Quantum phase transitions and disorder: Griffiths singularities, infinite randomness, and smearing

Transcription

1 Quantum phase transitions and disorder: Griffiths singularities, infinite randomness, and smearing Thomas Vojta Department of Physics, Missouri University of Science and Technology Phase transitions and quantum phase transitions Effects of impurities and defects: the common lore Rare regions, Griffiths singularities and smearing Experiments in disordered superconductors and itinerant magnets Classification of weakly disordered phase transitions Washington University St. Louis, Oct 5, 2017

2 Acknowledgements Recent graduates Manal Al Ali PhD 13 Hatem Barghathi PhD 16 Current group members Jack Crewse Ahmed Ibrahim (+ 3 undergraduate students) Fawaz Hrahsheh PhD 13 Chetan Kotabage PhD 11 Man Young Lee PhD 11 Collaborators: Experiment Almut Schroeder (Kent State) Istvan Kezsmarki (TU Budapest) David Nozadze PhD 13 Theory Jose Hoyos (Sao Paulo)

3 Phase transitions: the basics Phase transition: singularity in free energy occurs in macroscopic systems 1st order transition: phase coexistence, latent heat finite correlation length and time Continuous transition: no phase coexistence, latent heat diverging correlations solid liquid tripel point T t =273 K P t =6000 Pa critical point T c =647 K gaseous T(K) P c =2.2*10 8 Pa Critical behavior: 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 critical exponents are universal = independent of microscopic details

4 Quantum phase transitions occur at zero temperature as function of pressure, magnetic field,... driven by quantum rather than thermal fluctuations 2.0 paramagnet LiHoF 4 Transverse-field Ising model 1.5 H = J i,j σ z i σ z j h i σ x i ferromagnet transverse magnetic field induces spin flips via σ x = σ + + σ transverse magnetic field h(koe) h c transverse field suppresses magnetic order Quantum to classical mapping: maps QPT in d dimensions to classical PT in d + 1 dimensions imaginary time plays role of additional dimension

5 Phase transitions and quantum phase transitions Effects of impurities and defects: the common lore Rare regions, Griffiths singularities and smearing Experiments in disordered superconductors and itinerant magnets Classification of weakly disordered phase transitions

6 Phase transitions and disorder system undergoing classical, quantum, or nonequilibrium phase transition real systems always contain impurities, defects and other types of disorder Weak (random-t c, random-mass) disorder: spatial variation of coupling strength locally favors one phase over the other does not break order-parameter symmetries no change in character of the bulk phases Will the phase transition remain sharp or become smeared? Will the order of the transition change Will the critical behavior change?

7 Harris criterion Harris: stability of clean critical point variation of average local T c (i) between correlation volumes must be smaller than distance from global T c variation of average T c (i) in volume ξ d T c (i) ξ d/2 global distance from critical point T T c ξ 1/ν T c (i)/(t T c ) 0 dν > 2 at criticality +T C (1), ξ +T C (3), +T C (2), +T C (4), if clean critical point fulfills Harris criterion stable against disorder system is asymptotically clean as inhomogeneities vanish at large length scales macroscopic observables are self-averaging example: 3D classical Heisenberg magnet: ν = extension to general spatio-temporal disorder: T.V and R. Dickman, PRE 93, (2016)

8 Finite-disorder critical points if critical point violates Harris criterion unstable against disorder Common lore: new, different critical point which fulfills dν > 2 inhomogeneities finite at all length scales ( finite disorder ) macroscopic observables not self-averaging example: 3D classical Ising magnet: clean ν = dirty ν = frequency L R χ Distribution of critical susceptibilities of 3D dilute Ising model (Wiseman + Domany 98) χ(t c,l)/[χ(t c,l)]

9 Disorder and quantum phase transitions Disorder is quenched: impurities are time-independent disorder is perfectly correlated in imaginary time direction correlations increase the effects of disorder ( it is harder to average out fluctuations ) τ x Disorder generically has stronger effects on quantum phase transitions than on classical transitions

