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Use these free images for your websites, art projects, reports, and Powerpoint presentations! Chipotle Mexican Grill Inc on Thursday said better-than-expected sales at established restaurants boosted quarterly profits and signaled it may raise prices in the middle of 2014 to offset higher food costs, sending shares up 7.7 percent.
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Analysts say Chipotle needs mid-single-digit percentage growth of same-restaurant sales to protect profits. Luke Hemsworth first graced our television screens as the baby-faced drug-dealing footy player Nathan Tyson in Neighbours and went on to appear in a range of shows from All Saints to Blue Heelers as well as the television mini-series Bikie Wars: Brothers in Arms.
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Now, with so few people familiar with kohlrabi, the vegetable is often described as looking a lot like Sputnik, which is not far from the truth.
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The Denver-based chain previously said it expected low- to mid-single-digit percentage same-restaurant sales growth for 2013. In 2014, he made a successful transition to films and appeared in The Anomaly, Kill Me Three Times and recently Infini, which is screening as part of the Gold Coast Film Festival. The warehouse where we shot was 48-degrees on a good day and we were all wearing spacesuits! The cast were all so committed that if you stepped out of line and weren't prepared then you would cop a full-on assault from everyone. Shane Abbess (the director) is one of the most complex, brilliant, generous people I've ever met.
I couldn't hit my marks to save myself. I actually had one of my first ever days on set at Neighbours with Dan MacPherson and always loved what he did and how generous he was. Financially it's still a struggle but I think I enjoy acting far more now than I ever did, which makes the sacrifices easier. The flattened globe rests just above the surface of the earth, and long stems shoot up from the curved sides as well as from the top, giving it a spiky space-age look.
Sometimes its foliage will have been removed, like the kohlrabi pictures right, from Stout Oak Farm. Kohlrabi is an excellent source of vitamin C and potassium, and also contains vitamin B6, folic acid, magnesium and copper. Then cut the top and the root section off the kohlrabi, and work your way around with a paring knife or vegetable peeler.
Or you can cut it into wedges or matchsticks, and have it as part of a raw vegetable tray, with or without a sprinkle of salt, a light vinaigrette, or a veggie dip. If the kohlrabi is from a local farmer, it will almost certainly have fresh greens attached. Through advocacy, organizing and education, we work toward a sustainable local food system that meets the needs of both producers and consumers. But it hasn't all been smooth sailing for the oldest Hemsworth brother, he went on an indefinite hiatus from acting to pursue alternate endeavours, but much to the delight of audiences around the world, he is back and better than ever.
I don't want to give too much away but this is the kind of film that leaves you breathless. He has the ability to turn ideas on their heads and make the most of every single nuance and moment, all while providing a safe haven in which to express everything you need to. As an actor on the edge you are forced to do a lot of tough jobs, labouring, bar tending, night work in order to have time to go to auditions. But if you do get kohlbrabi with leaves and stems, be sure to eat them – they can be used as you would kale or any other cooking green. The kohlrabi leaves are rich in vitamin A, so don't forget to sautee them or use them into a soup or stir fry.
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The Weekend Edition Gold Coast caught up with Luke Hemsworth ahead of his visit to the Gold Coast to chat about the pitfalls and perks of acting.
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text:
Q: There is no measure such that $\mu(\lbrace f : (f(t_1), ... , f(t_n)) \in A\rbrace) = \mu_{(t_1,...,t_n)}(A)$ Let $C[0,\infty)$ denote the space of continuous functions $f : [0,\infty) \rightarrow \mathbb{R}$. Let $\mathcal{C} [0,\infty)$
denote the $\sigma$-algebra of subsets of $C[0, \infty)$ generated by sets of the form
$$
\lbrace f ∈ C[0,\infty) : f(t) ∈ A\rbrace,
$$
with $t\in [0,\infty)$ and $A \in \mathcal{B}(\mathbb{R})$. Find a consistent family of probability measures $\mu_{t}$
indexed
by finite sequences $t = (t_1,..., t_n)$ with $0\leq  t_1 < ... < t_n$ with the property that: there is no
measure $\mu$ on $C[0,\infty)$ satisfying
$$
\mu(\lbrace f : (f(t_1), ... , f(t_n)) \in A\rbrace) = \mu_{(t_1,...,t_n)}(A)
$$
for all $n\in\mathbb{N}, t_1 \leq ...\leq t_n$ and $A\in \mathcal{B}(\mathbb{R}^n)$
My approach was thinking about properties of $C[0,\infty)$, because if this space would have been changed to just all functions from $[0,\infty)$ to $\mathbb{R}$, then by Kolmogorov extension theorem there is no consistent family with the property above. Then it seems logical to think about sets like $\lbrace f\in C[0,\infty): f(q_i)=1 \mbox{ iff } q_i\neq 0, \mbox{ otherwise } 0\rbrace$, where $q_i$ is a sequence of all rationals and $r_i$ a sequence of real numbers.
This set is empty, but maybe this is a limit of a decreasing  sequence whose $\mu$ measure is not $0$ (for contradiction).
However, I cannot find such a family of measures. The one time it leads to non-consitency, the other time the limit set has $\mu$-measure $0$. Is there anyone who can find a good measure family? Thanks in advance.

redpajama_set_name: RedPajamaStackExchange


When fifty writers formed the Society of Midland Authors on April 24, 1915, their objectives included "a closer association among the writers of the Middle West" and "the stimulation of creative literary effort." A century later, the Society still accomplishes those goals — hosting literary events, honoring the region's best books with annual awards, and connecting its members with a newsletter, Twitter and other channels.
The Society's founding authors included Clarence Darrow, Harriet Monroe, Hamlin Garland, Edna Ferber, Vachel Lindsay, George Ade, Mary Hastings Bradley, Emerson Hough, Howard Vincent O'Brien, James Whitcomb Riley and William Allen White. Others who joined soon afterward were Jane Addams, Ring Lardner, Edgar Lee Masters, John T. McCutcheon and Lorado Taft.
Crediting John M. Stahl with sparking the idea of an authors organization, the Society's first president, Hobart Chatfield-Taylor, wrote: "None but a bold man would have sought to weld such individualistic — dare I say egotistic? — creatures as authors into a society of any sort."
Based in Chicago, the Society includes published authors from twelve states: Illinois, Indiana, Iowa, Kansas, Michigan, Minnesota, Missouri, Nebraska, North Dakota, Ohio, South Dakota and Wisconsin.
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See http://midlandauthors.com/winners_past.html for full list of past winners and runners-up. Also watch this website near the end of each year for details on how to enter the contest.
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Members must be current or former residents of the twelve Midland states. Each member is the author or co-author of a book demonstrating literary style and published by a recognized publisher, or the author of a published or professionally produced play. The Society also includes associate members — people with a connection to literature, such as librarians, teachers, publishers and editors.
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The banner for this web site was created from a photograph of Preston Bradley Hall, Chicago Cultural Center, formerly the main branch of the Chicago Public Library.
When: Tuesday, Oct. 9, 2018. Cocktail hour: 6-7 p.m.; Panel discussion: 7-8 p.m.
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Michele Weldon: �I chose writer because as a young reader the words on a page could grab me by the throat�or stroke my hair until I fell asleep.� Michele wrote�Escape Points: A Memoir�and�Writing To Save Your Life. She's a journalist, senior leader with�The OpEd Project,�emerita faculty at the Medill School of Journalism, and editorial director of�Take The Lead, and has delivered 200 keynotes worldwide.
R. Craig Sautter: �I love language and the sound and sense of words from which I can compose a kind of cosmic music.� Craig has written or co-written several books, including Inside the Wigwam and Wicked City: Kenna to Capone. He's a poet, philosopher and educator. Craig wrote and produced ads for Barack Obama and served as the 47th president of Midland Authors.
Linda Nemec Foster: �I write because the intangible rewards are gratifying and humbling.� Linda is the author of eleven collections of poetry, including�The Lake Michigan Mermaid and Talking Diamonds. She's also a literary presenter. Linda was selected to be the first Poet Laureate of Grand Rapids, Mich., (2003-05) and founded the Contemporary Writers Series at Aquinas College.
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— Free and open to the public.
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text:
Q: Maven build failing I am new to Java/Coding and release builds etc. Am faced with a problem currently and would appreciate any pointers to help solve. When I try to run a build using Maven on a Mac it fails with the error below. However if I run the same build on a Windows OS it runs fine. Any pointers?
Exception in thread "main" java.lang.StackOverflowError
at java.io.UnixFileSystem.getBooleanAttributes0(Native Method)
at java.io.UnixFileSystem.getBooleanAttributes(UnixFileSystem.java:242)
at java.io.File.exists(File.java:819)
at org.codehaus.plexus.components.io.resources.PlexusIoFileResource.isExisting(PlexusIoFileResource.java:126)
at org.codehaus.plexus.archiver.AbstractArchiver.asArchiveEntry(AbstractArchiver.java:296)
at org.codehaus.plexus.archiver.AbstractArchiver.asArchiveEntry(AbstractArchiver.java:329)
at org.codehaus.plexus.archiver.AbstractArchiver$1.hasNext(AbstractArchiver.java:448)
at org.codehaus.plexus.archiver.AbstractArchiver$1.hasNext(AbstractArchiver.java:467)
at org.codehaus.plexus.archiver.AbstractArchiver$1.hasNext(AbstractArchiver.java:467)
at org.codehaus.plexus.archiver.AbstractArchiver$1.hasNext(AbstractArchiver.java:467)
at org.codehaus.plexus.archiver.AbstractArchiver$1.hasNext(AbstractArchiver.java:467)
at org.codehaus.plexus.archiver.AbstractArchiver$1.hasNext(AbstractArchiver.java:467)


