ЯДЕРНАЯ ФИЗИКА, 2009, том 72, № 5, с. 804-810
= ЭЛЕМЕНТАРНЫЕ ЧАСТИЦЫ И ПОЛЯ
QUASIAVERAGES AND CLASSIFICATION OF EQUILIBRIUM STATES OF CONDENSED MEDIA WITH SPONTANEOUSLY BROKEN SYMMETRY
© 2009 N. N. Bogolyubov, Jr.1), D. A. Demyanenko2), M. Y. Kovalevsky2),3)*, N. N. Chekanova2)
Received October 14, 2008
Classification of equilibrium states of condensed media with spontaneously broken symmetry is carried out. Conditions of residual symmetry and spatial symmetry are formulated. The connection between these symmetry conditions and equilibrium states of various media with tensor order parameter is found out. Superfluid 3He, liquid crystals, quadrupolar magnetics are considered in detail. Possible homogeneous and heterogeneous states are found out. Discrete and continuous thermodynamic parameters, which define an equilibrium state, allowable form of order parameter, residual symmetry, and spatial symmetry generators are established.
PACS: 64.60.-i
1. INTRODUCTION
Second-order phase transitions are followed by a change of symmetry of equilibrium state of a medium when getting over critical temperature. Appropriate description of such states in condensed media below critical temperature needs an introduction of additional thermodynamic parameters, which are not concerned with the conservation laws, but resulting from physical nature of the new phase, into the theory. Under the phenomenological theory of equilibrium states classification [1] it is necessary to know free energy as an order parameter function in obvious form. Considerations of symmetry apply quite strong restrictions on the obvious form of free energy, but the attempts of connecting phenomenological parameters with intermolecular interaction parameters are confronted with considerable difficulties. Another difficulty of the mentioned approach is its correctness only in neighborhood of the point of phase transition, where one can neglect influence of fluctuation. Far from the critical temperature there are difficulties with the choice of obvious form of free energy as order parameter function and with solving corresponding nonlinear equations for it [2, 3]. Theoretical approach [4, 5], which does not depend on any definite mathematical model, uses idea of residual
1)SteklovMathematical Institute RAS, Moscow.
2)National S cience Center "Kharkov Institute of Physics and
Technology", Ukraine.
3)Belgorod State University, Russia.
E-mail: mikov@kharkov.ua
symmetry [6] of degenerate equilibrium state as a symmetry subgroup of a normal state.
In terms of this approach, effective equations for finding the equilibrium structure of order parameter, which do not include assumptions about the form of free energy and do not require closeness of temperature to the point of phase transition are found. Using the conception of quasiaveragues [7], confluent condensed media: superfluid 3He, liquid crystals, quadrupolar magnetics are studied, classification of equilibrium states is carried out and physical interpretation of the obtained solutions is given.
2. SYMMETRY PROPERTIES OF EQUILIBRIUM STATE OF NORMAL CONDENSED MEDIUM
Many-particle systems theory, which describes normal equilibrium states of condensed medium, in the context of statistical mechanics, can be built on basis of Gibbs operator
w = exp (Q - YaYa)■ (1)
Here, % = (JH, Pk, N, (a = 0, k, 4, a) — additive
motion integrals: H — Hamiltonian, Pk (k = 1, 2,3) — impulse operator, N — particle number operator, Sa (a = 1,2,3) — spin operator. Thermodynamic potential Q is determined from the condition of normalization Spw = 1. Generalized thermodynamic forces Ya, adjointed to additive motion integrals %, include: Y-1 = T is temperature, -Yk/Y0 = vk
is velocity, -Y4/Y0 = Hk is chemical potential and -Ya/Y0 = ha is effective magnetic field. Symmetry properties of statistic operator ( 1) are:
w,Pk
= 0,
w, £a(Y)
w, H
= 0,
0,
w,N
w, Lk(Y)
= 0, (2)
0.
First three relations in expression (2) represent spatially-temporal translational invariance and phase invariance of equilibrium state. The last two relations in (2) represent invariance of normal equilibrium state relative to the rotations in configurational and spin spaces. The expressions (2) include Èa(Y) and Li(Y) — generalized spin moment and orbital moment generators, which are determined as:
Li(Y) = Li + LY
è«(Y) = S a + S Y ,
LJ EE -iemYk(3)
d
dYi
SY = -i£aßY Yß
d dY~
they operate in Hilbert space and in space of thermodynamic functions, moreover for the differential operators relations have place: i
LY, Yj
= £ikjYk, i SY, Y/3 = Yj. Under the definition (3) operators ¿«(Y) and Lj(Y) satisfy commutative relations i Li(Y),Lk(Y) = —eikiLi(Y),
ta(Y), tp(Y) = —eaPlS7(Y). Symmetry conditions relative to the rotations in spin and config-urational spaces (2) mean neglect the weak dipole and spin-orbital interactions for the characteristic of an equilibrium state. Full symmetry group of normal state of condensed medium is given by
G = [SO (3)]s x [SO (3)]L x [U (1)]^ x x [T (3)] x [T (1)] .