10 Random transverse-field Ising model H = i,j J ij σ z i σ z j i h i σ x i nearest neighbor interactions J ij and transverse fields h i both random Strong-disorder renormalization group: Ma, Dasgupta, Hu (1979), Fisher (1992, 1995) in each step, integrate out largest of all J ij, h i cluster aggregation/annihilation process exact in the limit of large disorder h 1 h 2 h 3 J 1 J 2 ~ h 4 J 3 J 4 J 1 J=J 2 J 3 /h 3 J 4 h 5 Analytical solution in 1+1 dimensions: flow equations for entire probability distributions P (J), R(h) under renormalization, relative width of the distributions diverges disorder increases without limit

11 Infinite-disorder critical point distributions of macroscopic observables become infinitely broad average and typical values drastically different correlations: G av r η, log G typ r ψ averages dominated by rare events extremely slow dynamics log ξ τ ξ µ (activated dynamical scaling) Probability distribution of end-to-end correlations in a random quantum Ising chain (Fisher + Young 98)

12 Phase transitions and quantum phase transitions Effects of impurities and defects: the common lore Rare regions, Griffiths singularities and smearing Experiments in disordered superconductors and itinerant magnets Classification of weakly disordered phase transitions

13 Rare regions in a classical dilute ferromagnet critical temperature T c is reduced compared to clean value T c0 for T c < T < T c0 : no global order but local order on rare regions devoid of impurities rare region probability exponentially small p(l) e cld Thermodynamics of rare regions rare regions cannot order statically but act as large superspins very slow dynamics, enhanced thermodynamic response Can rare regions dominate thermodynamics of the entire system?

14 Griffiths region or Griffiths phase In generic classical systems: Griffiths: rare regions lead to singular free energy everywhere in the interval T c < T < T c0 Rare region susceptibility: susceptibility of single RR: χ L 2d /T sum over all RRs: χ RR dl e cld L 2d essential singularity large regions make negligible contribution Thermodynamic Griffiths effects are weak and essentially unobservable Long-time dynamics can be dominated by rare regions

15 Quantum Griffiths effects Quantum phase transitions: rare regions are finite in space but infinite in imaginary time fluctuations even slower than in classical case Griffiths singularities enhanced rare region at a quantum phase transition Transverse-field Ising systems: susceptibility of rare region: χ loc 1 e ald χ RR dl e cld e ald can diverge inside Griffiths region power-law quantum Griffiths singularities susceptibility: χ RR T d/z 1, specific heat: C RR T d/z z is continuously varying Griffiths dynamical exponent, diverges at criticality Connection between Harris criterion and Griffiths sing., T.V. + J.A. Hoyos, PRL 112, (2014)

16 Smeared phase transitions Randomly layered classical magnet: random layers of two different ferromagnets rare regions are thick 2d slabs of the material with higher Tc 2d (Ising) magnets have true phase transition global magnetization develops gradually as rare regions order independently global phase transition is smeared by disorder Smeared quantum phase transitions: if isolated rare region develops static order parameter transition smeared example: itinerant Ising magnet (order parameter damped by coupling to electrons, this prevents rare regions from tunneling) T.V., PRL 90, (2003), J.A. Hoyos and T.V., PRL 100, (2008)

17 Magnetization tail of smeared transition tail produced by largest rare regions (thickest slabs) m slab transition temperature T c (L) < T 0 c (T 0 c = higher of the two bulk T c ) finite size scaling: T c (L) T 0 c L ϕ (ϕ = clean shift exponent) probability for slab devoid of weak planes: w e cl Magnetization tail for T T 0 c m(t ) exp( B T T 0 c 1/ϕ ) R. Sknepnek and T.V., PRB 69, (2004) log 10 (m) p slope B exponential tail replaces Griffiths phase T c log 10 (1-p) T c -T - T

18 Phase transitions and quantum phase transitions Effects of impurities and defects: the common lore Rare regions, Griffiths singularities, and smearing Experiments in disordered superconductors and itinerant magnets Classification of weakly disordered phase transitions

19 Superconductor-metal quantum phase transition in nanowires ultrathin MoGe wires (width 10 nm) molecular templating using a single carbon nanotube [Bezryadin group, UIUC] pair breaking by surface magnetic impurities creates disorder and dissipation superconductor-metal QPT as function of wire thickness