Sentry is cross-platform crash reporting built with love.
如果不做这步,SENTRY_CONF=/var/www/sentry/conf/ sentry upgrade的时候,会报OperationalError: FATAL: Peer authentication failed for user "postgres"
如果提示sentry.exceptions.InvalidConfiguration: MISCONF Redis is configured to save RDB snapshots, but is currently not able to persist on disk. Commands that may modify the data set are disabled. Please check Redis logs for details about the error.
- Loading initial data for nodestore.
worker accepts messages serialized with pickle is a very bad idea!
environment variable (but please think about this before you do).
Background workers haven't checked in recently. This is likely an issue with your configuration or the workers aren't running.


text:
\section{Introduction}

The concept of the renormalization group (RG) is one of the cornerstones in the theory of phase transitions and critical phenomena.  It is based on the idea that a scale transformation can effectively be absorbed in changing the parameters of the system. If these parameters flow to a non-trivial fixed point under renormalization, the corresponding system is expected to be scale-invariant, exhibiting interesting universal features at large scales.

Renormalization as a continuous change of scale is a particularly natural concept in the context of continuum models or field theories. For discretized theories and lattice models, however, RG-transformations are much harder to implement since scale changes have to comply with the underlying lattice structure.  In fact, apart from very few exceptions, most renormalization schemes on lattices are approximate in the sense that  contributions of higher order are neglected.  One of the few exact RG schemes is the  block-spin renormalization group of the one-dimensional Ising model~\cite{nelson1975soluble} on an infinite lattice, where a coarse-graining in pairs of spins is \textit{exactly} equivalent to a parameter change in the Hamiltonian.  Unfortunately, this renormalization scheme is no longer exact in higher dimensions.

The present study is motivated by recent results obtained by Evenbly and White~\cite{evenbly2016entanglement} who demonstrated that discrete wavelet transforms~\cite{chui2016introduction} can be used to establish a real-space RG transformation for quantum systems on a lattice in the context of free-particle systems. Using finite-range Daubechies wavelets~\cite{daubechies1988orthonormal} in a multiscale entanglement renormalization ansatz (MERA)~\cite{vidal2008class} they were able to approximate the ground state of the critical Ising model. The approximative nature of their approach can be traced back to the fact that the employed wavelet filter functions have a finite spatial range. Therefore, the question arises whether it is possible to improve the accuracy by increasing the range of the wavelets and whether we can reach a limit in which such a MERA-inspired RG scheme becomes exact in the sense that the lattice system at the RG fixed point behaves exactly in the same way as its continuum counterpart.

In this paper we demonstrate that such a limit does indeed exist, giving explicit examples of exact lattice RG schemes for simple integrable systems. The RG procedures used here are based on long-range wavelets in the limit of perfect momentum separation,  known in wavelet theory as the Shannon limit~\cite{cattani2008shannon}.  Roughly speaking, such transformations divide the momentum space of the theory into two equal halves separating low and high wave numbers.  Studying various types of free theories (free bosons, free Dirac fermions, and the Ising model), we find that such a transformation completely decouples the low- and high-momentum modes, providing a consistent exact renormalization scheme  on a lattice. The exactness, however, comes at the price that the RG flow generates algebraic long-range interactions. In many situations the emergence of such long-range interactions is undesirable. Here  however, we would like to adopt the opposite viewpoint, studying long-range interactions as an interesting subject on its own and taking advantage of them. The main results are:

\begin{itemize}
 \item The filter functions in the limit of perfect momentum separation are unique and lead to non-local interactions which decay algebraically (i.e. with a power law) with distance.
 \item For free theories,  the RG scheme produces exactly the same energy excitation spectrum as in the continuum limit, bounded by a cut-off energy given by the inverse lattice spacing.
 \item The RG fixed point exhibits the same universal properties as the corresponding short-range model at criticality.
 \item At the long-range RG fixed point we find that various physical properties, e.g. the specific heat, are much less affected by lattice artifacts than in the short-range case.
 \item In fermionic models, the problem of fermion doubling, which emerges as an artifact in fermion lattice models with short-range interactions, is no longer present since the doubled UV-modes are eliminated already after the first RG step.
\end{itemize}

Wavelets may also provide a new approach for constructing higher-dimensional hyperbolic spaces, which is of  relevance for the AdS/CFT correspondence. It is a well-known fact that within AdS/CFT, the renormalization scale can be viewed as an extra dimension. In the context of MERA networks, this was first pointed out in \cite{Swingle:2009bg}. More recently, the approach of exact holographic mappings \cite{Qi:2013caa} has been connected to free fermion renormalization via a Haar wavelet  \cite{Lee:2015vla} and also extended to more general wavelets with a larger but finite range in \cite{Singh:2016mxd,PhysRevB.96.245103}. While previous applications of wavelets to AdS/CFT-inspired lattice models used wavelets with compact support, the main ingredient of the present work is that  in the Shannon limit, the wavelets have an infinite range and are therefore highly non-local. Since our goal is to achieve an exact decoupling of the IR and UV modes at every renormalization step, the `bulk' correlation functions in the holographic direction vanish, and therefore the standard methods for defining an emergent metric in the wavelet approach  \cite{Qi:2013caa,PhysRevB.96.245103} are not directly applicable. However, as we shall see, our approach provides an exact realization of Verma modules up to a certain energy scale and therefore may provide a new alternative approach to constructing  gravity duals.

As a concrete application to CFTs, we shall illustrate the usefulness of the Shannon limit by examining the spectrum of the quantum Ising model on the lattice at criticality. Although conformal symmetry is broken by the lattice, the non-local wavelet flow allows to recover the correct Verma module spectrum up to a scale of the inverse lattice space. Although this is not a strongly coupled theory, it provides a first step towards the better understanding of scale invariant theories on the lattice. Indeed, the Ising model was studied in connection to holography \cite{Castro:2011zq}.

The paper is organized as follows. In the following section we first demonstrate the key concept in the example of a single free quantum particle. In the third section the long-range filter functions are introduced on infinite as well as finite lattices. In the sections 4-6 this concept is applied to free bosons, free fermions, and the critical Ising model. After some concluding remarks, technical details are given in an appendix.

\section{Free quantum particle}

As the simplest example that outlines the main idea let us consider a free quantum particle in 1+1 dimensions which evolves according to the non-relativistic Schr\"odinger equation (setting $\hbar=m=1$)
\begin{equation}
\label{schroedinger}
i \frac{\partial}{\partial t}\psi(x,t) \;=\; -\frac12 \nabla^2 \psi(x,t) \,.
\end{equation}
This equation is invariant under the scale transformation $x \to \lambda x, \,\, t \to \lambda^z t$  with the dynamical exponent $z=2$, which is reflected in the quadratic dispersion relation
\begin{equation}
\label{quadraticdispersion}
\omega = \frac12 k^2\,.
\end{equation}
Let us now consider a discretized variant of the Schr\"odinger equation on an infinite lattice with equidistant sites positioned at $x_n = na$,  where $a$ is the lattice spacing and $n \in \mathbb Z$. Here it is customary to discretize the Laplacian on the r.h.s. of Eq.~(\ref{schroedinger}) by
\begin{equation}
\label{discreteSchroedinger}
i \frac{\partial}{\partial t} \psi_n \;=\; -\frac1{2a^2} \bigl( \psi_{n-1}-2\psi_n+\psi_{n+1}\bigr)\,,
\end{equation}
where $\psi_n:=\psi(x_n,t)$. It is easy to verify that the discretized Laplacian leads to a modified dispersion relation of the form
\begin{equation}
\omega = 1-\cos(a k)\,, \qquad k \in [-\tfrac{\pi}{a},\tfrac{\pi}{a}]
\end{equation}
which reproduces Eq. (\ref{quadraticdispersion}) in the limit $|k| \to 0$ but deviates from the quadratic law for all $k\neq 0$ due to the finite lattice spacing.

