Upload
others
View
3
Download
0
Embed Size (px)
Citation preview
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