Quantum phase transitions

Transcription

1 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

2 Acknowledgements at UMR Mark Dickison Bernard Fendler Ryan Kinney Chetan Kotabage Man Young Lee Rastko Sknepnek collaboration Dietrich Belitz Ted Kirkpatrick Jörg Schmalian Matthias Vojta Funding: National Science Foundation Research Corporation University of Missouri Research Board

3 Phase diagram of water solid liquid critical point T c =647 K P c =2.2*10 8 Pa tripel point T t =273 K P t =6000 Pa gaseous T(K) Phase transition: singularity in thermodynamic quantities as functions of external parameters

4 Phase transitions: 1st order vs. continuous 1st order phase transition: phase coexistence, latent heat, short range spatial and time correlations solid liquid critical point T c =647 K P c =2.2*10 8 Pa Continuous transition (critical point): no phase coexistence, no latent heat, infinite range correlations of fluctuations tripel point T t =273 K P t =6000 Pa gaseous T(K) Critical behavior at continuous transitions: diverging correlation length ξ T T c ν and time ξ τ ξ z T T c νz power-laws in thermodynamic observables: ρ T T c β, κ T T c γ Manifestation: critical opalescence (Andrews 1869) Universality: critical exponents are independent of microscopic details

5 Critical opalescence Binary liquid system: e.g. hexane and methanol T > T c 36 C: fluids are miscible T < T c : fluids separate into two phases T T c : length scale ξ of fluctuations grows When ξ reaches the scale of a fraction of a micron (wavelength of light): strong light scattering fluid appears milky 46 C 39 C Pictures taken from 18 C

6 Phase transitions and critical points Quantum phase transitions: How important is quantum mechanics? Quantum phase transitions in metallic systems Quantum phase transitions and disorder

7 How important is quantum mechanics close to a critical point? Two types of fluctuations: thermal fluctuations, energy scale k B T quantum fluctuations, energy scale ω c Quantum effects are unimportant if ω c k B T. Critical slowing down: ω c 1/ξ τ T T c νz 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

8 Quantum phase transitions occur at zero temperature as function of pressure, magnetic field, chemical composition,... driven by quantum rather than thermal fluctuations paramagnet LiHoF 4 Non-Fermi liquid ferromagnet AFM Pseudo gap SC Fermi liquid B c transversal magnetic field (koe) Doping level (holes per CuO 2 ) Phase diagrams of LiHoF 4 and a typical high-t c superconductor such as YBa 2 Cu 3 O 6+x

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

10 Phase diagrams close to quantum phase transition Quantum critical point controls nonzero-temperature behavior in its vicinity: Path (a): crossover between classical and quantum critical behavior Path (b): temperature scaling of quantum critical point T classical critical T 0 thermally disordered ordered quantum critical (a) B c (b) quantum critical quantum disordered QCP kt ~ 67 c B (b) kt -ST c thermally disordered quantum disordered 0 (a) B c B order at T=0 QCP

11 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 βĥ = lim N (e β ˆT /N e βû/n ) N = D[q(τ)] e S[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 classical transition in higher (d + z) dimension z = 1 if space and time enter symmetrically, but, in general, z 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

12 Toy model: transverse field Ising model Quantum spins S i on a lattice: (c.f. LiHoF 4 ) H = J i S z i S z i+1 h i S x i = J i S z i S z i+1 h 2 (S + i + S 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 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

13 Phase transitions and critical points Quantum phase transitions: How important is quantum mechanics? Quantum phase transitions in metallic systems Quantum phase transitions and disorder

14 Quantum phase transitions in metallic systems Ferromagnetic transitions: MnSi, UGe 2, ZrZn 2 pressure tuned Ni x Pd 1 x, URu 2 x Re x Si 2 composition tuned Antiferromagnetic transitions: CeCu 6 x Au x composition tuned YbRh 2 Si 2 magnetic field tuned dozens of other rare earth compounds, tuned by composition, field or pressure Other transitions: metamagnetic transitions superconducting transitions Temperature-pressure phase diagram of MnSi (Pfleiderer et al, 1997)

15 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-t C superconductors in normal phase heavy Fermion materials (rare earth compounds)

16 Non-Fermi liquid behavior in CeCu 6 x Au x 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 CeCu 6 x Au x (v. Löhneysen, 1996) 1.0 II C/T (J/mol K 2 ) x=0 x=0.05 x=0.1 x=0.15 x=0.2 x=0.3 T (K) 0.5 I x T (K)

17 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 quasi-particle dispersion: ɛ(p) p p F 3 log p p F specific heat: c V (T ) T 3 log T static spin susceptibility: χ (2) (q) = χ (2) (0) + c 3 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) + c d q d 1 + O( q 2 ) Long-range correlations due to coupling to soft particle-hole excitations

18 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 = tm 2 vm 4 log(1/m) + um 4 Ferromagnetic quantum phase transition turns 1st order general mechanism, leads to singular Landau theory for all zero wavenumber order parameters Phase diagram of MnSi.

19 Quantum phase transitions and exotic superconductivity Superconductivity in UGe 2 : phase diagram of UGe 2 has pocket of superconductivity close to quantum phase transition in this pocket, UGe 2 is ferromagnetic and superconducting at the same time superconductivity appears only in superclean samples Phase diagram and resistivity of UGe 2 (Saxena et al, Nature, 2000)

20 Character of superconductivity in UGe 2 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

21 Phase transitions and critical points Quantum phase transitions: How important is quantum mechanics? Quantum phase transitions in metallic systems Quantum phase transitions and disorder

22 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 ) Disorder generally has stronger effects on quantum phase transitions than on classical transitions τ x

23 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