Homework 1 — From Microscopic Hamiltonians to Emergent Spectra

Learning objectives¶

After completing this homework, you should be able to:

translate a Hamiltonian written in bra-ket notation into its matrix representation;
identify translational symmetry and use it to choose a convenient basis;
diagonalize a simple lattice Hamiltonian analytically using Fourier modes;
construct and diagonalize Hamiltonian matrices numerically;
compare analytical predictions with numerical results;
interpret finite-size spectra and understand how the level spacing changes with system size;
investigate how a local breaking of translational symmetry modifies collective spectral properties.

The main conceptual flow of this homework is

\boxed{ \text{Hamiltonian} \rightarrow \text{matrix} \rightarrow \text{symmetry} \rightarrow \text{eigenstates} \rightarrow \text{numerics} \rightarrow \text{physical interpretation} }

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Problem 1 — From an operator to a matrix¶

Although we have not yet formally introduced the tight-binding Hamiltonian, consider the Hamiltonian written in Eq. (11):

H = E_0\sum_{i=1}^{N}|i\rangle\langle i| - t\sum_{i=1}^{N} \left( |i\rangle\langle i+1| + |i+1\rangle\langle i| \right),

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with periodic boundary condition

|N+1\rangle = |1\rangle.

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Here, the basis states

\{|1\rangle, |2\rangle, \ldots, |N\rangle\}

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represent a particle localized on one of the $N$ sites.

Question¶

Write down the $N\times N$ matrix representation of $H$ in the basis

\{|1\rangle, |2\rangle, \ldots, |N\rangle\}.

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Your answer should make clear:

the diagonal matrix elements;
the nearest-neighbor hopping matrix elements;
the matrix elements that implement the periodic boundary condition.

Problem 2 — Use symmetry before computation¶

For the one-dimensional periodic chain, diagonalize the Hamiltonian analytically and show that the eigenvalues take the form

E(k)=E_0-2t\cos k.

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Also determine the allowed values of $k$ .

Hint 1¶

The system is translationally invariant. Consider restructuring the basis using the Fourier-transformed states

|k\rangle = \frac{1}{\sqrt N} \sum_{j=1}^{N} e^{ikj}|j\rangle.

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Apply the Hamiltonian to $|k\rangle$ and determine whether it is an eigenstate.

Hint 2¶

There are other ways to solve the same eigenvalue problem. For example, you may write the components of an eigenvector as

\psi_j \propto z^j

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and use the periodic boundary condition to determine the allowed values of $z$ .

Conceptual question¶

Why is translational symmetry useful here? In particular, explain why choosing a symmetry-adapted basis is more efficient than directly computing the determinant of a large $N\times N$ matrix.

Problem 3 — Numerical diagonalization and the thermodynamic limit¶

For simplicity, set

E_0=0, \qquad t=1.

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Construct the Hamiltonian numerically for

N=100

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and

N=200.

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Diagonalize the matrix numerically and sort the eigenvalues in ascending order,

E_1 \le E_2 \le \cdots \le E_N.

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For each system size, make a plot where:

the horizontal axis is the eigenvalue index $n$ ;
the vertical axis is the corresponding eigenvalue $E_n$ .

You may either plot the two spectra separately or show them together in a way that makes comparison easy.

Questions¶

Does the total bandwidth change significantly when $N$ is increased from 100 to 200?
What happens to the typical spacing between neighboring energy levels?
Compare your numerical result with the analytical dispersion relation

E(k)=-2\cos k.

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Are the numerical eigenvalues consistent with the analytical prediction?

In the lecture we argued that, if a fixed bandwidth contains an increasing number of states, the characteristic level spacing should decrease as the system size increases. Is your numerical result consistent with this argument?
Estimate how the characteristic level spacing changes when $N$ is doubled from 100 to 200.

Problem 4 — Breaking translational symmetry locally¶

We now introduce a local defect by weakening one hopping amplitude.

For the $N=100$ system, change the hopping between sites 50 and 51 from

t=1

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t'=0.1.

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That is,

t_{50,51}=t_{51,50}=0.1.

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For the $N=200$ system, make the corresponding modification between sites 100 and 101:

t_{100,101}=t_{101,100}=0.1.

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All other hopping amplitudes remain equal to 1.

Numerically diagonalize the modified Hamiltonians and compare the sorted spectra with those of the uniform systems.

Questions¶

How does the spectrum change after introducing the weak bond?
Is momentum $k$ still a good quantum number? Explain your answer in terms of translational symmetry.
Does the overall bandwidth change substantially?
Is the relative influence of this single modified bond stronger for $N=100$ or for $N=200$ ?
A single bond is modified out of approximately $N$ bonds. The fraction of modified bonds therefore scales as

\frac{1}{N}.

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What does this suggest about the influence of one local perturbation on global spectral properties in the thermodynamic limit

N\rightarrow\infty?

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Even if the effect on the global spectrum becomes relatively small as $N$ increases, does the defect still break translational symmetry exactly? Explain the distinction between these two statements.

Optional extension — Inspect the eigenstates¶

The eigenvalues tell us only part of the story.

For the uniform periodic chain, choose a few eigenstates and inspect the probability distribution

|\psi_i|^2

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over the lattice sites.

Repeat the same analysis for the system with the weak bond.

Questions¶

Are the eigenstates of the uniform system spatially extended?
How does the weak bond modify the spatial structure of the eigenstates?
Can the spectrum change only slightly while the eigenvectors change in a more visible way?

This optional exercise illustrates an important principle in condensed matter physics:

\boxed{ \text{The spectrum and the eigenstates contain different physical information.} }

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Summary¶

This homework connects several central ideas introduced in Week 1.

First, an abstract Hamiltonian can be represented concretely as a matrix,

\text{operator notation} \rightarrow \text{matrix representation}.

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Second, symmetry can reveal the natural basis of a problem,

\text{translation symmetry} \rightarrow \text{momentum basis}.

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Third, analytical and numerical approaches should agree,

\text{analytical solution} \leftrightarrow \text{numerical diagonalization}.

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Finally, increasing the number of sites makes the spectrum denser while the microscopic hopping scale remains fixed,

N\uparrow \qquad\Rightarrow\qquad \delta E\downarrow.

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The main lesson is that a condensed matter problem should not be approached only as a matrix to diagonalize. One should first ask what structure, symmetry, scale, and physical information are present in the Hamiltonian.