Universally diverging Grüneisen parameter and magnetocaloric effect close to quantum critical points

Transcription

1 united nations educational, scientific and cultural organization the abdus salam international centre for theoretical physics international atomic energy agency SMR Workshop on Novel States and Phase Transitions in Highly Correlated Matter July Universally diverging Grüneisen parameter and magnetocaloric effect close to quantum critical points Markus GARST Institut für Theorie der Kondensierten Materie Universität Karlsruhe Postfach Karlsruhe GERMANY These are preliminary lecture notes, intended only for distribution to participants strada costiera, trieste italy - tel fax sci_info@ictp.trieste.it -

2 Universally diverging Grüneisen parameter and magnetocaloric effect close to quantum critical points Markus Garst Achim Rosch (Universität zu Köln) Lijun Zhu and Qimiao Si (Rice University)

3 Outline Introduction quantum phase transitions Thermodynamical quantities Scaling analysis Expectations & experiment Summary workshop Trieste July 24 [1]

4 Phase diagram near a quantum critical point (QCP) Quantum fluctuations induce 2 nd order phase transition at zero temperature. controlled by pressure r = p p c p c doping r = x x c x c magnetic field r = H H c H c depending on the material. control parameter r quantum critical point <markus@tkm.uni-karlsruhe.de> workshop Trieste July 24 [2]

5 Why is the quantum phase transition interesting? Singular QCP determines the physics in its vicinity, even at finite temperatures! Thermodynamics can probe its properties! HOW? this talk! QCP control parameter r <markus@tkm.uni-karlsruhe.de> workshop Trieste July 24 [3]

6 Why is the quantum phase transition interesting? QCP may be the endpoint of a line, T c (r), of finite temperature, i.e., classical phase transitions. Finite temperature transition has its own criticality; differs qualitatively from the quantum phase transition at T =! belongs to different universality class. ordered phase T c (r) disordered QCP phase control parameter r <markus@tkm.uni-karlsruhe.de> workshop Trieste July 24 [4]

7 Can we nevertheless discern the QC contribution to thermodynamics? YES! Classical criticality is additive; sits on top of the quantum critical background. We are interested in this BACKGROUND! Classical critical divergencies dominate over the QC background only within a tiny wedge housing the phase boundary. classical regime ordered phase disordered phase QCP control parameter r <markus@tkm.uni-karlsruhe.de> workshop Trieste July 24 [5]

8 What does thermodynamics probe? Variations of the free energy F (r, T ) along the two directions of the phase diagram: control parameter r and. Derivatives probe the sensitivity n T n F (r, T ) m r m F (r, T ) n+m T n rmf (r, T ) wrt variation of T wrt variation of r wrt variation of both. control parameter r <markus@tkm.uni-karlsruhe.de> workshop Trieste July 24 [6]

9 QPT driven by pressure p magnetic field H 2 F T 2 specific heat coeff. γ specific heat coeff. γ 2 F r T thermal expansion α = 1 V V T p temperature dependence of magnetization: M T H 2 F r 2 compressibility κ = 1 V V p T differential susceptibility M H T <markus@tkm.uni-karlsruhe.de> workshop Trieste July 24 [7]

10 Important in the following: Single prefered direction to approach the classical phase transition. = e.g. specific heat, C p, and thermal expansion, α, diverge at the classical transition with the same exponent α! α C p T T c α control parameter r <markus@tkm.uni-karlsruhe.de> workshop Trieste July 24 [8]

11 Important in the following: Two (orthogonal) directions to approach the QCP. Specific heat C p and thermal expansion α yield complementary information about the QPT! control parameter r <markus@tkm.uni-karlsruhe.de> workshop Trieste July 24 [9]

12 Grüneisen parameter quotient of thermal expansion α = 1 V V T = 1 p V 2 F T p = 1 V S p T Γ = α C p and molar specific heat at constant pressure C p = T N A 2 F T 2 = T N A γ = T N A S T p <markus@tkm.uni-karlsruhe.de> workshop Trieste July 24 [1]

13 If physics is dominated by single energy scale E : scaling form of the molar entropy S N A = Ψ ( T E ) Γ = N A V T V m : molar volume S/ p T = 1 E S/ T p V m E p usually E varies slowly with pressure, for example for phonons E = ω D = Grüneisen law: Γ const. HOWEVER, violated near a quantum critical point! Typical energy scale vanishes: ω c ξ z. <markus@tkm.uni-karlsruhe.de> workshop Trieste July 24 [11]

14 Scaling analysis Upon scaling the unit length by l: scaling dimension Definition of a QCP: x x = x l 1-1 r r = r l 1/ν 1/ν T T = T l z z F F = F l φ φ = d + z free energy F : scale invariant physics independent of microscopic details F (r, T ) l d+z F(r, T)! = F(r l 1/ν, T l z ) <markus@tkm.uni-karlsruhe.de> workshop Trieste July 24 [12]

