ЖЭТФ, 2019, том 156, вып. 4 (10), стр. 738-741

ORDER PARAMETER IN ELECTRON SYSTEMS:

ITS FLUCTUATIONS AND OSCILLATIONS

K. B. Efetov^*

Ruhr University Bochum, Faculty of Physics and Astronomy

44780, Bochum, Germany

National University of Science and Technology “MISiS”

119049, Moscow, Russia

International Institute of Physics, UFRN

59078-400, Natal, Brazil

Received April 3, 2019,

revised version April 3, 2019

Accepted for publication April 5, 2019

Contribution for the JETP special issue in honor of I. M. Khalatnikov’s 100th anniversary

DOI: 10.1134/S004445101910016X

Although many thermodynamical physical quantities

could be calculated microscopically using rather so-

phisticated Bethe-Ansatz methods, one could not de-

The concept of the order parameter introduced by

Landau in his theory of phase transitions [1] plays the

termine the correlation functions using that technique.

Instead, it was demonstrated that it was sufficient to

central role in condensed matter and statistical physics.

compute correlations of superconducting or insulating

It has become clear later with development of the sca-

order parameters taking into account their fluctuations.

ling theory that, in order to describe a phase tran-

The publications [6, 7] have allowed us to demonstrate

sition, one should integrate over the order parameter

with a weight determined by a Ginzburg-Landau-Wil-

that exact results obtained in previous works for rather

special models [8-10] are general and one could obtain

son free energy functional [2]. This field of research

has attracted a lot of interest at the Landau Institute

results for realistic models using the general scheme.

Actually, this was one of the first steps in the subse-

in particular because scaling ideas had been proposed

previously by Patashinskii and Pokrovskii [3] and by

quent development of powerful bosonization techniques

[11] (for review, see, e. g. [12]).

Kadanoff [4], and first renormalization group study of

the phase transitions had been performed by Larkin

The idea that many interesting effects could be effi-

and Khmel’nitski [5].

ciently described by considering low energy fluctuations

Simultaneously, it was getting clear that fluctua-

of the order parameter motivated me later to study

tions of the order parameter could be very important

physics of granular superconductors [13]. In these ma-

not only near a phase transition but also in low-di-

terials, the superconductivity in a single grain can be

mensional systems. The dimension of the system was

well described by a phase of the order parameter fluc-

determined by geometry of the sample, while the di-

tuating in time. The modulus of the order parame-

mension of the electron bands was not always impor-

ter was assumed to be a constant and there was no

tant. It was demonstrated in 1975 in my works with

need to consider space variations of the phase inside

Anatoly Larkin [6, 7] that the behavior of two-point

the grain. In order to describe the macroscopic su-

correlation functions of one-dimensional electron sys-

perconductivity in the array of the grains one had to

tems at large distances or times was completely de-

account for the Josephson coupling between the grains.

termined by sound-like gapless quantum fluctuations.

Coulomb interaction turns out to be very important

enhancing the phase fluctuations in time and eventu-

^* E-mail: Konstantin.B.Efetov@ruhr-uni-bochum.de

ally leading to a superconductor-insulator transition

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ЖЭТФ, том 156, вып. 4 (10), 2019

Order parameter in electron systems. . .

dubbed later “Coulomb blockade”. Since then, I have

method really worked. This subject has been presented

made several other important works on granular su-

in numerous publications and reviews [33-37] and in a

perconductors and participated in writing a review on

book [38].

this subject. Physics of granular superconductors is

The supermatrix “order parameter” Q that appears

the same as physics of artificially designed Josephson

in the σ-model approach must be averaged with the free

networks and the description developed by me in 1980

energy functional, and it has no physical sense with-

[13] has been used and further developed later in a

out carrying out this procedure. Actually, the average

huge number of publications. Actually, the approach

〈Q〉 with the action of the σ-model is not an interest-

developed in this works allows one to describe not only

ing quantity because it is the average density of state

superconductors but also Coulomb blockade effects in

and the latter quantity is a smooth function of energy.

normal granular metals.

