11institutetext: 1: Theoretical Division, Los Alamos National Laboratory, Los
Alamos, New Mexico 87545, USA
2: Center for Integrated Nanotechnologies, Los Alamos National Laboratory, Los
Alamos, New Mexico 87545, USA
3: Department of Physics, Washington University, St. Louis, Missouri 63160,
USA
4: Department of Electrophysics, National Chiao Tung University, Hsinchu
30013, Taiwan

# Defects and glassy dynamics in solid 4He: Perspectives and current status

A. V. Balatsky1,2 M. J. Graf1 Z. Nussinov3 J.-J. Su4

(09.04.2012)

###### Abstract

We review the anomalous behavior of solid 4He at low temperatures with
particular attention to the role of structural defects present in solid. The
discussion centers around the possible role of two level systems and
structural glassy components for inducing the observed anomalies. We propose
that the origin of glassy behavior is due to the dynamics of defects like
dislocations formed in 4He. Within the developed framework of glassy
components in a solid, we give a summary of the results and predictions for
the effects that cover the mechanical, thermodynamic, viscoelastic, and
electro-elastic contributions of the glassy response of solid 4He. Our
proposed glass model for solid 4He has several implications: (1) The anomalous
properties of 4He can be accounted for by allowing defects to freeze out at
lowest temperatures. The dynamics of solid 4He is governed by glasslike
(glassy) relaxation processes and the distribution of relaxation times varies
significantly between different torsional oscillator, shear modulus, and
dielectric function experiments. (2) Any defect freeze-out will be accompanied
by thermodynamic signatures consistent with entropy contributions from
defects. It follows that such entropy contribution is much smaller than the
required superfluid fraction, yet it is sufficient to account for excess
entropy at lowest temperatures. (3) We predict a Cole-Cole type relation
between the real and imaginary part of the response functions for rotational
and planar shear that is occurring due to the dynamics of defects. Similar
results apply for other response functions. (4) Using the framework of glassy
dynamics, we predict low-frequency yet to be measured electro-elastic features
in defect rich 4He crystals. These predictions allow one to directly test the
ideas and very presence of glassy contributions in 4He.

###### Keywords:

Thermodynamics torsional oscillator shear modulus dielectric function glass
viscoelastic electro-elastic supersolid solid helium

###### pacs:

67.80.B-, 64.70.Q-, 67.80.bd

††journal: Journal of Low Temperature Physics

##  1 Introduction

The discovery of anomalous frequency and dissipation behavior seen in
torsional oscillators (TOs) Kim04a ; Kim04b  at low temperatures in 4He has
stimulated numerous investigations. These anomalies have been argued to
demonstrate a non-classical rotational inertia (NCRI) of the long ago
predicted supersolid quantum stateAndreev69 ; Chester67 ; Reatto69 ; Leggett70
; Anderson84 . Successive TO experiments Rittner06 ; Kondo07 ; Aoki07 ;
Clark07 ; Penzev07 ; Hunt09 ; Pratt11 ; Gadagkar2012  confirmed the finding of
the reported anomalous behavior. Hysteresis behavior and long equilibration
times have been observed Aoki07 ; Hunt09 ; Kim09 , which depend strongly on
growth history and annealing Rittner06 . In the same temperature range,
experiments including shear modulus Beamish05 ; Beamish06 , ultrasonic
Goodkind02 ; Burns93  and heat propagation Goodkind02  have also shown various
anomalies. The character and the existence of mass flow is still a matter of
intense investigation. Experiments designed to probe for mass flow by
squeezing the lattice report no such flow Beamish05 ; Beamish06 ; Greywall77 ;
Paalanen81 ; Sasaki06 ; Ray08 ; Bonfait89 ; Balibar08 . However, experiments
in which a chemical potential gradient was created via coupling to a
superfluid reservoir, suggest mass flow of an unusual type Ray08 ; Ray09 ;
Ray10 ; Ray11 ; Vekhov2012 . These and many other results have led to a flurry
of activity. Structural measurements Burns08 ; Blackburn07  suggest that solid
4He may be composed of a dynamic mosaic of crystals, highlight the importance
of defects, and report the absence of notable structural change in the
vicinity of the putative supersolid transition. The intricate structure and
dynamics of sold 4He make it a fascinating system. We note that when a mosaic
of 4He crystals with intervening liquid channels is placed in a metallic
container Ray08 ; BalibarComment , thermal expansion effects (in particular
the larger thermal expansion coefficient of helium vis a vis that of the
metallic container) may be at play in blocking a remnant superleak. Such a
thermal compression will effectively shut down any mass flow and close the
open channels as the temperature is raised. This scenario remains a viable
explanation for the anomalous mass flow and fountain effect reported by
Hallock’s group Ray08 ; Ray09 ; Ray10 ; Ray11 ; Vekhov2012 , until a superleak
can be ruled out.

After eight years of intense experimental and theoretical investigations one
must ask what are the established facts, what are the current expectations and
hypotheses, and what are the future directions of research in solid 4He. There
are numerous reviews and progress updates available describing the field
ProkofevAdvances ; Balibar2011 ; Boninsegni2012  and addressing some of these
questions. Most of the literature reviewing the subject deals with the notion
of supersolidity in 4He as established and proceeds with the discussion on the
status and future experiments with the goal of further “proving” the existence
of a supersolid phase transition at very low temperatures. Amongst the many
exciting proposals concerning 4He as well as supersolids, we briefly mention
Andreev’s proposal for superglass Andreev07 ; Andreev09 ; Korshunov09 ,
Anderson’s suggestion of vortex proliferation and flow Anderson2007 ; Kubota
and supersolid dislocation cores dislocation_core1 ; dislocation_core2 ;
dislocation_core3 ; Rossi . As will become evident in later sections, our
analysis centers on the dynamical effects of defects and as such may include
vortices, dislocations, or any other defects. Estimating the product of
typical dislocation core sizes ($\sim 3$ nm) and their density ($\sim 10^{10}$
m-2), it is seen that a direct NCRI origin from superfluid dislocation cores
in 4He cannot account for the magnitude of the TO anomaly (it is orders of
magnitude too small).

We believe that at this mature stage of research in 4He a somewhat more
general view on possible options is needed, where one asks the question: what
are the options for possible states that might form at low temperatures in 4He
and by implication in other solid bosonic matter? By taking a broader view one
explicitly allows for states other than pure supersolidity and includes
possible coexistence phases to form as well. This review is a contribution in
the spirit of broadening the conversation by explicitly allowing for other
components or “active ingredients” to be present in addition to, or perhaps
instead of, supersolidity in 4He. Thus, if one accepts the presence of defects
in a solid, then naturally the question arises about the dynamic signatures of
such crystal defects and whether they can dominate the response to an external
stimulus.

Figure 1: The phase diagram of 4He with the putative supersolid phase
transition below 0.3 K Kim04b  or an alternate order-disorder dominated
crossover region governed by impedance matching between the applied frequency
and internal relaxation processes.

We provide a brief critical analysis of some of the existing data and point
out that a significant fraction of the data on mechanical, thermodynamic and
dielectric properties of 4He can be analyzed in terms of the emergence of a
dissipative viscous component that we shall call glassy component Balatsky07 ;
Nussinov07 ; Graf08 ; Su10a ; Su10b . This extra component by itself is
sufficient to modify numerous properties of solid 4He and can be responsible
for anomalous thermodynamic, elastic and dielectric properties in solid 4He
observed in experiments. The glassy component is not a supersolid in the
classical sense ProkofevAdvances , yet it can coexist and couple to a
supersolid component as some of the proposed superglass phases indicate Hunt09
; Boninsegni06 ; wu2008 ; Biroli08 .

The precise nature of the state of solid 4He at lowest temperatures remains a
puzzle. In the past, investigation of 4He has played an important role in the
development of basic concepts in modern condensed matter physics like
superfluidity, order parameter, topological excitations, and critical
exponents. Given such a prominent role it played in the past it is paramount
to come up with the resolution of the puzzles that are clearly seen in
experiments at lowest temperatures. Yet there is one important difference that
might be key to a solution this time. We propose that precisely because the
compound is clean and very well characterized the enabling component for the
anomalies at lowest temperature are defects that undergo freeze-out and
constitute a glass like component.

To sharpen this point of the discussion, any supersolid component would imply
some sort of two-fluid hydrodynamics that schematically equates the total mass
current of helium atoms as a sum of normal component and superfluid component
determined by the normal and superfluid mass fractions $\rho_{n}$ and
$\rho_{s}$ and respective velocities. In this two-fluid picture the total mass
current is given as

${\bf j}=\rho_{n}{\bf v_{n}}+\rho_{s}{\bf v_{s}}$ (1)

In any superfluid or supersolid phase the coefficients $\rho_{n}(T,P)$ and
$\rho_{s}(T,P)$ are functions of thermodynamic variables like temperature $T$
and pressure $P$. There is no direct evidence of either dc or ac supersolid
mass current at lowest temperatures, where the putative supersolid state sets
in Rittner06 ; Beamish05 , except for the experiments by Hallock Ray08 ; Ray09
; Ray10 ; Vekhov2012 . However, it remains to be seen whether Hallock’s
results of mass flow and fountain effect cannot be explained through the
presence of superleaks connecting the superfluid leads through solid helium
with channels. Therefore, some even proposed that one cannot “squeeze a
superfluid component out of a stone” Dorsey06 . We know that these supersolid
expectations do not apply in the case of solid 4He at lowest temperatures,
because TO experiments clearly indicate that the period and damping exhibit
significant hysteresis effects and strong dependence on the history of sample
preparations and annealing. All these experimental observations combined would
suggest to an impartial observer that there is at the very least, in addition
to supersolidity, another physical component at play in 4He. Our analysis of
various experiments suggests that the crossovers, seen in the specific heat,
TO, shear modulus and dielectric function experiments Aoki07 ; LinChan07 ;
Beamish10 ; Yin11  are a reflection of the physics whose origin is not due to
supersolidity, but a consequence of the dynamics of defects in solid 4He .

Specifically, we propose the presence of a glassy component in solid 4He at
low temperatures in order to explain the observed anomalous linear temperature
dependence in the specific heat of an otherwise perfect Debye solid 4He
Balatsky07 ; Graf08 . We used similar ideas when we analyzed the TO, shear
modulus, and dielectric properties by assuming the presence of a glassy
component at the parts-per-million (ppm) concentration and asked what the
dynamic consequences should be. With the wealth of data available we do not
attempt to provide a complete overview of the field, but give a summary of the
work centered on the role of a glassy component in an otherwise nearly perfect
crystal.

The exact nature of the glassy component is not known. For example, it may be
caused by two-level systems of pinned dislocation lines, vortex excitations,
etc. It is however important to point out that the amplitude of period shift
can be changed dramatically and depends on growth history and annealing
procedures of the crystal. To explain the puzzling features of solid 4He we
_conjecture_ that structural defects like localized dislocation segments or
groups of displaced (out-of-equilibrium) atoms effectively form a set of two
level systems (TLSs) which are present at low temperatures. These immobile
crystalline defects will affect the thermodynamics of solid 4HeBalatsky07 ;
Graf08 , the mechanics of TOs Nussinov07  and shear modulus Su10a ; Su10b ,
and dielectric properties Su11 . Other mechanisms for TLS and glassy dynamics,
e.g., due to point defects and grain boundaries are possible as well. We are
at a stage where phenomenology allows us to make progress with testable
predictions, while a microscopic picture of crystal defects and interactions
is still missing.

