strong coupling problems in condensed matter and the ads...

Strong coupling problems in condensed matter

and the AdS/CFT correspondence

HARVARD

arXiv:0910.1139

Reviews:

Talk online: sachdev.physics.harvard.edu

arXiv:0901.4103

Monday, October 26, 2009

Frederik Denef, HarvardSean Hartnoll, Harvard

Christopher Herzog, PrincetonPavel Kovtun, VictoriaDam Son, Washington

Yejin Huh, Harvard

Max Metlitski, Harvard

HARVARDMonday, October 26, 2009

1. Quantum-critical transport Collisionless-t0-hydrodynamic crossover of CFT3s

2. Exact solution from AdS/CFT Constraints from duality relations

3. Quantum criticality of Dirac fermions “Vector” 1/N expansion

4. Quantum criticality of Fermi surfaces The genus expansion


The Superfluid-Insulator transition

Boson Hubbard model

M.P. A. Fisher, P.B. Weichmann, G. Grinstein, and D.S. Fisher, Phys. Rev. B 40, 546 (1989).


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

Ultracold 87Rbatoms - bosons

Superfluid-insulator transition


Insulator (the vacuum) at large U


Excitations:


Excitations of the insulator:


g

T

gc

0

InsulatorSuperfluid

Quantumcritical

TKT


g

T

gc

0

InsulatorSuperfluid

Quantumcritical

TKT

CFT3

!!" #= 0 !!" = 0

S =!

d2rd!"|"!#|2 + v2|$!#|2 + (g " gc)|#|2 +

u

2|#|4

#


g

T

gc

0

InsulatorSuperfluid

Quantumcritical

TKT


g

T

gc

0

InsulatorSuperfluid

Quantumcritical

TKT

Classical vortices and wave oscillations of the

condensate Dilute Boltzmann/Landau gas of particle and holes


g

T

gc

0

InsulatorSuperfluid

Quantumcritical

TKT

CFT at T>0


D. B. Haviland, Y. Liu, and A. M. Goldman, Phys. Rev. Lett. 62, 2180 (1989)

Resistivity of Bi films

M. P. A. Fisher, Phys. Rev. Lett. 65, 923 (1990)

Conductivity !

!Superconductor(T ! 0) = "!Insulator(T ! 0) = 0

!Quantum critical point(T ! 0) # 4e2

h


Quantum critical transport

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

Quantum “perfect fluid”with shortest possiblerelaxation time, !R

!R ! !kBT


Quantum critical transport Transport co-oe!cients not determined

by collision rate, but byuniversal constants of nature

Electrical conductivity

! =e2

h! [Universal constant O(1) ]

K. Damle and S. Sachdev, Phys. Rev. B 56, 8714 (1997).Monday, October 26, 2009

Quantum critical transport

P. Kovtun, D. T. Son, and A. Starinets, Phys. Rev. Lett. 94, 11601 (2005)

, 8714 (1997).

Transport co-oe!cients not determinedby collision rate, but by

universal constants of nature

Momentum transport!

s! viscosity

entropy density

=!

kB" [Universal constant O(1) ]


Quantum critical transport Euclidean field theory: Compute current correlations on R2 ! S1

with circumference 1/T

1/T

R2




2!T

4!T

!2!T

Complex ! plane

Direct 1/N or ! = 4! d expansion forcorrelators at "n = 2#nTi, with n integer




2!T

4!T

!2!T

Complex ! plane

Strong coupling problem:Correlators at ! ! 0, along the real axis.


Density correlations in CFTs at T >0

Two-point density correlator, !(k, ")

Kubo formula for conductivity #(") = limk!0

!i"

k2!(k, ")

For all CFT2s, at all !!/kBT

"(k, !) =4e2

hK

vk2

v2k2 ! !2; #(!) =

4e2

h

Kv

!i!

where K is a universal number characterizing the CFT2 (the levelnumber), and v is the velocity of “light”.

