1. The Ferromagnetic Quantum Phase Transition in Metals Dietrich Belitz, University of Oregon Ted Kirkpatrick, University of Maryland Kwan-yuet Ho Maria -Teresa Mercaldo Rajesh Narayanan Jörg Rollbühler Ronojoy Saha Yan Sang Sharon Sessions Sumanta Tewari Acknowledgments: Manuel Brando Malte Grosche Gil Lonzarich Christian Pfleiderer Greg Stewart Achim Rosch John Toner Thomas Vojta

2. Outline Lecture 1: Lecture 2: Lecture 3: 1.Motivation: Why Quantum Ferromagnets are Interesting 2.Classical Phase Transitions 1.Quantum FM Transitions: General Concepts 2.The Quantum FM Transition, Part I: History 3.The Quantum FM Transition, Part II: General Guidelines 4.The Fermi Liquid as an Ordered Phase 1.The Quantum FM Transition, Part III a.Ferromagnets b.Liquid-gas transition c.Superconductors, and liquid crystals a.Generalized Landau theory b.Order-parameter fluctuations c.Effects of quenched disorder a.A useful example: Classical 4 – theory b.Goldstone modes in a Fermi liquid

3. Lecture 4: 8.Exponents and Exponent Relations at Quantum Critical Points 9.“How Close is Close to the Critical Point?”, or How Hard is it to Measure Quantum Critical Exponents? 10. Phase Separation Away from the Coexistence Curve

4. Lecture 1 Chennai Lectures 2016

5. 1. Motivation: Why Quantum Ferromagnets are Interesting Q: What happens to a FM phase transition when the Curie temperature is very low? A: Lots of unexpected and strange behavior. For instance:  The transition changes from second order to first order ZrZn 2 (Uhlarz et al 2004) MnSi (Uemura et al 2007) Discontinuous magnetization Phase separation

6. Chennai Lectures 2016  The phase diagram develops an interesting wing structure UGe 2 (Kotegawa et al 2011) UCoAl (Aoki et al 2011)

7. Chennai Lectures 2016 (Sang et al 2014)  Quenched disorder suppresses these effects …  … leading to a continuous quantum phase transition … Ni 3 Al 1-x Ga 2 (Yang et al 2011)

8. Chennai Lectures 2016 … and for strong disorder to a quantum Griffiths region (Pikul 2012)(Westerkamp et al 2009)

9.  Non-Fermi-liquid transport behavior is observed in large regions of the phase diagram Similar behavior is seen in many other quantum FMs (Takashima et al 2007) Chennai Lectures 2016 These lectures discuss some of these interesting effects. For a recent review, see M. Brando, DB, F.M. Grosche, TRK arXiv:1502.02898 Rev. Mod. Phys., in press

10. 2. Classical Phase Transitions a. Ferromagnets  The ferromagnetic transition in Fe, Ni, and Co is one of the best known examples of a thermal phase transition.  The material is a paramagnet at high temperatures, but spontaneously develops long-range ferromagnetic order if cooled below the Curie temperature T c.if cooled below the Curie temperature  This transition in zero field is 2 nd order; i.e. the order parameter (= magnetization) is a continuous function of temperature, but not analytic at T = T c :  Mean-field theory (see below) qualitatively describes the data.  The transition at T < T c as a function of a magnetic field is first order, i.e., the order parameter changes discontinuously.  Phase diagram: Chennai Lectures 2016 Ni (Weiss & Forrer 1926)

11.  50 years later: The true critical behavior is M ~ (T c – T) β with β = 0.358 ± 0.003  Order-parameter fluctuations invalidate mean-field theory near criticality in d=3, but NOT in a hypothetical system in d > 4 (Ginzburg criterion, “upper critical dimension”, see below).  Other critical quantities:  Exponent values depend only on the dimensionality and general properties (e.g., Ising vs Heisenberg), NOT on microscopic details (“universality classes”). Chennai Lectures 2016 Ni (Weiss & Forrer 1926) Cohen & Carver 1977 o Susceptibility ~ |T – T c | -γ with γ ≈ 1.4 o Correlation length ξ ~ |T – T c | -ν with ν ≈ 0.7

