Undecidability of the spectral gap in rotationally symmetric Hamiltonians (2410.13589v1)

Abstract: The problem of determining the existence of a spectral gap in a lattice quantum spin system was previously shown to be undecidable for one [J. Bausch et al., "Undecidability of the spectral gap in one dimension", Physical Review X 10 (2020)] or more dimensions [T. S. Cubitt et al., "Undecidability of the spectral gap", Nature 528 (2015)]. In these works, families of nearest-neighbor interactions are constructed whose spectral gap depends on the outcome of a Turing machine Halting problem, therefore making it impossible for an algorithm to predict its existence. While these models are translationally invariant, they are not invariant under the other symmetries of the lattice, a property which is commonly found in physically relevant cases, posing the question of whether the spectral gap is still an undecidable problem for Hamiltonians with stronger symmetry constraints. We give a positive answer to this question, in the case of models with 4-body (plaquette) interactions on the square lattice satisfying rotation, but not reflection, symmetry: rotational symmetry is not enough to make the problem decidable.

• The paper demonstrates that spectral gap determination in rotationally symmetric Hamiltonians is undecidable by mapping it to the Turing halting problem.
• It employs a 4-body plaquette construction with aperiodic tiling to encode Turing machine computations while preserving 90-degree rotational symmetry.
• This result highlights a fundamental limitation on algorithmically analyzing quantum many-body systems, impacting our understanding of phase transitions and topological order.

This paper, "Undecidability of the spectral gap in rotationally symmetric Hamiltonians" (2410.13589), demonstrates that determining whether a quantum many-body system has a spectral gap above its ground state energy is an undecidable problem, even when the Hamiltonian describing the system possesses rotational symmetry. This result extends previous findings which established undecidability for translationally invariant but not fully symmetric Hamiltonians.

The spectral gap, defined as the energy difference between the ground state and the first excited state, is a fundamental property of quantum systems. Its presence or absence is crucial for understanding physical phenomena like phase transitions, topological order, and the stability of quantum information encoding. A positive gap often implies robust ground states and short-range correlations, while a zero gap typically signals critical phenomena or phase transitions.

Previous research showed that for general translationally invariant Hamiltonians on lattices, the existence of a spectral gap is undecidable. This means there is no algorithm that can take an arbitrary description of such a Hamiltonian and always determine in a finite amount of time whether it has a spectral gap. These undecidability proofs typically rely on constructing Hamiltonians whose ground state properties simulate the behavior of a Turing machine, specifically linking the existence of a gap to the halting problem (an established undecidable problem).

The key contribution of this paper is showing that adding a common physical symmetry – rotational invariance (specifically, 90-degree rotation invariance on a square lattice) – is not sufficient to restore decidability. This is significant because many physically relevant models in condensed matter physics exhibit such symmetries. The authors construct a family of 4-body (plaquette) interactions on a 2D square lattice. These interactions are carefully designed to be invariant under 90-degree rotations of the lattice but not under reflection.

The proof involves mapping the computation of a universal Turing machine onto the configuration space of the quantum system. The Hamiltonian is constructed such that configurations representing valid, non-halting computations have low energy, with the ground state corresponding to a "successful" history of the Turing machine computation. A spectral gap exists if and only if the Turing machine does not halt. Since the Halting problem is undecidable, the spectral gap problem for this class of rotationally symmetric Hamiltonians is also undecidable.

In practical terms, this undecidability result implies a fundamental limitation on our ability to computationally analyze certain quantum systems. For the specific class of rotationally symmetric Hamiltonians studied, there cannot exist a general-purpose algorithm that can reliably determine the gapped or gapless nature for all instances. This does not mean one cannot paper specific, well-behaved gapped or gapless systems (many of which are well understood and can be simulated or analyzed), but rather that the general problem is computationally intractable.

The construction used in the proof is complex and involves several steps:

1. 1D Encoding: Encoding the state and tape of a Turing machine into a 1D chain of qubits or spins.
2. Tiling: Using an aperiodic tiling (similar to Wang tiles) on the 2D lattice to enforce the flow of information representing the Turing machine computation history across the 2D plane. The tiling must respect the rotational symmetry.
3. Hamiltonian Construction: Defining local interaction terms (4-body plaquettes) based on the tiling rules and the encoded Turing machine states. These terms penalize invalid configurations or computational steps, ensuring that low-energy states correspond to valid computations. The rotational symmetry constraint imposes restrictions on the form of these local terms.
4. Mapping to Gap: Proving that the ground state energy is zero if and only if the Turing machine computation is "valid" and extends infinitely (doesn't halt), leading to a zero spectral gap. If the machine halts, the computation terminates at some finite step, which corresponds to a non-zero ground state energy and thus a positive spectral gap.

While this paper presents a theoretical undecidability result rather than an algorithm for implementation, it highlights the intrinsic difficulty of characterizing quantum phases and spectral properties in certain complex, symmetric systems. It suggests that for arbitrary systems within this class, researchers must rely on analytical insights or approximate numerical methods, acknowledging that a universally applicable decision procedure does not exist.