)| |2 + u�| |2

�2


Insulator (the vacuum) at large repulsion between bosons


Particles ⇠ †

Excitations of the insulator:


Holes ⇠

Excitations of the insulator:


g

T

gc

0

InsulatorSuperfluid

Quantumcritical

TKT

��c

h i 6= 0 h i = 0




hnSF=U~ 3!!!2p

{4" #

1zj=jc! "$ %1=2

j=jc{1! "1=2




hnMI=U~ 1z 12!!!2p

{17" #

j=jc$ %1=2

1{j=jc! "1=2




a1

2

V

Re( )Im( )

Higgs modeNambu–

Goldstonemode

j/jc 1

0 100 200 300 4000

5

10

15

20

Time (ms)

Lat

tice

dept

h (E

r)



V0

Ttot = 200 ms

A = 0.03V0

Tmod = 20WW

b

3

j/jc * 1

j/jc , 1

ΨΨ


LETTER RESEARCH

2 6 J U L Y 2 0 1 2 | V O L 4 8 7 | N A T U R E | 4 5 5





hnSF=U~ 3!!!2p

{4" #

1zj=jc! "$ %1=2

j=jc{1! "1=2




hnMI=U~ 1z 12!!!2p

{17" #

j=jc$ %1=2

1{j=jc! "1=2




a1

2

V

Re( )Im( )

Higgs modeNambu–

Goldstonemode

j/jc 1

0 100 200 300 4000

5

10

15

20

Time (ms)

Lat

tice

dept

h (E

r)



V0

Ttot = 200 ms

A = 0.03V0

Tmod = 20WW

b

3

j/jc * 1

j/jc , 1

ΨΨ


LETTER RESEARCH

2 6 J U L Y 2 0 1 2 | V O L 4 8 7 | N A T U R E | 4 5 5





hnSF=U~ 3!!!2p

{4" #

1zj=jc! "$ %1=2

j=jc{1! "1=2




hnMI=U~ 1z 12!!!2p

{17" #

j=jc$ %1=2

1{j=jc! "1=2




a1

2

V

Re( )Im( )

Higgs modeNambu–

Goldstonemode

j/jc 1

0 100 200 300 4000

5

10

15

20

Time (ms)

Lat

tice

dept

h (E

r)



V0

Ttot = 200 ms

A = 0.03V0

Tmod = 20WW

b

3

j/jc * 1

j/jc , 1

ΨΨ


LETTER RESEARCH

2 6 J U L Y 2 0 1 2 | V O L 4 8 7 | N A T U R E | 4 5 5





hnSF=U~ 3!!!2p

{4" #

1zj=jc! "$ %1=2

j=jc{1! "1=2




hnMI=U~ 1z 12!!!2p

{17" #

j=jc$ %1=2

1{j=jc! "1=2




a1

2

V

Re( )Im( )

Higgs modeNambu–

Goldstonemode

j/jc 1

0 100 200 300 4000

5

10

15

20

Time (ms)

Lat

tice

dept

h (E

r)



V0

Ttot = 200 ms

A = 0.03V0

Tmod = 20WW

b

3

j/jc * 1

j/jc , 1

ΨΨ


LETTER RESEARCH

2 6 J U L Y 2 0 1 2 | V O L 4 8 7 | N A T U R E | 4 5 5


Particles and holes correspond

to the 2 normal modes in the

oscillation of about = 0.

S =

Zd2rdt

⇥|@t |2 � c2|rr |2 � V ( )

⇤

V ( ) = (�� c)| |2 + u�| |2

�2


g

T

gc

0

InsulatorSuperfluid

Quantumcritical

TKT

��c

h i 6= 0 h i = 0




hnSF=U~ 3!!!2p

{4" #

1zj=jc! "$ %1=2

j=jc{1! "1=2




hnMI=U~ 1z 12!!!2p

{17" #

j=jc$ %1=2

1{j=jc! "1=2




a1

2

V

Re( )Im( )

Higgs modeNambu–

Goldstonemode

j/jc 1

0 100 200 300 4000

5

10

15

20

Time (ms)

Lat

tice

dept

h (E

r)



V0

Ttot = 200 ms

A = 0.03V0

Tmod = 20WW

b

3

j/jc * 1

j/jc , 1

ΨΨ


LETTER RESEARCH

2 6 J U L Y 2 0 1 2 | V O L 4 8 7 | N A T U R E | 4 5 5


Nambu-Goldstone mode is the

oscillation in the phase

at a constant non-zero | |.

S =

Zd2rdt

⇥|@t |2 � c2|rr |2 � V ( )

⇤

V ( ) = (�� c)| |2 + u�| |2

�2


g

T

gc

0

InsulatorSuperfluid

Quantumcritical

TKT

��c

A conformal field theory

in 2+1 spacetime dimensions:

a CFT3

h i 6= 0 h i = 0

S =

Zd2rdt

⇥|@t |2 � c2|rr |2 � V ( )

⇤

V ( ) = (�� c)| |2 + u�| |2

�2


g

T

gc

0

InsulatorSuperfluid

Quantumcritical

TKT

��c

Quantum state with

complex, many-body,

“long-range” quantum entanglement

h i 6= 0 h i = 0

S =

Zd2rdt

⇥|@t |2 � c2|rr |2 � V ( )

⇤

V ( ) = (�� c)| |2 + u�| |2

�2


g

T

gc

0

InsulatorSuperfluid

Quantumcritical

TKT

��c

No well-defined normal modes,

or particle-like excitations

h i 6= 0 h i = 0

S =

Zd2rdt

⇥|@t |2 � c2|rr |2 � V ( )

⇤

V ( ) = (�� c)| |2 + u�| |2

�2


g

T

gc

0

InsulatorSuperfluid

Quantumcritical

TKT

��c

h i 6= 0 h i = 0




hnSF=U~ 3!!!2p

{4" #

1zj=jc! "$ %1=2

j=jc{1! "1=2




hnMI=U~ 1z 12!!!2p

{17" #

j=jc$ %1=2

1{j=jc! "1=2




a1

2

V

Re( )Im( )

Higgs modeNambu–

Goldstonemode

j/jc 1

0 100 200 300 4000

5

10

15

20

Time (ms)

Lat

tice

dept

h (E

r)



V0

Ttot = 200 ms

A = 0.03V0

Tmod = 20WW

b

3

j/jc * 1

j/jc , 1

ΨΨ


LETTER RESEARCH

2 6 J U L Y 2 0 1 2 | V O L 4 8 7 | N A T U R E | 4 5 5


Higgs mode is theoscillation in theamplitude | |. This decaysrapidly by emitting pairsof Nambu-Goldstone modes.

S =

Zd2rdt

⇥|@t |2 � c2|rr |2 � V ( )

⇤

V ( ) = (�� c)| |2 + u�| |2

�2


g

T

gc

0

InsulatorSuperfluid

Quantumcritical

TKT

��c

h i 6= 0 h i = 0




hnSF=U~ 3!!!2p

{4" #

1zj=jc! "$ %1=2

j=jc{1! "1=2




hnMI=U~ 1z 12!!!2p

{17" #

j=jc$ %1=2

1{j=jc! "1=2




a1

2

V

Re( )Im( )

Higgs modeNambu–

Goldstonemode

j/jc 1

0 100 200 300 4000

5

10

15

20

Time (ms)

Lat

tice

dept

h (E

r)



V0

Ttot = 200 ms

A = 0.03V0

Tmod = 20WW

b

3

j/jc * 1

j/jc , 1

ΨΨ


LETTER RESEARCH

2 6 J U L Y 2 0 1 2 | V O L 4 8 7 | N A T U R E | 4 5 5


Despite rapid decay,there is a well-defined

Higgs “quasi-normal mode”.This is associated with a pole

in the lower-halfof the complex frequency plane.