20 Quantum Landau-Ginzburg-Wilson theory number of transport channels (states transverse to wire) is large, N 1 motion of (unpaired) electrons is three-dimensional wire width 10nm coherence length, wire length 500nm superconducting critical fluctuations are one-dimensional Free energy functional: S = T q,ω n ( r + ξ 2 0 q 2 + γ ω n ) φ(q, ω n ) 2 + u 2N d d xdτ φ 4 (x, τ) To apply strong-disorder RG, discretize space: S = T i,ω n (ϵ i + γ i ω n ) ϕ i (ω n ) 2 T i,ω n J i ϕ i ( ω n ) ϕ i+1 (ω n ) chain of coupled superconducting grains, coupled by Josephson interactions disorder: ϵ i, γ i, J i random variables

21 Strong-disorder renormalization group Competing local energies: local energy gaps ϵ i, favoring normal phase bonds J i (Josephson couplings), favoring superconducting phase Infinite-randomness critical point: distributions P (J) and R(ϵ) become infinitely broad universality class of random transverse-field Ising model critical exponents known exactly in 1D activated dynamical scaling, log ξ t ξ ψ (ψ = 1/2 in 1D) higher dimensions: same activated scaling scenario, exponents known numerically T c exp( const r νψ ) J.A. Hoyos, C. Kotabage and T.V., PRL 99, (2007), PRB 79, (2009)

22 Critical behavior and Griffiths singularities Specific heat: C(r, T ) = ( ln T ) d/ψ ( 0 Φ C r νψ ln T ) 0 T T Dynamical (optical) conductivity: calculated from Kubo formula include vector potential in SDRG Griffiths phase: C(r, T ) T d/z Griffiths dynamical exponent z r νψ diverges at criticality σ (r, ω) = 4e2 h L = 128 ( ln ω ) 1/ψ ( 0 Φ σ r νψ ln ω ) 0 ω ω δ Ordered Griffiths phase: long-range order but vanishing stiffness anomalous elasticity: f(θ) f(0) Θ 2 L (1+z) (z > 1) (h/4e 2 )σ ω/ω 0 P. Mohan, P.M. Goldbart, R. Narayanan, J. Toner and T.V, PRL 105, (2010) A. Del Maestro, B. Rosenow, J.A. Hoyos and T.V., PRL 105, (2010)

23 Experiment: ultrathin Ga films Xing et al., Science 350, 542 (2015) superconductivity below Tc 3.62K, suppressed by magnetic field field-driven QPT well described by 2D infinite-randomness critical point dynamical exponent diverges as z B Bc νψ with ν 1.2, ψ 0.5 Exponent values from MC simulations by T.V., A. Farquhar, J. Mast, PRE 79, (2009)

24 Ferromagnetic Griffiths singularities in Ni 1 x V x T c (K) and other T-scales µ sat (µ B ) PM Ni V 1-x x FM 0.6-5µ /V B GP T c T max T cross x(%) CG? x (%) " m (emu/mol) H=100G (a) x=11.4% 11.6% 12.07% 12.25% 13.0% 15.0%! " 1 + " dyn Ni V 1-x x T=2K (b) M (emu/mol) m T(K) H (G) S. Ubaid-Kassis, T. V., A. Schroeder, PRL 104, (2010) A. Schroeder, S. Ubaid-Kassis, T.V., JPCM 23, (2011) D. Nozadze + T.V., PRB 85, (2012) R. Wang et al., PRL 118, (2017)!, 1- " (a)! x=12.25% lot! (b) "! 0.4 dyn H=500G T=2K 0.2 Ni V T=14K - 1-x x - T=300K x (%) H/T (G/K) M m /H 0.5 (emu mol -1 G -0.5 )

25 Smeared quantum phase transition in Sr1 xcaxruo3 a B [mt] -150 I B/Ru M M [ B/Ru] T=2.9K 4.2K K 20K K b Composition, x 2 TC(x) Theory 1 (x M, Tc exp C x(1 x) x0c )2 d/ϕ ] L. Demko et al, PRL 108, (2012) F. Hrahsheh et al., PRB 83, (2011) C. Svoboda et al., EPL 97, (2012) ln(tc) Magnetization and[ Tc in tail: ln(m) 0 xc M(x); T=2.9K M(x); T=4.2K -5 M(B); T=4.2K -6 1 Theory Composition, x

26 Phase transitions and quantum phase transitions Effects of impurities and defects: the common lore Rare regions, Griffiths singularities, and smearing Experiments in disordered superconductors and itinerant magnets Classification of weakly disordered phase transitions

