NonLinear Sigma Model with Topological Term

Background and Methodology

Research Question

Can complex Langevin improve upon calculations of the mass gap for the $2D$ $\mathcal{O}(3)$ nonlinear sigma model with topological term $\theta \approx \pi$ ? We first attempt to recreate a method that calculates the critical exponents for the mass gap using standard Markov-Chain Monte Carlo (MCMC) with $\nu = i \theta$ and analytically continues to real $\theta$ .

Then, we perform the same calculation using complex Langevin to circumvent the sign problem created by a nonzero $\theta$ .

Lattice Action

In the continuum, the model has action

$S = \frac{1}{2g} \int d^{2} x \left( \partial_{\mu} \vec{\phi}(x)\right)^{2} - i \theta \int d^{2} x Q(x)$

with

$Q(x) = \frac{1}{8 \pi} \epsilon^{\mu \nu} \epsilon_{abc} \partial_{\mu} \phi^{b}(x) \partial_{\nu}\phi^{c}(x)$

with $\phi$ a 3-component unit vector.

In lattice action form, we have (see Alles et al, 2014):

$S_{L} = A_{L} - i \theta Q_{L}$

$A_{L} = -\frac{1}{g_{L}}\sum_{x,\mu}\left(\phi_{x}\cdot \phi_{x+\hat{\mu}}\right)$

$Q_{L} = \sum_{x}\sum_{\Delta} Q_{L} \Delta$

where $\vec{\phi}$ is a 3-component unit vector ( $\vec{\phi} \cdot\vec{\phi} = 1$ ) and $Q_{L}$ is the total topological charge on the lattice, calculated by summing the charge over all the triangles created by dividing each square on the lattice into two triangles (details on this method can be found in the section on Regularizing QL).

We want to determine the mass gap as a function of $\theta$ . As $\theta$ approaches $\pi$ from zero, the mass gap should approach zero according to the following equation:

$m(\theta) \propto (\pi - \theta)^{\epsilon_{WZNW}} \log \left(\frac{1}{\pi - \theta} \right)^{-\beta_{WZNW}}$

From Allés et al, “a direct detection of the power of the logarithmic correction in [this equation] requires an extremely accurate control of the statistics and error bars, an endeavor that seems to lie beyond present-day capabilities.” That is, $\beta_{WZNW}$ is very sensitive to errors on the calculation of $m$ , therefore one of the goals of this project is to use CL to improve the error on our calculation of the mass gap in order to determine the exponent of the logarithmic correction, $\beta_{WZNW}$ .

Monte Carlo with Analytical Continuation

This approach uses a conventional Monte Carlo simulation with a Metropolis step to simulate the $2D$ $\mathcal{O}(3)$ nonlinear sigma model where the topological term $\theta$ is replaced with an imaginary value $\nu = i \theta$ and observables are calculated with this value. The results are then analytically continued, i.e. the imaginary number is replaced with a real-valued number in a fit to project the results. The error on analytic continuation grows with the size of $\theta$ .

If you decide to actually try the analytical continuation, add a separate page here detailing all the algebra.

Complex Langevin

The complex Langevin method allows us to keep the complex-valued action by extending the fields $\phi_{i}$ into the complex plane and evolving the real and imaginary parts in parallel.

Add here the algorithm for CL – how to calculate the drift function and evolve the fields, etc. Pull from your notes for this.

Summary of Preliminary Results

Observables

Topological Charge

The topological charge, $Q_{L}$ , is a stable defect in the fields. It takes on quantized values (it is not a continuous number), and it characterizes phases of matter. Examples of other topological charges include the number of vortices formed in rotating Bose-Einstein condensates (as the rotation increases in a continuous way, the number of vortices experiences a quantized jump from 0 to 1, and so on). The vortices themselves would be examples of solitons, or topological defects, and the number of them is the topological charge of the system.

The creation or annihilation of these topological defects are hallmarks of phase transitions in 2D systems.

Fill in with more detail.

Is it extensive or intensive?

How does it relate to the mass gap?

Regularizing QL.

Since the topological charge must be an integer, the fields on our lattice must be restricted to configurations that only produce integer-valued charges. The process for ensuring this is called regularization, and we follow a schema introduced by Berg and Luscher. This process is detailed here.