Is it possible to find a modified Laplacian $\Delta^*$ that behaves on the lattice \textit{exactly} in the same way as in the continuum limit, exhibiting a clean quadratic dispersion? This is in fact an easy exercise. Because of linearity and translational invariance such a Laplacian has to be a convolution product of the form
\begin{equation}
(\Delta^* \psi)_n \;=\; \sum_{n'} \Delta^*_{n-n'} \,\psi_{n'}\,,
\end{equation}
with coefficients $\Delta^*_r$ given by the inverse Fourier transform of the dispersion relation:
\begin{equation}
\label{longrangelaplacian}
\Delta^*_r = -\frac{a}{2\pi} \int_{-\pi/a}^{\pi/a} k^2\,e^{i a k r}\, d k \;=\;
\left\{
\begin{array}{ll}
-\frac{\pi^2}{3a^2} & \textrm{ if } r=0\\
-\frac{2(-)^{r}}{r^2a^2} & \textrm{ otherwise }
\end{array}
\right.
\end{equation}
This modified discrete Laplacian, known in lattice quantum field theory as the SLAC-derivative~\cite{quinn1986new}, is advantageous in many ways. For example, the solutions of the discretized Schr\"odinger equation $i \dot \psi_n = \frac12 (\Delta \psi)_n$ will coincide \textit{exactly} with those of the continuous Schr\"odinger equation for all initial conditions in the common functional space, i.e., with Fourier components confined to the Brillouin zone $|k|\leq \pi/a$.

\begin{figure}
\centering\includegraphics[width=130mm]{laplacian}
\vspace{-2mm}
\caption{Left: Magnitude of the components of the short-range lattice Laplacian~$\Delta$ and its long-range counterpart $\Delta^*$ defined in  (\ref{longrangelaplacian}). Right: Corresponding dispersion relations.}
\label{fig:laplacian}
\end{figure}

However, the modified Laplacian \eqref{longrangelaplacian} is no longer local as \eqref{discreteSchroedinger}, but instead involves algebraically decaying long-range interactions with alternating signs (see Fig.~\ref{fig:laplacian}). Although this may seen as be a major drawback in various applications, the aim of the present work is to appreciate such long-range interactions and to investigate their properties. In particular, we are going to show that such interaction structures arise naturally as a fixed point under a certain kind of renormalization.

\section{Wavelet-inspired renormalization scheme}
\label{sec:RGScheme}

\begin{figure}[b]
\centering\includegraphics[width=100mm]{wavelettransform}
\vspace{-2mm}
\caption{Wavelet transformation, splitting the incoming signal by a cascade of high-pass filters (UV), low-pass filters (IR), and downsamplers ($\downarrow$). See text.}
\label{fig:wavelet}
\end{figure}

Recently Evenbly and White~\cite{evenbly2016entanglement} showed that discrete wavelet transforms can be used for entanglement renormalization, giving the first analytic example of a MERA that approximates the ground state of a critical Ising model. Wavelet transforms can be understood as localized Fourier transforms, providing a compromise between spectral and real-space resolution, and play an important role in signal processing.

The basic setup of a wavelet transformation is sketched in Fig.~\ref{fig:wavelet}. The incoming data is split into two identical copies which are passed to a high-pass (UV) and a low-pass (IR) filter. The filtered signals are then processed by downsamplers, which essentially pick up every other data point and therefore halve the degrees of freedom. As shown in the figure, this procedure is iterated several times. In wavelet theory the key point is to choose the filters in such a way that the transformation sequence as a whole is unitary.

There is a large variety of wavelet transformations with a different trade-off between filter quality and locality. As an example let us consider so-called Daubechies wavelets of order $N$, whose filter functions act on $N$ adjacent lattice sites~\cite{daubechies1992ten}. As demonstrated in Fig.~\ref{fig:filter}, the filter quality of these wavelets increases with $N$ on the expense of locality. In the so-called Shannon limit $N \to \infty$, the wavelet filters separate low and high momenta perfectly (sketched as solid lines in Fig.~\ref{fig:filter}), but then the corresponding filter functions are entirely non-local.

Although this non-locality may be seen as a disadvantage, the Shannon limit of perfect momentum separation has another significant advantage in the context of renormalization. One observes that for wavelets with a finite range, the overlapping filter functions effectively lead to couplings between the IR and the UV sector, but in the Shannon limit it turns out that both sectors decouple entirely, which is a prerequisite for a practical renormalization scheme.

If we decided to use finite-range wavelets, we could of course try to remove the remaining couplings between the IR and the UV sectors by additional downstream disentanglers, but it turns out that the resulting compound (wavelet plus disentangler) always reproduces the Shannon limit independent of the initial choice of the wavelet (see Appendix A for further details). Thus we can restrict our analysis to the Shannon limit from the beginning on.

The Shannon limit of perfect momentum separation can be implemented without explicit use of wavelets by a simple sequence of Fourier transformations, as sketched schematically in Fig.~\ref{fig:momentum-separation}. Suppose that we start with a signal or a data set consisting of $N$ discrete values. At first this data set is transformed to $k$-space by a simple discrete Fourier transformed (DFT) of size $N$. Then the output  is divided  'by hand' into two equal halves of low and high momenta, and finally each of the parts is transformed back to real space by an inverse DFT of size $N/2$. This implements a perfect Shannon filter without the explicit use of wavelets and downsamplers.

\begin{figure}
\centering\includegraphics[width=80mm]{filter}
\vspace{-2mm}
\caption{Filter transfer functions in the range $0 \leq |k| \leq \pi$ for the Daubechies-4 wavelet (dashed line), Daubechies-4 (dotted line), and in the Shannon limit of perfect momentum separation (solid line). }
\label{fig:filter}
\end{figure}

\begin{figure}
\centering\includegraphics[width=70mm]{momentum-separation}
\vspace{-2mm}
\caption{Perfect momentum separation of a data set. The data is first transformed to momentum space by means of a discrete Fourier transform DFT$_N$. Then the momenta are separated into two halves of low and high momenta, as indicated by the connections bands. These halves are then transformed back to real space by two inverse DFT$_{N/2}$. }
\label{fig:momentum-separation}
\end{figure}

As we are going to show below, the effect of perfect momentum separation applied to free bosonic or fermionic models amounts to cutting the dispersion relation of the excitation modes into two pieces. If we apply this procedure repeatedly to the IR part we obtain an RG scheme which iteratively `zooms' into the low-$k$ regime of the dispersion relation (see Fig.~\ref{fig:RG-scheme}). This process of repeated zooming defines the fixed point of the RG scheme. For example, if we start with a dispersion relation which is parabolic around $k=0$, and if we apply perfect momentum separation several times, the dispersion in the IR sector looks more and more like a parabola, approaching an RG fixed point with a perfectly quadratic dispersion. However, although the UV and IR parts decouple in each step, the couplings \textit{within} each of the parts become increasingly long-ranged under the renormalization flow, and it is intuitively clear that the RG fixed point of a perfectly quadratic dispersion is related to the modified Laplacian~(\ref{longrangelaplacian}) introduced in the previous Section. For the models studied in this paper, this allows us to construct a corresponding clean' variant with long-range interactions which nevertheless exhibits the same universal properties.

\begin{figure}
\centering\includegraphics[width=160mm]{RG-scheme}
\caption{Schematic illustration of the renormalization procedure investigated in this paper. The momentum range is divided in to two equal halves, containing the low momenta (IR) and the high momenta (UV). This separation cuts the dispersion relation (solid line) into two pieces. The RG step is iteratively applied to the IR part, magnifying the dispersion relation around $k=0$. This eventually leads to a fixed point where the dispersion relation is exactly quadratic.}
\label{fig:RG-scheme}
\end{figure}

