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,

$$\rho = \left\vert \Psi \right> \left< \Psi \right\vert \quad.$$ (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,

$$\rho = \sum_i \, p_i \, \left\vert \Psi_i \right> \left< \Psi_i \right\vert \quad.$$ (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,

$$\rho = \sum_i \, e^{-\beta E_i } \, \left\vert \Psi_i \right> \left< \Psi_i \right\vert = e^{- \beta {\mathcal{H}}} \quad.$$ (12)

The expectation of any operator ${\mathcal{O}}$ is given by,

$$\left< {\mathcal{O}}\right> = \frac{ {\rm Tr} \left[ {\math... ...ft< \Psi_i \right\vert {\mathcal{O}}\left\vert \Psi_i \right>$$ (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,

$$\rho({\bf R},{\bf R}';\beta) = \left< {\bf R}\right\vert e^... ...^{-\beta E_i } \, \Psi^*_i({\bf R}) \, \Psi_i({\bf R}') \quad.$$ (14)

For any hermitian Hamiltonian ${\mathcal{H}}$ the density matrix is symmetric in its two arguments,

$$\rho({\bf R},{\bf R}';\beta) = \rho({\bf R}',{\bf R};\beta) \quad.$$ (15)

The expectation value is given by,

$$\left< {\mathcal{O}}\right> = \frac{1}{Z} \int \! {\bf d}{\bf R}\, {\bf d}{\bf R}' \; \rho({\bf... ...}';\beta) \; \left< {\bf R}' \right\vert {\mathcal{O}}\left\vert {\bf R}\right>$$ (16)

$$Z = \;\;\;\; \int \! {\bf d}{\bf R}\; \rho({\bf R},{\bf R};\beta) \quad.$$ (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,

$$\Psi_\mathbf{n}({\bf r}) = \frac{1}{\sqrt{{V^{\!\!\!\!\!\!\:^\diamond}}}} \; e^{-i \, \mathbf{k}_\mathbf{n}{\bf r}}$$ (18)

with k-vector $\mathbf{k}_\mathbf{n}= 2 \pi \mathbf{n}/ L$, where $\mathbf{n}$ is a $D$-dimensional integer vector. Hence,

$$\rho({\bf r},{\bf r}';\beta) = \frac{1}{{V^{\!\!\!\!\!\!\:^\diamond}}} \sum_{\bf {n}} \: \exp \{... ...ambda \mathbf{k}_\mathbf{n}^2 + i \, \mathbf{k}_\mathbf{n}({\bf r}-{\bf r}') \}$$ (19)

$$= ( 4 \pi \lambda \beta )^{-D/2} \sum_{\mathbf{n}} \: \exp \left\{ -\frac{({\bf r}-{\bf r}'-\mathbf{n}{\bf L})^2}{4 \lambda \beta} \right\}$$ (20)

$$\approx ( 4 \pi \lambda \beta )^{-D/2} \exp \left\{ -\frac{({\bf r}-{\bf ... ...}{4 \lambda \beta} \right\} \quad {\rm {if}} \quad \lambda \beta \ll L^2 \quad,$$ (21)

where $\lambda = \hbar^2/ 2 m$ for a particle of mass $m$. Alternatively, this solution can be derived from the Bloch equation,

$$\frac{\partial \rho}{\partial \beta} = {\mathcal{H}}\rho \quad,$$ (22)

which is a diffusion equation in imaginary time $\beta$. The initial condition is provided by the known high temperature limit,

$$\rho({\bf R},{\bf R}';0) = \delta({\bf R}-{\bf R}') \quad.$$ (23)

For free particles, the Bloch equation simply reads,

$$\frac{\partial \rho}{\partial \beta} = - \lambda \Delta \rho \quad .$$ (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,

$$\Lambda = \frac{h}{\sqrt{2 \pi m k_B T}} \equiv \sqrt{ 4 \pi \lambda \beta} \quad.$$ (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,

$$n \Lambda^D \quad,$$ (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.