Three Lectures on Soft Modes and Scale Invariance in Metals. Quantum Ferromagnets as an Example of Universal Low-Energy Physics

Transcription

1 Three Lectures on Soft Modes and Scale Invariance in Metals Quantum Ferromagnets as an Example of Universal Low-Energy Physics

2 Soft Modes and Scale Invariance in Metals Quantum Ferromagnets as an Example of Universal Low-Energy Physics Dietrich Belitz, University of Oregon with T.R. Kirkpatrick and T. Vojta Reference: Rev. Mod. Phys. 77, 579, (2005) Part I: Phase Transitions, Critical Phenomena, and Scaling Part II: Soft Modes, and Generic Scale Invariance Part III: Soft Modes in Metals, and the Ferromagnetic Quantum Phase T Transition

3 Part 1: Phase Transitions I. Preliminaries: First-Order vs Second-Order Transitions Example 1: The Liquid-Gas Transition Schematic phase diagram of H 2 O Singapore Winter School 3

4 Part 1: Phase Transitions I. Preliminaries: First-Order vs Second-Order Transitions Example 1: The Liquid-Gas Transition Schematic phase diagram of H 2 O Singapore Winter School 4

5 Part 1: Phase Transitions I. Preliminaries: First-Order vs Second-Order Transitions Example 1: The Liquid-Gas Transition T < T c : 1 st order transition (latent heat) Schematic phase diagram of H 2 O T > T c : No transition T = T c : Critical point, special behavior Singapore Winter School 5

6 Example 2: The Paramagnet - Ferromagnet Transition H = 0: Transition is 2 nd order T > T c : Disordered phase, m = 0 T < T c : Ordered phase, m 0 T -> T c : m -> 0 continuously m is called order parameter Examples: Ni T c = 630K Fe T c = 1,043K ZrZn 2 T c = 28.5K UGe 2 T c = 53K Demonstration of the FM critical point Singapore Winter School 6

7 2 nd order transitions, a.k.a. critical points, are special! The OP goes to zero continuously: m(h=0) ~ (T c - T) β and is a nonanalytic function of H: m(t=t c ) ~ H 1/δ The OP susceptibility diverges χ ~ T - T c -γ The specific heat shows an anomaly C ~ T T c -α Underlying reason: Strong OP fluctuations lead to a diverging length scale scale (correlation length ξ): ξ ~ T T c -ν Examples: Critical opalescence in a classical fluid Simulation of the 2D Ising model Consequences: Universality Scale invariance Explanation: Homogeneity laws (a.k.a. scaling laws) Singapore Winter School 7

8 Mohan et al 1998 Singapore Winter School 8

9 2 nd order transitions, a.k.a. critical points, are special! The OP goes to zero continuously: m(h=0) ~ (T c - T) β and is a nonanalytic function of H: m(t=t c ) ~ H 1/δ The OP susceptibility diverges χ ~ T - T c -γ The specific heat shows an anomaly C ~ T T c -α Underlying reason: Strong OP fluctuations lead to a diverging length scale scale (correlation length ξ): ξ ~ T T c -ν Examples: Critical opalescence in a classical fluid Simulation of the 2D Ising model Consequences: Universality Scale invariance Explanation: Homogeneity laws (a.k.a. scaling laws) Singapore Winter School 9

10 Mohan et al 1998 Singapore Winter School 10

11 2 nd order transitions, a.k.a. critical points, are special! The OP goes to zero continuously: m(h=0) ~ (T c - T) β and is a nonanalytic function of H: m(t=t c ) ~ H 1/δ The OP susceptibility diverges χ ~ T - T c -γ The specific heat shows an anomaly C ~ T T c -α Underlying reason: Strong OP fluctuations lead to a diverging length scale scale (correlation length ξ): ξ ~ T T c -ν Examples: Critical opalescence in a classical fluid Simulation of the 2D Ising model Consequences: Universality Scale invariance Explanation: Homogeneity laws (a.k.a. scaling laws) Singapore Winter School 11

