Gaplessness is not generic for translation-invariant spin chains

@article{Lemm2019GaplessnessIN,
  title={Gaplessness is not generic for translation-invariant spin chains},
  author={Marius Lemm},
  journal={Physical Review B},
  year={2019}
}
  • M. Lemm
  • Published 28 February 2019
  • Physics
  • Physical Review B
The existence of a spectral gap above the ground state has far-reaching consequences for the low-energy physics of a quantum many-body system. A recent work of Movassagh [R. Movassagh, PRL 119 (2017), 220504] shows that a spatially random local quantum Hamiltonian is generically gapless. Here we observe that a gap is more common for translation-invariant quantum spin chains, more specifically, that these are gapped with a positive probability if the interaction is of small rank. This is in line… 
View PDF on arXiv
10 Citations
Highly Influential Citations
1
Background Citations
6

10 Citations

Citation Type

Citation Type

Quantitatively improved finite-size criteria for spectral gaps
TLDR
A new weighting scheme which has the benefit of adapting seamlessly to dimensions ≥ 3 where deriving the optimal threshold scaling l−2 has so far been inaccessible to the weighting approach of [9] and instead required sophisticated tools form quantum information theory [3].
Random translation-invariant Hamiltonians and their spectral gaps
We consider random translation-invariant quantum spin Hamiltonians on Z in which the nearest-neighbor interaction in every direction is randomly sampled and then distributed across the lattice. Our…
Existence of a Spectral Gap in the Affleck-Kennedy-Lieb-Tasaki Model on the Hexagonal Lattice.
TLDR
A mathematical finite-size criterion is proved which gives an analytical, size-independent bound on the spectral gap if the gap of a particular cut-out subsystem of 36 spins exceeds a certain threshold value, and this is verified numerically by performing state-of-the-art DMRG calculations on the subsystem.
Undecidability of the Spectral Gap in One Dimension
The spectral gap problem - determining whether the energy spectrum of a system has an energy gap above ground state, or if there is a continuous range of low-energy excitations - pervades quantum…
Improved local spectral gap thresholds for lattices of finite size
Knabe's theorem lower bounds the spectral gap of a one dimensional frustration-free local hamiltonian in terms of the local spectral gaps of finite regions. It also provides a local spectral gap…
The spin-one Motzkin chain is gapped for any area weight $t<1$
We prove a conjecture by Zhang, Ahmadain, and Klich that the spin-1 Motzkin chain is gapped for any area weight t < 1. Existence of a finite spectral gap is necessary for the Motzkin Hamiltonian to…
Correlation Length in Random MPS and PEPS
Tensor network states are used extensively as a mathematically convenient description of physically relevant states of many-body quantum systems. Those built on regular lattices, i.e. matrix product…
Numerical Methods for Many-Body Quantum Dynamics
TLDR
A new truncation for matrix product operators is given, which reproduces local properties faithfully without reproducing non-local properties, and the utility of many-body localization as a medium for a thermodynamic engine is studied.
Gapped PVBS models for all species numbers and dimensions
Product vacua with boundary states (PVBS) are cousins of the Heisenberg XXZ spin model and feature [Formula: see text] particle species on [Formula: see text]. The PVBS models were originally…
A tensor network approach to non-chiral topological order

References

SHOWING 1-10 OF 35 REFERENCES
Gapped and gapless phases of frustration-free spin- /1 2 chains
We consider a family of translation-invariant quantum spin chains with nearest-neighbor interactions and derive necessary and sufficient conditions for these systems to be gapped in the thermodynamic…
Generic Local Hamiltonians are Gapless.
TLDR
It is proved that generic quantum local Hamiltonians are gapless and there is a continuous density of states above the ground state, and it is proven that the groundstate has finite size degeneracies.
Automorphic Equivalence within Gapped Phases of Quantum Lattice Systems
Gapped ground states of quantum spin systems have been referred to in the physics literature as being ‘in the same phase’ if there exists a family of Hamiltonians H(s), with finite range interactions…
Spectral gaps of frustration-free spin systems with boundary
In quantum many-body systems, the existence of a spectral gap above the ground state has far-reaching consequences. In this paper, we discuss "finite-size" criteria for having a spectral gap in…
Undecidability of the spectral gap
TLDR
This work constructs families of quantum spin systems on a two-dimensional lattice with translationally invariant, nearest-neighbour interactions, for which the spectral gap problem is undecidable, and implies that there exists no algorithm to determine whether an arbitrary model is gapped or gapless.
Local gap threshold for frustration-free spin systems
We improve Knabe's spectral gap bound for frustration-free translation-invariant local Hamiltonians in 1D. The bound is based on a relationship between global and local gaps. The global gap is the…
Finitely correlated states on quantum spin chains
We study a construction that yields a class of translation invariant states on quantum spin chains, characterized by the property that the correlations across any bond can be modeled on a…
Undecidability of the Spectral Gap in One Dimension
The spectral gap problem - determining whether the energy spectrum of a system has an energy gap above ground state, or if there is a continuous range of low-energy excitations - pervades quantum…
Rigorous RG Algorithms and Area Laws for Low Energy Eigenstates in 1D
TLDR
A new algorithm is given, based on a rigorously justified RG type transformation, for finding low energy states for 1D Hamiltonians acting on a chain of n particles, which is natural and efficient.
A polynomial time algorithm for the ground state of one-dimensional gapped local Hamiltonians
TLDR
A polynomial time algorithm that provably finds the ground state of any 1D quantum system described by a gapped local Hamiltonian with constant ground-state energy is developed.
...
1
2
3
4
...