\section{Momentum-separating filter functions}
\label{MomentSeparating}

\subsection*{Infinite lattice:}
Let us consider a data set in the form of an infinite-dimensional square-normalizable complex vector $f=\{f_n \in\mathbbm C|n\in\mathbbm Z\}$. This data set can be encoded as a Fourier series
\begin{equation}
\tilde f(k) = \sum_{n\in \mathbbm Z} e^{i k n} f_n
\end{equation}
with a continuum of momenta in the range $k \in [-\pi,\pi]$. This interval is now divided into two equally-sized halves $|k|\leq \frac \pi 2$ (IR) and $\frac \pi 2 < |k| \leq \pi $ (UV), splitting the above Fourier series into
\begin{equation}
  {\tilde f}^{\mathrm{ IR}}(k) \;:=\; \frac{\tilde f\bigl(\tfrac k 2\bigr)} {\sqrt 2} , \qquad
  \tilde f^\mathrm{ UV}(k) \;:=\; \frac{\tilde f\bigl(\tfrac k 2 - \mathrm{sgn}(k) \pi\bigr)}{\sqrt 2}\,,
\end{equation}
where the prefactor of $1/\sqrt{2}$ is included to ensure unitarity. After separating and rescaling the momenta, we apply an inverse Fourier transform to these functions (cf. Fig.~\ref{fig:momentum-separation}), obtaining the real-space vectors
\begin{equation}
    f_m^{\mathrm{IR/UV}}
    = \frac 1{2\pi}\int_{-\pi}^{+\pi} \mathrm{d} k\, e^{- i k m } \tilde f^{\,\mathrm{IR/UV}}(k)
    = \sum_{n\in \mathbbm Z} S^{\mathrm{IR/UV}}_{2m-n} f_n,
\end{equation}
where
\begin{equation}
  S^{\mathrm{IR/UV}}_j
  = \frac{1}{\sqrt 2 \, \pi} \int_{-\pi}^\pi \mathrm{d} k\, e^{-i k j} \theta\big(\pm (\pi/2 - |k|)\big)
\end{equation}
are the perfect low- and highpass filters. Evaluating these integrals yields
\begin{equation}
\label{eq:Sinf}
S^{\mathrm{IR/UV}}_{j} = \begin{cases}
\frac{\delta_r}{\sqrt{2}} & \mathrm{if} \ j \ \mathrm{even} \\
\pm \frac{\sqrt 2 }{\pi j} \ (-1)^{(j-1)/2}& \mathrm{if} \ j \ \mathrm{odd}
\end{cases}
\end{equation}
or in table form:\\[-4mm]
\begin{center}
  \small
  \begin{tabular}{|c|cccccc|c|cccccc|} \hline
    $j$ & $\cdots$ &  -5 & -4 & -3 & -2 & -1 & \textbf 0 & 1 & 2 & 3 & 4 & 5 & $\cdots$ \\ \hline
    $S^{\mathrm{IR}}_j$ & $\cdots$ & $+\frac{\sqrt{2}}{5\pi}$ & 0 & $-\frac{\sqrt{2}}{3\pi}$ &  0 & $+\frac{\sqrt{2}}{\pi}$ & $\mathbf{\frac{1}{\sqrt{2}}}$ & $+\frac{\sqrt{2}}{\pi}$ & 0 & $-\frac{\sqrt{2}}{3\pi}$ &  0 & $+\frac{\sqrt{2}}{5\pi}$ & $\cdots$  \\
    $S^{\mathrm{UV}}_j$ & $\cdots$ &  $-\frac{\sqrt{2}}{5\pi}$ & 0 & $+\frac{\sqrt{2}}{3\pi}$ &  0 & $-\frac{\sqrt{2}}{\pi}$ & $\mathbf{\frac{1}{\sqrt{2}}}$ & $-\frac{\sqrt{2}}{\pi}$ & 0 & $+\frac{\sqrt{2}}{3\pi}$ &  0 & $-\frac{\sqrt{2}}{5\pi}$ & $\cdots$  \\\hline
  \end{tabular}
\end{center}
This transformation can be shown to be unitary in the sense that the block matrix
\begin{equation}
S := (S^{\mathrm{IR}} \oplus S^{\mathrm{UV}})_{n,m} = \binom{S^{\mathrm{IR}}_{2m-n}}{S^{\mathrm{UV}}_{2m-n}}
\end{equation}
satisfies
\begin{equation}
\sum_{l\in\mathbb Z}S^\alpha_{m,l}S^\beta_{n,l} = \delta_{\alpha,\beta}\delta_{m,n}\,.
\qquad \alpha,\beta \in\{\mathrm{IR,UV}\}
\end{equation}

\subsection*{Finite lattice:}
In analogy to the above, we can define a moment-separating transformation on a square-normalizable \emph{finite} complex vector $f=\{f_0,\ldots,f_{N-1}\} \in \mathbbm C^N $ with $N$ components (for simplicity we assume that $N$ is a power of $2$ and that $N\geq 4$). Here we start with a discrete Fourier transform (DFT)
\begin{equation}
\tilde f_q := \frac{1}{\sqrt{N}} \sum_{n=0}^{N-1} e^{\frac{2\pi \mathrm i n q}{N} } f_n\,,\qquad q\in\{-\tfrac N 2 ,\ldots, \tfrac N 2 -1\}
\end{equation}
where $q$ is the index of the wave numbers $k_q = \frac{2\pi q}{N}$. Again we would like to divide the range of $q$ into two equal halves, separating low and high momenta. However, the two modes $q=\pm \frac N 4$ are exactly on the edge between IR and UV, so it is not clear to which part they belong. We could of course make a choice and assign one of them to the IR part and the other one to the UV part, but then the resulting transformations in real space, $S^{\mathrm{IR}}$ and $S^{\mathrm{UV}}$, would be complex-valued. In order to obtain real transformation matrices, it turns out that we have to assign the symmetric combination of the two modes to the IR part and the antisymmetric contribution to the UV part, i.e., we define
\begin{equation}
\tilde f_{-N/4}^{\,\mathrm{IR}} = \frac 1{\sqrt 2} (\tilde f_{N/4} + \tilde f_{-N/4}), \qquad
\tilde f_{-N/4}^{\,\mathrm{UV}} = \frac 1{\sqrt 2 \ \mathrm i} (\tilde f_{N/4} - \tilde f_{-N/4}),
\end{equation}
and
\begin{equation}
\tilde f_q^{\,\mathrm{IR}} = \tilde f_q, \qquad
\tilde f_q^{\,\mathrm{UV}} = \tilde f_{q-\mathrm{sgn}(q)N/2} , \qquad
q \in \lbrace -\tfrac N 4 + 1, \ldots, \tfrac N 4 - 1 \rbrace.
\end{equation}
Finally, each of these halves is transformed back to real space, giving
\begin{equation}
f_m^{\mathrm{IR/UV}} = \frac 1{\sqrt{N/2}}\sum_{q=-N/4}^{N/4-1} e^{-\frac{4\pi \mathrm i m q}{N}}
\tilde f_q^{\,\mathrm{IR/UV}} = \sum_{n=0}^{N-1}  S^{\mathrm{IR/UV}}_{2m-n} f_n
\end{equation}
with the filters
\begin{equation}
\begin{split}
  S^{\mathrm{IR}}_j
  &= \frac{2}{N}\cos \frac{\pi j}{2} +
    \frac{\sqrt 2}N \sum_{q=-N/4+1}^{N/4-1} e^{-\frac{2\pi \mathrm i j q}{N}} . \\
  S^{\mathrm{UV}}_j
  &= \frac{2}{N}\sin \frac{\pi j}{2} +
    \frac{\sqrt 2}N \left(\sum_{q=-N/2}^{-N/4-1} e^{-\frac{2\pi \mathrm i j q}{N}} +\sum_{q=N/4+1}^{N/2-1} e^{-\frac{2\pi \mathrm i j q}{N}}\right) .
\end{split}
\end{equation}
Again, the sums can be evaluated, giving
\begin{equation}
S^{\mathrm{IR/UV}}_{j} =
\begin{cases}
\frac{\delta_j}{\sqrt 2} + \frac{1 \pm (1-\sqrt 2)}N & \mathrm{if} \ j \ \mathrm{even} \\[2mm]
\frac{\sqrt 2 \ (-1)^{(j-1)/2} }N \cot \frac{\pi j}N  & \mathrm{if} \ j \ \mathrm{odd}
\end{cases}
\label{eq:Sfin}
\end{equation}
and the block matrix $S^{\mathrm{IR}} \oplus S^{\mathrm{UV}}$ is unitary and real-valued. As expected, Eq.~(\ref{eq:Sfin}) reduces to Eq.~(\ref{eq:Sinf}) in the infinite lattice limit $N \to \infty$.

