Quantum Transport on the Bethe Lattice

Quantum transport is a study of how energy or particles move through different systems. One interesting type of system is the Bethe Lattice, which resembles a tree structure. In this article, we will look at a model that represents quantum transport on this type of lattice, especially when there are energy sources and drains involved.

#The Bethe Lattice

The Bethe lattice, also known as a Cayley tree, is a structure where each node connects to a fixed number of other nodes. This creates a branching effect. In the context of energy transport, it can be used to model how energy flows in molecules that capture light, like those found in plants.

In our model, we focus on a specific kind of Bethe lattice that has a limited number of generations, or layers, which helps us simplify our analysis.

#Adding Complex Potentials

To better understand transport in this model, we add complex potentials to represent sources and drains. The sources are placed on the outer edges of the lattice while the drain is located at the center. These complex potentials affect how energy flows through the structure.

When we analyze this model, we find that not all energy states can move freely from the outer edges to the central drain. In fact, only a few states are able to do so. Most of them remain Localized around the outer edges and do not reach the center.

#Eigenstates and Current

The states that can reach the central drain carry current, which reduces the overall analysis to simpler terms. When the connections between nodes are uniform across generations, the current reaches its peak value at a specific point where two energy states combine into a zero-energy state. This phenomenon occurs because of the added complex potentials.

However, when we introduce randomness to the connections, the maximum current does not occur at this exceptional point; it just occurs close to it.

#Understanding Transport in Non-Equilibrium Physics

Quantum transport is crucial in non-equilibrium physics, which deals with systems that are not in a stable state. One important tool in analyzing transport is the Landauer formula, which helps describe conductance in terms of how energy scatters through structures.

Our study is inspired by previous research in complex systems and networks, particularly tree-like networks. We aim to analyze how energy moves from sources to drains in our lattice model.

#Effective Potentials and Markov Dynamics

When we look at how energy behaves in this system, we notice that the effective potential that results from adding sources and drains changes based on the energy of the sources. This behavior can cause the dynamics to be non-Markovian, meaning they do not follow a simple memoryless process.

To simplify the model further, we can make an approximation that leads to a constant effective potential. This makes the problem more manageable and allows us to conduct our analysis more easily.

#Localized Eigenstates

We now focus on localized states that exist within our model without accessing the central drain. These states occur when we examine certain sites in the outer edges of the lattice.

When analyzing one branch of the lattice, particular eigenstates can be identified as localized. They do not penetrate towards the drain because they interfere destructively with each other.

The amplitudes of these localized states grow over time, indicating that energy is accumulating in the periphery rather than reaching the drain.

#Extended Eigenstates

On the other hand, we also identify extended eigenstates that can reach the central drain. We build these eigenstates by ensuring they can carry current from the outer sites to the center.

As we explore these extended states, we see that only a limited number can exist, showing that energy transport is not uniform across all states.

#The Role of Non-Hermiticity

The model we explore includes non-Hermitian aspects, which means that the Hamiltonian, a mathematical description of the system, does not possess standard symmetries. This non-Hermiticity arises due to the introduced complex potentials.

The presence of non-Hermitian features leads to particular eigenvalue behaviors, including potential coalescence. The eigenvalues associated with these states can take on complex values.

#Quantum Current and Its Calculation

As we evaluate the Currents carried by these eigenstates, we look at how the current expectation values change based on the model's parameters. The currents should ideally flow from the sources to the drain, but their behavior can differ with varying conditions.

When the model is uniformly structured, the highest current typically corresponds to the zero-energy eigenstates. But as we introduce randomness into the structure, this relationship becomes less definitive.

#Randomness in the Model

When we allow the number of connections to vary randomly within the lattice, we create a new scenario where the eigenvalues start to behave differently. The random distribution means that energy transport can vary significantly.

In systems where randomness is introduced, even though most states remain localized, we find that certain zero-energy states can emerge and allow for extended transport.

#Conclusion

In summary, our analysis of quantum transport on the Bethe lattice highlights the complexity and richness of energy movement in such systems. By introducing sources, drains, complex potentials, and randomness, we can explore how quantum states behave and how they impact the flow of energy.

This research sheds light on the dynamics of quantum systems, offering insight into how energy moves through intricate networks. As we continue to explore these aspects, we gain a deeper appreciation for the fundamental rules governing transport phenomena.