Degenerate ground state of Hamiltonian from analytical perspective

Question

Suppose I have a Hamiltonian that depends on the continuous vector parameter $\boldsymbol{\theta}$, and the ground state corresponds to line/plane or some other $1$ to $p-1$ dimensional subspace of the $p\,$ dimensional space of my parameter vector.

And suppose I wish to find the ground state energy using a statistical mechanics/analysis method:

1. I write my partition function $Z = \displaystyle\int d\boldsymbol{\theta}\,e^{-\beta H(\boldsymbol{\theta})}$
2. I use the formula for the internal energy $U = -\dfrac{\partial}{\partial \beta} \ln(Z)$
3. And I take the limit of $\beta \rightarrow \infty$ corresponding to $T\rightarrow 0$

This should give me the ground state energy. But, I am suspicious of whether the underlying mathematics suggests this is a fishy thing to do in the case of a degenerate ground state.

From my limited understanding of analysis:

I take:

$$U = \lim_{\beta \rightarrow \infty} \frac{\displaystyle \int d\boldsymbol{\theta}\,H(\theta)e^{-\beta H(\boldsymbol{\theta})}}{Z}$$

and this is handled with the saddle point method: Watson's lemma etc.

But while I understand how the saddle point method works (in terms of using a Taylor expansion of the Hamiltonian) for a single saddle point (this stack exchange post has a good overview) I don't think it works when there is a dimension $\dim \mathfrak g>0$ subspace for the ground state.

Would it be appropriate to do a Landau-type expansion of the Hamiltonian (though I have only come across Landau expansions of the free energy; and with macroscopic order parameters rather parameters on which the Hamiltonian depends!). But still, if I have a continuous region of the parameter space are ground states, then how to consider this analytical approach?

Answer 1

1. If the Hamiltonian is (continuously) degenerated, there is going to be a flat direction valley in the profile $\theta\mapsto H(\theta)$. Close to the bottom of the valley (=the ground state submanifold $\mathfrak{g}$), the equi-energy-surfaces/lines are locally almost aligned.

2. Let us for simplicity assume that we can globally choose adapted parameter coordinates $$\theta = \{x,y\}\in X\times Y$$ such that $$ \frac{\partial H}{\partial y}~\equiv~0; $$ such that the ground state submanifold $$\mathfrak{g}~=~\{x\in X \mid \frac{\partial H}{\partial x}=0\}\times Y $$ is parametrized by the $y$-coordinates; and such that the Hessian $$\left. \frac{\partial^2 H}{\partial x^2}\right|_{\mathfrak{g}}$$ is positive definite.

3. Change of coordinates produce a Jacobian factor $$d\theta~=~dy~ dx~J(x,y),$$ so that OP's partition function becomes $$Z~:=~\int \!d\theta~e^{-\beta H(x)}~=~\int_{Y}\!dy~ \int_{X}\!dx~J(x,y)~e^{-\beta H(x)}, \qquad \beta\text{ large}. $$

4. The integration over the $x$-parameters corresponds to a non-degenerate problem (with some external parameters $y$), which OP already knows how to calculate using the method of steepest descent.

5. There remains an ordinary integration over the $y$-parameters. [The $y$-integration may or may not be doable in analytic form. If the Jacobian does not depend on $y$, then the $y$-integration just gives a volume factor ${\rm Vol}(Y)={\rm Vol}(\mathfrak{g})$.]