15 Scaling dimension of the Grüneisen parameter: Γ(r, T) 1 T 2 F (r, T ) r T 2 F (r, T ) T 2! = 1 T 2 l (d+z) F (r l 1/ν, T l z ) r T 2 l (d+z) F (r l 1/ν, T l z ) T 2 l 1/ν Γ(r l 1/ν, T l z ) Scaling dimension of Γ equals minus the scaling dimension of the control parameter r: dim[γ] = dim[r] <markus@tkm.uni-karlsruhe.de> workshop Trieste July 24 [13]

16 Consequences: Γ diverges at the QCP! 2 low T regime quantum critical regime 1 control parameter r 2 T r νz low T regime 1. Quantum critical regime choose scale T l z = 1 Γ(r, T) = 1 T 1 νz 1 T 1 νz 2. Low-T regime choose scale r l 1/ν = 1 Γ(rT 1 νz, 1) Γ(, 1) Γ(r, T) = 1 r Γ(sign(r), T r νz ) 1 r Γ(sign(r), ) <markus@tkm.uni-karlsruhe.de> workshop Trieste July 24 [14]

17 Experiment I: Heavy fermion compound CeNi 2 Ge 2 QCP located at ambient pressure thermal expansion measurements feasible C / T (Jmol -1 K -2 ) Γ CeNi 2 Ge R. Küchler, P. Gegenwart et al. PRL (23) T (K) T (K) α / T (1-6 K -2 ) B (T) Γ cr 1 x= T (K) 5 CeNi 2 Ge 2 α, B II a γ cr T & α cr T 1 T = Γ cr 1 T = T 1 νz 3D-AF SDW in the quantum critical regime: νz = 1 T (K) <markus@tkm.uni-karlsruhe.de> workshop Trieste July 24 [15]

18 Experiment II: Heavy fermion compound YbRh 2 Si C el / T (J / K 2 mol) Γ cr 1 1 YbRh 2 (Si.95 Ge.5 ) 2 x =.7 x = 1.1 T (K) β / T (1-6 K -2 ) in the quantum critical regime: νz 1.4 (beware: energy scale 3mK) T (K) R. Küchler, P. Gegenwart et al. PRL (23) Does not conform to standard Hertz theory! <markus@tkm.uni-karlsruhe.de> workshop Trieste July 24 [16]

19 Critical Point vs. Critical Line Disorder might wash out QCP to a quantum critical line? critical behaviour along a finite pressure interval quantum critical line along the line: control parameter marginal zero scaling dimension Γ cr log T control parameter r Grüneisen can diverge at most logarithmically! Algebraic divergence exludes quantum critical line scenario. Divergence of Γ criterion for the existence of a QCP! <markus@tkm.uni-karlsruhe.de> workshop Trieste July 24 [17]

20 Universality in the low-t regime choose r l 1/ν = 1 in the scaling form, F (r, T ) = l (d+z) F (r l 1/ν, T l z ), for the free energy: F (r, T ) = r ν(d+z) f( T r νz ) entropy: S(r, T ) = r νd f ( T r νz ) constraint by the third law of thermodynamics: f (x) c x y no constant contribution and y > thermal expansion with r = (p p c )/p c specific heat α = 1 S V m p cν(d y z) V T m p c r ν(d yz) 1 T y C p = T S T p cy r ν(d yz) T y universal exponents, non universal prefactors <markus@tkm.uni-karlsruhe.de> workshop Trieste July 24 [18]

21 In the low-t regime: absence of a residual entropy leads to Universal Grüneisen parameter V m molar volume Γ = α C p = G r 1 V m (p p c ) prefactor G r given by critical exponents: G r = ν(d y z) y e.g. SDW transition G r = d z 2 for a gapped system, C p e /T, effectively y : G r = νz. Assumption of scaling predicts not only exponent, but also prefactor! <markus@tkm.uni-karlsruhe.de> workshop Trieste July 24 [19]

22 Constant entropy curves ds = S p = Γ = α T γ = 1 V m T dp + S T T dt = V m α dp + γ dt =! p dt dp S Grüneisen parameter measures pressure-caloric effect for a magnetic field driven QPT: role of the Grüneisen parameter is taken over by the magnetocaloric effect Γ = 1 T T H S <markus@tkm.uni-karlsruhe.de> workshop Trieste July 24 [2]

23 Constant entropy curves: Sign change of Γ! Near the QCP: system has to decide between two different groundstates. = Accumulation of entropy near the quantum critical point. Scenario I Scenario II Γ sign change at T c Γ sign change above QCP ds = ds= control parameter r control parameter r <markus@tkm.uni-karlsruhe.de> workshop Trieste July 24 [21]

