Reply to a comment "No robust phases in aerogel..." (cond-mat/0505281).
I.A.Fomin
ul. Kosygina 2
119334 Moscow
Russia
The arguments of Volovik are refuted.
cond-mat0505650.tex
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\begin{document}
\title{Reply to a comment "No robust phases in aerogel..." (cond-mat/0505281).}
\author{I.A.Fomin\\
P. L. Kapitza Institute for Physical Problems, \\
ul. Kosygina 2, 119334 Moscow,Russia}
\date{ }
\maketitle
\begin{abstract}
 The arguments of Volovik are refuted.
\end{abstract}
 \bigskip
% \pacs {PACS numbers: 67.57.Pq, 67.57.Bc, 67.57.De }

%\section {}
\textbf{1}. The estimations made in the Comment are based on the
assumption that ABM order parameter is the only relevant minimum
of the Ginzburg and Landau (GL) free energy and its energy is
smaller then that of other possible minima by the energy of the
order of the full condensation energy. This situation is opposite
to the situation considered in the criticized papers [C1], where
competition of nearly degenerate states is assumed (in what
follows references of the Comment are prefixed by a capital C).
Free energy of bulk (without aerogel) superfluid $^3$He has 18
extrema \cite{march} and the situation assumed in the Comment does
not seem to be very realistic.

For a present discussion relative energies of the states,
corresponding to nonferromagnetic equal-spin pairing phases are of
importance. Among the mentioned extrema  there are four minima
meeting this requirement \cite{mermin}. Two of them - ABM and
axiplanar state are so close in energy that identification of
A-phase as ABM-state has been contested in the literature
\cite{gould}. Axiplanar state unlike ABM contains in its vicinity
 robust states, as it was discussed earlier \cite{fomin1}. These
states are also close in energy to the ABM. For a crude estimation
of a relative difference of energies of competing states (to be
referred as $\gamma$ in what follows) weak coupling values of
$\beta_1,...,\beta_5$ parameters were used. With these values a
relative difference of energies of the robust state and ABM
corresponds to $\gamma\sim 1/20$. Strong coupling corrections to
parameters $\beta$ will change the difference, still $\gamma\sim
1/10$ is a fair estimation. Contribution of fluctuations to energy
has to be compared not with the full condensation energy $F_0$ but
with much smaller value $\gamma F_0$. The regular part of this
contribution, which comes from the gapped modes, is of the order
of $\alpha F_0$ in agreement with and in the notations of the
Comment. A value of parameter $\alpha\sim(\eta^2/\sqrt\tau)$  can
be estimated from the measured width of the specific heat jump
\cite{parpia}. According to this data $\alpha\sim 1$ when
$\tau\sim (1/30)$. Because of the weak dependence on $\tau$
everywhere in the GL region parameter $\alpha\approx 1/5$ is at
least comparable or greater then $\gamma$ and even a regular
contribution of fluctuations can mix-up energies of competing
states in a contrast to the statement of the Comment.




\textbf{2}. The main object of criticism in the Comment is a
contribution of fluctuations of Goldstone modes to the energy.
According to the Comment this contribution is of the order of
$\alpha^2F_0$ thus even smaller then the contribution of the
gapped modes so that the free energy is a regular function of
$\alpha$ and the original free fnergy $F_0({A}_{\mu j}^{(0)})$ is
a good starting point for expansion on a small $\alpha$. This
assertion is in a conflict with the Imry and Ma statement [C5]
that the ordered state can be destroyed by "arbitrarily small
random field". It indicates that new free energy $F(\bar{A}_{\mu
j})$ which includes the contribution of fluctuations has to be a
singular function of $\alpha$ and the argument based on continuity
has to be taken with a great care.

The standard procedure \cite{Larkin} of finding of $F(\bar{A}_{\mu
j})$ is based not on the direct averaging of the original free
energy (or of its parts as it is done in the Comment), but on a
derivation of equation for the average order parameter, which in
the present case has the following form:
$$
\tau\bar A_{\mu j}+\frac{1}{2}\sum_{s=1}^5
\beta_s\bigl[\frac{\partial I_s}{\partial A^*_{\mu j}}+
\frac{1}{2}\bigl( \frac{\partial^3 I_s}{\partial A^*_{\mu
j}\partial A_{\nu n}\partial A_{\beta l}} <a_{\nu n}a_{\beta l}>+
$$
$$ 2\frac{\partial^3 I_s}{\partial A^*_{\mu j}\partial A^*_{\nu
n} \partial A_{\beta l}}<a^*_{\nu n}a_{\beta l}>\bigr)\bigr]=
-<a_{\mu l}\eta_{lj}>.              \eqno(1)
$$
Corresponding free energy, if necessary, has to be constructed so
that it generates the derived equation. The averages of
fluctuations of the order parameter $<a_{\nu n}a_{\beta l}>$ in
"Goldstone" directions are proportional to a diverging integral,
i.e. are singular. It has been checked by a direct substitution
that coefficients in front of the singular averages are not
identical zeros. It means that GL equation contains singular
terms. There is no reason for a cancellation of singular terms in
the expression for free energy as well. It should be remarked
though that  free energy has not been used in the arguments of
Refs. [C1].

