Quantum mechanics without quasiparticles

Quantum mechanics without quasiparticles

HARVARD

New Directions in Theoretical PhysicsHiggs Centre, University of Edinburgh

January 9, 2014

Subir Sachdev

Talk online: sachdev.physics.harvard.eduFriday, January 10, 14

William Witczak-KrempaPerimeter

Erik SorensenMcMaster

Raghu MahajanStanford

Sean HartnollStanford

Matthias PunkInnsbruck

Friday, January 10, 14

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


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


Modern phases of quantum matterNot adiabatically connected

to independent electron states:many-particle

quantum entanglement,





Famous examples:

The fractional quantum Hall effect of electrons in two dimensions (e.g. in graphene) in the presence of a

strong magnetic field. The ground state is described by Laughlin’s wavefunction, and the excitations are

quasiparticles which carry fractional charge.





Famous examples:

Electrons in one dimensional wires form the Luttinger liquid. The quanta of density oscillations (“phonons”) are a quasiparticle basis of the low-energy Hilbert space. Similar comments apply to

magnetic insulators in one dimension.












quantum entanglement,and no quasiparticles


1. The simplest models without quasiparticles

A. Superfluid-insulator transition

of ultracold bosons in an optical lattice

B. Conformal field theories in

2+1 dimensions and

the AdS/CFT correspondence

2. Metals without quasiparticles

“Nematic” order in the high

temperature superconductors

Outline






2+1 dimensions and





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~ 3ffiffiffi2p

{4" #

1zj=jcð Þ$ %1=2

j=jc{1ð Þ1=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 12ffiffiffi2p

{17" #

j=jc$ %1=2

1{j=jcð Þ1=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





{4" #

1zj=jcð Þ$ %1=2

j=jc{1ð Þ1=2





{17" #

j=jc$ %1=2

1{j=jcð Þ1=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






{4" #

1zj=jcð Þ$ %1=2

j=jc{1ð Þ1=2





{17" #

j=jc$ %1=2

1{j=jcð Þ1=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






{4" #

1zj=jcð Þ$ %1=2

j=jc{1ð Þ1=2





{17" #

j=jc$ %1=2

1{j=jcð Þ1=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





{4" #

1zj=jcð Þ$ %1=2

j=jc{1ð Þ1=2





{17" #

j=jc$ %1=2

1{j=jcð Þ1=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






{4" #

1zj=jcð Þ$ %1=2

j=jc{1ð Þ1=2





{17" #

j=jc$ %1=2

1{j=jcð Þ1=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






{4" #

1zj=jcð Þ$ %1=2

j=jc{1ð Þ1=2





{17" #

j=jc$ %1=2

1{j=jcð Þ1=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






{4" #

1zj=jcð Þ$ %1=2

j=jc{1ð Þ1=2





{17" #

j=jc$ %1=2

1{j=jcð Þ1=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





{4" #

1zj=jcð Þ$ %1=2

j=jc{1ð Þ1=2





{17" #

j=jc$ %1=2

1{j=jcð Þ1=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


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

erflu

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

SuperfluidMott 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






{4" #

1zj=jcð Þ$ %1=2

j=jc{1ð Þ1=2





{17" #

j=jc$ %1=2

1{j=jcð Þ1=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).


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

h i 6= 0 h i = 0

S =

Zd2rdt

⇥|@t |2 � c2|rr |2 � V ( )

⇤

V ( ) = (�� c)| |2 + u�| |2

�2

No well-defined normal modes,

or quasiparticle excitations


g

T

gc

0

InsulatorSuperfluid

Quantumcritical

TKT

��c


g

T

gc

0

InsulatorSuperfluid

Quantumcritical

TKT

��c

“Boltzmann” theory of Nambu-

Goldstone and vortices

Boltzmann theory of

particles/holes


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 without

quasiparticles


�/T

�

Electrical transport in a free quasiparticle CFT3 for T > 0

⇠ T �(!)


Quasiparticle view of quantum criticality:Electrical transport for a (weakly) interacting CFT3

K. Damle and S. Sachdev, Phys. Rev. B 56, 8714 (1997).

~!kBT

1

; ⌃ ! a universal function

Re[�(!)]