S =

Zd2rdt

⇥|@t |2 � c2|rr |2 � V ( )

⇤

V ( ) = (�� c)| |2 + u�| |2

�2

D. Podolsky and S. Sachdev, Phys. Rev. B 86, 054508 (2012).





hnSF=U~ 3!!!2p

{4" #

1zj=jc! "$ %1=2

j=jc{1! "1=2




hnMI=U~ 1z 12!!!2p

{17" #

j=jc$ %1=2

1{j=jc! "1=2




a1

2

V

Re( )Im( )

Higgs modeNambu–

Goldstonemode

j/jc 1

0 100 200 300 4000

5

10

15

20

Time (ms)

Lat

tice

dept

h (E

r)



V0

Ttot = 200 ms

A = 0.03V0

Tmod = 20WW

b

3

j/jc * 1

j/jc , 1

ΨΨ


LETTER RESEARCH

2 6 J U L Y 2 0 1 2 | V O L 4 8 7 | N A T U R E | 4 5 5


!

S =

Zd2rdt

⇥|@t |2 � c2|rr |2 � V ( )

⇤

V ( ) = (�� c)| |2 + u�| |2

�2

D. Podolsky and S. Sachdev, Phys. Rev. B 86, 054508 (2012).


expected to diverge at low frequencies, if the probe in use coupleslongitudinally to the order parameter2,4,5,9 (for example to the real partof Y, if the equilibrium value of Y was chosen along the real axis), as isthe case for neutron scattering. If, instead, the coupling is rotationallyinvariant (for example through coupling to jYj2), as expected forlattice modulation, such a divergence could be avoided and the

response is expected to scale as n3 at low frequencies3,6,9,17.Combining this result with the scaling dimensions of the responsefunction for a rotationally symmetric perturbation coupling to jYj2,we expect the low-frequency response to be proportional to(1 2 j/jc)

22n3 (ref. 9 and Methods). The experimentally observed sig-nal is consistent with this scaling at the ‘base’ of the absorption feature(Fig. 4). This indicates that the low-frequency part is dominated byonly a few in-trap eigenmodes, which approximately show the genericscaling of the homogeneous system for a response function describingcoupling to jYj2.

In the intermediate-frequency regime, it remains a challenge toconstruct a first-principles analytical treatment of the in-trap systemincluding all relevant decay and coupling processes. Lacking such atheory, we constructed a heuristic model combining the discrete spec-trum from the Gutzwiller approach (Fig. 3a) with the line shape for ahomogeneous system based on an O(N) field theory in two dimen-sions, calculated in the large-N limit3,6 (Methods). An implicit assump-tion of this approach is a continuum of phase modes, which is

Sup

er!u

id d

ensi

ty

0.1

0.3

0.5

0.7

a

b1

0 0.5 1 1.5 2

0

0.02

0.04

d

0 0.5 0.6 0.7 0.8 0.9 1 1.1

Q0,G

0

0.5

1

Line

str

engt

h (a

.u.)

hQmod/U

hQ/U

j/jc

c

1 1.5 2 2.50

0.4

0.8

1.2V0 = 9.5Erj/jc = 1.4

k BΔT

/U, Δ

E/U

Q0,G

2

2

1

3

3

4

4

hQmod/U

Figure 3 | Theory of in-trap response. a, A diagonalization of the trappedsystem in a Gutzwiller approximation shows a discrete spectrum of amplitude-like eigenmodes. Shown on the vertical axis is the strength of the response to amodulation of j. Eigenmodes of phase type are not shown (Methods) and n0,G

denotes the gap as calculated in the Gutzwiller approximation. a.u., arbitraryunits. b, In-trap superfluid density distribution for the four amplitude modeswith the lowest frequencies, as labelled in a. In contrast to the superfluiddensity, the total density of the system stays almost constant (not shown).c, Discrete amplitude mode spectrum for various couplings j/jc. Each red circlecorresponds to a single eigenmode, with the intensity of the colour beingproportional to the line strength. The gap frequency of the lowest-lying modefollows the prediction for commensurate filling (solid line; same as in Fig. 2a)until a rounding off takes place close to the critical point due to the finite size ofthe system. d, Comparison of the experimental response at V0 5 9.5Er (bluecircles and connecting blue line; error bars, s.e.m.) with a 2 3 2 cluster mean-field simulation (grey line and shaded area) and a heuristic model (dashed line;for details see text and Methods). The simulation was done for V0 5 9.5Er (greyline) and for V0 5 (1 6 0.02) 3 9.5Er (shaded grey area), to account for theexperimental uncertainty in the lattice depth, and predicts the energyabsorption per particle DE.

0 0.2 0.4 0.6 0.8 1

0

0.01

0.02

0.03

Qmod/U

(1 –

j/j c)

2 kBΔT

/U

0 0.2 0.4 0.6 0.8 1

0

0.01

0.02

0.03

Qmod/U

k BΔT

/U

Figure 4 | Scaling of the low-frequency response. The low-frequencyresponse in the superfluid regime shows a scaling compatible with theprediction (1 2 j/jc)

22n3 (Methods). Shown is the temperature responserescaled with (1 2 j/jc)

2 for V0 5 10Er (grey), 9.5Er (black), 9Er (green), 8.5Er

(blue) and 8Er (red) as a function of the modulation frequency. The black line isa fit of the form anb with a fitted exponent b 5 2.9(5). The inset shows the samedata points without rescaling, for comparison. Error bars, s.e.m.

hQ0/U

j/jc

0 0.5 1 1.5 2 2.50

0.2

0.4

0.6

0.8

1

1.20 0.03 0.06 0.09 0.12 0.15

j

Super!uidMott Insulator

a b

3

1

2

V0 = 8Erj/jc = 2.2

k BT/U

1

2

3

V0 = 9Erj/jc = 1.6

V0 = 10Erj/jc = 1.2

Q0

0.11

0.13

0.15

0.17

0.12

0.14

0.16

0.18

0 400 8000.12

0.14

0.16

0.18

Qmod (Hz)

Q0

Q0

Figure 2 | Softening of the Higgs mode. a, The fitted gap values hn0/U(circles) show a characteristic softening close to the critical point in quantitativeagreement with analytic predictions for the Higgs and the Mott gap (solid lineand dashed line, respectively; see text). Horizontal and vertical error barsdenote the experimental uncertainty of the lattice depths and the fit error for thecentre frequency of the error function, respectively (Methods). Vertical dashedlines denote the widths of the fitted error function and characterize thesharpness of the spectral onset. The blue shading highlights the superfluid

region. b, Temperature response to lattice modulation (circles and connectingblue line) and fit with an error function (solid black line) for the three differentpoints labelled in a. As the coupling j approaches the critical value jc, the changein the gap values to lower frequencies is clearly visible (from panel 1 to panel 3).Vertical dashed lines mark the frequency U/h corresponding to the on-siteinteraction. Each data point results from an average of the temperatures over,50 experimental runs. Error bars, s.e.m.

RESEARCH LETTER

4 5 6 | N A T U R E | V O L 4 8 7 | 2 6 J U L Y 2 0 1 2





hnSF=U~ 3!!!2p

{4" #

1zj=jc! "$ %1=2

j=jc{1! "1=2




hnMI=U~ 1z 12!!!2p

{17" #

j=jc$ %1=2

1{j=jc! "1=2




a1

2

V

Re( )Im( )

Higgs modeNambu–

Goldstonemode

j/jc 1

0 100 200 300 4000

5

10

15

20

Time (ms)

Lat

tice

dept

h (E

r)



V0

Ttot = 200 ms

A = 0.03V0

Tmod = 20WW

b

3

j/jc * 1

j/jc , 1

ΨΨ


LETTER RESEARCH

2 6 J U L Y 2 0 1 2 | V O L 4 8 7 | N A T U R E | 4 5 5


Observation of Higgs quasi-normal mode across the superfluid-insulator transition of ultracold atoms in a 2-dimensional optical lattice:Response to modulation of lattice depth scales as expected from the LHP pole

Manuel Endres, Takeshi Fukuhara, David Pekker, Marc Cheneau, Peter Schaub, Christian Gross, Eugene Demler, Stefan Kuhr, and Immanuel Bloch, Nature 487, 454 (2012).