Note that the symmetric and antisymmetric combination of the edge modes may lead to unwanted couplings between the IR and the UV part, which have to be eliminated by appropriate transformations. We will come back to this point when discussing the Ising model and the $XY$ chain in Sect.~\ref{IsingSection}.

\section{Renormalization of free bosons}
\label{Free bosons}

\subsection*{Coarse-grained Hamiltonian:}
Let us consider uncharged free bosons of mass $m$ in one space dimension defined by the Hamiltonian
\begin{equation}
\label{eq:bosonicHamiltonian}
  H_B = \frac12 \int \mathrm{d}x \Big[ \Pi^2(x) + m^2 \Phi^2(x)
  + \Big(\frac{\mathrm{d} \Phi(x)}{\mathrm{d}x} \Big)^{\!2} \Big],
\end{equation}
where $\Phi(x)$ is a Hermitean scalar field and $\Pi(x)$ its conjugate, obeying the canonical
commutation relations
\begin{equation}
  [\Phi(x),\Phi(x')]=[\Pi(x),\Pi(x')]=0, \qquad [\Phi(x),\Pi(x')] = \mathrm i \delta(x-x').
\end{equation}
As a starting point for renormalization, we first have to UV-regularize the theory. Introducing the smeared field operators
\begin{equation}
  \Phi(f) = \int_{-\infty}^\infty \mathrm{d}x \, f(x) \Phi(x),\qquad
  \Pi(f) = \int_{-\infty}^\infty \mathrm{d}x \, f(x) \Pi(x)
\end{equation}
with the commutation relations
\begin{align}
  [\Phi(f),\Phi(g)] &= [\Pi(f),\Pi(g)]=0, \\
  [\Phi(f),\Pi(g)] &= \mathrm i\int_{-\infty}^\infty\mathrm{d}x\, f(x) g(x) = i\langle f|g\rangle
\end{align}
and defining an orthonormal set of equidistant real-valued test functions on $L^2(\mathbb{R})$ with $f_n(x) = f_0(x - n)$ and $\langle f_n | f_m \rangle = \delta_{mn}$ one obtains discrete
Hermitean field operators
\begin{equation}
\Phi_n:=\Phi(f_n)\,,\qquad \Pi_n:=\Pi(f_n)
\end{equation}
living on an infinite one-dimensional lattice with lattice spacing $a$. Orthonormality of the test functions implies that
\begin{equation}
  \Phi(x) \approx \sum_{n \in \mathbb Z} \Phi_n f_n(x), \qquad
  \Pi(x) \approx \sum_{n \in \mathbb Z} \Pi_n f_n(x)
\end{equation}
in the distributional sense, which can be used to coarse grain products of the field operators such as
\begin{equation}
  \int \mathrm{d}x\, \Pi^2(x)
  \approx \sum_{n,n' \in \mathbb Z} \Pi_n \Pi_{n'} \int \mathrm{d}x\, f_n(x) f_{n'}(x)
  = \sum_{n \in \mathbb Z} \Pi_n^2.
\end{equation}
If we choose simple rectangular test functions of the form $f_0(x)=\theta(x + 1/2)-\theta(x - 1/2)$, the scheme described above fails for terms involving derivatives of the fields because of discontinuity. However, as shown in the appendix, one can use a renormalization group argument to obtain a coarse-grained version of the Hamiltonian~(\ref{eq:bosonicHamiltonian}), reading
\begin{equation}\label{eq:H1}
  H = \frac 12 \sum_{n \in \mathbb Z} \Big[ \Pi_n^2 +   m^2 \Phi_n^2 \Big]
  - \frac 12 \sum_{n,n'\in \mathbb Z} \Phi_n \Delta_{n-n'} \Phi_{n'},
\end{equation}
where $\Delta$ denotes the standard discretized Laplacian in one dimension (cf. Eq.~(\ref{discreteSchroedinger})):
\begin{equation}\label{discreteLaplacian}
  \Delta_j=\begin{cases}
    -2 & \text{if } j=0 \\[-2mm]
    +1 &\text{if } j=\pm 1 \\[-2mm]
    0 & \text{otherwise} \end{cases}
\end{equation}
Similarly, we can define the bosonic Hamiltonian on a \textit{finite} ring of $N$ sites by
\begin{equation}\label{eq:H2}
  H = \frac 12 \sum_{n=0}^{N-1} \Big[ \Pi_n^2 +   m^2 \Phi_n^2 \Big]
  - \frac 12 \sum_{n,n'=0}^{N-1} \Phi_n \Delta_{n-n'} \Phi_{n'},
\end{equation}
assuming periodic boundary conditions for $\Delta_j$.

\subsection*{Renormalization scheme:}

Denoting by $\Phi=\{\Phi_n\}$ and $\Pi=\{\Pi_n\}$ the vectors of all field operators on the chain, the coarse-grained Hamiltonian that we use as the starting point for renormalization can be written in a compact form by
\begin{equation}
\label{initialBosonicHamiltonian}
H \;=\; \frac12\begin{bmatrix} \Pi \\ \phi \end{bmatrix}^\dagger
\begin{bmatrix} \mathbbm 1 & \\ & m^2 \mathbbm 1 - \Delta \end{bmatrix}
\begin{bmatrix} \Pi \\ \phi \end{bmatrix}
\end{equation}
with the short-range Laplacian $\Delta$ defined in Eq.~(\ref{discreteLaplacian}). At this point, we apply the moment-separating filters $S=S^{\mathrm{IR}} \oplus S^{\mathrm{UV}}$ introduced in Sect.~\ref{MomentSeparating}, and define the renormalized field operators via
\begin{equation}
\begin{bmatrix} \Pi_\text{IR} \\  \Pi_\text{UV} \end{bmatrix} = S \Pi\,,\qquad
\begin{bmatrix} \Phi_\text{IR} \\  \Phi_\text{UV} \end{bmatrix} = S \Phi \,.
\end{equation}
Since the transformation matrix $S$ is unitary, it is clear that these renormalized operators also obey bosonic commutation relations. With this definition we can rewrite $H$ as
\begin{equation}
H=
\begin{bmatrix} \Pi_\text{IR} \\  \Pi_\text{UV} \\ \Phi_\text{IR} \\  \Phi_\text{UV} \end{bmatrix} ^\dagger
\underbrace{\begin{bmatrix} S & \\ & S \end{bmatrix}
\begin{bmatrix} \mathbbm 1 & \\ & m^2 \mathbbm 1 - \Delta \end{bmatrix}
\begin{bmatrix} S & \\ & S \end{bmatrix} ^\dagger}_{{\scriptscriptstyle =\begin{bmatrix} \mathbbm 1 & \\ & m^2 \mathbbm 1 - S \Delta S^\dagger \end{bmatrix}}}
\begin{bmatrix} \Pi_\text{IR} \\  \Pi_\text{UV} \\ \Phi_\text{IR} \\  \Phi_\text{UV} \end{bmatrix}
\end{equation}
where the transformed Laplacian has the structure
\begin{equation}
S \Delta S^\dagger =
\begin{bmatrix}
S^\text{IR} \Delta {S^\text{IR}}^\dagger &
S^\text{UV} \Delta {S^\text{IR}}^\dagger \\
S^\text{IR} \Delta {S^\text{UV}}^\dagger &
S^\text{UV} \Delta {S^\text{UV}}^\dagger
\end{bmatrix}\,.
\end{equation}
A key point of this work is the observation that for perfect momentum separation this matrix has a block-diagonal form, i.e.,
\begin{equation}
S^\text{UV} \Delta {S^\text{IR}}^\dagger = S^\text{IR} \Delta {S^\text{UV}}^\dagger = 0.
\end{equation}
This can be verified most easily in momentum space, where the translation-invariant operator $\Delta$ is diagonal, and realizing that the two filters have no common overlap.