12. Chennai Lectures 2016  Theoretical explanation: Scaling, and the RG (Widom, Kadanoff, Wilson, Fisher, Wegner).  An important point: Scaling and the RG can be used to describe entire phases, not just critical points (“stable fixed points”, (e.g., Ma 1976) o Thermal fluctuations drive the critical singularities. o Observables obey homogeneity laws. E.g., with t = |T – T c |/T c and h the magnetic field, x x with b > 0 arbitrary vs curves for various T will collapse onto ONE function with two branches if the axes are scaled appropriately! o This actually works (see next slide):

13. Chennai Lectures 2016 CrBr 3 (Ho & Litster 1969)

14.  “Action” (= Hamiltonian/T), partition function, free energy density for a classical FM:  Mean-field approximation,, Landau free energy 2 nd order transition with mean-field exponents Chennai Lectures 2016

15. Focus on the susceptibility:  Homogeneous susceptibility:  Generalization to k > 0: x (Ornstein-Zernicke) x x correlation length  diverging length scale, LR correlations  Example of a “soft” or “massless” mode or excitation (here: a critical soft mode)  A time scale also diverges: (“critical slowing down”)  z is called “dynamical critical exponent”  Fluctuations 2 nd order transition with exponents in the appropriate universality class (Ising, XY, or Heisenberg)  Fluctuations are important only below an upper critical dimension (d < 4 in this case); Ginzburg criterion Chennai Lectures 2016

16. Lecture 2 Chennai Lectures 2016

17. b. Liquid-gas transition  The liquid-gas transition maps onto an Ising ferromagnet, but we usually get to see only the 1 st order transition.  Phase diagram:  The behavior near the critical point is in exact analogy to the ferromagnetic case. In particular, the correlation length diverges ( critical opalescence).  For T < T c one observes phase separation. (Magnetic analogy: MnSi experiment) Chennai Lectures 2016 CO 2 (Sengers & Levelt Sengers 1968)

18.  Now consider a superconductor with order parameter : x x same as XY ferromagnet  But, couples to the E&M vector potential (photons):  A comes with a gradient A-correlation function diverges as :  Photons by themselves are soft (“generic” soft mode), but a nonzero gives them a mass!  “Integrate out” the photons: x x x with an “effective” action Chennai Lectures 2016 c. Superconductors (and liquid crystals)

19.  Treat in mean-field approximation The effective Landau free energy is nonanalytic in : x x (Halperin, Lubensky, x Ma 1974) x  “Fluctuation-induced” 1 st order transition! (NB: Fluctuations = generic soft modes!)  Nematic-Smectic-A transition in liquid crystals: Photons -> nematic Goldstone modes  Superconductors: effect is too weak to be observable; liquid Xtals: situation messy  Particle physics: “Coleman-Weinberg mechanism” Chennai Lectures 2016

20.  Some ferromagnets have a low T c that is often susceptible to hydrostatic pressure.  This raises the prospect of a quantum critical point at T c = 0.  Quantum critical behavior is driven by quantum fluctuations must be different from classical critical behavior.  Crossovers ensure continuity. 3. Quantum Phase Transitions: General Concepts Q: How different is the description of QPTs in general from that of classical transitions? A: Very different, due to fundamental differences in statistical mechanics. Classical: QM: x x x x x statics and dynamics decouple! From DB, TRK, T. Vojta, Rev. Mod. Phys. 2005 Chennai Lectures 2016

21.  Solution: Trotter formula (Trotter 1959)  Coherent-state formalism (Casher, Lurie, Revzen 1969)  Divide the interval into infinitesimal slices, and integrate  End result:  is referred to as “imaginary time” (Wick rotation )  Analytic continuation yields real-time (or frequency) dynamics Statics and dynamics do indeed couple  If, then a quantum system in d spatial dimensions resembles a classical system in d + z dimensions! In general, (Hertz 1976)  Upper critical dimension of classical system implies upper critical dimension of quantum system  Mean-field theory more robust in quantum case!

