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The Hubbard Model & Bose-Hubbard Model

Connecting the dots between QC and condensed matter physics

As one of the keystone models in condensed matter physics, the Hubbard model provides crucial insights into electronic and magnetic properties of solid materials. In light of its’ practicality, a natural question to ask is whether we could employ a quantum computer to simulate the physics of its’ Hamiltonian, and thereby learn something of our material.

Given my forays into quantum computing — and not condensed matter physics — I had all the reason to follow my habitual procedure: just shut up and exponentiate the Hamiltonian. Especially when you’re dealing with a seemingly complicated many-body system.

Yet, a craving to peer behind the curtains emerged. Dare I ask, why is the Hamiltonian structured so? Good. What does it have to do with its sibling, the Bose-Hubbard Hamiltonian? Better. How can we witness the hard-core constraint through the lens of the Bose-Hubbard Hamiltonian, and why in the hell is the hard-core constraint referred to as the bosonic analogue of the Pauli Exclusion Principle? Elation. The deeper I went, the more illuminating the physics became.

Following this trail revitalized my intuition and formed stronger connections between what seemed like siloed concepts. In what follows, I guide you through it all, step-by-step, conveying a holistic view of the truly simplistic Hubbard family of models. All the while staying well clear of quantum computation, believe it or not.

Even if you haven’t heard a word on Hubbard models, or anything on condensed matter physics for that matter, you’ll walk away having rediscovered the binding webs of truth arising from these questions:

Allons-y!

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:

This, however, hides a nuance of creation and annihilation operators between fermions and bosons, which we will touch on in a moment.

But at the core, they share similar properties.

  • 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.

To gain some crude intuition, you can see how the following could be creation and annihilation operators if you define |1〉as the state with a fermion/boson.

The annihilation operator would “destroy” a fermion (a|1⟩ = |0⟩) if the initial state was |1⟩. Otherwise, it would destroy the vector space. The converse is clearly true for the creation operator. Although there’s a difference in commutativity between the fermionic and bosonic representations of these ladder operators, we will ignore that for our purposes.

A brief aside on commutativity relations

The differences between the fermionic and bosonic formalisms arise from a few simple facts that define fermions and bosons. Two fermions strictly cannot be at the same place if they have equivalent wave functions. Specifically, their wave functions must be antisymmetric if they are to occupy the same site! That’s however not a requirement for bosons, they are free to occupy the same site despite sharing equivalent wave functions (the reason behind Bose-Einstein condensation). With that binary fork, all quantum particles can then be characterized as either bosons or fermions.

Through those defining characteristics of bosons and fermions, you can derive a fundamental principle of quantum mechanics: the Pauli Exclusion Principle, which states that two indistinguishable fermions must have opposite spin to occupy the same site. This property promptly leads to the fermionic anticommutation relations which define strict rules for operators acting on a field of fermions. So these restrictions — which are different from the operators which act on bosonic fields — lead to the nuances you see between fermionic and bosonic ladder operators. By the same token, there exist bosonic commutation relations that must be satisfied for operators which act upon bosonic fields, including the ladder operators of course.

Yet, it turns out that these nuances don’t affect the intuition of the Hubbard model so we won’t focus on it from here onwards.

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.

It’s not necessary to understand the electron band structure theory, but you can think of it as a model that allows us to derive electrical properties of certain solids by analyzing it at a quantum level. The theory explains the energetic influences acting upon valence electrons throughout the solid’s crystalline lattice. Since conductivity is just the degree to which valence electrons can freely move through the material, band theory was usually successful at predicting the conductivity of solids.

Recall: the mission of physics is to uncover the rigorous principles that underlie physical phenomena. Naturally, this inconsistency motivated physicists at the time to find a better model relating the physics of these valence electrons to the macrosopic electrical attributes.

So in 1963, John Hubbard did exactly that. He presented a relatively simple model which effectively described the physics of interacting valence electrons in a lattice of orbitals. For reasons we’ll soon investigate, this model accounted for existence of Mott insulators! The Hubbard model outshined band theory in accounting for the repulsive interactions between electrons located at the same site of a lattice. Conventional band theory simply didn’t consider this on-site interaction.

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

Don’t feel intimidated by all the apparent noise, we’ll slowly peel back the layers to understand what this Hubbard Hamiltonian is truly conveying.