The block-diagonal form implies that the Hamiltonian \textit{decouples} into a direct sum
\begin{equation}
  H=H_\text{IR}\oplus H_\text{UV}
\end{equation}
with
\begin{equation}
H_\text{IR}=\begin{bmatrix} \Pi_\text{IR} \\  \Phi_\text{IR}  \end{bmatrix} ^\dagger
\begin{bmatrix} \mathbbm 1 & \\ & m^2 \mathbbm 1 - S^\text{IR}\Delta{S^\text{IR}}^\dagger \end{bmatrix}
\begin{bmatrix} \Pi_\text{IR} \\ \Phi_\text{IR} \end{bmatrix}
\end{equation}
and an analogous expression for $H_\text{UV}$. Note that this decoupling relies on perfect momentum separation as well as the preservation of the commutation relations.

We now discard the ultraviolet part and consider the infrared part $H^{(1)}:=H_\text{IR}$ as the renormalized Hamiltonian. Repeating this procedure many times we obtain a sequence of Hamil\-tonians
\begin{equation}
H=H^{(0)}\to H^{(1)}\to H^{(2)} \to \ldots \to H^{(r)}\to \ldots
\end{equation}
Each renormalization step divides the dimension of the Hamiltonian by $2$. Therefore, if we start with a finite lattice, it is useful to choose $N$ as a power of $2$. For example, if we start with $N=2^\ell$ lattice sites, the Hamiltonian $H^{(r)}$ after $r$ renormalization steps is a $2^{l-r}\times 2^{l-r}$ matrix, while on infinite lattices $H^{(r)}$ is an infinite-dimensional square matrix on all levels.

\subsection*{Exact renormalization of the discretized Laplacian:}
In order to analyze the RG flow, we have to understand how the discrete Laplacian $\Delta$ changes under repeated action of $S$. To this end we recursively define the $r$-fold renormalized Laplacian as
\begin{equation}
\Delta^{(r)} := S^\text{IR}\Delta^{(r-1)}{S^\text{IR}}^\dagger\,,\qquad \Delta^{(0)}:=\Delta\,,
\end{equation}
where the size of the transformation matrix $S_\text{IR}$ is different in each step. A straight-forward calculation based on Fourier techniques (not shown here) gives the exact result for the $r$-fold renormalized Laplacian acting on a lattice of $N$ sites:
\begin{equation}
\label{Deltar}
  \Delta^{(r)}_{j} =
  \begin{cases}
    \frac{\frac{2}{N} \sin \left( 2^{-r} \pi \right)}{\tan \left(  2^{-r}\pi/N\right)}-2
    & \text{if } j=0    \\[2mm]
    \frac{\frac{2}{N} (-1)^{j} \sin \left(  2^{1-r}\pi/N\right) \sin \left(  2^{-r} \pi\right)}
    {\cos \left( 2 j \pi/N\right)-\cos\left( 2^{1-r} \pi/N \right)}
    &\text{if } j\neq 0
\end{cases}
\end{equation}
On an infinite lattice this expression reduces to
\begin{equation}
  \lim_{N\to \infty} \Delta^{(r)}_{j} =
  \begin{cases}
    \frac{2^{1+r} \sin \left( 2^{-r} \pi \right)}{\pi} -2
    & \text{if } j=0    \\[2mm]
    \frac{2^{1+r} (-1)^{j}  \sin \left(  2^{-r} \pi\right)}
    {\pi(1-4^r j^2)}
    & \text{if } j\neq 0
\end{cases}
\end{equation}
Analyzing the solution (\ref{Deltar}) for large values of $r$ one finds that in each renormalization step, which essentially doubles the length scale, the Laplacian picks up the factor $1/4$. Thus, if we rescale the Laplacian by $4^r$ and take $r \to \infty$ we obtain the IR fixed point
\begin{equation}
\label{FixedPointLaplacian}
  \Delta^*_j = \lim_{r \to \infty} 4^r \Delta^{(r)}_{j} =
  \begin{cases}
    -\frac{\pi^2}{3} -\frac{2\pi^2}{3 N^2} & \text{if } j=0  \\[2mm]
    \frac{2 \pi^2 \, (-1)^{1+j}}
    {N^2 \sin^2\left(  \pi j/N\right)}  & \text{if } j\neq 0
  \end{cases}
\end{equation}
which in the limit $N \to \infty$ reduces to the Laplacian $\Delta^*_{j}$ on the infinite lattice defined in Eq.~(\ref{longrangelaplacian}). The expression given in (\ref{FixedPointLaplacian}) is the optimal' Laplacian on a finite lattice with $N$ sites which exhibits a clean quadratic dispersion. It can be understood as the ``correct'' discretization of the Laplacian, corresponding to our choice of RG scheme. Indeed, as shown in \ref{AA}, different choices of wavelet filters correspond to different fixed points, which can be directly found from the filter coefficients.

As for the renormalization of the Hamiltonian, a non-trivial RG fixed point can only be obtained if we rescale the Laplacian by a factor $4^r$. Therefore, the free boson Hamiltonian on a lattice with $N$ sites obtained by $r$-fold renormalization of the original Hamiltonian on $2^rL$ sites, is given by
\begin{equation}
H^{(r)}=\begin{bmatrix} \Pi \\  \Phi  \end{bmatrix} ^\dagger
\begin{bmatrix} \mathbbm 1 & \\ & m^2 \mathbbm 1 - 4^r \Delta^{(r)} \end{bmatrix}
\begin{bmatrix} \Pi \\ \Phi \end{bmatrix} \,.
\end{equation}
This Hamiltonian can be diagonalized via Fourier transformation, giving
\begin{equation}
H^{(r)}=\sum_{q=-N/2}^{N/2-1} \omega_q^{(r)} \Bigl(\eta_q^\dagger\eta_q + \tfrac12 \Bigr)
\end{equation}
with a bosonic number operators $\eta_q^\dagger\eta_q$ and the excitation energies
\begin{equation}
\label{wr}
 \omega_q^{(r)} = \sqrt{m^2 + 4^r\bigl[2-2\cos\bigl( 2^{-r} k_q\bigr)\bigr]}\,,
\end{equation}
where $k_q = 2\pi q/N $. In the IR limit, this expression converges to
\begin{equation}
\omega_q^* = \lim_{r \to \infty} \omega_q^{(r)} = \sqrt{m^2 + k_q^2}\,.
\end{equation}
which coincides \emph{exactly} with the continuum solution.

