UDC 536.75

PHYSICS

L. Ya. KOBELEV

ON THE QUESTION OF POINTS OF NONANALYTICITY OF THE GREEN FUNCTIONS OF A SYSTEM OF INTERACTING PARTICLES

(Presented by Academician N. N. Bogolyubov, 10 VI 1967)

1. Phase transitions (including those of the second kind) in nonrelativistic systems of interacting particles (see, for example, \((^{1-5})\)) testify to a nonanalytic dependence of the specific heat and the thermodynamic potential of the indicated systems on the temperature difference \(\Delta T = T_0 - T\) (\(T_0\) is the temperature of the phase transition). The nonanalytic dependence on \(\Delta T\) may be a consequence of the nonanalyticity of the exact one-particle (two-particle, etc.) thermodynamic Green functions of the system, defined by the corresponding equations with variational derivatives (Schwinger equations). It is shown below that such a dependence does indeed exist and can be extracted from the exact Green functions of the system, written with the aid of functional integrals over “external fields” \((^6)\) and “classical” trajectories \((^7)\), using a variant of perturbation theory consisting in expanding the functional integrals (without restrictions on the interaction parameter or the density of the number of particles) in a series in their deviation from Gaussian integrals over “external fields.” It is also shown that near the points of nonanalyticity, i.e., as \(\Delta T \to 0\), the “non-Gaussian” corrections to the Green function are small.

2. Consider a system of \(N\) interacting nonrelativistic particles (Bose or Fermi particles), described by the creation and annihilation operators from the ground state of particles and field quanta of the interaction \(\Psi(x)\), \(\Psi^*(x)\), and \(A(x)\), respectively, and by the Hamiltonian \(H = \Delta/2m + A(x)\) \((\hbar = c = 1)\). We define the interaction energy of two particles of the system by the relation (\(g\) is the interaction constant)

\[ D_0(x)=g^2\sum_n \int dp\, \frac{e^{ipx}}{L(p,w_n)}, \]

where \(L(p,w_n)\) characterizes the type of interaction and satisfies the condition \(L(0)<\infty\). Following \((^{6,8,9})\), it is not difficult to show that the exact solution of the thermodynamic Schwinger equation \((t=-i\beta;\ \beta=1/kT;\ x=\mathbf{x},t)\)

\[ \left[ -i\frac{\partial}{\partial t} -\frac{\Delta}{2m} -\langle A(x)\rangle +ig\frac{\delta}{\delta I_0(x)} \right]G(x,x')=\delta(x-x'), \]

\[ L(\Delta,\partial/\partial t)\langle A(x)\rangle = I_0(x)+igG(x,x) \tag{1} \]

for the thermodynamic one-particle causal Green function \(G(x,x')=\langle T(\Psi(x)\Psi^*(x'))\rangle\) (\(\langle \ldots\rangle\) denotes averaging over the statistical ensemble), which describes the behavior of the system under consideration, has the form

\[ G(x,x) = \frac{1}{\langle S\rangle} \int dA(x)\, \exp\left\{ -\frac{i}{2}\int A(x)D_0^{-1}(x-y)A(y)\,dx\,dy - \int_0^1 d\lambda \int G(x,x\mid \lambda A)A(x)\,dx \right\} G(x,x'\mid A); \tag{2} \]

\[ \langle S\rangle = \int dA(x)\, \exp\left\{ -\frac{i}{2}\int A(x)D_0^{-1}(x-y)A(y)\,dx\,dy - \int_0^1 d\lambda \int G(x,x\mid \lambda A)A(x)\,dx \right\}; \tag{3} \]

\[ [-i\partial/\partial t-\Delta/2m-\lambda A(x)]G(x,x'|\lambda A)=\delta(x-x'); \tag{4} \]

\[ [-i\partial/\partial t-(\Delta-\Delta')/2m-\lambda(A(x)-A^*(x'))]G(x,x')=0,\quad x'\to x. \tag{5} \]

In (2)—(3) the continual integral over \(A(x)\) of the functional \(\varphi(x|A)\) (the functional dependence is separated by a vertical bar) is defined as the limit of an \(n\)-fold integral of a function of \(n\) variables \(\varphi(x; A_1,\ldots,A_n)\) as \(n\to\infty\), \(A_{i+1}-A_i\to0\), \(\Delta x_i\to0\).

Integration over \(x_4\) and variational derivatives with respect to \(I_0(x)\), \(\langle A(x)\rangle\), etc., differ from zero only for \(0\le ix_4\le 1/T\); the function \(\delta(x_4)\) is defined at discrete “time” points \({}^{(10)}\). For retarded, advanced, commutator, etc., thermodynamic Green functions, the representation (2)—(4) remains valid when equation (4) for \(G(x,x'|A)\) is replaced by the equation for the retarded, advanced, etc., function \(G(x,x'|A)\). The “external source” \(I_0(x)\) in (2)—(3) is equal to zero.

