Quantum phase transitions

Thomas Vojta Department 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 collaboration

Mark Dickison Dietrich Belitz Bernard Fendler Ted Kirkpatrick Ryan Kinney J¨orgSchmalian Chetan Kotabage Matthias Vojta Man Young Lee Rastko Sknepnek

Funding:

National Science Foundation Research Corporation University of Missouri Research Board Phase diagram of water

10 9

10 8 critical point 7 liquid 10 Tc=647 K 8 solid Pc=2.2*10 Pa 10 6

10 5

10 4 tripel point gaseous

10 3 Tt=273 K

Pt=6000 Pa 10 2 100 200 300 400 500 600 700 T(K)

Phase transition: singularity in thermodynamic quantities as functions of external parameters Phase transitions: 1st order vs. continuous

10 9 1st order phase transition: 10 8 phase coexistence, latent heat, critical point 7 liquid 10 Tc=647 K 8 short range spatial and time correlations solid Pc=2.2*10 Pa 10 6

5 Continuous transition (critical point): 10 10 4 no phase coexistence, no latent heat, tripel point gaseous 10 3 Tt=273 K infinite range correlations of fluctuations Pt=6000 Pa 10 2 100 200 300 400 500 600 700 T(K) Critical behavior at continuous transitions: −ν z −νz diverging correlation length ξ ∼ |T − Tc| and time ξτ ∼ ξ ∼ |T − Tc| β −γ power-laws in thermodynamic observables: ∆ρ ∼ |T − Tc| , κ ∼ |T − Tc| • 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 46 C

T < Tc: fluids separate into two phases

T → Tc: length scale ξ of fluctuations grows

When ξ reaches the scale of a fraction of a micron (wavelength of light): strong light scattering 39◦C fluid appears milky

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

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 kBT quantum fluctuations, energy scale ~ωc

Quantum effects are unimportant if ~ωc ¿ kBT .

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

⇒ For any nonzero temperature, quantum fluctuations do not play a role close to the critical point ⇒ Quantum fluctuations do play a role a zero temperature

Thermal continuous phase transitions can be explained entirely in terms of classical physics Quantum phase transitions occur at zero temperature as function of pressure, magnetic field, chemical composition, ... driven by quantum rather than thermal fluctuations

2.0

paramagnet LiHoF 4 Non-Fermi 1.5 liquid

1.0 Pseudo Fermi 0.5 gap liquid ferromagnet AFM SC 0.0 0 0.2 0 20 40Bc 60 80

transversal magnetic field (kOe) Doping level (holes per CuO 2)

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 are quantum phase transitions more than an academic problem? Phase diagrams close to quantum phase transition

thermally Quantum critical point controls T disordered quantum critical kT ~ $% nonzero-temperature behavior c in its vicinity: (b) classical Path (a): crossover between critical classical and quantum critical quantum disordered behavior ordered (a) 0 B Path (b): temperature scaling of Bc quantum critical point QCP T quantum critical T -S k Tc (b)

thermally quantum disordered disordered (a) 0 Bc B order at T=0 QCP Quantum to classical mapping

Classical partition function: statics and dynamics decouple R R R R 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 R −βHˆ −βTˆ /N −βU/Nˆ N S[q(τ)] Z = Tre = limN→∞(e e ) = D[q(τ)] e 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 classical transition 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)

X X X h X H = −J SzSz −h Sx = −J SzSz − (S+ + S−) i i+1 i i i+1 2 i i i i i i

J: exchange energy, favors parallel spins, i.e., ferromagnetic state h: transverse magnetic field, induces quantum fluctuations between up and down states, 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 d dimensions 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, tuned by composition, field or pressure

Other transitions: Temperature-pressure phase diagram of MnSi (Pfleiderer • metamagnetic transitions et al, 1997) • superconducting transitions Fermi liquids versus non-Fermi liquids

Fermi liquid concept (Landau 1950s): interacting Fermions behave like almost non-interacting quasi-particles with renormalized parameters (e.g. effective mass) universal predictions for low-temperature properties specific heat C ∼ T magnetic susceptibility χ = const electric 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 diverges logarithmically 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 of 4 CeCu6−xAux (v. L¨ohneysen,1996) x=0 x=0.05 3 x=0.1

) x=0.15 2 x=0.2 x=0.3 2 II 1.0 C/T (J/mol K

T (K) 1 0.5 I

0 0.0 0.04 0.1 1.0 4 0.1 0.3 0.5 T (K) x Generic scale invariance metallic systems at T = 0 contain many soft (gapless) excitations such as particle-hole excitations (even away from any critical point)

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

3 quasi-particle dispersion: ∆²(p) ∼ |p − pF | log |p − pF |

3 specific heat: cV (T ) ∼ T log T

(2) (2) 2 1 2 static spin susceptibility: χ (q) = χ (0) + c3|q| log |q| + O(|q| ) in real space: χ(2)(r − r0) ∼ |r − r0|−5

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

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 scale invariance ⇒ singular Landau theory

Φ = tm2 − vm4 log(1/m) + um4

Ferromagnetic quantum phase transition turns 1st order general mechanism, leads to singular Landau theory 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 quantum phase 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 singlets not 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 for spin-triplet pairs

Ferromagnetic quantum phase transition promotes magnetically mediated 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 variation of 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”) τ Disorder generally has stronger effects on quantum phase transitions than on classical transitions

x Conclusions

• quantum phase transitions occur at zero temperature as a function of a parameter like pressure, chemical composition, disorder, magnetic field • quantum phase transitions are driven by quantum fluctuations rather than thermal fluctuations • quantum critical points control behavior in the quantum critical region at nonzero 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 principle in condensed matter physics