Early on it was recognized that pinned vibrating dislocation lines can account
for a plethora of anelastic damping phenomena in the ultrasound, TO, and shear
modulus experiments. Since most experiments are believed to be in the linear
or elastic strain-stress regime any plastic deformation due to the motion of
gliding dislocation loops has been neglected. However, this is not necessarily
so. We proposed Caizhi2012  that dislocation-induced anomalous softening of
solid 4He is possible due to the classical motion of gliding dislocation lines
in slip planes. This picture of dislocation motion is widely accepted in
conventional metals. Similar effects are at play in the quantum arena where
mobile dislocations (dislocation currents) lead to a screening of applied
shear via a Meissner-Higgs type effect Zaanen04 ; Cvetkovic08 . Such unpinned
dislocations that screen shear render the system more fluid and may, in line
with the framework that we advance in this article, trigger the 4He anomalies
as the mobile dislocations become quenched at low temperatures
dislocation_core3 ; Caizhi2012 . Recent experiments provide further impetus
for such a picture eyal12a ; eyal12b .

The technique of choice for interrogation of solid 4He has been the TO with
varying degrees of complexity. However, the TO does not provide any direct
information on the microstructure of samples. More direct structural x-ray and
neutron measurements do not have the adequate resolution to detect any changes
at the ppm level in the structure of 4He at lowest temperatures. Additional
challenges arise given how small the volume fraction of the glassy component
can be. We estimate it to be in the range of few hundreds of ppm in the
specific heat and pressure contribution. Therefore the precise
characterization of the microstructure of solid 4He, growth history,
annealing, and 3He dependence remain pressing issues in resolving the
hypothesis of the presence of a glassy component and TLS in solid 4He. The
notion of importance of the role of disorder in solid 4He received further
support over the years in observations of the strong dependence of TO results
on the specific design of sample cells, see work by the group of Chan
DYKim2012 . It is hard to imagine that an intrinsic material property like
supersolidity should depend strongly on the stiffness or geometry of the
torsional oscillator apparatus, while crystalline disorder can easily be
affected by those design properties.

In the analysis of the observed excess specific heat, we used a model of
independent TLS, which gives the canonical signature of a linear in
temperature contribution at lowest temperatures. Building on the presence of
TLS we evaluated the mechanical properties of the TO using a model of quenched
defects. This approach allows us to make predictions on the viscoelastic
properties of 4He and on the electro-elastic coupling that can be tested in a
setup that does not require the TO and hence can be tested over a much broader
frequency range. These predictions allow to directly verify the very presence
of quenched defects in 4He. We therefore would welcome any direct tests of our
ideas.

The remainder of this paper is organized as follows. We present the general
discussion on the role of quenched defects and disorder in solid 4He with
particular attention to the consequence of defects on the dynamics in Section
2. This is followed in Section 3 by an analysis of the thermodynamic
properties of solid 4He. In Section 4, we present a unified framework to
analyze dynamical response function that invokes arbitrarily high order
backaction effects of defects onto the solid bulk. We then invoke, in Section
5 this approach and summarize our analysis of the torsional oscillator. In
Section 6 we discuss the shear modulus analysis and the strain-stress
relations using a viscoelastic model designed to capture the anelastic
contribution from defects. In Section 7 we discuss predictions for the
dielectric properties that follow from the viscoelastic model with locally
frozen-in defect dynamics. Finally we conclude with a discussion and give our
view on future perspectives in the field.

##  2 Defect Quenching and Its Implications

In this review, we will examine the general consequences of a transition from
mobile defects (or dynamic fluid-like components) at high temperatures to
quenched immobile defects at low temperatures. Microscopically, these defects
may be dislocations, grain boundaries, vortices, or others. It may be posited
that in quantum solids such as 4He , zero point motion leads to larger
dynamics than is common in classical solids. To put the discussion in
perspective, we recall a few rudiments concerning annealing and quenching.
When quenched, systems fall out of equilibrium. En route to non-equilibrium
states, relaxation times increase until the dynamic components essentially
“freeze” (on pertinent experimental time scales) into an amorphous state. In
materials with (sufficiently large) external disorder, quenching may lead to
“spin-glass” characteristics. By contrast, in the absence of imposed external
disorder, when fluid components (either classical or quantum) fall out of
equilibrium by sufficient rapid cooling (so-called “super-cooling”) to low
enough temperatures, the resultant state is termed a “glass”. As liquids are
supercooled, their characteristic relaxation times and viscosity may increase
dramatically. If, instead, the temperature is lowered at a sufficiently slow
rate, the system does not quench and instead remains an “annealed” system in
thermal equilibrium. Notwithstanding exciting progress, exactly how the
dynamic components evolve in 4He crystals as the temperature is lowered is,
currently, an open problem. An initially surprising and undeniable feature
that has, by now, been seen in many experiments is that the solid 4He
anomalies depend dramatically on the growth history of the crystal and
diminish as the system is cooled down slowly and defect quenching is thwarted.
Memory effects reminiscent to those in glasses are further present. These
observations imply that “there is more to life” than static NCRI and other
annealed supersolid properties on their own– quenching plays a definitive role
in triggering the observed effects. To understand the experimental
observations and build a predictive theoretical framework, we invoke general
physical principles allowing a computation of response functions and using as
input known characteristics of quenching. Physically, as alluded to in the
Introduction, the quenching is that of the mobile defects (which may
constitute only a tiny fraction of the system) as they become arrested against
a crystalline background. Our analysis does not rule out the presence of a
small supersolid component. The observed large change in the TO dissipation
cannot be solely described by uniform Bose-Einstein condensation Nussinov07 ;
Graf08 ; Huse07 . It remains to be seen if nonuniform Bose-Einstein
condensation alone either along grain boundaries Clark06  or along the axis of
screw dislocations Boninsegni06 ; Pollet07 ; Boninsegni07  can explain the
dynamic response of TOs. Note that simple estimates of the supersolid fraction
of dislocation cores are orders of magnitude too small.

There exists a vast literature on defect quenching in systems that range from
vortices in superfluid Helium to cosmic strings Kibble76 ; Zurek85  and
countless others. Our particular focus is, of course, on defects in a
crystalline system (solid 4He). Quenched dynamics of such defects is
associated with a change of plasticity and related internal dissipation.
Dielectric (and other) response functions in systems of varying plasticity,
such as various glass formers as their temperature is lowered, indicate, in a
nearly universal fashion, the presence of a distribution of local relaxation
times. These lead to the canonical Cole-Cole or Cole-Davidson distribution
functions and related forms as we briefly elaborate. In an overdamped
dissipative system, an impulse (e.g., an external electric field or an elastic
deformation) at time $t=0$ leads a response $g(t)$ which at later times scales
as $g_{single}\sim\exp(-t/\tau)$ where $\tau$ is the (single) relaxation time.
When Fourier transformed to the complex frequency ($\omega$) plane, this leads
to the Debye relaxor $g_{single}(\omega)=g_{0}/(1-i\omega\tau)$. Now, in
systems that exhibit a distribution $P(\tau^{\prime})$ of local relaxation
events, the response functions attain the form $\int
d\tau^{\prime}P(\tau^{\prime})\exp(-t/\tau^{\prime})$. Empirically, in
dissipative plastic systems, relaxations scale as $(\exp(-t/\tau)^{c})$ with a
power $0<c<1$ that leads to a “stretching” (slower decay) of the response
function as compared to its single overdamped mode form of $\exp(-t/\tau)$.
This stretched exponential and other similar forms of the response function
capture the quintessence of the distribution of relaxation times. Two widely
used relaxation time distributions are the Cole-Cole (CC) and Davidson-Cole
(DC) functions that describe a superposition of overdamped oscillators (Debye
relaxors) Phase1 ; Phase2 . With $g(\omega)=g_{0}G(\omega)$, where $g_{0}$ is
a material specific constant, these two forms are given by different choices
for the function $G$,

$\displaystyle G_{CC}(\omega)=1/[1-(i\omega\tau)^{\alpha}],$ $\displaystyle
G_{DC}(\omega)=1/[1-i\omega\tau]^{\beta}.$ (2)

Values of $\alpha$ and $\beta$ that differ from unity qualitatively play the
role of the real-time stretching exponent $c$. In the dc limit the mechanical
motion of any mobile component will have ceased and there will be no relative
motion and no transients. Therefore the coefficient $g_{0}$ is generally a
function of frequency. In the case of the single TO its value is set by the
resonance, $g_{0}\approx g_{0}(\omega_{0})$.

These relaxation times can be associated with a distribution of TLSs
describing viable low temperature configurations of the defects. The simple
TLS analysis can account for thermodynamic measurements. Recent work Vural11
obtained results beyond the TLS model with fewer parameters for generic non-
uniform systems irrespective of specific microscopic origin. We will, for the
sake of simplicity, review our work on the low temperature properties of 4He
assuming TLSs. We conjectured Balatsky07  that structural defects, e.g.,
localized dislocation segments, form such a set of TLSs at low temperatures.
These immobile crystal defects affect the thermodynamics Balatsky07 ; Graf08
of bulk 4He and the mechanics Nussinov07  of the TO loaded with 4He. For the
analysis of the specific heat, we used independent TLS to obtain the universal
signature of a linear-in-temperature specific heat term at low temperatures.

##  3 Thermodynamics

Any true phase transition, including supersolid, is accompanied by a
thermodynamic signature. Therefore it was anticipated that thermodynamic
measurements will resolve the existing puzzles of supersolidity. The search
for such thermodynamic signatures proved to be challenging so far, see e.g.,
measurements of the specific heat Swenson62 ; Frank64 ; ClarkChan05 ;
LinChan07 ; LinChan09 ; WestChan09 , measurements of the pressure dependence
of the melting curve Todoshchenko06 ; Todoshchenko07 , and pressure-
temperature measurements of the solid phase Grigorev07 ; Grigorev07b ;
Lisunov2011 . The main difficulties lie in measuring small signals at low
temperatures in the presence of large backgrounds. With improving experiments
measurements were conducted down to 20 mK. While there is still no clear
evidence of a phase transition in the melting curve experiments, recent
pressure measurements and specific heat measurements have both shown
deviations from the expected pure Debye lattice behavior. Early on we proposed
that these deviations might be related to a glass transition and be described
by the contributions of two-level systems (TLS). Balatsky07 ; Graf08

We model the system of noninteracting TLS with a compact distribution of two-
level excitation spacings, see Fig. 2. Our results show that the low-
temperature deviations in the measured specific heat can be explained by
contributions from a glassy fraction and/or TLS of the solid.

###  3.1 Two level system model for the specific heat

Figure 2: Density of states (DOS) of the two-level tunneling system. The
black-dashed line represents the DOS for the standard glass model Anderson72 ;
Phillips72 ; Balatsky07 , while the blue-solid line is the truncated DOS used
to describing the TLS with a cutoff energy.

To avoid complications due to the presence of 3He atoms, we will compare the
effect of different growth processes on ultrapure 4He containing at most
(nominally) 1 ppb of 3He atoms. At such low levels of impurities, we expect to
see the intrinsic properties of solid.

We postulate that at temperature much below the lattice Debye temperature, the
specific heat of solid 4He is described by

$\displaystyle C(T)=C_{L}(T)+C_{g}(T),$ (3)

where the lattice contribution to the molar specific heat is given by
$C_{L}(T)=B_{L}T^{3}$, with coefficient $B_{L}=12\pi^{4}R/5\Theta_{D}^{3}$,
$R=8.314$ J/(mol K) is the gas constant, and $\Theta_{D}$ is the Debye
temperature. The second term describes the glass contribution due to the TLS
subsystem and is given by
$C_{g}(T)=k_{B}R\frac{d}{dT}\int_{0}^{\infty}dE\,{\cal D}_{g}(E)\,f(E),$ with
$k_{B}$ the Boltzmann constant and $f(E)$ the Fermi function. The density of
states (DOS) of the TLS may be modeled by the box distribution function (Fig.
2):

$\displaystyle{\cal D}_{g}(E)=\frac{1}{2}{\cal
D}_{0}\left[1-\tanh((E-E_{c})/W)\right].$ (4)

Here ${\cal D}_{0}$ is the zero-energy DOS, $E_{c}$ is a characteristic cutoff
energy, and $W$ is the width of the truncated density of states. For
$E_{c}\to\infty$, one obtains the standard hallmark result of glasses at low
temperatures: $C_{g}(T)=B_{g}T$, where $B_{g}=k_{B}R{\cal D}_{0}$. As we will
elaborate in the next section, the glass coefficient $B_{g}$ has an intrinsic
finite value at low temperature even for the purest 4He samples, independent
of 3He concentration.