\subsection*{Example: Heat capacity}
In order to demonstrate that the long-range Hamiltonian at the RG fixed point approximates the continuum limit more accurately that the unrenormalized short-range Hamiltonian we compute the heat capacity
\begin{figure}
\centering\includegraphics[width=150mm]{specific-heat}
\label{fig:specific-heat}
\caption{Left: Specific heat of free bosons in one dimension with the mass $m=1$ as a function of the inverse temperature $\beta=1/T$, calculated for the continuum model, the discretized short-range model, and the long-range model at the RG fixed point. Right: For better visualization of the deviations, we show the same data divided by the continuum solution. }
\end{figure}
\begin{equation}
C=\beta^2 \, \frac{\partial^2}{\partial \beta^2} \,\ln Z_\beta
\end{equation}
where
\begin{equation}
\ln Z_\beta = \ln \sum_{\{n_q\}} e^{-\beta H} = \sum_q \ln \sum_{n=0}^\infty e^{-\beta \omega_q (n+1/2)} =
\sum_q \ln\frac{1}{e^{\beta\omega_q/2}-e^{-\beta\omega_q/2}} .
\end{equation}
giving
\begin{equation}
C= \beta^2\sum_q \frac{\omega_q^2 }{(e^{\beta\omega_q/2}-e^{-\beta\omega_q/2})^2}\,.
\end{equation}
In the original continuum model, the specific heat is given by
\begin{equation}
C_\text{cont} =  \beta^2 \int_{-\infty}^{+\infty} \textrm d k \, \frac{m^2+k^2}{\Bigl(e^{\frac12\beta\sqrt{m^2+k^2}}-e^{-\frac12\beta\sqrt{m^2+k^2}}\Bigr)^2}
\end{equation}
On a lattice with short-range Laplacian, the corresponding non-renormalized expression in the thermodynamic limit $L\to\infty$ reads
\begin{equation}
C_\text{lattice} =  \beta^2 \int_{-\pi}^{+\pi} \textrm d k \, \frac{m^2+2-2\cos k}{\Bigl(e^{\frac12\beta\sqrt{m^2+k^2}}-e^{-\frac12\beta\sqrt{m^2+k^2}}\Bigr)^2}\,.
\end{equation}
At the renormalization fixed point, the deformed dispersion relation in the denominator turns into a quadratic one:
\begin{equation}
C^*_\text{lattice} =  \beta^2 \int_{-\pi}^{+\pi} \textrm d k \, \frac{m^2+k^2}{\Bigl(e^{\frac12\beta\sqrt{m^2+k^2}}-e^{-\frac12\beta\sqrt{m^2+k^2}}\Bigr)^2}
\end{equation}
In Fig.~\ref{fig:specific-heat} these expressions are plotted as functions of the inverse temperature $\beta$. As expected, both lattice approximations, the unrenormalized and the renormalized one, differ from the continuum model for high temperatures, where high momenta beyond the Brillouin zone become relevant. It is also clear that all curves converge in the limit of low temperatures, where the excitation modes are much longer than the lattice spacing. In the intermediate range, however, the figure demonstrates that the long-range Hamiltonian at the RG fixed point approximates the continuum model much better than the short-range model from where we started.
\subsection*{Example: Massless two-point correlation function}
As a second example, we study the two-point equal time correlator $G(x) = \langle \Omega | \Phi(n) \Phi(n+x) | \Omega \rangle$ for $m=0$. To this end, let us begin with the momentum space continuum Hamiltonian
\[
   H = \frac 12 \int \mathrm{d}k \, \Big[ \Pi(-k)\Pi(k) + k^2 \Phi(-k)\Phi(k) \Big].
\]
In the usual canon, we define annihilation operators
\[
  a_k \equiv \sqrt{\frac{|k|}2} \Big[\Phi(k) + i |k|^{-1} \Pi(k) \Big],
\]
annihilating the vacuum $a_k |\Omega\rangle = 0$. This allows to rewrite the Hamiltonian as
\begin{equation}\label{eq:diag-cont-hamiltonian}
  H = \int \mathrm{d}k \, |k| a_k^\dagger a_k^{} + \textrm{const.}
\end{equation}
and we see that $|\Omega\rangle$ is indeed the ground state. We directly obtain
\begin{equation}\label{eq:cont-kspace-corr}
  \langle \Omega | \Phi(k)\Phi(k') | \Omega \rangle = \frac 1{2\sqrt{|kk'|}} \langle \Omega | a^{}_{k} a^\dagger_{-k'} | \Omega \rangle = \frac {\delta(k+k')}{2 |k|},
\end{equation}
or, in real space,
\begin{equation} \label{eq:cont-realspace-corr}
  G_{\textrm{cont}}(x, \Lambda) = \frac 1{2\pi} \int_{1/\Lambda}^{\infty} \mathrm{d} k \, \frac{\cos k x}k \sim \frac 1{2\pi} [\log \Lambda - \log x - \gamma],
\end{equation}
where $\gamma$ is the Euler-Mascheroni constant and we introduced an IR cutoff $\Lambda$ to extract the overall divergent part of the integral. The UV divergence at $x \to 0$ remains.

The above has to be compared with the corresponding two-point correlation functions on the infinite lattice. Here, we can repeat the calculations in truncated momentum space $k \in [-\pi,\pi]$ for the lattice dispersion relations $\Delta^{(r)}(k)$. More specifically, in the short-range case $\Delta(k)=\sqrt{2-2\cos k}$, we get
\begin{equation} \label{eq:disc-realspace-corr}
G_{\textrm{lattice}}(x, \Lambda) = \frac 1{2\pi} \int_{1/\Lambda}^\pi \textrm d k \, \frac{\cos k x}{2 \sin k/2},
\end{equation}
while the same quantity at the RG fixed point $\Delta^*(k) = |k|$ is
\begin{equation} \label{eq:disc-realspace-corr-fp}
  G^*_{\textrm{lattice}}(x, \Lambda) = \frac 1{2\pi} \int_{1/\Lambda}^{\pi} \mathrm{d} k \, \frac{\cos k x}k.
\end{equation}
As we see in Fig.~\ref{fig:boson-correlator}, $G^*_{\textrm{lattice}}$ oscillates around the continuum solution -- we identify this as a ringing artifact, originating in the sharp cutoff at $|k|=\pi$.
\begin{figure}
\centering\includegraphics[width=80mm]{correlator}
\label{fig:boson-correlator}
\caption{Deviation of the two-point function $G(x, \Lambda)$ in the discrete theory from its continuum analog, according to Eqs.~(\ref{eq:cont-realspace-corr}),~(\ref{eq:disc-realspace-corr}),~and~(\ref{eq:disc-realspace-corr-fp}). The non local fixed point (yellow) oscillates around the continuum solution at the scale of a single lattice spacing, as is to be expected from the sharp cutoff at $k = \pm \pi$.}
\end{figure}

\section{Renormalization of free fermions}

As a second application, we consider massive Dirac fermions in 1+1 dimensions with the equal-time anticommutation relations
\begin{equation}
\begin{split}
\{\psi^a(t,x),\psi^b(t,y)\}&=\{{\psi^a}^\dagger(t,x),{\psi^b}^\dagger(t,y)\}=0\,,\\
\{\psi^a(t,x),{\psi^b}^\dagger(t,y)\} &=\delta^{ab}\delta(x-y)\,.
\end{split}
\end{equation}
The fermions evolve according to the free Lagrange density
\begin{equation}
\mathcal L = \bar \psi ( i \gamma^\mu \partial_\mu -m )\psi\,,
\end{equation}
where $m$ is the mass and $\bar\psi=\psi^\dagger \gamma^0$. Defining the conjugate momenta
\begin{equation}
\Pi(x) \;
i \psi^\dagger(x)
\end{equation}
with the equal-time anticommutation relations
\begin{equation}
\{\psi^a(t,x),\psi^b(t,y)\}=\{\Pi^a(t,x),\Pi^b(t,y)\}=0\,,\quad \{\psi^a(t,x),\Pi^b(t,y)\}=i \delta^{ab}\delta(x-y)
\end{equation}
the corresponding Hamiltonian density reads
\begin{equation}
\mathcal H \;=\; \Pi \partial_0\psi - \mathcal L \;=\; \Pi\bigl( -\gamma^0\gamma^1\partial_1 - i m \gamma^0)\psi
\end{equation}
In the chiral basis $\gamma^0=\sigma^x, \gamma^1=i\sigma^y$, the Hamiltonian is given by
\begin{align}\label{eq:fermion-hamiltonian}
\mathcal H = \Pi (\sigma^z\partial_1-im\sigma^x)\psi
\end{align}

As in the bosonic case we consider a discrete set of smeared field operators
\begin{equation}
\psi_n = \int_{-\frac12}^{\frac12} \mathrm{d}x\, \psi(x-n)\qquad
\Pi_n = \int_{-\frac12}^{\frac12} \mathrm{d}x\, \Pi(x-n) \, \textrm{d}x
\end{equation}
with the anticommutation relations
\begin{equation}
\lbrace \psi^a_n,\psi^b_{n'}\rbrace = \lbrace \Pi^a_{n},\Pi^b_{n'} \rbrace = 0\,, \,\qquad
\lbrace \psi^a_{n},\Pi^b_{n'} \rbrace = i\delta^{ab}\delta_{nn'}\,.
\end{equation}
In terms of these operators, the bare Hamiltonian, which serves as the starting point of the RG procedure, is given by
\begin{equation}
H \;=\; \sum_{n,n'}\Pi_n \sigma^z \partial_{n-n'} \psi_{n'} \,-\, i m \sum_n \Pi_n \sigma^x \psi_n\,.
\end{equation}
Similarly to for the lattice Laplacian \eqref{discreteLaplacian}, a naive lattice discretization of the first derivative can be given by
\begin{equation}
\label{discreteGradient}
\partial_n=\begin{cases}
\pm 1/2 &\text{if } n = \pm 1 \\
0 & \text{otherwise.} \end{cases}
\end{equation}

As is well known, its Fourier transform is
\begin{align}\label{grad}
\partial(k)= i \sin(k),
\end{align}
which becomes linear only for $|k| \rightarrow 0$, but otherwise introduces a number of inconvenient lattice effects.

