Measuring Complexity in Non-Hermitian Systems

In the world of physics, we often deal with complicated systems and their behavior, especially when many particles interact with each other. One such behavior we are interested in is called Many-body Localization (MBL). This is when particles get stuck in particular locations due to Disorder, rather than spreading out like jelly on a piece of toast.

There are two flavors of how we can think about these systems: Hermitian and non-Hermitian. You can think of Hermitian as the well-behaved cousin who always follows the rules. Non-Hermitian, on the other hand, is a bit more chaotic and doesn't always play by the same set of rules. This makes things interesting, yet slightly frustrating-like trying to get a cat into a bath.

In this write-up, we will explore how we can measure something called "Spread Complexity" in non-Hermitian systems, especially during the transition of many-body localization.

#What is Spread Complexity?

Spread complexity is a fancy term that refers to how complicated or "spread out" a state can become when particles interact. Imagine trying to tidy up a room filled with toys: if everything is scattered around, it looks chaotic. But if the toys are neatly put away, the room looks organized. Spread complexity helps us gauge just how organized or disorganized our particle system is.

So, how do we measure this spread complexity? We use mathematical tools that help us analyze the behavior of these systems, and it turns out that we can learn a lot by looking at certain numbers that represent the states of our system.

#Many-Body Localization (MBL) and Non-Hermitian Systems

Now let’s get into the nitty-gritty. In systems that show many-body localization, the presence of disorder-like having furniture scattered all over the room-prevents particles from moving around freely. Instead of the particles behaving like they are at a wild party, celebrating their freedom, they become like guests stuck in a corner, unable to mingle.

When we look at non-Hermitian models, things are a bit different. These systems can have particles that not only hop around but also gain or lose their "energy" (think of it like them losing their energy drinks at a party).

#Time-reversal Symmetry

Now, we also have a concept called time-reversal symmetry (TRS). It is a bit like if you could rewind a movie and everything would go back to how it was before. In models with TRS, if we run the system backward in time, we find that everything looks pretty much the same. However, in systems without TRS, the behavior can shift dramatically, like changing the plot of a movie halfway through.

#The Role of Disorder

Disorder in our systems acts as a kind of guest list gone wrong. Instead of orderly behavior, guests are left scrambling about, and this can lead to complex transitions when we look at how states evolve over time. As we crank up the disorder, we can observe transitions that help us separate chaotic behavior from well-behaved states.

#The Models We Use

We focus on two types of models to study these behaviors.

1. The first model is a disordered system that allows particles to hop while respecting time-reversal symmetry. It’s like a party where everyone can move around, but they still mostly follow the house rules.

2. The other model lacks that symmetry, which means the disorder allows for a bit more chaos-much like a party where people spill drinks and bump into each other.

#Singularity and Spreading

In our investigation, we introduce the notion of singular value spread complexity. This is a tool that helps us look at the singular values that tell us how chaotic things are. If we see a distinct peak in these numbers, it indicates that our system is behaving chaotically and is all over the place-like a party that has just hit the dance floor.

As the disorder increases, this peak tends to shrink or disappear, indicating a transition point where order replaces chaos.

#Thermofield Double States

We also examine something called thermofield double (TFD) states, which are idealized representations of systems in thermal equilibrium. These states act as our ideal party guests who know how to keep things together, and they're vital for analyzing spread complexity dynamics.

#Observing Changes

Through our analysis, we observed that particle behavior changes based on initial conditions and how disorder affects them. If we start with a nicely arranged state, the dynamics will be different compared to starting with a chaotic arrangement.

Picture it like starting a game of Jenga. If you start with a steady base, it’s easier to keep playing without any collapses. But if it’s all wobbly from the start, good luck keeping it together!

#Comparing Boundary Conditions

Next, we looked at boundary conditions, which is reminiscent of how crowds behave in open spaces versus confined rooms. When we compare our models under periodic boundary conditions (like a wrap-around party where you can leave through one door and come back through another) versus open boundary conditions (like a party where guests can only enter or leave through one door), we see fascinating differences in behavior.

In systems with TRS, the dynamics remain quite organized even under different boundary conditions, while the non-TRS models showcase more wild behavior, presenting unique challenges and a lot of surprises.

#Conclusions and Future Directions

In summary, we find that measuring spread complexity in non-Hermitian systems provides essential insights into the transitions between chaotic and orderly behavior. It acts as a key tool, helping us differentiate between various phases in our particle systems.

While we have unraveled much about these systems, we know there’s still more to explore. Just like every party has its surprises, so does the world of physics with countless questions still waiting to be answered. There’s a rich landscape of research ahead of us!

So, while we may not have found all the answers yet, we remain excited about unraveling new mysteries and understanding the complex dance of particles in the wild world of quantum mechanics. If only we could teach them to party a little better!