Yale University

Subir Sachdev•C. Buragohain•K. Damle•M. Vojta

1. Simple model of a quantum phase transition ---

coupled-ladder antiferromagnet

2. Collective dynamics at low temperatures

A. Quasiclassical wavesB. Quasiclassical particles

Application to one-dimensional gappedantiferromagnets

3. A global phase diagram for dopedantiferromagnets and application to thecuprate superconductors.

A. Magnetic order- paramagnettransitions in insulating square lattice antiferromagnets.

B. A phase diagram for dopedantiferromagnets.

C. Quantum phase transitions(a) Magnetic order(b) charge order

D. Experimental implications

4. Quantum impurity in a nearly-criticalantiferromagnet ---

localized excitation with an irrational spinirrational spin

Applications to double-layer quantum Hall systems and the superconductor-insulator transition.

1. A simple theoretical model

S=1/2 spins on coupled 2-leg ladders

jiij

ij SSJH��

⋅=�><

Follow ground state as a function of λ

10 ≤≤ λ

JλJ

1=λSquare lattice antiferromagnet

Experimental realization: 42CuOLa

Ground state has long-rangemagnetic (Neel) order

( ) 0 ≠−= + NS yx iii

�

0=λDecoupled 2-leg ladders

JK

J KAllow ≠

Quantum paramagnetground state for J K<<

Qualitatively similar ground state for allJ K

( )↓↑−↑↓=2

1

λ

T

0 1λcQuantum paramagnet

0=S�

Neel state

0≠S�

Quantum critical point

2. Collective dynamics for T>0

A. Neel stateExcitations: 2 spin wavesCorrelation length (ξ)

~ typical spin wave wavelength~ exp( )TkBs /2 ρπ

→sρ Spin stiffness

Typical spin wave energy~ ξ/c� TkB<<

Mode occupation number( )( ) 11/exp/1 >>−TkBε

Quasiclassical waves(Chakravarty et al, 1989)

Dynamic Structure Factorat antiferromagnetic wavevector

S(π,ω)

ω

3~ ξ

ξτϕ

c

QuasiclassicalWaves

Angular or “phase” relaxation

2. Collective dynamics for T>0

B. Quantum paramagnetExcited states

Triplet (S=1) particle

Energy dispersion away fromantiferromagnetic wavevector

∆+∆=

2

22kcε

→∆ Energy gap

All low T dynamic correlations are universal functions of ∆ and c

Key Observations

1. De Broglie wavelengthTk

c

B∆�~

<< interparticle spacing TkBe /~ ∆

2. Quantum mechanical S-matrixhas a super-universal form at low momenta

(in one dimension)

(-1)

(Quasiclassical particles)

Spins bounce while momenta exchange

x

t

x

t

Dynamic Structure Factorat antiferromagnetic wavevector

S(π,ω)

ω

~1

ϕτ

QuasiclassicalParticles

Particles are quantum lumps of “amplitude oscillation” of

anti-ferromagnetic order parameter

∆

collisionrate

TkB BeTk /~ ∆−

�

52BaNiOYfor linewidth andenergy Excitation Γ∆

C. Broholm et al. (unpublished)

Solid line for Γ – theory with no free parameters

λ

T

0 1λcQuantum paramagnet

0=S�

Neel state

0≠S�

Quasiclassical particles Quasiclassical

wavesTkB BeTk /~1 ∆−

�ϕτTkBse /2~1 πρ

ϕτ−

∆~T sT ρ~

η

ϕ

στ

TTh

QTkC B ~1;;1

1

2

Σ==�

QUANTUMCRITICAL

“Universalincoherentconductance”

Extended t-J model on the square lattice

[ ]�>

+⋅+−=ji

jiijjiijjiij nnVSSJcctH��

αα

Plot phase diagram of stable groundstates as a function of:

(1) Doping δ

(2) Frustration

OR

N, where spin symmetry

neighbor)(first neighbor) (second

1

2

JJ

( ) ( )NSpSU →2Method and partial results on phase diagram: S. Sachdev and N. Read, Int. J. Mod. Phys. B 5, 219 (1991).

