Zero Temperature Limit

In the preceding section, it was shown that for the hydrogen atom the Gaussian variational density matrix, as a function of ${\bf R}$ converges at low temperature to the Gaussian ground state wave function given by the Rayleigh-Ritz variational principle. It is generally true that the Rayleigh-Ritz ground state corresponds to the zero temperature limit of the VDM as we now show.

The Rayleigh-Ritz principle states that for any real parameterized wave function $\Psi({\bf R},q_{1},\ldots,q_{m})$ the variational energy

$$E(\{q\})=\begin{array}{c}\underline{\int \psi({\bf R}){\ma... ...}{\bf R}}\\ \int \psi({\bf R})^{2} \;{\bf d}{\bf R}\end{array}$$ (156)

is greater than or equal to the true ground state energy even at the minimum determined by

$$\begin{array}{c}\partial\\ \overline{\partial q_{k}}\end{array} E(\{q\})=0\;\;\forall k\;.$$ (157)

For the VDM ansatz, an amplitude parameter $D$ is assumed such that

$$\rho({\bf R},{\bf R'};\beta)=e^{D({\bf R'};\beta)}\tilde{\rho}({\bf R},\{q({\bf R'};\beta)\})\;.$$ (158)

As in the one particle example, it is expected that at low temperature, $\beta\to \infty$, the other $\dot{q}_{k}\to 0$ while $\dot{D}\to$ constant. From this assumption, Eq. 3.19 implies that as $\beta\to \infty$

$$\frac{\partial H}{\partial q_k} + \dot{D} \frac{\partial N}{\partial q_k} = 0$$ (159)

for all variational parameters, where we have defined $H\equiv\int \rho {\mathcal{H}}\rho\;{\bf d}{\bf R}$ and $N\equiv\int \rho^{2}\;{\bf d}{\bf R}$. Since $\partial H/\partial D=2H$ and $\partial N/\partial D=2N$, Eq. 3.41 for $q_{k} \equiv D$ implies $\dot{D}=-H/N\equiv -E_{0}$. So Eq. 3.41 may be rewritten as

$$\frac{\partial }{\partial q_k} \left( \frac{H}{N} \right) = 0$$ (160)

at the $\beta\to \infty$ fixed point. With the correspondence

$$\rho({\bf R},\{q({\bf R'},\beta)\})\to e^{D({\bf R'};\beta)}\psi({\bf R},\{q\})\quad,$$ (161)

this is equivalent to Eq. 3.39 and thus the Rayleigh-Ritz ground state corresponds to a zero temperature fixed point in the dynamics of the parameters.

$D$ is a function of ${\bf R'}$ and $\beta$, which is calculated by integrating from $\beta=0$ with Eq. 3.23 as initial conditions. The zero temperature limit of $\dot{D}$ is a constant, $-E_0$, which means in the low temperature limit $D$ can written as

$$D({\bf R'};\beta) = - \beta E_0 + f({\bf R'})\quad.$$ (162)

The function $f({\bf R'})$ can be rewritten as,

$$f({\bf R'}) = \ln \left\{ \psi_0({\bf R'}) \left[ \, 1 + \delta({\bf R'}) \, \right] \right\}\quad,$$ (163)

where the function $\delta({\bf R'})$ is introduced to describe the variational error in the solution of the Bloch equation. It is identical to zero if the variational ansatz includes the exact solution. It leads to loss of symmetry in ${\bf R}$ and ${\bf R'}$, which will discussed in the next section. Eq. 3.43 now reads,

$$\rho({\bf R},{\bf R'},\beta\to\infty) = e^{-\beta E_0} \ps... ...}) \psi_0({\bf R'}) \left[ 1 + \delta({\bf R'}) \right] \quad.$$ (164)

For certain potentials, several fixed points of the dynamics can exist. From Eq. 3.46, it follows that only the lowest energy state contributes to physical observables calculated from Eq. 2.4. This completes the argument that the zero temperature limit of the VDM corresponds to the Rayleigh-Ritz ground state.

In the case of an anti-symmetrized ansatz for the density matrix, it can be shown that the fixed point of the dynamics in imaginary time corresponds to the Rayleigh-Ritz ground state for an anti-symmetrized wave function.