This follows from the conformal mapping of the plane to the cylin-der, which relates correlators at T = 0 to those at T > 0.


Conformal mapping of plane to cylinder with circumference 1/T





!i"

k2!(k, ")

For all CFT2s, at all !!/kBT

"(k, !) =4e2

hK

vk2

v2k2 ! !2; #(!) =

4e2

h

Kv

!i!

where K is a universal number characterizing the CFT2 (the levelnumber), and v is the velocity of “light”.This follows from the conformal mapping of the plane to the cylin-der, which relates correlators at T = 0 to those at T > 0.

No hydrodynamics in CFT2s.Monday, October 26, 2009




!i"

k2!(k, ")

For all CFT3s, at !! ! kBT

"(k,!) =4e2

hK

k2

"v2k2 # !2

; #(!) =4e2

hK

where K is a universal number characterizing the CFT3, and v isthe velocity of “light”.





!i"

k2!(k, ")

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

However, for all CFT3s, at !! ! kBT , we have the Einstein re-lation

"(k, !) = 4e2"cDk2

Dk2 " i!; #(!) = 4e2D"c =

4e2

h!1!2

where the compressibility, "c, and the di!usion constant Dobey

" =kBT

(hv)2!1 ; D =

hv2

kBT!2

with !1 and !2 universal numbers characteristic of the CFT3



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

In CFT3s collisions are “phase” randomizing, and lead torelaxation to local thermodynamic equilibrium. So thereis a crossover from collisionless behavior for !! ! kBT , tohydrodynamic behavior for !! " kBT .

"(!) =

!"""#

"""$

4e2

hK , !! ! kBT

4e2

h!1!2 # "Q , !! " kBT

and in general we expect K $= !1!2 (verified for Wilson-Fisher fixed point).


SU(N) SYM3 with N = 8 supersymmetry

• Has a single dimensionful coupling constant, e0, which flowsto a strong-coupling fixed point e0 = e!0 in the infrared.

• The CFT3 describing this fixed point resembles “critical spinliquid” theories.

• This CFT3 is the low energy limit of string theory on anM2 brane. The AdS/CFT correspondence provides a dualdescription using 11-dimensional supergravity on AdS4!S7.

• The CFT3 has a global SO(8) R symmetry, and correlatorsof the SO(8) charge density can be computed exactly in thelarge N limit, even at T > 0.


SU(N) SYM3 with N = 8 supersymmetry

• The SO(8) charge correlators of the CFT3 are given by theusual AdS/CFT prescription applied to the following gaugetheory on AdS4:

S = ! 14g2

4D

!d4x"!ggMAgNBF a

MNF aAB

where a = 1 . . . 28 labels the generators of SO(8). Note thatin large N theory, this looks like 28 copies of an Abelian gaugetheory.


P. Kovtun, C. Herzog, S. Sachdev, and D.T. Son, Phys. Rev. D 75, 085020 (2007)

Im!(k, ")/k2 ImK!

k2 " !2

Collisionless to hydrodynamic crossover of SYM3


P. Kovtun, C. Herzog, S. Sachdev, and D.T. Son, Phys. Rev. D 75, 085020 (2007)

Im!(k, ")/k2

ImD!c

Dk2 ! i"

Collisionless to hydrodynamic crossover of SYM3


Universal constants of SYM3

!(") =

!"""#

"""$

4e2

hK , !" ! kBT

4e2

h!1!2 , !" " kBT

!c =kBT

(hv)2!1

D =hv2

kBT!2

K =!

2N3/2

3

!1 =8!2!

2N3/2

9

!2 =3

8!2

P. Kovtun, C. Herzog, S. Sachdev, and D.T. Son, Phys. Rev. D 75, 085020 (2007) C. Herzog, JHEP 0212, 026 (2002)


Electromagnetic self-duality

• Unexpected result, K = !1!2.

