Dietrich Belitz, University of Oregon with T.R. Kirkpatrick and T. Vojta

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

Soft Modes and Scale Invariance in MetalsQuantum 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

Part 1: Phase TransitionsI. Preliminaries: First-Order vs Second-Order Transitions • Example 1: The Liquid-Gas Transition Schematic phase diagram of H2O Singapore Winter School

Part 1: Phase TransitionsI. Preliminaries: First-Order vs Second-Order Transitions • Example 1: The Liquid-Gas Transition T < Tc: 1st order transition (latent heat) T > Tc: No transition T = Tc: Critical point, special behavior Schematic phase diagram of H2O Singapore Winter School

Example 2: The Paramagnet - Ferromagnet Transition • H = 0: Transition is 2nd order • T > Tc: Disordered phase, m = 0 • T < Tc: Ordered phase, m ≠ 0 • T -> Tc: m -> 0 continuously • m is called order parameter • Examples: • Ni Tc= 630K • Fe Tc= 1,043K • ZrZn2Tc= 28.5K • UGe2Tc = 53K Demonstration of the FM critical point Singapore Winter School

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

Mohan et al 1998 Singapore Winter School

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

Mohan et al 1998 Singapore Winter School

Source: Scientific American

T << Tc T > Tc Source: Ch. Bruder T ≈ Tc

Universality: All classical fluids share the same critical exponents: α = 0.113 ± 0.003; β = 0.321 ± 0.006; γ = 1.24 ± 0.01; ν = 0.625 ± 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 = 0.358 ± 0.003 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 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”). • dimensionality d • symmetry properties

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 Tc . The result looks like this: Now scale the axes, and plot h = H / |T – Tc| x Versus m = M/ |T – Tc| y If we choose y = β, and x = βδ, then the all of the curves collapse onto two branches, one for T > Tc, and one for T < Tc ! 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-simi- larity”), as demonstrated in this simulation of a 2-D Ising model. Mohan et al 1998

Source: J. V. Sengers

Measure the magnetization M of a FM as a function of the magnetic field H at a fixed temperature T very close to Tc . The result looks like this: Now scale the axes, and plot h = H / |T – Tc| x Versus m = M/ |T – Tc| y If we choose y = β, and x = βδ, then the all of the curves collapse onto two branches, one for T > Tc, and one for T < Tc ! 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-simi- larity”), as demonstrated in this simulation of a 2-D Ising model. Mohan et al 1998

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 xM(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 where t = |T – Tc| / Tc and therefore M(t, H) = b –β/ν M(t b 1/ν, H b δβ/ν) 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 βand 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) , with F(x) = M(t=1, x) an unknown scaling function. This explains the experimental observations!

II. Classical vs. Quantum Phase Transitions Critical behavior at 2nd order transitions is caused by thermal fluctuations. Question: What happens if Tc is suppressed to zero, which kills the thermal fluctuations? This can be achieved in many low-TcFMs, e.g., UGe2: 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

II. Classical vs. Quantum Phase Transitions Critical behavior at 2nd order transitions is caused by thermal fluctuations. Question: What happens if Tc is suppressed to zero, which kills the thermal fluctuations? This can be achieved in many low-TcFMs, e.g., UGe2: 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

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) • Hkin and Hpot commute • Hpot 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

Quantum system: H = H(a+, a) in second quantization Hkin and Hpot 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

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

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 2nd order with mean-field exponents (Hertz 1976) Singapore Winter School

III. The Quantum Ferromagnetic Transition Problem: The prediction does not agree with experiment ! When Tc is suppressed far enough, the transition (almost *) invariably becomes 1st 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

URhGe Huxley et al 2007 Singapore Winter School

III. The Quantum Ferromagnetic Transition Problem: The prediction does not agree with experiment ! When Tc is suppressed far enough, the transition (almost *) invariably becomes 1st 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

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

Part 2: Soft ModesI. Critical soft modes • Landau theory for a classical FM: • FL(m) = t m2 + u m4 + O(m6) • Assumptions: • 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 2nd order transition at t = 0. • NB: No m3 term for symmetry reasons => 2nd order transition ! In In a classical fluid there is a v m3 term that vanishes at the critical point t > 0 t = 0 t < 0 • m is small • The coefficients are finite • t ~ T – Tc Singapore Winter School

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

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 short 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 • 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. • 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) Singapore Winter School

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

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

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

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

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

massive massive massive massive soft massive

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 perpendi- perpendicular cular to the spontaneous magnetizaton) n n = dim(SO(3)/SO(2)) = 3 – 1 = 2 (2 transverse magnons) • The transverse susceptibility diverges as N 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

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 , or, equivalently, a complex scalar OP : • 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 III. Generic soft modes, Mechanism 2: Gauge invariance (““) Singapore Winter School

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

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

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: • Conclusion: 2 soft modes (“transverse photon”) 2 massive modes • 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

Ordered phase: < ϕ > ≠ 0 . • Write • Expand to second order in ϕ1, ϕ2, and A: Singapore Winter School

Dietrich Belitz, University of Oregon with T.R. Kirkpatrick and T. Vojta

Dietrich Belitz, University of Oregon with T.R. Kirkpatrick and T. Vojta

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