UDC 517.433

MATHEMATICS

Yu. M. SUKHOV

EXISTENCE AND REGULARITY OF THE LIMITING GIBBS STATE FOR ONE-DIMENSIONAL CONTINUOUS SYSTEMS OF QUANTUM STATISTICAL MECHANICS

(Presented by Academician A. N. Kolmogorov on 13 V 1970)

1. In a recent work of H. Araki (¹) the limiting Gibbs state was constructed for one-dimensional spin systems. In the present note the limiting Gibbs state is constructed for one-dimensional continuous quantum systems. We assume that the interaction is binary, with a finite potential \(V(r)\) possessing a hard core of radius \(d>0\).

2. Let \(\Omega=(a_1,a_2)\) be a finite interval on the line \(R^1\). Introduce the Fock spaces \(\mathscr H_B(\Omega)\) of bosons and \(\mathscr H_F(\Omega)\) of fermions and the particle-number operator \(N\) (see, for example, (²) *). The Hamiltonian \(H\) is a self-adjoint extension of the symmetric operator \(\hat H\) generated by the Schrödinger equation:

\[ \hat H f(x_1,\ldots,x_n) = -\sum_{j=1}^{n}\frac{\partial^2}{\partial x_j^2} f(x_1,\ldots,x_n) + \sum_{j\ne k} V(|x_j-x_k|) f(x_1,\ldots,x_n), \]

\[ x_j\in\Omega,\quad j=1,\ldots,n,\quad \min_{j\ne k}|x_j-x_k|>d,\quad n=1,2,\ldots \]

We consider self-adjoint extensions \(H\) of the operator \(\hat H\) determined by the following types of boundary conditions on the boundary \(\partial\Omega^n\) of the cube \(\Omega^n=\Omega\times\cdots\times\Omega\):

\[ f|_{\partial\Omega^n}=0 \quad\text{and}\quad \frac{\partial}{\partial \nu} f|_{\partial\Omega^n}=0, \]

where \(\nu\) is the outward normal.

The hard-core condition for \(V(r)\) means that both extensions are considered in the subspaces \(\mathscr H_B^{(d)}(\Omega)\subset \mathscr H_B(\Omega)\) and \(\mathscr H_F^{(d)}(\Omega)\subset \mathscr H_F(\Omega)\), consisting of chains of functions

\[ f=\{f_0,f_1(x),f_2(x_1,x_2),\ldots\}, \]

vanishing when \(\min_{j\ne k}|x_j-x_k|\le d\). The symbol \((d)\) will be omitted below. The limiting objects described in Theorems 1, 3, and 4 do not depend on the choice of extension.

The grand canonical Gibbs ensemble in \(\Omega\) generates a linear positive normalized functional (state) \(G_{\zeta,\beta,\Omega}\) on the \(C^*\)-algebras \(\mathfrak A_B(\Omega)\) and \(\mathfrak A_F(\Omega)\) of bounded operators in \(\mathscr H_B(\Omega)\) and \(\mathscr H_F(\Omega)\), respectively: for \(A\in\mathfrak A_B(\Omega)\;(\mathfrak A_F(\Omega))\)

\[ G_{\zeta,\beta,\Omega}(A)=\operatorname{tr} A\rho_{\zeta,\beta,\Omega}, \]

where

\[ \rho_{\zeta,\beta,\Omega} = \left[\operatorname{tr}\zeta^N\exp(-\beta H)\right]^{-1} \zeta_N\exp(-\beta H) \]

is a positive definite nuclear operator in \(\mathscr H_B(\Omega)\) (respectively in \(\mathscr H_F(\Omega)\)) with trace 1 (the density matrix). Under our assumptions on \(V(r)\), this functional exists for all \(\zeta>0\) and \(\beta>0\).

* The spaces \(\mathscr H_B(\Omega)\) and \(\mathscr H_F(\Omega)\) can be introduced for any Lebesgue-measurable subset of the line.

1. Let \(\Omega \to \infty\) be the set of finite intervals directed by inclusion. Then the \(c^*\)-algebras \(\mathfrak A_B(\Omega)\) and \(\mathfrak A_F(\Omega)\) form direct spectra of \(c^*\)-algebras (see \((^4)\)). Let \(\mathfrak A_B^{(0)}\) and \(\mathfrak A_F^{(0)}\) be the corresponding inductive limits, and let
   \[ \mathfrak A_B=\overline{\mathfrak A}_B^{(0)},\qquad \mathfrak A_F=\overline{\mathfrak A}_F^{(0)} \]
   (the bar denotes closure in the norm).