As shown in (4), our model goes beyond the standard glass model by introducing
a cutoff in the DOS of the TLS (Fig. 2). The cutoff could be due to the finite
barrier height of double-well potentials giving rise to the TLS because in
real materials the tunneling barrier has an upper bound set by lattice and
dislocation configurations Jaeckle72 . At high temperatures, the TLS
contribution is less important since the thermal energy easily overcomes the
barrier and effectively resembles a single harmonic degree of freedom.

Figure 3: $\delta C/T$ for experiments (squares) LinChan09  and the modified
glass model with a cutoff energy in the TLS DOS (blue line) of four samples
with different structural quality where $\delta C=C-C_{L}$. Note the large
deviation of data points at high temperatures in the highest purity crystal
SL34 of solid-liquid coexistence, which makes the subtraction of the Debye
contribution in this sample questionable.

####  3.1.1 Specific Heat

We compare our calculated specific heat with the experimental data by the Penn
State group LinChan09 ; LinChan07  for four different growth processes: BC20,
BC04, SL34 and SL31. BC20 (04) is the sample grown by blocked capillary (BC)
method over 20 (4) hours. SL 34 (31) represents the samples in solid-liquid
coexistence state with 34 (31) percents of solid ratio. Notice that sample
SL34 actually corresponds to their reported 75% solid-liquid coexistence
sample and SL31 corresponds to their constant pressure sample (CP) Su10a ;
LinChan09 . The data are described with three parameters: ${\cal D}_{0}$,
$E_{c}$ and the Debye temperature $\Theta_{D}$. We first determine
$\Theta_{D}$, or the lattice contribution, from the high-temperature data. The
lattice contribution is then subtracted from $C$ to obtain the difference
$\delta C=C-C_{L}$. We fit $\delta C/T$ with our specific heat formula for
TLS.

Next we plot $\delta C/T$ in Fig. 3. The TLS model with cutoff describes well
the data. In these plots we fixed the width of the cutoff to $W=1\,{\rm\mu
eV}$ since there is no qualitative difference when varying $W$ within
reasonable range. Notice that the shape of $\delta C(T)$ depends strongly on
the subtraction of the high-temperature lattice contribution.

The TLS behavior is mainly characterized by the zero-energy DOS and the cutoff
energy of the TLS, which are both noticeably larger in BC04, see Table 1. This
may be explained by the rapid growth process of a strained crystal, which
gives rise to both a larger TLS concentration, $n_{\rm TLS}$, and a smaller
cutoff energy, $E_{c}$, i.e., a smaller maximum tunneling barrier height.
Since the TLS concentrations of these samples range from $3.7$ to $21.5$ ppm,
which are at least 1000 times larger than the nominal 3He concentration, we
believe that 3He has no effect on the observed intrinsic heat capacity of
ultrapure solid 4He.

| $P$ | $V_{m}$ | $\Theta_{D}$ | ${\cal D}_{0}\times 10^{4}$ | $E_{c}\times 10^{2}$ | $n_{\rm TLS}$ | $\Delta S$
---|---|---|---|---|---|---|---
| (bar) | (cm3/ mol) | (K) | (1/meV) | (meV) | (ppm) | ($\mu$J/(mol K))
SL34 | 25 | 21.25 | 24.5 | 2.2 | 1.7 | 3.7 | 21.3
SL31 | 25 | 21.25 | 24.8 | 2.9 | 2.2 | 6.4 | 36.9
BC20 | 33 | 20.46 | 29.7 | 3.0 | 2.3 | 6.9 | 39.5
BC04 | 33 | 20.46 | 28.9 | 6.5 | 3.3 | 21.5 | 115.0
Table 1: Physical and model parameters: Debye temperature $\Theta_{D}$, zero
energy TLS DOS ${\cal D}_{0}$, cutoff energy $E_{c}$, concentration of TLS
$n_{\rm TLS}={\cal D}_{0}\times E_{c}$ and excess entropy $\Delta S$.

####  3.1.2 Entropy Analysis

Our analysis of the excess entropy supports the existence of a glassy
component or TLS. The excess entropy,

$\displaystyle\Delta S(T)=\int_{0}^{T}dT^{\prime}\,\delta
C(T^{\prime})/T^{\prime},$ (5)

is associated with an excess specific heat. We find consistently for specific
heat measurements ClarkChan05 ; LinChan07 ; LinChan09  that the obtained
entropy values $\Delta S\sim 10^{-4}$ J/(K mol) are 5 to 6 orders of magnitude
smaller compared to the theoretical prediction for a homogeneous supersolid if
the entire sample actually underwent Bose-Einstein condensation (BEC). In the
limit of a non-interacting BEC bosons with a quadratic dispersion one finds
the standard result $\Delta
S_{BEC}=15/4(\zeta(5/2)/\zeta(3/2))\,R(T/T_{c})^{3/2}\sim(5/4)\,R\sim 10.4$
J/(K mol). This means that if $\Delta S$ is indeed due to supersolidity, then
the supersolid volume fraction is at most 11 ppm or 0.0011% in the most
disordered or quenched sample of the four ultrapure samples studied in this
work, i.e., sample BC04. Such a supersolid fraction in the specific heat is
more than 100 to 1000 times smaller than is usually reported for the non-
classical rotational inertia fraction (NCRIF) in TO experiments. This enormous
discrepancy between supersolid fractions in specific heat and TO experiments
was already noticed in Refs. Balatsky07 ; Graf08 , while Lin et al. LinChan09
keep invoking a hyperscaling mechanism of unknown origin. Until to date, this
discrepancy remains a major puzzle that is hard to reconcile within a
supersolid scenario.

The validity of the analysis of the entropy in terms of a non-interacting BEC
is repeatedly questioned on grounds of how robust it is in the presence of
interactions. We discussed in our original study Balatsky07  the effects of
interactions on the entropy. Here it is important to realize that the entropy
is a total count of all low-energy states irrespective of a particular model
for the specific heat. Hence, we concluded that strong-coupling effects cannot
change the order of the effect. They may only change the magnitude. For
example, the well-known strongly correlated superfluid system 4He possesses a
superfluid entropy of $\sim 0.6R\sim 4.6$ J/(K mol) Ahlers1973 , which is only
half the value of the non-interacting BEC. With hindsight this justifies the
neglect of strong-coupling effects in the order of magnitude analysis. Clearly
the entropy is a reliable measure of any phase transition in 4He. No matter
how one looks at this puzzle, the reported excess entropy is either far too
small to explain observed NCRIF effects in torsional oscillators or far too
large to describe the boiling off of 3He atoms from dislocation lines, when
the nominal concentration of 3He is less than 1 ppb. For those reasons, we
argued in favor of two-level systems of low-lying states until a better
microscopic understanding of solid 4He emerges.

####  3.1.3 Comparison with Pressure Measurements

Figure 4: ($P-P_{0})/T^{2}$ vs. $T^{2}$ for $P\sim 33$ bar. The squares
represent the data reported by Grigor’ev et al. Grigorev07 ; Grigorev07b  The
blue-thick and black-thin lines are predictions for BC04 and BC20,
respectively. The black-dotted line has been shifted vertically by a constant
compared to BC04 to illustrate the capability of describing Grigor’ev’s data
by the TLS model. Predicted curves for BC04 and BC20 are shown for comparison.

Figure 5: Low-temperature pressure deviation from lattice contribution of
Debye solid (data by Yin et al. Yin11 ). The intercept of $\Delta P/T^{2}$ vs.
$T^{2}$ extrapolated from low-temperature data points is in agreement with the
TLS contribution of order 100 ppm ($\Delta P=P-P_{0}$). The arrow marks the
deviation from the Debye solid behavior. The large scatter in data at lowest
$T$ is due to the subtraction of $P_{0}$. Dashed lines are guides to the eyes.

Next we relate the pressure measurements with the specific heat measurement
through our model. The quantities to characterize the pressure measurement in
the combined lattice and glass models are $a_{L}$ and $a_{g}$ defined by

$\displaystyle P(T)\equiv
P_{0}+P_{L}(T)+P_{g}(T)=P_{0}+a_{L}T^{4}+a_{g}T^{2},$ (6)

where $P(T)$ is the pressure at temperature $T$. $P_{0}$, $P_{L}$, $P_{g}$ are
the corresponding pressure contributions of the ions at zero temperature,
lattice vibrations, and two-level excitations of the glass. On the other hand,
the thermodynamic Maxwell relations between pressure and specific heat give

$\displaystyle\left(\frac{\partial P}{\partial
T}\right)_{V}=\frac{\gamma_{g}}{V_{m}}\,C_{g,V}+\frac{\gamma_{L}}{V_{m}}\,C_{L,V},$
(7)

where $\gamma_{i}$ are the Grüneisen coefficients of the glass ($g$) and
lattice ($L$). Literature values for the Grüneisen coefficient of phonons in
solid hcp 4He range between $2.6<\gamma_{L}<3.0$ Grigorev07b ; Driessen86 ,
while nothing is known about $\gamma_{g}$ of glassy 4He. For simplicity we
assume $\gamma_{g}\sim\gamma_{L}=2.6$. Equation (7) is related to the first
Ehrenfest relation involving the compressibility, which was shown to be always
satisfied in glasses Nieuwenhuizen1997 . In Figs. 4 and 5 we show the
temperature dependence of the $(P-P_{0})/T^{2}$ data reported by Grigor’ev et
al. Grigorev07 ; Grigorev07b  and Yin et al.Yin11 . The curves for samples
BC04 and BC20 are derived from our specific heat analysis of the data by Lin
et al.LinChan09  The key result is the dependence $\sim T^{2}$ with finite
intercept at $T=0$. In the TLS model, the finite intercept describes the
glassy contribution, whereas the $T^{2}$ behavior is attributed to phonons. In
conclusion, the data by Grigor’ev et al. and Yin et al. are in agreement with
predicted $(P-P_{0})/T^{2}$ curves for a system of TLS.

##  4 General response function formalism and physical origin of dynamical
anomalies

To rationalize the TO experiments, we developed a general phenomenological
formalism in Ref. Nussinov07 . We have since extended and applied it to visco-
elastic and dielectric properties. With simple modifications, this predictive
approach can be used to study all measurable dynamical response functions.
Here, we summarize the essence of our approach. In later sections, we will
apply it in a self-contained way to the various quantities that we wish to
interrogate. We start with the equation of motion for generalized coordinates
$q_{i}$. These coordinates may be an angle, $q=\theta$, in the case of the TO
experiments, Cartesian components of atomic displacements , $q_{i}=u_{i}$ with
${\bf u}$ the local atomic displacement (as in our analysis of the visco-
elastic and dielectric response functions), or any other. Associated with
these coordinates are conjugate generalized momenta $p_{i}$ (e.g., angular
momentum in the TO analysis, linear momentum for Cartesian coordinates) and
their associated generalized forces $F_{i}$ (torques in the case of the TO,
and rectilinear forces for atomic displacements). Physically, these forces
correspond to a sum of two contributions: (i) Direct forces ($F_{direct}$).
These may originate from either externally applied forces on the solid
($F_{ext}$) or lowest order “direct” internal forces $(F_{int}$). By direct
forces we allude to forces on the coordinate $q$ that do not involve the
response of the system on $q$ as a result of its change.