• This is traced to a four -dimensional electromagneticself-duality of the theory on AdS4. In the large Nlimit, the SO(8) currents decouple into 28 U(1) cur-rents with a Maxwell action for the U(1) gauge fieldson AdS4.

• This special property is not expected for generic CFT3s.


K: a universal number analogous to the level number of the Kac-Moody algebra in 1+1 dimensions


C. Dasgupta and B.I. Halperin, Phys. Rev. Lett. 47, 1556 (1981)Monday, October 26, 2009

C. Herzog, P. Kovtun, S. Sachdev, and D.T. Son, hep-th/0701036Monday, October 26, 2009

The self-duality of the 4D abelian gauge fields leads to

C. Herzog, P. Kovtun, S. Sachdev, and D.T. Son, hep-th/0701036Monday, October 26, 2009

Electromagnetic self-duality

• Unexpected result, K = !1!2.

• This is traced to a four -dimensional electromagneticself-duality of the theory on AdS4. In the large Nlimit, the SO(8) currents decouple into 28 U(1) cur-rents with a Maxwell action for the U(1) gauge fieldson AdS4.

• This special property is not expected for generic CFT3s.


d-wave superconductivity in cuprates

!Electron states

occupied

H0 = !!

i<j

tijc†i!ci! "

!

k

!kc†k!ck!

• Begin with free electrons.


H =!

k

"!kc†k!ck! + !kc†k!c

†"k# + c.c.

#


• Add d-wave pairing interaction!k ! cos kx " cos ky which vanishes alongdiagonals


! ++

-

-



! ++

-

-H =

!

k

"!kc†k!ck! + !kc†k!c

†"k# + c.c.

#


• Add d-wave pairing interaction !k which van-ishes along diagonals

• Obtain Bogoliubov quasiparticles with dis-persion

$!2k + !2

kMonday, October 26, 2009


S! =!

d2k

(2!)2T

"

!n

!†1a (!i"n + vF kx#z + v"ky#x) !1a

+!

d2k

(2!)2T

"

!n

!†2a (!i"n + vF ky#z + v"kx#x) !2a.

S0" =

!d2xd#

#12($#%)2 +

c2

2("%)2 +

r

2%2 +

u0

24%4

$;

S!" =!

d2xd##&0%

%!†

1a#x!1a + !†2a#x!2a

& $,

4 two-component Dirac fermions



S! =!

d2k

(2!)2T

"

!n

!†1a (!i"n + vF kx#z + v"ky#x) !1a

+!

d2k

(2!)2T

"

!n

!†2a (!i"n + vF ky#z + v"kx#x) !2a.

S0" =

!d2xd#

#12($#%)2 +

c2

2("%)2 +

r

2%2 +

u0

24%4

$;

S!" =!

d2xd##&0%

%!†

1a#x!1a + !†2a#x!2a

& $,

Now consider a discrete spontaneous symmetry breaking, with Isingsymmetry, described by a real scalar field !.Two cases of experimental interest are:

• Time-reversal symmetry breaking: leads to a dx2!y2 + idxy

superconductor, in which the Dirac fermions are massive

• Break 4-fold lattice rotation symmetry to 2-fold lattice rota-tions: leads to a superconductor with nematic order.

We can write down the usual !4 theory for the scalar field:


r

dx2!y2 superconductor

rc

!!" = 0!!" #= 0

Time-reversal symmetry breaking

dx2!y2 ± idxy

superconductor


r

dx2!y2 superconductordx2!y2 superconductor+ nematic order

rc

!!" = 0!!" #= 0

Lattice rotation symmetry breaking


Tn

TcT

1/!1/!c

QuantumCritical

"/v!

