Quantum phase transitions Thomas Vojta Department of Physics, University of Missouri-Rolla ² Phase transitions and critical points ² Quantum phase transitions: How important is quantum mechanics? ² Quantum phase transitions in metallic systems ² Quantum phase transitions and disorder St. Louis, March 6, 2006 Acknowledgements at UMR collaboration Mark Dickison Dietrich Belitz Bernard Fendler Ted Kirkpatrick Ryan Kinney JÄorgSchmalian 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 in¯nite range correlations of fluctuations Pt=6000 Pa 10 2 100 200 300 400 500 600 700 T(K) Critical behavior at continuous transitions: ¡º z ¡ºz diverging correlation length » » jT ¡ Tcj and time »¿ » » » jT ¡ Tcj ¯ ¡γ power-laws in thermodynamic observables: ¢½ » jT ¡ Tcj ;· » jT ¡ Tcj ² 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 ¼ 36 C: fluids are miscible 46 C T < Tc: fluids separate into two phases T ! Tc: length scale » of fluctuations grows When » 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 ² Phase transitions and critical points ² Quantum phase transitions: How important is quantum mechanics? ² Quantum phase transitions in metallic systems ² 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 ~!c Quantum e®ects are unimportant if ~!c ¿ kBT . Critical slowing down: ºz !c » 1=»¿ » jT ¡ Tcj ! 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 ¯eld, 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¡¯H(p;q) = dp e¡¯T (p) dq e¡¯U(q) » dq e¡¯U(q) Quantum partition function: statics and dynamics are coupled R ¡¯H^ ¡¯T^ =N ¡¯U=N^ N S[q(¿)] Z = Tre = limN!1(e e ) = D[q(¿)] e imaginary time ¿ acts as additional dimension, at zero temperature (¯ = 1) the extension in this direction becomes in¯nite 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: ² mapping holds for thermodynamics only ² resulting classical system can be unusual and anisotropic ² extra complications without classical counterpart may arise, e.g., Berry phases Toy model: transverse ¯eld 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 ¯eld, induces quantum fluctuations between up and down states, favors paramagnetic state Limiting cases: jJj À jhj ferromagnetic ground state as in classical Ising magnet jJj ¿ jhj paramagnetic ground state as for independent spins in a ¯eld ) Quantum phase transition at jJj » jhj (in 1D, transition is at jJj = jhj) Quantum-to-classical mapping: QPT in transverse ¯eld Ising model in d dimensions maps onto classical Ising transition in d + 1 dimensions ² Phase transitions and critical points ² Quantum phase transitions: How important is quantum mechanics? ² Quantum phase transitions in metallic systems ² Quantum phase transitions and disorder Quantum phase transitions in metallic systems Ferromagnetic transitions: ² MnSi, UGe2, ZrZn2 { pressure tuned ² NixPd1¡x, URu2¡xRexSi2 { composition tuned Antiferromagnetic transitions: ² CeCu6¡xAux { composition tuned ² YbRh2Si2 { magnetic ¯eld tuned ² dozens of other rare earth compounds, tuned by composition, ¯eld or pressure Other transitions: Temperature-pressure phase diagram of MnSi (Pfleiderer ² metamagnetic transitions et al, 1997) ² 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. e®ective mass) universal predictions for low-temperature properties speci¯c heat C » T magnetic susceptibility  = const electric resistivity ½ = 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 ² high-TC superconductors in normal phase ² heavy Fermion materials (rare earth compounds) Non-Fermi liquid behavior in CeCu6¡xAux ² speci¯c heat coe±cient C=T at quantum critical point diverges logarithmically with T ! 0 ² cause by interaction between quasiparticles and critical fluctuations ² functional form presently not fully understood (orthodox theories do not work) Phase diagram and speci¯c heat of 4 CeCu6¡xAux (v. LÄohneysen,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) » jp ¡ pF j log jp ¡ pF j 3 speci¯c heat: cV (T ) » T log T (2) (2) 2 1 2 static spin susceptibility:  (q) =  (0) + c3jqj log jqj + O(jqj ) in real space: Â(2)(r ¡ r0) » jr ¡ r0j¡5 (2) (2) d¡1 2 in general dimension:  (q) =  (0) + cdjqj + O(jqj ) Long-range correlations due to coupling to soft particle-hole excitations Generic scale invariance and the ferromagnetic transition At critical point: ² critical modes and generic soft modes interact in a nontrivial way ² interplay greatly modi¯es 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 © = 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: ² phase diagram of UGe2 has pocket of superconductivity close to quantum phase transition ² in this pocket, UGe2 is ferromagnetic and superconducting at the same time ² 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 ² Phase transitions and critical points ² Quantum phase transitions: How important is quantum mechanics? ² Quantum phase transitions in metallic systems ² 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: ² impurities are time-independent ² disorder is perfectly correlated in imaginary time ) correlations increase the e®ects of disorder ("it