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Quantum Critical Behavior of Disordered Itinerant Ferromagnets D. Belitz – University of Oregon, USA T.R. Kirkpatrick – University of Maryland, USA M.T. Mercaldo – Università di Salerno, Italy S.L. Sessions – University of Oregon, USA

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Introduction: the relevance of quantum fluctuations phenomenology of QPT Quantum critical behavior of disordered itinerant FM Theoretical backgrounds: Hertz Theory Disordered Fermi Liquid The new approach Microscopic theory Derivation of the effective action Behavior of observables Experiments Conclusion

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QPT = Phase transitions at T=0, driven by a non- thermal control parameter quantum mechanical fluctuation are relevant Experiments have revealed that rich new physics occurs near quantum critical points

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Quantum criticality 2 yellowquantum critical region QCP In the yellow region (quantum critical region) the dymical properties fo the system are influenced by the presence of the QCP Dashed lines are crossover lines Fig. by T. Vojta from cond-mat/

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Quantum Critical Behavior of Itinerant Ferromagnets The FM transition in itinerant electron systems at T=0 was the first QPT to be studied in detail [Hertz PRB 14, 1165 (1976)] Hertz theory: due to the mapping d d+z, the transition in the physically interesting dimension is mean field like. This conclusion is now known to be incorrect The reason for the breakdown of the mean-field theory is the existence, in itinerant electron systems, of soft or massless modes other than the order parameter fluctuations.

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Disordered Fermi Liquid Disordered e-e correlations lead to non analyticities in electron systems. 3D The conductivity has a -temperature dependence [Altshluler, Aronov] The density of states has a -energy dependence [Altshuler, Aronov] The phase relaxation time has a 3/2 –energy dependence [Schmid] These effects are known as weak localization effects

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Starting Model We keep explicitly all soft modes Critical modes (magnetization ) Diffusive modes (particle-hole excitations which exist in itinerant electron systems at T=0) Belitz, Kirkpatrick, Mercaldo, Sessions, PRB 63, (2001); ibidem (2001)

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Interaction term S int =S int (s) + S int (t) Using the Hubbard-Stratonovich transformation we decouple the S int (t) term and introduce explicitly the magnetization in the problem

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Rewrite the fermionic degrees of freedom in terms of bosonic matrix fields This formulation is particularly well suited for a separation of soft and massive modes Write explicitly soft and massive modes Integrate out massive modes A eff = A GLW [M] + A NL M [q] + A c [M,q] details

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power counting analysis of the effective action Ma's method to identify simple fixed points We use physical arguments to determine which coupling constants should be marginal, and then check whether this choice leads self consistently to a stable fixed point Hertz fixed point is unstable New fixed point marginally stable N.B. there are 2 time scales in the problem : the diffusive time-scale (z d =2)and the critical one (z c =4 or d) pert. theory

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b RG length rescaling factor The values of critical exponents are reflected in the behavior of the single particle density of states and the electrical conductivity across the transition Power laws with scale dependent critical exponents Asymptotic critical behavior is not given by simple power laws

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Density of states At T=0 N( F + )=N F [1+ m i +i0 ] To obtain the dependence at T 0 one can use scaling theory: The leading correction to N, N, can be related to a correlation function that has scale dimension – ( d – 2 ) Expect scaling law: N(t, ,T) = b -(d-2) F N (tb 1/, b z c,Tb z c ) d = 3 n

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conductivity At T=0 =8 /G To obtain the dependence at T 0 one can use scaling theory: consist of a backgorund part that does not scale (since this quantity is unrelated to magnetism [ ] = 0 ) and a singular part that does [ ] = – (d – 2) unrelated to magnetism [ ] = 0 perturbation theory depends on critical dynamics ( z ) and on leading irrelevant operator u u is related to diffusive electrons [u] = d – 2 Expect scaling law: (t,T,u) = b -(d-2) F (tb 1/, Tb z d, hb z c ) d = 3

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Exp 1 Fe 1-x Co x S 2 DiTusa et al cond-mat/0306 Measure of conductivity for different applied fields For H=0, ~ T 0.3 The unusual T dependence of is a reflection of the critical behavior of the ordering spins This compound shows a PM-FM QPT for x~0.032 Disordered Itinerant FM

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Fe 1-x Co x S 2 DiTusa et al Scaling plot of the conductivity Exp 1

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Thermodynamics quantities The thermodynamic properties near the phase transition can all be obtained by a scaling ansatz for the free energy. Natural scaling ansatz for f (free energy density): f(t,T,h) = b -(d+z c ) f 1 (tb 1/, Tb z c, hb z c ) + b -(d+z d ) f 2 (tb 1/, Tb z d, hb z c ) m t =2 m h 1/ = z c /2 C T 1+ = –d/z c s t = 1

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Quantum critical behavior for disordered itinerant ferromagnets has been determined exactly Measurements of conductivity and density of states in the vicinity of the quantum critical point are the easiest way to experimentally probe the critical behavior Feedback of critical behavior on weak-localization corrections of relevance for (indirect) measurements of ferromagnetic quantum critical behavior understanding breakdown of Fermi liquid behavior in the vicinity of QCPs Conductivity and DOS acquire stronger corrections to Fermi-liquid behavior

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