[Paper Review] Stability of ground state degeneracy to long-range interactions

Matthew F. Lapa, Michael Levin|arXiv (Cornell University)|Jul 23, 2021

Cold Atom Physics and Bose-Einstein Condensates4 citations

TL;DR

This paper proves that certain gapped quantum many-body systems with exact ground state degeneracy remain stable under long-range (e.g., power-law) interactions, showing that the residual energy splitting between degenerate ground states is exponentially small in system size. The authors use a convergent polymer expansion adapted to long-range interactions to rigorously establish gap stability and exponentially small splitting, extending known results from short-range to long-range perturbations.

ABSTRACT

We show that some gapped quantum many-body systems have a ground state degeneracy that is stable to long-range (e.g., power-law) perturbations, in the sense that any ground state energy splitting induced by such perturbations is exponentially small in the system size. More specifically, we consider an Ising symmetry-breaking Hamiltonian with several exactly degenerate ground states and an energy gap, and we then perturb the system with Ising symmetric long-range interactions. For these models we prove (1) the stability of the gap, and (2) that the residual splitting of the low-energy states below the gap is exponentially small in the system size. Our proof relies on a convergent polymer expansion that is adapted to handle the long-range interactions in our model. We also discuss applications of our result to several models of physical interest, including the Kitaev p-wave wire model perturbed by power-law density-density interactions with an exponent greater than 1.

Motivation & Objective

• To establish the stability of ground state degeneracy in gapped quantum many-body systems under long-range interactions.
• To prove that the residual energy splitting between degenerate ground states remains exponentially small in system size despite long-range perturbations.
• To extend existing short-range stability results to long-range (e.g., power-law) interactions, which are common in physical systems like topological superconductors.
• To develop a mathematical framework—specifically, a convergent polymer expansion—for handling long-range interactions in systems with exact ground state degeneracy.

Proposed method

• Adapts the polymer expansion technique to handle long-range interactions by introducing a modified convergence condition that accounts for power-law decay.
• Uses a recursive construction of minimal weakly-connected polymers to bound the number of relevant configurations contributing to the perturbative expansion.
• Introduces a vertex-growing procedure that systematically builds connected polymers from disconnected components, preserving convergence through controlled summability.
• Applies a weighted sum over polymers with exponential and power-law factors to control the growth of contributions in the perturbative series.
• Employs a bound on the number of minimal weakly-connected polymers using a sum over subset counts and a factor of $( ext{const})^m$ to ensure convergence.
• Establishes the convergence of the expansion by showing the total contribution is bounded by $e^{c ilde{ ho} ho}$, ensuring stability of the gap and exponentially small splitting.

Experimental results

Research questions

• RQ1 Can the ground state degeneracy of a gapped quantum many-body system remain stable under long-range interactions?
• RQ2 Is the residual energy splitting between degenerate ground states exponentially small in system size when perturbed by long-range interactions?
• RQ3 Can the polymer expansion method be adapted to handle long-range interactions while preserving convergence and exponential bounds?
• RQ4 Does the gap remain open under Ising-symmetric long-range perturbations in systems with exact ground state degeneracy?

Key findings

• The energy gap remains open under long-range perturbations, proving stability of the spectral gap in the system.
• The residual splitting between degenerate ground states is exponentially small in the system size, specifically bounded by $e^{-cL}$ for some $c>0$.
• The polymer expansion converges under the modified conditions, allowing rigorous control over the perturbative corrections to the ground state energy.
• The method applies to models with power-law decaying interactions with exponent greater than 1, such as the Kitaev p-wave wire model.
• The result generalizes known short-range stability results to long-range interactions, filling a key gap in the understanding of robust ground state degeneracy.