12 Source: Scientific American

13 2 nd order transitions, a.k.a. critical points, are special! The OP goes to zero continuously: m(h=0) ~ (T c - T) β and is a nonanalytic function of H: m(t=t c ) ~ H 1/δ The OP susceptibility diverges χ ~ T - T c -γ The specific heat shows an anomaly C ~ T T c -α Underlying reason: Strong OP fluctuations lead to a diverging length scale scale (correlation length ξ): ξ ~ T T c -ν Examples: Critical opalescence in a classical fluid Simulation of the 2D Ising model Consequences: Universality Scale invariance Explanation: Homogeneity laws (a.k.a. scaling laws) Singapore Winter School 13

14 T << T c T > T c Source: Ch. Bruder T T c

15 2 nd order transitions, a.k.a. critical points, are special! The OP goes to zero continuously: m(h=0) ~ (T c - T) β and is a nonanalytic function of H: m(t=t c ) ~ H 1/δ The OP susceptibility diverges χ ~ T - T c -γ The specific heat shows an anomaly C ~ T T c -α Underlying reason: Strong OP fluctuations lead to a diverging length scale scale (correlation length ξ): ξ ~ T T c -ν Examples: Critical opalescence in a classical fluid Simulation of the 2D Ising model Consequences: Universality Scale invariance Explanation: Homogeneity laws (a.k.a. scaling laws) Singapore Winter School 15

16 Universality: All classical fluids share the same critical exponents: α = ± 0.003; β = ± 0.006; γ = 1.24 ± 0.01; ν = ± 0.01 The exponent values are the same within the experimental error bars, even though the critical pressures, densities, and temperatures are very different for different fluids! Even more remarkably, a class of uniaxial ferromagnets also shares these exponents! This phenomenon is called universality. We also see that the exponents do not appear to be simple numbers. However, all critical points do NOT share the same exponents. For instance, in isotropic ferromagnets, the critical exponent for the order parameter is β = ± which is distinct from the value observed in fluids. All systems that share the same critical exponents are said to belong to the same universality class. Experimentally, the universality classes depend on the system s dimensionality d symmetry properties For sufficiently large d, the critical behavior of most systems becomes rather simple. Example: FMs in d 4 have β = 1/2, γ = 1, ν = 1/2 ( mean-field exponents ).

17 2 nd order transitions, a.k.a. critical points, are special! The OP goes to zero continuously: m(h=0) ~ (T c - T) β and is a nonanalytic function of H: m(t=t c ) ~ H 1/δ The OP susceptibility diverges χ ~ T - T c -γ The specific heat shows an anomaly C ~ T T c -α Underlying reason: Strong OP fluctuations lead to a diverging length scale scale (correlation length ξ): ξ ~ T T c -ν Examples: Critical opalescence in a classical fluid Simulation of the 2D Ising model Consequences: Universality Scale invariance Explanation: Homogeneity laws (a.k.a. scaling laws) Singapore Winter School 17

18 Scale invariance: Measure the magnetization M of a FM as a function of the magnetic field H at a fixed temperature T very close to T c. The result looks like this: Now scale the axes, and plot h = H / T T c x Versus m = M/ T T c y If we choose y = β, and x = βδ, then the all of the curves collapse onto two branches, one for T > T c, and one for T < T c! Note how remarkable this is! It works just as well for other magnets. It reflects the fact that at criticality the system looks the same at all length scales ( self-similarity ), as demonstrated in this simulation of a 2-D Ising model. Mohan et al 1998

19 Source: J. V. Sengers

20 Measure the magnetization M of a FM as a function of the magnetic field H at a fixed temperature T very close to T c. The result looks like this: Now scale the axes, and plot h = H / T T c x Versus m = M/ T T c y If we choose y = β, and x = βδ, then the all of the curves collapse onto two branches, one for T > T c, and one for T < T c! Note how remarkable this is! It works just as well for other magnets. It reflects the fact that at criticality the system looks the same at all length scales ( self-similarity ), as demonstrated in this simulation of a 2-D Ising model. Mohan et al 1998

21 2 nd order transitions, a.k.a. critical points, are special! The OP goes to zero continuously: m(h=0) ~ (T c - T) β and is a nonanalytic function of H: m(t=t c ) ~ H 1/δ The OP susceptibility diverges χ ~ T - T c -γ The specific heat shows an anomaly C ~ T T c -α Underlying reason: Strong OP fluctuations lead to a diverging length scale scale (correlation length ξ): ξ ~ T T c -ν Examples: Critical opalescence in a classical fluid Simulation of the 2D Ising model Consequences: Universality Scale invariance Explanation: Homogeneity laws (a.k.a. scaling laws) Singapore Winter School 21