M. Vojta, Y. Zhang, and S. Sachdev, Phys. Rev. Lett. 85, 4940 (2000)E.-A. Kim, M. J. Lawler, P. Oreto, S. Sachdev, E. Fradkin, S.A. Kivelson,

Phys. Rev. B 77, 184514 (2008).Monday, October 26, 2009

http://arxiv.org/find/cond-mat/1/au:+Kim_E/0/1/0/all/0/1




http://arxiv.org/find/cond-mat/1/au:+Lawler_M/0/1/0/all/0/1

http://arxiv.org/find/cond-mat/1/au:+Lawler_M/0/1/0/all/0/1

http://arxiv.org/find/cond-mat/1/au:+Oreto_P/0/1/0/all/0/1

http://arxiv.org/find/cond-mat/1/au:+Oreto_P/0/1/0/all/0/1

http://arxiv.org/find/cond-mat/1/au:+Fradkin_E/0/1/0/all/0/1

http://arxiv.org/find/cond-mat/1/au:+Fradkin_E/0/1/0/all/0/1

http://arxiv.org/find/cond-mat/1/au:+Kivelson_S/0/1/0/all/0/1

http://arxiv.org/find/cond-mat/1/au:+Kivelson_S/0/1/0/all/0/1

S! =!

d2k

(2!)2T

"

!n

!†1a (!i"n + vF kx#z + v"ky#x) !1a

+!

d2k

(2!)2T

"

!n

!†2a (!i"n + vF ky#z + v"kx#x) !2a.

S0" =

!d2xd#

#12($#%)2 +

c2

2("%)2 +

r

2%2 +

u0

24%4

$;

S!" =!

d2xd##&0%

%!†

1a#x!1a + !†2a#x!2a

& $,

M. Vojta, Y. Zhang, and S. Sachdev, Physical Review Letters 85, 4940 (2000)

Ising order and Dirac fermions couple via a “Yukawa” term.

Nematic ordering

S!! =!

d2xd!""0#

#!†

1a!y!1a + !†2a!y!2a

$%

Time reversal symmetry breaking


S! =!

d2k

(2!)2T

"

!n

!†1a (!i"n + vF kx#z + v"ky#x) !1a

+!

d2k

(2!)2T

"

!n

!†2a (!i"n + vF ky#z + v"kx#x) !2a.

S0" =

!d2xd#

#12($#%)2 +

c2

2("%)2 +

r

2%2 +

u0

24%4

$;

S!" =!

d2xd##&0%

%!†

1a#x!1a + !†2a#x!2a

& $,

M. Vojta, Y. Zhang, and S. Sachdev, Physical Review Letters 85, 4940 (2000)

Ising order and Dirac fermions couple via a “Yukawa” term.

Nematic ordering

S!! =!

d2xd!""0#

#!†

1a!y!1a + !†2a!y!2a

$%

Time reversal symmetry breaking

For the latter case only, with vF = v! = c, theoryreduces to relativistic Gross-Neveu model


Integrating out the fermions yields an effective action for the scalar order parameter

Expansion in number of fermion spin components Nf

S! =Nf

v!vF!

!!0"(x, #);

v!

vF

"

+Nf

2

#d2xd#

$r"2(x, #)

%

+ irrelevant terms

where ! is a non-local and non-analytic functional of ".

The theory has only 2 couplings constants: r and v!/vF .

Y. Huh and S. Sachdev, Physical Review B 78, 064512 (2008).Monday, October 26, 2009

Integrating out the fermions yields an effective action for the scalar order parameter

Expansion in number of fermion spin components Nf

S! =Nf

v!vF!

!!0"(x, #);

v!

vF

"

+Nf

2

#d2xd#

$r"2(x, #)

%

+ irrelevant terms

where ! is a non-local and non-analytic functional of ".

The theory has only 2 couplings constants: r and v!/vF .There is a systematic expansion in powers of1/Nf for renormalization group equations

and all critical properties.Y. Huh and S. Sachdev, Physical Review B 78, 064512 (2008).


Fermi surface with full square lattice symmetry

Quantum criticality of Pomeranchuk instability

x

y


G(k,!) =Z

! ! vF (k ! kF )! i!2F!

k!kF!