3

finite-temperature kernel, N(⌧,!) = e�!⌧ + e�!(1/T�⌧):

�(⌧) =

Z +1

0N(⌧,!)S(!) . (4)

We employ the same protocol of collecting and analyzingdata as in Ref. [13]. More specifically, in the MC simu-lation we collect statistics for the correlation function atMatsubara frequencies !n = 2⇡Tn with integer n

�(i!n) = hK(⌧)K(0)ii!n + hKi (5)

which is related to �(⌧) by a Fourier transform. In thepath integral representation, �(i!n) has a direct unbi-ased estimator, |

Pk e

i!n⌧k |2, where the sum runs overall hopping transitions in a given configuration. Once�(⌧) is recovered from �(i!n), the analytical continuationmethods described in Ref. [13] are applied to extract thespectral function S(!). A discussion on the reproducibil-ity of the analytically continued results for this type ofproblems can also be found in Ref. [13].

We consider system sizes significantly larger than thecorrelation length by a factor of at least four to ensurethat our results are e↵ectively in the thermodynamiclimit. Furthermore, for the SF and MI phases, we setthe temperature T = 1/� to be much smaller than thecharacteristic Higgs energy, so that no details in the rel-evant energy part of spectral function are missed.

We consider two paths in the SF phase to approach theQCP: by increasing the interaction U ! Uc at unity fill-ing factor n = 1 (trajectory i perpendicular to the phaseboundary in Fig 1), and by increasing µ ! µc while keep-ing U = Uc constant (trajectory ii tangential to the phaseboundary in Fig 1). We start with trajectory i by consid-ering three parameter sets for (|g|, L,�): (0.2424, 20, 10),(0.0924, 40, 20, and (0.0462, 80, 40). The prime data areshown in the inset of Fig. 3. After rescaling results ac-cording to Eq. (1) we observe data collapse shown in themain panel of Fig.3. This defines the universal spectralfunction in the superfluid phase, �SF.

When approaching the QCP along trajectory ii,with (|gµ|, L,�) = (0.40, 25, 15), (0.30, 30, 15), and(0.20, 40, 20) we observe a similar data collapse and arriveat the same universal function �SF, see Fig. 4. The fi-nal match is possible only when the characteristic energyscale �(gµ) = C�(g(gµ)) involves a factor of C = 1.2.

The universal spectral function �SF has three distinctfeatures: a) A pronounced peak at !H/� ⇡ 3.3, whichis associated with the Higgs resonance. Since the peak’swidth �/� ⇡ 1 is comparable to its energy, the Higgsmode is strongly damped. It can be identified as awell-defined particle only in a moving reference frame;b) A minimum and another broad maximum between!/� 2 [5, 25] which may originate from multi-Higgs ex-citations [13]; c) The onset of the quantum critical quasi-plateau, in agreement with the scaling hypothesis (1),starting at !/� ⇡ 25. These features are captured by an

0

0.5

1

1.5

2

2.5

3

0 5 10 15 20 25 30 35 40

[Δ(-g

)/J]2/

ν−3 S SF

(ω)/J

ω/Δ(-g)

|g|=0.2424|g|=0.0924|g|=0.0462

0.5

1

1.5

2

2 4 6 8

S SF(ω

)/J

ω/J

FIG. 3. (Color online) Collapse of spectral functions for dif-ferent values of U along trajectory i in the SF phase, labeledby the detuning g = (U � Uc)/J . Inset: original data forSSF(!).

0

0.5

1

1.5

2

2.5

0 5 10 15 20 25 30 35 40

[(Δ(g

µ)/J

]2/ν−3 S SF

(ω)/J

ω/Δ(gµ)

|gµ|=0.40|gµ|=0.30|gµ|=0.20

0.5

1

1.5

2 4 6 8

S SF(ω

)/J

ω/J

FIG. 4. (Color online) Collapse of spectral functions for dif-ferent µ along trajectory ii in the SF phase, labeled by thedetuning gµ = (µ� µc)/J . Inset: original data for SSF(!).

approximate analytic expression with normalized �2 ⇠ 1,

�SF(x) =0.65x3

35 + x2/⌫

1 +

7 sin(0.55x)

1 + 0.02x3

�(6)

We only claim that a plateau is consistent with our imag-inary time data and emerges from the analytic continua-tion procedure which penalizes gradients in the spectralfunction; i.e., other analytic continuation methods mayproduce a di↵erent (oscillating) behavior in the same fre-quency range that is also consistent with the imaginarytime data.We now switch to the MI phase, where we approach the

QCP along trajectory iii in Fig 1. The scaling hypothesisfor the spectral function has a similar structure to the onein Eq. (1),

SMI(!) / �3�2/⌫�MI(!

�) . (7)

The low-energy behavior of �MI starts with the thresh-old singularity at the particle-hole gap value, �MI(x) ⇡

4

of the susceptibility was predicted [12, 13, 17] to be

�00� ⇠ (!/�)3 , ! ⌧ � ⌧ 1. (9)

The !3 rise is due to the decay of a Higgs mode into apair of Goldstone modes. On the other hand, Fig. 4 doesnot display a clear !3 low frequency tail. An alterna-tive method to look for this tail exists, without the needto analytically continue the numerical data to real time.Equation (9) transforms into the large imaginary timeasymptotics �s (⌧) ⇠ 1/⌧4.

For N = 3 we confirm the asymptotic behavior of 1/⌧4

within the numerical limitations. Interestingly, forN = 2we do not find a conclusive asymptotic fall-o↵ as 1/⌧4,Instead, the data fits better to an exponential decay, asin the disordered phase (see Eq. (7)), although the powerlaw might have a small amplitude below our statisticalerrors. In both cases, we can safely conclude that thespectral weight of the Higgs peak dominates over the lowfrequency !3 tail, enhancing its visibility. The large ⌧analysis is discussed elsewhere [23].

Discussion and Summary– Our results are directly ap-plicable to all experimental probes that couple to a func-tion of the order parameter magnitude. For example, thelattice potential amplitude in the trapped bosons sys-tem [16, 18], or pump-probe spectroscopy in Charge Den-sity Wave systems [6–8]. Such a probe can be expandednear criticality in terms of the order parameter fields andtheir derivatives,

⇥(x, ⌧) = ↵|~�|2 + �|@µ~�|2 + �(|~�|2)2 + . . . (10)

So long as ↵ 6= 0, the first term is more relevant than therest. Hence, the scalar susceptibility defined in Eq. (2)dominates the experimental response at low frequenciesand wave vectors.

In summary, we have calculated the scalar susceptibil-ity for relativistic O(2) and O(3) models in 2+1 dimen-

FIG. 4: Rescaled spectral function vs. !/� for N = 2, 3,at µ = 0.5. At low values of !/�, these curves collapse tothe universal scaling function �00

�(!/�), in accordance withEq. 5.

sions near criticality. We have demonstrated that theHiggs mode appears as a universal spectral feature sur-viving all the way to the quantum critical point. Sincethis is a strongly coupling fixed point, the existence ofa well defined mode that is not protected by symmetryis an interesting, not obvious, result. We presented newuniversal quantities to be compared with experimentalresults.

Acknowledgements.– We are very grateful to NikolayProkof’ev and Lode Pollet for helpful comments and wealso thank Dan Arovas, Netanel Lindner, Subir Sachdevand Manuel Endres for helpful discussions. AA and DPacknowledge support from the Israel Science Founda-tion, the European Union under grant IRG-276923, theU.S.-Israel Binational Science Foundation, and thank theAspen Center for Physics, supported by the NSF-PHY-1066293, for its hospitality. SG acknowledges the Clorefoundation for a fellowship.

[1] C. Varma, Journal of Low Temperature Physics 126, 901(2002).