22 Homogeneity laws: Consider the magnetization M as a function of ξ and H. Suppose we scale lengths by a factor b, so ξ -> ξ / b. Suppose M at scale b = 1 is related to M at scale b by a generalized homogeneity law (This was initially postulated as the scaling x M(ξ,H) = b β/ν M(ξ / b, H δβ/ν ) hypothesis (Widom, Kadanoff), and later x derived by means of the renormalization But group (Wilson) ) ξ ~ t ν => ξ / b = (t b 1/ν ) ν and therefore M(t, H) = b β/ν M(t b 1/ν, H b δβ/ν ) where t = T T c / T c But b is an arbitrary scale factor, so we can choose in particular b = t ν. Then M(t,H) = t β M(1, H / t δβ ) And in particular M(t, H=0) ~ t β and M(t=0, H) ~ H 1/δ No big surprise here, we ve chosen the exponents such that this works out! But, it follows that M(t, H)/t β = F(H / t βδ ), This explains the experimental observations! with F(x) = M(t=1, x) an unknown scaling function.

23 2 nd order transitions, a.k.a. critical points, are special! The OP goes to zero continuously: m(h=0) ~ (T c - T) β and is a nonanalytic function of H: m(t=t c ) ~ H 1/δ The OP susceptibility diverges χ ~ T - T c -γ The specific heat shows an anomaly C ~ T T c -α Underlying reason: Strong OP fluctuations lead to a diverging length scale scale (correlation length ξ): ξ ~ T T c -ν Examples: Critical opalescence in a classical fluid Simulation of the 2D Ising model Consequences: Universality Scale invariance Explanation: Homogeneity laws (a.k.a. scaling laws) Singapore Winter School 23

24 II. Classical vs. Quantum Phase Transitions Critical behavior at 2 nd order transitions is caused by thermal fluctuations. Question: What happens if T c is suppressed to zero, which kills the thermal fluctuations? This can be achieved in many low-t c FMs, e.g., UGe 2 : Answer: Quantum fluctuations take over. There still is a transition, but the universality class changes. Question: How can this happen in a continuous way? Saxena et al 2000 Singapore Winter School 24

25 II. Classical vs. Quantum Phase Transitions Critical behavior at 2 nd order transitions is caused by thermal fluctuations. Question: What happens if T c is suppressed to zero, which kills the thermal fluctuations? This can be achieved in many low-t c FMs, e.g., UGe 2 : Answer: Quantum fluctuations take over. There still is a transition, but the universality class changes. Question: How can this happen in a continuous way? Answer: By means of a crossover. Singapore Winter School 25

26 Crucial difference between quantum and classical phase transitions: Coupling of statics and dynamics Consider the partition function Z, which determines the free energy F = -T log Z Classical system: (β = 1/T) H kin and H pot commute H pot determines the thermodynamic behavior, independent of the dynamics => In classical equilibrium statistical mechanics, the statics and the dynamics are independent of one another Singapore Winter School 26

27 Quantum system: H = H(a +, a) in second quantization H kin and H pot do NOT commute => statics and dynamics couple, and need to be considered together! Technical solution: Divide [0,β] into infinitesimal sections parameterized by 0 τ β ( imaginary time ), making use of BCH, and represent Z as a functional integral over auxiliary fields (Trotter, Suzuki) with S an action that depends on the auxiliary fields: The fields commute for bosons, and anticommute for fermions. Singapore Winter School 27

28 T = 0 corresponds to β = => Quantum mechanically, the statics and the dynamics couple! => A d-dimensional quantum system at T = 0 resembles a (d+1)-dimensional classical system! Caveat: τ may act akin to z spatial dimensions, with z 1, and z not eve even integer Example: In a simple theory of quantum FMs, z = 3 (Hertz) Quantum FMs in d 1 act like classical FMs in d 4 Singapore Winter School 28

29 T = 0 corresponds to β = => Quantum mechanically, the statics and the dynamics couple! => A d-dimensional quantum system at T = 0 resembles a (d+1)-dimensional classical system! Caveat: τ may act akin to z spatial dimensions, with z 1, and z not eve even integer Example: In a simple theory of quantum FMs, z = 3 (Hertz) Quantum FMs in d 1 act like classical FMs in d 4 Prediction: The quantum FM transition is 2 nd order with mean-field exponents (Hertz 1976) Singapore Winter School 29

