Graduate Quantum statistical canonical formalism to find ground state at T

[Luqman Saleem]

TL;DR

As an example, calculating the ground state energy and wave function using canonical formalism of a simple fermionic model.

For my own understanding, I am trying to computationally solve a simple spinless fermionic Hamiltonian in Quantum Canonical Ensemble formalism . The Hamiltonian is written in the second quantization as

$$H = \sum_{i=1}^L c_{i+1}^\dagger c_i + h.c.$$

In the canonical formalism, the density matrix is given as
$$\rho = \frac{e^{-\beta H}}{Tr [e^{-\beta H}]}$$
where $\beta = 1/k_B T$ and the expectation value of any operator $A$ is given as
$$\langle A \rangle = Tr[\rho A]$$

Things that I want to calculate:
1. ground state energy at temperature $T$
2. ground state wave-function at $T$

My attempt to solution:
1. I first write $H$ in matrix form by using particle number basis i.e. 1100, 1010 $\cdots$ for 4 sites and 2 particles
2. then I diagonalize $H$ and find energy eigenvalues $E_i$ and eigenvectors $\psi_i$
3. then I write $\rho$ by using above given formula
4. finally, I try to calculate ground state energy using $\langle H \rangle = Tr[\rho H]$

Problems I am facing:
As Boltzmann constant $k_B$ is $1.36\times 10^{-23}$, so even at room room temperature $\beta$ is very large i.e. ~$10^{21}$. Due to which I can't calculate ground state energy at very low temperatures? Computer just give 'Infinity' values. And at very large temperatures, the energy seems to be decreasing with increasing $T$. ... totally wrong

Questions:
1. Is my algorithm correct? How can we calculate ground state energy at $T=0$ and at small temperatures?
2. How to calculate ground state wave function at $T$?

 

[atyy]

Is the temperature relevant to defining the ground state energy and wave function? Isn't the ground state defined at T=0?

 

[Luqman Saleem]

Is the temperature relevant to defining the ground state energy and wave function? Isn't the ground state defined at T=0?

At T=0, ground state is defined. I was just wondering how does statistical mechanics' formalism deal with this case. And I got the answer, I mean at T=0, $\rho = I$, which means $\langle H \rangle = \langle I H \rangle = \langle H \rangle$.

And I would say that ground state energy is relevant to temperature. Does not ground state energy increase with the increase in temperature?

 

[atyy]

And I would say that ground state energy is relevant to temperature. Does not ground state energy increase with the increase in temperature?

The ground state and the ground state energy do not change with temperature. The ground state is a property of the Hamiltonian, and is the state with the lowest energy.

The probability with which the ground state and other states are occupied changes with temperature.

In the canonical ensemble at finite T, each member of the notional canonical ensemble of systems is in a different state. The distribution of states across the members of the ensemble is given by the density matrix of the canonical ensemble. Roughly, one can think that the probability of a given state in the canonical ensemble is given by p(E) ∝ exp(-βE).

You can find an example in https://en.wikipedia.org/wiki/Canonical_ensemble. See the right side of the figure titled "Example of canonical ensemble for a quantum system consisting of one particle in a potential well."

You can also see section 6.5 of https://mcgreevy.physics.ucsd.edu/s12/lecture-notes/chapter06.pdf for an example of the distribution of states at finite T in the canonical ensemble for the quantum harmonic oscillator.