# t = |T – T c ```Three Lectures on Soft Modes
and Scale Invariance in Metals
Quantum Ferromagnets as an Example
of Universal Low-Energy Physics
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
Part 1: Phase Transitions
I. Preliminaries: First-Order vs Second-Order Transitions
 Example 1: The Liquid-Gas Transition
Schematic phase diagram of
H2O
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Part 1: Phase Transitions
I. Preliminaries: First-Order vs Second-Order Transitions
 Example 1: The Liquid-Gas Transition
Schematic phase diagram of
H2O
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Part 1: Phase Transitions
I. Preliminaries: First-Order vs Second-Order Transitions
 Example 1: The Liquid-Gas Transition
T &lt; Tc: 1st order transition (latent heat)
Schematic phase diagram of
H2O
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T &gt; 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 &gt; Tc: Disordered phase, m = 0
T &lt; Tc: Ordered phase, m ≠ 0
T -&gt; Tc: m -&gt; 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!
• The OP goes to zero continuously:
and is a nonanalytic function of H:
• The OP susceptibility diverges
m(H=0) ~ (Tc - T) β
m(T=Tc ) ~ H 1/δ
χ ~ |T - T | -γ
c
• The specific heat shows an anomaly C ~ |T – Tc| -α
Underlying reason: Strong OP fluctuations lead to a diverging length scale
scale (correlation length ξ): ξ ~ |T – Tc| -ν
Examples:
• Critical opalescence in a classical fluid
• Simulation of the 2D Ising model
Consequences: • Universality
• Scale invariance
Explanation:
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• Homogeneity laws (a.k.a. scaling laws)
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Mohan et al 1998
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2nd order transitions, a.k.a. critical points, are special!
• The OP goes to zero continuously:
and is a nonanalytic function of H:
• The OP susceptibility diverges
m(H=0) ~ (Tc - T) β
m(T=Tc ) ~ H 1/δ
χ ~ |T - T | -γ
c
• The specific heat shows an anomaly C ~ |T – Tc| -α
Underlying reason: Strong OP fluctuations lead to a diverging length scale
scale (correlation length ξ): ξ ~ |T – Tc| -ν
Examples:
• Critical opalescence in a classical fluid
• Simulation of the 2D Ising model
Consequences: • Universality
• Scale invariance
Explanation:
February 4-5, 2013
• Homogeneity laws (a.k.a. scaling laws)
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Mohan et al 1998
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2nd order transitions, a.k.a. critical points, are special!
• The OP goes to zero continuously:
and is a nonanalytic function of H:
• The OP susceptibility diverges
m(H=0) ~ (Tc - T) β
m(T=Tc ) ~ H 1/δ
χ ~ |T - T | -γ
c
• The specific heat shows an anomaly C ~ |T – Tc| -α
Underlying reason: Strong OP fluctuations lead to a diverging length scale
scale (correlation length ξ): ξ ~ |T – Tc| -ν
Examples:
• Critical opalescence in a classical fluid
• Simulation of the 2D Ising model
Consequences: • Universality
• Scale invariance
Explanation:
February 4-5, 2013
• Homogeneity laws (a.k.a. scaling laws)
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11
Source:
Scientific
American
2nd order transitions, a.k.a. critical points, are special!
• The OP goes to zero continuously:
and is a nonanalytic function of H:
• The OP susceptibility diverges
m(H=0) ~ (Tc - T) β
m(T=Tc ) ~ H 1/δ
χ ~ |T - T | -γ
c
• The specific heat shows an anomaly C ~ |T – Tc| -α
Underlying reason: Strong OP fluctuations lead to a diverging length scale
scale (correlation length ξ): ξ ~ |T – Tc| -ν
Examples:
• Critical opalescence in a classical fluid
• Simulation of the 2D Ising model
Consequences: • Universality
• Scale invariance
Explanation:
February 4-5, 2013
• Homogeneity laws (a.k.a. scaling laws)
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T &lt;&lt; Tc
T &gt; Tc
Source: Ch. Bruder
T ≈ Tc
2nd order transitions, a.k.a. critical points, are special!
