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Effective Hamiltonian with holomorphic variables
Physica A 271 (1999) 387–404
www.elsevier.com/locate/physa
E ective Hamiltonian with holomorphic variables Alessandro Cuccolia; c , Riccardo Giachettia; d , Riccardo Macioccoa; d; c , Valerio Tognettia; c , Ruggero Vaiab; c;∗ a Dipartimento
b Istituto
di Fisica dell’UniversitÃa di Firenze, Largo E. Fermi 2, I-50125 Firenze, Italy di Elettronica Quantistica del Consiglio Nazionale delle Ricerche, via Panciatichi 56=30, I-50127 Firenze, Italy c Istituto Nazionale di Fisica della Materia (INFM), Unit a di Firenze, Italy d Istituto Nazionale di Fisica Nucleare (INFN) Sezione di Firenze, Italy Received 1 March 1999
Abstract The pure-quantum self-consistent harmonic approximation (PQSCHA) permits to study a quantum system by means of an e ective classical Hamiltonian – depending on ˜ and temperature – and classical-like expressions for the averages of observables. In this work the PQSCHA is derived in terms of the holomorphic variables connected to a set of bosonic operators. The holomorphic formulation, based on the path integral for the Weyl symbol of the density matrix, makes it possible to approach directly general Hamiltonians given in terms of bosonic creation c 1999 Elsevier Science B.V. All rights reserved. and annihilation operators. PACS: 05.30.-d; 03.65.Sq; 05.30.Jp Keywords: Bosonic systems; Holomorphic variables; Classical e ective Hamiltonian
1. Introduction The notion of e ective potential in quantum statistical mechanics was introduced by Feynman [1] by means of a variational method for the path integral with imaginary time. The method was later on improved by Giachetti and Tognetti [2,3] and Feynman and Kleinert [4] in such a way to completely account for the quantum contribution of the harmonic part of the interaction. The new formulation, obtained by using the path integral in the Lagrangian form, has since been successfully applied to several condensed matter systems, usually providing an excellent agreement between theoretical results and experimental data [5]. The Lagrangian formulation, however, presents ∗
Corresponding author. Fax: +39-055414612. E-mail address: vaia@ieq. .cnr.it (R. Vaia)
c 1999 Elsevier Science B.V. All rights reserved. 0378-4371/99/$ - see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 9 9 ) 0 0 2 0 7 - 1
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some diculties for treating systems for which the kinetic energy and the potential energy are not separated: spin systems are remarkable instances of such a situation. In these cases, indeed, the Feynman–Jensen inequality fails to hold and the variational principle cannot be directly applied. A generalization that permits to overcome these problems has been found by using the path integral in the Hamiltonian formulation and the improved procedure for determining an e ective Hamiltonian is called the pure-quantum self-consistent harmonic approximation (PQSCHA) [6]. Several applications to condensed matter models have demonstrated its usefulness [5,7–9]. Nevertheless, this generalization cannot be applied in a natural way to eld-theory models, which are described in terms of creation and destruction operators: in the bosonic case the natural classical-like counterpart of these models requires the so called holomorphic variables z ∗ and z, connected to the creation and destruction operators. In order to calculate Hamiltonian path integrals we need a recipe to associate functions in the phase space to quantum observables [10]. This implies that some care is required to deal with ordering problems: of course, neglecting the cases where non-local topological terms are present, the ordering procedure must not a ect the nal results. It may, however, occur that some particular ordering turns out to be more suited to a given approach or to a speci c approximation, in the sense that it yields more directly the relevant results. The PQSCHA is a semiclassical expansion and therefore the Weyl ordering with the associated Wigner distribution [11] appears to be the most convenient choice. Weyl ordering, however, presents some drawbacks as well, for instance, it is somewhat cumbersome and its physical meaning is not immediately evident. Moreover, one can observe that a low-temperature eld theory is formulated using the normal ordering, since the related creation and destruction operators naturally de ne the vacuum (possibly close to the ground state) and it allows for an expansion in terms of Feynman diagrams. On the other hand, the e ective Hamiltonian approach has its main usefulness at intermediate and high temperatures, where perturbative approaches are useless while quanticity is still signi cant, since it fully accounts for the classical nonlinearity of the model under study, while the quantum character is accounted for at the one-loop level through suitable renormalization coecients; furthermore, it can be veri ed that whatever the ordering prescription one starts with, the nal result is such that the effective Hamiltonian, in the limit of low quantum coupling and=or in the limit of high temperature, reduces to the Weyl symbol of the original Hamiltonian operator. In this paper we present the derivation of the e ective Hamiltonian in the framework of the PQSCHA for Bose systems in terms of the holomorphic variables. In the next section we present the path integral formulation for calculating the density matrix and introduce the Weyl symbols and their composition properties. We nally establish a relation between Weyl and normal symbols that turns out to be particularly useful in the explicit calculation of the former. Section 3 shows the formulation of the PQSCHA in holomorphic variables both in one and in many degrees of freedom. We introduce the renormalization coecients that enter the expressions of the average values of the dynamical variables that, in turn, determine the e ective Hamiltonian. The additional diagonalization, necessary in the case of many degrees of freedom, is also given; the
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same section presents and uses the low-coupling approximation, and its specialization to deal with translation invariant systems, for which much more detailed results can be determined; eventually, we give an example of the application of the general theory to a model system with an interaction term expressed by a product of occupation numbers, thus demonstrating the advantages of the present formulation. We nally must add that, for the sake of a more clear exposition we have collected in the appendices the most relevant and less immediate calculations needed for the development of the subject. 2. Path integral for the density matrix in complex variables In order to x a well-de ned set of notations, we collect in this short section some basic properties of the Weyl and normal representations of operators in terms of complex variables. In the whole paper we set ˜ = 1. Consider a system of N degrees of freedom and let aˆ† = {aˆ† }=1;:::; N and aˆ = {aˆ }=1;:::; N be the creation and destruction operators satisfying the Bose commutation relations. We then introduce the corresponding classical coordinates (z ∗ ; z) ≡ {(z∗ ; z )}=1;:::; N . An operator O˜ can be represented either in the normal ordered or in the Weyl ordered form, namely Z t † t ∗ Oˆ = d(k∗ ; k) ei aˆ k ei k aˆ O˜ N (k∗ ; k) Z
t †
d(k∗ ; k) ei( aˆ
=
k+t k∗ a) ˆ
˜ ∗ ; k) : O(k
(1)
Here the notation t q is used to denote the transposed matrix and, for N pairs of complex variables q∗ = {q∗ }=1;:::; N and q = {q }=1;:::; N , we have introduced the integration measure Y dq∗ dq Y dRq dI q : (2) ≡ d(q∗ ; q) = 2i The normal and Weyl symbols, ON (z ∗ ; z) and O(z ∗ ; z), are then the inverse Fourier ˜ ∗ ; k), according to the general de nition transforms of O˜ N (k∗ ; k) and O(k Z t ∗ t ∗ ˜ ∗ ; k) : (3) f(z ∗ ; z) = d(k∗ ; k) ei( z k+ k z) f(k It is easily veri ed that t
O(z ∗ ; z) = e−(1=2) @z∗ @z ON (z ∗ ; z) :
(4)
For practical calculations it is sometimes useful to represent the above di erential operator as follows: t
e−(1=2) @z∗ @z (z ∗ ; z) = hh (z ∗ + Á∗ ; z − Á)ii2 =1=2 ; where hh·ii2 is the Gaussian average Z t ∗ 2 −2N d(Á∗ ; Á) (·) e− Á Á= : hh·ii2 =
(5)
(6)
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Weyl symbols have the remarkable property (see Appendix A) Z ˆ ˆ Tr(O1 O2 ) = d(z ∗ ; z) O1 (z ∗ ; z) O2 (z ∗ ; z) ;
(7) ˆ
that does not extend to products of more than two operators. In particular, if ˆ = e−ÿH is the density operator at the equilibrium temperature T = ÿ−1 , we have a classical-like expression of the statistical averages: Z 1 1 ˆ ˆ d(z ∗ ; z) (z ∗ ; z) O(z ∗ ; z) : (8) hOi = Tr(ˆO) = Z Z ˆ = If, for instance, we consider a single harmonic oscillator with Hamiltonian H
!(aˆ† aˆ + 12 ) at inverse temperature ÿ, it is straightforward to verify that the Weyl symbol for Hˆ is H(z ∗ ; z) = !z ∗ z, while for the density matrix, letting f = ÿ!=2, we rst determine the normal symbol N (z ∗ ; z) = exp[ − f − 2z ∗ z exp(−f)sinh f] and, using Eqs. (5) and (6), we nally get ∗ 1 e−2z z tanh f : (9) (z ∗ ; z) = cosh f Let us nally consider the standard form for the function (z ∗ ; z) in terms of path integral: Z ∗ (10) (z ∗ ; z) = D[z ∗ (u); z(u)] eS[z (u); z(u)] ; where D[z ∗ (u); z(u)] is the functional measure and S[z ∗ (u); z(u)] is an Euclidean action, depending on the functions z ∗ (u) and z(u) de ned for 06u6ÿ. The external variables z ∗ ; z may explicitly enter the expression of S through the boundary conditions on the functions z ∗ (u); z(u). In this work, however, we nd it convenient to use an expression of the path integral in which the boundary variables z ∗ (0); z(0), and z ∗ (ÿ); z(ÿ), are integrated over and the dependence on z ∗ ; z, is explicit in the form of S, namely Z ÿ 1 t ∗ ∗ t ∗ ∗ du ˙ − H(z (u); z(u)) [ z˙ (u)z(u) − z (u)z(u)] S[z (u); z(u)] = 2 0 1 − {t z ∗ (0)z(ÿ) − tz ∗ (ÿ)z(0)} 2 −{t [z ∗ (ÿ) − z ∗ (0)]z − tz ∗ [z(ÿ) − z(0)]} :