Volovik in construction of the free energy followed "physical"
argument, which does not take into account important features of
the problem. In particular, he overlooks a fact that Goldstone
directions depend on the average order parameter. As a result
variation of his free energy will not contain terms which have to
be present in the equation (1).

The singular terms in the Eq. (1) being proportional to the
diverging integral are much greater then the regular terms. That
determines a procedure of its solution. As a first step the
principal terms are set to be equal to zero. This condition
selects a degenerate class of robust order parameters. Remaining
terms in the equation are treated as a perturbation, lifting this
degeneracy. They have to be considered on a class of robust order
parameters. So, robust order parameters are asymptotic solutions
of GL equation in a limit $\gamma\rightarrow 0$, $\alpha\ll 1$ and
the ABM order parameter is not solution of this equation in the
considered limit in a contrast to the statement, made in the
Comment. Energies of two states were not compared directly. A
problem of comparison of different states does not arise here
because in the considered limit a family of robust phases is the
only nontrivial extremum of the free energy.

   Summing up one can say that the declared in the Comment error in
overestimation of fluctuations does not exist. The diverging terms
are present in the GL equation and this is sufficient for
selection of the robust phases.  The robust phases are extrema of
the proper free energy. The situation, considered in the Comment
and the one discussed in Refs. [C1] correspond to different
regions of parameters: $\gamma\sim 1$, $\alpha\ll 1$, so that
$\gamma\gg\alpha$ (Comment), $\gamma\leq\alpha\ll 1$ (Refs. [C1]).
For that reason a criticism presented in the first part of the
Comment has no relevance to the problem discussed in the
criticized papers.

About the situation in the real $^3$He it has to be remarked that
the present knowledge of coefficients $\beta_1, ...,\beta_5$ is
not sufficiently accurate for reliable reconstruction of
"topography" of the GL free energy. Even though the given above
estimations show that situation is favorable for realization of
robust phases the competing situation [C7] can not be ruled out
and it can realize in its range of parameters, for example when
aerogel is very dilute if macroscopic description still applies.



I thank A.F. Andreev and V.I. Marchenko for the stimulating
discussions.


\begin{thebibliography}{99}
\bibitem{march} V. I. Marchenko ZhETF, {\bf 93}, 141, (1987)
\bibitem{mermin} N. D. Mermin, G. Stare, Report {\bf 2186},
Materials Science Center, Cornell University, (1974)
\bibitem{gould} Y. H. Tang, Inseob Hahn, H. M. Bozler, and C. M.
Gould, Phys. Rev. Lett.,{\bf 67}, 1775 (1991)
\bibitem{fomin1} I. A. Fomin, Pis`ma ZhETF {\bf 77}, 285 (2003) [JETP Lett. {\bf 77},
240 (2003)]
\bibitem{Larkin}  A.I.Larkin, Yu.N.Ovchinnikov,ZhETF  {\bf 61} (1971) 1221
 [Sov.Phys.JETP {\bf 34} (1971) 651]
\bibitem{parpia} J. He, A. D. Corvin, J. M. Parpia \emph{et al.}, Phys. Rev. Lett.,{\bf
89}, 115301 (2002)

 \end{thebibliography}

  \end{document}

Aerogel induces local variations of the order parameter -
fluctuations $a_{\mu j}$. They depend on an averaged order
parameter $\bar A_{\mu j}$ ($A_{\mu j}=\bar A_{\mu j}+a_{\mu j}$).
When fluctuations are small $a_{\mu j}\ll A_{\mu j}$ their
contribution to free energy can be expressed in terms of $\bar
A_{\mu j}$. New expression for energy $F(\bar{A}_{\mu j})$ is
different from the original $F_0({A}_{\mu j}^{(0)})$ and their
extrema need not coincide. When fluctuations are induced by the
random tensor field  $\eta_{jl}({\bf r})$ their relative magnitude
is characterized by a  (in the notations of the Comment). There
are regular corrections to the free energy originating from the
gapped modes. This contribution is of the order of $\alpha$ of
$F_0({A}_{\mu j}^{(0)})$. Even this contribution can change
relative position of the minima of $F_0({A}_{\mu j}^{(0)})$ if
there exist a region $1\gg\alpha\sim\gamma$.
  At a choice of an equilibrium state a difference of their energies
is of importance.

 Main difficulty is that for ABM phase large
fluctuations of the order parameter along the Goldstone directions
bring the system out of a region of applicability of the standard
scheme.

