Adiabatic continuity in a partially reduced twisted Eguchi-Kawai model with one adjoint Dirac fermion

This paper provides numerical evidence from a partially reduced twisted Eguchi-Kawai model that the confined phase of large-$N$ $SU(N)$ gauge theory with one adjoint Dirac fermion persists under spatial compactification with periodic boundary conditions, supporting an adiabatic continuity scenario between large and small circles, whereas antiperiodic boundary conditions induce a clear deconfinement transition.

The Gist

The Big Picture: Shrinking the Universe

Imagine you are trying to understand how a massive, complex city (the universe) behaves. Usually, to study a city, you need to look at a huge map with millions of streets. But what if you could shrink that entire city down to a single, tiny room, and still know exactly how traffic flows, how people interact, and how the economy works?

In physics, this is called Volume Independence. It's a theory suggesting that if you have enough "colors" (a property of particles called $N$, where $N$ is a huge number), the physics of a giant universe is exactly the same as the physics of a tiny, single-point universe.

However, there's a catch. When you shrink the universe, the "traffic lights" (symmetries) sometimes break, and the simulation stops working. The city collapses into chaos.

The Problem: The "Adiabatic Continuity" Mystery

Physicists want to know: Can we shrink the universe all the way down to a tiny circle without the laws of physics breaking?

This is called Adiabatic Continuity. Think of it like a rubber band. If you stretch a rubber band slowly, it stays a rubber band. If you shrink a rubber band slowly, it should still be a rubber band, just smaller. But what if, at a certain point, the rubber band suddenly snaps or turns into a different material?

The researchers wanted to see if the "rubber band" of the universe (specifically, a theory called QCD with a special type of particle called an adjoint fermion) stays intact as the universe gets smaller and smaller.

The Experiment: The "Twisted" Toy Model

To test this, the authors built a digital toy model called the Twisted Eguchi-Kawai (TEK) model.

• The Setup: Instead of a 4D universe, they built a model that is 3D space plus 1 tiny circle (like a very short tube).
• The Twist: To keep the "traffic lights" (symmetries) from breaking when the universe gets small, they added a "twist" to the rules. Imagine a hallway where every time you take a step, the floor rotates slightly. This rotation prevents the particles from lining up in a way that breaks the rules.
• The Ingredients: They added one special type of particle (an adjoint fermion) to the mix. In the real world, these particles act like "glue" that holds the symmetries together, preventing the universe from collapsing.

They tested two types of "twists":

1. The Symmetric Twist: A standard, balanced rotation.
2. The Modified Twist: A slightly different, more complex rotation pattern.

The Results: What Happened?

1. The "Hot" Test (Antiperiodic Boundary Conditions)

First, they simulated a "hot" universe (like a summer day). In this scenario, the particles behave differently.

• Result: As they shrank the circle, the system suddenly "snapped." The particles stopped holding hands, and the universe went from a confined state (particles stuck together) to a deconfined state (particles flying free).
• Analogy: This is like heating a pot of water. Eventually, it hits a boiling point and turns to steam. This was expected and proved their computer model was working correctly.

2. The "Cold" Test (Periodic Boundary Conditions)

This is the main event. They simulated a "cold" universe where the particles are forced to stay in sync.

• The Heavy Particle Test: When the particles were heavy (slow and sluggish), the universe broke. The "rubber band" snapped, and the symmetry was lost.
• The Light Particle Test (The Big Discovery): When the particles were light (fast and energetic), something amazing happened.
  • They shrank the circle from large to tiny.
  • The "rubber band" never snapped.
  • The particles stayed confined (stuck together) the whole time.
  • The "traffic lights" (symmetries) stayed green.

Conclusion: For light particles, the universe can be shrunk down to a tiny dot, and the physics remains smooth and continuous. There is no sudden break. This confirms the Adiabatic Continuity scenario.

The "Modified Twist" vs. The "Symmetric Twist"

The researchers found that the Modified Twist was much better at keeping the simulation stable.

• Symmetric Twist: Like a wobbly table. At the size they tested ($N=36$), it started to shake, and the symmetry broke in some places.
• Modified Twist: Like a solid, heavy table. It stayed perfectly stable. The "traffic lights" never turned red, even when the universe got very small.

