Three Lectures on Soft Modes Three Lectures on Soft Modes and Scale Invariance in Metalsand Scale Invariance in Metals

Quantum Ferromagnets as an Example Quantum Ferromagnets as an Example of Universal Low-Energy Physicsof Universal Low-Energy Physics

Dietrich Belitz, University of Oregon

with T.R. Kirkpatrick and T. Vojta

Reference: Rev. Mod. Phys. 77, 579, (2005)

Soft Modes and Scale Invariance Soft Modes and Scale Invariance in Metalsin Metals

Quantum Ferromagnets as an Example Quantum Ferromagnets as an Example of Universal Low-Energy Physicsof Universal Low-Energy Physics

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 Transitions

I. Preliminaries: First-Order vs Second-Order Transitions

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Example 1: The Liquid-Gas Transition

Schematic phase diagram of H2O

Part 1: Phase Transitions

I. Preliminaries: First-Order vs Second-Order Transitions

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Example 1: The Liquid-Gas Transition

Schematic phase diagram of H2O

Part 1: Phase Transitions

I. Preliminaries: First-Order vs Second-Order Transitions

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Example 1: The Liquid-Gas Transition

Schematic phase diagram of H2O

T < Tc: 1st order transition (latent heat)

T > Tc: No transition

T = Tc: Critical point, special behavior

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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

•ZrZn2 Tc = 28.5K

•UGe2 Tc = 53K

Demonstration of the FM critical point

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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)

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Mohan et al 1998

Singapore Winter School 9February 4-5, 2013

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)

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Mohan et al 1998

Singapore Winter School 11February 4-5, 2013

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)

Source: Scientific American

Singapore Winter School 13February 4-5, 2013

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)

Source: Ch. Bruder

T << Tc T > Tc

T ≈ Tc

Singapore Winter School 15February 4-5, 2013

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)

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

Singapore Winter School 17February 4-5, 2013

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)

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

Singapore Winter School 21February 4-5, 2013

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)

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 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 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) , with F(x) = M(t=1, x) an unknown scaling function.

This explains the experimental observations!

Singapore Winter School 23February 4-5, 2013

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)

II. Classical vs. Quantum Phase Transitions

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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-Tc FMs, 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

II. Classical vs. Quantum Phase Transitions

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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-Tc FMs, 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.

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:

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

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(β = 1/T)

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!

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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.

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

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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)

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III. The Quantum Ferromagnetic Transition

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Problem: The prediction does not agree with experiment !

When Tc is suppressed far enough, the transition (almost *) invariably becomes 1st order!

Example: UGe2

* Some exceptions: • Strong disorder

• Quasi-1D systems

• Other types of order interfere

Taufour et al 2010

Many other examples

Generic phase diagram

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URhGe

Huxley et al 2007

III. The Quantum Ferromagnetic Transition

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Problem: The prediction does not agree with experiment !

When Tc is suppressed far enough, the transition (almost *) invariably becomes 1st order!

Example: UGe2

* Some exceptions: • Strong disorder

• Quasi-1D systems

• Other types of order interfere

Taufour et al 2010

Many other examples

Generic phase diagram

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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.

Part 2: Soft Modes

I. Critical soft modes

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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

• m is small

• The coefficients are finite

• t ~ T – Tc

t > 0 t = 0 t < 0

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• Qualitatively okay for d > 4 (Ginzburg)

• Qualitatively wrong for d < 4. In general,

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)

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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)

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II. Generic soft modes, Mechanism 1: Goldstone modes

Disordered phase:

Random orientation of spins

m = <m> = 0

So far we have been thinking of Ising magnets

Consider a classical planar magnet instead: Spins in a plane; OP m is a vector

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Ordered phase at T << Tc:

Near-perfect alignment of spins, m ≠ 0

NB: The direction of the spins is arbitrary!

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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.

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Suppose we rotate the spins by a slightly position dependent angle:

This will cost very little energy!

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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.

massive

massive

massive

massive

massive

soft

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”

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Singapore Winter School 46February 4-5, 2013

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 , 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

(“ “)

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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.

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• 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.)

Notes:

• 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:

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Now consider the soft modes in Gaussian approximation.

Disordered phase: < ϕ > = 0 .

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 !

• Write

• Expand to second order in ϕ1, ϕ2, and A:

Ordered phase: < ϕ > ≠ 0 .

Singapore Winter School 50February 4-5, 2013

• Write

• Expand to second order in ϕ1, ϕ2, and A:

• A acquires a mass ~ v2

• ϕ2 couples to the massive A, can be eliminated by shifting A:

Ordered phase: < ϕ > ≠ 0 .

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• Write

• Expand to second order in ϕ1, ϕ2, and A:

• A acquires a mass ~ v2

• ϕ2 couples to the massive A, can be eliminated by shifting A:

• This yields

with m2 ~ q2 v2, and

Ordered phase: < ϕ > ≠ 0 .

1 massive mode

3 massive modes (“transverse + longitudinal photons”)

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• No soft modes!

• The Goldstone mode candidate ϕ2 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

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

• Conclusion:

• Note: The same principle applies to more complicated gauge groups

Singapore Winter School 53February 4-5, 2013

Singapore Winter School 54February 4-5, 2013

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 55February 4-5, 2013

V. Fluctuation-induced 1st 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 !

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The resulting generalized MFT for 3-D systems is

• This describes a 1st order transition at some t1 > 0 !

• The fluctuating A - field changes the nature of the phase transition!

• This is called a fluctuation-induced 1st 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.

t>t1 t=t1 t<t1

• In superconductors, the effect is too small to be observed

• In liquid crystals, the transition is 1st order in some systems, but 2nd 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

with v > 0

Why do we care? See below.

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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 = kF, iωn = 0 => The leading properties of a Fermi liquid follow via scaling (Nayak & Wilczek, Shankar)

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Part 3: Soft Modes in Metals, and the Ferro- m magnetic Quantum Phase Transition

I. Soft modes in metals

Two-particle excitations

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+ T + m + H

• 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!)

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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

• First-order transition at T = 0 (always!), and at low T

• Second-order transition at higher T => tricritical point

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• 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

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• 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!

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• 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 64February 4-5, 2013

Singapore Winter School 65February 4-5, 2013

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 !