" + . . .

Electron Green’s function in Fermi liquid (T=0)


Electron Green’s function in Fermi liquid (T=0)

G(k,!) =Z

! ! vF (k ! kF )! i!2F!

k!kF!

" + . . .

Green’s function has a pole in the LHP at

! = vF (k ! kF )! i"(k ! kF )2 + . . .

Pole is at ! = 0 precisely at k = kF i.e. on a sphere ofradius kF in momentum space. This is the Fermi surface.

Re(!)

Im(!)


Fermi surface with full square lattice symmetry


x

y


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


x

y



x

y

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


!!c

Pomeranchuk instability as a function of coupling !

!!" = 0 !!" #= 0



!!c

T

!!" = 0 !!" #= 0


Phase diagram as a function of T and !

Quantumcritical Tc


!!c

T

!!" = 0 !!" #= 0



Quantumcritical Tc

Classicald=2 Isingcriticality


!!c

T

!!" = 0 !!" #= 0



Quantumcritical Tc

D=2+1 Ising

criticality ?



E!ective action for Ising order parameter

S! =!

d2rd!"(""#)2 + c2(!#)2 + ($" $c)#2 + u#4

#



E!ective action for Ising order parameter

S! =!

d2rd!"(""#)2 + c2(!#)2 + ($" $c)#2 + u#4

#

E!ective action for electrons:

Sc =!

d!

Nf"

!=1

#

$"

i

c†i!"" ci! !"

i<j

tijc†i!ci!

%

&

"Nf"

!=1

"

k

!d!c†k! ("" + #k) ck!



!!" > 0 !!" < 0

Coupling between Ising order and electrons

S!c = ! !

!d" #

Nf"

"=1

"

k

(cos kx ! cos ky)c†k"ck"

for spatially independent #



!!" > 0 !!" < 0

Coupling between Ising order and electrons

S!c = ! !

!d"

Nf"

"=1

"

k,q

#q (cos kx! cos ky)c†k+q/2,"ck!q/2,"

for spatially dependent #



S! =!

d2rd!"(""#)2 + c2(!#)2 + ($" $c)#2 + u#4

#

Quantum critical field theory

Z =!D!Dci! exp (!S" ! Sc ! S"c)

Sc =Nf!

!=1

!

k

"d!c†k! ("" + #k) ck!

S!c = ! !

!d"

Nf"

"=1

"

k,q

#q (cos kx! cos ky)c†k+q/2,"ck!q/2,"



Hertz theory

Integrate out c! fermions and obtain non-local correctionsto ! action

"S" ! Nf#2

!d2q

4$2

!d%

2$|!(q, %)|2

" |%|q

+ q2#

+ . . .

This leads to a critical point with dynamic critical expo-nent z = 3 and quantum criticality controlled by the Gaus-sian fixed point.



Hertz theory

Self energy of c! fermions to order 1/Nf

!c(k, !) ! i

Nf!2/3

This leads to the Green’s function

G(k, !) " 1! # vF (k # kF )# i

Nf!2/3

Note that the order 1/Nf term is more singular in the infrared thanthe bare term; this leads to problems in the bare 1/Nf expansionin terms that are dominated by low frequency fermions.

1Nf (q2 + |!|/q)



The infrared singularities of fermion particle-hole pairsare most severe on planar graphs: these all contribute at

leading order in 1/Nf .

1Nf (q2 + |!|/q)

1! ! vF (k ! kF )! i

Nf!2/3

Sung-Sik Lee, Physical Review B 80, 165102 (2009)



1Nf (q2 + |!|/q)

1! ! vF (k ! kF )! i

Nf!2/3

A string theory for the Fermi surface ?


