Abstract
We formulate an optimization problem of Hamiltonian design based on the variational principle. Given a variational ansatz for a Hamiltonian we construct a loss function to be minimised as a weighted sum of relevant Hamiltonian properties specifying thereby the search query. Using fractional quantum Hall effect as a test system we illustrate how the framework can be used to determine a generating Hamiltonian of a finite-size model wavefunction (Moore-Read Pfaffian and Read-Rezayi states), find optimal conditions for an experiment or "extrapolate" given wavefunctions in a certain universality class from smaller to larger system sizes. We also discuss how the search for approximate generating Hamiltonians may be used to find simpler and more realistic models implementing the given exotic phase of matter by experimentally accessible interaction terms.
Highlights
In quantum physics one often starts by postulating a Hamiltonian of the system and studying its properties
Because of the often non-trivial interplay of simple model terms, combining them we could as well find completely new phenomena as exemplified by the recent discovery of the many-body-localised phase [4, 14, 34] found for the disorder, hopping and interaction terms mixed in an appropriate proportion
Examples of the search queries may include: "What is the most stable Hamiltonian with certain quasiparticle braiding statistics?" or "How do I combine the limited number of experimentally accessible interaction terms such that the many-body ground state is in the desired universality class?" The applications we will illustrate are finding parent Hamiltonians for model states, "extrapolating" the model states to larger system sizes and identifying the optimal conditions for an experiment
Summary
In quantum physics one often starts by postulating a Hamiltonian of the system and studying its properties. Examples of the search queries may include: "What is the most stable Hamiltonian with certain quasiparticle braiding statistics?" or "How do I combine the limited number of experimentally accessible interaction terms such that the many-body ground state is in the desired universality class?" The applications we will illustrate are finding parent Hamiltonians for model states, "extrapolating" the model states to larger system sizes and identifying the optimal conditions for an experiment One of these tasks, finding the generating (entanglement) Hamiltonian given an eigenstate attracted interest in literature recently [7, 18, 41, 56, 62]. One might consider the spectral average of nearby gaps ratio as a measure of chaos [2], optimising which would require no reference wavefunction It is the flexibility in the loss function design that leads to the generality of the method being presented.
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