Preliminary Results

Preliminary simulation code is being run on small lattices ( $L = 10,40,80$ ) with the intent to scale to larger lattices in the next few months.

Early results look promising. For example, when comparing our results for the topological charge QL to those of Allés et al, we see a similar trend in the data:

We expect as the size of the lattice grows (to 180 x 180, as was used in the PRB paper), our $Q_L$ plot should match theirs.

Correlation Function and Correlation Length

The correlation function gives the statistical correlation between two lattice sites i and j. It’s a measure of the order in a system, and is useful for understanding how quantum fluctuations will affect sites on the lattice not immediately next to them. It is a field defined at all lattice sites, and is defined by the correlation between that site and the site $(i,j) = (0,0)$ .

The correlation length measures the average distance over which sites are correlated. A diverging correlation length is a mark of a system at or near a phase transition.

Add some plots of the correlation function and the correlation length as a function of itheta. Maybe also add the equations to calculate the correlation function/length?

The correlation length should be a real number, but our simulations are currently returning a complex result, due to the complexity of the correlation function. More investigation is required here.

Mass Gap

Add some content here about Haldane’s conjecture and the mass gap, how it’s defined, and how it relates to the correlation function

Our goal with this project is to determine the mass gap, which should vanish as $\theta \to \pi$ according to this equation:

$m(\theta) \propto (\pi - \theta)^{\epsilon_{WZNW}} \log \left(\frac{1}{\pi - \theta} \right)^{-\beta_{WZNW}}$

The mass gap is the inverse of the correlation length, which we compute in the simulation.

Here we provide a deeper dive into the mass gap and compare our results prior to analytical continuation to Alles et al

Magnetic Susceptibility

Add some content here - though this is not a high priority observable for us right now.

Simulation and Analysis Code

Simulation Code

Monte Carlo + Analytic Continuation

The first part of this project involves a standard Monte Carlo simulation at imaginary $\theta$ . We have thoroughly documented the MC Simulation Code here.

The simulation is run for imaginary values of $\theta$ , which means we must analytically continue our results for real $\theta$ . To do this, we fit our results to a curve and then substitute our imaginary $i \theta = \nu$ for a real $\theta = -i \nu$ . This work is in progress.

Complex Langevin

Another approach to systems with a sign problem is to use Complex Langevin. This has been demonstrated to be successful across disciplines and systems.

Cite at least 6 papers – condensed matter, HEP, QCD

The complex Langevin method, like all methods that tackle the sign problem, has challenges as well as benefits (see Berger et al, 2021

Cite more? E.g. “The complex langevin method: when can it be trusted?”

It can sometimes converge to mathematically correct but physically invalid solutions (see Guralnik and Pehlevan, 2008. It can sometimes diverge due to numerical instabilities, which can be controlled at the cost of computational time

Pick a few of the relevant citations here

his work is in progress, but the documentation for the CL simulation Code can be found here.

Systematics

Statistical simulations on a lattice are by definition approximate calculations. In order to account for the error introduced by scaling the system down to small numbers and discrete spacings, we have to account for the systematic effects due to these approximations.

We can do that by calculating our observables at multiple lattice lengths, Langevin step sizes, etc and extrapolating to the continuum limit.

[Here]((nonlinearsigma/Systematics.qmd) we conduct a thorough examination of the systematic effects.

Analysis

Analysis Pipeline

Current issues and open questions

Issues we are working on resolving:

The topological charge calculation ( $Q_{L}$ ) is currently returning non-integer values
The correlation function at the smallest nonzero lattice momentum $2\pi/L$ is a complex number, therefore so is our correlation length. Correlation length at imaginary $\theta$ should be real
Additionally, when this value is computed in C++, the results are significantly off from the calculation done in Python
Currently, the simulation returns the average value of the correlation function computed after thermalization. There is no way to change this after the simulation is run, and we don’t have the code set up yet to return any error on the correlation function. That may come in future versions of the code.

Research questions and next steps

What can a ML model learn from the configuration data we have so far?
How will Complex Langevin results differ from these if at all?