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arXiv:2111.08343 · SciPost Physics Lecture Notes

Fermionic Gaussian states: an introduction to numerical approaches

Jacopo Surace, Luca Tagliacozzo

Published May 16, 2022• 84 citations
arXivHTMLPDF

Abstract

This document is meant to be a practical introduction to the analytical and numerical manipulation of fermionic Gaussian systems. Starting from the basics, we move to relevant modern results and techniques, presenting numerical examples and studying relevant Hamiltonians, such as the transverse field Ising Hamiltonian, in detail. We finish introducing novel algorithms connecting fermionic Guassian states with matrix product states techniques. All the numerical examples make use of the free Julia package F_utilities.

DOI
10.21468/scipostphyslectnotes.54

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This document is meant to be a practical introduction to the analytical and numerical manipulation of fermionic Gaussian systems. Starting from the basics, we move to relevant modern results and techniques, presenting numerical examples and studying relevant Hamiltonians, such as the transverse field Ising Hamiltonian, in detail. We finish introducing novel algorithms connecting fermionic Guassian states with matrix product states techniques. All the numerical examples make use of the free Julia package F_utilities.
An introduction to the density matrix renormalization group is contained here, including coding examples. The focus of this code is on basic operations involved in tensor network computations, and this forms the foundation of the DMRjulia library. Algorithmic complexity, measurements from the matrix product state, convergence to the ground state, and other relevant features are also discussed. The present document covers the implementation of operations for dense tensors into the Julia language. The code can be used as an educational tool to understand how tensor network computations are done in the context of entanglement renormalization or as a template for other codes in low level languages. A comprehensive Supplemental Material is meant to be a "Numerical Recipes" style introduction to the core functions and a simple implementation of them. The code is fast enough to be used in research and can be used to make new algorithms.
We review existing classical simulation methods for performing fermionic Gaussian operations and develop new methods to address the gap by adhering to the fundamental theoretical framework established by Bravyi [Quantum Info. Comput. 5, 216 (2005)] for the most general fermionic Gaussian processes. Throughout this attempt, the focus remains on the unified approach that can be applied to generic fermionic Gaussian operations. This is beneficial since the selection of simulation methods has often been based on an ad hoc choice, heavily influenced by the specific model and circumstances, rather than on a systematic approach.
We study the structure of the continuous matrix product operator (cMPO)[1] for the transverse field Ising model (TFIM). We prove TFIM’s cMPO is solvable and has the form <?CDATA T=e−12H^FT={{\rm{e}}}^{-\displaystyle \frac{1}{2}{\hat{H}}_{{\rm{F}}}}T=e−21​H^F​?> T = e − 1 2 H ^ F . <?CDATA H^F{\hat{H}}_{{\rm{F}}}H^F​?> H ^ F is a non-local free fermionic Hamiltonian on a ring with circumference β, whose ground state is gapped and non-degenerate even at the critical point. The full spectrum of <?CDATA H^F{\hat{H}}_{{\rm{F}}}H^F​?> H ^ F is determined analytically. At the critical point, our results verify the state–operator-correspondence[2] in the conformal field theory (CFT). We also design a numerical algorithm based on Bloch state ansatz to calculate the low-lying excited states of general (Hermitian) cMPO. Our numerical calculations coincide with the analytic results of TFIM. In the end, we give a short discussion about the entanglement entropy of cMPO’s ground state.
This chapter is an introduction to some computation techniques for fermionic states. After defining Grassmann algebras it is possible to get a classical analogue for the fermionic degrees of freedom in a quantum system. Following the basic work of Berezin [23, 26], we show that we can compute with Grassmann numbers as we do with complex numbers: derivation, integration, Fourier transform. After that we show that we have quantization formula for fermionic observables. In particular there exists a Moyal product formula. As an application we consider explicit computations for propagators with quadratic Hamiltonians in annihilation and creation operators.