(11)
For a complete derivation we refer to the exhaustive analysis of phase space path integrals by Berezin [12], while a short but self-contained derivation is presented in Appendix A. 3. PQSCHA: one degree of freedom Although the main interest of the method is in the applications to problems with many degrees of freedom, it is much simpler to understand the method if we rst
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describe the case of a single degree of freedom. The idea is to decompose the pathintegral expression for (z ∗ ; z) into a rst sum over all paths sharing the same average point, de ned as the functional Z 1 ÿ ∗ du(z ∗ (u); z(u)) ; (12) (z ; z) = ÿ 0 and a second sum over average points. In order to do this we introduce in the path integral a resolution of the identity that xes the average point to (z ∗ ; z). It is then natural to de ne the reduced density, ! Z Z 1 ÿ ∗ ∗ ∗ du z(u) (z ; z; z ; z) = D[z (u); z(u)] z − ÿ 0 1 × z − ÿ ∗
Z 0
ÿ
! ∗
du z (u) eS[z
∗
(u); z(u)]
;
(13)
that collects all contributions coming from paths with the average point (z ∗ ; z), so that the full density reads Z ∗ ∗ ; z; z∗ ; z) : (14) (z ; z) = d(z ∗ ; z) (z We take (z ∗ ; z; z∗ ; z) as an non-normalized probability distribution in the variables (z ∗ ; z) and de ne its normalization constant as exp(−ÿHe (z ∗ ; z)), so that ∗
(z ∗ ; z; z∗ ; z) = e−ÿHe (z
; z)
P(z ∗ ; z; z∗ ; z) :
The thermal average of an observable Oˆ can then be written Z Z ∗ 1 ∗ ∗ ∗ ∗ ∗ ˆ d(z ; z) d(z ; z) O(z ; z) P(z ; z; z ; z) e−ÿHe (z ; z) hOi = Z
(15)
(16)
and it is natural to interpret exp[ − ÿHe (z ∗ ; z)] as a classical-like e ective density, whereas the probability distribution P(z ∗ ; z; z∗ ; z) describes the additional uctuations around the point (z ∗ ; z). In the classical limit it can be seen that P(z ∗ ; z; z∗ ; z) → (z ∗ − z∗ )(z − z) and exp[ − ÿHe (z ∗ ; z)] tends to the classical Boltzmann factor; it follows that the probability P describes the pure-quantum uctuations of the particle, thus ˆ providing a separation between classical-like and pure-quantum contribution to hOi. Observe that Eq. (16) is exact and provides an ideal starting point for approximations preserving the full classical nonlinear contribution. The explicit evaluation of the reduced density (z ∗ ; z; z∗ ; z) will be done in a selfconsistent approximation replacing H(z ∗ (u); z(u)) in action (11) with a trial Hamiltonian quadratic in the displacements from the average point, but depending upon the average points themselves. This means that the corresponding trial action results in being non-local and therefore cannot be obtained starting from any quantum operator: we are thus dealing with a larger class of actions. The parameters of H0 are optimized
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independently for any value of (z ∗ ; z). Explicitly we take H0 (z ∗ ; z; z∗ ; z) = w(z ∗ ; z) + E(z ∗ ; z)(z ∗ − z∗ )(z − z) + 12 [F(z ∗ ; z)(z − z)2 + c:c:] ;
(17)
where E(z ∗ ; z); F(z ∗ ; z) and w(z ∗ ; z) are functions to be optimized. Linear terms have obviously been neglected since they do not contribute to S0 . It is also evident that a quadratic H0 yields a Gaussian probability distribution P0 and it turns out that it is centred at the average point. We therefore introduce the uctuation variables (∗ ; ) ≡ (z ∗ − z∗ ; z − z) and denote by double brackets the averages over P0 : Z (18) hh·ii = d(∗ ; )( · )P0 (z ∗ + ∗ ; z + ; z∗ ; z) : Observe, in particular, that such averages can be easily evaluated when the moments hh∗ ii; hh2 ii and hh∗2 ii are known. As shown in Appendix B, the explicit result for 0 (z ∗ ; z; z∗ ; z) turns out to be 0 (z ∗ ; z; z∗ ; z) =
f 2 ˜∗ ˜ e−ÿw e−2 =L(f) : sinh f L(f)
(19)
∗ ˜ are connected to (∗ ; ) by a Bogoliubov transformation, Here (˜ ; ) ∗ ˜ ∗ = R∗ ˜ + S ;
∗ = S ∗ ˜ + R˜ ;
(20) ∗
˜ L(f) is the Langevin that diagonalizes the quadratic term, E∗ + 12 (F2 +c:c:)=!˜ : function L(f) = coth f − 1=f ;
(21)
where again f = ÿ!=2 while ! is given by !2 = !2 (z ∗ ; z) = E 2 (z ∗ ; z) − |F(z ∗ ; z)|2 :
(22)
Since no ambiguity can anymore arise, in the following we shall suppress the bar over z∗ and z. The optimization of the parameters w(z ∗ ; z); E(z ∗ ; z), and F(z ∗ ; z) is now in order. According to the PQSCHA method we require that P0 gives equal averages for the original and the trial Hamiltonian, as well as for their second derivatives [5], namely hhH(z ∗ + ∗ ; z + )ii = hhH0 (z ∗ + ∗ ; z + )ii = w(z ∗ ; z) + 12 !(z ∗ ; z)L(f(z ∗ ; z)) ;
(23)
hh@z∗ @z H(z ∗ + ∗ ; z + )ii = hh@z∗ @z H0 (z ∗ + ∗ ; z + )ii = E(z ∗ ; z) ; hh@2z H(z ∗ + ∗ ; z + )ii = hh@2z H0 (z ∗ + ∗ ; z + )ii = F(z ∗ ; z) ; hh@2z∗ H(z ∗ + ∗ ; z + )ii = hh@2z∗ H0 (z ∗ + ∗ ; z + )ii = F ∗ (z ∗ ; z) :
(24)
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Comparing Eqs. (19) and (15), the e ective Hamiltonian can be written as He (z ∗ ; z) = w(z ∗ ; z) +
1 sinh f(z ∗ ; z) ln ; ÿ f(z ∗ ; z)
and eventually Eq. (16) for thermal averages is approximated by Z ∗ ˆ = 1 d(z ∗ ; z) hhO(z ∗ + ∗ ; z + )ii e−ÿHe (z ; z) : hOi Z
(25)
(26)
In terms of the transformed variables the second moments of P0 , as de ned in ∗ ˜ = 0, so that in the original ones their ˜ = L(f)=2; hh˜∗ ˜∗ ii = hh˜ii Eq. (15), are hh˜ ii expressions turn out to be D(z ∗ ; z) = hh∗ ii = E
L(f) ; 2!
C(z ∗ ; z) = hh2 ii = −F ∗
L(f) ; 2!
L(f) ; (27) 2! that we call renormalization coecients. By means of them it is useful to introduce a di erential operator as follows. Letting C ∗ (z ∗ ; z) = hh∗2 ii = −F
(v; v∗ ) = D(v∗ ; v)@z∗ @z + 12 [C(v∗ ; v)@2z + C ∗ (v∗ ; v)@2z∗ ] :
(28)
we de ne the action of any function F() of the operator on a function a(z ∗ ; z) as F()a(z ∗ ; z) = F((v∗ ; v))a(z ∗ ; z) |(v∗ ;v)=(z∗ ; z) :
(29)
We can then give a useful representation of the double-bracket average hh·ii. Indeed, the Gaussian smearing of a generic function O(z ∗ ; z) can be represented as hhO(z ∗ + ∗ ; z + )ii = e O(z ∗ ; z) :
(30)
For practical calculations, the use of the right-hand side of (30) is particularly convenient when O(z ∗ ; z) is a low-degree polynomial. The function w(z ∗ ; z) is determined by Eq. (23), whose last term can be expressed as ∗ ∗ 1 2 !(z ; z)L(f(z ; z))
= (e )H(z ∗ ; z) ;
(31)
so that the e ective Hamiltonian can be written in a form with a more evident renormalization contribution 1 sinh f(z ∗ ; z) : (32) He (z ∗ ; z) = [(1 − ) e ]H(z ∗ ; z) + ln ÿ f(z ∗ ; z) It appears therefore that, besides the logarithmic term, He (z ∗ ; z) is given by the Weyl Hamiltonian corrected by terms of the second-order in the renormalization coecients. Moreover, as we shall show in the next section, the form (32) is an ideal starting point for a further approximation in order to deal with many degrees of freedom.
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4. PQSCHA: many degrees of freedom In this section we are concerned with the generalization of the method to a system with N degrees of freedom. The self-consistent solution of the counterpart of Eqs. (24) is obviously rather dicult: further simpli cation and (27) for arbitrary values of (z∗ ; z) is therefore in order. This is called the low-coupling approximation (LCA) and consists F(z∗ ; z), in expanding the parameters – which are N × N complex matrices E(z∗ ; z); ∗ ∗ – so as to make the renormalization coecients, and hence also the and F (z ; z) Gaussian averages hh·ii, independent of the con guration. Eventually, we describe this scheme of approximation and we focus, in particular, on translation invariant systems. Let us consider therefore a system of N degrees of freedom. Again the evaluation is done by replacing H with a trial Hamiltonian H0 , of (z ∗ ; z; z∗ ; z) = w(z∗ ; z) +t (z ∗ − z∗ )E(z∗ ; z)(z − z) H0 (z ∗ ; z; z∗ ; z) z∗ ; z)(z − z) + c:c:] ; + 12 [t (z − z)F( quadratic in the displacements from the average point Z 1 ÿ = du(z ∗ (u); z(u)) : (z∗ ; z) ÿ 0
(33)
(34)
As already mentioned, E; F, and F ∗ are a Hermitean and two symmetric N × N complex matrices, respectively. In expressing 0 we again introduce the pure-quantum uctuation variables (∗ ; ) = and the linear canonical transformation (z ∗ − z∗ ; z − z) X ∗ R∗k (z∗ ; z) ˜k + Sk (z∗ ; z) ˜k ; ∗ = k
=
X
∗
∗ Sk (z∗ ; z) ˜k + Rk (z∗ ; z) ˜k
;
(35)
k ∗ ˜ diagonalizes the quadratic term to new variabless (˜ ; ) X X ∗ [∗ E + 12 ( F + c:c:)] = !k (z∗ ; z) ˜k ˜k :
(36)
k
Here and in the subsequent discussion, transformed variables are labeled by Latin indices (k; l; : : :); Greek indices (; ; : : :) are used for the original ones. Diagonalization (36), that is not possible in the most general case, can be performed under suitable constraints on the matrices E; F, and F ∗ , (see e.g. [13]). However, this problem is not extremely relevant in the present context, since these matrices are not external data but parameters that must be optimized: this constraint just restricts the number of independent matrix elements. In addition, in most applications the canonical transformation (35) is mainly determined by symmetry considerations. In analogy to what we did in the previous section, there is no ambiguity in suppress The explicit calculation of 0 , given in Appendix B, leads to ing the bar over (z∗ ; z).
