Quantum phase transitions in condensed matter …qpt.physics.harvard.edu/talks/amherst11.pdfQuantum...

Quantum phase transitions in condensed matter physics,

with connections to string theory

HARVARD

Amherst College, October 27, 2011

sachdev.physics.harvard.eduFriday, October 28, 2011

Max Metlitski, Harvard

HARVARD

Eun Gook Moon, Harvard

Friday, October 28, 2011

1. Quantum critical points and string theory Entanglement and emergent dimensions

2. High temperature superconductors and strange metals Holography of compressible quantum phases

Outline

Square lattice antiferromagnet

H =�

�ij�

Jij�Si · �Sj

Examine ground state as a function of λ

S=1/2spins

H =�

�ij�

Jij�Si · �Sj

At large ground state is a “quantum paramagnet” with spins locked in valence bond singlets

=1√2

��↑↓�−

�� ↓↑��

H =�

�ij�

Jij�Si · �Sj

=1√2

��↑↓�−

�� ↓↑��

Nearest-neighor spins are “entangled” with each other.Can be separated into an Einstein-Podolsky-Rosen (EPR) pair.

H =�

�ij�

Jij�Si · �Sj

For λ ≈ 1, the ground state has antiferromagnetic (“Neel”) order,and the spins align in a checkerboard pattern

H =�

�ij�

Jij�Si · �Sj

For λ ≈ 1, the ground state has antiferromagnetic (“Neel”) order,and the spins align in a checkerboard pattern

No EPR pairs

=1√2

��↑↓�−

�� ↓↑��

Pressure in TlCuCl3

=1√2

��↑↓�−

�� ↓↑��

A. Oosawa, K. Kakurai, T. Osakabe, M. Nakamura, M. Takeda, and H. Tanaka, Journal of the Physical Society of Japan, 73, 1446 (2004).

TlCuCl3

An insulator whose spin susceptibility vanishes exponentially as the temperature T tends to zero.

TlCuCl3

Quantum paramagnet at ambient pressure

TlCuCl3

Neel order under pressureA. Oosawa, K. Kakurai, T. Osakabe, M. Nakamura, M. Takeda, and H. Tanaka, Journal of the Physical Society of Japan, 73, 1446 (2004).

=1√2

��↑↓�−

�� ↓↑��

λλcSpin S = 1“triplon”

Excitation spectrum in the paramagnetic phase

Spin S = 1“triplon”

Excitation spectrum in the Neel phase

Spin waves

Christian Ruegg, Bruce Normand, Masashige Matsumoto, Albert Furrer, Desmond McMorrow, Karl Kramer, Hans–Ulrich Gudel, Severian Gvasaliya,

Hannu Mutka, and Martin Boehm, Phys. Rev. Lett. 100, 205701 (2008)

Excitations of TlCuCl3 with varying pressure

0 0.5 1 1.5 2 2.5 30

Pressure [kbar]

0 0.5 1 1.5 2 2.5 30

Pressure [kbar]

Broken valence bondexcitations of the

quantum paramagnet

0 0.5 1 1.5 2 2.5 30

Pressure [kbar]

Spin wave

and longitudinal excitations

(similar to the Higgs particle)

of the Neel state.

0 0.5 1 1.5 2 2.5 30

Pressure [kbar]

Spin wave

and longitudinal excitations

(similar to the Higgs particle)

of the Neel state.

“Higgs” particle appears at theoretically predicted energy

=1√2

��↑↓�−

�� ↓↑��

Quantum critical point with non-local entanglement in spin wavefunction

=1√2

��↑↓�−

�� ↓↑��

depth ofentanglement

D-dimensionalspace

Tensor network representation of entanglement at quantum critical point

• Long-range entanglement

• Long distance and low energy correlations near thequantum critical point are described by a quantumfield theory which is relativistically invariant (wherethe spin-wave velocity plays the role of the velocityof “light”).

• The quantum field theory is invariant under scale andconformal transformations at the quantum criticalpoint: a CFT3

Characteristics of quantum critical point

• Allows unification of the standard model of particlephysics with gravity.

• Low-lying string modes correspond to gauge fields,gravitons, quarks . . .

String theory

• A D-brane is a D-dimensional surface on which strings can end.