(ii) Indirect “backaction” forces ($F_{BA}$). These allude to higher order
effects wherein a variation in the coordinate $q$ can lead to displacements in
other parts of the medium (e.g., those involving plastic regions or nearby
atoms) which then act back on the original coordinate $q$. To linear order

$\displaystyle
F_{BA}(t)=\int_{-\infty}^{t}g(t-t^{\prime})q(t^{\prime})dt^{\prime}.$ (8)

The backaction function $g(t-t^{\prime})$ captures how a displacement $q$ at
time $t^{\prime}$ can lead to a perturbation in the solid which then acts back
on the coordinate $q$ at time $t$. With these preliminaries in tow, we now
outline our standard linear response formalism which further takes into
account all higher order backaction effects.

$\bullet$ (1) Write down the Newtonian equation(s) of motion for the
generalized coordinate(s) $q$ that we wish to study. With $\chi_{0}$ a
suitable differential operator $\chi_{0}^{-1}$, these can be cast as

$\displaystyle\chi_{0}^{-1}q(t)=F_{direct}(t)+F_{BA}(t).$ (9)

This equation might seem a bit formal. To make it concrete, we note that in
the simplest case that we will discuss- that of the compact scalar angular
coordinate $\theta$ for the TO orientation, the operator

$\displaystyle\chi_{0}^{-1}=I_{osc}\frac{d^{2}}{dt^{2}}+\gamma_{osc}\frac{d}{dt}+\alpha_{osc}$
(10)

where $I_{osc},\gamma_{osc},$ and $\alpha_{osc}$ are, respectively, the
oscillator moment of inertia, dissipation, and torsion rod stiffness. A more
complicated tensorial operator involving the elastic modulus appears when
writing the equations of motion for the Cartesian components of the atomic
displacements.

$\bullet$ (2) When Fourier transformed to the complex frequency ($\omega$)
plane, $\chi_{0}(\omega)$ corresponds to the bare susceptibility. We will
denote the Fourier transforms of the various quantities (forces, backaction
function) by simply making it clear that the argument of the various
quantities is now the frequency $\omega$ and not the time $t$. We trivially
recast Eq. (9) as

$\displaystyle\chi^{-1}(\omega)q(\omega)=F_{direct}(\omega),$ (11)

where

$\displaystyle\chi^{-1}(\omega)=\chi_{0}^{-1}(\omega)-g(\omega).$ (12)

Equation (12) will be used in all of our upcoming analysis. The physical
content of Eq. (12) can be seen by writing its inverse as a geometric (or
Dyson) series

$\displaystyle\chi(t)=\chi_{0}(t)+\int
dt^{\prime}\chi_{0}(t)g(t-t^{\prime})\chi_{0}(t^{\prime})$ $\displaystyle+\int
dt^{\prime}\int
dt"\chi_{0}(t)g(t-t^{\prime})\chi_{0}(t^{\prime})g(t^{\prime}-t")\chi_{0}(t")+....$
(13)

The terms in this series correspond to (a) the direct contribution
($\chi_{0}$), (b) a lowest order backaction effect wherein a displacement at
an earlier time $t^{\prime}<t$ leads to a deformation of the solid which then
acts back on the coordinate at time $t$ (the term $\int
dt^{\prime}\chi_{0}(t)g(t-t^{\prime})\chi_{0}(t^{\prime})$), (c) a higher
order process in which a deformation at time $t^{\prime\prime}$ leads to a
backaction from the solid on the coordinate at time $t^{\prime}>t"$ which in
turn then acts back on its surroundings which then acts back on the original
coordinate at time $t$ (the term $\int dt^{\prime}\int
dt"\chi_{0}(t)g(t-t^{\prime})\chi_{0}(t^{\prime})g(t^{\prime}-t")\chi_{0}(t")$),
and so on ad infinitum. In Fourier space, the convolution integrals become
products and Eq. (4) becomes

$\displaystyle\chi(\omega)=\chi_{0}(\omega)+\chi_{0}(\omega)g(\omega)\chi_{0}(\omega)+\chi_{0}(\omega)g(\omega)\chi_{0}(\omega)g(\omega)\chi_{0}(\omega)+....$
(14)

The geometric series of Eq. (14) sums to Eq. (12). As simple as it is, Eq.
(12) combining standard linear response theory (step 1) with backaction
effects (or the Dyson equation) of step 2, leads to a very powerful tool that
allows us to investigate numerous systems while accounting for arbitrarily
high order backaction effects. As is well known and we will expand on and
employ in later sections, the real and imaginary parts of the poles of the
susceptibility $\chi(\omega)$ allow us to probe typical oscillation times and
dissipation. This will allow us to connect with TO and other measurements and
make precise statements about the backaction function $g$ which affords
information about the dynamics within the solid. It is important to stress
that Eq. (12) is general. In the derivation above no assumptions need to be
made concerning the precise physical origin of the backaction function $g$.
The adduced function $g$ captures all effects not present in the direct
equations of motion for the normal solid. If supersolid effects would be
present, they will directly appear in the function $g$.

Thus far, our sole assumption was that the deformations $q$ are small enough
to justify the linear (in $q$) order analysis of Equations (8, 9) for
measurements on the rigid solid. We now invoke additional assumptions (which
have been partially vindicated in a growing number of experiments since our
original proposal Nussinov07 ). These assumption relate to the form of
$g(\omega)$ and its dependence on temperature. They are motivated by our view
of defect quenching and characteristic relaxation times as the origin of the
4He anomalies.

$\bullet$ (3) As we will elaborate on in later sections, data for disparate
susceptibilites $\chi$ taken at different frequencies $\omega$ or temperatures
$T$collapse onto one curve. This indicates that $g$ is a function of only one
dimensionless argument ($\omega\tau(T)$) instead of both $\omega$ and $T$
independently. That is, there is only one dominant temperature dependent
relaxation time scale $\tau(T)$ for the backaction of the quenched solid on
the original coordinate $q$. As is well known and alluded to in Section (2),
exponential damping with a single relaxation time $\tau$, leads to a function
$g_{single}(\omega)=g_{0}(1-i\omega\tau)/(1+\omega^{2}\tau^{2})$ which, when
plotted with the real and imaginary parts of $g$ along the horizontal and
vertical axes describes a semi-circle as a function of the dimensionless
quantity $(\omega\tau)>0$. However, when plotted in this way, the 4He data for
the complex susceptibility measured by TO and other probes lead to a collapsed
curve which is more like that of a skewed semi-circle and there is a
distribution of relaxation times about a characteristic time scale $\tau$. For
ease of analysis, we will approximate the complex response functions by Eqs.
(2). The curve collapse allows for information about how the characteristic
transient relaxation times $\tau(T)$ increase as $T$ is decreased. We further
invoke the Vogel-Fulcher-Tammann form for glasses Rault00 ,

$\displaystyle\tau(T)=\left\\{\begin{array}[]{ll}\tau_{0}\,e^{\Delta/(T-T_{0})}&\mbox{
for $T>T_{0}$},\\\ \infty&\mbox{ for $T\leq T_{0}$}.\end{array}\right.$ (17)

Here, $T_{0}$ is the temperature at which the relaxation times would truly
diverge and $\Delta$ is an energy scale. In fitting the data in this way,
negative values $T_{0}<0$ were often found. That is, the typical dynamics as
adduced from the collapsed $\tau(T)$ is faster than simple activated dynamics
(one in which $T_{0}=0$ in Eq. (17)). This is consistent with the intrinsic
quantum character of the solid 4He crystal with large zero point motion as
compared to classical activated dynamics.

What is the physical content of this general formalism vis a vis the putative
supersolid transtition? Given perturbations of a typical frequency $\omega$,
the backaction response $g$ from the plastic components acting on $q$ may
either be sufficiently rapid or slow to respond. Just at the tipping point
when $\omega\tau(T_{X})\simeq 1$ different components will be maximally out of
synchrony with each other in being able to respond to the perturbation or not
and the dissipation (given by the reciprocal of the imaginary part of the zero
of $\chi^{-1}(\omega)$) is maximal. Similarly, there will be a change in the
typical periods of the system between a system which at high $T$ (i.e.,
$T<T_{X}$) contains rapidly equilibrating plastic components to those at low
$T$, which are too slow to respond and thus the system appears to have
undergone “a transition”.

In the sections that follow, we will apply and replicate anew and at great
length the considerations outlined above to the particular set of physical
quantities that we wish to investigate. We start, in Section 5 with the
investigation of the TO (for which the above formalism was first developed)
and then move to explore other arenas- the viscoelastic (Section 6) and
dielectric response (Section 7) functions where the above formalism leads to
experimentally testable predictions.

##  5 General susceptibility and response function of torsional oscillators

A formulation of the rotational susceptibility of the TO was given in Ref.
Nussinov07 . It is now often used as a basis for the linear response
discussion Dorsey08 ; Pratt11 ; Gadagkar2012 . In Section (4) we outlined the
key points of this formulation when written in its general form. Since its
derivation it has been applied by us and others to study other response
functions. The result of Eq. (20) below is none other than Eq. (12) when the
generalized coordinate $q$ corresponds to the TO angle. The bulk of this
section will be devoted to analyzing the experimental consequences of this
relation and its related counterpart for the double TO.

It is important to realize that the TO experiment measures the period and
dissipation of the entire apparatus by reporting the relationship between the
force and displacement (angle). Therefore a model is needed to determine the
relation between observable period and dissipation and the moment of inertia,
damping and effective stiffness of the media. We start with the general
equation of motion for a harmonic TO defined by an angular coordinate $\theta$
in the presence of an external and internal torque,Nussinov07

$\displaystyle
I_{osc}\ddot{\theta}(t)+\gamma_{osc}\dot{\theta}(t)+\alpha_{osc}\theta(t)={M}_{ext}(t)+{M}_{int}(t).$
(18)

Here, $I_{osc}$ is the moment of inertia of the (empty) TO apparatus,
$\alpha_{osc}$ is the restoring constant of the torsion rod, and
$\gamma_{osc}$ is its damping coefficient. $M_{ext}(t)$ is the externally
imposed torque by the drive.
${M}_{int}(t)=\int{g}(t-t^{\prime})\theta(t^{\prime})dt^{\prime}$ is the
internal torque exerted by solid 4He on the oscillator for a system with time
translation invariance. In general, the backaction $g(t-t^{\prime})$ is
temperature, $T$, dependent.

The external torque, $M_{ext}(t)=\dot{L}(t)$, is the derivative of the total
angular momentum of a rigid body, $L(t)=\frac{d}{dt}\int
d^{3}x~{}\rho(\vec{x})r^{2}~{}\dot{\theta}(\vec{x})$, where $r$ is the
distance to the axis of rotation, $\rho(\vec{x})$ is the mass density and
$\dot{\theta}(\vec{x})$ the local angular velocity about the axis of
rotation.Anderson08  The experimentally measured quantity is the angular
motion of the TO - not that of bulk helium, which is enclosed in it. Ab
initio, we cannot assume that the medium moves as one rigid body. If the non-
solid subsystem “freezes” into a glass, the medium will move with greater
uniformity and speed. This leads to an effect similar to that of the
nonclassical rotational moment of inertia, although its physical origin is
completely different. We argue for an alternate physical picture, namely that
of softening of the oscillator’s stiffness.

The angular coordinate $\theta(t)$ of the oscillator is a convolution of the
applied external torque ${M}_{ext}(t)$ with the response function
${\chi}(t,t^{\prime})$. Causality demands
${\chi}(t,t^{\prime})=\theta(t-t^{\prime}){\chi}(t,t^{\prime})$. Under Fourier
transformation, this leads to the Kramers-Kronig relations. In any time
translationally invariant system, the Fourier amplitude of the angular
response of the TO is

$\displaystyle\chi_{0}^{-1}(\omega)\theta(\omega)=M_{ext}(\omega)+M_{int}(\omega).$
(19)

Defining the total angular susceptibility as
$\chi^{-1}=\chi_{0}^{-1}-M_{int}$, we write the effective inverse
susceptibility as

$\displaystyle\chi^{-1}(\omega)=\alpha_{osc}-i\gamma_{osc}\omega-
I_{osc}\omega^{2}-g(\omega),$ (20)

where $g(\omega)$ is the Fourier transform of the backaction due to the added
solid 4He. In what follows, we will treat the backaction as a small
perturbation to the TO chassis.