�(!, T ) =e2

h⌃

✓~!kBT

◆

Universal conductivity ⇠ e2/hUniversal time scale ⇠ ~/kBT




~!kBT

1


Re[�(!)]

�(!, T ) =e2

h⌃

✓~!kBT

◆

O((u⇤)

2),

where u⇤is the

fixed point

interaction

Universal conductivity ⇠ e2/hUniversal time scale ⇠ ~/kBT




~!kBT

1


Re[�(!)]

�(!, T ) =e2

h⌃

✓~!kBT

◆

O((u⇤)

2),

where u⇤is the

fixed point

interaction

O(1/(u⇤)2)Universal conductivity ⇠ e2/hUniversal time scale ⇠ ~/kBT




~!kBT

1


Re[�(!)]

�(!, T ) =e2

h⌃

✓~!kBT

◆

O((u⇤)

2),

where u⇤is the

fixed point

interaction

O(1/(u⇤)2)




~!kBT

1


Re[�(!)]

�(!, T ) =e2

h⌃

✓~!kBT

◆

O((u⇤)

2),

where u⇤is the

fixed point

interaction

O(1/(u⇤)2)

Needed: a method for computing

the conductivity of strongly interacting CFT3s



Quantum Monte Carlo for lattice bosons

W. Witczak-Krempa, E. Sorensen, and S. Sachdev, arXiv:1309.2941

See also K. Chen, L. Liu, Y. Deng, L. Pollet, and N. Prokof’ev, arXiv:1309.5635

SuperfluidInsulator

Quantumcritical

CFTtêU

T

(a)

0 5 10 15 20ωn/(2πT)

0.35

0.36

0.37

0.38

0.39

0.40

σ(iω

n)/σQ

Quantum Rotor ModelVillain Model

(b)

FIG. 1. Probing quantum critical dynamics (a) Phase diagram of the superfluid-insulatorquantum phase transition as a function of t/U (hopping amplitude relative to the onsite repulsion)and temperature T at integer filling of the bosons. The conformal QCP at T = 0 is indicated by a bluedisk. (b) Quantum Monte Carlo data for the frequency-dependent conductivity, �, near the QCPalong the imaginary frequency axis, for both the quantum rotor and Villain models. The data has beenextrapolated to the thermodynamic limit and zero temperature. The error bars are statistical, and donot include systematic errors arising from the assumed forms of the fitting functions, which we estimateto be 5–10%.

0 2 4 6 8 10 12 14 16 18 20ωn/(2πT)

0

0.1

0.2

0.3

0.4

σ(iω

n)/σQ

T -> 0βU=110βU=100βU=80βU=70βU=60βU=50βU=40βU=30βU=20

(a)

0 2 4 6 8 10 12 14 16 18 20ωn/(2πT)

0

0.1

0.2

0.3

0.4

σ(iω

n)/σQ

T->0Lτ=160

Lτ=128Lτ=96

Lτ=80

Lτ=64Lτ=56

Lτ=48

Lτ=40Lτ=32

Lτ=24

Lτ=160 50 100 150L

τ

0.15

0.2

0.25

0.3

0.35

σ

ωn/(2πT)=7

(b)

FIG. 2. Quantum Monte Carlo data (a) Finite-temperature conductivity for a range of �U in theL ! 1 limit for the quantum rotor model at (t/U)

c

. The solid blue squares indicate the final T ! 0extrapolated data. (b) Finite-temperature conductivity in the L ! 1 limit for a range of L

⌧

for theVillain model at the QCP. The solid red circles indicate the final T ! 0 extrapolated data. The insetillustrates the extrapolation to T = 0 for !

n

/(2⇡T ) = 7. The error bars are statistical for both a) and b).

4


0 2 4 6 8 10 12 14

0.25

0.30

0.35

0.40

0.45

0.50

wê2pT

sHwLêsQ

Quantum Monte Carlo for lattice bosons�(!

