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Quantum phase transitions

Thomas VojtaDepartment of Physics, University of Missouri-Rolla

• Phase transitions and critical points

• Quantum phase transitions: How important is quantum mechanics?

• Quantum phase transitions in metallic systems

• Quantum phase transitions and disorder

St. Louis, March 6, 2006

Acknowledgements

at UMR

Mark DickisonBernard Fendler

Ryan KinneyChetan KotabageMan Young LeeRastko Sknepnek

collaboration

Dietrich BelitzTed KirkpatrickJorg SchmalianMatthias Vojta

Funding:

National Science FoundationResearch Corporation

University of Missouri Research Board

Phase diagram of water

solid

liquid

gaseoustripel point

critical point

100 200 300 400 500 600 70010 2

10 3

10 4

10 5

10 6

10 7

10 8

10 9

T(K)

Tc=647 K

Pc=2.2*108 Pa

Tt=273 K

Pt=6000 Pa

Phase transition:singularity in thermodynamic quantities as functions of external parameters

Phase transitions: 1st order vs. continuous

1st order phase transition:

phase coexistence, latent heat,short range spatial and time correlations

Continuous transition (critical point):

no phase coexistence, no latent heat,infinite range correlations of fluctuations

solid

liquid

gaseoustripel point

critical point

100 200 300 400 500 600 70010 2

10 3

10 4

10 5

10 6

10 7

10 8

10 9

T(K)

Tc=647 K

Pc=2.2*108 Pa

Tt=273 K

Pt=6000 Pa

Critical behavior at continuous transitions:

diverging correlation length ξ ∼ |T − Tc|−ν and time ξτ ∼ ξz ∼ |T − Tc|−νz

power-laws in thermodynamic observables: ∆ρ ∼ |T − Tc|β, κ ∼ |T − Tc|−γ

• Manifestation: critical opalescence (Andrews 1869)

Universality: critical exponents are independent of microscopic details

Critical opalescence

Binary liquid system:

e.g. hexane and methanol

T > Tc ≈ 36◦C: fluids are miscible

T < Tc: fluids separate into two phases

T → Tc: length scale ξ of fluctuations grows

When ξ reaches the scale of a fraction of amicron (wavelength of light):

strong light scatteringfluid appears milky

Pictures taken from http://www.physicsofmatter.com

46◦C

39◦C

18◦C

• Phase transitions and critical points

• Quantum phase transitions: How important is quantum mechanics?

• Quantum phase transitions in metallic systems

• Quantum phase transitions and disorder

How important is quantum mechanics close to a critical point?

Two types of fluctuations:

thermal fluctuations, energy scale kBTquantum fluctuations, energy scale ~ωc

Quantum effects are unimportant if ~ωc ¿ kBT .

Critical slowing down:

ωc ∼ 1/ξτ ∼ |T − Tc|νz → 0 at the critical point

⇒ For any nonzero temperature, quantum fluctuations do not play a roleclose to the critical point

⇒ Quantum fluctuations do play a role a zero temperature

Thermal continuous phase transitions can be explained entirely interms of classical physics

Quantum phase transitions

occur at zero temperature as function of pressure, magnetic field, chemicalcomposition, ...

driven by quantum rather than thermal fluctuations

paramagnet

ferromagnet

0 20 40 60 80transversal magnetic field (kOe)

0.0

0.5

1.0

1.5

2.0

Bc

LiHoF4

0

Doping level (holes per CuO2)

0.2

SCAFM

Fermiliquid

Pseudogap

Non-Fermiliquid

Phase diagrams of LiHoF4 and a typical high-Tc superconductor such as YBa2Cu3O6+x

If the critical behavior is classical at any nonzero temperature, why arequantum phase transitions more than an academic problem?

Phase diagrams close to quantum phase transition

Quantum critical point controlsnonzero-temperature behaviorin its vicinity:

Path (a): crossover betweenclassical and quantum criticalbehavior

Path (b): temperature scaling ofquantum critical point

quantumdisordered

quantum critical

(b)

(a)

Bc

QCP

B0

ordered

kT ~ ��c

T

classicalcritical

thermallydisordered

T

Bc

QCP

kT -STc

quantum critical

quantumdisordered

(b)

(a)

B

0

order at T=0

thermallydisordered

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: statics and dynamics are coupled

Z = Tre−βH = limN→∞(e−βT/Ne−βU/N)N =∫

D[q(τ)] eS[q(τ)]

imaginary time τ acts as additional dimension, at zero temperature (β = ∞)the extension in this direction becomes infinite

Quantum phase transition in d dimensions is equivalent to classicaltransition in higher (d + z) dimension

z = 1 if space and time enter symmetrically, but, in general, z 6= 1.

Caveats:• mapping holds for thermodynamics only• resulting classical system can be unusual and anisotropic• extra complications without classical counterpart may arise, e.g., Berry phases

Toy model: transverse field Ising model

Quantum spins Si on a lattice: (c.f. LiHoF4)

H = −J∑

i

SziS

zi+1−h

∑

i

Sxi = −J

∑

i

SziS

zi+1−

h

2

∑

i

(S+i + S−i )

J : exchange energy, favors parallel spins, i.e., ferromagnetic state

h: transverse magnetic field, induces quantum fluctuations between up and downstates, favors paramagnetic state

Limiting cases:

|J | À |h| ferromagnetic ground state as in classical Ising magnet

|J | ¿ |h| paramagnetic ground state as for independent spins in a field

⇒ Quantum phase transition at |J | ∼ |h| (in 1D, transition is at |J | = |h|)

Quantum-to-classical mapping: QPT in transverse field Ising model in ddimensions maps onto classical Ising transition in d + 1 dimensions

• Phase transitions and critical points

• Quantum phase transitions: How important is quantum mechanics?

