quantum phase transitions and disorder: rare regions, gri

Quantum phase transitions and disorder:Rare regions, Griffiths effects, and smearing

Thomas VojtaDepartment of Physics, University of Missouri-Rolla

• Phase transitions and quantum phase transitions• Quenched disorder and critical behavior: the common lore

• Rare regions and Griffiths effects• Smeared phase transitions

• An attempt of a classification

Santa Barbara, Jan 18, 2005

Acknowledgements

at UMR

Rastko SknepnekMark DickisonBernard Fendler

collaboration

Jorg SchmalianMatthias Vojta

Funding:

University of Missouri Research BoardNational Science Foundation CAREER Award

Classical and quantum phase transitions

classical phase transitions: at non-zero temperature, asymptotic criticalbehavior dominated by classical physics (thermal fluctuations)

quantum phase transitions: at zero temperature as function of pressure,magnetic field, chemical composition, ..., driven by quantum fluctuations

paramagnet

ferromagnet

0 20 40 60 80transversal magnetic field (kOe)

0.0

0.5

1.0

1.5

2.0

Bc

LiHoF4

phase diagrams of LiHoF4 (Bitko et al. 96) and MnSi (Pfleiderer et al. 97)

Imaginary time and quantum to classical mapping

Classical partition function: statics and dynamics decouple

Z =∫

dpdq e−βH(p,q) =∫

dp e−βT (p)∫

dq e−βU(q) ∼ ∫dq e−βU(q)

Quantum partition function:

Z = Tre−βH = limN→∞(e−βT/Ne−βU/N)N =∫

D[q(τ)] eS[q(τ)]

imaginary time τ acts as additional dimensionat T = 0, the extension in this direction becomes infinite

Caveats:• mapping holds for thermodynamics only• resulting classical system can be unusual and anisotropic (z 6= 1)• extra complications with no classical counterpart may arise, e.g., Berry phases

• Phase transitions and quantum phase transitions

• Quenched disorder and critical behavior: the common lore

• Rare regions and quantum Griffiths effects

• Smeared phase transitions


Weak disorder and Harris criterion

weak disorder: impurities lead to spatial variation of coupling strength g(x)Theory: random mass disorder Experiment: “good disorder” ???

Harris criterion: variation of average local gc(x) in correlation volume must besmaller than distance from global gc

variation of average gc in volume ξd

∆〈gc(x)〉 ∼ ξ−d/2

distance from global critical point t ∼ ξ−1/ν

∆〈gc(x)〉 < t ⇒ dν > 2

ξ

+gC(1),

+gC(4),

+gC(2),

+gC(3),

• if clean critical point fulfills Harris criterion ⇒ stable against disorder• inhomogeneities vanish at large length scales• macroscopic observables are self-averaging• example: 3D classical Heisenberg magnet: ν = 0.698

Finite-disorder critical points

if critical point violates Harris criterion ⇒ unstable against disorder

Common lore:

• system goes to new different critical point which fulfills dν > 2• inhomogeneities remain finite at all length scales (”finite disorder”)• macroscopic observables are not self-averaging• example: 3D classical Ising magnet: clean ν = 0.627 ⇒ dirty ν = 0.684

Distribution of criticalsusceptibilities of 3D dilute Isingmodel(Wiseman + Domany 98)

0.0 0.5 1.0 1.5 2.0 2.5χ(Tc,L)/[χ(Tc,L)]

0.00

0.02

0.04

0.06

0.08

0.10

freq

uenc

y

10.0 100.0L

0.10

0.12

0.14

0.16

0.18

0.20

Rχ

Disorder and quantum phase transitions

Disorder is quenched:

• impurities are time-independent

• disorder is perfectly correlated in imaginarytime direction

⇒ correlations increase the effects of disorder(”it is harder to average out fluctuations”)

x

τ

Disorder generically has stronger effects on quantum phase transitionsthan on classical transitions

Random quantum Ising model

H = −∑

〈i,j〉Jijσ

zi σ

zj −

∑

i

hiσxi

nearest neighbor interactions Jij and transverse fields hi both random

Exact solution in 1+1 dimensions:Ma-Dasgupta-Hu-Fisher real space renormalization group

• in each step, integrate out largest energy among all Jij and hi

• cluster aggregation/annihilation procedure• becomes exact in the limit of large disorder

Infinite-disorder critical point:

• under renormalization the disorder increases without limit• relative width of the distributions of Jij, hi diverges

Infinite-disorder critical point

• extremely slow dynamics log ξτ ∼ ξµ (activated scaling)• distributions of macroscopic observables become infinitely broad• average and typical values can be drastically different

correlations: − log Gtyp ∼ rψ Gav ∼ r−η

• averages are dominated by rare events

Probability distribution

of the end-to-end

correlations in a random

quantum Ising chain

(Fisher + Young 98)



• Rare regions and Griffiths effects



Griffiths effects in a classical dilute ferromagnet

• critical temperature Tc is reducedcompared to clean value Tc0

• for Tc < T < Tc0:no global order but local order on rareregions devoid of impurities

• probability: w(L) ∼ e−cLd:

rare regions have slow dynamics⇒ singular free energy everywhere in the Griffiths region (Tc < T < Tc0)

Classical Griffiths effects are generically weak and essentially unobservable

contribution to susceptibility: χRR ∼∫

dL e−cLdLγ/ν = finite

Quantum Griffiths effects

rare regions at a QPT are finite inspace but infinite in imaginary time

fluctuations of the rare regions areeven slower than in classical case ⇒Griffiths singularities are enhanced

t

rare region at a quantum phase

transition

Random quantum Ising systems

local susceptibility (inverse energy gap) of rare region: χloc ∼ ∆−1 ∼ eaLd

χRR ∼∫

dL e−cLdeaLd

can diverge inside Griffiths region

finite temperatures:χRR ∼ T d/z′−1 (z′ is continuously varying exponent)