)/(e

2/h

)








2+1 dimensions and





Outline






2+1 dimensions and





Outline


zr

AdS4R

2,1

Minkowski

CFT3

AdS/CFT correspondence

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


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



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











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

D. Chowdhury, S. Raju, S. Sachdev, A. Singh, and P. Strack, Phys. Rev. B 87, 085138 (2013)

AdS4 theory of quantum criticality

Most general e↵ective holographic theory for lin-

ear charge transport with 4 spatial derivatives:

Sbulk =

1

g

2M

Zd

4x

pg

1

4

FabFab

+ �L

2CabcdF

abF

cd

�

+

Zd

4x

pg

� 1

2

2

✓R+

6

L

2

◆�,

This action is characterized by 3 dimensionless parameters, whichcan be linked to data of the CFT (OPE coe�cients): 2-point cor-relators of the conserved current Jµ and the stress energy tensorTµ⌫ , and a 3-point T , J , J correlator. Constraints from both theCFT and the gravitational theory bound |�| 1/12 = 0.0833..


Good agreement between high precision Monte Carlo for imaginary frequencies,

and holographic theory after rescaling e↵ective T and taking �Q = 1/g2M .



AdS4 theory of quantum criticality

0 2 4 6 8 10 12 14

0.25

0.30

0.35

0.40

0.45

0.50

wê2pT

sHiwLês Q

g = 0.08

g = -0.08


0 2 4 6 8 10 12 14

0.25

0.30

0.35

0.40

0.45

0.50

wê2pT

sHwLêsQ

Predictions of holographic theory,

after analytic continuation to real frequencies



AdS4 theory of quantum criticality�(!

)/(e

2/h

)






2+1 dimensions and





Outline






2+1 dimensions and





Outline


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)Friday, January 10, 14

TSDW Tc

T0

2.0

0

α"

1.0 SDW

Superconductivity

BaFe2(As1-xPx)2

AF



Physical Review B 81, 184519 (2010)

Neel (AF) and “nematic” order


TSDW Tc

T0

2.0

0

α"

1.0 SDW

Superconductivity

BaFe2(As1-xPx)2

AF




Neel (AF) and “nematic” order


TSDW Tc

T0

2.0

0

α"

1.0 SDW

Superconductivity

BaFe2(As1-xPx)2

AF




SuperconductorBose condensate of pairs of electrons


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




StrangeMetal

no quasiparticles,Landau-Boltzmann theory

does not apply


LETTERdoi:10.1038/nature11178

Electronic nematicity above the structural andsuperconducting transition in BaFe2(As12xPx)2S. Kasahara1,2, H. J. Shi1, K. Hashimoto1{, S. Tonegawa1, Y. Mizukami1, T. Shibauchi1, K. Sugimoto3,4, T. Fukuda5,6,7, T. Terashima2,Andriy H. Nevidomskyy8 & Y. Matsuda1

Electronic nematicity, a unidirectional self-organized state thatbreaks the rotational symmetry of the underlying lattice1,2, hasbeen observed in the iron pnictide3–7 and copper oxide8–11 high-temperature superconductors. Whether nematicity plays anequally important role in these two systems is highly controversial.In iron pnictides, the nematicity has usually been associated withthe tetragonal-to-orthorhombic structural transition at temper-ature Ts. Although recent experiments3–7 have provided hints ofnematicity, they were performed either in the low-temperatureorthorhombic phase3,5 or in the tetragonal phase under uniaxialstrain4,6,7, both of which break the 906 rotational C4 symmetry.Therefore, the question remains open whether the nematicity canexist above Ts without an external driving force. Here we reportmagnetic torque measurements of the isovalent-doping systemBaFe2(As12xPx)2, showing that the nematicity develops well aboveTs and, moreover, persists to the non-magnetic superconductingregime, resulting in a phase diagram similar to the pseudogapphase diagram of the copper oxides8,12. By combining these resultswith synchrotron X-ray measurements, we identify two distincttemperatures—one at T*, signifying a true nematic transition,and the other at Ts (,T*), which we show not to be a true phasetransition, but rather what we refer to as a ‘meta-nematic trans-ition’, in analogy to the well-known meta-magnetic transition inthe theory of magnetism.