A. Cuccoli et al. / Physica A 271 (1999) 387–404
the following expression for He : 1 X sinh fk (z ∗ ; z) ln He (z ∗ ; z) = w(z ∗ ; z) + ÿ fk (z ∗ ; z)
395
(37)
k
∗ ∗ with fk (z ∗ ; z)=ÿ!k (z ∗ ; z)=2, while the moments of P0 are hh˜k ˜l ii=kl L(fk (z ∗ ; z))=2 ∗ ∗ and hh˜k ˜l ii = hh˜k ˜l ii = 0. The PQSCHA conditions that determine the optimized parameters w(z ∗ ; z); E(z ∗ ; z); F(z ∗ ; z), and F ∗ (z ∗ ; z), are
hhH(z ∗ + ∗ ; z + )ii = hhH0 (z ∗ + ∗ ; z + )ii = w(z ∗ ; z) +
1X !k (z ∗ ; z) L(fk (z ∗ ; z)) ; 2
(38)
k
hh@z∗ @z H(z ∗ + ∗ ; z + )ii = hh@z∗ @z H0 (z ∗ + ∗ ; z + )ii = E (z ∗ ; z) ; hh@z @z H(z ∗ + ∗ ; z + )ii = hh@z @z H0 (z ∗ + ∗ ; z + )ii = F (z ∗ ; z) ; ∗ (z ∗ ; z) : hh@z∗ @z∗ H(z ∗ + ∗ ; z + )ii = hh@z∗ @z∗ H0 (z ∗ + ∗ ; z + )ii = F
(39)
Generalizing the procedure of the previous section for the case of one degree of freedom, we de ne the renormalization coecients 1X ∗ ∗ (Rk Rk + Sk Sk ) L(fk ) ; D (z ∗ ; z) = hh∗ ii = 2 k
C (z ∗ ; z) = hh ii = ∗ (z ∗ ; z) = hh∗ ∗ ii = C
1X 2
k
1X 2
∗ ∗ (Rk Sk + Rk Sk ) L(fk ) ;
(R∗k Sk + R∗k Sk ) L(fk ) ;
(40)
k
and, starting from X 1 ∗ C (v∗ ; v)@z @z + C D (v∗ ; v)@z∗ @z + (v∗ ; v)@z∗ @z∗ (v∗ ; v) = 2
(41)
and de ning the action of operator in analogy with Eq. (29), we nd that the Gaussian smearing by double-bracket average can be put into the form hhO(z ∗ + ∗ ; z + )ii = e O(z ∗ ; z) :
(42)
As in the preceding section, it can be easily seen that the e ective Hamiltonian reads 1 X sinh fk (z ∗ ; z) (43) ln He (z ∗ ; z) = [(1 − ) e ] H(z ∗ ; z) + ÿ fk (z ∗ ; z) k
and the general expression for calculating thermal averages takes the form: Z ∗ 1 ˆ d(z ∗ ; z)hhO(z ∗ + ∗ ; z + )iie−ÿHe (z ; z) : hOi = Z
(44)
We now describe the low-coupling approximation, whose main purpose is to make the averages hh·ii independent of the con guration [5]. The self-consistent equations therefore need to be solved only once, with a great simpli cation for implementing the
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method. One can think of several ways to obtain this goal: their choice substantially depends on the physics of the problem [6]. The simplest way, however, consists in expanding the matrices E(z ∗ ; z) and F(z ∗ ; z) around a self-consistent minimum (z0∗ ; z0 ) of He : E(z ∗ ; z) = E + E(z ∗ ; z) ; F(z ∗ ; z) = F + F(z ∗ ; z) ;
(45)
where E = E(z0∗ ; z0 ), F = F(z0∗ ; z0 ), thus using the convention to drop the arguments of functions evaluated at (z0∗ ; z0 ). As a consequence the frequencies will be split as, !k (z ∗ ; z) = !k + !k (z ∗ ; z) and for the evaluation of the averages we shall only take D and C as renormalization coecients. It is explicitly shown in Appendix C that the e ective Hamiltonian becomes 1 X sinh fk ln ; (46) He (z ∗ ; z) = e0 H(z ∗ ; z) − 0 e0 H(z0∗ ; z0 ) + ÿ fk k
where fk = ÿ!k =2 and X 1 ∗ @z∗ @z∗ ) : D @z∗ @z + (C @z @z + C 0 = 2
(47)
The general formula for thermal averages (44) still holds provided that the Gaussian smearing of O(z ∗ ; z) is calculated with the LCA renormalization coecients, i.e. as exp(0 )O(z ∗ ; z). It often occurs that the indices ; ; : : : refer to the sites of a lattice, whose symmetries can sometimes be very helpful in order to simplify the analysis, provided that the minimum con guration of He (z ∗ ; z) shares the same property. The calculations are very easy in the case of translation symmetry. In particular, for a one-dimensional lattice E = E− ;
F = F− ;
(48)
where the hermiticity of E and the symmetry of F imply that the components E and F satisfy ∗ ; E = E−
F = F− :
(49)
Performing a Fourier transformation, X X k eik ; ∗ = N −1=2 ∗k e−ik ; = N −1=2 k
the left-hand-side of (36) becomes X X ∗ 1 !k ˜k ˜k ; Ek ∗k k + (Fk k −k + c:c:) = 2 k
where Ek =
(50)
k
(51)
k
X
E e−ik ;
Fk =
X
F e−ik :
(52)
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As a consequence of (49) Ek is real and Fk = F−k . If the lattice has more than one dimension the generalization is obvious and, in particular, the summation over wave vectors is performed over the rst Brillouin zone. If there are internal degrees of freedom for each lattice site, the Fourier transform must be followed by a diagonalization in the internal space. Finally, the diagonalization of the quadratic form (51) is completed by a Bogoliubov transformation ∗ ∗k = Ak ˜k − Bk ˜−k ; ∗
∗ ˜ k + A∗−k ˜−k ; −k = −B−k
(53)
where the canonical conditions imply A∗k Ak − Bk∗ Bk = 1;
Ak = A−k ;
Bk = B−k ;
(54)
and it turns out that !k (A∗k Ak + Bk∗ Bk ) = Ek ;
!k Ak Bk = Fk =2 :
(55)
Finally, !k2 = Ek2 − Fk∗ Fk :
(56)
By combining transformations (50) and (53) we obtain the LCA approximation of the matrices R and S de ned in (35): Rk = N −1=2 A∗k eik ;
∗ Sk = −N −1=2 Bk∗ e−ik
(57)
Using then (40) we also can specify the LCA renormalization coecients: 1 X L(fk ) Ek cos k( − ) ; D = N 2!k k
C = −
1 X ∗ L(fk ) Fk cos k( − ) ; N 2!k k
∗ =− C
1 X L(fk ) Fk cos k( − ) : N 2!k
(58)
k
In many cases only a reduced set of these coecients will be explicitly needed in He . For instance, any model with on-site nonlinearity entails only on-site coecients D = D , C = C which are independent of due to translation invariance. 5. An application As an example of a system that is more conveniently treated by means of holomorphic variables, we shall consider a model Hamiltonian with quartic on-site interaction, X X † ˆ =− aˆi aˆi+d + (V1 nˆi + V2 nˆ2i ) ; (59) H id
i
where i runs over sites of a Bravais lattice, d denotes the displacements towards the Z nearest neighbour, nˆi = aˆ†i aˆi , and V1 and V2 ¿ 0 are constants. One can recognize
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in (59) the Hamiltonian of the well-known Bose–Hubbard model [14,15]. Its Weyl symbol is easily found to be X X N zi∗ zi+d + [ (V1 − V2 )zi∗ zi + V2 (zi∗ zi )2 ] : (60) H(z ∗ ; z) = − V1 − 2 id
i
H is invariant under global phase transformations, and provided that the LCA e ec∗ ; z0i )= tive Hamiltonian shares the same property, the translation invariant minimum (z0i ∗ (z0 ; z0 ) of He is degenerate; therefore we can choose as representative the real minimum with z0 = z0∗ . It is easy to verify that Eq. (39) for the matrices E and F becomes X i; j+d ; Eij = [(V1 − V2 ) + 4V2 (z02 + D)]ij − d
Fij = 2V2 (z02 + C)ij ;
(61)
and Fij∗ = Fij . They yield the following Fourier transforms: X cos(k · d) ; Ek = (V1 − V2 ) + 4V2 (z02 + D) − d
Fk = F = 2V2 (z02 + C) ;
(62)
The renormalization coecients read now 1 X L(fk ) Ek ; D= N 2!k k F X L(fk ) C =− N 2!k
(63)
k
with !k2 = Ek2 − F 2 . Using (46), the e ective Hamiltonian can be eventually written as X [4Dzi∗ zi + C(zi∗2 + zi2 )] ; He (z ∗ ; z) = G(ÿ) + H(z ∗ ; z) + V2
(64)
i
where G(ÿ) = −V2 N [2z02 (2D + C) + 2D2 + C 2 ] +
1 X sinh fk ln ÿ fk
(65)
k
is a uniform contribution that a ects the partition function but cancels out in the expression of thermal averages. The (real) translation-invariant stationary points of He are given by z0;1 = 0 ; 2 z0;2 =
V2 − V1 + Z − 2D − C : 2V2
(66) (66)
2 ¡ 0 the minimum is z0;1 and in the above formulas One can check that when z0;2 2 ¿ 0 the minimum is z0;2 . Since the one should replace z0 = z0;1 ; conversely, for z0;2 calculation of the renormalization coecients depends on the chosen minimum, both cases can simultaneously occur: if this is the case, the relevant minimum to be used at a given temperature is the one which gives the smaller free energy.
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6. Summary We have presented a construction of the e ective Hamiltonian suitable for the study of the thermodynamics of eld-theory models, with Hamiltonian expressed in terms of creation and destruction operators. The construction is done by a path-integral approach, in terms of the holomorphic variables z ∗ and z. The main result we presented consists of Eq. (26) for one degree of freedom, and of Eq. (44) for many degrees of freedom: in the latter case the LCA e ective Hamiltonian (46) allows one to perform practical calculations. As in the previous formulation [4] with respect to phase-space conjugate variables, p and q, the framework retains all classical nonlinear e ects embodied in the classical-like formulas for the thermal averages and the partition function. The quantum e ects related to the quadratic part of the Hamiltonian are completely accounted for, while the pure-quantum nonlinearity is treated at one-loop (Hartree–Fock) level. The low-coupling approximation permits actually to deal with many degrees of freedom, so that our results can be used for a direct application to nite-temperature eld theory. We nally observe that this approach can be extended to deal with the thermodynamics of Fermi systems [16].
