The logarithmic term in Szego theorem

.gitignore

@@ -10,3 +10,4 @@ data*.dat
 determinant.pdf
 fig?.jpg
 perturbative.pdf
+determinant-v1.1.pdf

determinant.tex

@@ -17,9 +17,9 @@
 
 
 %\def\fileversion{\InputIfFileExists{.git/current_version}{\it version~}{\it unversioned}%
-\def\fileversion{\it version~0.4\,%
+\def\fileversion{\it version~1.1\,%
 %(\pdffilesize{\jobname.tex} bytes)}
-\if\pdfstrcmp{\pdffilesize{\jobname.tex}}{18119}0{(original)}\else{(changed)}\fi}
+\if\pdfstrcmp{\pdffilesize{\jobname.tex}}{21446}0{(original)}\else{(changed)}\fi}
 
 \DeclareMathOperator{\Tr}{Tr}
 \DeclareGraphicsExtensions{.pdf,.ai}
@@ -41,7 +41,7 @@
 
 
 \maketitle
-{\parskip-\baselineskip\par\noindent {\hfill\raisebox{-\baselineskip}[0pt][0pt]{\fileversion}\hfill}}
+{\parskip-\baselineskip\par\noindent {\hfill\raisebox{-\baselineskip}[0pt][10pt]{\fileversion}\hfill}}
 
 
 
@@ -457,11 +457,78 @@ In this case the solution is easy and equal
 x_{0} = \left(1+\gamma c + \frac{i}{\pi} \gamma \log\frac{1-e^{i\pi c}}{1-e^{-i\pi c}} \right)^{-1}
 \end{equation}
 
-\section{Hubbart-Stratanovich with auxiliary fermionic field}
+%\section{Hubbart-Stratanovich with auxiliary fermionic field}
+%
+%\begin{equation}
+%\mathrm{Tr}_{d} \left[e^{i \lambda d^{+} d} e^{i \frac{\pi}{2} (d^{+} a + a^{+} d)}\right] = e^{i\lambda a^{+}a}
+%\end{equation}
+
+
+%\section{The finite temperature case}
+%
+%For finite temperature the function $f(e^{i\theta})$ is analytic: 
+%\begin{multline}
+%-\log f(e^{i\theta}) = -\log \frac{1}{e^{2\beta\cos\theta - \alpha}} = 
+%\sum_{n=1}^{\infty}\frac{1}{n}e^{-\alpha n} e^{2n\beta\cos\theta} = 
+%\sum_{n=1}^{\infty}\frac{e^{-\alpha n}}{n}\sum_{m=0}^{\infty} \frac{n^{m}\beta^{m}}{m!}  (2\cos\theta)^{m}
+%=\\=
+%\sum_{n=1}^{\infty}\frac{e^{-\alpha n}}{n}\sum_{m=0}^{\infty} \frac{n^{m}\beta^{m}}{m!} \sum_{k=0}^{m} C^{k}_{m} e^{i(2k-m)\theta}
+%=
+%\sum_{n=1}^{\infty}\frac{e^{-\alpha n}}{n}\sum_{m=0}^{\infty} \sum_{k=0}^{m} \frac{n^{m}\beta^{m}}{k!(m-k)!}  e^{i(2k-m)\theta}
+%\end{multline}
+%so that for $\log f(z)=\sum_{l=-\infty}^{\infty}V_{l} z^{l} $ we get
+%\begin{equation}
+%V_{l} = \sum_{n=1}^{\infty}\sum_{s=0}^{\infty} \frac{n^{|l|+2s-1}\beta^{|l|+2s}e^{-\alpha n}}{s!(|l|+s)!}
+%=\sum_{n=1}\frac{e^{-\alpha n}}{n}I_{l}(2\beta n)
+%\end{equation}
+%where $I_{l}(z)$ is modified Bessel function of the first kind.
+
+
+\section{The logarithmic term in Szeg\"o theorem}
+
+Let us consider more accurately the Fisher-Hartwig formula.
+We follow the designations given in \cite{Deift2011} and \cite{Abanov2011}
+In our case
+\begin{equation}
+f(e^{i\theta}) = 1 + (e^{i\lambda}-1)\frac{1}{e^{2\beta\cos\theta - \alpha}+1}
+\end{equation}
+It is crucial to be able to expand into Laurent series:
+\begin{equation}
+\log f(z) = \sum_{l=-\infty}^{\infty}V_{l}z^{l}
+\end{equation}
+For finite $\beta$ the $V_{l}$ gets a cutoff $l\sim \beta$ and Szeg\"o theorem gives us a formula
+for the determinant of $L\times L$ matrix
+\begin{equation}
+\exp\left( V_{0} L + \sum_{l=1}^{\infty} l V_{l}V_{-l}\right)
+\end{equation}
+Note the infinity in the sum.
 
