φ3, φ4 model with quenched impurities

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  • Short Notes

    phys. stat. sol. (b) - 134, K89(1986) Subject classification: 64.50; 61 .70

    OR Z der Karl-Marx-Univer sitiit Leipzig

    cp", q4 Model with Quenched Impurities 1)

    BY K. SCHIELE

    To study the influence of quenched impurities on the critical behaviour of a system undergoing a first-order phase transition the (p3, 2 model is used. Quenched impurities are acting like local fields /I/. They are described by additional stochastic variables (e(x)). Therefore the Hamiltbnian has the form

    z=- = d x - 0 A 2 + - ( V Q ) Q 2 b 3 +-a +--$+wk+%'Z2}. c (1) C T sd (2 2 3! 4! Due to the presence of impurities the parameters A(x) T - To(x) , c(x), g(x) are random fields. We may write b(x) = [b] - 6b(x) , b [b] and analogous for g and c where [. . .] denotes the average on the impurity variable. In Landau theory only non-universal quantities a s transition temperatures o r

    transition heat are influenced by the impurities. The use of the renormalization

    group enables us to go beyond this theory. A scaling procedure yields the ex-

    pressions 6b(x) and 6c(x) as irrelevant values. Considering static proper- t ies the variable can be integrated out,

    zeff(m> = -1nJb-c exp(-a@,z>) . ( 2) The only remaining random contribution comes from the integration (2) as shown by Lubensky /2/. We assume

    [Y(x)Y(x')] = A ~ ( x - x') ,

    Using the replica-trick

    non-overlapping impurities defined by (3):

    ( 3)

    +y(x))a 2 +,(W g 2 + b a 3

    t o n \

    1) Karl-Marx-Platz, DDR-7010 Leipzig, GDR.

  • K9 0 physica status solidi (b) 134

    one obtains a translation invariant Hamiltonian

    The following renormalization group equations can be derived:

    3 K 6 c b 3K6Ab 2

    + dc d l - = - (4 -d - 2 g ) c + (1 +AI3 (1 + A)3

    9

    (A = a(T - To), Kd = 21-d9c-di%(d/2), ln(h/h ) = 1, ho represents the cut off), Equations (7) to (9) take into consideration the most divergent corrections of

    the perturbation series /3/. Since the renormalization group equations have the

    same fixed points as the model of the pure system (A = 0) the static universal and asymptotic behaviour remains unchanged. This is in correspondence with the Har r i s criterion. For the pure system the cri t ical exponent OL is calculate< to be -1 + (2/5)& . According to H a r r i s for positive a one expects a n impurity- influenced crit ical behaviour. In our case this holds for d < 3.5. For d < 4 one obtains additional divergent terms. Though they are less divergent their p re s - ence leads to further contributions in the renormalization group equations. Hence a full extrapolation E +3 has to be considered cautiously.

    To study dynamics one usually uses Langevin equations

    0

    == o t -y*. + S ( x , t) ; (r , r , and F(x, t), S(x, t) are kinetic coefficients and white noises, re- spectively).

    Following /4, 5/ both equations can be solved approximately. Using Fourier

    transformation and inserting (11)- into (10) one obtains a nonlinear stochastic equaticn which can be treated by Green' s functions in l inear response,

  • Short Notes K91

    The dynamic correlation function is calculated by the fluctuation-dissipation theorem. In lowest o rde r i t shows the same qualitative behaviour as the pure system. Further we r emark that no additional impurity caused divergence oc-

    curs i n the perturbation series. This can be shown by a decomposition of the Green 's function into a contribution of the pure system and a remaining par t leading only to finite corrections in d = 6 - E . Therefore quenched impurit ies do not change the dynamic cri t ical behaviour of the model system. In differ-

    ence to the pure system the asymptotic cri t ical region may be broadened for certain values of the overlap A .

    This cri t ical region can be estimated by the Ginzburg criterion. Further

    as in Landau theory a change of the non-universal parameters is proposed. Fo r slowly relaxing impurit ies the kinetic coefficient is replaced by

    l?-L r-' + w /y and the transition temperature is shifted by Tc-+ Tc + 2 + [Wx)] .

    References

    /1/ B.I. HALPARIN and C.M. VARMA, Phys. Rev. B - 14, 4030 (1976). /2/ T. C. LUBENSKY, Phys. Rev. B - 11, 3573 (1975). /3/ K. SCHIELE, phys. stat. sol. (b) - 134, K1 (1986). /4/ A. Z. FATASHINSKII and V. L. POKROVSKII, Fluktuatsionnaya teorya

    fazovykh perekhodov, Nauka, Moscow 1975 and 1980. /5/ L. SASVARI and F. SCHWABL, Z. Fhys. B - 46, 269 (1982).

    (Received January 14, 1986)