Appendix A: Path integral for in terms of complex variables We give here a short derivation of the expression for (z ∗ ; z) as a path integral. Although the subject is rather well known, we have chosen to include it since we want to stress those aspects connected with Weyl symbols. We start from the Trotter’s formula for a discrete imaginary time u → i (i = 1; : : : ; M , ≡ M=ÿ) and we take then the continuum limit: ˆ
ˆ M: ˆ = e−ÿH = lim (1 − H) M →∞
(A.1)
In order to introduce the Weyl formulation, it is necessary to write the expression of the Weyl symbol of the product of M operators. Let us rst derive the product rule for two operators Oˆ and Oˆ 1 , i.e. the Weyl symbol O ∗ O1 (z ∗ ; z) for Oˆ Oˆ 1 . Using Eq. (1) and the Baker-Campbell-Hausdor identity, we have Z Z t † t ∗ ∗ Oˆ Oˆ 1 = d(k0∗ ; k0 ) d(k1∗ ; k1 )ei[ aˆ (k0 +k1 )+ (k0 +k1 )a]ˆ t ∗
t ∗
˜ 0∗ ; k0 )O˜ 1 (k1∗ ; k1 ) : ×e(1=2)( k1 k0 − k0 k1 ) O(k
(A.2)
Using now the expression of the inverse Fourier transform and the representation of the -function Z t ∗ t ∗ (A.3) (k∗ ) (k) = d(z ∗ ; z)ei( z k+ k z) ;
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we can determine the product rule for the Weyl symbols that is traditionally called star product: Z t ∗ t ∗ ∗ t ∗ t ∗ ∗ O ∗ O1 (z ; z) = d(h1∗ ; h1 ) d(k1∗ ; k1 )ei[ z (h1 +k1 )+ (h1 +k1 )z] e(1=2)( k1 h1 − h1 k1 ) Z ×
t ∗
t ∗
d(C∗1 ; C1 ) d(z1∗ ; z1 )e−i( C1 h1 + h1 C1 ) t ∗
t ∗
t
∗
×e−i( z1 k1 + k1 z1 ) O(C∗1 ; C1 )O1 (z1∗ ; z1 ) Z = 22N d(C∗1 ; C1 ) d(z1∗ ; z1 )O(C∗1 ; C1 )O1 (z1∗ ; z1 ) ∗
×e−2[ (C1 −z
∗ )(C1 −z1 )−t (C∗ 1 −z1 )(C1 −z)]
:
(A.4)
The integration of this expression, using (A.3), immediately yields Eq. (7). By iterating ˆ Oˆ 1 ; Oˆ 2 : Eq. (A.4), we obtain the star product of three operators O; Z ∗ 4N d(C∗1 ; C1 ) d(z1∗ ; z1 ) d(C∗2 ; C2 ) d(z2∗ ; z2 ) O ∗ O1 ∗ O2 (z ; z) = 2 ×O(C∗1 ; C1 )O1 (z1∗ ; z1 )O2 (z2∗ ; z2 ) t
∗
∗
t
∗
∗
t
∗
∗
×e−2[ (C1 −C2 )(C1 −z1 )− (C1 −z1 )(C1 −C2 )] ×e−2[ (C2 −z
∗ )(C2 −z2 )−t (C∗ 2 −z2 )(C2 −z)]
:
(A.5)
It is now easy to see that in general, if we take Oˆ = 1ˆ and therefore O = 1, we can write # Z "Y M M ∗ 2MN ∗ ∗ ∗ ∗ Oi (z ; z) = 2 d(Ci ; Ci ) d(zi ; zi )Oi (zi ; zi ) i=1
i=1
( ×exp −2
M X
) [t (C∗i − C∗i+1 )(Ci − zi ) −t (C∗i − zi∗ )(Ci − Ci+1 )]
;
i=1
(A.6) where CM +1 ≡ z. Letting Ci = zi + i (with M +1 = z) and zi = zi − zi−1 , the previous expression becomes M
∗
2MN
∗ Oi (z ; z) = 2
Z "Y M
i=1
# d(∗i ; i ) d(zi∗ ; zi )Oi (zi∗ ; zi )
i=1
( ×exp 2
M X i=1
) t
[
(∗i+1
+
∗ zi+1 )i
−t ∗i (i+1
+ zi+1 )]
:
(A.7)
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The integrations over the variables i are performed taking into account a di erent representation of the -function, namely Z t ∗ t ∗ d(z ∗ ; z)e±( z k− k z) ; (A.8) (k)(k∗ ) = Since no generality is lost in assuming M to be an even number, we eventually obtain # Z "Y M M ∗ MN ∗ ∗ ∗ Oi (z ; z) = 2 d(zi ; zi )Oi (zi ; zi ) i=1
i=1
M=2 k X X ∗ ∗ (tz2k+1 z2j −t z2j z2k+1 ) ×exp 2 − k=1 j=1
+
M=2 X j=1
∗ (tz2j z −t z ∗ z2j ) :
(A.9)
Finally, according to Eq. (A.1), we set Oi = 1 − jH ' e−jH , and in the continuum limit the integrand becomes the exponential of Z ÿ=2 Z u Z ÿ ∗ du H(z (u); z(u)) − 2 du du0 [t z˙∗ (2u)z(2u ˙ 0 ) − tz˙∗ (2u0 )z(2u)] ˙ − 0
0
Z +2
ÿ=2 0
0
du0 [t z ∗ (ÿ)z(2u ˙ 0 ) −t z˙∗ (2u0 )z(ÿ)] + 2
Z
ÿ=2
0
du0 [t z˙∗ (2u0 )z −t z ∗ z(2u ˙ 0 )] : (A.10)
The integration over u0 is trivial and, by the change of variable u → ÿ − u, we obtain the path integral given by Eqs. (10) and (11), with the measure D[z ∗ (u); z(u)] formally de ned by ∗
MN
D[z (u); z(u)] = lim 2 M →∞
M Y
d(zi∗ ; zi ) :
(A.11)
i=1
Appendix B: The PQSCHA method Let us evaluate the reduced density 0 (13) with the trial Hamiltonian H0 , Eq. (17). Representing the -function as in Eq. (A.8) one gets Z Z ∗ ∗ d(v∗ ; v)ez v−v z D[z ∗ (u); z(u)] 0 (z ∗ ; z; z∗ ; z) = ×e
S0 [z ∗ (u);z(u)]−(1=ÿ)
Rÿ 0
du[z ∗ (u)v−v∗ z(u)]
;
(B.1)
where S0 is given by Eq. (11) with H0 . We change the integration variables in the path integral to (∗ (u); (u)) = (z ∗ (u) − z∗ ; z(u) − z) and we make the Bogoliubov
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transformation ∗ ˜ ∗ (u) = R∗ ˜ (u) + S (u); ∗
∗ ˜ (u) = S ∗ ˜ (u) + R(u) ;
(B.2)
∗
with R R − S S = 1. This permits to diagonalize the quadratic term ∗ ˜ : E∗ (u)(u) + 1 [F2 (u) + c:c:] = !˜ (u)(u)
(B.3)
2
It is straightforward to verify that this transformation preserves at the same time the functional measure and the form of the exponent in the path integral. Performing the same transformation on (z ∗ ; z), (z ∗ ; z) and (v∗ ; v) we obtain Z Z ∗ ˜ d(v˜∗ ; v) ˜ D[˜ (u); (u)] 0 (z ∗ ; z; z∗ ; z) = e−ÿw (Z ×exp
ÿ
0
1 ∗ 1 ∗ ˜ − ˜∗ (u)(u)] ˜˙ ˜ du − [˜ (u)v˜ − v˜∗ (u)] + [˜˙ (u)(u) ÿ 2
i h ∗ ∗ 1 ∗ ˜ ˜ ˜ + − [˜ (0)(ÿ) − ˜ (ÿ)(0)] − !˜ (u)(u) 2 ∗
∗
∗
)
˜ ˜ ˜ − (z˜ − z˜ )[(ÿ) − (0)]] − [[˜ (ÿ) − ˜ (0)](z˜ − z) ∗
: (B.4)
The shift
v˜ v˜∗ ; (u) − (B.5) ( (u); (u)) → (u) + ÿ! ÿ! eliminates the linear term, and the resulting expression of 0 contains the path integral of a single harmonic oscillator: Z Z ∗ ∗ ˜ ˜ ˜ −v˜ v=ÿ! d(v˜∗ ; v)e D[˜ (u); (u)] 0 (z ∗ ; z; z∗ ; z) = e−ÿw ∗
∗
(Z
ÿ
∗ 1 ˜˙∗ ˜ − ˜∗ (u) (u)] ˜˙ ˜ [ (u) (u) − !˜ (u)(u) 2 0 ∗ ∗ v˜ − [˜ (ÿ) − ˜ (0)] z˜ − z˜ + ÿ! ∗ v˜ ∗ ˜ ˜ [(ÿ) − (0)] − z˜∗ − z˜ − ÿ! ) ∗ 1 ˜∗ ˜ ˜ ˜ : − [ (0)(ÿ) − (ÿ) (0)] 2
×exp
du
(B.6)
Therefore, using Eq. (9) we obtain Z ∗ ˜ ˜ −v˜ v=ÿ! d(v˜∗ ; v)e 0 (z ∗ ; z; z∗ ; z) = e−ÿw
1 cosh f v˜ v˜∗ ∗ z˜ − z˜ + tanh f ; ×exp −2 z˜∗ − z˜ − ÿ! ÿ!
(B.7)
A. Cuccoli et al. / Physica A 271 (1999) 387–404
403
where f = ÿ!=2. Finally, performing the integration over (v˜∗ ; v), ˜ we arrive at Eq. (19). Eq. (32) can be derived from Eq. (31); the latter, in turn, can be easily veri ed in this way: e H(z ∗ ; z) = hhH(z ∗ + ∗ ; z + )ii = D(z ∗ ; z)hh@z∗ @z H(z ∗ + ∗ ; z + )ii + 12 [C(z ∗ ; z)hh@2z H(z ∗ + ∗ ; z + )ii + c:c:] :
(B.8)
Using Eqs. (24) and (27) the right-hand side becomes D(z ∗ ; z)E(z ∗ ; z) + 12 [C(z ∗ ; z)F(z ∗ ; z) + c:c:] L(f(z ∗ ; z)) ; =[E 2 (z ∗ ; z) − F ∗ (z ∗ ; z)F(z ∗ ; z)] 2!(z ∗ ; z)
(B.9)
and nally, Eq. (31) follows from Eq. (22). In order to deal with the case of many degrees of freedom we use representation (A.8) of the -function for implementing constraint (34), so that the expression for becomes 0 (z ∗ ; z; z∗ ; z) Z Z t ∗ t ∗ 0 (z ∗ ; z; z∗ ; z) = d(C∗ ; C)e z C− C z D[z ∗ (u); z(u)] (
1 ×exp S0 [z (u); z(u)] − ÿ ∗
Z 0
ÿ
) t ∗
t ∗
du[ z (u)C − C z(u)]
; (B.10)
where S0 is the action (11) with the trial Hamiltonian (33). This expression is evaluated by making the linear canonical transformation (35) which permits to decouple the path integral into a product of one dimensional harmonic path integrals. The same transformation, done over all the pairs of conjugate variables that appear in (B.10), Q yields the form 0 = exp(−ÿw) k 0 k for the reduced density. Taking into account (36) we can use the result for one degree of freedom Eq. (19) and we obtain Y fk 2 ˜∗ ˜ 2 ∗ ∗ −ÿw exp − ; (B.11) =e 0 (z ; z; z ; z) sinh fk L(fk ) L(fk ) k k k
where fk = ÿ!k =2.
Appendix C: Low-coupling approximation We give here the steps which lead to the LCA expression (46) for He , when the LCA is made by expanding the renormalization coecients around the self-consistent minimum (z0∗ ; z0 ) of He . We start by expanding Eq. (43) to the lowest order in the di erence !k (z ∗ ; z) = !k (z ∗ ; z) − !k , recalling the convention to drop the arguments of functions evaluated
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at (z0∗ ; z0 ). The expansion of the logarithmic term is 1 X sinh fk (z ∗ ; z) 1 X sinh fk X L(fk ) = !k (z ∗ ; z) ln ln + ÿ fk (z ∗ ; z) ÿ fk 2 k
k
=
k
1 X sinh fk ln + [e 0 H(z ∗ ; z) − e0 0 H(z0 ; z0∗ )] ; ÿ fk k
(C.1) where we have taken into account the fact that Eq. (31) generalizes to X L(fk (z ∗ ; z)) !k (z ∗ ; z) = [e ]H(z ∗ ; z) : 2
(C.2)
k
Letting = 0 + , we have the expansion [(1 − )e ]H(z ∗ ; z) = [(1 − )e ]e0 H(z ∗ ; z) − e 0 H(z ∗ ; z) = e0 H(z ∗ ; z) − e 0 H(z ∗ ; z) + O(()2 ) ;
(C.3)
so that, eventually, Eq. (46) is obtained. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
R.P. Feynman, A.R. Hibbs, Quantum Mechanics and Path Integrals, McGraw-Hill, New York, 1965. R. Giachetti, V. Tognetti, Phys. Rev. Lett. 55 (1985) 912. R. Giachetti, V. Tognetti, Phys. Rev. B 33 (1986) 7647. R.P. Feynman, H. Kleinert, Phys. Rev. A 34 (1986) 5080. A. Cuccoli et al., J. Phys.: Condens. Matter 7 (1995) 7891. A. Cuccoli, V. Tognetti, P. Verrucchi, R. Vaia, Phys. Rev. A 45 (1992) 8418. A. Cuccoli, V. Tognetti, P. Verrucchi, R. Vaia, Phys. Rev. Lett. 77 (1996) 3439. A. Cuccoli, V. Tognetti, P. Verrucchi, R. Vaia, Phys. Rev. B 56 (1997) 14 456. A. Cuccoli, V. Tognetti, P. Verrucchi, R. Vaia, Phys. Rev. B 58 (1998) 14 151. W.H. Louisell, Quantum Statistical Properties of Radiation, Wiley, New York, 1973. M. Hillery, R.F. O’Connell, M.O. Scully, E.P. Wigner, Phys. Rep. 106 (1984) 122. F.A. Berezin, Sov. Phys. Usp. 23 (1980) 763. J.H.P. Colpa, Physica 93A (1978) 327. M.P.A. Fisher, P.B. Weichman, G. Grinstein, D.S. Fisher, Phys. Rev. B 40 (1989) 546. V.A. Kashurnikov, B.V. Svistunov, Phys. Rev. B 53 (1996) 11 776. A. Cuccoli, R. Giachetti, R. Maciocco, V. Tognetti, R. Vaia, unpublished.