Magnetic torque measurements provide a stringent test of nematicityfor systems with tetragonal symmetry13. The torque t 5 m0VM 3 H is athermodynamic quantity, a differential of the free energy with respect toangular displacement. Here m0 is the permeability of vacuum, V is thesample volume, and M is the magnetization induced in the magneticfield H. When H is rotated within the tetragonal a–b plane (Fig. 1a, b), tis a periodic function of 2w, where w is the azimuthal angle measuredfrom the a axis:

t2w~12

m0H2V xaa{xbbð Þ sin 2w{2xab cos 2w½ $ ð1Þ

where the susceptibility tensor xij is defined by Mi 5SjxijHj. In a systemmaintaining tetragonal symmetry, t2w should be zero, because xaa 5 xbband xab 5 0. Finite values of t2w appear if a new electronic or magneticstate emerges that breaks the C4 tetragonal symmetry. In such a case,rotational symmetry breaking is revealed by xaa ? xbb and/or xab ? 0,depending on the direction of the nematicity.

BaFe2(As1–xPx)2 is a prototypical family of iron pnictides14–18, whosephase diagram is displayed in Fig. 1c. The temperature evolution of thetorque t(w) for the optimally doped compound (x 5 0.33) is depicted inthe upper panels of Fig. 1d. The two- and four-fold oscillations, t2w andt4w, obtained from the Fourier analysis are shown respectively in themiddle and lower panels of Fig. 1d. The distinct two-fold oscillationsappear at low temperatures, whereas they are absent at high temperatures

1Department of Physics, Kyoto University, Kyoto 606-8502, Japan. 2Research Center for Low Temperature and Materials Sciences, Kyoto University, Kyoto 606-8501, Japan. 3Research and UtilizationDivision, JASRI SPring-8, Sayo, Hyogo 679-5198, Japan. 4Structural Materials Science Laboratory, RIKEN SPring-8, Sayo, Hyogo 679-5148, Japan. 5Quantum Beam Science Directorate, JAEA SPring-8,Sayo, Hyogo 679-5148, Japan. 6Materials Dynamics Laboratory, RIKEN SPring-8, Sayo, Hyogo 679-5148, Japan. 7JST, Transformative Research-Project on Iron Pnictides (TRIP), Chiyoda, Tokyo 102-0075,Japan. 8Department of Physics and Astronomy, Rice University, 6100 Main Street, Houston, Texas 77005, USA. {Present address: Institute for Materials Research, Tohoku University, Sendai 980-8577,Japan.

90 K

0 180

35 K

0 180 360

65 K

0 180

75 K

0 180

b c

a

HSingle crystal

I (deg.) W

(a.u

.)cd

Paramagnetic(tetragonal)

(100)T

Antiferromagnetic(orthorhombic)

(010)T

Fe As/P

T > T* T < T*T *

Electronic nematic

Superconducting

0 0.2 0.4 0.60

100

200

x

T (K

)

W2I

(a.u

.) W

4I (a

.u.)

I (deg.) I (deg.) I (deg.)