We will now apply our formalism to the study of the ingle TO, which is
described by Eq. (20), and then turn to the double TO. Very recently, Beamish
Beamish2012  and Maris Maris2012  employed the same general linear response
formalism to explain some of the TO results in terms of purely mechanical
effects due to either the changing stiffness of the torsion rod or floppiness
of the sample cell flange (lid).

###  5.1 A model for the single torsional oscillator

In what follows, we analyze the experimental consequences of Eq. (20).

####  5.1.1 Rotational susceptibility - period and dissipation

We can now calculate specific consequences of the phenomenological model
introduced above. The effective oscillator parameters are defined as the sum
of parameters describing the apparatus, $\chi_{0}^{-1}$, and the added solid
4He given by

$g(\omega)=i\gamma_{He}\omega+I_{He}\omega^{2}+g_{0}G(\omega).$ (21)

It is convenient to introduce a net moment of inertia $I=I_{osc}+I_{He}$ and
net dissipation $\gamma=\gamma_{osc}+\gamma_{He}$. The transient dynamic
response function $G(\omega)$ can be approximated by a distribution of
overdamped oscillators with relaxation time $\tau$ as discussed in Section (2)
[see Eqs. (2), in particular].

The resonant frequency of the TO with a backaction is given by the root of

$\displaystyle\chi^{-1}(\omega)=\alpha-i\gamma\omega-I\omega^{2}-g_{0}G(\omega)\equiv
0.$ (22)

As discussed in Section 4, we anticipate that when the relaxation time is
similar to the period of the underdamped oscillator, the dissipation will be
maximal, sometimes referred to as “$\omega\tau=1$” physics.

In linear response theory, the homogeneous Eq. (22) is scale invariant. Thus,
we normalize all oscillator quantities by the effective moment of inertia $I$,
i.e., $\bar{\alpha}=\alpha/I$, $\bar{\gamma}=\gamma/I$, and
$\bar{g}_{0}=g_{0}/I$. As can be seen from Eq. (22), for an ideal
dissipationless oscillator ($\bar{\gamma}=0$), the resonant frequency
$\omega_{0}=\sqrt{\bar{\alpha}}$ is the pole of $\chi(\omega)$ in the limit
$1/\tau\to 0$. If we expand $\chi^{-1}$ about this root,
$\omega=\omega_{0}+\delta\omega$, we find to leading order in $\delta\omega$

$\displaystyle\delta\omega\approx-\frac{i\bar{\gamma}\omega_{0}+\bar{g}_{0}G(\omega_{0})}{i\bar{\gamma}+2\omega_{0}}.$
(23)

It follows that the shift in dissipation with respect to high temperatures is

$\displaystyle\Delta Q^{-1}\equiv
Q^{-1}-Q^{-1}_{0}\approx\frac{\bar{g}_{0}}{\omega_{0}^{2}}{\rm Im\
}G(\omega_{0}),$ (24)

whereas the shift in resonant frequency is

$\displaystyle\Delta\omega\equiv 2\pi(f_{0}-f)$ $\displaystyle\approx$
$\displaystyle\frac{\bar{g}_{0}}{4\omega_{0}^{2}}\Big{(}2\omega_{0}\,{\rm Re\
}G(\omega_{0})+\bar{\gamma}\,{\rm Im\ }G(\omega_{0})\Big{)},$ (25)

which increases monotonically when $T$ is lowered. Combining Eqns. (24) and
(25) for the strongly underdamped oscillator, we arrive at

$\frac{\Delta Q^{-1}}{\Delta\omega}=\frac{4{\rm Im\
}G(\omega_{0})}{2\omega_{0}{\rm Re\ }G(\omega_{0})+\bar{\gamma}{\rm Im\
}G(\omega_{0})}\approx\frac{2}{\omega_{0}}\frac{{\rm Im\ }G(\omega_{0})}{{\rm
Re\ }G(\omega_{0})}.$ (26)

It is this general relationship for the response function of the damped
oscillator that was successfully applied in the TO analysis by Pratt et al.
Pratt11 ; Gadagkar2012  to demonstrate the interplay of rotational,
relaxation, and shear dynamics in solid 4He. For a Debye relaxor the ratio of
Eq. (26) reduces to $2\tau$ and provides a direct means to measure the
relaxation time. Similar results for the ratio were obtained for other
phenomenological models.Huse07 ; Dorsey08  For example, Huse and Khandker
Huse07  assumed a simple phenomenological two-fluid model, where the
supersolid is dissipatively coupled to a normal solid resulting in a ratio of
${\Delta Q^{-1}}\frac{\omega_{0}}{\Delta\omega}\approx 1$, while Yoo and
DorseyDorsey08  developed a viscoelastic model, and KorshunovKorshunov09
derived a two-level system glass model that captures the results of the
general model originally proposed by Nussinov et al.Nussinov07  To make
further progress we assume that $\tau$ follows the phenomenological Vogel-
Fulcher-Tammann (VFT) equation of Eq. (17).

Figure 6: The resonant frequency (black, left axis) and dissipation (red,
right axis) vs. temperature. The experimental data from Hunt et al.Hunt09  are
well described by a Cole-Cole (CC) distribution function. Graf10 ; Graf11

Figure 6 provides a fit to the measured data by Hunt et al.Hunt09  assuming a
CC distribution of relaxation times. As shown, an excellent fit is obtained.
For comparison, we also tried a DC distribution for relaxation times, but
found only fair agreement. It is worth mentioning that unlike in the Debye
relaxation analysis by Hunt and coworkers, i.e., a single overdamped mode, we
do not require a supersolid component to simultaneously account for frequency
shift and dissipation peak Graf10 ; Graf11 . Our model leads to a universal
scaling of period change vs. dissipation in a Cole-Cole or Davidson-Cole plot
as seen in Ref. Pratt11 ; Gadagkar2012 . Indeed similar viscoelastic behavior
may have already been observed in solid hydrogen Clark2006 .

###  5.2 A model for the double oscillator

Figure 7: Cartoon of the double torsional oscillator modeled in Eqns. (5.2).
The upper moment of inertia ($I_{1}$) is the dummy bob, while the lower moment
of inertia ($I_{2}$) is the cylindrical pressure cell that can be loaded with
4He. The stiffness of the torsion rods is given by $k_{1}$ and $k_{2}$ with
$k_{1}\approx k_{2}$ by design.

The double TO results of the Rutgers group by Kojima have proved difficult to
explain when simply extrapolating from the single TO model Dorsey08 . Here we
model the coupled double TO, sketched in Fig. 7, by the following system of
equations:

$\displaystyle\left(-I_{1}\omega^{2}-i\gamma_{1}\omega+k_{1}+k_{2}\right)\Theta_{1}(\omega)-k_{2}\Theta_{2}(\omega)$
$\displaystyle=$ $\displaystyle F(\omega),$
$\displaystyle\left(-I_{2}\omega^{2}-i\gamma_{2}\omega+k_{2}-g(\omega)\right)\Theta_{2}(\omega)-k_{2}\Theta_{1}(\omega)$
$\displaystyle=$ $\displaystyle 0.$ (27)

where $\Theta_{i}$ are torsion angles, $\gamma_{i}$ are damping coefficients,
$k_{i}$ are torsion rod stiffnesses, $g(\omega)$ is the glass backaction term,
and $F(\omega)$ is the applied external torque. The subindex “$1$” refers to
the upper or dummy bob in the experiment, while “$2$” refers to the lower
oscillator with the pressure cell that can be loaded with solid 4He. For a
strongly underdamped oscillator and a small backaction, it suffices to solve
first for the bare resonant frequencies and later include perturbatively
damping and backaction terms, for details see Ref. Graf10 . More recently this
approach has been extended to a triple TO Mi2011 .

Figure 8: Frequency and dissipation in double TO by Aoki et al.Aoki07
(symbols) compared with glass theory (lines). Panel (a): Temperature
dependence of resonant frequency shifts $\Delta f_{1}$ (black, left axis) and
$\Delta f_{2}$ (blue, right axis). Panel (b): Temperature dependence of
dissipation $Q_{1}^{-1}$ (black, left axis) and $Q_{2}^{-1}$ (blue, right
axis). The experimental data were corrected for the significant temperature-
dependent background of the empty cell.

Figure 8 shows good agreement between our phenomenological model of the
coupled double TO and experiment. The TO parameters $I_{i}$ and $k_{i}$ can be
determined from the bare resonant frequencies $f_{i}^{0}=\omega_{i}^{0}/2\pi$.
In addition, the damping coefficients $\gamma_{i}$ can be extracted from the
high-temperature dissipation $Q_{i,\infty}^{-1}$. Finally, the backaction
${g}(\omega)$ accounts through $\tau(T)$ for the temperature dependence of
$\Delta f_{i}$ and $Q_{i}^{-1}$. Our phenomenological theory of the double
oscillator explained for the first time both frequency shift and dissipation
peak for in-phase and out-of-phase torsional response Aoki07 . Data for in-
phase frequency $f_{1}=496$ Hz and out-of-phase $f_{2}=1173$ Hz are shown in
Fig. 8, plotted against the temperature. The obtained values for moment of
inertia and rod stiffness agree well with other estimates Aoki08 .

Remarkably an anomalous damping coefficient $\gamma_{1}\sim-\gamma_{2}$ is
required to explain the behavior of increased dissipation with increased
frequency. Such anomalous damping is already required to describe the unloaded
pressure cell. Thus it is unrelated to the properties of solid 4He and an
intrinsic property of the composite TO. After loading the cell with solid 4He
the dissipation ratio becomes $Q_{2}^{-1}/Q_{1}^{-1}=2.5$ at 300 mK with
frequency ratio $f_{2}/f_{1}=2.37$. Our fit results in a negative parameter
$T_{0}=-32.73$ mK. This effective negative value of $T_{0}$ is in line with
earlier comment in Section 4, concerning the quantum character of solid 4He.
This value may be indicative of strong zero point quantum fluctuations that
thwart a glass transition.

Finally, the comparison in Fig. 8 shows that an explicit frequency-dependent
backaction must be used with
$g_{0}(\omega)=g_{0}\left({\omega}/{\omega_{1}^{0}}\right)^{p}$ and $p=1.77$
to account for the experimental fact of $\Delta f_{1}/f_{1}\approx\Delta
f_{2}/f_{2}$, i.e., the relative frequency shift is unaffected by the changing
resonant frequency. In contrast, various theories describing solid 4He in
torsional oscillators as viscoelastic material Dorsey08  or two-level systems
moving through a solid matrix Andreev07 ; Andreev09 ; Korshunov09  predict an
exponent of $p=4$ for the backaction term.

##  6 Shear and stiffness of a viscoelastic medium

Another aspect of the dynamic response of 4He crystals was revealed through a
series of elasticity studies.Paalanen81 ; Goodkind02 ; Burns93 ; Beamish07 ;
Beamish09 ; Beamish10  In particular, Beamish and coworkers demonstrated the
qualitative similarity between shear modulus and the TO measurements. In the
shear modulus experiment solid helium is grown in between and around two
closely spaced sound transducers. When one of the transducers applies an
external strain, the other transducer measures the induced stress from which
the shear modulus of the sample is deduced.Beamish07  In this way, the
experiment provides a direct measurement of the elastic response to the
applied force within a broad and tunable frequency range. The frequency
dependence is especially crucial in determining the nature of the relaxation
processes and complements current TO experiments with their limited frequency
range.