www.elsevier.com/locate/physa
E ective Hamiltonian with holomorphic variables Alessandro Cuccolia; c , Riccardo Giachettia; d , Riccardo Macioccoa; d; c , Valerio Tognettia; c , Ruggero Vaiab; c;∗ a Dipartimento
b Istituto
di Fisica dell’UniversitÃa di Firenze, Largo E. Fermi 2, I-50125 Firenze, Italy di Elettronica Quantistica del Consiglio Nazionale delle Ricerche, via Panciatichi 56=30, I-50127 Firenze, Italy c Istituto Nazionale di Fisica della Materia (INFM), Unit a di Firenze, Italy d Istituto Nazionale di Fisica Nucleare (INFN) Sezione di Firenze, Italy Received 1 March 1999
Abstract The pure-quantum self-consistent harmonic approximation (PQSCHA) permits to study a quantum system by means of an e ective classical Hamiltonian – depending on ˜ and temperature – and classical-like expressions for the averages of observables. In this work the PQSCHA is derived in terms of the holomorphic variables connected to a set of bosonic operators. The holomorphic formulation, based on the path integral for the Weyl symbol of the density matrix, makes it possible to approach directly general Hamiltonians given in terms of bosonic creation c 1999 Elsevier Science B.V. All rights reserved. and annihilation operators. PACS: 05.30.-d; 03.65.Sq; 05.30.Jp Keywords: Bosonic systems; Holomorphic variables; Classical e ective Hamiltonian
1. Introduction The notion of e ective potential in quantum statistical mechanics was introduced by Feynman [1] by means of a variational method for the path integral with imaginary time. The method was later on improved by Giachetti and Tognetti [2,3] and Feynman and Kleinert [4] in such a way to completely account for the quantum contribution of the harmonic part of the interaction. The new formulation, obtained by using the path integral in the Lagrangian form, has since been successfully applied to several condensed matter systems, usually providing an excellent agreement between theoretical results and experimental data [5]. The Lagrangian formulation, however, presents ∗
Corresponding author. Fax: +39-055414612. E-mail address: vaia@ieq. .cnr.it (R. Vaia)
c 1999 Elsevier Science B.V. All rights reserved. 0378-4371/99/$ - see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 9 9 ) 0 0 2 0 7 - 1
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some diculties for treating systems for which the kinetic energy and the potential energy are not separated: spin systems are remarkable instances of such a situation. In these cases, indeed, the Feynman–Jensen inequality fails to hold and the variational principle cannot be directly applied. A generalization that permits to overcome these problems has been found by using the path integral in the Hamiltonian formulation and the improved procedure for determining an e ective Hamiltonian is called the pure-quantum self-consistent harmonic approximation (PQSCHA) [6]. Several applications to condensed matter models have demonstrated its usefulness [5,7–9]. Nevertheless, this generalization cannot be applied in a natural way to eld-theory models, which are described in terms of creation and destruction operators: in the bosonic case the natural classical-like counterpart of these models requires the so called holomorphic variables z ∗ and z, connected to the creation and destruction operators. In order to calculate Hamiltonian path integrals we need a recipe to associate functions in the phase space to quantum observables [10]. This implies that some care is required to deal with ordering problems: of course, neglecting the cases where non-local topological terms are present, the ordering procedure must not a ect the nal results. It may, however, occur that some particular ordering turns out to be more suited to a given approach or to a speci c approximation, in the sense that it yields more directly the relevant results. The PQSCHA is a semiclassical expansion and therefore the Weyl ordering with the associated Wigner distribution [11] appears to be the most convenient choice. Weyl ordering, however, presents some drawbacks as well, for instance, it is somewhat cumbersome and its physical meaning is not immediately evident. Moreover, one can observe that a low-temperature eld theory is formulated using the normal ordering, since the related creation and destruction operators naturally de ne the vacuum (possibly close to the ground state) and it allows for an expansion in terms of Feynman diagrams. On the other hand, the e ective Hamiltonian approach has its main usefulness at intermediate and high temperatures, where perturbative approaches are useless while quanticity is still signi cant, since it fully accounts for the classical nonlinearity of the model under study, while the quantum character is accounted for at the one-loop level through suitable renormalization coecients; furthermore, it can be veri ed that whatever the ordering prescription one starts with, the nal result is such that the effective Hamiltonian, in the limit of low quantum coupling and=or in the limit of high temperature, reduces to the Weyl symbol of the original Hamiltonian operator. In this paper we present the derivation of the e ective Hamiltonian in the framework of the PQSCHA for Bose systems in terms of the holomorphic variables. In the next section we present the path integral formulation for calculating the density matrix and introduce the Weyl symbols and their composition properties. We nally establish a relation between Weyl and normal symbols that turns out to be particularly useful in the explicit calculation of the former. Section 3 shows the formulation of the PQSCHA in holomorphic variables both in one and in many degrees of freedom. We introduce the renormalization coecients that enter the expressions of the average values of the dynamical variables that, in turn, determine the e ective Hamiltonian. The additional diagonalization, necessary in the case of many degrees of freedom, is also given; the
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same section presents and uses the low-coupling approximation, and its specialization to deal with translation invariant systems, for which much more detailed results can be determined; eventually, we give an example of the application of the general theory to a model system with an interaction term expressed by a product of occupation numbers, thus demonstrating the advantages of the present formulation. We nally must add that, for the sake of a more clear exposition we have collected in the appendices the most relevant and less immediate calculations needed for the development of the subject. 2. Path integral for the density matrix in complex variables In order to x a well-de ned set of notations, we collect in this short section some basic properties of the Weyl and normal representations of operators in terms of complex variables. In the whole paper we set ˜ = 1. Consider a system of N degrees of freedom and let aˆ† = {aˆ† }=1;:::; N and aˆ = {aˆ }=1;:::; N be the creation and destruction operators satisfying the Bose commutation relations. We then introduce the corresponding classical coordinates (z ∗ ; z) ≡ {(z∗ ; z )}=1;:::; N . An operator O˜ can be represented either in the normal ordered or in the Weyl ordered form, namely Z t † t ∗ Oˆ = d(k∗ ; k) ei aˆ k ei k aˆ O˜ N (k∗ ; k) Z
t †
d(k∗ ; k) ei( aˆ
=
k+t k∗ a) ˆ
˜ ∗ ; k) : O(k
(1)
Here the notation t q is used to denote the transposed matrix and, for N pairs of complex variables q∗ = {q∗ }=1;:::; N and q = {q }=1;:::; N , we have introduced the integration measure Y dq∗ dq Y dRq dI q : (2) ≡ d(q∗ ; q) = 2i The normal and Weyl symbols, ON (z ∗ ; z) and O(z ∗ ; z), are then the inverse Fourier ˜ ∗ ; k), according to the general de nition transforms of O˜ N (k∗ ; k) and O(k Z t ∗ t ∗ ˜ ∗ ; k) : (3) f(z ∗ ; z) = d(k∗ ; k) ei( z k+ k z) f(k It is easily veri ed that t
O(z ∗ ; z) = e−(1=2) @z∗ @z ON (z ∗ ; z) :
(4)
For practical calculations it is sometimes useful to represent the above di erential operator as follows: t
e−(1=2) @z∗ @z (z ∗ ; z) = hh (z ∗ + Á∗ ; z − Á)ii2 =1=2 ; where hh·ii2 is the Gaussian average Z t ∗ 2 −2N d(Á∗ ; Á) (·) e− Á Á= : hh·ii2 =
(5)
(6)
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Weyl symbols have the remarkable property (see Appendix A) Z ˆ ˆ Tr(O1 O2 ) = d(z ∗ ; z) O1 (z ∗ ; z) O2 (z ∗ ; z) ;
(7) ˆ
that does not extend to products of more than two operators. In particular, if ˆ = e−ÿH is the density operator at the equilibrium temperature T = ÿ−1 , we have a classical-like expression of the statistical averages: Z 1 1 ˆ ˆ d(z ∗ ; z) (z ∗ ; z) O(z ∗ ; z) : (8) hOi = Tr(ˆO) = Z Z ˆ = If, for instance, we consider a single harmonic oscillator with Hamiltonian H
!(aˆ† aˆ + 12 ) at inverse temperature ÿ, it is straightforward to verify that the Weyl symbol for Hˆ is H(z ∗ ; z) = !z ∗ z, while for the density matrix, letting f = ÿ!=2, we rst determine the normal symbol N (z ∗ ; z) = exp[ − f − 2z ∗ z exp(−f)sinh f] and, using Eqs. (5) and (6), we nally get ∗ 1 e−2z z tanh f : (9) (z ∗ ; z) = cosh f Let us nally consider the standard form for the function (z ∗ ; z) in terms of path integral: Z ∗ (10) (z ∗ ; z) = D[z ∗ (u); z(u)] eS[z (u); z(u)] ; where D[z ∗ (u); z(u)] is the functional measure and S[z ∗ (u); z(u)] is an Euclidean action, depending on the functions z ∗ (u) and z(u) de ned for 06u6ÿ. The external variables z ∗ ; z may explicitly enter the expression of S through the boundary conditions on the functions z ∗ (u); z(u). In this work, however, we nd it convenient to use an expression of the path integral in which the boundary variables z ∗ (0); z(0), and z ∗ (ÿ); z(ÿ), are integrated over and the dependence on z ∗ ; z, is explicit in the form of S, namely Z ÿ 1 t ∗ ∗ t ∗ ∗ du ˙ − H(z (u); z(u)) [ z˙ (u)z(u) − z (u)z(u)] S[z (u); z(u)] = 2 0 1 − {t z ∗ (0)z(ÿ) − tz ∗ (ÿ)z(0)} 2 −{t [z ∗ (ÿ) − z ∗ (0)]z − tz ∗ [z(ÿ) − z(0)]} :