V

c

a bM

HI

TorqueW = P0M × H≠ 0

a b

Figure 1 | Torque magnetometry and the doping–temperature phasediagram of BaFe2(As12xPx)2. a, b, Schematic representations of theexperimental configuration for torque measurements under in-plane fieldrotation. In a nematic state, domain formation with different preferreddirections in the a–b plane (‘twinning’) will occur. We used very small singlecrystals with typical size ,70mm 3 70mm3 30mm, in which a significantdifference in volume between the two types of domains enables the observationof uncompensated t2w signals. The equation given in the figure for t assumesunit volume; see text for details. A single-crystalline sample (brown block) ismounted on the piezo-resistive lever which is attached to the base (blue block)and forms an electrical bridge circuit (orange lines) with the neighbouringreference lever. A magnetic field H can be rotated relative to the sample, asillustrated by a blue arrow on a sphere. In this experiment, the field is preciselyapplied in the a–b plane. c, Phase diagram of BaFe2(As1–xPx)2. This system isclean and homogeneous14,16,17, as demonstrated by the quantum oscillationsobserved over a wide x range16. The antiferromagnetic transition at TN (filledcircles)15 coincides or is preceded by the structural transition at Ts (opentriangles)18. The superconducting dome extends over a doping range0.2 , x , 0.7 (open squares), with maximum Tc 5 31 K. Crosses indicate thenematic transition temperature T* determined by the torque and synchrotronX-ray diffraction measurements. The insets illustrate the tetragonal FeAs/Player. xab 5 0 above T* yielding an isotropic torque signal (green-shadedcircle), whereas xab ? 0 below T*, indicating the appearance of the nematicityalong the [110]T (Fe–Fe bond) direction, illustrated with the green-shadedellipse. d, The upper panels depict the temperature evolution of the raw torquet(w) at m0H 5 4 T for BaFe2(As0.67P0.33)2 (Tc 5 30 K). All torque curves arereversible with respect to the field rotation. t(w) can be decomposed as t(w) 5 t2w 1 t4w 1 t6w 1 ???, where t2nw 5 A2nw sin 2n(w 2 w0) has 2n-foldsymmetry with integer n. The middle and lower panels display the two- andfour-fold components obtained from Fourier analysis. The four-foldoscillations t4w (and higher-order terms) arise primarily from the nonlinearsusceptibilities13. a.u., arbitrary units.

3 8 2 | N A T U R E | V O L 4 8 6 | 2 1 J U N E 2 0 1 2


BaFe2(As1�x

Px

)2

S. Kasahara, H.J. Shi, K. Hashimoto, S. Tonegawa, Y. Mizukami,T. Shibauchi, K. Sugimoto, T. Fukuda, T. Terashima, A.H. Nevidomskyy, and

Y. Matsuda, Nature 486, 382 (2012).

LETTERdoi:10.1038/nature11178

Electronic nematicity above the structural andsuperconducting transition in BaFe2(As12xPx)2S. Kasahara1,2, H. J. Shi1, K. Hashimoto1{, S. Tonegawa1, Y. Mizukami1, T. Shibauchi1, K. Sugimoto3,4, T. Fukuda5,6,7, T. Terashima2,Andriy H. Nevidomskyy8 & Y. Matsuda1

Electronic nematicity, a unidirectional self-organized state thatbreaks the rotational symmetry of the underlying lattice1,2, hasbeen observed in the iron pnictide3–7 and copper oxide8–11 high-temperature superconductors. Whether nematicity plays anequally important role in these two systems is highly controversial.In iron pnictides, the nematicity has usually been associated withthe tetragonal-to-orthorhombic structural transition at temper-ature Ts. Although recent experiments3–7 have provided hints ofnematicity, they were performed either in the low-temperatureorthorhombic phase3,5 or in the tetragonal phase under uniaxialstrain4,6,7, both of which break the 906 rotational C4 symmetry.Therefore, the question remains open whether the nematicity canexist above Ts without an external driving force. Here we reportmagnetic torque measurements of the isovalent-doping systemBaFe2(As12xPx)2, showing that the nematicity develops well aboveTs and, moreover, persists to the non-magnetic superconductingregime, resulting in a phase diagram similar to the pseudogapphase diagram of the copper oxides8,12. By combining these resultswith synchrotron X-ray measurements, we identify two distincttemperatures—one at T*, signifying a true nematic transition,and the other at Ts (,T*), which we show not to be a true phasetransition, but rather what we refer to as a ‘meta-nematic trans-ition’, in analogy to the well-known meta-magnetic transition inthe theory of magnetism.

