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The Thermal Density Matrix

A quantum mechanical system in a pure state can be described by single wave function $\left\vert\Psi \right>$, which can be expressed in terms of eigenvalues $E_i$ and eigenfunctions $\left\vert \Psi_i \right>$ of the Hamiltonian ${\mathcal{H}}$. The corresponding density matrix operator is given by,

\rho = \left\vert \Psi \right> \left< \Psi \right\vert
\end{displaymath} (10)

The density matrix provides a convenient way to extend the study to finite temperature. Following the principles of statistical mechanics, one puts the system in contact with a heat bath and assigns classical probabilities $p_i$ to the quantum mechanical states $\left\vert \Psi_i \right>$, which leads to a thermal density matrix,
\rho = \sum_i \, p_i \, \left\vert \Psi_i \right> \left< \Psi_i \right\vert
\end{displaymath} (11)

In the canonical ensemble at temperature $T$, the probabilities are proportional to the Boltzmann factor $p_i \propto {\rm {exp}}(-\beta
E_i)$, where $k_B T = 1/ \beta$. The density matrix now reads,
= \sum_i \, e^{-\beta E_i } \, \left\vert \Psi_i \right> \left< \Psi_i \right\vert
= e^{- \beta {\mathcal{H}}}
\end{displaymath} (12)

The expectation of any operator ${\mathcal{O}}$ is given by,
\left< {\mathcal{O}}\right> =
\frac{ {\rm Tr} \left[ {\math...
...ft< \Psi_i \right\vert {\mathcal{O}}\left\vert \Psi_i \right>
\end{displaymath} (13)

and the canonical partition function is $Z = \sum_{i} e^{-\beta E_i
}$. For sake of numerical simulations, it is convenient to change to a position-space representation. Introducing the set of coordinates for a system of $N$ particles in $D$ dimensions ${\bf R}= \{ {\bf r}_1 \ldots
{\bf r}_N \}$, the density matrix becomes,
\rho({\bf R},{\bf R}';\beta)
= \left< {\bf R}\right\vert e^...
...^{-\beta E_i } \, \Psi^*_i({\bf R}) \, \Psi_i({\bf R}')
\end{displaymath} (14)

For any hermitian Hamiltonian ${\mathcal{H}}$ the density matrix is symmetric in its two arguments,
\rho({\bf R},{\bf R}';\beta) = \rho({\bf R}',{\bf R};\beta)
\end{displaymath} (15)

The expectation value is given by,
$\displaystyle \left< {\mathcal{O}}\right>$ $\textstyle =$ $\displaystyle \frac{1}{Z}
\int \! {\bf d}{\bf R}\, {\bf d}{\bf R}' \; \rho({\bf...
...}';\beta) \; \left< {\bf R}' \right\vert {\mathcal{O}}\left\vert {\bf R}\right>$ (16)
$\displaystyle Z$ $\textstyle =$ $\displaystyle \;\;\;\; \int \! {\bf d}{\bf R}\; \rho({\bf R},{\bf R};\beta)
\quad.$ (17)

For a free particle in a periodically repeated box of size $L$ and volume ${V^{\!\!\!\!\!\!\:^\diamond}}=L^D$, the density matrix can derived from the exact eigenfunctions of the Hamiltonian given by plane waves,
\Psi_\mathbf{n}({\bf r}) = \frac{1}{\sqrt{{V^{\!\!\!\!\!\!\:^\diamond}}}} \; e^{-i \, \mathbf{k}_\mathbf{n}{\bf r}}
\end{displaymath} (18)

with k-vector $\mathbf{k}_\mathbf{n}= 2 \pi \mathbf{n}/ L$, where $\mathbf{n}$ is a $D$-dimensional integer vector. Hence,
$\displaystyle \rho({\bf r},{\bf r}';\beta)$ $\textstyle =$ $\displaystyle \frac{1}{{V^{\!\!\!\!\!\!\:^\diamond}}}
\sum_{\bf {n}} \: \exp \{...
...ambda \mathbf{k}_\mathbf{n}^2 + i \, \mathbf{k}_\mathbf{n}({\bf r}-{\bf r}') \}$ (19)
  $\textstyle =$ $\displaystyle ( 4 \pi \lambda \beta )^{-D/2}
\sum_{\mathbf{n}} \: \exp \left\{ -\frac{({\bf r}-{\bf r}'-\mathbf{n}{\bf L})^2}{4 \lambda \beta} \right\}$ (20)
  $\textstyle \approx$ $\displaystyle ( 4 \pi \lambda \beta )^{-D/2} \exp \left\{ -\frac{({\bf r}-{\bf ...
...}{4 \lambda \beta} \right\}
\quad {\rm {if}} \quad \lambda \beta \ll L^2
\quad,$ (21)

where $\lambda = \hbar^2/ 2 m$ for a particle of mass $m$. Alternatively, this solution can be derived from the Bloch equation,
\frac{\partial \rho}{\partial \beta} = {\mathcal{H}}\rho
\end{displaymath} (22)

which is a diffusion equation in imaginary time $\beta $. The initial condition is provided by the known high temperature limit,
\rho({\bf R},{\bf R}';0) = \delta({\bf R}-{\bf R}')
\end{displaymath} (23)

For free particles, the Bloch equation simply reads,
\frac{\partial \rho}{\partial \beta} = - \lambda \Delta \rho
\quad .
\end{displaymath} (24)

The $\Delta$ operator can be applied either to the first or to the second argument in $\rho ({\bf r},{\bf r}';\beta )$. In any case, this equation describes the diffusion of paths in imaginary time. The exact solution is given by Eq. 2.11. The diffusion constant $\lambda$ is determined by the mass of the particle. It is large for light particles leading to a fast diffusion in imaginary time, and small for heavy and therefore classical particles. The width of the density matrix is given by $\sqrt{4 \lambda \beta}$, which is related to the frequently used thermal de Broglie wave length defined as,
\Lambda = \frac{h}{\sqrt{2 \pi m k_B T}} \equiv \sqrt{ 4 \pi \lambda \beta}
\end{displaymath} (25)

If this length reaches the order of the inter-particle spacing, many body quantum effects become important. This relation is usually discussed in terms of the degeneracy parameter,
n \Lambda^D
\end{displaymath} (26)

which relates the volume per particle ${V^{\!\!\!\!\!\!\:^\diamond}}/N\equiv n^{-1}$ to the volume occupied by an individual path $\Lambda^D$. This parameter defines a temperature scale for the emergence of quantum statistical effects such as Bose condensation in Bosonic systems and the formation of a Fermi surface in Fermion systems. The latter process will be discussed in detail in sections 2.6 and 5.3.

next up previous contents
Next: Imaginary Time Path Integrals Up: Path Integral Monte Carlo Previous: Path Integral Monte Carlo   Contents
Burkhard Militzer 2003-01-15