Publications by Marius Christopher Lemm

A class of two-dimensional AKLT models with a gap

The AKLT spin chain is the prototypical example of a frustration-free quantum spin system with a spectral gap above its ground state. Affleck, Kennedy, Lieb, and Tasaki also conjectured that the two-dimensional version of their model on the hexagonal latti ...

American Mathematical Society2020

Finite-size criteria for spectral gaps in 𝐷-dimensional quantum spin systems

Marius Christopher Lemm

We generalize the existing finite-size criteria for spectral gaps offrustration-free spin systems to D>2 dimensions. We obtain a local gapthreshold of 3/n, independent of D, for nearest-neighbor interactions. The 1/n scaling persists for arbitrary finite-r ...

American Mathematical Society2020

Quantitative lower bounds on the Lyapunov exponent from multivariate matrix inequalities

Marius Christopher Lemm

The Lyapunov exponent characterizes the asymptotic behavior of long matrix products. Recognizing scenarios where the Lyapunov exponent is strictly positive is a fundamental challenge that is relevant in many applications. In this work we establish a novel ...

2020

Existence of a Spectral Gap in the Affleck-Kennedy-Lieb-Tasaki Model on the Hexagonal Lattice

Marius Christopher Lemm

The S=1 Affleck-Kennedy-Lieb-Tasaki (AKLT) quantum spin chain was the first rigorous example of an isotropic spin system in the Haldane phase. The conjecture that the S=3/2 AKLT model on the hexagonal lattice is also in a gapped phase has remained open, de ...

2020

A Local Law for Singular Values from Diophantine Equations

Marius Christopher Lemm

We introduce the

N\times N

random matrices

X_{j,k}=\exp\left(2\pi i \sum_{q=1}^d\ \omega_{j,q} k^q\right) \quad \text{with } \{\omega_{j,q}\}_{\substack{1\leq j\leq N\\ 1\leq q\leq d}} \text{ i.i.d. random variables},

and

d

a fixed integer. We pr ...

2020

A central limit theorem for integrals of random waves

Marius Christopher Lemm, Matthew De Courcy-Ireland

We derive a central limit theorem for the mean-square of random waves in the high-frequency limit over shrinking sets. Our proof applies to any compact Riemannian manifold of arbitrary dimension, thanks to the universality of the local Weyl law. The key te ...

2019

Global eigenvalue distribution of matrices defined by the skew-shift

Marius Christopher Lemm

We consider large Hermitian matrices whose entries are defined by evaluating the exponential function along orbits of the skew-shift

\binom{j}{2} \omega+jy+x \mod 1

for irrational

\omega

. We prove that the eigenvalue distribution of these matrices conv ...

2019

On the finite-size Lyapunov exponent for the Schröedinger operator with skew-shift potential

Marius Christopher Lemm

It is known that a one-dimensional quantum particle is localized when subjected to an arbitrarily weak random potential. It is conjectured that localization also occurs for an arbitrarily weak potential generated from the nonlinear skew-shift dynamics: $v_ ...

2019

Spectral gaps of frustration-free spin systems with boundary

Marius Christopher Lemm

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 frustration-free spin systems and their applications. We exten ...

2019

Gaplessness is not generic for translation-invariant spin chains

Marius Christopher Lemm

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, Phys. Rev. Lett. 119, 220504 (2017)] shows that a spatially random loca ...

2019

Marius Christopher Lemm

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A class of two-dimensional AKLT models with a gap

Finite-size criteria for spectral gaps in 𝐷-dimensional quantum spin systems

Quantitative lower bounds on the Lyapunov exponent from multivariate matrix inequalities

Existence of a Spectral Gap in the Affleck-Kennedy-Lieb-Tasaki Model on the Hexagonal Lattice

A Local Law for Singular Values from Diophantine Equations

A central limit theorem for integrals of random waves

Global eigenvalue distribution of matrices defined by the skew-shift

On the finite-size Lyapunov exponent for the Schröedinger operator with skew-shift potential

Spectral gaps of frustration-free spin systems with boundary

Gaplessness is not generic for translation-invariant spin chains

Global eigenvalue distribution of matrices defined by the skew-shift

On the finite-size Lyapunov exponent for the Schröedinger operator with skew-shift potential

Gaplessness is not generic for translation-invariant spin chains

Quantitative lower bounds on the Lyapunov exponent from multivariate matrix inequalities

A central limit theorem for integrals of random waves

Spectral gaps of frustration-free spin systems with boundary

A class of two-dimensional AKLT models with a gap

Existence of a Spectral Gap in the Affleck-Kennedy-Lieb-Tasaki Model on the Hexagonal Lattice

A Local Law for Singular Values from Diophantine Equations

Finite-size criteria for spectral gaps in 𝐷-dimensional quantum spin systems