(11)
For a complete derivation we refer to the exhaustive analysis of phase space path integrals by Berezin [12], while a short but self-contained derivation is presented in Appendix A. 3. PQSCHA: one degree of freedom Although the main interest of the method is in the applications to problems with many degrees of freedom, it is much simpler to understand the method if we rst
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describe the case of a single degree of freedom. The idea is to decompose the pathintegral expression for (z ∗ ; z) into a rst sum over all paths sharing the same average point, de ned as the functional Z 1 ÿ ∗ du(z ∗ (u); z(u)) ; (12) (z ; z) = ÿ 0 and a second sum over average points. In order to do this we introduce in the path integral a resolution of the identity that xes the average point to (z ∗ ; z). It is then natural to de ne the reduced density, ! Z Z 1 ÿ ∗ ∗ ∗ du z(u) (z ; z; z ; z) = D[z (u); z(u)] z − ÿ 0 1 × z − ÿ ∗
Z 0
ÿ
! ∗
du z (u) eS[z
∗
(u); z(u)]
;
(13)
that collects all contributions coming from paths with the average point (z ∗ ; z), so that the full density reads Z ∗ ∗ ; z; z∗ ; z) : (14) (z ; z) = d(z ∗ ; z) (z We take (z ∗ ; z; z∗ ; z) as an non-normalized probability distribution in the variables (z ∗ ; z) and de ne its normalization constant as exp(−ÿHe (z ∗ ; z)), so that ∗
(z ∗ ; z; z∗ ; z) = e−ÿHe (z
; z)
P(z ∗ ; z; z∗ ; z) :
The thermal average of an observable Oˆ can then be written Z Z ∗ 1 ∗ ∗ ∗ ∗ ∗ ˆ d(z ; z) d(z ; z) O(z ; z) P(z ; z; z ; z) e−ÿHe (z ; z) hOi = Z
(15)
(16)
and it is natural to interpret exp[ − ÿHe (z ∗ ; z)] as a classical-like e ective density, whereas the probability distribution P(z ∗ ; z; z∗ ; z) describes the additional uctuations around the point (z ∗ ; z). In the classical limit it can be seen that P(z ∗ ; z; z∗ ; z) → (z ∗ − z∗ )(z − z) and exp[ − ÿHe (z ∗ ; z)] tends to the classical Boltzmann factor; it follows that the probability P describes the pure-quantum uctuations of the particle, thus ˆ providing a separation between classical-like and pure-quantum contribution to hOi. Observe that Eq. (16) is exact and provides an ideal starting point for approximations preserving the full classical nonlinear contribution. The explicit evaluation of the reduced density (z ∗ ; z; z∗ ; z) will be done in a selfconsistent approximation replacing H(z ∗ (u); z(u)) in action (11) with a trial Hamiltonian quadratic in the displacements from the average point, but depending upon the average points themselves. This means that the corresponding trial action results in being non-local and therefore cannot be obtained starting from any quantum operator: we are thus dealing with a larger class of actions. The parameters of H0 are optimized
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independently for any value of (z ∗ ; z). Explicitly we take H0 (z ∗ ; z; z∗ ; z) = w(z ∗ ; z) + E(z ∗ ; z)(z ∗ − z∗ )(z − z) + 12 [F(z ∗ ; z)(z − z)2 + c:c:] ;
(17)
where E(z ∗ ; z); F(z ∗ ; z) and w(z ∗ ; z) are functions to be optimized. Linear terms have obviously been neglected since they do not contribute to S0 . It is also evident that a quadratic H0 yields a Gaussian probability distribution P0 and it turns out that it is centred at the average point. We therefore introduce the uctuation variables (∗ ; ) ≡ (z ∗ − z∗ ; z − z) and denote by double brackets the averages over P0 : Z (18) hh·ii = d(∗ ; )( · )P0 (z ∗ + ∗ ; z + ; z∗ ; z) : Observe, in particular, that such averages can be easily evaluated when the moments hh∗ ii; hh2 ii and hh∗2 ii are known. As shown in Appendix B, the explicit result for 0 (z ∗ ; z; z∗ ; z) turns out to be 0 (z ∗ ; z; z∗ ; z) =
f 2 ˜∗ ˜ e−ÿw e−2 =L(f) : sinh f L(f)
(19)
∗ ˜ are connected to (∗ ; ) by a Bogoliubov transformation, Here (˜ ; ) ∗ ˜ ∗ = R∗ ˜ + S ;
∗ = S ∗ ˜ + R˜ ;
(20) ∗
˜ L(f) is the Langevin that diagonalizes the quadratic term, E∗ + 12 (F2 +c:c:)=!˜ : function L(f) = coth f − 1=f ;
(21)
where again f = ÿ!=2 while ! is given by !2 = !2 (z ∗ ; z) = E 2 (z ∗ ; z) − |F(z ∗ ; z)|2 :
(22)
Since no ambiguity can anymore arise, in the following we shall suppress the bar over z∗ and z. The optimization of the parameters w(z ∗ ; z); E(z ∗ ; z), and F(z ∗ ; z) is now in order. According to the PQSCHA method we require that P0 gives equal averages for the original and the trial Hamiltonian, as well as for their second derivatives [5], namely hhH(z ∗ + ∗ ; z + )ii = hhH0 (z ∗ + ∗ ; z + )ii = w(z ∗ ; z) + 12 !(z ∗ ; z)L(f(z ∗ ; z)) ;
(23)
hh@z∗ @z H(z ∗ + ∗ ; z + )ii = hh@z∗ @z H0 (z ∗ + ∗ ; z + )ii = E(z ∗ ; z) ; hh@2z H(z ∗ + ∗ ; z + )ii = hh@2z H0 (z ∗ + ∗ ; z + )ii = F(z ∗ ; z) ; hh@2z∗ H(z ∗ + ∗ ; z + )ii = hh@2z∗ H0 (z ∗ + ∗ ; z + )ii = F ∗ (z ∗ ; z) :
(24)
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Comparing Eqs. (19) and (15), the e ective Hamiltonian can be written as He (z ∗ ; z) = w(z ∗ ; z) +
1 sinh f(z ∗ ; z) ln ; ÿ f(z ∗ ; z)
and eventually Eq. (16) for thermal averages is approximated by Z ∗ ˆ = 1 d(z ∗ ; z) hhO(z ∗ + ∗ ; z + )ii e−ÿHe (z ; z) : hOi Z
(25)
(26)
In terms of the transformed variables the second moments of P0 , as de ned in ∗ ˜ = 0, so that in the original ones their ˜ = L(f)=2; hh˜∗ ˜∗ ii = hh˜ii Eq. (15), are hh˜ ii expressions turn out to be D(z ∗ ; z) = hh∗ ii = E
L(f) ; 2!
C(z ∗ ; z) = hh2 ii = −F ∗
L(f) ; 2!
L(f) ; (27) 2! that we call renormalization coecients. By means of them it is useful to introduce a di erential operator as follows. Letting C ∗ (z ∗ ; z) = hh∗2 ii = −F
(v; v∗ ) = D(v∗ ; v)@z∗ @z + 12 [C(v∗ ; v)@2z + C ∗ (v∗ ; v)@2z∗ ] :
(28)
we de ne the action of any function F() of the operator on a function a(z ∗ ; z) as F()a(z ∗ ; z) = F((v∗ ; v))a(z ∗ ; z) |(v∗ ;v)=(z∗ ; z) :
(29)
We can then give a useful representation of the double-bracket average hh·ii. Indeed, the Gaussian smearing of a generic function O(z ∗ ; z) can be represented as hhO(z ∗ + ∗ ; z + )ii = e O(z ∗ ; z) :
(30)
For practical calculations, the use of the right-hand side of (30) is particularly convenient when O(z ∗ ; z) is a low-degree polynomial. The function w(z ∗ ; z) is determined by Eq. (23), whose last term can be expressed as ∗ ∗ 1 2 !(z ; z)L(f(z ; z))
= (e )H(z ∗ ; z) ;
(31)
so that the e ective Hamiltonian can be written in a form with a more evident renormalization contribution 1 sinh f(z ∗ ; z) : (32) He (z ∗ ; z) = [(1 − ) e ]H(z ∗ ; z) + ln ÿ f(z ∗ ; z) It appears therefore that, besides the logarithmic term, He (z ∗ ; z) is given by the Weyl Hamiltonian corrected by terms of the second-order in the renormalization coecients. Moreover, as we shall show in the next section, the form (32) is an ideal starting point for a further approximation in order to deal with many degrees of freedom.
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4. PQSCHA: many degrees of freedom In this section we are concerned with the generalization of the method to a system with N degrees of freedom. The self-consistent solution of the counterpart of Eqs. (24) is obviously rather dicult: further simpli cation and (27) for arbitrary values of (z∗ ; z) is therefore in order. This is called the low-coupling approximation (LCA) and consists F(z∗ ; z), in expanding the parameters – which are N × N complex matrices E(z∗ ; z); ∗ ∗ – so as to make the renormalization coecients, and hence also the and F (z ; z) Gaussian averages hh·ii, independent of the con guration. Eventually, we describe this scheme of approximation and we focus, in particular, on translation invariant systems. Let us consider therefore a system of N degrees of freedom. Again the evaluation is done by replacing H with a trial Hamiltonian H0 , of (z ∗ ; z; z∗ ; z) = w(z∗ ; z) +t (z ∗ − z∗ )E(z∗ ; z)(z − z) H0 (z ∗ ; z; z∗ ; z) z∗ ; z)(z − z) + c:c:] ; + 12 [t (z − z)F( quadratic in the displacements from the average point Z 1 ÿ = du(z ∗ (u); z(u)) : (z∗ ; z) ÿ 0
(33)
(34)
As already mentioned, E; F, and F ∗ are a Hermitean and two symmetric N × N complex matrices, respectively. In expressing 0 we again introduce the pure-quantum uctuation variables (∗ ; ) = and the linear canonical transformation (z ∗ − z∗ ; z − z) X ∗ R∗k (z∗ ; z) ˜k + Sk (z∗ ; z) ˜k ; ∗ = k
=
X
∗
∗ Sk (z∗ ; z) ˜k + Rk (z∗ ; z) ˜k
;
(35)
k ∗ ˜ diagonalizes the quadratic term to new variabless (˜ ; ) X X ∗ [∗ E + 12 ( F + c:c:)] = !k (z∗ ; z) ˜k ˜k :
(36)
k
Here and in the subsequent discussion, transformed variables are labeled by Latin indices (k; l; : : :); Greek indices (; ; : : :) are used for the original ones. Diagonalization (36), that is not possible in the most general case, can be performed under suitable constraints on the matrices E; F, and F ∗ , (see e.g. [13]). However, this problem is not extremely relevant in the present context, since these matrices are not external data but parameters that must be optimized: this constraint just restricts the number of independent matrix elements. In addition, in most applications the canonical transformation (35) is mainly determined by symmetry considerations. In analogy to what we did in the previous section, there is no ambiguity in suppress The explicit calculation of 0 , given in Appendix B, leads to ing the bar over (z∗ ; z).