Magnetic torque measurements provide a stringent test of nematicityfor systems with tetragonal symmetry13. The torque t 5 m0VM 3 H is athermodynamic quantity, a differential of the free energy with respect toangular displacement. Here m0 is the permeability of vacuum, V is thesample volume, and M is the magnetization induced in the magneticfield H. When H is rotated within the tetragonal a–b plane (Fig. 1a, b), tis a periodic function of 2w, where w is the azimuthal angle measuredfrom the a axis:

t2w~12

m0H2V xaa{xbbð Þ sin 2w{2xab cos 2w½ $ ð1Þ

where the susceptibility tensor xij is defined by Mi 5SjxijHj. In a systemmaintaining tetragonal symmetry, t2w should be zero, because xaa 5 xbband xab 5 0. Finite values of t2w appear if a new electronic or magneticstate emerges that breaks the C4 tetragonal symmetry. In such a case,rotational symmetry breaking is revealed by xaa ? xbb and/or xab ? 0,depending on the direction of the nematicity.

BaFe2(As1–xPx)2 is a prototypical family of iron pnictides14–18, whosephase diagram is displayed in Fig. 1c. The temperature evolution of thetorque t(w) for the optimally doped compound (x 5 0.33) is depicted inthe upper panels of Fig. 1d. The two- and four-fold oscillations, t2w andt4w, obtained from the Fourier analysis are shown respectively in themiddle and lower panels of Fig. 1d. The distinct two-fold oscillationsappear at low temperatures, whereas they are absent at high temperatures

1Department of Physics, Kyoto University, Kyoto 606-8502, Japan. 2Research Center for Low Temperature and Materials Sciences, Kyoto University, Kyoto 606-8501, Japan. 3Research and UtilizationDivision, JASRI SPring-8, Sayo, Hyogo 679-5198, Japan. 4Structural Materials Science Laboratory, RIKEN SPring-8, Sayo, Hyogo 679-5148, Japan. 5Quantum Beam Science Directorate, JAEA SPring-8,Sayo, Hyogo 679-5148, Japan. 6Materials Dynamics Laboratory, RIKEN SPring-8, Sayo, Hyogo 679-5148, Japan. 7JST, Transformative Research-Project on Iron Pnictides (TRIP), Chiyoda, Tokyo 102-0075,Japan. 8Department of Physics and Astronomy, Rice University, 6100 Main Street, Houston, Texas 77005, USA. {Present address: Institute for Materials Research, Tohoku University, Sendai 980-8577,Japan.

90 K

0 180

35 K

0 180 360

65 K

0 180

75 K

0 180

b c

a

HSingle crystal

I (deg.)

W (a

.u.)c

d

Paramagnetic(tetragonal)

(100)T

Antiferromagnetic(orthorhombic)

(010)T

Fe As/P

T > T* T < T*T *

Electronic nematic

Superconducting

0 0.2 0.4 0.60

100

200

x

T (K

)

W2I

(a.u

.) W

4I (a

.u.)

I (deg.) I (deg.) I (deg.)

V

c

a bM

HI

TorqueW = P0M × H≠ 0

a b

Figure 1 | Torque magnetometry and the doping–temperature phasediagram of BaFe2(As12xPx)2. a, b, Schematic representations of theexperimental configuration for torque measurements under in-plane fieldrotation. In a nematic state, domain formation with different preferreddirections in the a–b plane (‘twinning’) will occur. We used very small singlecrystals with typical size ,70mm 3 70mm3 30mm, in which a significantdifference in volume between the two types of domains enables the observationof uncompensated t2w signals. The equation given in the figure for t assumesunit volume; see text for details. A single-crystalline sample (brown block) ismounted on the piezo-resistive lever which is attached to the base (blue block)and forms an electrical bridge circuit (orange lines) with the neighbouringreference lever. A magnetic field H can be rotated relative to the sample, asillustrated by a blue arrow on a sphere. In this experiment, the field is preciselyapplied in the a–b plane. c, Phase diagram of BaFe2(As1–xPx)2. This system isclean and homogeneous14,16,17, as demonstrated by the quantum oscillationsobserved over a wide x range16. The antiferromagnetic transition at TN (filledcircles)15 coincides or is preceded by the structural transition at Ts (opentriangles)18. The superconducting dome extends over a doping range0.2 , x , 0.7 (open squares), with maximum Tc 5 31 K. Crosses indicate thenematic transition temperature T* determined by the torque and synchrotronX-ray diffraction measurements. The insets illustrate the tetragonal FeAs/Player. xab 5 0 above T* yielding an isotropic torque signal (green-shadedcircle), whereas xab ? 0 below T*, indicating the appearance of the nematicityalong the [110]T (Fe–Fe bond) direction, illustrated with the green-shadedellipse. d, The upper panels depict the temperature evolution of the raw torquet(w) at m0H 5 4 T for BaFe2(As0.67P0.33)2 (Tc 5 30 K). All torque curves arereversible with respect to the field rotation. t(w) can be decomposed as t(w) 5 t2w 1 t4w 1 t6w 1 ???, where t2nw 5 A2nw sin 2n(w 2 w0) has 2n-foldsymmetry with integer n. The middle and lower panels display the two- andfour-fold components obtained from Fourier analysis. The four-foldoscillations t4w (and higher-order terms) arise primarily from the nonlinearsusceptibilities13. a.u., arbitrary units.