Upon first glance of a Hamiltonian, look at the relative signs of each term. It’s what intuitively encodes the dynamics at play. Following that, the core revelation of the Hubbard model is the sign difference between the first term (referred to as the “hopping” term for reasons we’ll delve into) and the second term (referred to as the on-site interaction term). Let’s first understand what each term means though.

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).

Given the linearity and Hermiticity of the Hamiltonian, we can extend this to describe not only the energy of individual configurations but also the energy of superpositions of different configurations. So instead of one hill, the Hamiltonian is actually defining a broader landscape of hills, with the bottom of each hill being an eigenstate.

With that in mind, the negative sign preceding the hopping term alludes to us that a large hopping term is energetically favourable (lowers the overall energy of the quantum state in the lattice). But what does this mean physically?

Suitably named, the hopping term encourages the particles in the lattice to “hop” or tunnel between adjacent sites, with the attractiveness to do so proportional to the scalar t. To see this, try applying one of the terms to a simple two-level system, focusing on one coupling:

Example of the tunneling term acting on one coupling

Recall that the action of the creation and annihilation operators on a two-level system is

respectively.

Observe how the particle has tunnelled to the adjacent site (from |01> to |10>), collecting a favourable reduction to the overall energy t. So we now mathematically see why t is proportional to the willingness of electrons to tunnel between adjacent sites. The bigger the t, the more electrons “wish” to hop around.

As mentioned before, the Hamiltonian is an energy operator, so it must measure both kinetic energy and potential energy. The hopping/tunnelling term accounts for the kinetic energy of the system, but how is the potential energy described? For that, we have the second term.

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”.

Pardon the anthropomorphization here, but you get the point.

Though rare, it’s worth noting that U can be negative, implying the opposite of what I said above: double-occupation is energetically encouraged.

Some may ask: why are double occupations even allowed? Hell, why isn’t three the site-specific upper bound for that matter? Recall that the origins of the model come from describing lattices of valence orbitals. From chemistry, we know that each orbital (a site) can be occupied by at most 2 electrons. Further, recall that electrons are fermions. By the definition of fermions and the Pauli Exclusion Principle, it’s impossible for two completely equivalent fermions to occupy the same site. If they did, then they are by definition bosons and not fermions! So, if two electrons do occupy the same site, then it must be that they have opposite spin (an indistinguishable property, but which ensures that their wave fns. are antisymmetric).

Mathematically, the n_{i↑} and n_{i↓} in the on-site interaction term are called spin-density operators, defined as

which returns an eigenvalue of zero if their corresponding spin does not exist at site i. For this operator, each level of the system is an eigenstate (|0⟩, |1⟩, … , |n⟩) of it, corresponding to eigenvalues of √i (where i is the level number), this operator is often called the “number” operator. It effectively measures the number of occupations at a given site. What a convenient tool to check for double-occupations!

So when you inspect this on-site interaction term, it contributes potential energy only once two electrons of opposite spin occupy the same site (a double-occupation), as measured through the number operators, which lines up perfectly with what we’ve discussed hitherto.

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.

The physics of the Hubbard model (interacting particles in a lattice) is determined by the competition between the strength of the hopping coefficient t, which characterizes the system’s kinetic energy, and the strength of the interaction term U.

Systems described by large U relative to t (U t) discourage the tunneling of electrons to instead quell the repulsive force of a double occupation. This is how Hubbard explained the existence of Mott insulators! For a solid substance to be conductive, it must by definition allow loosely bound electrons to tunnel freely through the valence orbitals of the crystalline lattice. But a large U/t has the opposite effect: it broadly discourages the free tunnelling of electrons. So Mott insulators are simply described by Hubbard models with U/t ≫ 1!

In a similar sense, we can describe the emergence of conductivity as the ratio U/t decreases below 1 (t ≫ U). In the limit as U/t → 0, the substance becomes a superconductor!

Ergo, the Hubbard model effectively captures the essence of the metal-insulator transition for certain interacting lattices of fermions by encoding the push-and-pull dynamics between the tunneling and on-site interaction terms.

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.

It turns out that most of these assumptions are actually quite reasonable and still allow the Hubbard model to effectively describe lattices of fermions. To learn more about these assumptions, you can check out this great resource.

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.

From a physical perspective, it’s also often used to model bosonic atoms arranged in an optical lattice.

Bosonic atoms in an optical lattice, source

To pull it apart, let’s analyze the Bose-Hubbard Hamiltonian.

