Initial commit

.gitattributes

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+* text=auto

.gitignore

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+.idea
+dist
+
+*.aux
+*.fdb_latexmk
+*.fls
+*.tdo
+*.toc
+*.log
+*.pdf
+*.bbl
+*.bcf
+*.blg
+*.run.xml
+*.out
+
+*.nb

Makefile

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+### Build tools
+#
+LATEXMK := latexmk -pdflatex="luahblatex %O %S" -pdf -dvi- -ps- -quiet -logfilewarninglist
+WS := wolframscript -f
+
+### Directory variables
+#
+PDF_DIR := pdfs
+FIG_DIR := figures
+CALC_DIR := calc
+
+### Here we go
+#
+OUT_PDF:= $(PDF_DIR)/notes.pdf
+
+.PHONY: all
+all: $(OUT_PDF)
+
+### How we do that
+#
+
+## setup main pdf deps as variable that subdirs can add to
+MAIN_PDF_DEPS := bibliography.bib
+
+## Defining common directory recipes
+$(PDF_DIR):
+	mkdir $(PDF_DIR)
+$(FIG_DIR):
+	mkdir -p $(FIG_DIR)
+$(CALC_DIR):
+	mkdir -p $(CALC_DIR)
+
+## Figures
+#
+
+FIGURES := 
+## Making main.pdf and other pdfs
+#
+$(PDF_DIR)/notes.pdf: notes.tex $(MAIN_PDF_DEPS) | $(PDF_DIR) $(FIGURES)
+	$(LATEXMK) $(<F)
+	cp $(@F) $@
+
+### Convenience scripts for tidying tex
+.PHONY: declutter
+declutter:
+	@rm -f *.tdo
+	@rm -f *.run.xml
+	@rm -f *.bbl
+
+.PHONY: tidy
+tidy: declutter
+	@latexmk -c
+.PHONY: clean
+clean: declutter
+	rm -rf $(PDF_DIR)
+	@latexmk -C

README.md

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+# Notes on the lack of convergence for different tau values for sufficiently low T.
+
+Why is this happening?!