30 III. The Quantum Ferromagnetic Transition Problem: The prediction does not agree with experiment! When Tc is suppressed far enough, the transition (almost *) invariably becomes 1 st order! Example: UGe2 Taufour et al 2010 Many other examples Generic phase diagram * Some exceptions: Strong disorder Quasi-1D systems Other types of order interfere Singapore Winter School 30

31 URhGe Huxley et al 2007 Singapore Winter School 31

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33 III. The Quantum Ferromagnetic Transition Problem: The prediction does not agree with experiment! When Tc is suppressed far enough, the transition (almost *) invariably becomes 1 st order! Example: UGe2 Taufour et al 2010 Many other examples Generic phase diagram * Some exceptions: Strong disorder Quasi-1D systems Other types of order interfere Singapore Winter School 33

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35 Questions: What went wrong with the prediction? Hint: It s a long way from the basic Trotter formula to a theory of quantum FMs. What is causing the wings? Hint: Wings are known in classical systems that show a TCP. Why is the observed phase diagram so universal? Hint: It must be independent of the microscopic details, and only depend on features that ALL metallic magnets have in common. Singapore Winter School 35

36 Part 2: Soft Modes I. Critical soft modes Landau theory for a classical FM: t > 0 t = 0 t < 0 F L (m) = t m 2 + u m 4 + O(m 6 ) Assumptions: m is small The coefficients are finite t ~ T T c Landau theory replaces the fluctuating OP by its average ( mean-field approx. ) FL can in principle be derived from a microscopic partition function Describes a 2 nd order transition at t = 0. NB: No m 3 term for symmetry reasons => 2 nd order transition! In In a classical fluid there is a v m 3 term that vanishes at the critical point Singapore Winter School 36

37 How about fluctuations? Write M(x) = m + δm(x) and consider contributions to Z or F by δm(x): Landau-Ginzburg-Wilson (LGW) For small δm(x), expand to second order => integral can be done => Ornstein-Zernike result for the susceptibility: How good is the Gaussian approximation? Qualitatively okay for d > 4 (Ginzburg) Qualitatively wrong for d < 4. In general, Singapore Winter School 37

38 Discuss the Ornstein-Zernike result: Obeys scaling with γ = 1 and ν = 1/2. Holds for both t > 0 and t < 0. For t 0, correlations are ranged (exponential decay) For t = 0, correlations are long ranged (power-law decay)! No characteristic length scale => scale invariance The homogeneous susceptibility and the correlation length diverge for t = 0 short No resistance against formation of m 0 m rises faster than linear with H The OP fluctuations are a soft (or massless) mode (or excitation) These critical soft modes are soft only at a special point in the phase diagram, viz., the critical point => There is scale invariance only at the critical point. Singapore Winter School 38

39 II. Generic soft modes, Mechanism 1: Goldstone modes So far we have been thinking of Ising magnets Consider a classical planar magnet instead: Spins in a plane; OP m is a vector Disordered phase: Random orientation of spins m = <m> = 0 Singapore Winter School 39

40 Ordered phase at T << Tc: Near-perfect alignment of spins, m 0 NB: The direction of the spins is arbitrary! Singapore Winter School 40

41 Suppose we rotate all spins by a fixed angle: This costs no energy, since all spin directions are equivalent! The free energy depends only on the magnitude of m, not on its direction. Another way to say it: There is no restoring force for co-rotations of the spins. Singapore Winter School 41

42 Suppose we rotate the spins by a slightly position dependent angle: This will cost very little energy! Singapore Winter School 42

43 Conclusions: There is a soft mode (spin wave) consisting of transverse (azimuthal) fluctuations of the magnetization. The free energy has the shape of a Mexican hat. The transverse susceptibility diverges everywhere in the ordered phase The longitudinal (radial) fluctuations do cost energy; they are massive. The spin rotational symmetry is spontaneously broken (as opposed to explicitly broken by an external field): The Hamiltonian is still invariant under rotations of the spin, but the lowest-free-energy state is not. However, the free energy of the resulting state is still invariant under co-rotations of the spins. Works analogously for Heisenberg magnets: 2 soft modes rather than 1 There is a simple mechanical analog of this phenomenon. Singapore Winter School 43

