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The Hubbard Model & Bose-Hubbard Model

Connecting the dots between QC and condensed matter physics

Preliminaries

Ladder operators

Creation and annihilation operators are a useful tool when working with quantum many-body systems. To understand them, a background with quantum harmonic oscillators won’t be required since we’ll focus on two-level systems where the creation and annihilation operators can be represented as:

  • We call a^dag the fermionic creation operator and athe fermionic annihilation operator. We can further specify the spin and site number that the operator acts upon and say that a^{†}_{iσ} creates a fermion at site i with spin σ, while a_{iσ} destroys a fermion at site i with spin σ.
  • Similarly, we call b^dag a bosonic creation operator (letter doesn’t matter) and b a bosonic annihilation operator. With a specified index, we say that b^{†}_{i} creates a boson at site i and b destroys a boson at site i.

What is the Hubbard Hamiltonian?

Historical background

Quantum physicists pre-1963 faced a stark inconsistency. Their tried-and-true band structure theory seemed to contain edge cases. Namely, in light of a class of solids called “Mott insulators”, the theory predicts that they should conduct electricity (valence electrons are loosely bound) when in fact they do not.

The Hamiltonian

To continue our trek through the Hubbard model, let’s introduce the Hamiltonian. It neatly packages the contents of the Hubbard model in a mathematical form for us to analyze. We use the convenient language of creation and annihilation operators, and you will soon understand why.

Hubbard Hamiltonian
Sketch of a 2D Hubbard model on a square lattice, source

The tunnelling term

Recall that the Hamiltonian is a measurement operator which tells us the energy of a given quantum particle with respect to the Hamiltonian. Note how all natural systems “desire” to be in their lowest energy state. Think of a ball on a hill. It rolls down to release all of its kinetic and potential energy into its environment. Analogously, you can think of this Hamiltonian as the hill which ascribes certain energy levels to each possible configuration of electrons in our lattice (think of the configuration of electrons in the lattice as the position of the ball).

Example of the tunneling term acting on one coupling

The on-site interaction term

The role of the on-site interaction term is to encode how energetically taxing it can be for an electron to hop into a site that’s already occupied by another electron. The degree to which it’s energetically unfavourable for a double-occupation to occur scales proportionally to the scalar U. So a system with very large U would abhor a double occupation — it would cause a steep spike in the system’s potential energy. In that system, the electrons at each site will exhibit a repulsive force towards any neighbouring electrons so as to avoid such an energetic “catastrophe”.

How does the difference in sign affect the dynamics?

Equipped with our understanding of each term, wrapping it all together to achieve a broader understanding of the Hubbard model is straightforward.

Key assumptions

Despite how accurate the Hubbard model may be in predicting certain electrical properties, it is indeed an approximate model. Here are the key assumptions that underpin it, put briefly:

  1. We assume that that nuclei of each atom in the lattice remain fixed.
  2. We assume that two electrons interact with each other only if they are occupying the same site/orbital.
  3. We restrict ourselves to nearest-neighbor couplings, meaning that we assume electrons can only hop to another atom if it’s adjacent to its current site.

The Bose-Hubbard model

Although Hubbard’s original work aimed at capturing the physics of fermions, it turns out that the underlying principles generalize well to describe the physics of interacting bosons as well. Just as the Hubbard model effectively characterizes the metal-insulator transition for fermions, the Bose-Hubbard model effectively characterizes the superfluid-insulator transition for bosons.

Bosonic atoms in an optical lattice, source
Bose-Hubbard Hamiltonian (with chemical potential term)

The hard-core boson model

A neat fact resulting from the Bose-Hubbard model is that if you take the limit U → , you’ll arrive at the bosonic analogue of the Pauli Exclusion Principle! That is, in the position basis.

Matching theory with simulations

A theoretical understanding of the relationship between U/t and the dynamics of our system may well be sufficient, but seeing these results come to life in simulations lit a spark within me that I’d like to reignite within you.

Source

Discussion

Hubbard’s initial intentions behind his paper were simply to provide a fermionic lattice model to account for the existence of Mott insulators, yet this approximate model was eventually extended to successfully describe a plethora of lattice structures of interacting quantum particles, bosons and fermions alike. It became the go-to template for studying the physics of such systems!

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pavanjay.com | Invested in QC + ML | EECS @UWaterloo | Seeker of rigorous truth

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