Rare regions at quantum phase transitions with overdampeddynamics

itinerant Ising quantum antiferromagnet

magnetic fluctuations are damped due to coupling to electrons

Γ(q, ωn) = t + q2 + |ωn|

in imaginary time: long-range power-law interaction ∼ 1/(τ − τ ′)2

1D Ising model with 1/r2 interaction is known to have an ordered phase

⇒ isolated rare region can develop a static magnetization, i.e.,large islands do not tunnel (c.f. Millis, Morr, Schmalian and Castro-Neto)

⇒ conventional quantum Griffiths behavior does not existmagnetization develops gradually on independent rare regions

quantum phase transition is smeared by disorder

Phase diagram at a smeared transition

T

x

Magneticphase

exponentialtail

possible realization: ferromagnetic quantum phase transition in NixPd1−x

Universality of the smearing scenario

Condition for disorder-induced smearing:

isolated rare region can develop a static order parameter⇒ rare region has to be above lower critical dimension

Examples:

• quantum phase transitions of itinerant electrons(disorder correlations in imaginary time + long-range interaction 1/τ2)

• classical Ising magnets with planar defects(disorder correlations in 2 dimensions)

• classical non-equilibrium phase transitions in the directed percolationuniversality class with extended defects(disorder correlations in at least one dimension)

Disorder-induced smearing of a phase transition is aubiquitous phenomenon

Isolated islands – Lifshitz tail arguments

probability to find rare region of size L devoid of defects: w ∼ e−cLd

region has transition at distance tc(L) < 0 from the clean critical pointfinite size scaling: |tc(L)| ∼ L−φ (φ = clean shift exponent)

Consequently:probability to find a region which becomes critical at tc:

w(tc) ∼ exp(−B |tc|−d/φ)

total magnetization at coupling t is given by the sum over all rare regions havingtc > t:

m(t) ∼ exp(−B |t|−d/φ) (t → 0−)

Computer simulation of a model system

Classical Ising model in 2+1 dimensions

H = −1Lτ

∑

〈x,y〉,τ,τ ′Sx,τSy,τ ′ −

1Lτ

∑

x,τ,τ ′JxSx,τSx,τ ′

Jx: binary random variable, P (J) = (1− c) δ(J − 1) + c δ(J)totally correlated in the time-like direction

• short-range interactions in the two space-like directions• infinite-range interaction in the time-like direction

(static magnetization on the rare regions is retained, but time directioncan be treated exactly, permitting large sizes)

Smeared transition in the infinite-range model

4.75 4.8 4.85 4.9 4.95 5 5.05T

0

50

100

χ

0

0.05

0.1

0.15

0.2

0.25

m

susceptibilitymagnetization

4.87 4.88 4.89 4.9T

0

50

100

χ

4.87 4.88 4.89 4.9T

0.00

0.01

0.02

0.03

m

L=100L=200L=400L=1000

Magnetization + susceptibilityof the infinite-range model

seeming transition close toT = 4.88

phase transition is smeared

(m and χ are independent of L)

Lifshitz magnetization tailtowards disordered phaselog(m) ∼ − 1/(Tc0 − T )

Local magnetization distribution

0 50 100 150 200 250 300 350 400x 0

50100

150200

250300

350400

y

00.020.040.060.080.1

0.120.140.160.18

m

-25 -20 -15 -10 -5 0log(m)

10-4

10-3

10-2

10-1

100

P[lo

g(m

)]

4.914.904.894.88

4.874.864.854.84

Local magnetization in thetail region (T = 4.8875)

global magnetization startsto form on isolated islands

very inhomogeneous system

Distribution of the localmagnetization values

very broad, even onlogarithmic scale

ln(mtyp) ∼ 〈m〉−1/2

Classification of dirty phase transitions according toimportance of rare regions

Dimensionality Griffiths effects Dirty critical point Examplesof rare regions (classical PT, QPT)

dRR < d−c weak exponential conv. finite disorder class. magnet with point defects

dilute bilayer Heisenberg model

dRR = d−c strong power-law infinite randomness Ising model with linear defects

random quantum Ising model

itin. quantum Heisenberg magnet?

dRR > d−c RR become static smeared transition Ising model with planar defects

itinerant quantum Ising magnet

Dimer-diluted 2d Heisenberg quantum antiferromagnet

H = J‖∑〈i,j〉

a=1,2

εiεjSi,a · Sj,a + J⊥∑

i

εiSi,1 · Si,2,

Large scale Monte-Carlo simulations:

conventional finite-disorder critical point with power-law scalingcritical exponents are universal, dynamical exponent z = 1.31(after accounting for corrections to scaling)

0 0.1 0.2 0.3 0.4p

0

0.5

1

1.5

2

2.5

J ⊥ /

J ||

AF ordered

disordered

J

J2

z

5 10 20 50 100L

1

1.2

1.5

2

2.5

3

4

Ltm

ax/L

p=1/3p=2/7p=1/5p=1/8

Conclusions

• even weak disorder can have surprisingly strong effects on a quantum phasetransition

• rare regions play a much bigger role quantum phase transitions than aclassical transitions

• effective dimensionality of rare regions determines overall phenomenologyof phase transitions in disordered systems

• Ising systems with overdamped dynamics: sharp phase transition is destroyedby smearing because static order forms on rare spatial regions

• at a smeared transition, system is extremely inhomogeneous, even on alogarithmic scale

Griffiths effects at quantum phase transitions leads to a rich variety ofnew and exotic phenomena

quantum phase transitions and disorder: rare regions, gri

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