bibliography.bib

@@ -0,0 +1,245 @@
+@article{Nam1967,
+	doi = {10.1103/physrev.156.470},
+	url = {https://doi.org/10.1103/physrev.156.470},
+	year = {1967},
+	month = apr,
+	publisher = {American Physical Society ({APS})},
+	volume = {156},
+	number = {2},
+	pages = {470--486},
+	author = {Sang Boo Nam},
+	title = {Theory of Electromagnetic Properties of Superconducting and Normal Systems. {I}},
+	journal = {Physical Review}
+}
+
+@book{SolyomV3,
+	doi = {10.1007/978-3-642-04518-9},
+	url = {https://doi.org/10.1007/978-3-642-04518-9},
+	year = {2010},
+	publisher = {Springer Berlin Heidelberg},
+	author = {Jen{\H{o}} S{\'{o}}lyom},
+	title = {Fundamentals of the Physics of Solids},
+	subtitle = {Normal, Broken-Symmetry, and Correlated Systems},
+	volume = {3},
+}
+
+@article{Henkel1999,
+	doi = {10.1007/s003400050823},
+	url = {https://doi.org/10.1007/s003400050823},
+	year = {1999},
+	month = dec,
+	publisher = {Springer Science and Business Media {LLC}},
+	volume = {69},
+	number = {5-6},
+	pages = {379--387},
+	author = {C. Henkel and S. P\"{o}tting and M. Wilkens},
+	title = {Loss and heating of particles in small and noisy traps},
+	journal = {Applied Physics B}
+}
+
+@article{Henkel2006,
+	doi = {10.1007/s00340-006-2219-9},
+	url = {https://doi.org/10.1007/s00340-006-2219-9},
+	year = {2006},
+	month = apr,
+	publisher = {Springer Science and Business Media {LLC}},
+	volume = {84},
+	number = {1-2},
+	pages = {61--68},
+	author = {C. Henkel and K. Joulain},
+	title = {Electromagnetic field correlations near a surface with a nonlocal optical response},
+	journal = {Applied Physics B}
+}
+
+@article{QubitRelax,
+	doi = {10.1103/physreva.86.010301},
+	url = {https://doi.org/10.1103/physreva.86.010301},
+	year = {2012},
+	month = jul,
+	publisher = {American Physical Society ({APS})},
+	volume = {86},
+	number = {1},
+	author = {Luke S. Langsjoen and Amrit Poudel and Maxim G. Vavilov and Robert Joynt},
+	title = {Qubit relaxation from evanescent-wave Johnson noise},
+	journal = {Physical Review A}
+}
+
+@incollection{Pippard,
+	doi = {10.1016/s0065-2539(08)60130-4},
+	url = {https://doi.org/10.1016/s0065-2539(08)60130-4},
+	year = {1954},
+	publisher = {Elsevier},
+	pages = {1--45},
+	author = {A.B. Pippard},
+	title = {Metallic Conduction at High Frequencies and Low Temperatures},
+	booktitle = {Advances in Electronics and Electron Physics Volume 6}
+}
+
+@article{Churchill2016,
+	doi = {10.1103/physrevb.94.235422},
+	url = {https://doi.org/10.1103/physrevb.94.235422},
+	year = {2016},
+	month = dec,
+	publisher = {American Physical Society ({APS})},
+	volume = {94},
+	number = {23},
+	author = {R. J. Churchill and T. G. Philbin},
+	title = {Electromagnetic reflection,  transmission,  and energy density at boundaries of nonlocal media},
+	journal = {Physical Review B}
+}
+
+@book{AGD,
+	author    = {Abrikosov, A A and Dzyaloshinskii, I and Gorkov, L P and Silverman, Richard A},
+	title     = {Methods of quantum field theory in statistical physics},
+	publisher = {Dover},
+	address   = {New York, NY},
+	year      = {1975},
+}
+
+@article{Zimmermann1991,
+	doi = {10.1016/0921-4534(91)90771-p},
+	url = {https://doi.org/10.1016/0921-4534(91)90771-p},
+	year = {1991},
+	month = nov,
+	publisher = {Elsevier {BV}},
+	volume = {183},
+	number = {1-3},
+	pages = {99--104},
+	author = {W. Zimmermann and E.H. Brandt and M. Bauer and E. Seider and L. Genzel},
+	title = {Optical conductivity of {BCS} superconductors with arbitrary purity},
+	journal = {Physica C: Superconductivity}
+}
+
+@article{Ford1984,
+	doi = {10.1016/0370-1573(84)90098-x},
+	url = {https://doi.org/10.1016/0370-1573(84)90098-x},
+	year = {1984},
+	month = nov,
+	publisher = {Elsevier {BV}},
+	volume = {113},
+	number = {4},
+	pages = {195--287},
+	author = {G.W. Ford and W.H. Weber},
+	title = {Electromagnetic interactions of molecules with metal surfaces},
+	journal = {Physics Reports}
+}
+
+@article{Agarwal1975,
+	doi = {10.1103/physreva.11.230},
+	url = {https://doi.org/10.1103/physreva.11.230},
+	year = {1975},
+	month = jan,
+	publisher = {American Physical Society ({APS})},
+	volume = {11},
+	number = {1},
+	pages = {230--242},
+	author = {G. S. Agarwal},
+	title = {Quantum electrodynamics in the presence of dielectrics and conductors. I. Electromagnetic-field response functions and black-body fluctuations in finite geometries},
+	journal = {Physical Review A}
+}
+
+@article{Nam1967_Part2,
+	doi = {10.1103/physrev.156.487},
+	url = {https://doi.org/10.1103/physrev.156.487},
+	year = {1967},
+	month = apr,
+	publisher = {American Physical Society ({APS})},
+	volume = {156},
+	number = {2},
+	pages = {487--493},
+	author = {Sang Boo Nam},
+	title = {Theory of Electromagnetic Properties of Strong-Coupling and Impure Superconductors. {II}},
+	journal = {Physical Review}
+}
+
+@article{Tenberg2019,
+	doi = {10.1103/physrevb.99.205306},
+	url = {https://doi.org/10.1103/physrevb.99.205306},
+	year = {2019},
+	month = may,
+	publisher = {American Physical Society ({APS})},
+	volume = {99},
+	number = {20},
+	author = {Stefanie B. Tenberg and Serwan Asaad and Mateusz T. M{\k{a}}dzik and Mark A. I. Johnson and Benjamin Joecker and Arne Laucht and Fay E. Hudson and Kohei M. Itoh and A. Malwin Jakob and Brett C. Johnson and David N. Jamieson and Jeffrey C. McCallum and Andrew S. Dzurak and Robert Joynt and Andrea Morello},
+	title = {Electron spin relaxation of single phosphorus donors in metal-oxide-semiconductor nanoscale devices},
+	journal = {Physical Review B}
+}
+
+@article{Kolkowitz2015,
+	doi = {10.1126/science.aaa4298},
+	url = {https://doi.org/10.1126/science.aaa4298},
+	year = {2015},
+	month = jan,
+	publisher = {American Association for the Advancement of Science ({AAAS})},
+	volume = {347},
+	number = {6226},
+	pages = {1129--1132},
+	author = {S. Kolkowitz and A. Safira and A. A. High and R. C. Devlin and S. Choi and Q. P. Unterreithmeier and D. Patterson and A. S. Zibrov and V. E. Manucharyan and H. Park and M. D. Lukin},
+	title = {Probing Johnson noise and ballistic transport in normal metals with a single-spin qubit},
+	journal = {Science}
+}
+
+@book{llv9,
+	place={Oxford},
+	title={Statistical Physics. Part 2},
+	volume={9},
+	publisher={Pergamon Press},
+	author={Lifshitz, Evgenii Mikhailovich. and Pitaevskii, Lev Petrovich.},
+	year={1980}
+}
+
+@book{Tinkham,
+	title={Introduction to Superconductivity},
+	author={Michael Tinkham},
+	edition = 2,
+	isbn={9780486435039},
+	lccn={95022378},
+	series={Dover Books on Physics},
+	year={2004},
+	publisher={Dover Publications},
+	location={Mineola, N.Y.},
+}
+
+@article{Mattis,
+	doi = {10.1103/physrev.111.412},
+	url = {https://doi.org/10.1103/physrev.111.412},
+	year = {1958},
+	month = jul,
+	publisher = {American Physical Society ({APS})},
+	volume = {111},
+	number = {2},
+	pages = {412--417},
+	author = {D. C. Mattis and J. Bardeen},
+	title = {Theory of the Anomalous Skin Effect in Normal and Superconducting Metals},
+	journal = {Physical Review}
+}
+
+@book{FetterWalecka,
+	address = {Boston},
+	author = {Fetter, A. L. and Walecka, J. D.},
+	publisher = {McGraw-Hill},
+	title = {Quantum Theory of Many-Particle Systems},
+	year = 1971
+}
+
+@article{Lindhard,
+	title = {ON THE PROPERTIES OF A GAS OF CHARGED PARTICLES},
+	author = {Lindhard, J},
+	abstractNote = {},
+	doi = {},
+	journal = {Kgl. Danske Videnskab. Selskab Mat.-fys. Medd.},
+	number = 8,
+	volume = 28,
+	year = {1954},
+	month = {1}
+}
+
+@book{LandauLifshitzElectrodynamics,
+	author        = "Landau, Lev Davidovich and Lifshitz, Evgenii Mikhailovich and Pitaevskii, Lev Petrovich",
+	title         = "{Electrodynamics of continuous media; 2nd ed.}",
+	publisher     = "Butterworth",
+	address       = "Oxford",
+	series        = "Course of theoretical physics",
+	year          = "1984",
+	url           = "https://cds.cern.ch/record/712712",
+}