1. Let us now represent \(G(x,x'|A)\) and \(G(x,x|A)\) through continual integrals over all continuous trajectories \(\mathbf{x}(\nu)\), beginning at “time” \(\nu=t'\) at the point \(\mathbf{x}'\) and ending at “time” \(t\) at the point \(\mathbf{x}\) \({}^{(11,12)}\):

\[ \begin{aligned} G(x,x'|A) &=\int_{G_{x,x'}} d_G\mathbf{x}(\nu)\exp\left\{i\int_{t'}^{t} A(x(\nu))\,d\nu\right\}\equiv\\ &\equiv \lim_{\substack{\Delta\nu=\nu_{i+1}-\nu_i\to0\\ \Delta A=A(x_{i+1})-A(x_i)\to0\\ \Delta x=x_{i+1}-x_i\to0\\ n\to\infty}} \int dx_1\ldots\int dx_n\,G_0(\mathbf{x}-\mathbf{x}_1;t-\nu_1)\ldots G_0(\mathbf{x}_n-\mathbf{x}';\nu_n-t')\times\\ &\quad\times \exp\left\{i\sum_{i=1}^{n}A(\mathbf{x}_i)(\nu_{i+1}-\nu_i)\right\},\qquad it'<it,\quad t=-i\beta; \end{aligned} \tag{6} \]

\[ (-i\partial/\partial t-\Delta/2m)G_0(x,x')=\delta(x-x'); \tag{7} \]

\[ G(x,x)=\lim_{x'\to x}\int_{G_{x,x'}} d_G\mathbf{x}(\nu) \exp\left\{i\int_{x_4^0}^{x_4}[dA_1(x(\nu))+2iA_2(x(\nu))]\,d\nu\right\}. \tag{8} \]

Here \(x_4^0\) is an arbitrary parameter, which in what follows, for convenience, is taken equal to \(x_4\); \(dA_1(x)=A_1(x+\Delta x)-A_1(x)+d_{A_1}A_1(x)\) (\(d_{A_1}A_1(x)\) is the variation of \(A_1(x)\) at the points \(x\)); \(A=\operatorname{Re}A+i\operatorname{Im}A=A_1+iA_2\). In the last integrals in (6)—(8) one should replace the solutions (7) \(G_0(p)\) by \(G_0(p)n(p)\). Taking (6) into account, \(G(x,x')\) takes the form (\(G(x,x')=0\) for \(ix_4'>ix_4\))

\[ \begin{aligned} G(x,x') &=\frac{1}{\langle S\rangle}\int dA(x)\int_{G_{x,x'}} d_G\mathbf{x}(\nu)\, \exp\Bigg\{-\frac{i}{2}\int A(x)D_0^{-1}(x-y)A(y)\,dx\,dy\\ &\quad-\int_0^1 d\lambda\int G(x,x|\lambda A)A(x)\,dx +i\int_{x_4^0}^{x_4} A(x(\nu))\,d\nu\Bigg\},\qquad ix_4'<ix_4. \end{aligned} \tag{9} \]

1. Let us isolate in (9) the “Gaussian” part in \(A(x)\) by means of a nonlinear transformation of \(A(x)\) of the form

\[ A(x)=A'(x)+\int D_0(x-y)\widetilde F(y)\,dy, \]

where

\[ \widetilde F(y)=F(y)+\varepsilon(x(\nu))= i\int_0^1 d\lambda\,G(yy|\lambda A)+\int_{x_4^0}^{x_4} d\nu\,\delta(x(\nu)-y)\delta(\nu-y^0). \]

This transformation eliminates from the exponent (9) the second and third terms and brings (9) to the form

\[ \begin{aligned} G(x,x') &=\frac{1}{\langle S\rangle}\int dA'(x)\int_{G_{x,x'}} d_G\mathbf{x}(\nu)\,D(|A) \exp\Bigg\{-\frac{i}{2}\int A'(x)D_0^{-1}(x-y)\times\\ &\quad\times A'(y)\,dx\,dy+\frac{i}{2}\int \widetilde F(x)D_0(x-y)\widetilde F(y)\,dx\,dy\Bigg\}, \end{aligned} \tag{10} \]

where \(D(|A)\) is the Jacobian of the transformation, coinciding with the Fredholm determinant of the integral equation (see (13)), \(A(x)=A'(x)+\int K(x,y)A(y)\,dy\), for \(K(x,y)=\int D_0(x-y_1)\dfrac{\delta F(y_1)}{\delta A(y)}\,dy_1\).