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the following expression for He : 1 X sinh fk (z ∗ ; z) ln He (z ∗ ; z) = w(z ∗ ; z) + ÿ fk (z ∗ ; z)
395
(37)
k
∗ ∗ with fk (z ∗ ; z)=ÿ!k (z ∗ ; z)=2, while the moments of P0 are hh˜k ˜l ii=kl L(fk (z ∗ ; z))=2 ∗ ∗ and hh˜k ˜l ii = hh˜k ˜l ii = 0. The PQSCHA conditions that determine the optimized parameters w(z ∗ ; z); E(z ∗ ; z); F(z ∗ ; z), and F ∗ (z ∗ ; z), are
hhH(z ∗ + ∗ ; z + )ii = hhH0 (z ∗ + ∗ ; z + )ii = w(z ∗ ; z) +
1X !k (z ∗ ; z) L(fk (z ∗ ; z)) ; 2
(38)
k
hh@z∗ @z H(z ∗ + ∗ ; z + )ii = hh@z∗ @z H0 (z ∗ + ∗ ; z + )ii = E (z ∗ ; z) ; hh@z @z H(z ∗ + ∗ ; z + )ii = hh@z @z H0 (z ∗ + ∗ ; z + )ii = F (z ∗ ; z) ; ∗ (z ∗ ; z) : hh@z∗ @z∗ H(z ∗ + ∗ ; z + )ii = hh@z∗ @z∗ H0 (z ∗ + ∗ ; z + )ii = F
(39)
Generalizing the procedure of the previous section for the case of one degree of freedom, we de ne the renormalization coecients 1X ∗ ∗ (Rk Rk + Sk Sk ) L(fk ) ; D (z ∗ ; z) = hh∗ ii = 2 k
C (z ∗ ; z) = hh ii = ∗ (z ∗ ; z) = hh∗ ∗ ii = C
1X 2
k
1X 2
∗ ∗ (Rk Sk + Rk Sk ) L(fk ) ;
(R∗k Sk + R∗k Sk ) L(fk ) ;
(40)
k
and, starting from X 1 ∗ C (v∗ ; v)@z @z + C D (v∗ ; v)@z∗ @z + (v∗ ; v)@z∗ @z∗ (v∗ ; v) = 2
(41)
and de ning the action of operator in analogy with Eq. (29), we nd that the Gaussian smearing by double-bracket average can be put into the form hhO(z ∗ + ∗ ; z + )ii = e O(z ∗ ; z) :
(42)
As in the preceding section, it can be easily seen that the e ective Hamiltonian reads 1 X sinh fk (z ∗ ; z) (43) ln He (z ∗ ; z) = [(1 − ) e ] H(z ∗ ; z) + ÿ fk (z ∗ ; z) k
and the general expression for calculating thermal averages takes the form: Z ∗ 1 ˆ d(z ∗ ; z)hhO(z ∗ + ∗ ; z + )iie−ÿHe (z ; z) : hOi = Z
(44)
We now describe the low-coupling approximation, whose main purpose is to make the averages hh·ii independent of the con guration [5]. The self-consistent equations therefore need to be solved only once, with a great simpli cation for implementing the
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method. One can think of several ways to obtain this goal: their choice substantially depends on the physics of the problem [6]. The simplest way, however, consists in expanding the matrices E(z ∗ ; z) and F(z ∗ ; z) around a self-consistent minimum (z0∗ ; z0 ) of He : E(z ∗ ; z) = E + E(z ∗ ; z) ; F(z ∗ ; z) = F + F(z ∗ ; z) ;
(45)
where E = E(z0∗ ; z0 ), F = F(z0∗ ; z0 ), thus using the convention to drop the arguments of functions evaluated at (z0∗ ; z0 ). As a consequence the frequencies will be split as, !k (z ∗ ; z) = !k + !k (z ∗ ; z) and for the evaluation of the averages we shall only take D and C as renormalization coecients. It is explicitly shown in Appendix C that the e ective Hamiltonian becomes 1 X sinh fk ln ; (46) He (z ∗ ; z) = e0 H(z ∗ ; z) − 0 e0 H(z0∗ ; z0 ) + ÿ fk k
where fk = ÿ!k =2 and X 1 ∗ @z∗ @z∗ ) : D @z∗ @z + (C @z @z + C 0 = 2
(47)
The general formula for thermal averages (44) still holds provided that the Gaussian smearing of O(z ∗ ; z) is calculated with the LCA renormalization coecients, i.e. as exp(0 )O(z ∗ ; z). It often occurs that the indices ; ; : : : refer to the sites of a lattice, whose symmetries can sometimes be very helpful in order to simplify the analysis, provided that the minimum con guration of He (z ∗ ; z) shares the same property. The calculations are very easy in the case of translation symmetry. In particular, for a one-dimensional lattice E = E− ;
F = F− ;
(48)
where the hermiticity of E and the symmetry of F imply that the components E and F satisfy ∗ ; E = E−
F = F− :
(49)
Performing a Fourier transformation, X X k eik ; ∗ = N −1=2 ∗k e−ik ; = N −1=2 k
the left-hand-side of (36) becomes X X ∗ 1 !k ˜k ˜k ; Ek ∗k k + (Fk k −k + c:c:) = 2 k
where Ek =
(50)
k
(51)
k
X
E e−ik ;
Fk =
X
F e−ik :
(52)
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As a consequence of (49) Ek is real and Fk = F−k . If the lattice has more than one dimension the generalization is obvious and, in particular, the summation over wave vectors is performed over the rst Brillouin zone. If there are internal degrees of freedom for each lattice site, the Fourier transform must be followed by a diagonalization in the internal space. Finally, the diagonalization of the quadratic form (51) is completed by a Bogoliubov transformation ∗ ∗k = Ak ˜k − Bk ˜−k ; ∗
∗ ˜ k + A∗−k ˜−k ; −k = −B−k
(53)
where the canonical conditions imply A∗k Ak − Bk∗ Bk = 1;
Ak = A−k ;
Bk = B−k ;
(54)
and it turns out that !k (A∗k Ak + Bk∗ Bk ) = Ek ;
!k Ak Bk = Fk =2 :
(55)
Finally, !k2 = Ek2 − Fk∗ Fk :
(56)
By combining transformations (50) and (53) we obtain the LCA approximation of the matrices R and S de ned in (35): Rk = N −1=2 A∗k eik ;
∗ Sk = −N −1=2 Bk∗ e−ik
(57)
Using then (40) we also can specify the LCA renormalization coecients: 1 X L(fk ) Ek cos k( − ) ; D = N 2!k k
C = −
1 X ∗ L(fk ) Fk cos k( − ) ; N 2!k k
∗ =− C
1 X L(fk ) Fk cos k( − ) : N 2!k
(58)
k
In many cases only a reduced set of these coecients will be explicitly needed in He . For instance, any model with on-site nonlinearity entails only on-site coecients D = D , C = C which are independent of due to translation invariance. 5. An application As an example of a system that is more conveniently treated by means of holomorphic variables, we shall consider a model Hamiltonian with quartic on-site interaction, X X † ˆ =− aˆi aˆi+d + (V1 nˆi + V2 nˆ2i ) ; (59) H id
i
where i runs over sites of a Bravais lattice, d denotes the displacements towards the Z nearest neighbour, nˆi = aˆ†i aˆi , and V1 and V2 ¿ 0 are constants. One can recognize
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in (59) the Hamiltonian of the well-known Bose–Hubbard model [14,15]. Its Weyl symbol is easily found to be X X N zi∗ zi+d + [ (V1 − V2 )zi∗ zi + V2 (zi∗ zi )2 ] : (60) H(z ∗ ; z) = − V1 − 2 id
i
H is invariant under global phase transformations, and provided that the LCA e ec∗ ; z0i )= tive Hamiltonian shares the same property, the translation invariant minimum (z0i ∗ (z0 ; z0 ) of He is degenerate; therefore we can choose as representative the real minimum with z0 = z0∗ . It is easy to verify that Eq. (39) for the matrices E and F becomes X i; j+d ; Eij = [(V1 − V2 ) + 4V2 (z02 + D)]ij − d
Fij = 2V2 (z02 + C)ij ;
(61)
and Fij∗ = Fij . They yield the following Fourier transforms: X cos(k · d) ; Ek = (V1 − V2 ) + 4V2 (z02 + D) − d
Fk = F = 2V2 (z02 + C) ;
(62)
The renormalization coecients read now 1 X L(fk ) Ek ; D= N 2!k k F X L(fk ) C =− N 2!k
(63)
k
with !k2 = Ek2 − F 2 . Using (46), the e ective Hamiltonian can be eventually written as X [4Dzi∗ zi + C(zi∗2 + zi2 )] ; He (z ∗ ; z) = G(ÿ) + H(z ∗ ; z) + V2
(64)
i
where G(ÿ) = −V2 N [2z02 (2D + C) + 2D2 + C 2 ] +
1 X sinh fk ln ÿ fk
(65)
k
is a uniform contribution that a ects the partition function but cancels out in the expression of thermal averages. The (real) translation-invariant stationary points of He are given by z0;1 = 0 ; 2 z0;2 =
V2 − V1 + Z − 2D − C : 2V2
(66) (66)
2 ¡ 0 the minimum is z0;1 and in the above formulas One can check that when z0;2 2 ¿ 0 the minimum is z0;2 . Since the one should replace z0 = z0;1 ; conversely, for z0;2 calculation of the renormalization coecients depends on the chosen minimum, both cases can simultaneously occur: if this is the case, the relevant minimum to be used at a given temperature is the one which gives the smaller free energy.
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6. Summary We have presented a construction of the e ective Hamiltonian suitable for the study of the thermodynamics of eld-theory models, with Hamiltonian expressed in terms of creation and destruction operators. The construction is done by a path-integral approach, in terms of the holomorphic variables z ∗ and z. The main result we presented consists of Eq. (26) for one degree of freedom, and of Eq. (44) for many degrees of freedom: in the latter case the LCA e ective Hamiltonian (46) allows one to perform practical calculations. As in the previous formulation [4] with respect to phase-space conjugate variables, p and q, the framework retains all classical nonlinear e ects embodied in the classical-like formulas for the thermal averages and the partition function. The quantum e ects related to the quadratic part of the Hamiltonian are completely accounted for, while the pure-quantum nonlinearity is treated at one-loop (Hartree–Fock) level. The low-coupling approximation permits actually to deal with many degrees of freedom, so that our results can be used for a direct application to nite-temperature eld theory. We nally observe that this approach can be extended to deal with the thermodynamics of Fermi systems [16].
