Transcript Slides

The Ferromagnetic Quantum Phase
Transition in Metals
Dietrich Belitz, University of Oregon
Ted Kirkpatrick, University of Maryland
Acknowledgments:
Kwan-yuet Ho
Maria -Teresa Mercaldo
Rajesh Narayanan
Jörg Rollbühler
Ronojoy Saha
Yan Sang
Sharon Sessions
Sumanta Tewari
Achim Rosch
John Toner
Thomas Vojta
Manuel Brando
Malte Grosche
Gil Lonzarich
Christian Pfleiderer
Greg Stewart
Outline
Lecture 1:
1. Motivation: Why Quantum Ferromagnets are Interesting
2. Classical Phase Transitions
a. Ferromagnets
b. Liquid-gas transition
c. Superconductors, and liquid crystals
Lecture 2:
1. Quantum FM Transitions: General Concepts
2. The Quantum FM Transition, Part I: History
3. The Quantum FM Transition, Part II: General Guidelines
Lecture 3:
4. The Fermi Liquid as an Ordered Phase
a. A useful example: Classical 𝟇4 – theory
b. Goldstone modes in a Fermi liquid
1. The Quantum FM Transition, Part III
a. Generalized Landau theory
b. Order-parameter fluctuations
c. Effects of quenched disorder
Lecture 4:
8. Exponents and Exponent Relations at Quantum Critical
Points
9. “How Close is Close to the Critical Point?”, or
How Hard is it to Measure Quantum Critical Exponents?
10. Phase Separation Away from the Coexistence Curve
Lecture 1
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1. Motivation: Why Quantum Ferromagnets are Interesting
Q: What happens to a FM phase transition when the Curie temperature is
very low?
A: Lots of unexpected and strange behavior. For instance:
 The transition changes from
second order to first order
ZrZn2 (Uhlarz et al 2004)
Discontinuous magnetization
MnSi (Uemura et al 2007)
Phase separation
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 The phase diagram develops
an interesting wing structure
UGe2 (Kotegawa et al 2011)
UCoAl (Aoki et al 2011)
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 Quenched disorder
suppresses these effects …
 … leading to a continuous
quantum phase transition …
(Sang et al 2014)
Ni3 Al1-x Ga2 (Yang et al 2011)
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… and for strong disorder to a
quantum Griffiths region
(Pikul 2012)
(Westerkamp et al 2009)
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 Non-Fermi-liquid transport behavior is
observed in large regions of the phase
diagram
(Takashima et al 2007)
Similar behavior is seen in many other quantum FMs
These lectures discuss some of these interesting effects.
For a recent review, see
M. Brando, DB, F.M. Grosche, TRK
arXiv:1502.02898
Rev. Mod. Phys., in press
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2. Classical Phase Transitions
a. Ferromagnets
 The ferromagnetic transition in Fe, Ni, and Co is one of the best known examples of
a thermal phase transition.
 The material is a paramagnet at high temperatures, but spontaneously develops
long-range ferromagnetic order if cooled below the Curie temperature Tc .
 This transition in zero field is 2nd order; i.e. the order parameter (= magnetization) is
a continuous function of temperature, but not analytic at T = Tc :
 Mean-field theory (see below) qualitatively describes the data.
 The transition at T < Tc as a function of a magnetic field is
first order, i.e., the order parameter changes
discontinuously.
 Phase diagram:
Ni (Weiss &
Forrer 1926)
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 50 years later: The true critical behavior is
M ~ (Tc – T) β
with
β = 0.358 ± 0.003
Ni (Weiss &
Forrer 1926)
Cohen & Carver 1977
 Order-parameter fluctuations invalidate mean-field theory near criticality in d=3, but
NOT in a hypothetical system in d > 4 (Ginzburg criterion, “upper critical dimension”,
see below).
 Other critical quantities:
o Susceptibility
𝜒 ~ |T – Tc|-γ with γ
o Correlation length
ξ ~ |T – Tc|-ν
≈ 1.4
with ν ≈ 0.7
 Exponent values depend only on the dimensionality and general properties (e.g., Ising
vs Heisenberg), NOT on microscopic details (“universality classes”).
