Asymptotic freedom, lost: Complex conformal field theory in the two-dimensional $O(N>2)$ nonlinear sigma model and its realization in Heisenberg spin chains

This paper demonstrates that the two-dimensional $O(N>2)$ nonlinear sigma model possesses a generic complex conformal field theory fixed point in the complex coupling plane, which is numerically confirmed in non-Hermitian Heisenberg spin chains and can be realized through engineered dissipation to prepare long-range entangled states.

The Gist

The Big Picture: Breaking the Rules of Physics

Imagine you are playing a game of "magnetic marbles" on a flat, two-dimensional table. In the world of standard physics (specifically for $N > 2$), there is a famous rule called Asymptotic Freedom.

Think of this rule like a strict referee who says: "No matter how you try to arrange these marbles, they will never settle into a perfect, ordered pattern. They will always be chaotic, and if you try to zoom in to see the details, the chaos just gets stronger and stronger."

For decades, physicists believed this rule was unbreakable for these specific types of magnetic systems. They thought there was no "middle ground" or "sweet spot" where the system could be perfectly balanced and critical (on the edge of order and chaos).

This paper says: "Not so fast."

The authors discovered that if you change the rules of the game slightly—by allowing the "coupling" (the strength of the interaction between marbles) to become a complex number (a mix of real and imaginary numbers)—the referee's rule breaks. Suddenly, a beautiful, stable, and exotic "sweet spot" appears.

The Key Concepts, Explained with Analogies

1. The "Complex" Twist

In normal math, numbers are like points on a line (1, 2, 3...). In complex numbers, you have a second dimension (imaginary numbers, like $i$).

• The Analogy: Imagine you are driving a car on a straight road (real physics). You can only go forward or backward. The paper suggests that if you allow the car to drive on a parallel highway in a different dimension (the complex plane), you can find a hidden parking spot (a fixed point) that doesn't exist on the main road.
• The Result: In this complex world, the system finds a "Complex Conformal Field Theory" (CCFT). This is a state of perfect, scale-invariant balance, but it lives in this imaginary dimension.

2. The Spiral Dance (Renormalization Group Flow)

Usually, when you zoom in on these magnetic systems, the interactions get stronger and stronger until they break (this is "flowing to strong coupling").

• The Analogy: Imagine a dancer spinning on a stage. In the old world, the dancer would spin faster and faster until they flew off the stage into chaos.
• The New World: In this complex world, the dancer enters a spiral. They spin around a specific point (the CCFT) in a beautiful, outward spiral. They don't fly off immediately; they hover in a complex, swirling dance around this new "fixed point."

3. The "Watermelon" Operators

The paper talks about "watermelon operators." This sounds weird, but it's just a name physicists give to specific patterns of connections between particles.

• The Analogy: Imagine a watermelon cut into slices. If you have 4 slices, they all meet at the center. In the quantum world, these "slices" are strands of magnetic influence meeting at a point.
• The Discovery: The authors found that in this complex world, these watermelon patterns behave in a very specific, predictable way. They act like a "fingerprint" that proves the system has found this new exotic state.

4. The Spin-1 Chain: The Real-World Lab

The authors didn't just do math; they built a model using Spin-1 Heisenberg Chains.

• The Analogy: Think of a chain of beads, where each bead is a tiny magnet that can point in three directions (Up, Down, or Sideways).
• The Experiment: They took a computer simulation of this chain and tweaked the knobs (the coupling constants) until they turned them into complex numbers.
• The Result: The chain stopped being chaotic. Instead, it settled into a state where the magnets were entangled over long distances, behaving exactly like the "Complex Conformal Field Theory" they predicted.

5. The "No-Click" Magic (Dissipation)

This is the most exciting part for the future. How do we actually make this happen in a real lab?

• The Analogy: Imagine you are watching a movie, but every time a character makes a mistake (a "click"), you rewind the tape. You only keep the scenes where nothing goes wrong (the "no-click" trajectories).
• The Physics: In quantum mechanics, if you constantly monitor a system and only keep the results where no "error" occurs, the system is forced to evolve under a special "non-Hermitian" rule.
• The Payoff: The authors showed that if you do this "no-click" monitoring on a spin chain, the system naturally relaxes into this exotic, long-range entangled state. It's like engineering the chaos to create a perfect, entangled crystal.