27 Disorder at phase transitions: two frameworks fate of average disorder strength under coarse graining importance of rare regions and strength of Griffiths singularities Recently: general relation between Harris criterion and rare region physics T.V. + J.A. Hoyos, Phys. Rev. Lett. 112, (2014), Phys. Rev. E 90, (2014) below d + c, same inequality, dν > 2, governs relevance or irrelevance of disorder and fate of the Griffiths singularities above d + c, behavior is even richer relevance of rare regions depends on inequality d + c ν > 2

28 Conclusions even weak disorder can have surprisingly strong effects on a phase transition rare regions play a much bigger role at quantum phase transitions than at classical transitions classification of Griffiths phenomena according to effective dimensionality of rare regions experimental evidence for quantum Griffiths singularities and smeared phase transitions has been found at magnetic and superconducting quantum phase transitions in disordered metals Quenched disorder at quantum phase transitions leads to a rich variety of new effects and exotic phenomena Reviews: T.V., J. Phys. A 39, R143 (2006); J. Low Temp. Phys. 161, 299 (2010); AIP Conf. Proc. 1550, 188 (2013); Ann. Rev. Cond. Mat. Phys., to appear 2018

29 Imaginary time and 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 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 T = 0, the extension in this direction becomes infinite Caveats: mapping holds for thermodynamics only resulting classical system can be unusual and anisotropic (z 1) if quantum action is not real, extra complications may arise, e.g., Berry phases

30 Strong-disorder renormalization group introduced by Ma, Dasgupta, Hu (1979), further developed by Fisher (1992, 1995) asymptotically exact if disorder distribution becomes broad under RG Basic idea: Successively integrate out the local high-energy modes and renormalize the remaining degrees of freedom. Discretized large-n action: (ϵ i, γ i, J i : random variables) S = T i,ω n (ϵ i + γ i ω n ) ϕ i (ω n ) 2 T i,ω n J i ϕ i ( ω n ) ϕ i+1 (ω n ) the competing local energies are: bonds (Josephson couplings) J i, favoring ordered phase local energy gaps ϵ i, favoring disordered phase in each RG step, integrate out largest among all J i and ϵ i J.A. Hoyos, C. Kotabage and T.V., PRL 99, (2007), PRB 79, (2009)

31 RG recursions and flow equations ε 1 ε 2 ε 3 ε 4 ε 5 if largest energy is a gap, e.g., ϵ 3 J 2, J 3 : J 1 J 2 ~ J 3 J 4 J 1 J=J 2 J 3 /ε 3 J 4 site 3 is removed from the system bonds treated in 2nd order perturbation theory renormalized bond J = J 2 J 3 /ϵ 3 ε 1 ε 2 ε 3 ε 4 ε 5 if largest energy is a bond, e.g., J 2 ϵ 2, ϵ 3 : J 1 J 2 J 3 ~ ε=ε 2 ε 3 /J 2 J 4 rotors of sites 2 and 3 are parallel replaced by single rotor, moment µ = µ 2 + µ 3 renormalized gap ϵ = ϵ 2 ϵ 3 /J 2 J 1 J 3 flow equations for probability distributions P (J) and R(ϵ) P Ω R Ω = [P (Ω) R(Ω)] P + R(Ω) = [R(Ω) P (Ω)] R + P (Ω) ( dj 1 dj 2 P (J 1 )P (J 2 ) δ J J ) 1J 2 Ω ( dϵ 1 dϵ 2 R(ϵ 1 )R(ϵ 2 ) δ ϵ ϵ ) 1ϵ 2 Ω

32 Fixed points If bare distributions do not overlap: ln ϵ > ln J : no clusters formed disordered phase ln ϵ < ln J : all sites connected ordered phase If bare distributions do overlap: ln ϵ > ln J : rare clusters disordered Griffiths phase ln ϵ < ln J : rare holes ordered Griffiths phase ln ϵ = ln J : cluster aggregation and decimation balance at all energies critical point P( J) P( J) P( J) R( R( P( J) R( R( Disordered J, Disordered Griffiths J, Critical point J, P(ζ) = 1 Γ e ζ/γ, R(β) = 1 Γ e β/γ Ordered Griffiths log. variables ζ = ln(ω/j), β = ln(ω/ϵ), Γ = ln(ω 0 /Ω) Distributions become infinitely broad infinite-randomness critical point R( P( J) J, Ordered J,