do.sh

@@ -0,0 +1,18 @@
+#!/usr/bin/env sh
+# Do - The Simplest Build Tool on Earth.
+# Documentation and examples see https://github.com/8gears/do
+
+set -Eeuo pipefail # -e "Automatic exit from bash shell script on error"  -u "Treat unset variables and parameters as errors"
+
+build() {
+   echo "I am ${FUNCNAME[0]}ing"
+   make pdfs/notes.pdf
+}
+
+all() {
+   build
+}
+
+"$@" # <- execute the task
+
+[ "$#" -gt 0 ] || printf "Usage:\n\t./do.sh %s\n" "($(compgen -A function | grep '^[^_]' | paste -sd '|' -))"

notes.tex

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+\documentclass{article}
+
+%other packages
+\usepackage{amsmath}
+\usepackage{amssymb}
+\usepackage{physics}
+
+% \usepackage[
+% 	style=phys, articletitle=false, biblabel=brackets, chaptertitle=false, pageranges=false, url=true
+% ]{biblatex}
+
+\usepackage{graphicx}
+\usepackage{todonotes}
+\usepackage{siunitx}
+
+\usepackage{cleveref}
+
+\title{Cliff Notes}
+
+% \addbibresource{./bibliography.bib}
+
+\graphicspath{{./figures/}}
+
+\newcommand{\vf}{v_{\mathrm{F}}}
+\newcommand{\qf}{q_{\mathrm{F}}}
+
+\begin{document}
+
+\maketitle
+
+We wanted to see what happened for sufficiently low temperatures to the noise from varying mean free paths.
+We should expect that the effect of impurities is to broaden the cliff, but for sufficiently cold or warm systems, the effect of impurities should be minimal.
+However, in \cref{fig:nc}, we see that for low $T$, the gap between curves does not decrease.
+
+It's unclear why that is.
+The lower limit in the figure is the smallest $T$ such that the code works, so it's possible that that convergence could occur for even lower $T$.
+However, the curves do not even begin to approach each other.
+Also, this range of $T$ values covers a couple orders of magnitude in $\Delta$, so it seems that some effect on $\tau$ should be visible, at least in the values closest to $\tau T_c \approx 1$.
+Varying aspects of the numerical calculation like the cutoff for interpolation don't change the relationship between the curves.
+
+I think the most likely explanation is that somewhere a numerical error proportional to $\tau$ gets incorporated somewhere, which is not unreasonable given how $\tau$ gets added to the frequency (so there's likely a sum of two terms of different order of magnitude).
+More investigation would likely be good.
+
+\begin{figure}[htp]
+	\centering
+	\includegraphics[width=\linewidth]{low_t_no_converge}
+	\caption{$\chi(T)$ for low temperatures, for different $\tau$.} \label{fig:nc}
+\end{figure}
+
+% \printbibliography
+
+\end{document}