Effect of slowly decaying long-range interactions on topological qubits

1. Problem Statement

The paper investigates the robustness of topological ground state degeneracy in quantum many-body systems when subjected to slowly decaying long-range interactions (specifically power-law interactions scaling as $1/r^\alpha$).

• Context: Topological qubits rely on a ground state splitting ($\delta$) that is exponentially small in system size ($L$), i.e., $\delta \sim e^{-cL}$. This property ensures protection against errors and decoherence.
• The Gap: Existing stability theorems (e.g., Refs. 14, 15) guarantee this exponential protection only for short-range interactions or long-range interactions that satisfy a summability condition ($\sum |f(r)| < c$).
• The Challenge: Many physical systems (e.g., fractional quantum Hall states with Coulomb interactions) exhibit non-summable power-law interactions where $0 < \alpha < d$ (spatial dimension). In these cases, standard quasi-adiabatic continuation methods only yield a polynomial bound on the splitting, leaving open the possibility that long-range interactions could destroy the topological protection entirely.
• Goal: Determine the scaling behavior of the ground state splitting $\delta$ for systems with power-law interactions where $0 < \alpha < 1$ (violating the summability condition).

2. Methodology

Since a direct analytical solution for the 1D Ising spin chain with true power-law interactions is intractable, the authors employ path integral techniques (specifically the instanton method introduced by Coleman) to compute the splitting in three distinct "toy models" that capture the essential physics.

A. Model 1: Ising Spin Chain with All-to-All Interactions

• Hamiltonian: $H = -\sum \sigma^z_j \sigma^z_{j+1} + \frac{\lambda}{4L^\alpha} (\sum \sigma^x_j)^2$.
• Approach: The authors calculate the free energy difference between symmetry sectors at finite temperature $\beta$ and take the limit $\beta \to \infty$.
• Technique: They use a Hubbard-Stratonovich transformation to decouple the interaction, converting the problem into a path integral over a single auxiliary field $\phi(\tau)$. A saddle-point approximation is then applied.
• Key Insight: The dominant contribution comes from "instanton" paths (tunneling events) connecting the two degenerate ground states. The action of these paths is evaluated to find the leading order behavior of the splitting.

B. Model 2: Quantum Rotor Chain with True Power-Law Interactions

• Hamiltonian: A chain of quantum rotors with a potential breaking $U(1)$ to $\mathbb{Z}_2$ symmetry, coupled via true power-law interactions $f(r) \sim 1/r^\alpha$.
• Approach: This model allows for a semi-classical limit ($g \to 0$, where $g$ acts as $\hbar$).
• Technique: The standard instanton method is applied directly to the multi-body path integral. The authors assume translational symmetry for the instanton solution to reduce the problem to a single-particle effective mass equation.

C. Model 3: Non-Local Toy Model

• Hamiltonian: A model where the short-range Ising term is replaced by a projector onto the two ferromagnetic "cat" states, coupled with an all-to-all interaction $(\sum O_j)^2$.
• Approach: This model is non-local but allows for an exact algebraic solution using the matrix determinant lemma, avoiding path integrals. It tests the sensitivity of the scaling to the specific form of the operators $O_j$.

3. Key Contributions and Results

Main Result: Stretched Exponential Scaling

For all three models (and specifically for the Ising chain with all-to-all interactions and the rotor chain with true power-law interactions), the ground state splitting $\delta$ scales as a stretched exponential rather than a simple exponential or polynomial:

$$\delta \sim \exp\left( -C L^{\frac{1+\alpha}{2}} \right)$$

where $C$ is a constant depending on parameters like $\lambda$ and $\beta$, and $0 < \alpha < 1$.

• Comparison:
  • Short-range / Summable ($\alpha > 1$): $\delta \sim e^{-cL}$ (Standard topological protection).
  • Non-summable Power-law ($0 < \alpha < 1$): $\delta \sim e^{-cL^{(1+\alpha)/2}}$.
  • Significance: The protection is weakened but not destroyed. The splitting remains exponentially small in a power of $L$, meaning the system retains robustness, albeit with a slower decay rate than short-range systems.

Specific Findings per Model

1. Ising Chain (All-to-All): The saddle-point analysis yields the instanton action $S \propto L^{(1+\alpha)/2}$. The prefactor involves a determinant ratio that scales sub-dominantly compared to the exponential term.
2. Quantum Rotor Chain: The effective mass of the instanton scales as $m_0 \sim L^{\alpha-1}$. This leads to an action $S \propto L \sqrt{m_0} \sim L^{(1+\alpha)/2}$, confirming the same scaling law as the Ising chain.
3. Non-Local Toy Model:
   • For $O_j = \sigma^x_j$, the result matches the stretched exponential scaling.
   • Sensitivity: For $O_j = \sigma^x_j \sigma^x_{j+1}$, the splitting scales as $\delta \sim e^{-cL}$ (standard exponential) because the matrix elements between ground states vanish exponentially. This suggests that while the stretched exponential is a generic feature for "standard" long-range couplings, the precise scaling can depend on the detailed structure of the interaction operators.

4. Significance and Implications

• Robustness of Topological Qubits: The results suggest that topological qubits are not fundamentally fragile to slowly decaying long-range interactions (like Coulomb interactions). While the error suppression is less efficient (stretched exponential vs. exponential), it remains strong enough to potentially support fault-tolerant quantum computing if the system size is sufficiently large.
• Theoretical Framework: The paper provides a rigorous calculation for a regime where standard stability theorems fail. It bridges the gap between short-range stability proofs and the physical reality of long-range forces in condensed matter systems.
• Conjecture: The authors conjecture that the original 1D Ising spin chain with true power-law interactions $f(r) \sim 1/r^\alpha$ exhibits the same stretched exponential scaling, despite the lack of a direct calculation for that specific case.
• Future Directions: The paper highlights the need to extend these results to 2D systems (e.g., fractional quantum Hall states) and to explore continuum field theory perspectives to determine if this scaling is universal.

Summary Table of Scaling Laws (from Table I)

Model	Interaction Type	Scaling of $\delta$
Summable Ising Chain	$\sum |f(r)| \leq c$	$\exp(-CL)$
All-to-All Ising Chain	$f(r) \sim 1/L^\alpha$	$\exp(-CL^{\frac{1+\alpha}{2}})$
Quantum Rotor Chain	$f(r) \sim 1/r^\alpha$	$\exp(-CL^{\frac{1+\alpha}{2}})$
Non-local Toy Model	$f(r) \sim 1/L^\alpha$	$\exp(-CL^{\frac{1+\alpha}{2}})$ (or $\exp(-CL)$ depending on $O_j$)

In conclusion, the paper demonstrates that topological protection survives long-range interactions, transitioning from a standard exponential decay to a stretched exponential decay characterized by the exponent $\frac{1+\alpha}{2}$.