24 Scenario I: Dilute Bose Gas in d = 3 S = dx ( φ (x) shows QPT as a function of µ [ τ 2 µ ] φ(x) + u2 φ(x) 4 ) scaling exponents: z=2, d=3, ν = 1 2 condensed phase symmetric phase symmetric phase: gapped spectrum condensed phase: Goldstone boson linear spectrum chemical potential µ applicable to TlCuCl 3 Bose-Einstein condensation of magnons <markus@tkm.uni-karlsruhe.de> workshop Trieste July 24 [22]

25 Hartree-Fock-Bogoliubov approximation:,5 µ =.2,6 µ =.1 µ =.2 specific heat coeff. γ,4,3,2,1 µ = µ =.1 µ =.2 µ =.1 µ =.2 µ =.2 e µ/t T 2,2,4,6,8,1 thermal expansion α,4,2 -,2 -,4 µ =.2 µ =.1 µ =,2,4,6,8,1 Pronounced sign change of the thermal expansion at T c 1 st order jump at finite T c : artefact of the approx. <markus@tkm.uni-karlsruhe.de> workshop Trieste July 24 [23]

26 Grüneisen parameter of the dilute Bose gas µ Γ 1 µ =.2 1 Grüneisen parameter Γ 5-5 µ =.1 µ =.2 µ =.1 µ = ,2,4,6,8,1 rescaled parameter saturate at the universal values: symmetric phase: gapped spectrum y = G r = νz = 1 condensed phase: linear spectrum y = 3 = G r = ν(y z d) y = 1 2 <markus@tkm.uni-karlsruhe.de> workshop Trieste July 24 [24]

27 Scenario I: Heavy fermion compound CeCu 6 x Au x T (K) 1..5 PM disordered phase AF ordered phase x A. de Visser, K. Grube, H. v. Loehneysen <markus@tkm.uni-karlsruhe.de> workshop Trieste July 24 [25]

28 Scenario II: Ising Chain in a Transverse Field H = J i σ z i σ z i+1 + h i σ x i as a function of magnetic field h: QPT between a magnetic and a paramagnetic ground state continuum theory: fermions with relativistic spectrum ɛ k = r 2 + k 2 where r = h h c h c free energy: F = T dk [ 2π log 2 cosh ɛ ] k 2T scaling exponents: z = d = ν = 1 control parameter r <markus@tkm.uni-karlsruhe.de> workshop Trieste July 24 [26]

29 Temperature sweeps at constant magnetic field: r =.1 r =.2 r =.3 1 r =.1 r Γ 1.8 γ 3/π 2.5 e r /T Γ 5 r =.2 r = dm dt h r =.3 r =.2 r =.1 r r.5 1 r =.1 r =.2 r = <markus@tkm.uni-karlsruhe.de> workshop Trieste July 24 [27]

30 Magnetic Field sweeps at constant temperature: γ 3/π T =.5 T =.1 T =.15 5 T =.5 T =.1 T = control parameter r 1 Γ.5 dm dt h.5 T =.15 T =.1 T = control parameter r control parameter r <markus@tkm.uni-karlsruhe.de> workshop Trieste July 24 [28]

31 Scenario II: Heavy fermion compound CeRu 2 Si 2 C. Paulsen JLTP (199). <markus@tkm.uni-karlsruhe.de> workshop Trieste July 24 [29]

32 Restrictions A simple scaling ansatz might fail. 1. If there are more critical time scales e.g. spin fluctuations and fermionic quasiparticles. local QCP 2. above the upper critical dimension dangerously irrelevant operator may spoil scaling explicit calculation necessary! 3. scaling exponent resonances 4. Scaling analysis yields only critical part! CAUTION: possibly non-critical contributions <markus@tkm.uni-karlsruhe.de> workshop Trieste July 24 [3]

33 Hertz theory of itinerant magnetism describes spin density wave (SDW) instability of certain heavy fermions S = 1 βv ω n,k 1 2 ΦT [ δ + ξ 2 k 2 + ω ] n T k z 2 Φ + u Φ 4 e.g. in the quantum critical regime for d=3 d = 3, z = 2 d = 3, z = 3 α cr T 1/2 T 1/3 c cr T 3/2 T log 1 T logarithmic corrections to scaling for d=z=3 resonance Γ cr T 1 T 2/3 log 1 T 1 <markus@tkm.uni-karlsruhe.de> workshop Trieste July 24 [31]

34 Summary: PRL 91 (23) Grüneisen parameter and magnetocaloric effect diverge at the QCP in the low-t regime universal behaviour thermal expansion more divergent than specific heat criterion for the existence of a simple QCP characterization of universality class: ν, z provides mapping of the entropy landscape <markus@tkm.uni-karlsruhe.de> workshop Trieste July 24 [32]