22.  Stoner (1937): Mean-field theory for both classical and quantum case Susceptibility: RPA (“spin screening”) with non-magnetic electrons and the relevant interaction constant Transition at (non-thermal control parameter)  Moriya et al (early 1970s): Self-consistent one-loop theory (aka self-consistent spin fluctuation theory)  Hertz (1976): Developed RG framework for QPTs in general, used metallic FMs as a prime example.  Millis (1993): Used Hertz’s RG framework to determine temperature scaling. Plausibility arguments for Hertz’s action (with a lot of hindsight): 4. The Quantum FM Transition in Metals, Part I: History Landau free energy (again): t is the inverse physical (dressed) susceptibility 1/ x Chennai Lectures 2016

23. RPA again: Generalize everything to nonzero wavevectors (Fourier trafo of ) and frequencies (Fourier trafo of ) x holds for noninteracting electrons x (Lindhard fct) AND for interacting ones action describes the dynamics of clean electrons (“Landau damping”) Represents coupling of conduction electrons dynamics to the magnetization At for classical magnets implies for quantum magnets  Conclusion (as of 1976): Mean-field theory yields exact critical behavior for all d > 1 ( QPTs not very interesting as far as critical behavior goes)

24. Chennai Lectures 2016  As mentioned before, this is not what is observed experimentally.  In most clean systems, the transition becomes first order if T c is suppressed far enough  Examples:  There are many others: ZrZn 2 Uemura et al 2007 ZrZn 2 (Uhlarz et al 2004) MnSi (Pfleiderer et al 1997, Uemura et al 2007) UGe 2 (Aoki et al 2011) Uhlarz et al 2004

25. Chennai Lectures 2016  There are many more examples (Brando et al, Rev Mod Phys in press)

26. Observations: A tricritical point separates the 2 nd order transitions from the 1 st order ones. In a magnetic field, tricritical wings appear: A quantum critical point is eventually realized, but only at a nonzero magnetic field! This behavior is seen in systems that are very different with respect to electronic structure, type of magnet (Ising, XY, Heisenberg), etc. The explanation must be universal, i.e., a lie in stat. mech., not in solid-state effects. Look for an explanation that involves x low-lying excitations (aka soft modes) These materials are all metals Conduction electrons likely important Study Fermi liquids first. Q: Don’t we know everything about Fermi liquids, and wasn’t that built into Hertz’s x action? A: NO! (Kotegawa et al 2011) Chennai Lectures 2016 5. The Quantum FM Transition, Part II: General Guidelines

27. Spoiler: The answer in a nutshell: A novel type of fluctuation-induced 1 st x order transition The leading wavenumber dependence of the inverse susceptibility that enters the action is x for 1 < d < 3 and x for d = 3 This is a consequence of soft modes in a Fermi liquid, see below Scaling suggests h ~ k, so this nonanalyticity is also present for at k = 0 as a function of h. The magnetization is seen by the conduction electrons as an effective field. In a generalized Landau theory, translates into This leads to a generalized Landau free energy x x with x which leads to a 1 st -order transition. Conclusion: In clean metals, generic soft modes couple to the ferromagnetic order parameter and make the quantum FM transition generically 1 st order. Chennai Lectures 2016 (DB, TRK, T. Vojta 1999)

28. Lecture 3 Chennai Lectures 2016

29.  O(3) ϕ 4 – theory in d spatial dimensions with a field in 1-direction:  Saddle-point solution for the ordered phase: with x  Fluctuations: 6. The Fermi Liquid as an Ordered Phase a. A useful example: Classical ϕ 4 – theory (again)

30.  Properties of the ordered phase: Magnetization Transverse susceptibility Longitudinal and transverse modes couple x nonanalyticities First derived in perturbation theory (Vaks, Larkin, Pikin 1967, x Brézin & Wallace 1973) o Exact result, governed by Ward identity o Transverse fluctuations are soft (Goldstone modes of the spontaneously broken SO(3) ) o Longitudinal fluctuations are massive Reduction due to fluctuations Potential for SO(2) ≅ U(1) (planar magnet) Chennai Lectures 2016 Long-ranged correlations!