[2] S. D. Huber, E. Altman, H. P. Buechler, and G. Blatter,Physical Review B 75, 085106 (2007).

[3] R. Sooryakumar and M. Klein, Physical Review Letters45, 660 (1980).

[4] P. Littlewood and C. Varma, Physical Review B 26, 4883(1982).

[5] C. Ruegg et al., Physical Review Letters 100, 205701(2008).

[6] Y. Ren, Z. Xu, and G. Lupke, The Journal of ChemicalPhysics 120, 4755 (2004).

[7] J. P. Pouget, B. Hennion, C. Escribe-Filippini, and M.Sato, Phys. Rev. B 43, 8421 (1991).

[8] R. Yusupov et al., Nature Physics (2010).[9] I. A✏eck and G. F. Wellman, Physical Review B 46,

8934 (1992).[10] K. G. Wilson and M. E. Fisher, Phys. Rev. Lett. 28, 240

(1972).[11] N. Lindner and A. Auerbach, Physical Review B 81,

054512 (2010).[12] D. Podolsky, A. Auerbach, and D. P. Arovas, Physical

Review B 84, 174522 (2011).[13] S. Sachdev, Physical Review B 59, 14054 (1999).[14] W. Zwerger, Physical Review Letters 92, 027203 (2004).[15] N. Dupuis, Phys. Rev. A 80, 043627 (2009).[16] M. Endres et al., Nature 487, 454 (2012).[17] D. Podolsky and S. Sachdev, Phys. Rev. B 86, 054508

(2012).[18] L. Pollet and N. Prokof’ev, Physical Review Letters 109,

10401 (2012).[19] S. Sachdev, Quantum Phase Transitions, second edition

ed. (Cambridge University Press, New York, 2011).[20] M. P. A. Fisher, P. B. Weichman, G. Grinstein, and D. S.

Fisher, Physical Review B 40, 546 (1989).[21] F. D. M. Haldane, Physical Review Letters 50, 1153

(1983).[22] S. Chakravarty, B. I. Halperin, and D. R. Nelson, Physi-

cal Review B 39, 2344 (1989).

4

of the susceptibility was predicted [12, 13, 17] to be

�00� ⇠ (!/�)3 , ! ⌧ � ⌧ 1. (9)

The !3 rise is due to the decay of a Higgs mode into apair of Goldstone modes. On the other hand, Fig. 4 doesnot display a clear !3 low frequency tail. An alterna-tive method to look for this tail exists, without the needto analytically continue the numerical data to real time.Equation (9) transforms into the large imaginary timeasymptotics �s (⌧) ⇠ 1/⌧4.

For N = 3 we confirm the asymptotic behavior of 1/⌧4

within the numerical limitations. Interestingly, forN = 2we do not find a conclusive asymptotic fall-o↵ as 1/⌧4,Instead, the data fits better to an exponential decay, asin the disordered phase (see Eq. (7)), although the powerlaw might have a small amplitude below our statisticalerrors. In both cases, we can safely conclude that thespectral weight of the Higgs peak dominates over the lowfrequency !3 tail, enhancing its visibility. The large ⌧analysis is discussed elsewhere [23].

Discussion and Summary– Our results are directly ap-plicable to all experimental probes that couple to a func-tion of the order parameter magnitude. For example, thelattice potential amplitude in the trapped bosons sys-tem [16, 18], or pump-probe spectroscopy in Charge Den-sity Wave systems [6–8]. Such a probe can be expandednear criticality in terms of the order parameter fields andtheir derivatives,

⇥(x, ⌧) = ↵|~�|2 + �|@µ~�|2 + �(|~�|2)2 + . . . (10)

So long as ↵ 6= 0, the first term is more relevant than therest. Hence, the scalar susceptibility defined in Eq. (2)dominates the experimental response at low frequenciesand wave vectors.

In summary, we have calculated the scalar susceptibil-ity for relativistic O(2) and O(3) models in 2+1 dimen-

FIG. 4: Rescaled spectral function vs. !/� for N = 2, 3,at µ = 0.5. At low values of !/�, these curves collapse tothe universal scaling function �00

�(!/�), in accordance withEq. 5.

sions near criticality. We have demonstrated that theHiggs mode appears as a universal spectral feature sur-viving all the way to the quantum critical point. Sincethis is a strongly coupling fixed point, the existence ofa well defined mode that is not protected by symmetryis an interesting, not obvious, result. We presented newuniversal quantities to be compared with experimentalresults.

Acknowledgements.– We are very grateful to NikolayProkof’ev and Lode Pollet for helpful comments and wealso thank Dan Arovas, Netanel Lindner, Subir Sachdevand Manuel Endres for helpful discussions. AA and DPacknowledge support from the Israel Science Founda-tion, the European Union under grant IRG-276923, theU.S.-Israel Binational Science Foundation, and thank theAspen Center for Physics, supported by the NSF-PHY-1066293, for its hospitality. SG acknowledges the Clorefoundation for a fellowship.

[1] C. Varma, Journal of Low Temperature Physics 126, 901(2002).

[2] S. D. Huber, E. Altman, H. P. Buechler, and G. Blatter,Physical Review B 75, 085106 (2007).

[3] R. Sooryakumar and M. Klein, Physical Review Letters45, 660 (1980).

[4] P. Littlewood and C. Varma, Physical Review B 26, 4883(1982).

[5] C. Ruegg et al., Physical Review Letters 100, 205701(2008).

[6] Y. Ren, Z. Xu, and G. Lupke, The Journal of ChemicalPhysics 120, 4755 (2004).

[7] J. P. Pouget, B. Hennion, C. Escribe-Filippini, and M.Sato, Phys. Rev. B 43, 8421 (1991).

[8] R. Yusupov et al., Nature Physics (2010).[9] I. A✏eck and G. F. Wellman, Physical Review B 46,

8934 (1992).[10] K. G. Wilson and M. E. Fisher, Phys. Rev. Lett. 28, 240

(1972).[11] N. Lindner and A. Auerbach, Physical Review B 81,

054512 (2010).[12] D. Podolsky, A. Auerbach, and D. P. Arovas, Physical

Review B 84, 174522 (2011).[13] S. Sachdev, Physical Review B 59, 14054 (1999).[14] W. Zwerger, Physical Review Letters 92, 027203 (2004).[15] N. Dupuis, Phys. Rev. A 80, 043627 (2009).[16] M. Endres et al., Nature 487, 454 (2012).[17] D. Podolsky and S. Sachdev, Phys. Rev. B 86, 054508

(2012).[18] L. Pollet and N. Prokof’ev, Physical Review Letters 109,

10401 (2012).[19] S. Sachdev, Quantum Phase Transitions, second edition

ed. (Cambridge University Press, New York, 2011).[20] M. P. A. Fisher, P. B. Weichman, G. Grinstein, and D. S.

Fisher, Physical Review B 40, 546 (1989).[21] F. D. M. Haldane, Physical Review Letters 50, 1153

(1983).[22] S. Chakravarty, B. I. Halperin, and D. R. Nelson, Physi-

cal Review B 39, 2344 (1989).