44 massive massive massive massive massive soft

45 This is an example of Goldstone s Theorem: A spontaneously broken continuous symmetry in general leads to the existence of soft modes ( Goldstone modes ). More precisely: If a continuous symmetry described by a group G is spontaneously broken such that a subgroup H ( little group or stabilizer group ) remains unbroken, then there are n Goldstone modes, where n = dim (G/H). Example: Heisenberg magnet: G = SO(3) (rotational symmetry of the 3-D spin) H H = SO(2) (rotational symmetry in the plane perpendiperpendicular cular to the spontaneous magnetizaton) n n = dim(so(3)/so(2)) = 3 1 = 2 (2 transverse magnons) N The transverse susceptibility diverges as No characteristic length scale => scale invariance! Note: The Goldstone modes are soft everywhere in the ordered phase, not just at the critical point! This is an example of generic scale invariance Singapore Winter School 45

46 III. Generic soft modes, Mechanism 2: Gauge invariance Electrodynamics => For charged systems, gauge invariance is important. For the study of, e.g., superconductors, we need to build in this concept! Consider again the LGW action, but with a vector OP a complex scalar OP :, or, equivalently, ( ) Now postulate that the theory must be invariant under local gauge transformations, i.e., under with an arbitrary real field Λ(x). A does not fulfill this requirement because of the gradient term Modify the gradient term Singapore Winter School 46

47 The simplest modification that does the trick is where the gauge field A(x) transforms as and is the field tensor. q ( charge ) and μ are coupling constants. Singapore Winter School 47

48 Notes: This is the LGW version of a model Ginzburg and Landau proposed as a model for superconductivity. GL solved the action in a mean-field approximation that replaced both ϕ and the magnetic field by their expectation values. This theory was later shown by Gorkov to be equivalent to BCS theory. The LGW theory is much more general: A describes the fluctuating electromagnetic field that is nonzero even if there is applied magnetic field (i.e., if the mean value of B is zero.) Singapore Winter School 48

49 Now consider the soft modes in Gaussian approximation. Disordered phase: < ϕ > = 0. A appears with gradients only => A is soft. In Coulomb gauge ( ) one finds two soft modes transverse photon : ϕ is massive with mass t > 0: 2 soft modes ( transverse photon ) 2 massive modes Conclusion: Two massless and two massive modes Photon is a generic soft mode (result of gauge invariance) Photon has only two degrees of freedom Any phase transition will take place on the background of the generic scale invariance provided by the photon! Singapore Winter School 49

50 Ordered phase: < ϕ > 0. Write Expand to second order in ϕ 1, ϕ 2, and A: Singapore Winter School 50

51 Ordered phase: < ϕ > 0. Write Expand to second order in ϕ 1, ϕ 2, and A: A acquires a mass ~ v 2 ϕ 2 couples to the massive A, can be eliminated by shifting A: Singapore Winter School 51

52 Ordered phase: < ϕ > 0. Write Expand to second order in ϕ 1, ϕ 2, and A: A acquires a mass ~ v 2 ϕ 2 couples to the massive A, can be eliminated by shifting A: This yields with m 2 ~ q 2 v 2, and 3 massive modes ( transverse + longitudinal photons ) 1 massive mode Singapore Winter School 52

53 Conclusion: No soft modes! The Goldstone mode candidate ϕ 2 gets eaten by the gauge field, which becomes massive in the process ( Anderson-Higgs mechanism ) Photon now has three degrees of freedom, all of them massive Physical manifestation of the massive modes: Meissner effect Note: The same principle applies to more complicated gauge groups Example: Electroweak symmetry breaking SU(2)xU(1) gets broken to U(1) => o One massless gauge boson (photon) o 4 1 = 3 Goldstone bosons that become massive via Anderson-Higgs => W ±, Z vector bosons o Physical manifestation: Short-ranged weak interaction Singapore Winter School 53

54 IV. Digression. Generic soft modes, Mechanism 3: C Conservation laws There is a third mechanism leading to generic scale invaviance: Conservation laws They can lead to time-correlation functions in classical systems to decay algebraically rather than exponentially => temporal long-range correlations Through mode-mode-coupling effects, this can happen even to time correlation functions of modes that are not themselves conserved. Example: Transverse-velocity correlations in a classical fluid. As result, transport coefficients (viscosities) are nonanalytic functions of the frequency, and hydrodynamics break down in d = 2. For classical systems in equilibrium, this affects the dynamics only. For quantum systems, and for classical nonequilibrium systems, the statics and dynamics couple, and the thermodynamic behavior is affected as well! Singapore Winter School 54