31.  Expand in powers of fluctuations and gradients, assign scale dimensions, and look for a stable fixed point. The above behavior is exact for all d > 2 and can be described by a  Fixed-point action x dimensionless plus least irrelevant operators, e.g.  This suffices for deriving all scaling behaviors, e.g.  Nonanalyticities are leading corrections to scaling at the stable fixed point  d = 2 is lower critical dimension (Mermin-Wagner ✔ )  Derivation via RG methods: Chennai Lectures 2016

32. The shear viscosity in classical fluids has a nonanalytic frequency dependence since it couples to the diffusive transverse-velocity fluctuations. This is an example of classical long-time tails, i.e., non-exponential decay of correlation functions. The conductivity at T = 0 in a disordered metal is a nonanalytic function of the frequency This is an example of what are called weak-localization and Altshuler-Aronov effects in disordered metals. We will now apply analogous arguments to a clean Fermi liquid.  Note: Similar arguments can be used to exactly characterize nonanalyticities in various other systems, both classical and quantum. For instance,

33. the soft modes (analogous to π ) that These soft modes are the clean analogs of what are called “(spin)-diffusons” in disordered systems, where they lead to weak-localization and Altshuler-Aronov effects. (They are NOT density fluctuations.) o are controlled by a Ward identity o are Goldstone modes of a spontaneously broken symmetry o have a linear dispersion relation o acquire a mass at T > 0 and h > 0 Chennai Lectures 2016 (i) The stable Fermi-liquid fixed point  A long story (Wegner 1970s, TRK & DB 1997, 2012) made very short:  In a Fermi system at T = 0, there are two-particle excitations that are  The Fermi liquid is an ordered phase characterized by b. Goldstone modes in a Fermi liquid

34. an order parameter given by the DOS at the Fermi surface (more precisely: By the spectrum of the Green’s function): an order-parameter susceptibility (analogous to ) a stable RG fixed point with fixed-point action all other terms are x RG irrelevant! a Scale dimensions Least irrelevant operators with scale dimension Chennai Lectures 2016  This allows for the derivation of homogeneity laws that yield the exact leading nonanalyticities in a Fermi liquid!

35.  DOS:  Spin susceptibility: analogous to agrees with perturbative results (Khveshchenko & Reizer 1998) Jiang et al 2006 Chennai Lectures 2016 (ii) Universal properties of the Fermi liquid  first derived in perturbation theory (DB, TRK, T Vojta 1997; Betouras et al 2005)  NB the sign of the effect!  Soft modes long-ranged correlations nonanalytic behavior

36.  Ordinary Landau theory: with t = 1/ the inverse susceptibility  Now recall the argument given earlier:  The FL Goldstone modes lead to a generalized Landau theory: acts as an effective field in a metal, the FL Goldstone modes couple to via a Zeeman term via the nonanalytic h – dependence of ! x Chennai Lectures 2016 7. The Quantum Ferromagnetic Transition, Part III a. Generalized Landau theory The quantum ferromagnetic transition in clean metals in zero field is generically 1 st order !

37.  T > 0 gives Goldstone modes a mass x tricritical point  Magnetic field h > 0 tricritical wings  Generic phase diagram:  Predicted to hold for all clean metallic  Third Law General constraints on the shape of the phase diagram (see my Colloquium talk)  Pre-asymptotic region: Crossover to Hertz-Millis-Moriya behavior. Example: MnSi  Comparison with experiments: Excellent qualitative agreement. The transition in clean materials is generically 1 st order. In some systems (e.g., Ni x Pd 1-x ) the transition needs to be followed to lower T. Chennai Lectures 2016 Ferromagnets (isotropic or anisotropic, itinerant or not, and even Kondo lattices) Ferrimagnets (only requirement is a homogeneous magnetization component) Magnetic nematics (DB, TRK, J Rollbühler 2005) (Pfleiderer et al 1997) Other consequences:

38.  Technical derivation: Coupled field theory for fermions and OP fluctuations. Q: How good or bad is the mean-field approximation? A: Nobody really knows, but: Treat conduction electrons in a tree approximation Hertz’s action 1-loop order nonanalyticities appear Higher order: Only prefactors change Chennai Lectures 2016 OP fluctuations are above their upper critical dimension In liquid crystals, they are below the upper critical dimension This makes it plausible that the 1 st order transition in quantum ferromagnets is much more robust than in liquid crystals. b. Order-parameter fluctuations Other suggestions for avoiding a quantum critical point : Textured phases (spiral, etc), with the length scale set by the maximum in (Chubukov et al, Green et al)

