Nonlinear response in thermo field dynamics

Volume 100A, number 3

PHYSICS LETTERS

16 January 1984

NONLINEAR RESPONSE IN THERMO FIELD DYNAMICS

H. MATSUMOTO, Y. NAKANO and H. UMEZAWA Department of Physics, The University of Alberta, Edmonton, Alberta T6G 2,I1, Canada Received 18 October 1983

Nonlinear response theory at f'mitetemperature is studied in the frameworkof thermo field dynamics.With the operator algebra in thermo field dynamics,the retarded formulation of repsonse theory can be put in a simple time ordered formulation, leadingto the usual Feynman diagrammethod.

Thermo field dynamics (TFD) [1] has been developed to describe a quantum field system at finite temperature in a formalism similar to quantum field theory at zero temperature. Field operators are defined in real space-time and all the operator formalisms of quantum field theory are straightforwardly extended. The relationship to axiomatic statistical mechanics ,1 was clarified in ref. [3] and the relationship to the path order formalism [4] was studied in a recent paper [5]. The purpose of this paper is to show a particular merit of TFD in its application to response theory. Since a time-dependent external source effect is accumulated over the past, the response relation is usually expressed in terms of retarded Green functions. Calculation of the retarded form of the perturbative expansion is quite tedious since the Feyuman diagram technique is not applicable. In the following, we show that in TFD the retarded form of perturbative expansions can be easily put into the time ordered form of expansions in the operator formalism, thus making the Feyuman diagram method usable. This shows a particular usefulness of TFD in its application to the non-equilibrium system. In TFD the operator space is constructed by duplicating that at zero temperature. The duplication is carried out by the tilde conjugation. It is defined for operators (O) and c-numbers (c) as O10 2 = O 1 0 2,

ClO 1 + c 2 0 2 = c~O 1 + c202, *~

(~ = O,

(1)

where * denotes complex conjugation. (For simplicity we restrict operators to bosonlike ones; for fermionlike operators and for further details, see the textbook in ref. [1] .) Now, the operator space is a direct product of that of zero temperature (H) and its tilde conjugation (~):

=0

(A

(2)

A statistical average is given by the expectation value between thermal vacua IO(fl)):

(A) a = (O(fl)lA tO (fl))

(A E H),

(3)

where/3 = 1/keT with k e being the Boltzmann constant and ( )a denoting a statistical average. The time-evolution of any operator O in the duplicated space is generated by the thermal hamiltonian/t: O = i t & O]

(.q = n - ~ ,

(4)

with H being the hamiltonian of the system at zero temperature. ,1 See, for example, ref. [2]. 0.375-9601/84/$ 03.00 © Elsevier Science Publishers BN. (North.Holland Physics Publishing Division)

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Volume 100A, number 3

PHYSICS LETTERS

16 January 1984

The thermal vacuum is required to satisfy the tilde substitution rule A (t + ½i3)1 0(3)) = Al'(t) IO(3)),

(5a)

or equivalently

(O(J3)Ia(t - ½i3) = (0(3) IA~?(t),

(5b)

for any Heisenberg operator A (t) E H. (t means hermitian conjugation.) This rule is shown to be equivalent to the Kubo-Martin-Schwinger (KMS) condition [6],

(A(t)B(t'))~ = (B(t')A(t + i~) )~ .

(6)

The following arguments are based upon the algebraic properties (2)-(5). For Heisenberg operators A, B E H, the relations (2) and (5) lead to a remarkable identity between the retarded Green functions and the time ordered ones:

O(t - t') (O(33) 1[A(t), B(t')] IO(3)) = (O(3) ITA(t)[B(t') - ~t(t' + ½i3)] I O(3)),

(7)

where the time ordering denoted by T should operate only on the real part of time variables. Therefore, in the response theory of TFD, the deviation of an expectation value ofA (t) caused by a time-dependent external source [controlling the Heisenberg operator field B(t)] is calculated, in a linear approximation, by

6 ( A(t))3 = i f at'(O(J3)ITA(t) [B(t') - l~t (t' + ~-i3)] 10(3))J(t'),

(8)

when the hamiltonian of the external source J(t) is specified as Hext(t ) = -

J(t)B(t).