The state \(G_{\zeta,\beta,\Omega}\), for fixed \(\Omega\), may be regarded as given on families of \(c^*\)-algebras \(\{\mathfrak A_B(\Omega')\}\) and \(\{\mathfrak A_F(\Omega')\}\), where \(\Omega'\subset \Omega\).*

Theorem 1. For all \(\zeta>0\) and \(\beta>0\), for any \(A\in \mathfrak A_B^{(0)}\) \((\mathfrak A_F^{(0)})\), there exists the limit
\[ \lim_{\Omega\to\infty} G_{\zeta,\beta,\Omega}(A)=G_{\zeta,\beta}^{(0)}(A). \]

From Theorem 1 and theorems on the extension of positive linear functionals it follows that on the \(c^*\)-algebra \(\mathfrak A_B\) \((\mathfrak A_F)\) there exists a unique state \(G_{\zeta,\beta}\) coinciding on \(\mathfrak A_B^{(0)}\) \((\mathfrak A_F^{(0)})\) with \(G_{\zeta,\beta}^{(0)}\). The state \(G_{\zeta,\beta}\) is called the limiting Gibbs state.

1. In the \(c^*\)-algebras \(\mathfrak A_B\) and \(\mathfrak A_F\) there acts a one-parameter group \(\{T_\tau,-\infty<\tau<+\infty\}\) of spatial shifts (translations) \((^1)\). The limiting state \(G_{\zeta,\beta}\) is invariant under the action of this group and has the regularity property described in the following theorem:

Theorem 2. Let \(A_1\in\mathfrak A_B(\Omega_1)\), \(A_2\in\mathfrak A_B(\Omega_2)\) \((A_1\in\mathfrak A_F(\Omega_1),\ A_2\in\mathfrak A_F(\Omega_2))\). Then
\[ \left|G_{\zeta,\beta}(A_1\cdot(T_\tau A_2))-G_{\zeta,\beta}(A_1)G_{\zeta,\beta}(A_2)\right| \le \|A_1\|\cdot\|A_2\|\gamma(\tau,\Omega_1,\Omega_2), \tag{1} \]
where (for any fixed \(\zeta>0\) and \(\beta>0\))
\[ \lim_{|\tau|\to\infty}\gamma(\tau,\Omega_1,\Omega_2)=0. \]

1. The theorem given below establishes the impossibility of phase transitions (in the sense of Ehrenfest \((^6)\)) with respect to \(\zeta\) in the systems under consideration.

Theorem 3. The limit
\[ \lim_{\Omega\to\infty}\frac{1}{a_2-a_1}\ln\operatorname{tr}\zeta^N\exp(-\beta H)=\varphi(\zeta,\beta) \]
defines, for all \(\beta>0\), an infinitely differentiable function of the variable \(\zeta>0\).

Similar results are contained in \((^5)\).

1. Scheme of the proof of Theorems 1 and 2. Let \(\Omega_0\subset\Omega\). The reduced density matrix \(\rho_{\zeta,\beta,\Omega}^{(\Omega_0)}\) is defined by the condition
   \[ G_{\zeta,\beta,\Omega}\left(\pi_{\Omega}^{\Omega_0}A\right)=\operatorname{tr} A\rho_{\zeta,\beta,\Omega}^{(\Omega_0)} \tag{2} \]
   and is a positive-definite nuclear operator in \(\mathscr H_B(\Omega_0)\) \((\mathscr H_F(\Omega_0))\) with trace \(1\).** We denote the kernel of this operator by \(\rho_{\zeta,\beta,\Omega}^{(\Omega_0)}(\bar x,\bar y)\). Here
   \[ \bar x=(x_1,\ldots,x_n),\qquad \bar y=(y_1,\ldots,y_n);\qquad x_j,\ y_j\in\Omega_0,\qquad j=1,\ldots,n; \]
   \[ \min\bigl[|x_j-x_k|,\ |y_j-y_k|\bigr]>d;\qquad n=1,2,\ldots \]

* In this case, in both instances, if \(\{\pi_\Omega^{\Omega'}\}\) are homomorphisms of the direct spectrum \((\Omega'\subset\Omega)\), then for \(A\in\mathfrak A(\Omega')\)
\[ G_{\zeta,\beta,\Omega}\left(\pi_\Omega^{\Omega'}A\right)=G_{\zeta,\beta,\Omega}(A\otimes E), \]
where \(E\) is the identity operator in \(\mathscr H(\Omega\setminus\Omega')\) (see \((^4)\)).