The opposite conclusion of the Comment is based on illustrative
argument of direct evaluation of energy of Goldstone modes. This
argument does not take into account that the space of degeneracy
of $A_{\mu j}$ is curved and an increment in a direction of a
tangent to this space with account of nonlinear terms brings order
parameter out of its space of degeneracy. Directions in which
order parameter is degenerate depend on the order parameter
itself. For the standard procedure, used in Refs. [C1] this is
taken into account automatically.


are not yet properly evaluated. Previously made "common sense"
estimation of the diverging integrals is very crude even thou it
is approved in the Comment. The infinite coefficient in expansion
of the free energy on $\alpha$ means that the expansion starts
with a larger singular term. That can be $\alpha^{\kappa}$ with
$\kappa<1$ ("common sense" estimation is equivalent to
$\kappa=0$). For a finite $\kappa$ singular terms can still be
principal if $\alpha^{\kappa}\gg\gamma$ but they vanish in a limit
$\alpha\rightarrow 0$.


In real $^3$He difference of energies even for A and B-phases does
not exceed 1/5 of the condensation energy. There are four
different minima .  Contribution of fluctuations has to be
compared with this small difference of energies of the competing
states (which is referred as  and not with the full condensation
energy. Existence of the small parameter $\gamma$ opens a region
where relative fluctuations of the order parameter are small, but
still essential for a choice of the average order parameter.  The
regular contribution to energy, coming from gapped modes, is of
the order of $\alpha\sim(\eta^2/\sqrt\tau)$.


The random tensor  introduced in references [5-7] of the "Comment"
gives rise to spacial fluctuations of the order parameter
Coordinate dependent order parameter can be represented as a sum:
$A_{\mu j}({\bf r})=\bar{A}_{\mu j}+a_{\mu j}({\bf r})$, where
$\bar{A}_{\mu j}\neq 0$ is the average value of the order
parameter. Free energy of superfluid $^3$He can be also split in
two parts:
$$
F=F_0(\bar{A}_{\mu j})+F_{fl}(\bar{A}_{\mu j},a_{\mu j}),
$$
where $F_0(\bar{A}_{\mu j})$ is obtained by a substitution of
$\bar{A}_{\mu j}$ in the Ginzburg and Landau functional. The
remaining part of the free energy $F_{fl}$ contains averages of
the fluctuating increments $\overline{a_{\mu j}a_{\nu l}}$ etc..
Mineev and Zhitomirsky ignore the existence of $F_{fl}$ and
following the argument of Imry and Ma [C4] and Volovik [C3] arrive
at the conclusion that the quasi-isotropic A-like phases proposed
in Ref. [C5-C7] are energetically unfavorable. This conclusion is
erroneous since contribution of fluctuations to the free energy
can not be neglected as one can see from comparison of possible
gains of energy coming from the two parts.

rage order parameter $\bar\psi$ and fluctuating increment $\zeta$
and using  result of Larkin and Ovchinnikov \cite{Larkin}
$\bar{\zeta^2}\sim\bar\psi^2(\eta^2/\sqrt\tau)$ one can estimate
the corresponding contribution to the free energy as $\delta
F_{\bot}\sim N(0)\tau^{3/2}T_c^2\eta^2$. This contribution is
already greater then $\delta F_0$ to the extent $\sqrt\tau/\eta^2$
($\eta^2/\sqrt\tau\ll 1$ is requied for fluctuations to remain
small in comparison with the average value of the order
parameter). -order approximation for the observed A-like phase.


 At the strength of the above estimations the argument presented in
 the "Comment" of Mineev and Zhitomirsky gives no reason for a
 revision of results of Refs.[C5-C7].

 I thank E.I.Kats and D.E.Khmelnitskii for useful discussion and
 the Institute of Laue and Langevin  for  hospitality and support
 of my visit to Grenoble.




\begin{thebibliography}{99}

\bibitem{Larkin}  A.I.Larkin, Yu.N.Ovchinnikov,ZhETF  {\bf 61} (1971) 1221
 [Sov.Phys.JETP {\bf 34} (1971) 651]


 \end{thebibliography}

  \end{document}


$$
A_{\mu j}=\hat x_{\mu} a_j+\hat y_{\mu} b_j , \eqno(1)
$$

cond-mat0505650.extracted_bib
\begin{thebibliography}{99}
\bibitem{march} V. I. Marchenko ZhETF, {\bf 93}, 141, (1987)
\bibitem{mermin} N. D. Mermin, G. Stare, Report {\bf 2186},
Materials Science Center, Cornell University, (1974)
\bibitem{gould} Y. H. Tang, Inseob Hahn, H. M. Bozler, and C. M.
Gould, Phys. Rev. Lett.,{\bf 67}, 1775 (1991)
\bibitem{fomin1} I. A. Fomin, Pis`ma ZhETF {\bf 77}, 285 (2003) [JETP Lett. {\bf 77},
240 (2003)]
\bibitem{Larkin}  A.I.Larkin, Yu.N.Ovchinnikov,ZhETF  {\bf 61} (1971) 1221
 [Sov.Phys.JETP {\bf 34} (1971) 651]
\bibitem{parpia} J. He, A. D. Corvin, J. M. Parpia \emph{et al.}, Phys. Rev. Lett.,{\bf
89}, 115301 (2002)

 \end{thebibliography}