• The low-energy theory on a D-brane is an ordinary quantumfield theory with no gravity.

• In D = 2, we obtain strongly-interacting CFT3s. These are“dual” to string theory on anti-de Sitter space: AdS4.

• A D-brane is a D-dimensional surface on which strings can end.

• The low-energy theory on a D-brane is an ordinary quantumfield theory with no gravity.

• In D = 2, we obtain strongly-interacting CFT3s. These are“dual” to string theory on anti-de Sitter space: AdS4.

CFTD+1

D-dimensionalspace

M. Levin and C. P. Nave, Phys. Rev. Lett. 99, 120601 (2007)F. Verstraete, M. M. Wolf, D. Perez-Garcia, and J. I. Cirac, Phys. Rev. Lett. 96, 220601 (2006)

String theory near a D-brane

D-dimensionalspace

Emergent directionof AdS4

D-dimensionalspace

Emergent directionof AdS4 Brian Swingle, arXiv:0905.1317

ρA = TrBρ = density matrix of region A

Entanglement entropy SEE = −Tr (ρA ln ρA)

Entanglement entropy

D-dimensionalspace

Draw a surface which intersects the minimal number of links

Emergent directionof AdS4

The entanglement entropy of a region A on the boundary equals the minimal area of a surface in the higher-dimensional

space whose boundary co-incides with that of A.

This can be seen both the string and tensor-network pictures

S. Ryu and T. Takayanagi, Phys. Rev. Lett. 96, 18160 (2006).Brian Swingle, arXiv:0905.1317

Field theories in D + 1 spacetime dimensions are

characterized by couplings g which obey the renor-

malization group equation

du= β(g)

where u is the energy scale. The RG equation is

local in energy scale, i.e. the RHS does not depend

upon u.

uKey idea: ⇒ Implement u as an extra dimen-sion, and map to a local theory in D + 2 spacetimedimensions.

At the RG fixed point, β(g) = 0, the D + 1 di-mensional field theory is invariant under the scaletransformation

xµ → xµ/b , u → b u

At the RG fixed point, β(g) = 0, the D + 1 di-mensional field theory is invariant under the scaletransformation

xµ → xµ/b , u → b u

This is an invariance of the metric of the theory in

D + 2 dimensions. The unique solution is

�2dxµdxµ + L2 du

Or, using the length scale z = L2/u

ds2 = L2 dxµdxµ + dz2

This is the space AdSD+2, and L is the AdS radius.

J. McGreevy, arXiv0909.0518

AdSD+2RD,1

Minkowski

CFTD+1

J. McGreevy, arXiv0909.0518

Quantum critical point with non-local entanglement in spin wavefunction

=1√2

��↑↓�−

�� ↓↑��

Classicalspin

Dilutetriplon

Quantumcritical

Neel orderPressure in TlCuCl3

S. Sachdev and J. Ye, Phys. Rev. Lett. 69, 2411 (1992).A. V. Chubukov, S. Sachdev, and J. Ye, Phys. Rev. B 49, 11919 (1994).

A 2+1 dimensional system at its

quantum critical point

AdS4-Schwarzschild black-brane

AdS/CFT correspondence at non-zero temperatures

Black-brane at temperature of

2+1 dimensional quantum critical

system

Friction of quantum criticality = waves

falling into black brane Friday, October 28, 2011

system

Provides successful description of many properties of quantum

critical points at non-zero temperatures

Outline

The cuprate superconductors

DavisFriday, October 28, 2011

Ground state has long-range Néel order

H =�

�ij�

Jij�Si · �Sj

Hole-doped

Electron-doped

Hole-doped

Electron-doped

Electron-doped cuprate superconductors

Hole-doped

Electron-doped

Resistivity∼ ρ0 +ATn

Figure prepared by K. Jin and and R. L. Greenebased on N. P. Fournier, P. Armitage, and

R. L. Greene, Rev. Mod. Phys. 82, 2421 (2010).

Hole-doped

Electron-doped

Hole-doped

Electron-doped

StrangeMetal

Ishida, Nakai, and HosonoarXiv:0906.2045v1

0 0.02 0.04 0.06 0.08 0.10 0.12

AFM Ort/

S. Nandi, M. G. Kim, A. Kreyssig, R. M. Fernandes, D. K. Pratt, A. Thaler, N. Ni, S. L. Bud'ko, P. C. Canfield, J. Schmalian, R. J. McQueeney, A. I. Goldman,

Physical Review Letters 104, 057006 (2010).

