Symmetry-resolved Krylov Complexity and the Uncoloured Tensor Model

This paper investigates symmetry-resolved Krylov complexity in the Uncoloured Tensor Model, establishing conditions under which charge subspaces mirror the full-space complexity and demonstrating that the averaged complexity within symmetry subspaces is bounded by that of the full operator.

The Gist

The Big Picture: Tracking a Messy Room

Imagine you have a very complicated, messy room (this represents a Quantum System). You drop a single, simple object into the room, like a red ball (this is your Initial Operator).

Over time, the ball bounces around, hits things, and eventually, the whole room gets messy. The "red ball" has spread out, mixing with everything else. In physics, we call this Chaos.

The scientists in this paper are trying to answer two main questions:

1. How fast does the mess spread? (This is called Krylov Complexity).
2. Can we predict the mess by looking at just one corner of the room? (This is Symmetry-Resolved Complexity).

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Part 1: The "Krylov Chain" (Measuring the Mess)

To measure how chaotic the system is, the authors use a tool called Krylov Complexity.

The Analogy:
Imagine the "mess" spreading out on a long, straight chain of dominoes.

• At the start (Time = 0), the red ball is on the very first domino.
• As time passes, the ball jumps to the next domino, then the next.
• Krylov Complexity is simply asking: "On average, how far down the chain has the ball traveled?"

If the ball stays near the start, the system is calm (not chaotic). If the ball races to the end of the chain very quickly, the system is chaotic.

Part 2: The "Symmetry" Shortcut (The Magic Filter)

The paper focuses on systems that have Symmetries. In our room analogy, imagine the room is divided into separate, identical zones by invisible walls. For example, maybe the room has a "Left Side" and a "Right Side" that look exactly the same.

The authors ask: If we only look at the "Left Side" of the room, can we predict how the ball moves in the whole room?

Usually, the answer is no. The ball might bounce from the Left to the Right, and if you only watch the Left, you miss half the action.

The Discovery:
The authors found a special "Golden Rule." They discovered that if the red ball (the operator) is distributed in a very specific, balanced way across all the zones, then watching just one zone is exactly the same as watching the whole room.

• The Analogy: Imagine the ball is made of 100 tiny specks of dust. If 50 specks are in the Left Zone and 50 are in the Right Zone, and they move in perfect sync, you can just count the specks in the Left Zone and know exactly what the whole room is doing. This saves a massive amount of work!

Part 3: The "Uncoloured Tensor Model" (The Test Lab)

To test their "Golden Rule," the scientists used a specific, complex mathematical model called the Uncoloured Tensor Model.

The Analogy:
Think of this model as a giant, 3D Rubik's Cube made of fermions (tiny quantum particles).

• The "Coloured" version: Imagine the cube has red, blue, and green faces. It's very structured.
• The "Uncoloured" version (their focus): Imagine the cube is all one color. It looks simpler, but it actually has a huge amount of hidden repetition (degeneracy). Many different moves lead to the exact same result.

Because this model is so repetitive, it's a perfect test lab. The scientists ran computer simulations to see if their "Golden Rule" worked here.

Part 4: What They Found

They tested two types of "zones" (symmetries) in their model:

1. The "Perfect Match" Zones:
   They found some zones where the "Golden Rule" worked perfectly. In these specific sub-sections of the model, the complexity grew exactly the same way as in the full model.

   • Takeaway: We can do the hard math on a tiny, simple piece of the puzzle and get the answer for the whole puzzle. This saves huge amounts of computer power.

2. The "Mismatch" Zones:
   They also found other zones where the rule didn't work. In these areas, the complexity grew differently than in the full room.

   • Takeaway: You can't just pick any corner of the room to look at; you have to pick the right corner.

They also confirmed a guess made by other scientists: The average complexity of all the little zones is always less than or equal to the complexity of the whole room. The whole is always more chaotic than the sum of its parts.