Here, [SO (3)]S, [SO (3)]L are symmetry groups relative to the rotations in spin and configurational spaces, [T (3)], [T (1)] are translational groups in space and time, [U (1)]^ are phase symmetry group. Each element of a group is a unitary operator U = = exp iGg (g is real transformation parameters), which leaves the Gibbs distribution invariant: UwU + = w. Linear combinations of operators G e
ta, Li, N, pP| are generators of these transformations. This property is right for any voluntary transformation parameters by virtue of symmetry
relations (2). Averages like SpW G, b(x) turn into
zero at voluntary quasilocal operator b(x) for G e e |£a,Li,N,pP|. This is also right for operators
b(x) = A(x), which have a physical meaning of order parameter operator and do not commute with motion integrals G. Since order parameter operators are linear or bilinear on creation and annihilation
operators, averages SpW G, A(x) are linear on
order parameters A(x). This causes equilibrium averages of order parameters to turn into zero in normal state SpWAa(x) = 0.
3. EQUILIBRIUM STATES OF CONFLUENT CONDENSED MEDIA AND THE PROBLEM OF THEIR CLASSIFICATION
Theoretical footing for description of equilibrium states of condensed media with spontaneously broken symmetry is formed by quasiaverage conception of Bogolyubov [7]. Constructive feature of this conception is introduction into the statistic operator of an infinitely small perturbation or a source vF, which decreases the symmetry of statistic equilibrium state in comparison with the symmetry of Hamiltonian and allows to generalize the Gibbs distribution on condensed media in conditions of spontaneously broken symmetry. The Gibbs distribution for degenerate media, according to the conception of quasiaverages, is given by
Wv = exp ( Qv — YaYa — vF
The source F, which breaks the equilibrium state symmetry, is a linear functional of the order parameter operator Aa(x):
F = / d3x /0(x)A0(x) + h.c.
In this state the equilibrium average of order parameter does not equal zero (Aa(x)) = SpWAa(x) = = lim lim SpWvAa(x) = 0. To obtain effective
v^0 V^cx
equations that specify equilibrium values of order parameter it is necessary to know transformation properties of these physical values. The translation invariance condition is given by
Pk, Aa(x)j = —VkAa(x).
The generator of phase transformation group is the particle number operator N. Order parameter operator Aa(x) is transformed according to the relation
N, Aa(x)l = —gAa(x).
Value of constant g depends on tensor dimension and internal structure of order parameter operator. In the case of spin rotations with generators Sa, operators Aa(x) transform according to the relation:
Sa, Aa(x) = -gaabAb(x),
where gaab are certain constants. In case of spatial rotations order parameter operators transform in accordance with formula
L i, A a (x) = -giab A b(x) - £ijk Xk Vj A a(x),
where Li is orbital moment coefficient and values giab are constants. Linearity of the written switching relations by order parameter results in the linearity of equations for its equilibrium structure determination.
For translationally invariant equilibrium states, that satisfy the relation w,Pk =0, let us determine the residual symmetry condition of equilibrium state [8]
W,T(£, Y)
0,
where the residual symmetry generator T (£, Y) = = aiLi(Y) + baSa(Y) + cN is a linear combination of motion integrals with real parameters ai,ba,c = = £; Yi,Ya and reflects presence of residual symmetry in the condensed medium, which is less than in the initial, more symmetric, state.
The conditions of residual symmetry result in a system of linear differential equations in partial derivatives for the order parameter:
ai (giabAb + SikiYkdAa/dYi) + + ba (gaabAb + Ea^YpdAa/BY.y) + igcAa = 0.
These equations are considerably simplified in case when vectors Ya = Yk = 0, and become linear homogeneous algebraic equations
Tab (£, 0)Ab = 0, (4)
Tab (£, 0) = aigiab + ba gaab + igcSab.
The condition of nontrivial solution Aa = 0 results in equation on allowable values of residual symmetry generator det \Tab (£, 0)| =0 and classifies equilibrium states of degenerate condensed media. In this case Gibbs operator depends on thermodynamic parameters and parameters of residual symmetry generator w = w(Y,£).
For spatially-heterogeneous equilibrium states of degenerate condensed media residual and spatial symmetry generators are characterized by equalities [8]
T ({, Y) = aiLi(Y) + basa(Y) + cN + d%P%, (5)
Pk (n, Y) = Pk - PkN - QkaSa(Y) - tkjLj (Y) and lead to the relations
iSp [w,T(t, Y^^a(x) = 0, (6)
iSp \w,Pk (n, Y^ Aa(x) = 0.
Here, ai,ba,c,di = £ and n = (Pk,qka,tkj) are real parameters, which characterize residual and spatial symmetry generators. Relations (6) lead to the order parameter dependence on coordinate and on parameters of residual and spatial symmetry £, n. To specify the structure of orde
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