Conformal field theoryin 2+1 dimensions at T = 0

Einstein gravityon AdS4


Conformal field theoryin 2+1 dimensions at T > 0

Einstein gravity on AdS4

with a Schwarzschildblack hole


Conformal field theoryin 2+1 dimensions at T > 0,

with a non-zero chemical potential, µand applied magnetic field, B

Einstein gravity on AdS4

with a Reissner-Nordstromblack hole carrying electric

and magnetic chargesMonday, October 26, 2009

AdS4-Reissner-Nordstrom black hole

ds2 =L2

r2

!f(r)d!2 +

dr2

f(r)+ dx2 + dy2

",

f(r) = 1!!

1 +(r2

+µ2 + r4+B2)

"2

" !r

r+

"3

+(r2

+µ2 + r4+B2)

"2

!r

r+

"4

,

A = iµ

#1! r

r+

$d! + Bx dy .

T =1

4#r+

!3!

r2+µ2

"2!

r4+B2

"2

".


T. Faulkner, H. Liu, J. McGreevy, and D. Vegh, arXiv:0907.2694F. Denef, S. Hartnoll, and S. Sachdev, arXiv:0908.1788

Examine free energy and Green’s function of a probe particle


Short time behavior depends uponconformal AdS4 geometry near boundary



Long time behavior depends uponnear-horizon geometry of black hole



Radial direction of gravity theory ismeasure of energy scale in CFT



AdS4-Reissner-Nordstrom black hole

ds2 =L2

r2

!f(r)d!2 +

dr2

f(r)+ dx2 + dy2

",

f(r) = 1!!

1 +(r2

+µ2 + r4+B2)

"2

" !r

r+

"3

+(r2

+µ2 + r4+B2)

"2

!r

r+

"4

,

A = iµ

#1! r

r+

$d! + Bx dy .

T =1

4#r+

!3!

r2+µ2

"2!

r4+B2

"2

".


AdS2 x R2 near-horizongeometry

r ! r+ " 1!

ds2 =R2

!2

!!d"2 + d!2

"+

r2+

R2

!dx2 + dy2

"


T. Faulkner, H. Liu, J. McGreevy, and D. Vegh, arXiv:0907.2694

Infrared physics of Fermi surface is linked tothe near horizon AdS2 geometry of

Reissner-Nordstrom black hole



AdS4

Geometric interpretation of RG flow



AdS2 x R2

Geometric interpretation of RG flow


Green’s function of a fermion

T. Faulkner, H. Liu, J. McGreevy, and

D. Vegh, arXiv:0907.2694

G(k,!) ! 1! " vF (k " kF )" i!!(k)

See also M. Cubrovic, J Zaanen, and K. Schalm, arXiv:0904.1993


Green’s function of a fermion

T. Faulkner, H. Liu, J. McGreevy, and

D. Vegh, arXiv:0907.2694

G(k,!) ! 1! " vF (k " kF )" i!!(k)

Similar to non-Fermi liquid theories of Fermi surfaces coupled to gauge fields, and at quantum critical points


Free energy from gravity theoryThe free energy is expressed as a sum over the “quasinor-mal frequencies”, z!, of the black hole. Here ! representsany set of quantum numbers:

Fboson = !T!

!

ln

"|z!|2"T

####!$

iz!

2"T

%####2&

Ffermion = T!

!

ln

"####!$

iz!

2"T+

12

%####2&

Application of this formula shows that the fermions ex-hibit the dHvA quantum oscillations with expected pe-riod (2"/(Fermi surface ares)) in 1/B, but with an ampli-tude corrected from the Fermi liquid formula of Lifshitz-Kosevich.

F. Denef, S. Hartnoll, and S. Sachdev, arXiv:0908.1788Monday, October 26, 2009

General theory of finite temperature dynamics and transport near quantum critical points, with

applications to antiferromagnets, graphene, and superconductors

Conclusions


The AdS/CFT offers promise in providing a new understanding of

strongly interacting quantum matter at non-zero density

Conclusions


strong coupling problems in condensed matter and the ads...

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