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Theoretical explanation: Scaling, and the RG (Widom, Kadanoff, Wilson, Fisher,
Wegner).
o Thermal fluctuations drive the critical singularities.
o Observables obey homogeneity laws. E.g., with t = |T – Tc|/Tc and h the
magnetic field,
x
x
with b > 0 arbitrary
vs
curves for various T will collapse
onto ONE function with two branches if the axes
are scaled appropriately!
o This actually works (see next slide):
An important point: Scaling and the RG
be used to describe entire phases,
critical points (“stable fixed points”,
Ma 1976)
can
not just
(e.g.,
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CrBr3
(Ho & Litster 1969)
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 “Action” (= Hamiltonian/T), partition function, free energy density for a classical FM:
 Mean-field approximation,
,
Landau free energy
2nd order transition with mean-field exponents
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Focus on the susceptibility:
Homogeneous susceptibility:
Generalization to k > 0:
x
x

(Ornstein-Zernicke) x
correlation length
diverging length scale, LR correlations
Example of a “soft” or “massless” mode or excitation (here: a critical soft mode)
A time scale also diverges:
(“critical slowing down”)
z is called “dynamical critical exponent”
Fluctuations
2nd order transition with exponents in the appropriate universality
class (Ising, XY, or Heisenberg)
Fluctuations are important only below an upper critical dimension (d < 4 in this case);
Ginzburg criterion
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Lecture 2
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b. Liquid-gas transition
 The liquid-gas transition maps onto an Ising ferromagnet,
but we usually get to see only the 1st order transition.
 Phase diagram:
 The behavior near the critical point is in exact analogy
to the ferromagnetic case. In particular, the correlation
length diverges (
critical opalescence).
 For T < Tc one observes
phase separation. (Magnetic
analogy: MnSi experiment)
CO2
(Sengers & Levelt Sengers 1968)
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c. Superconductors (and liquid crystals)
 Now consider a superconductor with order parameter
x
x
 But,
:
same as XY ferromagnet
couples to the E&M vector potential (photons):
 A comes with a gradient
A-correlation function diverges as
 Photons by themselves are soft (“generic” soft mode), but a nonzero
mass!
:
gives them a
 “Integrate out” the photons:
x
x
x
with an “effective” action
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 Treat
in mean-field approximation
nonanalytic in
x
x
Lubensky, x
Ma 1974)
The effective Landau free energy is
:
(Halperin,
x
 “Fluctuation-induced” 1st order transition! (NB: Fluctuations = generic soft modes!)
 Nematic-Smectic-A transition in liquid crystals: Photons -> nematic Goldstone
modes
 Superconductors: effect is too weak to be observable; liquid Xtals: situation messy
 Particle physics: “Coleman-Weinberg mechanism”
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3. Quantum Phase Transitions: General Concepts
 Some ferromagnets have a low Tc that is often susceptible to hydrostatic pressure.
 This raises the prospect of a quantum critical point at
Tc = 0.
 Quantum critical behavior is driven by quantum
fluctuations
must be different from classical
critical behavior.
 Crossovers ensure continuity.
Q: How different is the description of QPTs in general
from that of classical transitions?
From DB, TRK, T. Vojta,
Rev. Mod. Phys. 2005
A: Very different, due to fundamental differences in
statistical mechanics.
Classical:
x
x
x
statics and dynamics decouple!
QM:
x
x
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 Solution: Trotter formula
(Trotter 1959)
 Coherent-state formalism (Casher, Lurie, Revzen 1969)
 Divide the interval
into infinitesimal slices, and integrate
 End result:

is referred to as “imaginary time” (Wick rotation
)
 Analytic continuation yields real-time (or frequency) dynamics
Statics and dynamics do indeed couple
 If
, then a quantum system in d spatial dimensions resembles a
classical system in d + z dimensions! In general,
(Hertz 1976)
 Upper critical dimension of classical system
implies
upper critical dimension of quantum system
 Mean-field theory more robust in quantum case!
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4. The Quantum FM Transition in Metals, Part I: History
 Stoner (1937): Mean-field theory for both classical and quantum case
Susceptibility:
RPA (“spin screening”)
with
and
non-magnetic electrons
the relevant interaction constant
Transition at
(non-thermal control parameter)
 Moriya et al (early 1970s): Self-consistent one-loop theory (aka self-consistent spin
fluctuation theory)
 Hertz (1976): Developed RG framework for QPTs in general, used metallic FMs as a
prime example.
