Undecidability of the spectral gap in rotationally symmetric Hamiltonians

• URL: http://arxiv.org/abs/2410.13589v1
• Date: Thu, 17 Oct 2024 14:23:58 GMT
• Title: Undecidability of the spectral gap in rotationally symmetric Hamiltonians
• Authors: Laura Castilla-Castellano, Angelo Lucia,
• Abstract summary: We show that the problem of determining the existence of a spectral gap in a lattice quantum spin system is not undecidable 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.
• Score: 0.30693357740321775
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

Related papers

• Meson spectroscopy of exotic symmetries of Ising criticality in Rydberg atom arrays [39.58317527488534]
  Coupling two Ising chains in a ladder leads to an even richer $mathcalD(1)_8$ symmetries.<n>Here, we probe these emergent symmetries in a Rydberg atom processing unit, leveraging its geometry to realize both chain and ladder configurations.
  arXiv  Detail & Related papers  (2025-06-26T14:19:30Z)
• Solving graph problems using permutation-invariant quantum machine learning [35.99391901074448]
  In quantum machine learning, the ansatz can be tuned to correspond to the specific symmetry of the problem.<n>We show how the symmetry can be included in the quantum circuit in a straightforward constructive method.
  arXiv  Detail & Related papers  (2025-05-19T06:44:03Z)
• Observation of non-Hermitian bulk-boundary correspondence in non-chiral non-unitary quantum dynamics of single photons [31.05848822220465]
  In non-Hermitian systems, preserved chiral symmetry is one of the key ingredients, which plays a pivotal role in determining non-Hermitian topology. We theoretically predict and experimentally demonstrate the bulk-boundary correspondence of a one-dimensional (1D) non-Hermitian system with chiral symmetry breaking.
  arXiv  Detail & Related papers  (2025-04-07T09:43:43Z)
• Disentangling anomaly-free symmetries of quantum spin chains [0.0]
  We prove that any finite, internal, anomaly-free symmetry in a 1+1d lattice Hamiltonian system can be disentangled into an on-site symmetry.
  arXiv  Detail & Related papers  (2025-03-12T18:08:22Z)
• Oscillators with imaginary coupling: spectral functions in quantum mechanics and quantum field theory [0.0]
  Non-hermitian hamiltonians with $calPT$-symmetry can have both real spectra and unitary time evolution. We analyze two-point correlation functions and their associated K"allen-Lehmann representation. We conjecture that positivity violation of some spectral functions of the theory could be a generic sign of the existence of complex pairs of energy eigenvalues.
  arXiv  Detail & Related papers  (2024-12-18T17:09:30Z)
• Hilbert space geometry and quantum chaos [39.58317527488534]
  We consider the symmetric part of the QGT for various multi-parametric random matrix Hamiltonians. We find for a two-dimensional parameter space that, while the ergodic phase corresponds to the smooth manifold, the integrable limit marks itself as a singular geometry with a conical defect.
  arXiv  Detail & Related papers  (2024-11-18T19:00:17Z)
• Quantifying non-Hermiticity using single- and many-particle quantum properties [14.37149160708975]
  The non-Hermitian paradigm of quantum systems displays salient features drastically different from Hermitian counterparts. We propose a formalism that quantifies the (dis-)similarity of these right and left ensembles, for single- as well as many-particle quantum properties. Our findings can be instrumental in unveiling new exotic quantum phases of non-Hermitian quantum many-body systems.
  arXiv  Detail & Related papers  (2024-06-19T13:04:47Z)
• Dimension matters: precision and incompatibility in multi-parameter quantum estimation models [44.99833362998488]
  We study the role of probe dimension in determining the bounds of precision in quantum estimation problems. We also critically examine the performance of the so-called incompatibility (AI) in characterizing the difference between the Holevo-Cram'er-Rao bound and the Symmetric Logarithmic Derivative (SLD) one.
  arXiv  Detail & Related papers  (2024-03-11T18:59:56Z)
• Renormalization group and spectra of the generalized P\"oschl-Teller potential [0.0]
  We study the P"oschl-Teller potential $V(x) = alpha2 g_s sinh-2(alpha x) + alpha2 g_c cosh-2(alpha x)$, for every value of the dimensionless parameters $g_s$ and $g_c singularity. We show that supersymmetry of the potential, when present, is also spontaneously broken, along with conformal symmetry.
  arXiv  Detail & Related papers  (2023-08-08T21:44:55Z)
• Refuting spectral compatibility of quantum marginals [0.0]
  The main idea of this work is a symmetry-reduced semidefinite programming hierarchy for detecting incompatible spectra. The hierarchy can provide refutations that are dimension-free, certifying incompatibility in all local dimensions.
  arXiv  Detail & Related papers  (2022-11-11T16:48:58Z)
• Penrose dodecahedron, Witting configuration and quantum entanglement [55.2480439325792]
  A model with two entangled spin-3/2 particles based on geometry of dodecahedron was suggested by Roger Penrose. The model was later reformulated using so-called Witting configuration with 40 rays in 4D Hilbert space. Two entangled systems with quantum states described by Witting configurations are discussed in presented work.
  arXiv  Detail & Related papers  (2022-08-29T14:46:44Z)
• Spectral form factor in a minimal bosonic model of many-body quantum chaos [1.3793594968500609]
  We study spectral form factor in periodically-kicked bosonic chains. We numerically find a nontrivial systematic system-size dependence of the Thouless time.
  arXiv  Detail & Related papers  (2022-03-10T15:56:24Z)
• Spectral Properties of Symmetric Quantum States and Symmetric Entanglement Witnesses [0.0]
  We introduce and explore two questions concerning spectra of operators that are of interest in the theory of entanglement in symmetric quantum systems. First, we investigate the inverse eigenvalue problem for symmetric entanglement witnesses -- that is, we investigate what their possible spectra are. Second, we investigate the problem of characterizing which separable symmetric quantum states remain separable after conjugation by an arbitrary unitary acting on symmetric space -- that is, which states are separable in every orthonormal symmetric basis.
  arXiv  Detail & Related papers  (2021-08-23T20:57:11Z)
• Information retrieval and eigenstates coalescence in a non-Hermitian quantum system with anti-$\mathcal{PT}$ symmetry [15.273168396747495]
  Non-Hermitian systems with parity-time reversal ($mathcalPT$) or anti-$mathcalPT$ symmetry have attracted a wide range of interest owing to their unique characteristics and counterintuitive phenomena. We implement a Floquet Hamiltonian of a single qubit with anti-$mathcalPT$ symmetry by periodically driving a dissipative quantum system of a single trapped ion.
  arXiv  Detail & Related papers  (2021-07-27T07:11:32Z)
• $\PT$ Symmetry and Renormalisation in Quantum Field Theory [62.997667081978825]
  Quantum systems governed by non-Hermitian Hamiltonians with $PT$ symmetry are special in having real energy eigenvalues bounded below and unitary time evolution. We show how $PT$ symmetry may allow interpretations that evade ghosts and instabilities present in an interpretation of the theory within a Hermitian framework.
  arXiv  Detail & Related papers  (2021-03-27T09:46:36Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.