3. A global phase diagram for dopedantiferromagnets and application to thecuprate superconductors

(M. Vojta and S.S., Phys. Rev. Lett. in press)

Ground states are fully characterizedby the manner in which symmetries are broken:

1.S - electromagnetic U(1)

2.M – magnetic Sp(N)

3.C – lattice translations are reflections(must be broken by observables

(like charge density) which areinvariant under S and M)

e.g.Neel state - breaks only M

Incommensurate, collinear spin-density-wave -breaks M and C

Spin-Peierls stateC broken;Bond-centered chargedensity wave (“stripe”)

A. Magnetic order- paramagnettransitions in insulating square lattice antiferromagnets. δ=0

(Kotov et al cond-mat/9903154Singh et al cond-mat/9904064)

12 / JJ0 4.0≈

Neel stateM broken

Second-orderquantum phase transition (?)

N (number of spincomponents)

S (length of spin)

Quantumdimer model

SemiclassicalNon-linear σ model

Schwingerbosons

N/S

M order parameter

C order parameter

N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989)G. Murthy and S. Sachdev, Nucl. Phys. B344, 557 (1990)

Quantum “entropic” effects prefer one-dimensional ordered structures

B. A phase diagram for doped antiferromagnets

length0.5/unit density hole ≈

S broken for all δ>0 , large N

Plateaus at 1/(even integer)

p=4

p=2 p=4

p=1

8/1at energy excitation Fermion =δ

2=p 4=p

−π −π/2 0 π/2 π

kx

−π

−π/2

0

π/2

π

k y

0 0.25 0.5 0.75 1

Energy [J]

−π −π/2 0 π/2 π

(a) Onset of M orderingLow energy excitations:• Magnetic order parameter• Gapless Fermi quasiparticles

QGapless FermiPoints in a d-wavesuperconductor

If (magnetic-ordering wavevector),or if fermion excitations are fully gapped, then magnetic ordering transition is identical to that at δ=0, and described by the O(3) non-linear σ model:

( ) 1 ; 21 222 =�

�

���

�∂= � nnµτ

gdxdS

C. Quantum phase transitions

GQ ≠

NMR measurements by Imai et alPhys. Rev. Lett. 70, 1002 (1993).

Quasiclassicalwaves

Quasiclassicalparticles

Particle “resonance” peak in neutron scattering ?

(Rossat-Mignod, Mason, Keimer…)

(b) Onset of C ordering

H Gapless FermiPoints in a d-wavesuperconductor

If H = K (charge-ordering wavevector), quantum transition is described by a theory of

4 “Dirac” fermions: ,2 complex scalars (amplitude of charge density

wave at wavevector K): ,

1

1

2

2

αψ 1 αψ 2

xΦ yΦ

22

22112

yxdxdS Φ∂+Φ∂+∂+∂= � µµαααα ψψψψτ

( )H.c.1212 +Φ+Φ+ βααβαα ψσψεψσψλ yyyxi

( ) ( ) ��

�ΦΦ+Φ+Φ+Φ+Φ+2222222

yxyxyx vur

Renormalization group analysisExpansion in ε=3-d

(Similar to analysis by L. Balents, M.P.A. Fisher, and C. Nayak Int. J. Mod. Phys. B 12, 1033 (1998))

332

λλελ −=�d

d

Scale-invariant, interacting fixed pointwith dynamic exponent z=1.