 Millis (1993): Used Hertz’s RG framework to determine temperature scaling.
Plausibility arguments for Hertz’s action (with a lot of hindsight):
•
Landau free energy (again):
•
t is the inverse physical (dressed) susceptibility 1/
x
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•
RPA again:
•
Generalize everything to nonzero wavevectors
and frequencies
(Fourier trafo of
(Fourier trafo of
)
)
x
holds for noninteracting electrons
x
(Lindhard fct) AND for interacting ones
action
•

describes the dynamics of clean electrons (“Landau damping”)
•
Represents coupling of conduction electrons dynamics to the magnetization
•
At
•
for classical magnets implies
•
for quantum magnets
Conclusion (as of 1976): Mean-field theory yields exact critical behavior for
all d > 1 (
QPTs not very interesting as far as critical behavior goes)
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 As mentioned before, this is not what
is observed experimentally.
 In most clean systems, the transition
becomes first order if Tc is
suppressed far enough
ZrZn2
Uhlarz et al 2004
 Examples:
•
ZrZn2 (Uhlarz et al 2004)
•
MnSi
(Pfleiderer et al 1997, Uemura et al 2007)
•
UGe2
(Aoki et al 2011)
Uemura et al 2007
 There are many others:
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 There are many more examples (Brando et al, Rev Mod Phys in press)
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5. The Quantum FM Transition, Part II: General Guidelines
Observations:
•A tricritical point separates the 2nd order transitions from the 1st order ones.
•In a magnetic field, tricritical wings appear:
•A quantum critical point is eventually realized,
but only at a nonzero magnetic field!
•This behavior is seen in systems that are very
with respect to electronic structure,
magnet (Ising, XY, Heisenberg), etc.
a
The explanation must be universal, i.e.,
lie in stat. mech., not in solid-state effects.
x
Look for an explanation that involves
low-lying excitations (aka soft modes)
•These materials are all metals
different
type of
(Kotegawa et al 2011)
Conduction electrons likely important
Study Fermi liquids first.
Q: Don’t we know everything about Fermi liquids, and wasn’t that built into Hertz’s
x
action?
A: NO!
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Spoiler: The answer in a nutshell: A novel type of fluctuation-induced 1st
x
order transition
•The leading wavenumber dependence of the inverse susceptibility that enters the
action is
x
for 1 < d < 3
and
for d = 3
x
•This is a consequence of soft modes in a Fermi liquid, see below
•Scaling suggests h ~ k, so this nonanalyticity is also present for
function of h.
at k = 0 as a
•The magnetization is seen by the conduction electrons as an effective field.
In a generalized Landau theory,
•This leads to a generalized Landau free energy
x
which leads to a 1st-order transition.
translates into
with
x
x
•Conclusion: In clean metals, generic soft modes couple to the ferromagnetic
order1999)
(DB, TRK, T. Vojta
st
parameter and make the quantum FM transition generically 1 order.
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Lecture 3
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6. The Fermi Liquid as an Ordered Phase
a. A useful example: Classical ϕ4 – theory (again)
 O(3) ϕ4 – theory in d spatial dimensions with a field in 1-direction:
 Saddle-point solution for the ordered phase:
with
x
 Fluctuations:
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 Properties of the ordered phase:
•
Magnetization
•
Transverse susceptibility
Reduction due to
fluctuations
o Exact result, governed by Ward identity
o Transverse fluctuations are soft (Goldstone
modes of the spontaneously broken SO(3) )
o Longitudinal fluctuations are massive
•
Longitudinal and transverse modes couple
x
nonanalyticities
Potential for SO(2) ≅ U(1)
(planar magnet)
Long-ranged
correlations!
•
First derived in perturbation theory (Vaks, Larkin, Pikin 1967,
x
Brézin & Wallace 1973)
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 Derivation via RG methods:
 Expand in powers of fluctuations and gradients, assign scale dimensions,
and look for a stable fixed point.
The above behavior is exact for all d > 2 and can be described by a
 Fixed-point action
x
dimensionless
plus least irrelevant operators, e.g.