Iron pnictides: a new class of high temperature superconductors

TSDW Tc

1.0 SDW

Superconductivity

BaFe2(As1-xPx)2

Temperature-doping phase diagram of the iron pnictides:

Resistivity∼ ρ0 +ATα

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)

TSDW Tc

1.0 SDW

Superconductivity

BaFe2(As1-xPx)2

TSDW Tc

1.0 SDW

Superconductivity

BaFe2(As1-xPx)2

StrangeMetal

Temperature-pressure phase diagram of an organic superconductor

N. Doiron-Leyraud, P. Auban-Senzier, S. Rene de Cotret, A. Sedeki, C. Bourbonnais, D. Jerome, K. Bechgaard, and Louis Taillefer, Physical Review B 80, 214531 (2009)

Generally, the ground state of a Ce heavy-fermion sys-tem is determined by the competition of the indirect Ru-derman Kittel Kasuya Yosida (RKKY) interaction whichprovokes magnetic order of localized moments mediatedby the light conduction electrons and the Kondo interac-tion. This last local mechanism causes a paramagneticground state due the screening of the local moment ofthe Ce ion by the conduction electrons. Both interac-tions depend critically on the hybridization of the 4felectrons with the conduction electrons. High pressure isan ideal tool to tune the hybridization and the positionof the 4f level with respect to the Fermi level. Thereforehigh pressure experiments are ideal to study the criti-cal region where both interactions are of the same orderand compete. To understand the quantum phase transi-tion from the antiferromagnetic (AF) state to the para-magnetic (PM) state is actually one of the fundamen-tal questions in solid state physics. Di!erent theoreticalapproaches exist to model the magnetic quantum phasetransition such as spin-fluctuation theory of an itinerantmagnet [10–12], or a new so-called ’local’ quantum criti-cal scenario [13, 14]. Another e"cient source to preventlong range antiferromagnetic order is given by the va-lence fluctuations between the trivalent and the tetrava-lent configuration of the cerium ions [15].

The interesting point is that in these strongly corre-lated electron systems the same electrons (or renormal-ized quasiparticles) are responsible for both, magnetismand superconductivity. The above mentioned Ce-115family is an ideal model system, as it allows to studyboth, the quantum critical behavior and the interplay ofthe magnetic order with a superconducting state. Espe-cially, as we will be shown below, unexpected observa-tions will be found, if a magnetic field is applied in thecritical pressure region.

PRESSURE-TEMPERATURE PHASE DIAGRAM

In this article we concentrate on the compoundCeRhIn5. At ambient pressure the RKKY interactionis dominant in CeRhIn5 and magnetic order appears atTN = 3.8 K. However, the ordered magnetic moment ofµ = 0.59µB at 1.9 K is reduced of about 30% in com-parison to that of Ce ion in a crystal field doublet with-out Kondo e!ect [17]. Compared to other heavy fermioncompounds at p = 0 the enhancement of the Sommerfeldcoe"cient of the specific heat (! = 52 mJ mol!1K!2) [18]and the cylotron masses of electrons on the extremal or-bits of the Fermi surface is rather moderate [19, 20]. Thetopologies of the Fermi surfaces of CeRhIn5 are cylin-drical and almost identical to that of LaRhIn5 which isthe non 4f isostructural reference compound. From thisit can be concluded that the 4f electrons in CeRhIn5

are localized and do not contribute to the Fermi volume[19, 20].

By application of pressure, the system can be tunedthrough a quantum phase transition. The Neel temper-

FIG. 2. Pressure–temperature phase diagram of CeRhIn5 atzero magnetic field determined from specific heat measure-ments with antiferromagnetic (AF, blue) and superconduct-ing phases (SC, yellow). When Tc < TN a coexistence phaseAF+SC exist. When Tc > TN the antiferromagnetic order isabruptly suppressed. The blue square indicate the transitionfrom SC to AF+SC after Ref. 16.