Snir Gazit, Daniel Podolsky,

and Assa Auerbach,

arXiv:1212.3759

Kun Chen, Longxiang Liu,

Youjin Deng, Lode Pollet,

and Nikolay Prokof’ev,

arXiv:1301.3139

Observation of Higgs quasi-normal mode in quantum Monte Carlo

Scaling of spectral response

functions predicted in

D. Podolsky and S. Sachdev,

Phys. Rev. B 86, 054508 (2012).


g

T

gc

0

InsulatorSuperfluid

Quantumcritical

TKT

��c

A conformal field theory

in 2+1 spacetime dimensions:

a CFT3

h i 6= 0 h i = 0

S =

Zd2rdt

⇥|@t |2 � c2|rr |2 � V ( )

⇤

V ( ) = (�� c)| |2 + u�| |2

�2


g

T

gc

0

InsulatorSuperfluid

Quantumcritical

TKT

��c


g

T

gc

0

InsulatorSuperfluid

Quantumcritical

TKT

��c

Boltzmann theory of

particles/holes


g

T

gc

0

InsulatorSuperfluid

Quantumcritical

TKT

��c

“Boltzmann” theory of Nambu-

Goldstone and vortices


g

T

gc

0

InsulatorSuperfluid

Quantumcritical

TKT

��c

CFT3 at T>0


g

T

gc

0

InsulatorSuperfluid

Quantumcritical

TKT

��c

CFT3 at T>0

Boltzmann theory of particles/holes/vortices

does not apply


g

T

gc

0

InsulatorSuperfluid

Quantumcritical

TKT

��c

CFT3 at T>0

Needed: Accurate theory of

quantum critical dynamics


Quantum critical dynamics

S. Sachdev, Quantum Phase Transitions, Cambridge (1999).

Quantum “nearly perfect fluid”

with shortest possible local equilibration time, ⌧eq

⌧eq = C ~kBT

where C is a universal constant.

Response functions are characterized by poles in LHPwith ! ⇠ kBT/~.

These poles (quasi-normal modes) appear naturally inthe holographic theory.

(Analogs of Higgs quasi-normal mode.)


M.P.A. Fisher, G. Grinstein, and S.M. Girvin, Phys. Rev. Lett. 64, 587 (1990) K. Damle and S. Sachdev, Phys. Rev. B 56, 8714 (1997).

� =

Q2

h⇥ [Universal constant O(1) ]

(Q is the “charge” of one boson)

Quantum critical dynamics

Transport co-oe�cients not determined

by collision rate of quasiparticles, but by

fundamental constants of nature

Conductivity







Outline







Outline


J. McGreevy, arXiv0909.0518

xir

Renormalization group: ) Follow coupling

constants of quantum many body theory as a func-

tion of length scale r


Key idea: ) Implement r as an extra dimen-

sion, and map to a local theory in d + 2 spacetime

dimensions.

r xi

Renormalization group: ) Follow coupling

constants of quantum many body theory as a func-

tion of length scale r

J. McGreevy, arXiv0909.0518


r

Holography

xi

For a relativistic CFT in d spatial dimensions, the

metric in the holographic space is fixed by de-

manding the scale transformation (i = 1 . . . d)

xi ! ⇣xi , t ! ⇣t , ds ! ds


r

Holography

xi

This gives the unique metric

ds

2=

1

r

2

��dt

2+ dr

2+ dx

2i

�

This is the metric of anti-de Sitter space AdSd+2.


zr

AdS4R

2,1

Minkowski

CFT3

AdS/CFT correspondence

xi


zr

AdS4R

2,1

Minkowski

CFT3


SE =

Zd

4x

p�g

1

22

✓R+

6

L

2

◆�

xi

This emergent spacetime is a solution of Einstein gravity with a negative cosmological constant







Outline







Outline


Objects so massive that light is gravitationally bound to them.

Black Holes


Horizon radius R =2GM

c2

Objects so massive that light is gravitationally bound to them.

Black Holes

In Einstein’s theory, the region inside the black hole horizon is disconnected from

the rest of the universe.


Around 1974, Bekenstein and Hawking showed that the application of the

quantum theory across a black hole horizon led to many astonishing

conclusions

Black Holes + Quantum theory


_

Quantum Entanglement across a black hole horizon


_



_


Black hole horizon


_

Black hole horizon



Black hole horizon


There is a non-local quantum entanglement between the inside

and outside of a black hole


Black hole horizon








This entanglement leads to ablack hole temperature

(the Hawking temperature)and a black hole entropy (the Bekenstein entropy)


zrThis emergent spacetime is a solution of Einstein gravity

with a negative cosmological constant

AdS4R

2,1

Minkowski

CFT3


SE =

Zd

4x

p�g

1

22

✓R+

6

L

2

◆�

xi


There is a family of

solutions of Einstein

gravity which describe non-zero

temperatures

r

Gauge-gravity duality at non-zero temperatures

SE =

Zd

4x

p�g

1

22

✓R+

6

L

2

◆�



A 2+1 dimensional

system at T > 0with

couplings at its quantum critical point

r

SE =

Zd

4x

p�g

1

22

✓R+

6

L

2

◆�


A “horizon”, similar to the surface of a black hole !


r

SE =

Zd

4x

p�g

1

22

✓R+

6

L

2

◆�

A 2+1 dimensional

system at T > 0with



The temperature and entropy of the

horizon equal those of the quantum

critical point


r A 2+1 dimensional

system at T > 0with



Quasi-normal modes of quantum criticality = waves

falling into black hole




critical point

r A 2+1 dimensional

system at T > 0with



Characteristic damping timeof quasi-normal modes:

(kB/~)⇥ Hawking temperature




critical point

r A 2+1 dimensional

system at T > 0with



R. C. Myers, S. Sachdev, and A. Singh, Physical Review D 83, 066017 (2011)

h

Q2�

0.0 0.5 1.0 1.5 2.00.0

0.5

1.0

1.5

⇥

4�T

� = 0

� =1

12

� = � 1

12

AdS4 theory of quantum criticality

�(!)

�(1)


h

Q2�

0.0 0.5 1.0 1.5 2.00.0

0.5

1.0

1.5

⇥

4�T

� = 0

� =1

12

� = � 1

12

R. C. Myers, S. Sachdev, and A. Singh, Physical Review D 83, 066017 (2011)

�(!)

�(1)


• Stability constraints on the e↵ective theory (|�| < 1/12)allow only a limited !-dependence in the conductiv-

ity. This contrasts with the Boltzmann theory in which

�(!)/�1 becomes very large in the regime of its validity.



Universal Scaling of the Conductivity at the Superfluid-Insulator Phase Transition

Jurij Smakov and Erik SørensenDepartment of Physics and Astronomy, McMaster University, Hamilton, Ontario L8S 4M1, Canada

(Received 30 May 2005; published 27 October 2005)

The scaling of the conductivity at the superfluid-insulator quantum phase transition in two dimensionsis studied by numerical simulations of the Bose-Hubbard model. In contrast to previous studies, we focuson properties of this model in the experimentally relevant thermodynamic limit at finite temperature T. Wefind clear evidence for deviations from !k scaling of the conductivity towards !k=T scaling at lowMatsubara frequencies !k. By careful analytic continuation using Pade approximants we show that thisbehavior carries over to the real frequency axis where the conductivity scales with !=T at smallfrequencies and low temperatures. We estimate the universal dc conductivity to be !! " 0:45#5$Q2=h,distinct from previous estimates in the T " 0, !=T % 1 limit.

DOI: 10.1103/PhysRevLett.95.180603 PACS numbers: 05.60.Gg, 02.70.Ss, 05.70.Jk

The nontrivial properties of materials in the vicinity ofquantum phase transitions [1] (QPTs) are an object ofintense theoretical [1–3] and experimental studies. Theeffect of quantum fluctuations driving the QPTs is espe-cially pronounced in low-dimensional systems, such ashigh-temperature superconductors and two-dimensional(2D) electron gases, exhibiting the quantum Hall effect.Particularly valuable are theoretical predictions of thebehavior of the dynamical response functions, such as theoptical conductivity and the dynamic structure factor, sincethey allow for direct comparison of the theoretical resultswith experimental data. It was pointed out by Damle andSachdev [2] that at the quantum-critical coupling thescaled dynamic conductivity T#2&d$=z!#!; T$ at low fre-quencies and temperatures is a function of the singlevariable @!=kBT:

!#!=T; T ! 0$ " #kBT=@c$#d&2$=z!Q!#@!=kBT$: (1)

Here !Q " Q2=h is the conductivity ‘‘quantum’’ (Q " 2efor the models we consider), !#x ' @!=kBT$ is a universaldimensionless scaling function, c a nonuniversal constant,and z the dynamical critical exponent. For d " 2 the ex-ponent vanishes, leading to a purely universal conductivity[4], depending only on frequency !, measured against acharacteristic time @" set by finite temperature T as@!=kBT. Once @!=kBT % 1, for fixed T, the system nolonger ‘‘feels’’ the effect of finite temperature and it isnatural to expect that at such high ! a crossover to atemperature-independent regime will take place [3], sothat !#!; T$ ( !#!$ with !#!$ decaying at high frequen-cies as 1=!2 [2]. Deviations from scaling of ! with !therefore signal that temperature effects have become im-portant. Note that the predicted universal behavior occursfor fixed !=T as T ! 0. The physical mechanisms oftransport are predicted [2] to be quite distinct in the differ-ent regimes determined by the value of the scaling variablex: hydrodynamic, collision dominated for x) 1, and col-lisionless, phase coherent for x% 1 with ! " !#1$

largely independent of x in d " 2 and ! independent ofT [2,5].