Why Does This Matter?

1. New Physics: It proves that "Asymptotic Freedom" isn't an absolute law; it's just a rule for the "real" world. In the "complex" world, new universality classes (new types of physics) exist.
2. Entanglement: These states are highly entangled. Entanglement is the fuel for quantum computers. This paper suggests a new way to create these states using dissipation (letting energy leak out in a controlled way) rather than fighting against it.
3. The "Walking" Behavior: In the real world, some materials seem to be "stuck" on the edge of a phase transition (they "walk" but don't decide). This paper suggests that these materials might actually be trying to reach this complex fixed point but are being pulled back by the fact that they are stuck on the "real" axis.

Summary

The authors found a hidden "parallel universe" of physics where the coupling constants are complex numbers. In this universe, the chaotic magnetic systems we thought were doomed to disorder actually find a stable, beautiful, and exotic state of balance. They proved this exists in computer models of spin chains and showed that we might be able to create these states in the lab by carefully watching the system and only keeping the "perfect" outcomes.

It's like discovering that a messy room, if viewed through a special complex-colored lens, is actually a perfectly organized masterpiece.

Technical Summary

1. Problem Statement

The two-dimensional $O(N)$ nonlinear sigma model (NLSM) is a cornerstone of quantum field theory. For $N > 2$, the model is famously asymptotically free: the coupling constant flows to strong coupling in the infrared (IR), resulting in a mass gap, the absence of spontaneous symmetry breaking, and no non-trivial fixed point at finite coupling. This behavior is consistent with the Mermin-Wagner theorem, which forbids continuous symmetry breaking in 2D.

The authors investigate whether this conclusion holds in non-Hermitian settings, specifically where the coupling constant $g$ is allowed to be complex ($g \in \mathbb{C}$). Such scenarios arise in monitored quantum dynamics and open quantum systems. The central questions are:

1. Do non-trivial fixed points exist in the complex coupling plane for $N > 2$?
2. Are these fixed points generic (i.e., accessible by tuning a single parameter)?
3. Can they be realized in experimentally relevant microscopic models (e.g., spin chains)?

2. Methodology

The authors employ a multi-faceted approach combining field theory, statistical mechanics, and numerical simulation:

• Field Theory & Analytic Continuation:

  • They analyze the $O(N)$ NLSM with a complex coupling $g$.
  • They utilize the connection between the NLSM and $O(N)$ loop models. In loop models, two real fixed points (dense and dilute phases) annihilate at $N=2$ and move into the complex plane for $N>2$, forming a pair of complex conjugate fixed points.
  • They argue that for $N>2$, "loop crossings" (which distinguish the NLSM from the loop model) correspond to an irrelevant operator (specifically the $\ell=4$ watermelon operator). Therefore, the complex fixed points of the loop model should describe the generic criticality of the non-Hermitian NLSM.

• Numerical Simulation (Exact Diagonalization & DMRG):

  • Model: They study non-Hermitian spin-1 Heisenberg chains and spin-1/2 ladders. The Hamiltonian includes nearest-neighbor ($J_1$), next-nearest-neighbor ($J_2$), and biquadratic ($K$) terms, with $J_2$ and $K$ allowed to be complex.
  • Optimization: To overcome slow finite-size convergence caused by a weakly irrelevant operator, they use a gradient descent algorithm to tune two parameters ($J_2, K$) to minimize the deviation of scaling dimensions from theoretical predictions, effectively locating the finite-size critical point.
  • Spectrum Analysis: They map the low-energy spectrum of the non-Hermitian Hamiltonian to Conformal Field Theory (CFT) operators using the operator-state correspondence. They extract the complex central charge ($c$) and scaling dimensions ($\Delta$).
  • Entanglement: They calculate the generalized entanglement entropy using biorthogonal ground states to verify the logarithmic scaling characteristic of CFTs and extract the complex central charge.

• Open Quantum Dynamics:

  • They construct a Lindblad master equation where the "no-click" trajectories (post-selected dynamics) are governed by the non-Hermitian Hamiltonian. This establishes a physical protocol for preparing the critical state.