Appendix A: Path integral for in terms of complex variables We give here a short derivation of the expression for (z ∗ ; z) as a path integral. Although the subject is rather well known, we have chosen to include it since we want to stress those aspects connected with Weyl symbols. We start from the Trotter’s formula for a discrete imaginary time u → i (i = 1; : : : ; M , ≡ M=ÿ) and we take then the continuum limit: ˆ
ˆ M: ˆ = e−ÿH = lim (1 − H) M →∞
(A.1)
In order to introduce the Weyl formulation, it is necessary to write the expression of the Weyl symbol of the product of M operators. Let us rst derive the product rule for two operators Oˆ and Oˆ 1 , i.e. the Weyl symbol O ∗ O1 (z ∗ ; z) for Oˆ Oˆ 1 . Using Eq. (1) and the Baker-Campbell-Hausdor identity, we have Z Z t † t ∗ ∗ Oˆ Oˆ 1 = d(k0∗ ; k0 ) d(k1∗ ; k1 )ei[ aˆ (k0 +k1 )+ (k0 +k1 )a]ˆ t ∗
t ∗
˜ 0∗ ; k0 )O˜ 1 (k1∗ ; k1 ) : ×e(1=2)( k1 k0 − k0 k1 ) O(k
(A.2)
Using now the expression of the inverse Fourier transform and the representation of the -function Z t ∗ t ∗ (A.3) (k∗ ) (k) = d(z ∗ ; z)ei( z k+ k z) ;
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we can determine the product rule for the Weyl symbols that is traditionally called star product: Z t ∗ t ∗ ∗ t ∗ t ∗ ∗ O ∗ O1 (z ; z) = d(h1∗ ; h1 ) d(k1∗ ; k1 )ei[ z (h1 +k1 )+ (h1 +k1 )z] e(1=2)( k1 h1 − h1 k1 ) Z ×
t ∗
t ∗
d(C∗1 ; C1 ) d(z1∗ ; z1 )e−i( C1 h1 + h1 C1 ) t ∗
t ∗
t
∗
×e−i( z1 k1 + k1 z1 ) O(C∗1 ; C1 )O1 (z1∗ ; z1 ) Z = 22N d(C∗1 ; C1 ) d(z1∗ ; z1 )O(C∗1 ; C1 )O1 (z1∗ ; z1 ) ∗
×e−2[ (C1 −z
∗ )(C1 −z1 )−t (C∗ 1 −z1 )(C1 −z)]
:
(A.4)
The integration of this expression, using (A.3), immediately yields Eq. (7). By iterating ˆ Oˆ 1 ; Oˆ 2 : Eq. (A.4), we obtain the star product of three operators O; Z ∗ 4N d(C∗1 ; C1 ) d(z1∗ ; z1 ) d(C∗2 ; C2 ) d(z2∗ ; z2 ) O ∗ O1 ∗ O2 (z ; z) = 2 ×O(C∗1 ; C1 )O1 (z1∗ ; z1 )O2 (z2∗ ; z2 ) t
∗
∗
t
∗
∗
t
∗
∗
×e−2[ (C1 −C2 )(C1 −z1 )− (C1 −z1 )(C1 −C2 )] ×e−2[ (C2 −z
∗ )(C2 −z2 )−t (C∗ 2 −z2 )(C2 −z)]
:
(A.5)
It is now easy to see that in general, if we take Oˆ = 1ˆ and therefore O = 1, we can write # Z "Y M M ∗ 2MN ∗ ∗ ∗ ∗ Oi (z ; z) = 2 d(Ci ; Ci ) d(zi ; zi )Oi (zi ; zi ) i=1
i=1
( ×exp −2
M X
) [t (C∗i − C∗i+1 )(Ci − zi ) −t (C∗i − zi∗ )(Ci − Ci+1 )]
;
i=1
(A.6) where CM +1 ≡ z. Letting Ci = zi + i (with M +1 = z) and zi = zi − zi−1 , the previous expression becomes M
∗
2MN
∗ Oi (z ; z) = 2
Z "Y M
i=1
# d(∗i ; i ) d(zi∗ ; zi )Oi (zi∗ ; zi )
i=1
( ×exp 2
M X i=1
) t
[
(∗i+1
+
∗ zi+1 )i
−t ∗i (i+1
+ zi+1 )]
:
(A.7)
A. Cuccoli et al. / Physica A 271 (1999) 387–404
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The integrations over the variables i are performed taking into account a di erent representation of the -function, namely Z t ∗ t ∗ d(z ∗ ; z)e±( z k− k z) ; (A.8) (k)(k∗ ) = Since no generality is lost in assuming M to be an even number, we eventually obtain # Z "Y M M ∗ MN ∗ ∗ ∗ Oi (z ; z) = 2 d(zi ; zi )Oi (zi ; zi ) i=1
i=1
M=2 k X X ∗ ∗ (tz2k+1 z2j −t z2j z2k+1 ) ×exp 2 − k=1 j=1
+
M=2 X j=1
∗ (tz2j z −t z ∗ z2j ) :
(A.9)
Finally, according to Eq. (A.1), we set Oi = 1 − jH ' e−jH , and in the continuum limit the integrand becomes the exponential of Z ÿ=2 Z u Z ÿ ∗ du H(z (u); z(u)) − 2 du du0 [t z˙∗ (2u)z(2u ˙ 0 ) − tz˙∗ (2u0 )z(2u)] ˙ − 0
0
Z +2
ÿ=2 0
0
du0 [t z ∗ (ÿ)z(2u ˙ 0 ) −t z˙∗ (2u0 )z(ÿ)] + 2
Z
ÿ=2
0
du0 [t z˙∗ (2u0 )z −t z ∗ z(2u ˙ 0 )] : (A.10)
The integration over u0 is trivial and, by the change of variable u → ÿ − u, we obtain the path integral given by Eqs. (10) and (11), with the measure D[z ∗ (u); z(u)] formally de ned by ∗
MN
D[z (u); z(u)] = lim 2 M →∞
M Y
d(zi∗ ; zi ) :
(A.11)
i=1
Appendix B: The PQSCHA method Let us evaluate the reduced density 0 (13) with the trial Hamiltonian H0 , Eq. (17). Representing the -function as in Eq. (A.8) one gets Z Z ∗ ∗ d(v∗ ; v)ez v−v z D[z ∗ (u); z(u)] 0 (z ∗ ; z; z∗ ; z) = ×e
S0 [z ∗ (u);z(u)]−(1=ÿ)
Rÿ 0
du[z ∗ (u)v−v∗ z(u)]
;
(B.1)
where S0 is given by Eq. (11) with H0 . We change the integration variables in the path integral to (∗ (u); (u)) = (z ∗ (u) − z∗ ; z(u) − z) and we make the Bogoliubov
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A. Cuccoli et al. / Physica A 271 (1999) 387–404
transformation ∗ ˜ ∗ (u) = R∗ ˜ (u) + S (u); ∗
∗ ˜ (u) = S ∗ ˜ (u) + R(u) ;
(B.2)
∗
with R R − S S = 1. This permits to diagonalize the quadratic term ∗ ˜ : E∗ (u)(u) + 1 [F2 (u) + c:c:] = !˜ (u)(u)
(B.3)
2
It is straightforward to verify that this transformation preserves at the same time the functional measure and the form of the exponent in the path integral. Performing the same transformation on (z ∗ ; z), (z ∗ ; z) and (v∗ ; v) we obtain Z Z ∗ ˜ d(v˜∗ ; v) ˜ D[˜ (u); (u)] 0 (z ∗ ; z; z∗ ; z) = e−ÿw (Z ×exp
ÿ
0
1 ∗ 1 ∗ ˜ − ˜∗ (u)(u)] ˜˙ ˜ du − [˜ (u)v˜ − v˜∗ (u)] + [˜˙ (u)(u) ÿ 2
i h ∗ ∗ 1 ∗ ˜ ˜ ˜ + − [˜ (0)(ÿ) − ˜ (ÿ)(0)] − !˜ (u)(u) 2 ∗
∗
∗
)
˜ ˜ ˜ − (z˜ − z˜ )[(ÿ) − (0)]] − [[˜ (ÿ) − ˜ (0)](z˜ − z) ∗
: (B.4)
The shift
v˜ v˜∗ ; (u) − (B.5) ( (u); (u)) → (u) + ÿ! ÿ! eliminates the linear term, and the resulting expression of 0 contains the path integral of a single harmonic oscillator: Z Z ∗ ∗ ˜ ˜ ˜ −v˜ v=ÿ! d(v˜∗ ; v)e D[˜ (u); (u)] 0 (z ∗ ; z; z∗ ; z) = e−ÿw ∗
∗
(Z
ÿ
∗ 1 ˜˙∗ ˜ − ˜∗ (u) (u)] ˜˙ ˜ [ (u) (u) − !˜ (u)(u) 2 0 ∗ ∗ v˜ − [˜ (ÿ) − ˜ (0)] z˜ − z˜ + ÿ! ∗ v˜ ∗ ˜ ˜ [(ÿ) − (0)] − z˜∗ − z˜ − ÿ! ) ∗ 1 ˜∗ ˜ ˜ ˜ : − [ (0)(ÿ) − (ÿ) (0)] 2
×exp
du
(B.6)
Therefore, using Eq. (9) we obtain Z ∗ ˜ ˜ −v˜ v=ÿ! d(v˜∗ ; v)e 0 (z ∗ ; z; z∗ ; z) = e−ÿw
1 cosh f v˜ v˜∗ ∗ z˜ − z˜ + tanh f ; ×exp −2 z˜∗ − z˜ − ÿ! ÿ!
(B.7)
A. Cuccoli et al. / Physica A 271 (1999) 387–404
403
where f = ÿ!=2. Finally, performing the integration over (v˜∗ ; v), ˜ we arrive at Eq. (19). Eq. (32) can be derived from Eq. (31); the latter, in turn, can be easily veri ed in this way: e H(z ∗ ; z) = hhH(z ∗ + ∗ ; z + )ii = D(z ∗ ; z)hh@z∗ @z H(z ∗ + ∗ ; z + )ii + 12 [C(z ∗ ; z)hh@2z H(z ∗ + ∗ ; z + )ii + c:c:] :
(B.8)
Using Eqs. (24) and (27) the right-hand side becomes D(z ∗ ; z)E(z ∗ ; z) + 12 [C(z ∗ ; z)F(z ∗ ; z) + c:c:] L(f(z ∗ ; z)) ; =[E 2 (z ∗ ; z) − F ∗ (z ∗ ; z)F(z ∗ ; z)] 2!(z ∗ ; z)
(B.9)
and nally, Eq. (31) follows from Eq. (22). In order to deal with the case of many degrees of freedom we use representation (A.8) of the -function for implementing constraint (34), so that the expression for becomes 0 (z ∗ ; z; z∗ ; z) Z Z t ∗ t ∗ 0 (z ∗ ; z; z∗ ; z) = d(C∗ ; C)e z C− C z D[z ∗ (u); z(u)] (
1 ×exp S0 [z (u); z(u)] − ÿ ∗
Z 0
ÿ
) t ∗
t ∗
du[ z (u)C − C z(u)]
; (B.10)
where S0 is the action (11) with the trial Hamiltonian (33). This expression is evaluated by making the linear canonical transformation (35) which permits to decouple the path integral into a product of one dimensional harmonic path integrals. The same transformation, done over all the pairs of conjugate variables that appear in (B.10), Q yields the form 0 = exp(−ÿw) k 0 k for the reduced density. Taking into account (36) we can use the result for one degree of freedom Eq. (19) and we obtain Y fk 2 ˜∗ ˜ 2 ∗ ∗ −ÿw exp − ; (B.11) =e 0 (z ; z; z ; z) sinh fk L(fk ) L(fk ) k k k
where fk = ÿ!k =2.
Appendix C: Low-coupling approximation We give here the steps which lead to the LCA expression (46) for He , when the LCA is made by expanding the renormalization coecients around the self-consistent minimum (z0∗ ; z0 ) of He . We start by expanding Eq. (43) to the lowest order in the di erence !k (z ∗ ; z) = !k (z ∗ ; z) − !k , recalling the convention to drop the arguments of functions evaluated
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A. Cuccoli et al. / Physica A 271 (1999) 387–404
at (z0∗ ; z0 ). The expansion of the logarithmic term is 1 X sinh fk (z ∗ ; z) 1 X sinh fk X L(fk ) = !k (z ∗ ; z) ln ln + ÿ fk (z ∗ ; z) ÿ fk 2 k
k
=
k
1 X sinh fk ln + [e 0 H(z ∗ ; z) − e0 0 H(z0 ; z0∗ )] ; ÿ fk k
(C.1) where we have taken into account the fact that Eq. (31) generalizes to X L(fk (z ∗ ; z)) !k (z ∗ ; z) = [e ]H(z ∗ ; z) : 2
(C.2)
k
Letting = 0 + , we have the expansion [(1 − )e ]H(z ∗ ; z) = [(1 − )e ]e0 H(z ∗ ; z) − e 0 H(z ∗ ; z) = e0 H(z ∗ ; z) − e 0 H(z ∗ ; z) + O(()2 ) ;
(C.3)
so that, eventually, Eq. (46) is obtained. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
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