Part 5: The Computer Glitch (A Technical Warning)

The paper also mentions a headache they faced with their computers. Because the model has so many identical parts (degeneracy), the computer algorithm they used (the Lanczos Algorithm) started to get confused.

The Analogy:
Imagine trying to walk a tightrope. Usually, you balance perfectly. But because the tightrope is made of identical-looking planks, your feet start to slip, and you accidentally step off the rope into the "wrong" area of the air.

• The computer started making small errors that grew bigger and bigger, eventually making the results unreliable.
• The authors had to stop their calculations early because the computer was "hallucinating" numbers.

Summary: Why Does This Matter?

1. Efficiency: If we can find the "Perfect Match" zones in complex quantum systems (like black holes or future quantum computers), we can simulate them much faster. We don't need supercomputers to solve the whole puzzle; we just need to solve a tiny, symmetrical piece.
2. Understanding Chaos: It helps us understand how chaos spreads in the universe. Does it spread evenly, or does it get stuck in certain areas?
3. New Tools: They provided a mathematical checklist (the "Golden Rule") for physicists to know before they start calculating whether they can use this shortcut.

In a nutshell: The authors found a way to use symmetry to simplify the math of quantum chaos, proved it works on a specific complex model, and warned us about the computer glitches that happen when things get too repetitive.

Technical Summary

1. Problem Statement

The paper addresses the computational challenges associated with calculating Krylov Complexity ($C_K$) in quantum systems with high degrees of freedom and significant symmetries.

• Context: Krylov complexity measures the growth of an operator in the Krylov basis (generated by the Liouvillian $L = [H, \cdot]$) and serves as a probe for quantum chaos. However, for large systems, the dimension of the operator space scales exponentially, making full-space calculations intractable.
• The Gap: While Symmetry-resolved Krylov Complexity (calculating $C_K$ within specific charge sectors) offers a way to reduce computational cost, it is not always equivalent to the full-space complexity.
• Core Question: Under what specific conditions does the Krylov complexity of an invariant operator in a specific symmetry subspace (charge sector) exactly match the Krylov complexity of the operator in the full Hilbert space? Furthermore, how does this behave in the Uncoloured Tensor Model, a disorder-free system with high degeneracy and rich symmetry?

2. Methodology

The authors employ a combination of analytical derivation and explicit numerical simulation.

A. Analytical Framework

• Krylov Complexity Review: They define $C_K(t)$ as the average position of the wavefunction in the Krylov chain, governed by Lanczos coefficients ($b_n$).
• Symmetry Resolution: They consider a conserved charge $Q$ commuting with the Hamiltonian ($[H, Q]=0$). The Hilbert space decomposes into charge sectors $\mathcal{H} = \oplus_q \mathcal{H}_q$.
• Derivation of Equipartition Conditions:
  • They derive the necessary and sufficient conditions for the Lanczos coefficients in a specific charge sector $q_0$ ($b_n^{(q_0)}$) to be identical to those in the full space ($b_n$).
  • This requires the ratio of the operator's norm in a specific Liouvillian eigenspace $(A, q_0)$ to the total norm in that Liouvillian eigenspace $A$ to be independent of the eigenspace index $A$.
  • Key Condition (Eq. 3.9): For equipartition to hold, the operator must satisfy:
    $$\frac{\text{Tr}(\Pi_{b,q_0} O^\dagger \Pi_{a,q_0} O)}{\text{Tr}(\Pi_{q_0} O^\dagger O)} = \frac{\text{Tr}(\Pi_b O^\dagger \Pi_a O)}{\text{Tr}(O^\dagger O)}$$
    for all energy pairs $(a,b)$ and charge sectors. This implies the operator must have non-trivial projections across all relevant energy pairs in the subspace, proportional to their presence in the full space.