3. Key Contributions

A. Discovery of Complex Conformal Field Theories (CCFTs)

The paper demonstrates that the $O(N)$ NLSM for $N > 2$ possesses a pair of complex conjugate fixed points in the complex coupling plane. These are described by Complex Conformal Field Theories (CCFTs) with:

• A complex central charge $c(N) \in \mathbb{C}$. For $N=3$, $c \approx 1.51 \pm 0.158i$.
• Complex scaling dimensions for operators.
• A spiral Renormalization Group (RG) flow in the complex plane, where trajectories spiral out from the fixed point before eventually flowing to strong coupling. This implies that asymptotic freedom is "lost" in the sense that a large portion of the phase diagram flows to a non-trivial CCFT rather than the trivial Gaussian fixed point.

B. Proof of Genericity

The authors prove that these CCFTs are generic for any non-Hermitian system with $O(N)$ symmetry ($N>2$).

• The fixed point has only one relevant singlet operator (the energy operator $\epsilon$).
• Consequently, the CCFT can be reached by tuning a single complex parameter in the Hamiltonian.
• The "loop crossing" operator, which is relevant in the dense phase for $N<2$, becomes irrelevant for $N>2$, ensuring the universality class of the NLSM matches that of the $O(N)$ loop model.

C. Microscopic Realization

The paper provides the first concrete microscopic realization of these CCFTs:

• Spin-1 Heisenberg Chain: They locate the CCFT at specific complex values of couplings $(J_2, K) \approx (0.0660 \pm 0.338i, 0.176 \pm 0.335i)$.
• Spin-1/2 Ladder: They identify lines of CCFT points in a spin-1/2 ladder model that possesses an exact non-invertible symmetry, which forbids loop crossings and allows for precise numerical identification of the fixed point.

D. Dissipative State Preparation

A significant contribution is the proposal of a protocol to prepare these highly entangled critical states:

• By engineering a Lindbladian with specific jump operators, the system's "no-click" dynamics are governed by the non-Hermitian Hamiltonian.
• The CCFT vacuum is the longest-lived state (lowest decay rate $\gamma$) and the ground state (lowest real energy).
• Therefore, under dissipative dynamics, the system naturally relaxes into the CCFT vacuum, providing a route to prepare long-range entangled critical states via engineered dissipation.

4. Key Results

• Spectral Agreement: The numerically extracted scaling dimensions for the spin-1 chain at the critical point show excellent agreement with analytic predictions derived from the $O(N)$ loop model (e.g., $\Delta_\epsilon \approx 1.67 - 1.1i$ vs. predicted $1.66 - 1.12i$).
• Central Charge: The calculated complex central charge for the spin-1 chain is $c = 1.529 - 0.161i$, matching the theoretical prediction $c_+ = 1.51 - 0.158i$ within 2%.
• Operator Identification: The spectrum correctly identifies:
  • The energy operator ($\epsilon$, singlet).
  • The stress tensor ($T$, spin 0).
  • The $O(N)$ current ($J$, spin 1).
  • Watermelon operators ($\ell$-leg operators), appearing as clusters of nearly degenerate states with specific spin multiplicities, confirming the presence of the emergent non-invertible symmetry.
• RG Flow: The flow in the complex $g$-plane forms a spiral, confirming the loss of asymptotic freedom in the non-Hermitian regime.

5. Significance

• New Universality Class: The work establishes CCFTs as a new universality class for criticality in dissipative quantum systems, distinct from unitary CFTs and non-unitary CFTs with real negative central charges.
• Resolution of "Walking" Behavior: It provides a theoretical explanation for "walking" behavior (quasi-fixed points) and conformality loss observed in various condensed matter and high-energy systems, attributing them to the influence of complex fixed points.
• Experimental Relevance: By linking the theory to spin-1 chains and spin-1/2 ladders, and proposing a Lindblad protocol, the paper bridges the gap between abstract complex field theory and potential experimental realization in quantum simulators (e.g., cold atoms, trapped ions) using continuous monitoring and post-selection.
• Theoretical Consistency: It challenges the standard perturbative understanding of the NLSM, suggesting that perturbative beta functions must admit zeros in the complex plane, offering a new consistency check for field theory expansions.

In summary, the paper fundamentally alters the understanding of the 2D $O(N)$ NLSM for $N>2$ by revealing a hidden landscape of complex critical points, proving their genericity, and demonstrating a practical pathway to realize and study them in open quantum systems.