39.  The generic soft modes are now diffusive  The susceptibility now is x x (Altshuler et al early 1980s) NB the sign! x and by the same arguments as before we have, in d = 3, (TRK & DB 1996)  This predicts a 2 nd order transition with non-mean-field exponents.  For instance,, and, consistent with many observed phase diagrams (see below for other interpretations)  Order-parameter fluctuations lead to log-normal modifications of the power laws (DB et al 2001)  Increasing the disorder from the clean limit decreases the tricritical temperature continuously (Pikul et al 2012) Quenched disorder changes things drastically: Chennai Lectures 2016 c. Effects of quenched disorder

40.  Prediction for evolution of phase diagram:  Quantum Griffiths effects (McCoy & Wu 1968, D.S. Fisher 1995)  Based on the idea of the classical Griffiths region below the clean T c. Diverging susceptibilities without long-range order.  Quantum version may coexist with, and be superimposed on, quantum critical behavior (Millis et al, Randeria et al, T. Vojta, …)  Experiments have been interpreted using these concepts, e.g. Chennai Lectures 2016 (Pikul 2012) (Westerkamp et al 2009) (Sang et al 2014) Complications for strong disorder:

41. Lecture 4 Chennai Lectures 2016

42. 8. Exponents and Exponent Relations at Quantum Critical x Points Now consider a continuous QPT, e.g. the FM one in Ni 3 Al 1-x Ga 2 : A classical critical point is characterized by power-law behavior of observables, and corresponding critical exponents: Order parameter: Order-parameter susceptibility: Correlation length: Specific heat: Chennai Lectures 2016

43. A quantum critical point we can approach at T = 0 by varying the non-thermal control parameter t, or at t = 0 by letting T -> 0.  We need more critical exponents! Order parameter: Order-parameter susceptibility: Correlation length: The specific heat vanishes at T = 0 => Consider the Specific-heat coefficient: Chennai Lectures 2016

44. The T- exponents will in general be different from the t- exponents! What is the relation? Various points towards an answer:  In quantum statistical mechanics, inverse temperature is related to (imaginary) time. => T- scaling is governed by a dynamical exponent z that describes how the relaxation time diverges as a function of the relaxation length:  Power laws result from generalized homogeneity laws for observables : with b an arbitrary length rescaling factor and a scaling function x (and r = t, sorry!). Put b = r -ν => At r = 0 we have with  Conclusion: If describes the static scaling behavior, then describes the temperature scaling behavior  Caveat: This is true only under special circumstances. More generally, the T- scaling of may be described by A z, rather than by THE z, due to multiple time scales and/or dangerous irrelevant variables. Chennai Lectures 2016

45. That’s a lot of exponents. How many are independent? First consider a classical transition. Look at the homogeneity law for the order parameter: => ✓ ✓ Now, the susceptibility is a thermodynamic derivative of : This yields => => “Widom’s equality” Similar arguments lead to, e.g., “Essam-Fisher equality” “Fisher equality” Chennai Lectures 2016

46. Some exponent relations depend explicitly on the dimensionality, e.g. “hyperscaling” They are not valid if the dimensionality is larger than the “upper critical dimension”. For many classical transitions, that’s d = 4. Classically, only two exponents are independent. These exponent relations all depend on various forms of scaling being valid. While that’s usually the case at critical points, there is no guarantee. Weaker statements that depend only on thermodynamic stability take the form of rigorous inequalities. For instance, “Rushbrooke inequality” In disordered systems, a rigorous lower bound on the correlation length exponent is known: Chayes, Chayes, Spencer, Fisher (1986) (See also the “Harris criterion”).