 This suffices for deriving all scaling behaviors, e.g.
 Nonanalyticities are leading corrections to scaling at the stable fixed point
 d = 2 is lower critical dimension (Mermin-Wagner ✔ )
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 Note: Similar arguments can be used to exactly characterize nonanalyticities in
various other systems, both classical and quantum. For instance,
•
The shear viscosity in classical fluids has a nonanalytic frequency dependence
since it couples to the diffusive transverse-velocity fluctuations.
•
This is an example of classical long-time tails, i.e., non-exponential decay of
correlation functions.
•
The conductivity at T = 0 in a disordered metal is a nonanalytic function of the
frequency
•
This is an example of what are called weak-localization and Altshuler-Aronov
effects in disordered metals.
We will now apply analogous arguments to a clean Fermi liquid.
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b. Goldstone modes in a Fermi liquid
(i) The stable Fermi-liquid fixed point
 A long story (Wegner 1970s, TRK & DB 1997, 2012) made very short:
 In a Fermi system at T = 0, there are two-particle excitations that are
 The Fermi liquid is an ordered phase characterized by
•
the soft modes
(analogous to π ) that
o are controlled by a Ward identity
o are Goldstone modes of a spontaneously broken symmetry
o have a linear dispersion relation
o acquire a mass at T > 0 and h > 0
•
These soft modes are the clean analogs of what are called
“(spin)-diffusons” in disordered systems, where they lead to weak-localization
and Altshuler-Aronov effects. (They are NOT density fluctuations.)
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•
an order parameter given by the DOS at the Fermi surface (more precisely: By the
spectrum of the Green’s function):
•
an order-parameter susceptibility (analogous to
•
a stable RG fixed point with fixed-point action
)
all other terms are
RG irrelevant!
x
a
•
Scale dimensions
•
Least irrelevant operators
with scale dimension
 This allows for the derivation of homogeneity laws that yield the exact leading
nonanalyticities in a Fermi liquid!
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(ii) Universal properties of the Fermi liquid
 DOS:
Jiang et al 2006
•
analogous to
•
agrees with perturbative results (Khveshchenko & Reizer
1998)
 Spin susceptibility:
 first derived in perturbation theory (DB, TRK, T Vojta 1997; Betouras et al 2005)
 NB the sign of the effect!
 Soft modes
long-ranged correlations
nonanalytic behavior
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7. The Quantum Ferromagnetic Transition, Part III
a. Generalized Landau theory
 Ordinary Landau theory:
with t = 1/
the inverse susceptibility
 Now recall the argument given earlier:
•
•
acts as an effective field
in a metal, the FL Goldstone modes couple to
via a Zeeman term
via the nonanalytic h – dependence of
x
 The FL Goldstone modes lead to a generalized Landau theory:
The quantum ferromagnetic transition in clean metals in zero
field is generically 1st order !
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!
Other consequences:
 T > 0 gives Goldstone modes a mass
x
tricritical point
 Magnetic field h > 0
tricritical wings
 Generic phase diagram:
 Predicted to hold for all clean metallic
•
Ferromagnets (isotropic or anisotropic,
itinerant or not, and even Kondo lattices)
•
Ferrimagnets (only requirement is a
homogeneous magnetization component)
•
Magnetic nematics
(DB, TRK,
J Rollbühler 2005)
 Third Law
General constraints on the shape
of the phase diagram (see my Colloquium talk)
 Pre-asymptotic region: Crossover to
Hertz-Millis-Moriya behavior. Example: MnSi
 Comparison with experiments:
Excellent qualitative agreement. The transition in
clean materials is generically 1st order. In some
systems (e.g., NixPd1-x) the transition needs to be
followed to lower T.
(Pfleiderer et al 1997)
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b. Order-parameter fluctuations
 Technical derivation: Coupled field theory for fermions and OP fluctuations.
•
Treat conduction electrons in a tree approximation
•
1-loop order
•
Higher order: Only prefactors change
Hertz’s action
nonanalyticities appear
Q: How good or bad is the mean-field approximation?
A: Nobody really knows, but:
•
OP fluctuations are above their upper critical dimension
•
In liquid crystals, they are below the upper critical dimension
•
This makes it plausible that the 1st order transition in quantum
ferromagnets is much more robust than in liquid crystals.