Similar to the TO analysis in the previous section, we analyzed the shear
modulus within the general linear response function framework, where the
amplitude of the shear modulus increases (stiffens) upon lowering $T$, because
of the freezing out of low-energy excitations. This change is accompanied by a
prominent dissipation peak, indicative of anelastic relaxation processes. We
calculated the complex shear modulus $\mu(\omega;T)$ of a viscoelastic
material and predicted: (a) the maximum of the shear modulus change and the
height of the dissipation peak are independent of frequency and (b) the
inverse crossover temperature $1/T_{X}$ vs. the applied frequency $\omega$
obeys the form $\omega\tau(T_{X})=1$ characteristic of dynamic crossover.

###  6.1 Model of dynamic shear modulus

As in our analysis of the TO, we start with the same general linear response
function formulation outlined in Section 4. Our final result of Eq. (31) will,
once again, reflect the general relation of Eq. (12). Here we replace the
angular coordinate of the TO with a displacement coordinate and the restoring
force with a stress tensor Su10b . For ease, our notation in the below will
differ slightly from that in Section 4. The equation of motion for
displacement $u_{i}$ in the $i$-th direction of a volume element in the
presence of an external driving force is

$\displaystyle-\rho\,\omega^{2}u_{i}+\partial_{j}\,\sigma_{ij}^{\rm He}\ \
=f_{i}^{\rm EXT}(\omega)+f_{i}^{\rm BA}(\omega),$ (28)

where $\rho$ is the mass density, $f_{i}^{\rm EXT}$ and $f_{i}^{\rm BA}$ are
the external force density and the backaction force density, and
$\sigma_{ij}^{\rm He}$ is the elastic stress tensor of solid helium. In
general, $\sigma_{ij}^{\rm He}=\lambda_{ijkl}\,u_{kl}$, with the elastic
modulus tensor $\lambda_{ijkl}$ LL_elastic . In the case of a homogeneous
solid with shear wave propagation along the $z$ axis and wave polarization in
the $x$-$y$ plane, the backaction takes on the form

$\displaystyle f_{i}^{\rm
BA}(\omega)={\overline{G}}(\omega;T)\,\partial_{z}^{2}\,u_{i}(\omega),$ (29)

where $\overline{G}$ describes the strength of the backaction on the solid
(viscoelastic response) and $i=x,y$. Although $f_{i}^{\rm BA}(\omega)$ is much
smaller than the purely elastic restoring force
$\partial_{j}\,\sigma_{ij}^{\rm He}$, it is this term that is responsible for
the stiffening of shear modulus with decreasing temperature.

Polycrystalline and amorphous materials are nearly isotropic, hence the
elastic modulus tensor becomes
$\lambda_{ijkl}=\lambda_{0}\delta_{ij}\delta_{kl}+\tilde{\mu}_{0}(\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk})$.
Note the stress tensor in Eq. (28) is finite only for orientations $j=z$ and
either $k$ or $l$ equal to $z$. With $k,l$ being interchangeable, the relevant
element will be $\lambda_{iziz}$, which gives the purely elastic shear modulus
$\tilde{\mu}_{0}$. Finally the fully dressed shear modulus (dressed by the
backaction) relates the displacement to an external force, or
$[-\rho\,\omega^{2}+\mu\,\partial_{z}^{2}]u_{i}(\omega)=f_{i}^{\rm
EXT}(\omega)$. Comparing this expression with Eq. (28), we obtain for the
dynamic shear modulus in a viscoelastic material the general response function

$\displaystyle\mu(\omega;T)=\tilde{\mu}_{0}(T)-\overline{G}(\omega;T).$ (30)

Next we employ for $\overline{G}(\omega;T)$ the Cole-Cole distribution
function [specifically, by reference to Eq. (2), we set
$\overline{G}(\omega;T)=g_{0}\mu_{0}G(\omega,T)$ to obtain the specific form

$\displaystyle\mu(\omega;T)$ $\displaystyle=$
$\displaystyle\tilde{\mu}_{0}\left[1-\frac{g_{0}}{1-(i\omega\tau)^{\alpha}}\right],$
(31)

with the sample dependent parameter $g_{0}$. The experimentally measurable
quantities are the amplitude of the shear modulus, $|\mu|$, and the phase
delay between the input and read-out signal, $\phi\equiv{\rm arg}\,(\mu)$;
$\phi$ measures the dissipation of the system, which is related to the inverse
of the quality factor $Q^{-1}\equiv\tan\phi$.

Several interesting results follow immediately from the general response
function in Eq. (31): (1) The change in shear modulus $\Delta\mu$ between zero
and infinite relaxation time is $\Delta\mu/\tilde{\mu}_{0}=g_{0}$. It measures
the strength of the backaction as well as the concentration of defects. (2) At
fixed $T$, the shear modulus amplitude $|\mu(\omega;T)|$ decreases with
increasing $\omega$. (3) The parameter $\omega\tau$ is the only scaling
parameter. (4) The peak height $\Delta\phi$ of the phase angle is proportional
to $g_{0}$. When $\omega\tau\sim 1$, then $\Delta\phi\approx
g_{0}\,{\cot(\alpha\pi/4)/(2-g_{0})}$ for $g_{0}\ll 1$. In the limit
$1<\alpha\leq 2$, this simplifies further to

$\displaystyle\Delta\phi\approx(1-\alpha/2)(\Delta\mu/\tilde{\mu}_{0})\ll 1,$
(32)

where $\Delta\phi$ is in units of radians. Quite remarkably, the peak height
$\Delta\phi$ depends only on the phenomenological parameters $\alpha$ and
$g\propto\Delta\mu$. At fixed temperature the maximum change of both
$\Delta\mu$ and $\Delta\phi$ is independent of frequency.

###  6.2 Results

Let us compare our model calculations with the experimental shear modulus
measurements by Beamish and coworkers Beamish10  for a transducer driven at
2000 Hz, 200 Hz, 20 Hz, 2 Hz, and 0.5 Hz Comment1 . Specifically, for the
$T$-dependent $\tau(T)$ in Eq. (31) we consider Vogel-Fulcher-Tammann (VFT)
and power-law (PL) relaxation processes. In our model parameter search, we are
not constraining $T_{0}$ to be positive (where $\tau$ diverges). Fair
agreement between calculations with a single set of parameters and experiments
is obtained with $T_{0}=-69.3$ mK, see Fig. 9. We refer to these calculations
as “VFT<”. Similar to the TO results, a negative $T_{0}$ means that no real
phase transition occurs in solid 4He. The agreement between theory and
experiment at highest (2000 Hz) and lowest (0.5 Hz) frequencies is of poorer
quality. This discrepancy may be related to very different backgrounds at
these frequencies and may not be intrinsic to 4He. Another reason may be the
presence of additional dissipation mechanisms, not accounted for. Furthermore,
our calculations confirm that the change of amplitude and peak height of phase
angle are nearly frequency independent between 2 Hz and 200 Hz. By defining
the crossover temperature $T_{X}$ as the point where the phase angle peaks, we
find as predicted that $T_{X}$ decreases with decreasing $\omega$.

Figure 9: Experimental data and theoretical calculations of the shear modulus
vs. temperature assuming a VFT relaxation time. The red and blue squares are
the experimental data for the amplitude and dissipation of shear modulus. The
black-solid lines show the theoretical calculations. We used the set of
parameters $\alpha=1.31$, $g_{0}=1.44\times 10^{-1},\tilde{\mu}_{0}=0.47$ pA
Hz-1, $\tau_{0}=50.0$ ns, $\Delta=1.92$K, and $T_{0}=-69.3$ mK. Notice a
negative $T_{0}$ means that there is no true phase transition occurring at
finite temperatures, probably because of strong quantum fluctuations of helium
atoms. The shear modulus amplitude and the phase angle were shifted by 0.1 pA
Hz-1 and 3 degrees with respect to the 2 Hz data.

Finally, when we set $T_{0}=0$ K the VFT expression (“VFT0”) reduces to an
activated Arrhenius form. While it describes reasonably well the position
$T_{X}$, it gives a much narrower linewidth for $\Delta\phi$ than does VFT<,
which is not in accord with the data and thus not shown. Notice that the VFT
relaxation is not the only possible relaxation process that can describe the
data; power-law or other types of relaxation can give similar level of
agreement to the experiment Beamish10 . Iwasa proposed a relaxation process
Iwasa10  similar to our phenomenological one, which is based on the theory of
pinned vibrating dislocation lines by Granato and Lücke GranatoLuecke56 .
Clearly further experiments at lower frequencies and lower temperatures are
required to determine the exact type of relaxation processes in solid 4He.

Figure 10: The Cole-Cole plots for experimental data and for VFT calculation.
For given form of $\tau$, all different frequency curves collapse onto one
single master curve reflecting that $\omega\tau$ is the only scaling
parameter. The Cole-Cole plots show reflection symmetry about
Re[$|\mu-\tilde{\mu}_{0}|/\Delta\mu$]=0.5, which is a consequence of the Cole-
Cole distribution function.

Figure 10 shows the Cole-Cole plot for experiments and calculations. The
experimental curves for different frequencies collapse roughly onto one curve
confirming our theoretical assumption that $\omega\tau(T)$ is a universal
scaling parameter. This behavior was also seen in TO experiments Hunt09 . In
addition, the data are symmetric about
Re$[|\mu-\tilde{\mu}_{0}|/\Delta\mu]=0.5$ as expected for the Cole-Cole
distribution. A more detailed data analysis may resolve the remaining
discrepancy between theory and experiment. The discrepancy may be due to
either the presence of additional relaxation processes at temperatures above
$T_{X}$, i.e., a more complicated form for $\tau(T)$, or by a modified
functional form of the defect distribution function.

Figure 11: Prediction for the inverse crossover temperature vs. applied
frequency. The green squares correspond to $T_{X}$ in Ref. Beamish10 . For the
power-law process with phase transition occurring at 40 mK, we used
$\tau=\tau_{0}(|T_{g}|/(T-T_{0}))^{p}$ for $T>T_{0}$ and $\tau=\infty$ for
$T\leq T_{0}$.

The pertinent question about a true phase transition at zero frequency vs.
crossover dynamics can be addressed by investigating the asymptotic limit of
$\omega\tau(T)=1$. From this expression we estimate $T_{X}$ as a function of
the applied frequency $f=\omega/2\pi$. Figure. 11 shows $1/T_{X}$ vs. $f$. The
VFT< and VFT0 calculations give significantly better agreement than the PL
calculations with phase transitions occurring at 0 K (PL0) and 40 mK (PL40).
For positive $T_{0}$ (see PL40), we find a true freeze-out transition, which
would indicate arrested dynamics for $f\to 0$ Hz. For both VFT and PL
relaxation times our calculations demonstrate that in the low-frequency limit
the existence of a phase transition shows clear signatures of $T_{X}$
converging toward the ideal glass temperature $T_{0}$. Therefore the absence
of arrested behavior may serve as experimental evidence against a true phase
transition.

###  6.3 Viscoelastic model

The viscoelastic model successfully describes composite materials with
anelastic properties. In fact, Yoo and Dorsey Dorsey08  applied the
viscoelastic model to the TO experiments. More generally, a distribution of
viscous components, embedded in an otherwise elastic solid, may be treated
through a generalized Maxwell model Su10b ; Su11 . Conceptually one may
subdivide the material into many elements and solve the coupled materials
equations for stress and strain. Here we use constitutive materials equations
to show that our results for a glass are equivalent to the generalized Maxwell
model with parallel connections of an elastic component with an infinite set
of Debye relaxors, see Fig. 12.