47. Chennai Lectures 2016 At quantum critical points, some of the classical relations still hold, e.g., Widom Fisher Analogous relations hold for the T- exponents: Others change. For instance, a generalization of Essam-Fisher becomes where and are the dynamical exponents that govern the T – dependence of the order parameter and the specific heat, respectively. These two are in general NOT the same! The rigorous Rushbrooke inequality holds for the T – exponents: but gets modified for the t – exponents:

48. Hyperscaling relations, e.g., again hold only below some upper critical dimensionality, which tends to be lower for quantum phase transitions than for classical ones. Overall, at a quantum critical point there are at least two independent static critical exponents, plus the dynamical ones. Depending on what form of scaling holds (which depends on the critical point and the dimensionality) there may be as many as five independent static exponents. For details, see Chennai Lectures 2016

49. 9. “How Close is Close to the Critical Point?”, or x How Hard is it to Measure Quantum Critical Exponents? Q: How close to the critical point does one have to go in order to observe asymptotic critical behavior? A: It depends on the critical point and the observable, but in general very close. At many classical critical points one needs to be closer than 0.1%, and then one needs two or three decades to convincingly see the power laws! Plus, the extracted exponent values can depend strongly on the value of T c ! Chennai Lectures 2016

50. Farther away from criticality one often observes effective power laws with exponents that are controlled by unstable RG fixed points. There is no reason to believe that requirements at quantum critical points are less stringent. Various issues:  Many quantum critical points are controlled by chemical composition. This is hard to control, and the critical concentration is typically not known very precisely.  Additional physics can mask quantum criticality, e.g., quantum Griffiths effects in many disordered quantum ferromagnets. Chennai Lectures 2016

51. Some examples:  Ni 3 Al 1-x Ga 2 again: The observed behavior is in agreement with Hertz-Millis theory. There are good theoretical reasons to believe that this is not the asymptotic critical behavior. For instance,, which violates the rigorous bound. Chennai Lectures 2016

52. Butch & Maple (2009)  URu 2-x Re x Si 2 : Critical concentration known to not better than 10%, observed exponents almost certainly not critical exponents. Additional physics is known to be present (Hidden Order, Quantum Griffiths effects?) Chennai Lectures 2016

53. 10. Phase Separation Away from the Coexistence Curve Return to the observed phase separation in systems that show a 1 st order transition. PS away from the coexistence curve! Observed by various techniques (μSR, NMR, NQR) Observed in many different systems: QFMs, heavy-fermion systems, rare-earth nickelates (Mott transition), etc. Schematic phase diagrams: Chennai Lectures 2016 Conceivable explanations: Non-equilibrium effects (unlikely, see MnSi samples) Droplet formation due to quenched disorder TRK & DB arXiv:1602.01447

54. Chennai Lectures 2016 Summary  The quantum ferromagnetic transition in clean metals is generically 1 st order.  This can be understood as a fluctuation-induced 1 st order transition due to the coupling of the magnetization to generic soft modes in metals (clean versions of diffusons).  The Fermi liquid can be understood as an ordered phase with a broken symmetry, with the generic soft modes as Goldstone modes.  At nonzero temperature there is a tricritical point, and tricritical wings.  Good agreement between theory and experiments.  Quenched disorder changes a crucial sign, leading to a 2 nd order transition. Strong disorder leads to additional complications.  Additional critical exponents are needed for QCPs.  True critical behavior is hard to observe. In classical systems it took decades to obtain reliable exponents. For QCPs, experiments are probably still far from the necessary level of precision.  Phase separation away from a coexistence curve requires an explanation. Suggestion: Droplet formation due to quenched disorder.

55. Chennai Lectures 2016 Recommended Reading Quantum Ferromagnets: M. Brando et al, arXiv:1502.02898, Rev. Mod. Phys., in press Generic Scale Invariance: DB, TRK, Thomas Vojta, Rev. Mod. Phys. 77, 579 (2005) Quantum Phase Transitions: J. Hertz, Phys. Rev. B 14, 1165 (1976) S. Sachdev, Quantum Phase Transitions (Cambridge Univ. Press 1999) Scaling, and Renormalization Group: H.E. Stanley, Introduction to Phase Transitions and Critical Phenomena (Oxford Univ. Press 1971) S.-K. Ma, Modern Theory of Critical Phenomena (Perseus 1976) M.E. Fisher, in Advanced Course on Critical Phenomena, F.W. Hahne (ed.) (Springer 1983) J. Cardy, Scaling and Renormalization in Statistical Physics (Cambridge Univ. Press 1996)