Intriguingly, early numerical studies [6–9] of QPTs inmodel systems have failed to observe scaling with@!=kBT. The results of the experiments seeking to verifythe scaling hypothesis are ambiguous as well. Some ofthem, performed at the 2D quantum Hall transitions [10]and 3D metal-insulator transitions [11], appear to supportit. Others either note the absence of scaling [12] or suggesta different scaling form [13]. While the discrepancy be-tween theory and experiment may be attributed to theunsuitable choice of the measurement regime [2], typicallyleading to @!=kBT % 1, there is no good reason why thepredicted scaling would not be observable in numericalsimulations if careful extrapolations first to L! 1 andthen T ! 0 for fixed !=T are performed.

Our primary goal is to resolve this controversy by per-forming precise numerical simulations of the frequency-dependent conductivity at finite temperatures in the vicin-ity of the 2D QPT, exploiting recent algorithmic advancesto access larger system sizes and a wider temperaturerange. After the extrapolation of the results to the thermo-dynamic and T " 0 limits and careful analytic continu-ation, we are able to demonstrate how the predicteduniversal behavior of the conductivity may indeed berevealed.

We consider the 2D Bose-Hubbard (BH) model with theHamiltonian H BH "H 0 *H 1, where the first termdescribes the noninteracting soft core bosons hopping viathe nearest-neighbor links of a 2D square lattice, and thesecond one includes the Hubbard-like on-site interactions:

H 0 " &tX

r;##byr br*# * byr*#br$ &$

Xrnr; (2)

H 1 "U2

Xrnr#nr & 1$: (3)

Here # " x; y, nr " byr br is the particle number operatoron site r, and byr ; br are the boson creation and annihilation

PRL 95, 180603 (2005) P H Y S I C A L R E V I E W L E T T E R S week ending28 OCTOBER 2005

0031-9007=05=95(18)=180603(4)$23.00 180603-1 ! 2005 The American Physical Society

The SSE data also display a high narrow peak at very lowfrequencies, whose position and shape are unstable withrespect to the choice of the initial image and MaxEntparameters. This is clearly an artifact of the method; how-ever, its presence is indicative of the tendency to accumu-late the weight at very low frequencies, in qualitativeagreement with H V result. The subsequent falloff in theconductivity at high frequencies is physically consistent,but its functional form depends on the Pade approximantused. For !=!c * 1=2, we expect the analytic continu-ation of the data for H V to become sensitive to the order ofthe approximant used and we therefore indicate the resultsin this regime by dotted lines only. We note that results atall temperatures yield the same dc conductivity !? !0:45"5#!Q, theoretically predicted [4] to be universal.Because of the very different scaling procedure this resultdiffers from previous numerical result !? ! 0:285"20#!Qon the same model [6] in the T ! 0 limit. It also differssignificantly from a theoretical estimate [2], !$ !1:037!Q, valid to leading order in " ! 3% d. Remark-ably, our result for the dc conductivity is very close to theone obtained in Ref. [8] for the phase transition in thedisordered Bose-Hubbard model. Experimental results in-dicate a value close to unity [26]; however, it was previ-ously observed [7] that long-range Coulomb interactions,impossible to include in the present study, tend to increase! considerably. The same data are shown versus !=T inFig. 4(b). Notably, when using this parametrization !ccancels out and all our data follow the same functionalform. The scaling with !=T at low frequencies is nowimmediately apparent, with a surprisingly wide low !=Tpeak. The width of this peak is consistent with the data inFig. 3(d). Furthermore, on the same !=T scale the con-tinuous time SSE data for H BH and the results for H Vqualitatively agree.

In summary, we have demonstrated that by doing a verycareful data analysis it is possible to observe the theoreti-cally predicted universal !=T scaling at the 2D superfluid-

insulator transition. We have also estimated the universaldc conductivity at this transition and found that it differssignificantly from existing numerical and theoreticalestimates.

We thank S. Sachdev, S. Girvin, and A. P. Young forvaluable comments and critical remarks. Financial sup-port from SHARCNET, NSERC, and CFI is gratefullyacknowledged. All calculations were done at theSHARCNET facility at McMaster University.

[1] J. Hertz, Phys. Rev. B 14, 1165 (1976); A. J. Millis, Phys.Rev. B 48, 7183 (1993); S. Sachdev, Quantum PhaseTransitions (Cambridge University Press, Cambridge,England, 2001); M. Vojta, Rep. Prog. Phys. 66, 2069(2003).

[2] K. Damle and S. Sachdev, Phys. Rev. B 56, 8714 (1997).[3] S. L. Sondhi et al., Rev. Mod. Phys. 69, 315 (1997).[4] M. P. A. Fisher, G. Grinstein, and S. M. Girvin, Phys. Rev.

Lett. 64, 587 (1990).[5] S. Sachdev, Phys. Rev. B 55, 142 (1997).[6] M.-C. Cha et al., Phys. Rev. B 44, 6883 (1991).[7] E. S. Sørensen et al., Phys. Rev. Lett. 69, 828 (1992);

M. Wallin et al., Phys. Rev. B 49, 12 115 (1994).[8] G. G. Batrouni et al., Phys. Rev. B 48, 9628 (1993).[9] R. T. Scalettar, N. Trivedi, and C. Huscroft, Phys. Rev. B

59, 4364 (1999).[10] L. W. Engel et al., Phys. Rev. Lett. 71, 2638 (1993).[11] H.-K. Lee et al., Phys. Rev. Lett. 80, 4261 (1998); Science

287, 633 (2000).[12] N. Q. Balaban, U. Meirav, and I. Bar-Joseph, Phys. Rev.

Lett. 81, 4967 (1998).[13] Y. S. Lee et al., Phys. Rev. B 66, 041104(R) (2002).[14] M. P. A. Fisher et al., Phys. Rev. B 40, 546 (1989).[15] A. W. Sandvik, J. Phys. A 25, 3667 (1992).[16] O. F. Syljuasen and A. W. Sandvik, Phys. Rev. E 66,

046701 (2002); O. F. Syljuasen, ibid. 67, 046701 (2003).[17] M. P. A. Fisher and D. H. Lee, Phys. Rev. B 39, 2756

(1989).[18] F. Alet and E. S. Sørensen, Phys. Rev. E 67, 015701(R)

(2003); 68, 026702 (2003).[19] W. Krauth and N. Trivedi, Europhys. Lett. 14, 627 (1991).[20] J. K. Freericks and H. Monien, Phys. Rev. B 53, 2691

(1996); N. Elstner and H. Monien, ibid. 59, 12 184 (1999).[21] G. D. Mahan, Many-Particle Physics (Plenum, New York,

1990), 2nd ed..[22] D. J. Scalapino, S. R. White, and S. Zhang, Phys. Rev.