B. Numerical Study: Uncoloured Tensor Model

• Model Selection: The authors study the $n=3, d=3$ Uncoloured Tensor Model (Klebanov-Tarnopolsky model).
  • Hamiltonian: Constructed from fermionic fields $\psi_{ijk}$ with $O(3)^3$ symmetry.
  • Scale: The Hilbert space dimension is $2^{27}$ (16,384), but due to high degeneracy, there are only 34 distinct energy eigenvalues.
• Algorithm: They use the Lanczos algorithm with Partial Re-Orthogonalization (PRO) to generate the Krylov basis and coefficients.
• Challenges: They address numerical instability caused by the extreme degeneracy of the Liouvillian spectrum. In degenerate systems, Krylov vectors can "leak" out of the true subspace due to finite precision errors, causing the algorithm to fail prematurely.

3. Key Contributions

1. General Conditions for Equipartition:
   The paper provides a rigorous derivation of the conditions (Eq. 3.9) under which the complexity in a symmetry subspace equals the full-space complexity. This allows researchers to identify the smallest computationally feasible subspace that yields exact results for the full system.

2. Validation in Tensor Models:
   They apply these conditions to the Uncoloured Tensor Model, a system known for its high degeneracy and lack of disorder averaging (unlike the SYK model).

3. Analysis of Symmetry Types:

   • Discrete Symmetries ($Z$ and $N_1$): They demonstrate that for operators commuting with discrete symmetries (like the product of all Gamma matrices), the equipartition condition is met. The Lanczos coefficients in the subspaces match the full space exactly.
   • Continuous Noether Symmetries ($O(3)$ charges): They show that for continuous symmetries, the energy spectra in different charge sectors are not identical. Consequently, the equipartition condition fails. In these cases, the symmetry-resolved complexity differs from the full complexity, and in some sectors (e.g., charge $\pm 1$), the complexity can even exceed the full-space value.

4. Conjecture Verification:
   They numerically verify the conjecture from [45] that the average symmetry-resolved complexity is bounded above by the full-space complexity ($C_K(t) \geq \sum p_q C_K^{(q)}(t)$) over the time scales accessible to their numerical methods.

4. Results

• Lanczos Coefficients ($b_n$):
  • In the infinite temperature limit, $b_n$ shows an initial linear growth (characteristic of chaos), followed by a plateau and decay due to finite system size.
  • At finite temperatures, the initial growth rate slows down, and the plateau is reached at larger $n$.
• Krylov Complexity ($C_K$):
  • $C_K(t)$ exhibits initial exponential growth followed by approximate linear growth.
  • Equipartition Cases: For discrete symmetries, $C_K^{(q)}(t)$ is identical to $C_K(t)$ for all $t$.
  • Non-Equipartition Cases: For continuous symmetries, $C_K^{(q)}(t)$ deviates from $C_K(t)$. The average of the resolved complexities remains below the full complexity, supporting the conjecture.
• Numerical Instability:
  The study highlights that for highly degenerate matrices, the Lanczos algorithm terminates early (around $n \approx 65$ in their 16k-dimensional system) because the Krylov vectors lose orthogonality and leak into the larger computational space, creating spurious degenerate eigenvalues in the tridiagonal matrix.

5. Significance and Future Directions

• Computational Efficiency: The derived conditions allow for the reduction of complex many-body problems to smaller symmetry subspaces without losing information about the operator growth, provided the operator satisfies the specific projection criteria.
• Understanding Chaos in Tensor Models: The results confirm that the Uncoloured Tensor Model exhibits chaotic signatures (linear growth of $b_n$, exponential growth of $C_K$) similar to the SYK model, despite being disorder-free.
• Limitations of Symmetry Resolution: The work clarifies that symmetry resolution is not a universal shortcut; it only works if the operator's structure aligns perfectly with the symmetry decomposition (equipartition). If the energy spectra differ across sectors, the complexity dynamics differ.
• Future Work: The authors suggest extending these conditions to infinite-dimensional systems and developing "bottom-up" approaches to construct operators in subspaces that guarantee equipartition, avoiding the need for full-system diagonalization.

In summary, this paper bridges the gap between theoretical symmetry resolution and practical numerical computation, providing a concrete framework for when and how symmetry subspaces can be used to accurately study quantum chaos in complex tensor models.