Other suggestions for avoiding a quantum critical point :
•
Textured phases (spiral, etc), with the length scale set by the maximum in
(Chubukov et al, Green et al)
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c. Effects of quenched disorder
Quenched disorder changes things drastically:
 The generic soft modes are now diffusive
 The susceptibility now is
x
x
(Altshuler et al early 1980s)
x
and by the same arguments as before we have, in d = 3,
NB the sign!
(TRK & DB 1996)
 This predicts a 2nd order transition with non-mean-field exponents.
 For instance,
, and
,
consistent with many observed phase diagrams
(see below for other interpretations)
 Order-parameter fluctuations lead to log-normal
modifications of the power laws (DB et al 2001)
 Increasing the disorder from the clean limit
decreases the tricritical temperature continuously
(Pikul et al 2012)
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 Prediction for evolution of phase diagram:
Complications for strong disorder:
 Quantum Griffiths effects (McCoy & Wu 1968,
D.S. Fisher 1995)
 Based on the idea of the classical Griffiths region
below the clean Tc. Diverging susceptibilities
without long-range order.
(Sang et al 2014)
 Quantum version may coexist with, and be superimposed on, quantum critical
behavior (Millis et al, Randeria et al, T. Vojta, …)
 Experiments have been interpreted using these concepts, e.g.
(Westerkamp
et al 2009)
(Pikul 2012)
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Lecture 4
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8. Exponents and Exponent Relations at Quantum Critical
x Points
Now consider a continuous QPT, e.g. the FM one in Ni3 Al1-x Ga2:
A classical critical point is characterized by power-law behavior of observables,
and corresponding critical exponents:
Order parameter:
Order-parameter susceptibility:
Correlation length:
Specific heat:
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A quantum critical point we can approach at T = 0 by varying the non-thermal
control parameter t, or at t = 0 by letting T -> 0.
 We need more critical exponents!
Order parameter:
Order-parameter susceptibility:
Correlation length:
The specific heat vanishes at T = 0 => Consider the
Specific-heat coefficient:
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The T- exponents will in general be different from the t- exponents! What is the
relation? Various points towards an answer:
In quantum statistical mechanics, inverse temperature is related to (imaginary)
time. => T- scaling is governed by a dynamical exponent z that describes how
the relaxation time diverges as a function of the relaxation length:
Power laws result from generalized homogeneity laws for observables
with b an arbitrary length rescaling factor and
x (and r = t, sorry!).
:
a scaling function
Put b = r-ν =>
At r = 0 we have
with
Conclusion: If
describes the static scaling behavior, then
describes the temperature scaling behavior
Caveat: This is true only under special circumstances. More generally, the Tscaling of
may be described by A z, rather than by THE z, due to multiple
time scales and/or dangerous irrelevant variables.
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That’s a lot of exponents. How many are independent?
First consider a classical transition. Look at the homogeneity law for the order
parameter:
✓
=>
✓
Now, the susceptibility
is a thermodynamic derivative of
:
This yields
=>
=>
“Widom’s equality”
Similar arguments lead to, e.g.,
“Essam-Fisher equality”
“Fisher equality”
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Some exponent relations depend explicitly on the dimensionality, e.g.
“hyperscaling”
They are not valid if the dimensionality is larger than the “upper critical
dimension”. For many classical transitions, that’s d = 4.
Classically, only two exponents are independent.
These exponent relations all depend on various forms of scaling being valid.
While that’s usually the case at critical points, there is no guarantee. Weaker
statements that depend only on thermodynamic stability take the form of
rigorous inequalities. For instance,
“Rushbrooke inequality”
In disordered systems, a rigorous lower bound on the correlation length
exponent is known:
Chayes, Chayes, Spencer, Fisher (1986)
(See also the “Harris criterion”).
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At quantum critical points, some of the classical relations still hold, e.g.,
Widom
Fisher
Analogous relations hold for the T- exponents:
Others change. For instance, a generalization of Essam-Fisher becomes
where
and
are the dynamical exponents that govern the T –
dependence of the order parameter and the specific heat, respectively.
These two are in general NOT the same!