Lett. 68, 2830 (1992); Phys. Rev. B 47, 7995 (1993).[23] R. Fazio and D. Zappala, Phys. Rev. B 53, R8883 (1996).[24] M. Jarrell and J. E. Gubernatis, Phys. Rep. 269, 133

(1996).[25] W. H. Press et al., Numerical Recipes in C (Cambridge

University Press, Cambridge, England, 1992).[26] See Y. Liu et al., Physica (Amsterdam) 83D, 163 (1995),

and references therein.[27] Finite Size Scaling and Numerical Simulations of Statis-

tical Systems, edited by V. Privman (World Scientific,Singapore, 1990).

0

0.2

0.4

0.6

0.8

!(’"

)

0 0.5 1 1.5"/"c

L#=32L#=40L#=48L#=64SSE

0

0.2

0.4

0.6

0.8

!(’"

)T/

0 25 50 75"/T

L#=32L#=40L#=48L#=64SSE

(a) (b)

FIG. 4. The real part of the conductivity !0 at the criticalcoupling in units of !Q. The data marked L#, plotted vs!=!c, were obtained using H V , combined with the analyticcontinuation of $"!=!c# as explained in the text. Results for!=!c * 1=2 are denoted by dotted lines. The data marked SSE,plotted vs !=10, were obtained by direct SSE simulations ofH BH with L ! 20, % ! 10 and subsequent maximum entropyanalysis (a). Results as a function of !=T (b).


180603-4

!!i!k"=!2"!Q" #h$kxi$!xx!i!k"

!k% #!i!k"

!k: (7)

Here h$kxi is the kinetic energy per link and #!i!k" is thefrequency-dependent stiffness. To measure !xx!i!k" wenote that !xx!$" may be expressed in terms of the correla-tion functions !%&

xx !r; $" # hK%x !r; $"K&

x !0; 0"i of operatorsK&x !r; $" # tbyr&x!$"br!$" and K$x !r; $" # tbyr !$"br&x!$",which may be estimated efficiently in SSE [15].Remarkably, it is possible to analytically perform theFourier transform with respect to $ yielding

!%&xx !r; !k" #

!1

'

Xn$2

m#0

"amn!!k"N!&;%;m""; (8)

where N!&;%;m" is the number of times the operatorsK%!r" and K&!0" appear in the SSE operator sequenceseparated by m operator positions, and n is the expansionorder. The coefficients "amn!!k" are given by the degeneratehypergeometric (Kummer) function: "amn!!k"#1 F1!m&1; n;$i'!k". This expression and (8) allow us to directlyevaluate !xx!r; !k" as a function of Matsubara frequencies,eliminating any errors associated with the discretization ofthe imaginary time interval. Analogously, in the link-current representation #!i!k" can be calculated [7], andthe conductivity can be obtained from Eq. (7).

In Fig. 3 we show results for !!i!k" versus !k obtainedusing the geometrical worm algorithm on H V at Kc[Fig. 3(a)] and by SSE simulations at tc;(c of H BH[Fig. 3(c)]. In both cases the results have been extrapolatedto the thermodynamic limit L! 1 at fixed '. As evidentfrom Fig. 3(a), the results deviate from scaling with !k atsmall !k and more significantly so at higher temperatures(small L$). These deviations are also visible in the con-tinuous time SSE data in Fig. 3(c), demonstrating that theycannot be attributed to time discretization errors. Similardeviations have been noted previously [6,7] but were notanalyzed at fixed '. Since the deviations persist in the L!1 limit at fixed ', they may only be interpreted as finite Teffects. Expecting a crossover to!k=T scaling at small!k,we plot our results versus !k=T in Fig. 3(d). For L$ ' 32,!!!1=T" is already independent of T (L$). In fact, asshown in Fig. 3(d), for !1...5, !!!k=T; T" can unambigu-ously be extrapolated to a finite !!!k=T; T ! 0" ( #!x"limit. This fact is a clear indication that !k=T scalingindeed occurs as T ! 0. Tentatively, for increasing!k=T, !!!k=T; T ! 0" appears to reach a constant valueof roughly 0:33!Q (#!1" in excellent agreement withtheoretical estimates [2,23]. We note that deviations from!k scaling appear to be largely absent in simulations ofH BH with disorder [7,8]. However, at this QCP the dy-namical critical exponent is different (z # 2). As is evidentfrom the size of the error bars in Fig. 3, simulations of H Vare much more efficient than the SSE simulations directlyon H BH. In the following analytic continuation we there-fore use the SSE data mostly as a consistency check.

Our results on the imaginary frequency axis are limitedby the lowest Matsubara frequency, !1 # 2"[email protected], the information about the behavior of !0!!" %Re!!!" at low! is embedded in values of the CCCF at allMatsubara frequencies, allowing us to determine it. Inorder to study the !=T scaling predicted for the hydro-dynamic collision-dominated regime [2] @!=kBT ) 1, wehave attempted analytic continuations of #!i!k" to obtain!0!!" at real frequencies. SSE results for H BH wereanalytically continued using the Bryan maximum entropy(ME) method [24] with flat initial image. For the resultsobtained for the link-current model H V we use a methodthat should be most sensitive to low frequencies !=!c < 1or #!x) 1". We fit the extrapolated low frequency part(first 10–15 Matsubara frequencies) of #!i!k" to a 6th-order polynomial. The resulting 6 coefficients are thenused to obtain a !3; 3" Pade approximant using standardtechniques [25]. This approximant is then used for theanalytic continuation of # by i!k ! !& i). Resultingreal frequency conductivities !0!!" are displayed inFig. 4(a) versus !=!c. The typical SSE data are plottedversus !=10 and are only shown for L # 20, ' # 10.

The results for H V show a broadened peak as !! 0,due to inelastic scattering, followed by a second peaknicely consistent in height and width with the SSE data.

0.2

0.3

0.4

!(i"

k)

0 0.1 0.2 0.3 0.4 0.5"k/(2#"c)

L$=16

L$=24

L$=32

L$=40

L$=48

L$=64

0 25 50 75 100"k/T

L$=16

L$=24

L$=32

L$=40

L$=48

T=0 Ext.

0.1

0.2

0.3

0.4!(

i"k)

0 10 20 30"k

%=2%=4%=6%=8%=10

0.006

0.007

0.008

0.009

0.010

&("

1)

100 200 300L

L$=64

Fit

(a)(b)

(c) (d)

FIG. 3 (color online). The conductivity !!!k" in units of !Qvs Matsubara frequency !k=!c as obtained from H V (a). Allresults have been extrapolated to the thermodynamic limit L!1 using the scaling form f!L" # a& b exp!$L=*"=

####Lp

[27] bycalculating #!!k" at fixed L$ using 9 lattice sizes from L #L$ . . . 4L$ as shown in (b). !!!k" in units of !Q vs Matsubarafrequency !k as obtained from SSE calculations of H BH, withsome typical error bars shown. All results have been extrapolatedto the thermodynamic limit by calculating !xx!!k" for fixed 'using 5 lattice sizes L # 12; . . . ; 30 (c). Scaling plot of theconductivity data from (a) vs !k=T. ! denotes extrapola-tions to T ! 0 (L$ ! 1) at fixed !k=T using: f!L$" # c&d exp!$L$=*$"=

######L$p

[27] (d).


180603-3



Universal Scaling of the Conductivity at the Superfluid-Insulator Phase Transition

Jurij Smakov and Erik SørensenDepartment of Physics and Astronomy, McMaster University, Hamilton, Ontario L8S 4M1, Canada

(Received 30 May 2005; published 27 October 2005)

The scaling of the conductivity at the superfluid-insulator quantum phase transition in two dimensionsis studied by numerical simulations of the Bose-Hubbard model. In contrast to previous studies, we focuson properties of this model in the experimentally relevant thermodynamic limit at finite temperature T. Wefind clear evidence for deviations from !k scaling of the conductivity towards !k=T scaling at lowMatsubara frequencies !k. By careful analytic continuation using Pade approximants we show that thisbehavior carries over to the real frequency axis where the conductivity scales with !=T at smallfrequencies and low temperatures. We estimate the universal dc conductivity to be !! " 0:45#5$Q2=h,distinct from previous estimates in the T " 0, !=T % 1 limit.