• The OP goes to zero continuously:
and is a nonanalytic function of H:
• The OP susceptibility diverges
m(H=0) ~ (Tc - T) β
m(T=Tc ) ~ H 1/δ
χ ~ |T - T | -γ
c
• The specific heat shows an anomaly C ~ |T – Tc| -α
Underlying reason: Strong OP fluctuations lead to a diverging length scale
scale (correlation length ξ): ξ ~ |T – Tc| -ν
Examples:
• Critical opalescence in a classical fluid
• Simulation of the 2D Ising model
Consequences: • Universality
• Scale invariance
Explanation:
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• Homogeneity laws (a.k.a. scaling laws)
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Universality: All classical fluids share the same critical exponents:
α = 0.113 &plusmn; 0.003; β = 0.321 &plusmn; 0.006; γ = 1.24 &plusmn; 0.01; ν = 0.625 &plusmn; 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 &plusmn; 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
•
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”).
2nd order transitions, a.k.a. critical points, are special!
• The OP goes to zero continuously:
and is a nonanalytic function of H:
• The OP susceptibility diverges
m(H=0) ~ (Tc - T) β
m(T=Tc ) ~ H 1/δ
χ ~ |T - T | -γ
c
• The specific heat shows an anomaly C ~ |T – Tc| -α
Underlying reason: Strong OP fluctuations lead to a diverging length scale
scale (correlation length ξ): ξ ~ |T – Tc| -ν
Examples:
• Critical opalescence in a classical fluid
• Simulation of the 2D Ising model
Consequences: • Universality
• Scale invariance
Explanation:
February 4-5, 2013
• Homogeneity laws (a.k.a. scaling laws)
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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 &gt; Tc, and one for T &lt; 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-similarity”), 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 &gt; Tc, and one for T &lt; 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-similarity”), as demonstrated in this simulation of a 2-D
Ising model.
Mohan et al 1998
2nd order transitions, a.k.a. critical points, are special!
• The OP goes to zero continuously:
and is a nonanalytic function of H:
• The OP susceptibility diverges
m(H=0) ~ (Tc - T) β
m(T=Tc ) ~ H 1/δ
χ ~ |T - T | -γ
c
• The specific heat shows an anomaly C ~ |T – Tc| -α
Underlying reason: Strong OP fluctuations lead to a diverging length scale
scale (correlation length ξ): ξ ~ |T – Tc| -ν
Examples:
• Critical opalescence in a classical fluid
• Simulation of the 2D Ising model
Consequences: • Universality
• Scale invariance
Explanation:
February 4-5, 2013
• Homogeneity laws (a.k.a. scaling laws)
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Homogeneity laws: Consider the magnetization M as a function of ξ and H.
Suppose we scale lengths by a factor b, so ξ -&gt; ξ / b. Suppose M at scale b = 1 is
related to M at scale b by a generalized homogeneity law
x M(ξ ,H) = b –β/ν M(ξ / b, H δβ/ν)
x
But
ξ ~ t –ν =&gt; ξ / b = (t b 1/ν) –ν
(This was initially postulated as the “scaling
derived by means of the renormalization
group (Wilson) )
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!
2nd order transitions, a.k.a. critical points, are special!
• The OP goes to zero continuously:
and is a nonanalytic function of H:
• The OP susceptibility diverges
m(H=0) ~ (Tc - T) β
m(T=Tc ) ~ H 1/δ
χ ~ |T - T | -γ
c
• The specific heat shows an anomaly C ~ |T – Tc| -α
Underlying reason: Strong OP fluctuations lead to a diverging length scale
scale (correlation length ξ): ξ ~ |T – Tc| -ν
Examples:
• Critical opalescence in a classical fluid
• Simulation of the 2D Ising model
Consequences: • Universality
• Scale invariance
Explanation:
February 4-5, 2013
• Homogeneity laws (a.k.a. scaling laws)
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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-Tc
FMs, e.g., UGe2:
over. There still is a transition, but
the universality class changes.
Question: How can this happen in a
continuous way?
Saxena et al 2000
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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-Tc
FMs, e.g., UGe2:
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.
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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
=&gt; In classical equilibrium statistical mechanics, the
statics and the dynamics are independent of one another
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Quantum system:
H = H(a+, a) in second quantization
Hkin and Hpot do NOT commute =&gt; 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.
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T = 0 corresponds to β = ∞
=&gt; Quantum mechanically, the statics and the dynamics couple!
=&gt; 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 = ∞
=&gt; Quantum mechanically, the statics and the dynamics couple!
=&gt; 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
Problem: The prediction does not agree with experiment !
When Tc is suppressed far enough, the transition (almost *) invariably
becomes 1st order!