Bose-Hubbard Hamiltonian (with chemical potential term)

It essentially describes the same dynamics as the Hubbard Hamiltonian but within a bosonic context. So, b^† and b are bosonic creation and annihilation operators, respectively. Similar to our fermionic case, the number operator defined as

tells us the number of bosons at a given site i.

Despite the on-site interaction term looking a bit different, it serves the exact same purpose as the Hubbard Hamiltonian in discouraging the double-occupation of a site by bosons. The Bose-Hubbard model, akin to its’ parent Hamiltonian, describes the transition from a Mott insulating state to a superfluid state by varying the power dynamics between t and U. A very large U relative to t results in a Mott insulating state and the bosons in the lattice are generally discouraged from tunnelling around. This is again due to the energetic tax of a would-be double-occupation. Conversely, a very large t relative to U induces a superfluid state in which bosons tunnel without an ounce of care! It seems that the Bose-Hubbard model is a completely reasonable extension of the Hubbard model, pulling many of the same concepts.

But what’s the point of the third term? Where did that come from? It turns out that the third term, characterizing the total chemical potential energy, doesn’t affect the dynamics of the system (in other words, not important). In fact, you could even include so in our prior Hubbard Hamiltonian, and upon simulating the lattice, you’d observe that the system evolves invariant to it. In essence, it accumulates µ amount of chemical potential energy for each boson in the lattice, and since the number of total bosons in the lattice remain constant over time (all bosons are contained), so too is the term’s contribution meaning it has no effect on the dynamics of the system.

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.

If we think about this intuitively, considering such large U in a system just means that no two bosons will ever occupy the same site, it just couldn’t afford the near-infinite energetic tax. That sounds like the bosonic version of the Pauli Exclusion Principle, which states that no two identical fermions can occupy the same site!

We call this the hard-core constraint on bosons, and such a Bose-Hubbard Hamiltonian with very large U/t is known as a hard-core boson model. Suitably named since it accurately depicts the behaviour of hard-core bosons in a lattice structure.

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.

For simplicity, let’s consider a 1D chain of 7 sites. If you’d like to retain some physical connection, think of a slice of the following optical lattice with 7 sites along a straight line:

Source

In the interest of brevity, we’ll do just two simulations that will hopefully shed a clear picture of the differences in our system’s evolution. I encourage you to walk through my code on GitHub used to produce these yourself and gain intuition by manipulating the Hamiltonian. Recall, the construction of the Bose-Hubbard Hamiltonian

Our chain will initially contain two bosons at each end, and with a constant system size of 7, we’re interested in investigating how the probability for both bosons to occupy the same site increases or decreases as we change U/t.

On a computational note, we could simulate this Hamiltonian using a quantum computer, but that would require us to apply the Suzuki-Trotter decomposition and consider many other burdens which we could circumvent by simply evolving the system using exact diagonalization. It’s a technique to calculate the exact matrix exponential (unitary) by diagonalizing the Hamiltonian, but only feasible with a low system size. Hey, I did say we’d steer clear of QC!

Alas, the plots below show the results of our two simulations. Each horizontal slice of time corresponds to the probabilities of finding either zero, one, or two bosons at each site.

We can observe the hard-core constraint in effect as the probability of two bosons occupying the same site vanishes as we increase U/t from 0 to 10. To further this point, I ran several more simulations at different values of U/t and we can clearly observe that all sites — especially the center site — rapidly decline in the probability of containing 2 bosons at any period as you increase U/t linearly.

Our theory matches simulation! If you felt like these plots came out of nowhere, do read the code in the repo and experiment around with it yourself.

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!

Let’s condense our core findings. The Hubbard Hamiltonian is a rather simple generalization to describe the physics of fermions (later extended to bosons) in a lattice, giving us a complete description behind the transition between Mott insulators and other conductors. Scoping that down from general particles to bosons gives rise to the Bose-Hubbard model, which inherits many of its properties from its’ parent. Further increasing U→∞ in the Bose-Hubbard Hamiltonian brings you to the hard-core constraint which can be thought of as the bosonic analogue of the Pauli Exclusion Principle.

Thank you Soham Pal, Kent Ueno, and Alexandre Cooper-Roy for such helpful discussions, couldn’t have compiled it all without your insights!

If you enjoyed the article, learned something new, or found a mistake, feel free to reach me at pavanjay.com.

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