DOI: 10.1103/PhysRevLett.95.180603 PACS numbers: 05.60.Gg, 02.70.Ss, 05.70.Jk

The nontrivial properties of materials in the vicinity ofquantum phase transitions [1] (QPTs) are an object ofintense theoretical [1–3] and experimental studies. Theeffect of quantum fluctuations driving the QPTs is espe-cially pronounced in low-dimensional systems, such ashigh-temperature superconductors and two-dimensional(2D) electron gases, exhibiting the quantum Hall effect.Particularly valuable are theoretical predictions of thebehavior of the dynamical response functions, such as theoptical conductivity and the dynamic structure factor, sincethey allow for direct comparison of the theoretical resultswith experimental data. It was pointed out by Damle andSachdev [2] that at the quantum-critical coupling thescaled dynamic conductivity T#2&d$=z!#!; T$ at low fre-quencies and temperatures is a function of the singlevariable @!=kBT:

!#!=T; T ! 0$ " #kBT=@c$#d&2$=z!Q!#@!=kBT$: (1)

Here !Q " Q2=h is the conductivity ‘‘quantum’’ (Q " 2efor the models we consider), !#x ' @!=kBT$ is a universaldimensionless scaling function, c a nonuniversal constant,and z the dynamical critical exponent. For d " 2 the ex-ponent vanishes, leading to a purely universal conductivity[4], depending only on frequency !, measured against acharacteristic time @" set by finite temperature T as@!=kBT. Once @!=kBT % 1, for fixed T, the system nolonger ‘‘feels’’ the effect of finite temperature and it isnatural to expect that at such high ! a crossover to atemperature-independent regime will take place [3], sothat !#!; T$ ( !#!$ with !#!$ decaying at high frequen-cies as 1=!2 [2]. Deviations from scaling of ! with !therefore signal that temperature effects have become im-portant. Note that the predicted universal behavior occursfor fixed !=T as T ! 0. The physical mechanisms oftransport are predicted [2] to be quite distinct in the differ-ent regimes determined by the value of the scaling variablex: hydrodynamic, collision dominated for x) 1, and col-lisionless, phase coherent for x% 1 with ! " !#1$

largely independent of x in d " 2 and ! independent ofT [2,5].

Intriguingly, early numerical studies [6–9] of QPTs inmodel systems have failed to observe scaling with@!=kBT. The results of the experiments seeking to verifythe scaling hypothesis are ambiguous as well. Some ofthem, performed at the 2D quantum Hall transitions [10]and 3D metal-insulator transitions [11], appear to supportit. Others either note the absence of scaling [12] or suggesta different scaling form [13]. While the discrepancy be-tween theory and experiment may be attributed to theunsuitable choice of the measurement regime [2], typicallyleading to @!=kBT % 1, there is no good reason why thepredicted scaling would not be observable in numericalsimulations if careful extrapolations first to L! 1 andthen T ! 0 for fixed !=T are performed.

Our primary goal is to resolve this controversy by per-forming precise numerical simulations of the frequency-dependent conductivity at finite temperatures in the vicin-ity of the 2D QPT, exploiting recent algorithmic advancesto access larger system sizes and a wider temperaturerange. After the extrapolation of the results to the thermo-dynamic and T " 0 limits and careful analytic continu-ation, we are able to demonstrate how the predicteduniversal behavior of the conductivity may indeed berevealed.

We consider the 2D Bose-Hubbard (BH) model with theHamiltonian H BH "H 0 *H 1, where the first termdescribes the noninteracting soft core bosons hopping viathe nearest-neighbor links of a 2D square lattice, and thesecond one includes the Hubbard-like on-site interactions:

H 0 " &tX

r;##byr br*# * byr*#br$ &$

Xrnr; (2)

H 1 "U2

Xrnr#nr & 1$: (3)

Here # " x; y, nr " byr br is the particle number operatoron site r, and byr ; br are the boson creation and annihilation


0031-9007=05=95(18)=180603(4)$23.00 180603-1 ! 2005 The American Physical Society

QMC yields �(0)/�1 ⇡ 1.36

Holography yields �(0)/�1 = 1 + 4� with |�| 1/12.

Maximum possible holographic value �(0)/�1 = 1.33

W. Witzack-Krempa and S. Sachdev, arXiv:1302.0847Wednesday, April 10, 13

Traditional CMT

Identify quasiparticles and their dispersions

Compute scattering matrix elements of quasiparticles (or of collective modes)

These parameters are input into a quantum Boltzmann equation

Deduce dissipative and dynamic properties at non-zero temperatures


Start with strongly interacting CFT without particle- or wave-like excitations

Compute OPE co-efficients of operators of the CFT

Relate OPE co-efficients to couplings of an effective graviational theory on AdS

Solve Einsten-Maxwell equations. Dynamics of quasi-normal modes of black branes.

Traditional CMT Holography and black-branes










Traditional CMT





Holography and black-branes






Traditional CMT







Traditional CMT
















Outline







Outline


TSDW Tc

T0

2.0

0

α"

1.0 SDW

Superconductivity

BaFe2(As1-xPx)2

AF




StrangeMetal

no quasiparticles,Landau-Boltzmann theory

does not apply


Holographic theory of a strange metal

Electric flux

The density of particles Q creates an electric flux Erwhich modifies the metric of the emergent spacetime.



The density of particles Q creates an electric flux Erwhich modifies the metric of the emergent spacetime.

Hidden Fermi surfaces

of “quarks” ?

Electric flux



Electric flux

The general metric transforms under rescaling as

xi ! ⇣ xi, t ! ⇣

zt, ds ! ⇣

✓/dds.

Recall: conformal matter has ✓ = 0, z = 1, and the metric is

anti-de Sitter


of “quarks” ?



Electric flux


xi ! ⇣ xi, t ! ⇣

zt, ds ! ⇣

✓/dds.


anti-de Sitter

The value ✓ = d�1 reproduces all the essential characteristicsof the entropy and entanglement entropy of a strange metal.

L. Huijse, S. Sachdev, B. Swingle, Physical Review B 85, 035121 (2012)

N. Ogawa, T. Takayanagi, and T. Ugajin, JHEP 1201,

125 (2012).


of “quarks” ?



Electric flux


xi ! ⇣ xi, t ! ⇣

zt, ds ! ⇣

✓/dds.


anti-de Sitter

L. Huijse, S. Sachdev, B. Swingle, Physical Review B 85, 035121 (2012)

N. Ogawa, T. Takayanagi, and T. Ugajin, JHEP 1201,

125 (2012).

The null-energy condition of gravity yields z � 1 + ✓/d. In d = 2, this

leads to z � 3/2. Field theory on strange metal yields z = 3/2 to 3 loops!

M. A. Metlitski and S. Sachdev, Phys. Rev. B 82, 075127 (2010)


of “quarks” ?


New insights and solvable models for diffusion and transport of strongly interacting systems near quantum critical points

The description is far removed from, and complementary to, that of the quantum Boltzmann equation which builds on the quasiparticle/vortex picture.

Good prospects for experimental tests of frequency-dependent, non-linear, and non-equilibrium transport

Conclusions

Conformal quantum matter


Conclusions

More complex examples in metallic states are experimentally

ubiquitous, but pose difficult strong-coupling problems to conventional methods of field

theory


Conclusions

String theory and gravity in emergent dimensions

offer a remarkable new approach to describing states with many-particle quantum entanglement.

Much recent progress offers hope of a holographic description of “strange metals”


r

anti-de Sitter space

Emergent holographic direction

R

d,1

Minkowski

AdSd+2


anti-de Sitter space


Documents

The quantum phases of matter and gauge-gravity dualityqpt.physics.harvard.edu/talks/fortaleza13.pdf · Fortaleza, Brazil, April 10, 2013 Subir Sachdev Scientific American 308, 44