** The operator \(\rho_{\zeta,\beta,\Omega}^{(\Omega_0)}\) is determined uniquely by condition (2). It can be shown that it has the form
\[ \rho_{\zeta,\beta,\Omega}^{(\Omega_0)} = \left[\operatorname{tr}\zeta^N\exp(-\beta H)\right]^{-1} \operatorname{tr}_{\mathscr H(\Omega\setminus\Omega_0)}\zeta^N\exp(-\beta H). \]

Theorem 4. For all \(\zeta>0\) and \(\beta>0\), for any \(\Omega_0\) there exists the limit
\[ \lim_{\Omega\to\infty}\rho^{(\Omega_0)}_{\zeta,\beta,\Omega}(\bar{x},\bar{y}) = \rho^{(\Omega_0)}_{\zeta,\beta}(\bar{x},\bar{y}), \]
uniformly for fixed \([l_1,l_2]\) in \(\Omega_0\subset [l_1,l_2]\) and \(\bar{x},\bar{y}\in\Omega_0^n\), \(n=1,2,\ldots\).

The limiting kernel \(\rho^{(\Omega_0)}_{\zeta,\beta}(\bar{x},\bar{y})\) defines in \(\mathscr H_B(\Omega_0)\), \(\mathscr H_F(\Omega_0)\) a positive definite kernel operator \(\rho^{(\Omega_0)}_{\zeta,\beta}\) with trace 1. The operators \(\rho^{(\Omega_0)}_{\zeta,\beta}\) and \(\rho^{(\Omega'_0)}_{\zeta,\beta}\), for \(\Omega_0\subset\Omega'_0\), are related by
\[ \rho^{(\Omega_0)}_{\zeta,\beta} = \operatorname{tr}_{\mathscr H(\Omega'_0\setminus\Omega_0)} \rho^{(\Omega'_0)}_{\zeta,\beta}. \]

The method of proof of Theorem 4 is a modification of the well-known matrix method (transfer-matrix method) in classical statistical mechanics.

Theorem 1 is a consequence of Theorem 4 and Lemma 1.

Lemma 1. Let \(\rho_n(x,y)\) be a sequence of kernels defining positive definite kernel operators \(\rho_n\) with trace 1 in the Hilbert space \(L^2(M,\mu)\), \(\mu(M)<\infty\). Suppose that, uniformly in \(x,y\in M\), there exists the limit
\[ \lim_{n\to\infty}\rho_n(x,y)=\rho(x,y), \]
which defines a positive definite kernel operator \(\rho\) with trace 1. Then
\[ \lim_{n\to\infty}\|\rho_n-\rho\|_1=0, \]
where \(\|A\|_1=\operatorname{tr}\sqrt{AA^*}\).

Proof. Let \(\lambda_1\geq \lambda_2\geq\cdots\geq0\) be the sequence of eigenvalues of the operator \(\rho\), and \(e_i(x)\), \(i=1,2,\ldots\), the corresponding eigenvectors. From (3) it follows that
\[ \lim_{n\to\infty}\sum_{i,j}\bigl((\rho_n e_i,e_j)-\lambda_i\delta_{ij}\bigr)^2=0. \]

We shall now show that the sequence \(\{\rho_n\}\) is compact in the Banach space \(\mathfrak P\) of kernel operators in \(L^2(M,\mu)\) with norm \(\|\cdot\|_1\). To this end we use the compactness criterion in Banach spaces with basis (3). As a basis in \(\mathfrak P\) it is natural to take the system \(\{E_{ij}\}\) of matrix units for the basis \(\{e_i(x),\, i=1,2,\ldots\}\). Let
\[ \rho_n^{(i_0)} = \sum_{i,j<i_0}(\rho_n e_i,e_j)E_{ij}, \qquad \bar{\rho}_n^{(i_0)} = \sum_{i,j\geq i_0}(\rho_n e_i,e_j)E_{ij}, \]
\[ \widetilde{\rho}_n^{(i_0)} = \sum_{i=1}^{i_0-1}\sum_{j=i_0}^{\infty}(\rho_n e_i,e_j)E_{ij}. \]