The rigorous Rushbrooke inequality holds for the T – exponents:
but gets modified for the t – exponents:
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Hyperscaling relations, e.g.,
again hold only below some upper critical dimensionality, which tends to
be lower for quantum phase transitions than for classical ones.
Overall, at a quantum critical point there are at least two independent
static critical exponents, plus the dynamical ones. Depending on what
form of scaling holds (which depends on the critical point and the
dimensionality) there may be as many as five independent static
exponents. For details, see
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9. “How Close is Close to the Critical Point?”, or
x How Hard is it to Measure Quantum Critical Exponents?
Q: How close to the critical point does one have to go in order to observe
asymptotic critical behavior?
A: It depends on the critical point and the observable, but in general very close.
At many classical critical points one needs to be closer than 0.1%, and then one
needs two or three decades to convincingly see the power laws!
Plus, the extracted exponent values can depend strongly on the value of Tc!
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Farther away from criticality one often observes effective power laws with
exponents that are controlled by unstable RG fixed points.
There is no reason to believe that requirements at quantum critical points are
less stringent. Various issues:
 Many quantum critical points are controlled by chemical composition. This is
hard to control, and the critical concentration is typically not known very
precisely.
 Additional physics can mask quantum criticality, e.g., quantum Griffiths
effects in many disordered quantum ferromagnets.
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Some examples:
 Ni3 Al1-x Ga2 again:
The observed behavior is in agreement with Hertz-Millis theory. There are good
theoretical reasons to believe that this is not the asymptotic critical behavior.
For instance,
, which violates the rigorous bound
.
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 URu2-x Rex Si2 :
Critical concentration
known to not better than
10%, observed exponents
almost certainly not
critical exponents.
Additional physics is
known to be present
(Hidden Order, Quantum
Griffiths effects?)
Butch & Maple (2009)
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10. Phase Separation Away from the Coexistence Curve
Return to the observed phase separation in
systems that show a 1st order transition.
•PS away from the coexistence curve!
•Observed by various techniques (μSR, NMR,
NQR)
•Observed in many different systems: QFMs,
heavy-fermion systems, rare-earth nickelates
(Mott transition), etc.
•Schematic phase diagrams:
Conceivable explanations:
• Non-equilibrium effects
(unlikely, see MnSi samples)
• Droplet formation due to
quenched disorder
TRK & DB arXiv:1602.01447
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Summary
 The quantum ferromagnetic transition in clean metals is generically 1st order.
 This can be understood as a fluctuation-induced 1st order transition due to
the coupling of the magnetization to generic soft modes in metals (clean
versions of diffusons).
 The Fermi liquid can be understood as an ordered phase with a broken
symmetry, with the generic soft modes as Goldstone modes.
 At nonzero temperature there is a tricritical point, and tricritical wings.
 Good agreement between theory and experiments.
 Quenched disorder changes a crucial sign, leading to a 2nd order transition.
Strong disorder leads to additional complications.
 Additional critical exponents are needed for QCPs.
 True critical behavior is hard to observe. In classical systems it took
decades to obtain reliable exponents. For QCPs, experiments are probably
still far from the necessary level of precision.
 Phase separation away from a coexistence curve requires an explanation.
Suggestion: Droplet formation due to quenched disorder.
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Recommended Reading
Quantum Ferromagnets:
•
M. Brando et al, arXiv:1502.02898, Rev. Mod. Phys., in press
Generic Scale Invariance:
•
DB, TRK, Thomas Vojta, Rev. Mod. Phys. 77, 579 (2005)
Quantum Phase Transitions:
•
J. Hertz, Phys. Rev. B 14, 1165 (1976)
•
S. Sachdev, Quantum Phase Transitions (Cambridge Univ. Press 1999)
Scaling, and Renormalization Group:
•
H.E. Stanley, Introduction to Phase Transitions and Critical Phenomena (Oxford Univ. Press 1971)
•
S.-K. Ma, Modern Theory of Critical Phenomena (Perseus 1976)
•
M.E. Fisher, in Advanced Course on Critical Phenomena, F.W. Hahne (ed.) (Springer 1983)
•
J. Cardy, Scaling and Renormalization in Statistical Physics (Cambridge Univ. Press 1996)
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