+Aristov \cite{Aristov1998} proposes formally cut off this series by $l=L$ (referring to work \cite{Luttinger1963}) and indeed, it gives a result in the limit of $T=0$
 \begin{equation}
-\mathrm{Tr}_{d} \left[e^{i \lambda d^{+} d} e^{i \frac{\pi}{2} (d^{+} a + a^{+} d)}\right] = e^{i\lambda a^{+}a}
+-\frac{\lambda^{2}}{2\pi^{2}}\left(\log (2L\sin p_{F})+\gamma_\text{Euler}\right)
 \end{equation}
+Nevertheless, the Fisher-Hartwig formula given in \cite{Deift2011}, Eq.~(1.12) does not imply the dependence on $L$ (notation in paper $n$) for non-singular $f$.
+The cutoff is made automatically for $l\sim\beta$, or in other words, the Eq.~(1.12) does not imply the finite $L$ if $T>0$.
+
+For zero temperature the discontinuity forces us to this more general formula in Eq.~(1.12).
+The particular case that interests us is described in \cite{Abanov2011} in Eq.~(B.9) and afterwards.
+The substitution into Eq.~(1.12) of \cite{Deift2011} gives us Eq.~(B.10) in \cite{Abanov2011}.
+Expansion of Eq.~(B.10) in \cite{Abanov2011} for small $\lambda$ gives us very similar result
+\begin{equation}
+-\frac{\lambda^{2}}{2\pi^{2}}\left(\log (2L\sin p_{F})+\gamma_\text{Euler}+1\right)
+\end{equation}
+This result is also mentioned in \cite{LeHur2012}, right after Eq.~(2.49).
+Taking into account the interaction-dependence obtained from the perturbative approach, we get
+\begin{equation}
+-\frac{\lambda^{2}}{2\pi^{2}}\left(K\left[\log (2L\sin p_{F})+\gamma_\text{Euler}\right]+1\right)
+\end{equation}
+The numerical result given by Andreas speaks in favour of this formula:
+\begin{figure}[h]
+\centering
+\includegraphics[scale=.5]{figKG}
+\caption{Analytics and numerics comparison for $p_{F}=\pi/2$ (half-filling)}
+\label{fig:figKG}
+\end{figure}
 
 \bibliographystyle{amsplain}
 \bibliography{../../literature/quantinfo, localcites}

localcites.bib

 %% This BibTeX bibliography file was created using BibDesk.
 %% http://bibdesk.sourceforge.net/
 
-%% Created for Alex Kashuba at 2022-06-03 16:14:16 +0200 
+%% Created for Alex Kashuba at 2022-10-17 21:18:58 +0200 
 
 
 %% Saved with string encoding Unicode (UTF-8) 
@@ -61,10 +61,10 @@
 	url = {https://en.wikipedia.org/wiki/Baker–Campbell–Hausdorff_formula}}
 
 @inbook{McCoyWu1973,
-	author = {B. M. McCoy and T. T. Wu,},
+	author = {B. M. McCoy and T. T. Wu},
 	chapter = {X},
 	date-added = {2022-02-02 15:14:29 +0100},
-	date-modified = {2022-06-03 16:12:10 +0200},
+	date-modified = {2022-10-17 21:18:57 +0200},
 	publisher = {Harvard University Press, Cambridge},
 	title = {The two-dimensional {I}sing model},
 	year = {1973}}