Why Does This Matter? (The Anomaly Check)

You might wonder: "Is this allowed by the laws of physics?"
The paper checks this using something called Anomaly Constraints. Think of this as a cosmic rulebook.

• The rulebook says: "If you shrink the universe, you can't just make everything disappear or change the rules arbitrarily. The 'debt' of the universe (anomalies) must be paid."
• The researchers showed that their "smooth shrinking" scenario fits perfectly with this rulebook. The universe stays confined, but the internal "clocks" of the particles shift in a way that satisfies the cosmic laws. It's like a magic trick that looks impossible but actually follows all the rules of physics.

Summary in a Nutshell

Imagine you have a giant, complex machine. You want to know if you can shrink it down to the size of a marble without it breaking.

• The authors built a tiny, twisted version of this machine.
• They found that if the machine's parts are light and fast, you can shrink the machine all the way down, and it keeps working perfectly.
• If the parts are heavy, it breaks.
• This proves that for certain types of matter, the universe is "adiabatically continuous"—meaning the physics of a huge universe and a tiny universe are smoothly connected, with no sudden explosions or breaks in the middle.

This gives physicists a powerful new tool: they can study the complex physics of the whole universe by simulating it on a tiny, manageable computer model, knowing that the results will be accurate.

Technical Summary

1. Problem Statement

The paper addresses the adiabatic continuity conjecture in large-$N$ $SU(N)$ gauge theories with adjoint fermions. Specifically, it investigates whether the confined, center-symmetric phase of $N_f=1$ adjoint QCD on $\mathbb{R}^3 \times S^1$ persists smoothly as the circumference of the compact circle ($L_4$) is reduced from large to small values, without undergoing a phase transition (deconfinement).

• Context: In large-$N$ gauge theories, Volume Independence (Eguchi-Kawai reduction) implies that physics on a large volume is equivalent to that on a single-site (or reduced) lattice, provided the center symmetry remains unbroken.
• The Challenge: While pure Yang-Mills theory undergoes a deconfinement transition at small $L_4$, theories with adjoint fermions (which carry no center charge) are expected to stabilize the center symmetry, potentially allowing the confined phase to exist at arbitrarily small $L_4$.
• Specific Gap: Previous studies focused on $N_f=2$ adjoint fermions, which risk entering a conformal window in the massless limit, obscuring confinement dynamics. This paper focuses on $N_f=1$, a theory that is asymptotically free and expected to remain confining even in the massless limit, making it an ideal candidate to test adiabatic continuity.

2. Methodology

The authors employ a Partially Reduced Twisted Eguchi-Kawai (TEK) model to simulate the theory efficiently.

• Model Setup:
  • Lattice: A $1^3 \times L_4$ lattice, where three spatial directions are reduced to a single site, and the fourth direction ($S^1$) has $L_4$ sites.
  • Parameters: $N=36$ (color number), $L_4=2$ (circumference), coupling $b = 1/(g^2N) \in [0.30, 0.46]$, and hopping parameter $\kappa \in [0.03, 0.16]$ (controlling fermion mass).
  • Fermions: One flavor of adjoint Wilson-Dirac fermion.
  • Boundary Conditions:
    • Periodic: To preserve center symmetry and test adiabatic continuity.
    • Antiperiodic: To simulate thermal deconfinement as a benchmark.
• Twist Configurations: Two types of twists are compared to ensure robustness:
  1. Symmetric Twist: The standard twist used in previous literature.
  2. Modified Twist: A specific twist configuration (defined by parameters $l=3$) proposed to better preserve volume independence at finite $N$.
• Observables:
  • Polyakov Loop ($P$): The order parameter for deconfinement along the $S^1$ direction. $P \approx 0$ indicates confinement (center symmetry unbroken).
  • Volume Independence Order Parameters ($W$ and $W_{123}$): Wilson loops in the reduced directions. $W=0$ and $W_{123}=0$ indicate that the $(Z_N)^3$ center symmetry in the reduced directions is unbroken, validating the equivalence between the reduced model and the full large-$N$ theory.
• Simulation Technique: Rational Hybrid Monte Carlo (RHMC) algorithm with multi-shift Conjugate Gradient solvers.