Conductivity, level correlations, density-density corre-

The idea of fluctuations of the order parameter

lations, etc. can be written in terms of a product of

turned out to be fruitful in problems of Anderson loca-

several Q like, e. g., 〈QQ〉. Therefore, the matrix Q is

lization and mesoscopics. Originally, it was conjectured

not an order parameter in its usual sense. In order

in a “bosonic” replica reformulation of models with dis-

to obtain an interesting physical quantity, one should

order by Wegner [14] but soon it has been followed

integrate a product of several Q over all configurations.

by “fermionic” replica representation in my work with

Recently, I have encountered a rather similar situ-

Larkin and Khmelnitskii [15]. Within this approach

ation in my investigation of a possibility of existence

one reduces summation of certain classes of diagrams

of a thermodynamically stable “Time-crystal”. The ti-

(so called “diffusons” and “cooperons”) to study of fluc-

me-crystal is expected to demonstrate an oscillating be-

tuations of a matrix. The matrix looked formally as an

havior of physical quantities in time. The concept of

order parameter and a “free energy functional” looked

a time-crystal has been proposed several years ago by

very similar to the one describing fluctuations of the

Wilczek [39] in a simple model but later it was realized

phase in superconductor. The free energy functional

that the time crystal state proposed there was not the

had a form of a non-linear σ-model. The “fermionic”

ground state and therefore could not be stable. More-

σ-model [15] has been later modified for studying dis-

over, it was even argued that the thermodynamically

ordered electron systems with electron-electron inter-

stable quantum crystal could not exist at all [40].

action [16] and quantum Hall effect in [17].

However, in a recent preprint [41] I have suggested

Although the σ-model allowed one to perform very

and investigated a model that can undergo a transition

efficiently perturbation theory and renormalization

into a state with an order parameter b depending on

group calculations, it was not possible to do non-per-

both real and imaginary times. As a result, effective

turbative calculations. In order to circumvent this dif-

wave functions also depend on the real and imaginary

ficulty, I have derived in 1982 [18] a supermatrix σ-mo-

time. The position of this non-trivial order parameter

del that was free of these problems and allowed one to

in time is arbitrary and the averaging over the positions

perform essentially non-perturbative calculations. The

gives zero. At the same time, the average of a product

first application of the zero-dimensional σ-model to

of the order parameters can be finite and can be mea-

studying level-level correlations in small metal particles

sured experimentally. For example, the average 〈bb〉

was successful, and I demonstrated [19] that level-le-

can be measured in quantum scattering experiments.

vel correlation functions agreed with those obtained

This situation resembles the one with fluctuation of or-

from the Wigner-Dyson random matrix theory [20,21].

der parameters considered previously but now the order

These works were followed by a review where the su-

parameter oscillates in time rather than fluctuates.

persymmetry method was applied to problems of com-

The present paper is not a review of my publications

pound nuclei [22]. Somewhat later the random matrix

in several different fields of physics. I simply wanted to

theory has been applied in study of quantum chaos [23]

use an opportunity to emphasize considering several

and the supersymmetry turned out to be useful there.

examples that many new results in several fields of re-

Since then, a huge amount of works has been published

search can be obtained with the help of the generalized

where the zero-dimensional σ-model was used for meso-

concept of the order parameter. Using this approach,

scopic and quantum chaos problems [24-26]. Somewhat

one can considerably simplify calculations because it is

later the supersymmetric σ-model has been successfully

sufficient to consider large distances without going into

applied to study of localization in disordered quantum

details of band structures and interactions. Very of-

wires and on the Bethe lattice [27-32]. I continued to

ten this route gives a possibility to solve problems that

work in this direction for quite a long time because the

have not been solved before and predict new physical

739

11*

K. B. Efetov

ЖЭТФ, том 156, вып. 4 (10), 2019

phenomena. I have realized the efficiency of this ap-

L. P. Kadanoff, Physics 2, 263 (1966).

proach during my years at the Landau Institute and

A. I. Larkin and D. E. Khmelnitskii, Zh. Exp. Teor.

used it later in many works.

