Extracting average properties of disordered spin chains with translationally invariant tensor networks

This paper introduces a tensor network method that leverages statistical translation invariance to efficiently calculate disorder-averaged properties of random spin chains directly in the thermodynamic limit, bypassing the need for explicit disorder sampling.

The Gist

The Big Problem: The "Noisy Room"

Imagine you are trying to understand how a crowd of people behaves in a room. In a normal physics experiment (a "clean" system), everyone is identical and follows the same rules. It's like a choir singing in perfect harmony; you can predict the sound easily.

But in the world of disordered spin chains (the subject of this paper), the "people" in the room are all different. Some are loud, some are quiet, some are shy, and some are aggressive. This is called randomness or disorder.

To understand the average behavior of this chaotic crowd, physicists usually have to run the simulation thousands of times, each time with a different random arrangement of people, and then average the results. This is like trying to predict the weather by simulating every single possible storm pattern one by one. It takes a massive amount of computer power and time.

The New Solution: The "Universal Translator"

The authors of this paper, Kevin, Wei, and Nick, developed a clever shortcut. Instead of simulating thousands of different random rooms, they built a single, super-smart model that represents all the possible random arrangements at once.

They call this model a Tensor Network (specifically, a Matrix Product Operator). Think of it like a universal translator or a master recipe.

• The Old Way: You write a unique recipe for every single variation of a cake (chocolate with nuts, vanilla with sprinkles, etc.), bake them all, and taste them to find the average flavor.
• The New Way: You write one "Master Recipe" that contains instructions for every variation simultaneously. When you follow this one recipe, it automatically accounts for all the different possibilities without you having to bake them individually.

How It Works: The "Control Panel"

To make this "Master Recipe" work, the scientists introduced a clever trick using ancilla qudits.

• Imagine every person in the crowd has a control panel (a small screen) next to them.
• This screen doesn't change the person; it just labels them. It says, "This person is a 'Type A' loud person," or "This person is a 'Type B' quiet person."
• The scientists created a single system where these control panels run through every possible combination of labels at the same time.

Because the rules of the game are the same for every spot in the line (statistical translation invariance), this single "Master Recipe" can be stretched out infinitely. It doesn't matter if the line is 10 people long or a billion people long; the recipe stays the same size and efficient.

The "Normalization" Step: Keeping the Score Straight

There was a tricky part. When you mix all these different random scenarios together, the math gets messy. Some scenarios are very rare but very important, while others are common but weak. If you just average them, you might lose the rare, important ones.

The authors added a special "Scorekeeper" step to their algorithm.

• Imagine you are mixing different soups. Some are very salty, some are very bland. If you just pour them all into one pot, the flavor gets lost.
• The "Scorekeeper" (a normalization operator) constantly adjusts the volume of each soup so that the final mixture represents the true average, ensuring that the rare, strong flavors aren't drowned out by the common, weak ones.
• This step is crucial. Without it, the computer would throw away the most interesting parts of the data to save space.

The Test: The "Random Magnet"

To prove their method works, they tested it on a famous, difficult puzzle called the Random Transverse Field Ising Model.

• Think of this as a row of tiny magnets that are randomly strong or weak, and randomly pointing up or down.
• This system is known to be extremely hard to solve because it has "rare regions"—spots where the magnets behave in a very strange, unique way that dominates the whole system.
• The Result: The new method successfully predicted the average behavior of these magnets at different temperatures. It matched the known answers perfectly, even though it used a relatively small amount of computer memory (a "bond dimension" of about 100).

Why It Matters

This paper proves that you don't need to simulate thousands of random worlds to understand the average behavior of a disordered system. You can build one efficient, infinite model that captures the essence of all the chaos.

It's like realizing you don't need to watch every single episode of a TV show with a different plot twist to understand the main character's personality; you just need one "super-episode" that summarizes all the possibilities perfectly.

Key Takeaway: The authors created a mathematical tool that acts as a "universal average," allowing physicists to study messy, random quantum systems directly in the infinite limit without needing to run thousands of separate simulations.

Technical Summary

Technical Summary: Extracting Average Properties of Disordered Spin Chains with Translationally Invariant Tensor Networks

Problem Statement
Simulating quantum many-body systems with quenched randomness presents two primary challenges for tensor network methods. First, the lack of translational symmetry prevents the direct application of standard sweeping optimization procedures used in Density Matrix Renormalization Group (DMRG) algorithms, often requiring excessive sweeps due to inhomogeneous entanglement distribution. Second, calculating disorder-averaged physical observables traditionally necessitates simulating a large number of independent disorder realizations, which is computationally prohibitive. While recent progress has addressed the optimization of single samples via numerical renormalization groups, the challenge of efficiently averaging over disorder configurations in the thermodynamic limit remains. This work addresses the calculation of finite-temperature disorder-averaged expectation values for random spin chains without explicitly sampling over disorder configurations.

