New Method to Study Energy Levels in Particles

This article talks about a new way to study Energy Levels in a particle system using a method called Tensor Renormalization Group combined with a Transfer Matrix approach. This new scheme helps us find energy levels and other important information about the particles and their behavior in a structured way.

#The Basics of the Method

The new method starts by changing the arrangement of the tensor network related to a specific particle model. This process is known as coarse-graining. After this, we create a matrix called a transfer matrix from the modified tensors. The eigenvalues of this transfer matrix help us find the energy levels we are interested in.

Next, we can determine the properties of these energy states. Each energy state has specific Quantum Numbers, which we can find out using a rule based on symmetry. This means that certain conditions must be met for these quantum numbers to appear. By using the right tensor network representations, we can calculate the properties we want.

Moreover, we can also work out the wave function of each energy state. By examining how this wave function changes in different positions, we can gain insights into the momentum of the states.

#Application to the D Ising Model

To display how this new method works, we apply it to a system known as the d Ising model. This model is a standard example used in physics to understand phase transitions and critical phenomena. After applying our spectroscopy method to this model, we compare our findings with known exact results to see how accurate our new approach is.

#The Challenges in Studying Quantum Systems

Studying energy levels and properties in quantum systems is vital for better understanding what happens at the smallest scales in nature. In certain fields like quantum chromodynamics, researchers rely on methods such as Monte Carlo simulations to find energy levels. However, this method can encounter problems. For example, it often needs a long period of time during calculations to accurately capture the lowest energy levels. Moreover, to uncover information about excited levels, a lot of statistical data is required to balance out noise.

Given these difficulties, researchers are looking for alternative methods that can provide better results without these issues. One promising option is the tensor network method, which can be organized into two categories: Hamiltonian and Lagrangian formalisms.

#Differences Between the Two Formulations

The Hamiltonian approach focuses on energy and how it changes, while the Lagrangian approach looks at the entire system and its dynamics over time. In the context of the d Ising model, our new approach falls within the Lagrangian framework, and we aim to enhance our understanding of the energy levels using this method.

#How Our Method Works

Our new spectroscopy scheme starts with the transfer matrix formalism. This approach has the advantage of not needing long time periods for calculations, unlike the Monte Carlo method. However, building the transfer matrix directly involves working with very large matrices, which is challenging.

To handle this, we apply the tensor renormalization group method. This method compresses the information from the large matrix, making it easier to manage. Several algorithms exist for this compression process, but we choose the Higher Order Tensor Renormalization Group (HOTRG) method due to its accuracy and capability to function in higher dimensions.

Once we compress the tensors, we can both find the energy levels of the system and identify the quantum numbers associated with each energy eigenstate. By applying the symmetry properties of the system, we can derive a selection rule that helps us figure out the quantum numbers.

#Extracting Quantum Numbers

The selection rule is connected to the matrix element of an interpolating operator, which is derived from the symmetry of the system. In our method, we need to compute these matrix elements effectively. This computation can be done using an impurity tensor network, a special type of tensor network that includes a specific factor.

After calculating the matrix elements, we can determine the quantum numbers for the eigenstates. For instance, we can use the ground state as a reference point. If the matrix element shows specific behavior, we can classify the quantum numbers of other states based on that information.

#Results for the d Ising Model

We apply our new method to the d Ising model and compare the results with known exact values. Our findings demonstrate that we can successfully determine energy levels and quantum numbers for multiple states. More specifically, we achieve accurate results for up to twenty energy levels in a disordered phase.

The lowest energy gap is particularly interesting. We find that the accuracy improves around critical points, which is in contrast to the free energy. However, as the size of the system increases and we move to higher energy states, errors tend to increase.

In terms of quantum number classification, we use a single spin field to work out the quantum numbers for the energy eigenstates. Using our method, we reach up to twenty accurate classifications based on matrix elements and the selection rule. However, the method struggles with higher excited states, making classification harder.

#Momentum Identification

Another essential aspect is identifying the momentum of each state. We find that the wave functions of different eigenstates can be analyzed in terms of their position. This information allows us to draw conclusions about their momentum.

To confirm our results, we also compute matrix elements related to momentum fields. By linking these calculations with the selection rules, we clarify the momentum identification process. Overall, the computed momentum values align well with expectations from physical principles.

#Checking Energy and Momentum Relationship

With the momentum identified, we can also study the relationship between energy and momentum. We find that energy levels for certain momentum values behave as expected. In our analysis, both the continuum and lattice versions of the dispersion relation closely match the observed values from simulations, particularly at higher momenta.

#Scattering Phase Shift

To understand interactions between particles, we also look at two-particle channels. By studying matrix elements for two-field operators, we ascertain that total momentum remains consistent across states. This analysis leads us to determine scattering phase shifts using results from earlier calculations.

#Conclusion and Future Directions

To sum up, our new spectroscopy scheme that employs tensor renormalization group methods alongside the transfer matrix approach shows promise for studying energy levels in quantum systems. The application of this method to the d Ising model demonstrates its effectiveness in determining energy levels, quantum numbers, and momentum classifications.

Looking ahead, we aim to apply this method to other quantum systems to broaden its applicability and gain further insights into quantum behaviors across various models. This development could lead to a deeper understanding of fundamental phenomena in physics and enhance our toolkit for studying complex particle systems.