T>0 fermion Green’s function:

δ+8222 OCaCuSrBi

( ) ( ) ���

����

�ΧΛ= −

−

TkTkck

TkkG

BBBF F

F ωω η

η�,, 11

Anomalous exponent �+= 6/εηF

Proposal: this quantum-criticality is the origin of photoemission line broadening observed by Valla,Fedorov,Johnson,Wells et al (Science, in press) inoptimally doped

���

����

�Χ=∆

TkTkkc

BB

ω�2

(Bare ε expansion for scaling function fails at verylow momenta and frequency. Self energy has term

. Self-consistent, quasi-classical theoryis necessary to compute fermion damping)

1Χ

( ) ( ) ( )ωδεω �2~,0 Tki B−Σ

-5-2.5

02.5

px

-5

-2.5

0

2.5

ω

0

5

10

-4 -2 0 2

px

-5

-4

-3

-2

-1

0

1

2

3

ω

Im Φf-1(ω, p) * nf(ω)

for p=(px ,0,0), ε=0.2,ω −> ω + iη, η=0.2

0

1

2

3

4

5

6

7

8

9

10

Scaling function for photoemission probability.

4. Quantum impurity in a nearly-criticalantiferromagnet

Make anyany localized deformation of antiferromagnet; e.g. remove a spin

X

Susceptibility impbA χχχ +=(A = area of system)

In paramagnetic phase as 0→T

Tkb

Bec

/22

∆−��

���

� ∆=π

χ� Tk

SSB

imp 3)1(; +=χ

Y

( ) ( ) 00 2 ≠=⋅∞→

mSS YY

��

ττ

For a general impurity impχ defines the value of S

cλλ =At

( ) ( ) '10 ητ

τ =⋅ YY SS��

)(and 'νηλλ cm −=

However '1

1ηχ −≠

Timp

This last relationship holds in the multi-channel Kondo problem because the magnetic response of the screening cloud is negligible due to an exact “compensation” property. There is no such property here, and naïve scaling applies. This leads to

TkBimp

numberUniversal=χ

Curie response of an irrational spin

In the Neel phase

stiffnessspinnumberUniversal=⊥impχ

( ) 2/1 stiffnessspin sysxs ρρρ =Bulk susceptibility vanishes while impurity susceptibility diverges as 0→sρ

32

3

toleadsaveragingthermal,0 At

imp

2

imp ⊥+=

>

χχTk

ST

B

Boundary quantum field theory

parameterorder Neel φ�

spinimpurity ofn Orientatio n�

d−= 3 ; functions εβ

Bulk excitations in quantum paramagnet

At T=0 there is a magnon pole

ωω

−∆= 1),(GS

Impurities broaden this to

Γ−−∆=

iGS

ωω 1),(

We obtained (exponents are exact)dd

i cn )(1�

−∆≈Γ

meV 5 eV 2.0

meV 41 005.0

(1999)) 1939 , PRL , (Fong OCuYBa doped-on Zn sexperimentFor 732

=Γ�

==∆=

c

nalet

i

�

82

Measured value = 4.25 meV

Conclusions

1. Described T>0 crossovers near a simple magnetic quantum phase transition; foundrelaxation rates and transport coefficientswhich are universal functions of fundamental constants and thermodynamicvariables (“universal incoherent conductance”).

2. Theory of T>0 spin dynamics in gapped quasi-one-dimensional antiferromagnets-quantitative comparisons with experiments.

3. Global phase diagram for transitions in doped antiferromagnets compared with many experiments on the high temperature superconductors.A. Proposed a phase diagram for doped

antiferromagnets on the square lattice. Allground states are “conventional”, and theirexcitations can be described by electron Hartree-Fock/ BCS theory.

B. Described quantum-critical points betweenphases: these control anomalous behaviorin T>0 crossovers.

C. Experimental issues:- Bond-centered vs. site-centered stripes- States with 2x1 unit cells ?- Correspondence between wavevectors of Fermi points and spin/charge fluctuations ?

D. Theoretical issues:- Theory for T>0 fermion damping in quantum-critical region

- Charge transport for

4. Theory of quantum impurity in a nearly critical antiferromagnet – irrational spin excitations and a new boundary quantum field theory.

cTT >