Figure 12: Lump circuit of the generalized Maxwell model. The elastic spring
$\mu_{0}$ represents the purely elastic shear modulus at high temperature.
Each Debye relaxor is made out of a series of elastic spring $\mu_{\rm RS}$
and dissipative dash-pot $\eta^{(n)}$.

The anomalous stiffening of the shear modulus can be described within the
viscoelastic model, though other defect mechanisms like dislocation glide are
possible too Friedel ; Caizhi2012 . The equivalent lump circuit model is
sketched in Fig. 12, where the basic dissipative element is the Debye relaxor.
It is comprised of a serial connection of a rigid solid (RS), characterized by
an elastic shear modulus $\mu_{\rm RS}$, and a Newtonian liquid (NL) or dash-
pot, characterized by a viscosity $\eta$. The RS component describes the ideal
elastic solid helium of this volume element, while NL represents the anelastic
component, which gives rise to viscous damping. The two parts are connected in
series, so that both share the same magnitude of stress, while the net strain
is additive. The strain rate equation for both constituents of the Debye
relaxor is

$\displaystyle\dot{\epsilon}={\dot{\sigma}}/{\mu_{\rm RS}}+{\sigma}/{\eta},$
(33)

where $\epsilon$ is the net strain of the Debye relaxor (DR) and $\sigma$ is
the magnitude of stress shared by the components RS and NL. In order to obtain
the above equation, we used the constitutive materials equations for strain
and stress: $\epsilon_{\rm RS}=\sigma_{RS}/\mu_{RS}$ and $\dot{\epsilon}_{\rm
NL}=\sigma_{NL}/\eta$. After performing the Fourier transformation we obtain
for a single Debye relaxor (DR) the shear modulus $\mu_{\rm
DR}=\sigma/\epsilon$,

$\displaystyle\mu_{DR}(\omega)=\frac{\mu_{\rm RS}}{1+\frac{i}{\omega\tau_{\rm
DR}}}=\mu_{\rm RS}\left[1-\frac{1}{1-i\omega\tau_{\rm DR}}\right],$ (34)

with relaxation time $\tau_{\rm DR}\equiv\eta/\mu_{\rm RS}$. For a
viscoelastic material with a single relaxation time, the solid behaves like a
parallel connection between the elastic part and the Debye relaxor

$\displaystyle\mu(\omega)$ $\displaystyle\equiv$
$\displaystyle\mu_{0}+\mu_{\rm
DR}(\omega)=\tilde{\mu}_{0}\left[1-\frac{g_{0}}{1-i\omega\tau_{\rm
DR}}\right],$ (35)

with $g_{0}=\mu_{\rm RS}/\tilde{\mu}_{0}$ and
$\tilde{\mu}_{0}=\mu_{0}+\mu_{\rm RS}$ is the dressed elastic shear modulus.
To consider the general case of many components, we introduce Debye relaxors
with different relaxation times connected in parallel as shown in Fig. 12. The
total anelastic contribution from $n$ such constituents is given by a weighted
sum. The corresponding continuous version of this expression with a
distribution $P(s)$ of relaxation times is

$\displaystyle\mu_{ae}(\omega)=\mu_{\rm RS}\int_{0}^{\infty}ds\
P(s)\,\left[1-\frac{1}{1-i\omega\tau s}\right]=\mu_{\rm RS}-\frac{\mu_{\rm
RS}}{1-(i\omega\,\tau)^{\alpha}}.$ (36)

Here the Cole-Cole distribution,

$\displaystyle
P(s)=\frac{t^{-(1-\alpha)}\sin\alpha\pi}{1+s^{2\alpha}+2s^{\alpha}\cos\alpha\pi},$
(37)

was used. Similar to our response functions elsewhere, the net shear modulus
of the composite system is given by the sum of two terms- the purely elastic
component and the anelastic contribution

$\displaystyle\mu(\omega)$ $\displaystyle=$
$\displaystyle\mu_{0}+\mu_{ae}(\omega)=\tilde{\mu}_{0}\left[1-\frac{g_{0}}{1-(i\omega\tau)^{\alpha}}\right].$
(38)

Indeed, this expression is identical to the one obtained previously using the
general response function formalism with a backaction (12). This is no
coincidence, since the backaction term accounts for damping and thus describes
the anelastic response of defects to the external stress.

##  7 Dielectric properties of a viscoelastic medium

The measurements of $\epsilon(\omega,T)$ by Yin et al. Yin11  showed the
anomalous increase of the dielectric function of solid 4He at low
temperatures. A similar test experiment in liquid helium showed no such
effect. We propose that these results may be explained by an electro-elastic
coupling of a quenched solid with frozen-in internal stress. Such behavior
cannot be described by the standard Clausius-Mossotti equation via a change in
mass density or polarizability (due to, e.g., dipole-induced dipole
interactions). Neither the measured change of the mean mass density
$\delta\rho/\rho\sim 10^{-6}$, nor the predicted correction in polarizability,
which actually leads to a decrease of $\epsilon(\omega,T)$ at low temperatures
Kirkwood36 ; Chan77 , can account for the reported anomalous change of the
dielectric function of order $\delta\epsilon/\epsilon\sim 10^{-5}$, while a
model with frozen-in stress and electro-elastic coupling can explain the data.

###  7.1 Model for electro-elastic properties

The relation of Eq. (47) that we derive below constitutes yet another
realization of our general relation of Eq. (12). We now turn to the specifics
of the electro-elastic coupling describing the interaction between the
electromagnetic and the strain fields. The coupling may be obtained by
expanding the dipole moment, $\bf p(r_{\it a})$, of a nonpolar atom around its
equilibrium value:

$\displaystyle{\bf p(r_{\it a})\approx p(R_{\it a})+(u_{\it a}\cdot\nabla)\
p\left.\right|_{R_{\it a}}},$ (39)

where $\bf r_{\it a}=R_{\it a}+u_{\it a}$ is the position of atom $a$, $\bf
R_{\it a}$ is its equilibrium position, and $\bf u_{\it a}$ is its
displacement. The polarization is obtained by averaging ${\bf p}$ over a
macroscopic volume element $v$, ${\bf P(r)}=({1}/{v})\sum_{\it a}{\bf
p(r_{a})}$. In the continuum limit (when the macroscopic length scale is far
larger than the atomic length) $\sum_{a}{\bf p(r_{\it
a})}\approx(1/v)\int_{v}d{\bf r}^{\prime}\ {\bf p(r^{\prime})}$ . In the
presence of a local strain field, to linear order in the displacement, Eq.
(39) reads

$\displaystyle{\bf P(r)}\approx(1/v^{2})\int_{v}d\,{\bf R\
[\,p(R)+(u\cdot\nabla)\ p(R)\,]}.$ (40)

Integration of the first term yields the macroscopic polarization for zero
internal strain (a solid in equilibrium), ${\bf P}_{0}$. It is related to the
macroscopic electric field ${\bf E}$ by ${\bf P}_{0}\equiv\chi_{0}{\bf E}$
where $\chi_{0}=(\epsilon_{0}-1)/4\pi$ is the zero-strain susceptibility and
$\epsilon_{0}$ is the permittivity. The second term in the integration
describes the polarization change $\bf\delta P$ due to atomic displacements.
Neglecting surface contributions the second term modifies the polarization
${\bf P}={\bf P}_{0}+\delta{\bf P}$ by

$\displaystyle\delta{\bf P}=-(1/v^{2})\int d{\bf R\ (\nabla\cdot u)\
p(R)}\approx-e_{ii}\,{\bf P}_{0},$ (41)

with $e_{ii}=(1/v)\int_{v}d{\bf R}\,(\nabla\cdot\bf u)$ the macroscopic
frozen-in dilatory strain (we use the repeated indices summation convention
$e_{ii}\equiv e_{11}+e_{22}+e_{33}$). In obtaining Eq. (41), we assumed that
${\bf P}$ is slowly varying. This long-wave length approximation holds for
wavelengths of microwave ($\lambda>10^{-3}$m) and above. The polarization
change alters the dielectric function
$\epsilon_{ii}=(\epsilon_{0})_{ii}+\delta\epsilon_{ii}$ by

$\displaystyle\delta\epsilon_{ii}\,(\omega,T)=-4\pi\chi_{0}\,e_{ii}\,(\omega,T)$
(42)

with $\epsilon_{ii}-1=4\pi P_{i}/E_{i}$. Only the diagonal components of the
strain tensor play a role in a leading order expansion of the electro-elastic
coupling. In principle, the shear strain can couple to the electric field by
considering dipole-induced dipole interactions (van der Waals), which is a
higher order effect. The same coupling mechanism between photon (electric
field) and phonons (strain) gives rise to the acousto-optical effect. Our
derivation of Eq. (42) is, to leading order, equivalent to the Pockels
coefficient for acousto-optical coupling in isotropic dielectrics,
$\delta\epsilon_{ij}=-\epsilon^{2}\,[2P_{44}\,e_{ij}+P_{12}\,e_{kk}\delta_{ij}],$
where $P_{kl}$ are the reduced Pockels coefficients (of order unity)
Werthamer69 ; Kisliuk91 ; Landau_Lifshitz .

We next turn to the locally frozen-in strains and write the equation of motion
within the general response function theory. As before in an isotropic medium
the elastic stress tensor $\sigma_{ij}^{\rm He}=\lambda_{ijkl}\,\partial
u_{k}/\partial x_{l}$ with
$\lambda_{ijkl}=\lambda_{0}\delta_{ij}\delta_{kl}+\mu_{0}(\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk})$.
If the electric field couples to local density fluctuations only through
dilatory strain, then the important matrix element is the Lamé parameter
$\lambda_{0}$. We write the displacement to an out-of-equilibrium internal
(INT) force in the presence of the backaction as

$\displaystyle-\rho\omega^{2}\,u_{i}(\omega)+\lambda_{0}\,\partial_{i}^{2}u_{i}(\omega)=f_{i}^{\rm
INT}(\omega)+f_{i}^{\rm BA}(\omega),$ (43)

where $u_{i}$ is a displacement in the $i$th direction and $\rho$ is the mass
density. The backaction force density of nearby atoms is given by

$\displaystyle f_{i}^{\rm
BA}(\omega)=\overline{G}(\omega;T)\,\partial_{i}^{2}\,u_{i}(\omega),$ (44)

where ${\overline{G}}$ is the strength of the pertinent backaction.
$f_{i}^{\rm INT}$ is the out-of-equilibrium (frozen-in) internal force density
in the $i$th direction at the defect. Integrating Eq. (43) yields the strain
due to the internal dilatory stress $\sigma_{ii}$:

$\displaystyle{e}_{ii}(\omega,T)={\sigma}_{ii}\left(\lambda_{0}-\overline{G}(\omega;T)\right)^{-1}.$
(45)

Again, we assume that the backaction can be described by a Cole-Cole
distribution of Debye relaxors,
$\overline{G}(\omega;T)={g_{0}}\lambda_{0}/{\big{[}1-(i\omega\,\tau)^{\alpha}\big{]}}$.
The corresponding local dilatory strain reads

$\displaystyle{e}_{ii}(\omega,T)$ $\displaystyle=$ $\displaystyle
e_{0}\left(1-{g_{0}}/\big{[}{1-(i\omega\tau)^{\alpha}}\big{]}\right)^{-1},$
(46)

where $e_{0}\equiv\sigma_{ii}/\lambda_{0}$ at $T=0$. From Eqns. (42) and( 46)
the change in the dielectric function due to local strain fluctuations is

$\displaystyle\delta\epsilon_{ii}$ $\displaystyle=$
$\displaystyle-4\pi\chi_{0}\,e_{0}\left(1-{g_{0}}/\big{[}{1-(i\omega\tau)^{\alpha}}\big{]}\right)^{-1}.$
(47)

This result is similar to the one for the TO and shear modulus discussed in
the earlier sections. At low temperatures, $\tau\rightarrow\infty$ and
$e_{ii}\rightarrow e_{0}$, hence the strain is minimal and the reduction of
the dielectric function due to local strain fluctuations is small. At high
temperatures, $\tau\rightarrow 0$ and $e_{ii}\rightarrow e_{0}(1-g_{0})^{-1}$
reaches its maximum resulting in the largest reduction of strain, where solid
4He is softest. The main result is that the dielectric function reflects the
arrested dynamics of the glassy components at low temperatures through the
acousto-optical (electro-elastic) coupling.

