Homework 1: Algorithmic Perspective

Although I've tried to make things correct/clear there are occassionally typos/confusing wording/etc. When there is a discrepancy between the problem I clearly mean to be asking and the one I've asked, please solve the former. The piazza website is a good place to help us clarify the problem statements.

When answering questions please write out in a complete and explanatory way what you're doing. Someone who doesn't know what's going on should be able to read it and understand (so in addition to equations, use words!).

Problem 1: Anderson Tower of States

In this problem you're going to show for a two-dimensional solid that the number of states which become degenerate in the thermodynamic limit and the number of spontaneously symmetry-broken states are the same.

Consider a two-dimensional system of size $ L \times L$ where space is discretized on a fine grid with grid-spacing $\Delta \times \Delta$. Assume that there are electrons on this grid with spacing $a \times a$. (So there are $a/l$ grid-points between each electron.)

(a) Count the number of spontaneously broken degenerate states which exist in the thermodynamic limit.

(b) Count (and explicitly write out how you counted) the number of states which have a finite energy gap above the ground state for a finite system but which will become degenerate in the thermodynamic limit.


Problem 2: Hubbard and Heisenberg Models by Hand

There will be a number of models we will consider in this class. One of these is the Hubbard model; in this assignment you will work out some properties of the Hubbard model by hand. In future problems, you will use these results to compare against code you write.

In this problem, you will work out the Hubbard model by hand.

The Hubbard model is $H= -t \sum_{\langle ij \rangle \sigma} c_{i\sigma}^\dagger c_{j\sigma} + U \sum_{j} n_{j\uparrow}n_{j\downarrow} - \mu \sum_j (n_{j\uparrow} + n_{j\downarrow})$

Problem 3: An exact diagonalization code

Your goal in this problem is to produce a working exact diagonalization code which uses lin tables and works in arbitrary $Sz$ sectors. It should be set up to work for two-dimensional Heisenberg models (another model we will see recur throughout class)

$$H=\sum_{\langle i,j \rangle} s_i \cdot s_j$$

with the ability in the future to modify it for other Hamiltonians (you will find this useful when developing over codes to compare against)


Problem 4: Anderson Tower of States

Using your exact diagonalization code, compute the ground state of the anti-ferromagnetic, two-dimensional square nearest neighbor Heisenberg model for different systems sizes and identify the tower of states that correspond to the spontaneous symmetry breaking of the anti-ferromagnetic.