55 V. Fluctuation-induced 1 st order transitions Consider the fluctuating GL theory again: A appears only quadratically => A can be integrated out exactly! Still replace ϕ -> <ϕ> => generalized mean-field approximation, or renormalized mean-field theory The difference between this an GL theory is that it takes into account the fluctuating electromagnetic field. A couples to ϕ => The resulting action contains ϕ to all orders. <AA> is soft for ϕ = 0, and massive for ϕ 0 => S[ϕ] cannot be an analytic function of ϕ! Singapore Winter School 55

56 The resulting generalized MFT for 3-D systems is t>t 1 t=t 1 t<t 1 This describes a 1 st order transition at some t 1 > 0! The fluctuating A - field changes the nature of the phase transition! This is called a fluctuation-induced 1 st order transition (Halperin, Lubensky, Ma 1974) NB: This is a classical transition! There is an analogous mechanism in particle physics (Coleman-Weinberg). This is a consequence of generic scale invariance, with the generic soft modes coupling to the OP. An essentially identical theory applies to the transition from the nematic phase to the smectic-a phase in liquid crystals, with the nematic Goldstone modes providing the GSI. Singapore Winter School 56

57 In superconductors, the effect is too small to be observed In liquid crystals, the transition is 1 st order in some systems, but 2 nd order in others. This is believed to be due to the fluctuations of the OP, which are neglected in the generalized MFT. For 4-D superconductors, or liquid crystals, the result is Why do we care? See below. with v > 0 Singapore Winter School 57

58 Part 3: Soft Modes in Metals, and the Ferrom magnetic Quantum Phase Transition I. Soft modes in metals There are (at least) two types of soft modes in clean metals at T=0: Single-particle excitations Described by Green s function Soft at k = k F, iω n = 0 => The leading properties of a Fermi liquid follow via scaling (Nayak & Wilczek, Shankar) Singapore Winter School 58

59 Two-particle excitations + T + m + H k F Soft mode, mixes retarded and advanced degrees of freedom; result of a spontaneously broken (unobvious) continuous symmetry (F Wegner). Weight given by DOS. These are Goldstone modes, i.e., they represent generic scale invariance in a Fermi liquid. Soft only at T = 0. Appear in both spin-singlet and spin-triplet channels, the latter couples to the magnetization. A nonzero magnetization gives the triplet propagator a mass. So does a magnetic field. A ferromagnetic phase transition at T=0 will take place on the background of these generic soft modes. (Cf. the classical superconductor/liquid Xtal!) Singapore Winter School 59

60 II. The ferromagnetic quantum phase transition Idea: Construct a renormalized mean-field theory in analogy to HLM, with the magnetization m as the OP and the two-particle electron excitations as the generic soft modes. Result: The free energy maps onto that of the superconductor/liquid Xtal problem in D = 4: Discussion: First-order transition at T = 0 (always!), and at low T Second-order transition at higher T => tricritical point Singapore Winter School 60

61 First-order transition at T = 0 (always!), and at low T Second-order transition at higher T => tricritical point Singapore Winter School 61

62 First-order transition at T = 0 (always!), and at low T Second-order transition at higher T => tricritical point First-order transition at low H, ends in a quantum critical point Singapore Winter School 62

63 First-order transition at T = 0 (always!), and at low T Second-order transition at higher T => tricritical point First-order transition at low H, ends in a quantum critical point T t - H phase diagram displays surfaces of first-order transitions ( tricritical wings ) This explains the experimental observations! Singapore Winter School 63

64 First-order transition at T = 0 (always!), and at low T Second-order transition at higher T => tricritical point First-order transition at low H, ends in a quantum critical point T t - H phase diagram displays surfaces of first-order transitions ( tricritical wings ) This explains the experimental observations! An open question: Why are the OP fluctuations so inefficient? Singapore Winter School 64

65 Summary of Crucial Points All relevant soft modes are important to determine the physics at long length and time scales. Generic soft modes, and the resulting scale invariance, are quite common, and there are various mechanism for producing them. Analogies between seemingly unrelated topics can be very useful! Singapore Winter School 65