We shall show that \(D(|A)\) contains a nonanalytic factor due to the dependence of \(G(x,x|A)\) on the real part \(A\). Considering the transformation only for \(A_1\) and using the relations \(G(x,y)-G(y,x)=\delta_{x,y}\), \(\delta F(x)/\delta A_1(y)=\delta_{y,x}\cdot\widetilde{\Pi}_{\lambda}(0)=\delta_{y,x}\cdot\Pi_{\lambda}(0)(1-\Pi_{\lambda}(0)D_0(0))^{-1}\), where

\[ \Pi_{\lambda}(0)=-\frac{T}{(2\pi)^2}\sum_{w_n}\int dp\int_0^1 \lambda\,d\lambda\,G^2(p,w_n|\lambda A),\quad D_0(0)=g^2L^{-1}(0) \tag{11} \]

and \(G(p,w_n|A)\) satisfies (5), and restricting ourselves to the study of the case \(F(x)\) independent of \(x\) (the analogue of a Bose-particle condensate), we find, separating from \(\Pi_{\lambda}(0|A)\) the term with \(d_{A_1}A_1(x)\) in the linear approximation \((\widetilde D_0(0)=D_0(0)(1-\Pi_{\lambda}(0)D_0(0))^{-1}\):

\[ D(|A)=D(|A_2)(1-\Pi_{\lambda}(0)\widetilde D_0(0))^{-1} \exp\left\{\frac{\Pi_{\lambda}(0)D_0(0)}{1-2\Pi_{\lambda}(0)D_0(0)}\,d_AA_1\,dv\,\eta\right\}, \]

\[ \eta=\pm 1. \tag{12} \]

The appearance of the nonanalytic term is due to the retention in \(G(x,x|A)\) of first-order terms with respect to the variation of \(A_1\). The ambiguity of the integral over \(A_1\) is removed by the choice of the sign of \(d_AA_1\) (determined by \(\eta\)) from the normalization condition of \(G(x,x')\).

1. To eliminate the remaining linear dependence on \(dA_1(x)\) (and \(A_2(x)\)), we separate in \(D(|A)\) and in one of the functionals \(F(x)\) in the expression \(\int \overline F(x)D_0F(y)\,dx\,dy\) the terms linear in \(A_1\) and \(A_2\), and eliminate them by means of the transformation

\[ A'(x)=A''(x)+\int D_1(x-y)F_1(y)\,dy, \tag{*} \]

where \(F_1(x)\) is a functional of \(A_1\) (and \(A_2\)), chosen from the condition of compensation of the linear terms (\(F_1\) has the form \(F_1=\int FD_0\Pi_{\lambda}\,dx\,dy-\int_{x_4^0}^{x_4'}D_0(x(y)-y)\Pi_{\lambda}(y-x)\,dy\,dx\,dy\)). In this case, in (10) there will appear the Jacobian \(D_1'(|A'')\) and the factor \(\exp\left\{\dfrac{i}{2}\int F_1(x)D_1(x-y)F_1(y)\,dx\,dy\right\}\). Omitting the intermediate calculations, we give the expression for \(G(x,x')\) after application of an infinite number of transformations of the form (*) and complete elimination from the subintegral expression (10) of the terms linear in \(A_1(x)\) (it is not difficult to show that elimination of the terms linear in \(A_2(x)\) does not lead to the appearance of new nonanalyticities):

\[ G(x,x')=\langle S\rangle^{-1}\int dA^{\infty}(x)\int_{G_{x,x'}}d_G\mathbf{x}(y)\,D(|A)\prod_{i\geqslant1}^{\infty}D_i(|A^{\infty})\times \]

\[ \times\exp\left\{-\frac{i}{2}\int A^{\infty}D_{\infty}^{-1}(x-y)A^{\infty}\,dx\,dy +\frac{i}{2}\int\left[F(x)D_0(x-y)F(y)+ \sum_{i=1}^{\infty}F_i(x)D_i(x-y)F_i(y)\right]\,dx\,dy\right\}, \tag{13} \]

where \(A^{\infty}=A^i|_{i=\infty}\); \(D_i(x)\) is the function \(D_0(x)\) renormalized by the inclusion of terms \(\sim AA\) at each of the transformations; \(F_i\) is a functional of \(A^i\), eliminating the linear terms in the \((i+1)\)-st transformation; \(D_i(|A^i)\) are the Jacobians of the \((i+1)\)-st transformation. For \(i=\infty\) the quantities \(F\)

and \(F_i\), and also the variational derivatives of \(F\) and \(F_i\) with respect to \(A^i\), do not contain terms linear and quadratic in \(A(x)\). The parts \(D_i(|A)\) that depend on \(A_1^i\) have the symbolic form

\[ D_i(|A_1^i)=\exp\left\{\ln\left[1+D_i(0)\frac{\delta F_i}{\delta A^i}\left(1+d_{A_1}A_1dv\,\eta\right)\right]\right\}, \tag{14} \]