Methodology
The authors propose a tensor network-based algorithm that exploits statistical translation invariance—the condition where the probability distribution of disorder variables is identical across all lattice sites. The core strategy involves representing the disorder-averaged density matrix, $\rho = \sum_{\{R_n\}} [\prod_n P(R_n)] \rho_G[\{R_n\}]$, as a single, translationally invariant Matrix Product Operator (MPO).

The algorithm proceeds as follows:

1. Ancilla Introduction: To handle the disorder, ancilla qudits $|R_n\rangle$ of dimension $N_D$ (the number of possible disorder values) are introduced on every site. A translationally invariant Hamiltonian $H = \sum_n h_n$ is constructed, where local terms $h_n$ act as controlled operations: $h_n = \sum_{R_n} \tilde{h}_n[R_n] \otimes |R_n\rangle\langle R_n|$. The ancillas act as controls, selecting the specific local Hamiltonian term based on the disorder value.
2. Modified Imaginary-Time Evolution: The algorithm constructs the Gibbs state via imaginary-time evolution using a modified Time-Evolving Block Decimation (TEBD) approach. Instead of simply evolving $e^{-\tau H}$, the method evolves an unnormalized operator $\tilde{\rho}(\tau) = N(\tau)e^{-\tau H}$. Here, $N(\tau)$ is a diagonal operator acting on the ancillas that contains the inverse partition functions $1/Z[\{R_n\}]$ for each disorder configuration. This ensures that tracing out the physical spins yields an identity matrix on the disorder qudits, preserving the correct normalization for the mixture of Gibbs states.
3. Normalization and Inversion: A critical step involves updating the normalization operator $N(\tau)$ at each time step. This requires inverting an MPO $\Lambda$ derived from the trace of the physical spins. Since $\Lambda$ is close to the identity for small time steps, the authors employ a variational optimization (a generalization of the VOMPS algorithm) to find a low-bond-dimension MPO approximation $\Lambda_M^{-1}$ that maximizes the fidelity with the identity operator.
4. Truncation: After updating the normalization, the MPO is truncated back to a target bond dimension $D$ using standard Schmidt value truncation. The cycle of evolution, normalization adjustment, and truncation is repeated until the target inverse temperature $\beta$ is reached. The final disorder-averaged density matrix is obtained by tracing out the ancilla qudits.

Key Results
The method was benchmarked on the random transverse field Ising model (RTFIM) near its infinite-randomness critical point ($\delta=0$).

• Efficiency: The algorithm successfully reproduced average properties of the RTFIM with relatively small bond dimensions (order $\sim 100$) down to low temperatures.
• Critical Scaling: The extracted correlation length $\xi$ exhibits the theoretically predicted activated dynamical scaling $\xi \sim (\ln \beta)^2$ characteristic of the infinite-randomness fixed point. The results showed convergence with respect to bond dimension and time-step size.
• Disorder Distribution: By varying the number of discrete disorder values $N_D$, the authors confirmed that the scaling of $\xi$ depends on the variance of the disorder distribution, consistent with theoretical predictions.
• Rare Regions: By analyzing the transfer matrices of individual disorder configurations (without tracing out ancillas), the authors extracted the distribution of correlation lengths. They observed a heavy tail in the distribution, confirming the presence of rare regions. The typical correlation length ($\xi_{typ} \approx 35.4$) was found to be significantly smaller than the average correlation length ($\xi_{av} \approx 47.5$), in agreement with theoretical expectations for infinite-randomness physics.
• Phase Behavior: Additional results confirmed the algebraic growth of the correlation length $\xi \sim 1/T^{\alpha(\delta)}$ on the ordered side of the transition, distinct from the exponential growth in clean systems.

Significance and Claims
The paper presents a proof-of-principle that a mixture of Gibbs states corresponding to different disorder configurations can be efficiently represented as a single, translationally invariant MPO. The authors claim that their modified TEBD algorithm, which explicitly handles normalization via ancilla qudits and variational inversion, allows for the direct calculation of disorder-averaged finite-temperature properties in the thermodynamic limit without explicit sampling.

The work demonstrates that despite the theoretical uncertainty regarding the efficiency of MPO approximations for disordered systems, such approximations exist and can capture non-trivial physics, including the activated scaling and rare-region effects of the infinite-randomness fixed point. The authors note that while the current implementation uses Suzuki-Trotter decomposition, future improvements could utilize cluster expansions to reduce time-step errors. They also identify the development of a variational formulation for direct access to disorder-averaged ground states as a promising direction for future work, distinct from the adiabatic evolution methods previously used. The code for the algorithm is provided as an open-source package, DisorderKit.jl.