3 8 2 | N A T U R E | V O L 4 8 6 | 2 1 J U N E 2 0 1 2



Quantum criticality of Ising-nematic ordering in a metal

x

yOccupied states

Empty states

A metal with a Fermi surfacewith full square lattice symmetry


x

y

Spontaneous elongation along y direction:Ising order parameter � < 0.



Spontaneous elongation along x direction:Ising order parameter � > 0.

x

yQuantum criticality of Ising-nematic ordering in a metal


Ising-nematic order parameter

� ⇠Z

d2k (cos kx

� cos ky

) c†k�

ck�

Measures spontaneous breaking of square latticepoint-group symmetry of underlying Hamiltonian


Spontaneous elongation along x direction:Ising order parameter � > 0.

x

yQuantum criticality of Ising-nematic ordering in a metal


x

y

Spontaneous elongation along y direction:Ising order parameter � < 0.



rrc

Pomeranchuk instability as a function of coupling r

��⇥ = 0⇥�⇤ �= 0

or



T

Phase diagram as a function of T and r

��⇥ = 0

Quantumcritical

⇥�⇤ �= 0rrc

TI-n

Quantum criticality of Ising-nematic ordering


T


��⇥ = 0

Quantumcritical

⇥�⇤ �= 0rrc

TI-n Classicald=2 Isingcriticality



T


��⇥ = 0

Quantumcritical

⇥�⇤ �= 0rrc

TI-n

D=2+1 Ising

criticality ?



T


��⇥ = 0

Quantumcritical

⇥�⇤ �= 0rrc

TI-n

D=2+1 Ising

criticality ?



T


��⇥ = 0

Quantumcritical

⇥�⇤ �= 0rrc

TI-n

Strongly-coupled“non-Fermi liquid”

metal with no quasiparticles



T


��⇥ = 0⇥�⇤ �= 0rrc

TI-n





T


��⇥ = 0⇥�⇤ �= 0rrc

TI-n



StrangeMetal



• Formulate a continuum theory with a conserved momentum.

• Determine the relaxation rate of momentum due to small

perturbations that violate the conservation of momentum,

such as s-wave (V0) and d-wave (h0) impurity scattering.

• We can then relate the momentum relaxation rate to the

resistivity via hydrodynamic/memory matrix methods.

• All steps above can also be implemented in holographic mod-

els, and consistent results are obtained i.e. solution of grav-

itational equations provides results consistent with hydrody-

namics, and with the breakdown of hydrodynamics due to

perturbations that violate momentum conservation.


Theory of transport without quasiparticles (inspired by holography):

S. Hartnoll, R. Mahajan, M. Punk, and S. Sachdev, to appearFriday, January 10, 14











































Strongly-coupled quantum criticality leads to a novel regime of quantum dynamics without quasiparticles.

The simplest examples are conformal field theories in 2+1 dimensions, realized by ultracold atoms in optical lattices.

Holographic theories provide an excellent quantitative description of quantum Monte studies of quantum-critical boson models

Exciting recent progress on the description of transport in metallic states without quasiparticles, via field theory and holography

















Documents

Quantum mechanics without quasiparticles