• Quantum phase transitions in metallic systems

• Quantum phase transitions and disorder

Quantum phase transitions in metallic systems

Ferromagnetic transitions:

• MnSi, UGe2, ZrZn2 – pressure tuned

• NixPd1−x, URu2−xRexSi2 – composition tuned

Antiferromagnetic transitions:

• CeCu6−xAux – composition tuned

• YbRh2Si2 – magnetic field tuned

• dozens of other rare earth compounds, tunedby composition, field or pressure

Other transitions:

• metamagnetic transitions

• superconducting transitions

Temperature-pressure phasediagram of MnSi (Pfleidereret al, 1997)

Fermi liquids versus non-Fermi liquids

Fermi liquid concept (Landau 1950s):

interacting Fermions behave like almost non-interacting quasi-particles withrenormalized parameters (e.g. effective mass)

universal predictions for low-temperature properties

specific heat C ∼ Tmagnetic susceptibility χ = constelectric resistivity ρ = A + BT 2

Fermi liquid theory is extremely successful, describes vast majority of metals.

In recent years:

systematic search for violations of the Fermi liquid paradigm• high-TC superconductors in normal phase• heavy Fermion materials (rare earth compounds)

Non-Fermi liquid behavior in CeCu6−xAux

• specific heat coefficient C/T at quantum critical point divergeslogarithmically with T → 0

• cause by interaction between quasiparticles and critical fluctuations• functional form presently not fully understood (orthodox theories do not

work)

Phase diagram and specific heat ofCeCu6−xAux (v. Lohneysen, 1996)

0.1 0.3 0.5x

0.0

0.5

1.0

T (

K)

II

I

0.1 1.0T (K)

0

1

2

3

4

C/T

(J/

mol

K2 )

x=0x=0.05x=0.1x=0.15x=0.2x=0.3

0.04 4 0

1

2

3

4

Generic scale invariance

metallic systems at T = 0 contain many soft (gapless) excitations such asparticle-hole excitations (even away from any critical point)

⇒ soft modes lead to long-range spatial and temporal correlations

quasi-particle dispersion: ∆ε(p) ∼ |p− pF |3 log |p− pF |specific heat: cV (T ) ∼ T 3 log T

static spin susceptibility: χ(2)(q) = χ(2)(0) + c3|q|2 log 1|q| + O(|q|2)

in real space: χ(2)(r− r′) ∼ |r− r′|−5

in general dimension: χ(2)(q) = χ(2)(0) + cd|q|d−1 + O(|q|2)

Long-range correlations due to coupling to soft particle-hole excitations

Generic scale invariance and the ferromagnetic transition

At critical point:

• critical modes and generic soft modes interact in a nontrivial way• interplay greatly modifies critical behavior of the quantum phase transition

(Belitz, Kirkpatrick, Vojta, Rev. Mod. Phys., 2005)

Example:Ferromagnetic quantum phase transition

Long-range interaction due to generic scaleinvariance ⇒ singular Landau theory

Φ = tm2 − vm4 log(1/m) + um4

Ferromagnetic quantum phase transitionturns 1st order

general mechanism, leads to singular Landautheory for all zero wavenumber order parameters Phase diagram of MnSi.

Quantum phase transitions and exotic superconductivity

Superconductivity in UGe2:

• phase diagram of UGe2 has pocket of superconductivity close to quantumphase transition

• in this pocket, UGe2 is ferromagnetic and superconducting at the same time

• superconductivity appears only in superclean samples

Phase diagram and resistivity of UGe2 (Saxena et al, Nature, 2000)

Character of superconductivity in UGe2

Conventional (BCS) superconductivity:

Cooper pairs form spin singletsnot compatible with ferromagnetism

Theoretical ideas:

phase separation (layering or disorder): NO!

partially paired FFLO state: NO!

spin triplet pairing with odd spatial symmetry(p-wave)

⇒ magnetic fluctuations are attractive forspin-triplet pairs

Ferromagnetic quantum phasetransition promotes magneticallymediated spin-triplet superconductivity

• Phase transitions and critical points

• Quantum phase transitions: How important is quantum mechanics?

• Quantum phase transitions in metallic systems

• Quantum phase transitions and disorder

Quantum phase transitions and disorder

In real systems:

Impurities and other imperfections lead to spatial variationof coupling strength

Questions:

Will the phase transition remain sharp or become smeared?Will the critical behavior change?

Disorder and quantum phase transitions:

• impurities are time-independent• disorder is perfectly correlated in imaginary time⇒ correlations increase the effects of disorder

(”it is harder to average out fluctuations”)

Disorder generally has stronger effects on quantumphase transitions than on classical transitions

x

τ

Conclusions

• quantum phase transitions occur at zero temperature as a function of aparameter like pressure, chemical composition, disorder, magnetic field

• quantum phase transitions are driven by quantum fluctuations rather thanthermal fluctuations

• quantum critical points control behavior in the quantum critical region atnonzero temperatures

• quantum phase transitions in metals have fascinating consequences:non-Fermi liquid behavior and exotic superconductivity

• quantum phase transitions are very sensitive to disorder

Quantum phase transitions provide a new ordering principlein condensed matter physics