\[ D_i(0)\frac{\delta F_i}{\delta A^i} =\frac{a_i}{1-b_i} = D_i(0)\frac{\delta A F_i}{\delta A^i} \left[ 1-\int_{x_4^0}^{x_4} D_i\bigl(x(\nu)-y'\bigr) \frac{\delta F_i(y')}{\delta F_i(y)} \,dy'\,dy\,d\nu \right]^{-1}, \]

where \(\delta F_i/\delta A^i\) and \(\delta A F_i/\delta A^i\) are the total and “partial” (with respect to the explicit dependence on \(A\)) variational derivatives. In computing \(F_i\) and \(D_i\) one must take into account that \(\delta_F F_i/\delta F_i\) no longer depends on \(A_1\), and the dependence on \(A_1\) remains only in \(\int F_iD_iF_i\,dx\,dy\) and in \(\delta A F_i/\delta A^i\). Each of the Jacobians \(D_i(|A_1^i)\) has a point of nonanalyticity determined by the relation \(1-b_iT=0\).

1. The perturbation-theory method used below consists in separating out in (13) (when representing \(\langle S\rangle\) by means of analytic transformations \(A'\), \(A''\), (*) etc.) the Gaussian quadratic form in \(A_1(x)\) (i.e., in separating out in the exponent the terms proportional to \(A_1^\infty\)) and expanding the remaining terms in powers of \(A_1^\infty\). Taking account of the normalization

\[ \lim_{j\to\infty,\ dA_1\to0,\ \rho x\to0}\sum d_{A_1}A_1(x_j)\,d\nu=1, \]

we then obtain \(\bigl(G(x,x')=G_{\Gamma}(x,x')+G_{\mathrm{nG}}(x,x')\bigr)\)

\[ G_{\Gamma}(x,x') = \prod_{i\geq0}^{\infty} \left(\frac{D_{\infty}}{D_{\infty}^{s}}\right)^{1/2} \left(1+\frac{a_i}{1-b_i}\right) \left(1+\frac{a_i^{s}}{1-b_i^{s}}\right)^{-1} \times \]

\[ \times \exp\left\{ \frac{a_i}{1-b_i} - \frac{a_i^{s}}{1-b_i^{s}} + \frac{i}{2}\int \bigl(F_i^{0}D_iF_i^{0}-F_i^{0s}D_i^{s}F_i^{0s}\bigr)\,dx\,dy \right\} G_{\Gamma}^{0}(x,x'). \tag{15} \]

In (15), \(a_0=\widetilde D_0(0)\Pi_\lambda(0)\); \(F_i^0\) does not depend on \(x(\nu)\); \(b_0=\widetilde D_0(0)\Pi_\lambda(0)\); the index \(s\) means that the quantity was obtained under the transformation \(\langle S\rangle\);

\[ D_i^{-1}=\widetilde D_i^{-1}(0) + \sum_{n=0}^{i} \frac{\delta^2 a_n}{\delta A^i\delta A^i} (1-b_n)^{-1} \simeq D_0^{-1}(0) + \sum_{n=0}^{i} \frac{\widetilde a_n}{1-b_n}; \]

\[ F_j\simeq \frac{i}{2}\int F_{j-1}D_j \frac{\delta F_{j-1}}{\delta A^j} \,dx\,dy - \frac{\delta a_j}{\delta A^j}(1-b_j)^{-1}; \qquad F_0\equiv\widetilde F \quad (i,j=1,2,\ldots,\infty). \]

The function \(G_{\Gamma}^{0}\) satisfies an approximate Dyson equation. The non-Gaussian part of the Green function \(G_{\mathrm{nG}}\) near \(1-b_i\simeq\Delta T_i/T\to0\) has order \((1-b_i)^2\widetilde a_i^{-2}G_{\Gamma}\), since \(D_{\infty}^{-1}\sim \widetilde a_iT/\Delta T_i\); therefore, near the points of nonanalyticity, \(G\simeq G_{\Gamma}+O(G_{\Gamma}\Delta T_i^2/T^2)\). It can be shown that the presence of nonanalytic singularities in \(\Delta T_i\) in the thermodynamic potential \(\Omega\) of the particle system, continuous at the point \(T_i\), leads to \(\partial^2\Omega/\partial T^2\sim\ln(T/\Delta T_i)\), which makes it possible to regard the smallest of the \(T_i\) as the temperature of a second-order phase transition. In this case the dependence of \(\partial\Omega/\partial\mu\) (\(\mu\) is the chemical potential) on \(\Delta T\) contains both an analytic and a nonanalytic term.

Ural State University
named after A. M. Gorky

Received
25 IV 1967

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