(9)

Here, we have assumed that, in the limit t ~ _0% the external source vanishes and that an equilibrium state remains. The expression (8) suggests that, in the time ordered formalism, the effective hamfltonian of the external source is given by /S/ext(t) = -J(t)[B(t)

- Bt(t + ½i3)] •

(10)

One can actually verify that the effect of the external source on the expectation value is given, to all orders, by the following formula: oo

(A(t))Y~=(O(3)lTexp(i f dt'J(t')[B(t')-t~(t' + ½i3)])A(t)lO(3)).

(11)

--oo

Using (2) and (5), it can be proven that the ths of (11) is equivalent to the usual expression for the nonlinear response, as follows: (O(O)I[T exp(i

f dt'J(t')[B(t')-]Jt(t'+ ½i3)])]A(t)[Texp(i f dt'J(t')[B(t')-ri?(t'+ ½i3)])]'0(3)) t

--oo

I

t

\

=(O(3)lA(t) T expti_ f dr' J(t')[B(t')- ~'~ (t'+ ½i3)])I 0(3)) --oo

126

(12)

Volume 100A, number 3

PHYSICS LETTERS

16 January 1984

t t = (O(/3)I IT exp ( - i f ~ dt'J(t')t~t (t'+ ½i/~))]A(t)IT exp (i f ~ dt' J(t')B(t'))1 [O(fl))

= (O(~)1

exp - i

(-,

dt'J(t')B(t')

A(t)

exp i

(,/

¢

dt'J(t')B(t')

10([3))

t

where Tis the anti-time ordering operator. Since, in (12), the time ordered product is transformed into the antitime ordered one, the result induces the retarded form. Now, one finds that the Feynman diagram method of TFD is applicable to the estimation of(8) or (11). For such a formulation, one should take the total hamfltonian as /=/tot -- N - i~ - J(t)[B(t) - ff~ (t + ~-i~)].

(13)

There has been a similar work [7] to formulate a non-linear response system in terms of the path-ordered Feynman.type rules, which are different from those of the present paper. In TFD, Feynman rules are extracted from the formulation in terms of the time ordered operator products. The present analysis shows that the framework of TFD is also useful to describe time-dependent nonlinear response sytems. It is worthwhile to mention that (11) is a generating functional for the retarded Green functions of all orders in the Heisenberg representation, which can be calculated by the causal Green functions in equilibrium. In fact, if one notices

f dtf~t(t + ½i/~)J(t) =

7 dtf~t(t)J(t - ~-i13)= J dt/~t(t) exp(-~ifJa/at)J(t),

(14)

it is easy to see that each coefficient in the power series of (11) with respect to J(t) is a combination of the time ordered Green functions in equilibrium. The present formalism in the limit ~ -~ o~ can be regarded as the time ordered formal/sin (and therefore the Feynman diagram method) of response theory at zero temperature. This work was supported by the Natural Sciences and Engineering Research Council, Canada and the Dean of Faculty of Science, University of Alberta, Edmonton, Alberta, Canada.

References [1] L. Leplae, F. Mancini and H. Umezawa, Phys. Rep. 10 (1974) 151; Y. Takahashi and H. Umezawa, Collect. Phenom. 2 (1975) 55; H. Matsumoto, Fortschx. Phys. 25 (1977) 1; H. Umezawa,H. Matsumoto and M. Tachiki, Thetmo field dynamicsand condensed states (North-Holland,Amsterdam, 1982). [2] D. Kastler, Constructive statistical mechanics and structural field theory, Catg~se Lecture in Physics, Vol. 4, 1969 session (Gordon and Breach, New York, 1970). [3] I. Ojima, Ann. Phys. (NY) 137 (1981) 1. [4] J. Schwinger,J. Math. Phys. 2 (1961) 407. [5] H. Matsumoto, Y. Nakano, H. Umezawa, F. Mancini and F. Maxinaxo,Ptog. Theor. Phys. 70 (1983) 599. [6] R. Kubo, J. Phys. Soc. Japan 12 (1957) 570; P. Martin and J. Schwinger, Phys. Rev. 115 (1959) 1342. [7] L.V. Keldysh, Soy. Phys. JETP 20 (1965) 1018. 127