arXiv subjects

P. N. Thomas Lloyd

Publications and source records attributed to P. N. Thomas Lloyd.

5 recordsLinked to original sources

arXiv2026

Successes and challenges of using Semidefinite Programming for the study of Spin Chain Hamiltonians

We study semidefinite programming (SDP) methods to analyze spin chain Hamiltonians. We examine the ground state energy, the first excited charged states and ground state correlators in two simple models: the Ising model in a transverse magnetic field and the closely related 3-state Potts model. Our goal is to understand precisely what the SDP program is doing and when it works well, why it does so. We focus on the following novel ingredients: using charge constraints to obtain excited states and to see if additional constraints from integrable models are effective at improving the method. At criticality we also explore to what extent we can use approximate Virasoro correlators to extract conformal data: the central charge and some critical exponents of charged states. We also use these to identify the location of the phase transition. In the special case where the system is made of free fermions we prove that the SDP finds the exact energy of the ground state and produces the correct two point functions of the fermions. Away from free fermion theories, the SDP gets progressively worse at estimating data beyond the value of the ground state energy (like correlation functions), although it qualitatively matches these. In order to be effective, the SDP seems to run into scaling issues where the amount of input needed scales poorly with the lattice volume.

hep-th↗

arXiv2024

Numerical exploration of the bootstrap in spin chain models

We analyze the bootstrap approach (a dual optimization method to the variational approach) to one-dimensional spin chains, leveraging semidefinite programming to extract numerical results. We study how correlation functions in the ground state converge to their true values at and away from criticality and at relaxed optimality. We consider the transverse Ising model, the three state Potts model, and other non-integrable spin chains and investigate to what extent semidefinite methods can reliably extract numerical emergent physical data, including conformal central charges, correlation lengths and scaling dimensions. We demonstrate procedures to extract these data and show preliminary results in the various models considered. We compare to exact analytical results and to exact diagonalization when the system volume is small enough. When we attempt to go to the thermodynamic limit, the semidefinite numerical method with translation invariance imposed as a constraint finds the solution with periodic boundary conditions even if these have not been specified. This implies that the determination of all conformal data in correlators has to be handled at finite volume. Our investigation reveals that the approach has practical challenges. In particular, the correlation functions extracted from the optimal solution, which function as slack variables in the optimization, have convergence issues that suggest an underlying exponential complexity in the system size.

hep-th↗

arXiv2024

Staggered bosons and Kahler-Dirac bosons

We describe a novel way to think about bosonic lattice theories in Hamiltonian form where each lattice site has only a half boson degree of freedom. The construction requires a non-trivial Poisson bracket between neighboring sites and leads to gapless theories with non-invertible symmetries. We also describe a bosonic version of Kahler-Dirac fermions, dubbed Kahler-Dirac bosons that can be performed on any triangulation of a manifold. This also leads to a straightforward implementation of supersymmetry on the lattice and one immediately deduces the Dirac equation of the corresponding Kahler-Dirac fermions.

hep-th↗

arXiv2023

One dimensional Staggered Bosons, Clock models and their non-invertible symmetries

We study systems of staggered boson Hamiltonians in a one dimensional lattice and in particular how the translation symmetry by one unit in these systems is in reality a non-invertible symmetry closely related to T-duality. We also study the simplest systems of clock models derived from these staggered boson Hamiltonians. We show that the non-invertible symmetries of these lattice models together with the discrete ${\mathbb Z}_N$ symmetry predict that these are critical points with a $U(1)$ current algebra at $c=1$ and radius $\sqrt{2N}$ whenever $N>4$.

hep-th↗

arXiv2022

Changes in electric-field noise due to thermal transformation of a surface ion trap

We aim to illuminate how the microscopic properties of a metal surface map to its electric-field noise characteristics. In our system, prolonged heat treatments of a metal film can induce a rise in the magnitude of the electric-field noise generated by the surface of that film. We refer to this heat-induced rise in noise magnitude as a thermal transformation. The underlying physics of this thermal transformation process is explored through a series of heating, milling, and electron treatments performed on a single surface ion trap. Between these treatments, $^{40}$Ca$^+$ ions trapped 70~$μ$m above the surface of the metal are used as detectors to monitor the electric-field noise at frequencies close to 1~MHz. An Auger spectrometer is used to track changes in the composition of the contaminated metal surface. With these tools we investigate contaminant deposition, chemical reactions, and atomic restructuring as possible drivers of thermal transformations.

cond-mat.mtrl-sci↗