Obviously, \(\rho_n^{(i_0)}\) and \(\bar{\rho}_n^{(i_0)}\) are positive definite operators,
\[ \|\rho_n^{(i_0)}\|_1+\|\bar{\rho}_n^{(i_0)}\|_1=1, \]
and also
\[ \rho_n=\rho_n^{(i_0)}+\bar{\rho}_n^{(i_0)}+\widetilde{\rho}_n^{(i_0)*}+\widetilde{\rho}_n^{(i_0)}. \]
Take an arbitrary \(\varepsilon>0\) and choose \(i_0=i_0(\varepsilon)\) and \(n_0=n_0(\varepsilon)\) so that
\[ \sum_{i\geq i_0}\lambda_i<\varepsilon/4, \]
and, for \(n>n_0(\varepsilon)\),
\[ \left|\|\rho_n^{(i_0)}\|_1-\sum_{i=1}^{i_0-1}\lambda_i\right|<\frac{\varepsilon}{4}, \qquad \sum_{i\ne j}\bigl((\rho_n e_i,e_j)\bigr)^2<\frac{\varepsilon^2}{16i_0^2}. \]

* A similar assertion was obtained by I. D. Novikov.

Then, for \(n>n_0\),

\[ \left\|\rho_n-\rho_n^{(i_0)}\right\|_1 \leq \left\|\rho_n^{(i_0)}\right\|_1 + 2\left\|\widetilde{\rho}_n^{(i_0)}\right\|_1 = 1-\left\|\rho_n^{(i_0)}\right\|_1 + 2\left\|\widetilde{\rho}_n^{(i_0)}\right\|_1 \leq \]

\[ \leq 1-\sum_{i=1}^{i_0-1}\lambda_i + \frac{\varepsilon}{4} + 2\left\|\widetilde{\rho}_n^{(i_0)}\right\|_1 < \frac{\varepsilon}{2} + 2\left\|\widetilde{\rho}_n^{(i_0)}\right\|_1 . \]

The proof of the lemma is completed by the obvious observation that

\[ \left\|\widetilde{\rho}_n^{(i_0)}\right\|_1 \leq \sum_{i=1}^{i_0-1} \left[ \sum_{j=i_0}^{\infty}\bigl((\rho_n e_i,e_j)\bigr)^2 \right]^{1/2} < i_0 \left[ \sum_{i\ne j}\bigl((\rho_n e_i,e_j)\bigr)^2 \right]^{1/2}. \]

Theorem 2 is a consequence of Lemma 1 and of Theorem 5 stated below, which establishes the so-called cluster property of the limiting kernels
\(\rho_{\xi,\beta}^{(\Omega_0,\Omega'_0)}(\bar{x},\bar{y})\).
Let \(\bar{x},\bar{y}\in\Omega^l\); \(\bar{x}',\bar{y}'\in{\Omega'_0}^{\,l'}\);
\(T_\tau\bar{x}'=(x'_1+\tau,\ldots,x'_l+\tau)\);
\(T_\tau\bar{y}'=(y'_1+\tau,\ldots,y'_l+\tau)\).

Theorem 5. The inequality holds

\[ \left| \rho_{\xi,\beta}^{(\Omega_0\cup T_\tau\Omega'_0)} (\bar{x}\cup T_\tau\bar{x}',\bar{y}\cup T_\tau\bar{y}') - \rho_{\xi,\beta}^{(\Omega_0)}(\bar{x},\bar{y})\, \rho_{\xi,\beta}^{(\Omega'_0)}(\bar{x}',\bar{y}') \right| \leq \bar{\gamma}(\tau,\Omega_0,\Omega'_0), \]

where

\[ \lim_{|\tau|\to\infty} |\tau|^\alpha \bar{\gamma}(\tau,\Omega_0,\Omega'_0)=0 \quad \text{for all } \alpha>0 . \]

The proof of Theorem 3 rests on Theorems 4 and 5 and is carried out by the standard method of the so-called correlation functions.

The author thanks Ya. G. Sinai for useful discussions of the results and comments during the preparation of the note.

Moscow State University
named after M. V. Lomonosov

Received
29 IV 1970

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