The derivation of Eq. (47) for the dielectric response is extremely general.
We illustrate how a Cole-Cole form for the elastic relaxation implies a
similar dielectric response (and vice versa). An identical result holds for
other forms of the local elastic relaxation dynamics- these will leave a
similar imprint on the dielectric response. Historically (since the 1940s) the
Cole-Cole response function Phase1  was found to be valid for the dielectric
response. In our initial works on 4He in trying to capture the local quenched
dynamics, we first assumed this form for the TO and then for the general
elastic response. By virtue of their inter-relation and coupling, the local
relaxation dynamics is the same for all of these quantities. Therefore
measurements of the dielectric response may inform about local dynamics and
vice versa. Practically, our prediction of Eq. (47) applies to any
nonpolarized system with a local distribution of stress relaxations. In
polarized materials, detecting a change in the dielectric function due to
atoms sensing different local fields is challenging because the large
intrinsic polarizability of the material will overwhelm the contributions
derived above. Among the nonpolarized solids, solid 4He favors the observation
of this phenomenon especially because of its softness. Rapid cooling of solid
helium allows a large local strain build-up, which is proportional to the size
of the effect. Similarly, we expect the effect to be seen via delicate effects
in solid 3He, hydrogen or xenon.

###  7.2 Results

The results of our electro-elastic predictions for the dielectric function are
shown in Fig. 13. We obtain excellent agreement with experiment Yin11  for an
applied alternating voltage at 500 Hz. Our analysis predicts, that similar to
the TO and shear modulus, a dissipation peak appears in the dielectric
function. The phase lag angle $\phi=arg(\epsilon)$ records the lag between the
real and imaginary part of $\epsilon(\omega;T)$. Future observation of the
dissipation peak will provide an important test of our picture concerning
quenched dynamics in solid 4He. Consistent with earlier sections in this
review, we assumed a Vogel-Fulcher-Tammann (VFT) form for the defect
relaxation time $\tau(T)=\tau_{0}\,e^{\Delta/(T-T_{0})}$. As in the previous
sections, we obtain from our fits a negative $T_{0}=-119$ mK, in accordance
with an avoided dynamic arrest of defect motion.

Figure 13: Experimental data and calculations of the dielectric function vs.
temperature. The red circles are the experimental data of the dielectric
function (data by Yin et al. Yin11 ). The black lines are the calculated
amplitude and phase lag (dissipation) of $\epsilon(\omega;T)$. We used
parameters $\alpha=1.49$, $e_{0}=8.88\times 10^{-4},g=0.21$, $\tau_{0}=10.4$
ns, $\Delta=1.92$ K, and $T_{0}=-119$ mK.

A rough estimate of the electro-elastic coupling can be obtained from mass
flow measurements in bulk solid 4He, Ray10  where a pressure difference of
$\Delta P_{L}\sim 0.1$ bar across a centimeter-sized pressure cell was
reported. The estimated local strain, with a bulk modulus $B=320$ bar, is
accordingly $\Delta P_{L}/B=3\times 10^{-4}$. This is consistent with the
value we used for the fit in Fig. 13, namely $e_{0}=8.88\times 10^{-4}$. In
addition, the $P(T)$ measurement by Yin clearly deviates from a purely Debye
lattice behavior at around $T=0.4$ K with a large positive intercept
corresponding to the order of 100 ppm of TLS, see Fig. 5. This number is
roughly five times larger than the most disordered sample in Lin’s Lin09
specific heat experiments on ultrapure 4He with less than 1 ppb of 3He
impurities Lin09 ; Su10 . Thus the crystals grown by Yin are strongly
disordered and harbor sufficiently many defects to support centers of local
strain fields.

Thus far, we assumed that both local and global stress are constant at low
temperatures. From Fig. 5 we can read off that the global pressure change
between 300 mK and 40 mK is less than $\Delta P_{T}=0.18$ mbar. This is more
than three orders of magnitude smaller than the local stress
$\sigma_{L}=8.88\times 10^{-4}\times 320\ {\rm bar}\sim 250$ mbar inferred
from the dilatory strain $e_{0}$ used in the fit, as well as the static
pressure difference $\Delta P_{L}=0.1$ bar measured at two pressure gauges in
mass flow experiments Ray10 . Putting all the pieces together, we find that
the change of the dielectric function based on global density changes in the
Clausius-Mossotti equation is negligible. This is because the corresponding
density change, $\Delta\rho/\rho=\Delta P_{T}/B<10^{-6}$, leads to a change in
the dielectric function of only
$\delta\epsilon\approx(\epsilon-1)\Delta\rho/\rho=0.065\Delta
P_{T}/B<10^{-7}$, which is more than two orders of magnitude too small to
account for the observed effect of order $10^{-5}$. Whereas the model of local
stress and electro-elastic coupling can account for the magnitude and
temperature dependence of the dynamic dielectric function. More recently Yin
et al. Yin2012  redesigned their dielectric function experiment with a
simplified capacitor geometry and found no measurable anomaly at low
temperatures. Within our theory of disorder such a null result is consistent
with a negligible amount of frozen-in stress in the solid.

##  8 Conclusions and future directions

In this review we provided a general overview of the role of defects in solid
4He. We suggested that defect dynamics and freeze-out provides a rich ground
to account for a significant fraction of the data, while at the same time they
allow enough flexibility to accomodate sample and history dependence and
hysteretic behavior ubiquitously seen in experiments. The general response
function approach presented covers a wide range of observed effects. We
provide a brief synopsis of these below.

###  8.1 Thermodynamics

We started our discussion with the notion that any true phase transition,
including supersolid, is accompanied by a thermodynamic signature. Therefore
thermodynamic measurements like specific heat can reveal the signature one
would naturally associate with a phase transition. However, we found that the
excess specific heat and corresponding entropy are consistent with the
contribution from noninteracting two level systems (TLSs). The estimated
fraction of these TLS to the specific heat is at the level of tens to hundreds
of parts-per-million. Consequently, the corresponding entropy contribution
$\Delta S\sim 10^{-4}$ J/(K mol) is inconsistent by several orders of
magnitude with reported values of the superfluid fraction (NCRIF) in the TO
and shear modulus experiments. We also point out that some of the most
disordered samples demonstrated ”supersolid” fraction up to 20%, while there
is little evidence for any excess entropy on the scale of the gas constant $R$
times the fraction of defects in any measured sample.

###  8.2 Single and double torsional oscillators

We have shown that a phenomenological glass (viscoelastic) model for quenched
defects accounts for the experimentally observed change in resonant frequency
and the concomitant peak in dissipation. Our analysis of torsional oscillator
(TO) experiments revealed that most are well described by a Cole-Cole
distribution for relaxation times. In addition, we derived a simple relation
for the ratio of change in dissipation and change in resonant frequency that
can explain the large ratios observed in experiments. The values for the glass
exponents in the distribution function of the backaction required to fit the
experimental data point toward broad distributions of glassy relaxation times,
which clearly invalidate any attempt to describe these experiments by a single
overdamped mode, i.e., a single Debye relaxation process.

We also applied these ideas to understand the double oscillator data. Our
studies of the coupled oscillator showed that the observed shifts in resonant
frequencies and dissipation are in agreement with a glassy backaction
contribution provided one includes anomalous damping in the dummy bob and an
explicit frequency dependence in the backaction term. As a side comment, it
came as a surprise that already the unloaded double TO (no 4He) required a
negative damping coefficient for the dummy bob to accurately describe resonant
frequencies and dissipation at 300 mK.

Finally, one should keep in mind that a significant difference between glassy
and supersolid dynamics is that a glassy contribution to the TO grows with
frequency, while a superfluid component decreases with frequency. This could
be another differentiating factor for separating very different relaxation
mechanisms.

###  8.3 Shear modulus

We showed that the shear modulus anomaly of solid 4He is strongly affected by
the dynamics of defects. The freezing out of defects leads to stiffening of
the solid concomitant with a peak in dissipation. By studying the glass
susceptibility due to the backaction, we found that both the amplitude change
and $T$-dependence of the shear modulus are well captured by this model. An
important consequence of the dynamic response analysis was the description of
the dissipation or phase angle. We found that the peak height of the
dissipation is independent of the applied frequency and linearly proportional
to the Cole-Cole exponent $\alpha$ as well as the backaction strength $g_{0}$.
As $g_{0}$ depends on the concentration of the TLS, we predicted that
increasing disorder will result in larger amplitude changes of the shear
modulus. Additionally, we extracted a universal scaling behavior proportional
to $\omega\tau(T)$ using the Cole-Cole plot. In this plot all curves of the
shear modulus collapsed onto a single curve over a wide range of frequencies.
Furthermore, we have shown that the glass contribution can be described by a
viscoelastic model through the incorporation of anelastic elements in
constitutive equations of stress and strain.

###  8.4 Dielectric function

We have shown that the arrested glass dynamics causes the low-temperature
anomaly in strained solid 4He through the acousto-optical (electro-elastic)
coupling and proposed that the temperature behavior of the dielectric function
is coupled to local strain fields near crystal defects. It records the glassy
dynamics and freeze-out of the hypothesized TLS excitations, which also lead
to a stiffening of the solid with decreasing temperature. This effect is not
captured by the standard Clausius-Mossotti relation, which attributes
dielectric function changes to a change in mass density or polarizability of
the nonpolar 4He atom. An important consequence of the phenomenological glass
susceptibility is the decrease of the dielectric function at high
temperatures, accompanied by a broad dissipation peak that can be measured by
the imaginary part of the dielectric function. We hypothesized that the
cooperative motion of atoms forming the TLS along dislocation segments is the
relevant process contributing to the reported anomaly. In our model, both the
change in $\epsilon(\omega;T)$ and the dissipation peak are to leading order
independent of the applied frequency. Since the coefficient $g_{0}$ of the
backaction depends on the concentration of defects, we predicted that the
change in dielectric function will be larger in quench-cooled or shear-
stressed samples, while it vanishes in defect-free single crystals. Beyond the
specific application to solid 4He, which we invoked here, our formalism
allowed for a direct link between elastic and dielectric properties which
could prove fruitful in many other systems.

In summary we encourage and welcome experiments that will allow a more precise
structural characterization of 4He. Also one needs to sharpen and provide a
more detailed analysis of the structural aspects of solid 4He in order to be
able to investigate any arrested dynamics of the postulated glassy regions to
separate it from a simple crossover phenomenon. Finally, we believe that more
dynamic studies probing the frequency or time response to a stimulus are
necessary to investigate the differences between small subsystems of glassy,
supersolid or superglassy origin in solid 4He.

###### Acknowledgements.

We are grateful to colleagues and collaborators who provided encouragement and
constructive criticism of the ideas presented here, A.F. Andreev, P. W.
Anderson, I. Beyerlein, J. C. Davis, A. Dorsey, C. Reichardt, B. Hunt, E.
Pratt, V. Gadagkar, J. Reppy, M. Chan, N. Prokof’ev, B. Svistunov, D.
Schmeltzer, A. Kuklov and E. Rudavsky. This work was supported by the US Dept.
of Energy at Los Alamos National Laboratory under contract No. DE-
AC52-06NA25396 and the Basic Energy Sciences Office. Z. N. was partially
supported by the NSF grant Award No. DMR-1106293. We also acknowledge steady
and generous support by the Aspen Center for Physics where parts of this
review were written.

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