Example: UGe2
 Many other examples
 Generic phase diagram
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Taufour et al 2010
* Some exceptions: •
•
•
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Strong disorder
Quasi-1D systems
Other types of order interfere
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URhGe
Huxley et al 2007
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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
 Many other examples
 Generic phase diagram
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Taufour et al 2010
* Some exceptions: •
•
•
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Strong disorder
Quasi-1D systems
Other types of order interfere
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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.
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Part 2: Soft Modes
I. Critical soft modes
 Landau theory for a classical FM:
t&gt;0
t=0 t&lt;0
FL(m) = t m2 + u m4 + O(m6)
Assumptions: • m is small
• The coefficients are finite
• t ~ T – Tc
• 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 =&gt; 2nd order transition !
In
In a classical fluid there is a v m3 term that vanishes at the critical point
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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
=&gt; integral can be done
=&gt; Ornstein-Zernike result for the susceptibility:
How good is the Gaussian approximation?
• Qualitatively okay for d &gt; 4 (Ginzburg)
• Qualitatively wrong for d &lt; 4. In general,
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Discuss the Ornstein-Zernike result:
Obeys scaling with γ = 1 and ν = 1/2.
Holds for both t &gt; 0 and t &lt; 0. For |t| ≠ 0, correlations are
ranged (exponential decay)
short
For t = 0, correlations are long ranged (power-law decay) !
No characteristic length scale =&gt; scale invariance
The homogeneous susceptibility and the correlation length diverge for t = 0
• 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 =&gt; There is scale invariance only at the critical point.
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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 = &lt;m&gt; = 0
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Ordered phase at T &lt;&lt; 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
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.
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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 perpendiperpendicular
cular to the spontaneous magnetizaton)
n = dim(SO(3)/SO(2)) = 3 – 1 = 2 (2 transverse magnons)
N
n
The transverse susceptibility diverges as
No characteristic length scale =&gt; 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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III. Generic soft modes, Mechanism 2: Gauge invariance
Electrodynamics =&gt; 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
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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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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.)
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Now consider the soft modes in Gaussian approximation.
Disordered phase: &lt; ϕ &gt; = 0 .
• A appears with gradients only =&gt; A is soft. In Coulomb gauge
(
) one finds two soft modes “transverse photon”:
2 soft modes
(“transverse photon”)
• ϕ is massive with mass t &gt; 0:
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 !
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 Ordered phase: &lt; ϕ &gt; ≠ 0 .
• Write
• Expand to second order in
ϕ1, ϕ2, and A:
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 Ordered phase: &lt; ϕ &gt; ≠ 0 .
• 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:
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 Ordered phase: &lt; ϕ &gt; ≠ 0 .
• 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
3 massive modes
(“transverse + longitudinal
photons”)
with m2 ~ q2 v2, and
1 massive mode
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• Conclusion:
• 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
• Note: The same principle applies to more complicated gauge groups
Example: Electroweak symmetry breaking
SU(2)xU(1) gets broken to U(1)
=&gt; o One massless gauge boson (photon)
o 4 – 1 = 3 Goldstone bosons that become massive via
Anderson-Higgs =&gt; W&plusmn;, Z vector bosons
o Physical manifestation: Short-ranged weak
interaction
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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 =&gt; 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 !
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V. Fluctuation-induced 1st order transitions
Consider the fluctuating GL theory again:
• A appears only quadratically =&gt; A can be integrated out exactly!
• Still replace ϕ -&gt; &lt;ϕ&gt; =&gt; “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 =&gt; The resulting action contains ϕ to all orders.
• &lt;AA&gt; is soft for ϕ = 0, and massive for ϕ ≠ 0 =&gt; S[ϕ] cannot be an
analytic function of ϕ !
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The resulting generalized MFT for 3-D systems is
t&gt;t1
t=t1 t&lt;t1
• This describes a 1st order transition
at some t1 &gt; 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.
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• 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 &gt; 0
Why do we care? See below.
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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 = kF, iωn = 0 =&gt; The leading properties of a Fermi liquid follow via
scaling (Nayak &amp; Wilczek, Shankar)
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 Two-particle excitations
+T+m+H
kF
• 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 =&gt; tricritical point
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• First-order transition at T = 0 (always!), and at low T
• Second-order transition at higher T
=&gt; tricritical point
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• First-order transition at T = 0 (always!), and at low T
• Second-order transition at higher T
=&gt; 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
=&gt; 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
=&gt; 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?
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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 !
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