Fiz. 56, 2067 (1969) [Sov. Phys. JETP 29,

1123

In contrast to the standard notion of a static long

(1969)].

range order in an ordered phase, one may encounter si-

tuations when there is no static long-range order. One

K. B. Efetov and A. I. Larkin, Zh. Eksp. Teor. Fiz.

can see from the results of the investigation of seve-

66, 2290 (1974) [Sov. Phys. JETP 39, 1129 (1974)].

ral models of electrons with interaction or moving in

K. B. Efetov and A. I. Larkin, Zh. Eksp. Teor. Fiz.

a random potential that there can be interesting non-

69, 764 (1975) [Sov. Phys. JETP 42, 390 (1976)].

trivial physics. The properties of the models have been

understood considering either fluctuations or oscilla-

I. E. Dzyaloshinskii and A. I. Larkin, Zh. Exp. Teor.

Fiz. 65, 411 (1973) [Sov. Phys. JETP 38, 202 (1974)].

tions in space and time of a generalized order param-

eter. Coulomb blockade, Anderson localization, spa-

A. Luther and I. Peschel, Phys. Rev. B 9, 2911 (1974).

ce-time quantum crystals, etc., are clearly quite dif-

ferent phenomena but their theoretical description has

10.

A. Luther and V. Emery, Phys. Rev. Lett. 33, 589

(1974).

many common features. Of course, it is always pleasant

to obtain new physical results and have a possibility to

11.

F. D. M. Haldane, J. Phys. C 14, 2585 (1981).

compare them with experiments. However, unification

of the description of many rather different phenomena

12.

A. O. Gogolin, A. A. Nersesyan, and A. M. Tsvelik,

Bosonization and Strongly Correlated Systems, Cam-

using a single concept gives also a strong esthetic sat-

bridge Univ. Press, New York (2004).

isfaction.

An essential part of my results presented in this

13.

K. B. Efetov, Zh. Eksp. Teor. Fiz. 78, 2017 (1980)

paper has either been done at the Landau Institute or

[Sov. Phys. JETP 51, 1015 (1980)].

followed from ideas developed there. I have started

14.

F. Wegner, Z. Phys. B 35, 207 (1979).

my scientific carrier and worked for many years at the

Landau Institute in its best time, and I am personally

15.

K. B. Efetov, A. I. Larkin, and D. E. Khmelnit-

very grateful to Isaac Markovich for the creation of

skii, Zh. Eksp. Teor. Fiz. 79, 1120 (1980) [Sov. Phys.

the Institute, for the support of my research, and for

JETP 52, 568 (1980)].

giving me the possibility to work at the Institute.

16.

A. M. Finkelstein, Zh. Eksp. Teor. Fiz. 84, 168 (1983)

[Sov. Phys. JETP 57, 97 (1983)].

Acknowledgements.

Financial support

Deutsche Forschungsgemeinschaft (Projekt

17.

A. M. M. Pruisken, Nucl. Phys. B 235, 277 (1984).

EF 11/10-1) and of the Ministry of Science and

18.

K. B. Efetov, Zh. Eksp. Teor. Fiz. 82, 872 (1982)

Higher Education of the Russian Federation in the

[Sov. Phys. JETP 55, 514 (1982)].

framework of Increase Competitiveness Program of

NUST “MISiS” (№ K2-2017-085) is greatly appreciated.

19.

K. B. Efetov, Zh. Eksp. Teor. Fiz. 83, 33 (1982) [Sov.

Phys. JETP 56, 467 (1982)]; J. Phys. C 15, L909

(1982).

The full text of this paper is published in the English

20.

E. Wigner, Ann. Math. 53, 36 (1951); 67, 325 (1958).

version of JETP.

21.

F. J. Dyson, J. Math. Phys. 3, 140, 157, 166, 1199

(1962).

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