ature shows a smooth maximum around 0.8 GPa andis monotonously suppressed for higher pressures. How-ever, CeRhIn5 is also a superconductor in a large pres-sure region from about 1.3 to 5 GPa. It has been shownthat when the superconducting transition temperatureTc > TN the antiferromagnetic order is rapidly sup-pressed (see figure 2) and vanishes at a lower pressurethan that expected from a linear extrapolation to T = 0.Thus the pressure where Tc = TN defines a first criticalpressure p!

c and clearly just above p!c anitferromagnetism

collapses. The intuitive picture is that the opening of asuperconducting gap on large parts of the Fermi surfaceabove p!

c impedes the formation of long range magneticorder. A coexisting phase AF+SC in zero magnetic fieldseems only be formed if on cooling first the magnetic or-der is established. We will discuss below the microscopicevidence of an homogeneous AF+SC phase.

At ambient pressure CeRhIn5 orders in an incommen-surate magnetic structure [21] with an ordering vectorof qic=(0.5, 0.5, ") and " = 0.297 that is a magneticstructure with a di!erent periodicity than the one of thelattice. Generally, an incommensurate magnetic struc-ture is not favorable for superconductivity with d wavesymmetry, which is realized in CeRhIn5 above p!

c [22].Neutron scattering experiments under high pressure donot give conclusive evidence of the structure under pres-sures up to 1.7 GPa which is the highest pressure studiedup to now [23–25]. The result is that at 1.7 GPa the in-commensurability has changed to " ! 0.4. The main dif-ficulty in these experiments with large sample volume isto ensure the pressure homogeneity. Near p!

c the controlof a perfect hydrostaticity is a key issue as the materialreacts quite opposite on uniaxial strain applied along thec and a axis.

From recent nuclear quadrupol resonance (NQR) data

G. Knebel, D. Aoki, and J. Flouquet, arXiv:0911.5223.Tuson Park, F. Ronning, H. Q. Yuan, M. B. Salamon, R. Movshovich, J. L. Sarrao, and J. D. Thompson, Nature 440, 65 (2006)

Temperature-pressure phase diagram of an heavy-fermion superconductor

Fermi surface+antiferromagnetism

The electron spin polarization obeys�

�S(r, τ)�

= �ϕ(r, τ)eiK·r

where K is the ordering wavevector.

Metal with “large” Fermi surface

��ϕ� = 0

Metal with electron and hole pockets

Increasing SDW order

��ϕ� �= 0

Increasing interactionFermi surface reconstruction and

onset of antiferromagnetism

Nd2−xCexCuO4

T. Helm, M. V. Kartsovnik, M. Bartkowiak, N. Bittner,

M. Lambacher, A. Erb, J. Wosnitza, and R. Gross,

Phys. Rev. Lett. 103, 157002 (2009).

Quantum oscillations

��ϕ� = 0

��ϕ� �= 0

��ϕ� = 0

��ϕ� �= 0

Quantum critical point realizes a strong-coupled non-Fermi liquid

compressible phaseAF

Fermi liquid with “large”

Fermi surface

AF metal with “small”

Fermi pocketsIncreasing SDW order

Fermi liquid with “large”

Fermi surface

AF metal with “small”

Fermi pocketsIncreasing SDW order

Strange metal ?

D.J. Scalapino, E. Loh, and J.E. Hirsch, Phys. Rev. B 34, 8190 (1986)Ar. Abanov, A. V. Chubukov, and A. M. Finkel'stein, Europhys. Lett. 54, 488 (2001)

S. Raghu, S.A. Kivelson, and D.J. Scalapino, Phys. Rev. B 81, 224505 (2010)M. A. Metlitski and S. Sachdev, Phys. Rev. B 85, 075127 (2010)

Superconductor ✓

DavisFriday, October 28, 2011

Challenge to string theory:

Describe quantum critical points and the phases of a compressible metallic systems

Can we obtain holographic theories of superfluids and Fermi liquids?

Are there any other compressible phases like strange metals?

Are there any other compressible phases like strange metals? Yes....

Holographic representation: AdS4

A 2+1 dimensional

CFTat T=0

+ +Electric flux

There is “charge” density on the boundary, which sources an electric field in the bulk

Holographic theory of a metalS. SachdevarXiv:1107.5321

Conclusions

Quantum phase transitions in condensed matter …qpt.physics.harvard.edu/talks/amherst11.pdfQuantum...

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