3. Key Contributions

1. First Numerical Study of $N_f=1$ Adiabatic Continuity: The paper provides the first lattice evidence for adiabatic continuity in the $N_f=1$ adjoint QCD case, avoiding the conformal window issues associated with $N_f=2$.
2. Validation of the Modified Twist: The study demonstrates that the Modified Twist is superior to the Symmetric Twist for maintaining volume independence at $N=36$. The Symmetric Twist showed signs of residual symmetry breaking in the composite Wilson loop $W_{123}$, whereas the Modified Twist maintained zero values for all order parameters.
3. Anomaly Consistency Check: The authors provide a theoretical consistency check using 't Hooft anomalies. They argue that the observed adiabatic continuity is compatible with the mixed center-chiral anomaly of the underlying 4D theory, provided the discrete chiral symmetry is broken in a specific pattern ($Z_{4N} \to Z_4$) while the center symmetry remains intact.

4. Key Results

A. Volume Independence (Validity of the Reduced Model)

• Symmetric Twist: The order parameter $W$ was near zero, but the composite loop $W_{123}$ showed non-zero values, particularly for heavy fermions ($\kappa=0.03$). This suggests that volume independence is not robust at $N=36$ for this twist.
• Modified Twist: Both $W$ and $W_{123}$ remained consistent with zero across the entire parameter range (both heavy and light fermions). This confirms that the Modified Twist successfully preserves $(Z_N)^3$ center symmetry, validating the use of the reduced model for this study.

B. Adiabatic Continuity (Periodic Boundary Conditions)

• Heavy Fermions ($\kappa=0.03$): The Polyakov loop $P$ exhibited a sharp rise around $b \approx 0.36$, indicating a deconfinement phase transition. This mimics pure Yang-Mills behavior where the center symmetry breaks at small circle sizes.
• Light Fermions ($\kappa=0.16$): The Polyakov loop $P$ remained near zero throughout the entire range of $b$ (from strong to weak coupling).
  • Conclusion: There is no phase transition as the circle size is reduced. The confined phase at large $L_4$ is smoothly connected to the confined phase at small $L_4$. This is the numerical evidence for adiabatic continuity.

C. Thermal Benchmark (Antiperiodic Boundary Conditions)

• With antiperiodic boundary conditions, the system exhibited a clear deconfinement transition for both twists, consistent with standard finite-temperature expectations. This confirms the simulation setup correctly reproduces known physics.

D. Comparison with $N_f=2$

• Unlike the $N_f=2$ case (where the Polyakov loop fluctuated, suggesting partial breaking of center symmetry), the $N_f=1$ case showed a clean transition from a fully symmetric phase to a fully broken phase (or no transition in the light regime). This is attributed to the weaker center-stabilizing force of a single adjoint flavor compared to two.

5. Significance

• Theoretical Validation: The results strongly support the conjecture that $N_f=1$ adjoint QCD on $\mathbb{R}^3 \times S^1$ exhibits adiabatic continuity. This implies that non-perturbative physics (confinement) at small scales can be understood via semi-classical methods (like bion condensation) that are valid at weak coupling, bridging the gap between weak and strong coupling regimes.
• Methodological Advancement: The paper establishes the Modified Twist as a reliable tool for partially reduced TEK simulations at moderate $N$, overcoming limitations of the symmetric twist.
• Anomaly Matching: The work demonstrates that the adiabatic continuity scenario is not in conflict with the 't Hooft anomaly constraints of the 4D theory. The anomaly requires a specific pattern of symmetry breaking (chiral symmetry breaking) which is compatible with a center-symmetric (confined) vacuum at small $L_4$.
• Future Directions: The authors suggest extending these simulations to larger $N$ and approaching the continuum limit ($b \to \infty$) to solidify the results, as well as exploring the resurgent structure of the theory (connection between perturbative and non-perturbative sectors) using this framework.

In summary, the paper provides robust numerical evidence that for $N_f=1$ adjoint QCD, the confined phase is stable against spatial compactification, provided the fermions are light and the correct twist (Modified Twist) is used to maintain volume independence.