\subsection*{Renormalization via Shannon wavelet:}

Written in matrix form, the Hamiltonian is
\begin{equation}
\label{initialFermionicHamiltonian}
H \;=\; \frac12\begin{bmatrix} \Pi \\ \psi \end{bmatrix}^T
\begin{bmatrix}  & -im\sigma^x \mathbbm 1 + \sigma^z \partial \\ &  \end{bmatrix}
\begin{bmatrix} \Pi \\ \psi \end{bmatrix}
\end{equation}
By the same reasoning as in the bosonic section, an RG step that decouples exactly the UV and IR components of the fields at each iteration will consequently decouple the Hamiltonian, the IR contribution being
\begin{equation}
H_\text{IR}=\begin{bmatrix} \Pi_\text{IR} \\  \Psi_\text{IR}  \end{bmatrix} ^T
\begin{bmatrix}  & -im \sigma^x \mathbbm 1 + \sigma^z S^{\text{IR}}\partial S^{{\text{IR}}\, T}  \\ &  \end{bmatrix}
\begin{bmatrix} \Pi_\text{IR} \\ \Psi_\text{IR} \end{bmatrix}
\end{equation}

Let us now analyze the behavior of the lattice operator $\partial(k)$ in \eqref{grad} under the action of $S^{\text {IR}}$. By construction, after $r$ RG steps, the Fourier transformed derivative operator is the original operator `zoomed' by a factor of $2^r$,
\begin{align}\label{}
\partial^{(r)}(k):= i \sin \left( 2^{-r}k \right)
\end{align}
To get it in real space, we simply Fourier transform back to obtain
\begin{equation}\label{partialr}
\partial^{(r)}_{n} = \frac i{2\pi} \int_{-\pi}^\pi \mathrm{d} k\, e^{-i k n} \sin \left( 2^{-r}k \right) = \frac{(-1)^n}{2^r n - (2^r n)^{-1}} \frac{\sin \left( 2^{-r} \pi \right)}{2^{-r} \pi}.
\end{equation}
As we go to the IR $r\rightarrow \infty$, $\eqref{partialr}$ scales as $2^{-r}$, as it should for a first derivative operator of scaling dimension one. The fixed point is
\[
\partial^*_n=\lim_{r\rightarrow \infty} 2^r \partial^{(r)}_n = \frac{(-1)^n}n.
\]

Again, we see that the improved lattice Shannon fixed point operator does not have compact support, but falls linearly with distance and is thus highly non-local. The dispersion relation for the renormalized fermion is given by (the absolute value of) the eigenvalues of the matrix
\[
  -im\sigma^x \mathbbm 1 + \sigma^z \partial^{(r)}(k),
\]
which evaluates to
\[
E^{(r)}(k) = \sqrt{m^2 +\left| 2^r \partial_k^{(r)} \right|^2} = \sqrt{m^2 + \left(\frac{\sin 2^{-r}k}{2^{-r}}\right)^{\!\!2}}.
\]
In the IR limit $r \to \infty$, this yields the exact continuum dispersion
\begin{align}\label{gradFP}
E^*(k)=\sqrt{m^2+k^2}
\end{align}
in analogy to the bosonic case \eqref{wr}.

\subsection*{On Fermion doubling and non-locality}
One of the most fundamental and ubiquitous results in lattice field theory is the Nielsen-Ninomiya theorem\cite{Nielsen:1981hk,Kaplan:2009yg}. It states that, given the following action for massless fermions in momentum space,
\[
\int_{-\pi/a}^{\pi/a} d^{2d}k\, \bar \psi(-k) D(k) \psi_k
\]
on a lattice of spacing $a$, the operator $D$ cannot satisfy the following four properties simultaneously:
\begin{enumerate}
\item $D(k)$ is a periodic analytic function of $k^\mu$
\item $D(k) \sim \gamma^\mu k_\mu$ for $a|k|\ll 1$
\item $D(k)$ is invertible for $k_\mu\neq 0$
\item $\{ \gamma^5,D(k) \}=0$
\end{enumerate}

In $1+1$ dimensions, chiral symmetry implies that the two components of the Dirac spinor $\psi$ decouple as
\[
H = \frac1{2\pi} \int_{-\pi/a}^{\pi/a} \mathrm{d}k\, D(k) \Big[[\psi^1(k)]^\dagger \psi^1(k) - [\psi^2(k)]^\dagger \psi^2(k) \Big],
\]
where $D(k)$ now is a \emph{scalar} function. This Hamiltonian is easily diagonalized: We define annihilation operators
\[
  c_k =
  \begin{cases}
    \psi^1(k) &\text{for } D(k) > 0 \\
    [\psi^1(k)]^\dagger &\text{for } D(k) < 0
  \end{cases}, \qquad
  b_k =
  \begin{cases}
    [\psi^2(k)]^\dagger &\text{for } D(k) > 0 \\
    \psi^2(k) &\text{for } D(k) < 0
  \end{cases}
\]
so that
\begin{equation}\label{eq:diagonal-hamiltonian}
  H = \frac1{2\pi} \int_{-\pi/a}^{\pi/a} \mathrm{d}k\, |D(k)| \Big[c^\dagger_k c^{}_k + b^\dagger_k b^{}_k\Big] + \mathrm{const.}
\end{equation}
As in the bosonic case, the vacuum is annihilated by the $c_k$ and $b_k$. The low-energy excitations are created by $c^\dagger_k$ and $b^\dagger_k$, where $D(k) \approx 0$.

The fermion doubling problem is merely the statement that if $D(k)$ is periodic in the Brillouin zone $[-\pi/a,\pi/a]$, then for every time that $D(k)$ crosses $0$ from below, it must cross it somewhere else from above. This implies that on a lattice, the number of low-energy excitations (i.e., particles) of types $c$ and $b$ must be \emph{even}. In contradistinction, a continuum theory with, e.g., $D(k) = k$ allows for a \emph{single} particle of type $c$ and $b$, respectively. There are many well known methods to eliminate the unwanted doublers on the lattice, by avoiding any combination of the above assumptions, e.g., staggered, Wilson domain wall and overlap fermions \cite{Kogut:1974ag,Chandrasekharan:1998wg,Jansen:1994ym,Creutz:1984mg}. Let us now examine this issue within our approach.

\subsection*{Mean occupation number (and heat capacity)}
As is clear from the above subsections, our construction clearly violates the first assumption of the Nielsen-Ninomiya theorem: the RG fixed point of the lattice operator $\partial^*_{\textrm{lattice}}(k)$ is non-analytic in the Brillouin zone. To verify that it is also free of doubler fermions, we compute the mean occupation number in the canonical ensemble. To this end, consider the partition function
\[
  Z[\beta, \mu_c, \mu_b] = \mathrm{Tr}\, \exp \Big[ -\beta H + \mu_c N_c + \mu_b N_b \Big],
\]
where
\[
  N_c = \frac{1}{2\pi} \int_{-\pi}^\pi \mathrm{d}k\, c^\dagger_k c^{}_k \qquad N_b = \frac{1}{2\pi} \int_{-\pi}^\pi \mathrm{d}k\, b^\dagger_k b^{}_k.
\]
Imposing a finite size $L$ and using the diagonal form of the Hamiltonian~(\ref{eq:diagonal-hamiltonian}), the above trace reduces to
\[
  Z[\beta, \mu_c, \mu_b] = \prod_{e^{i k L} = 1} \mathrm{Tr}\, \big[ e^{(\mu_c - \beta |D(k)|) c^\dagger_k c^{}_k} \big] \mathrm{Tr}\, \big[ e^{(\mu_b - \beta |D(k)|) b^\dagger_k b^{}_k} \big].
\]
We can therefore evaluate the mean occupation number $n_c = N_c/L$ in the thermodynamic limit $L \to \infty$ as
\[
  \langle n_c \rangle_\beta = \left.\frac{\mathrm d}{\mathrm d \mu_c}\right|_{\mu_c,\mu_b=0} \!\!\!\!\!\!\! \frac{\log Z[\beta, \mu_c, \mu_b]}L
  \sim \frac 1{2\pi} \int_{-\pi}^\pi \mathrm{d}k\, \left.\frac{\mathrm d}{\mathrm d \mu_c}\right|_{\mu_c=0} \!\!\!\! \log \mathrm{Tr}\, \big[ e^{(\mu_c - \beta |D(k)|) c^\dagger_k c^{}_k} \big]
\]
and similarly for  $n_b = N_b/L$. By the Pauli exclusion principle,
\[
  \mathrm{Tr}\, \big[ e^{(\mu_c - \beta |D(k)|) c^\dagger_k c^{}_k} \big] = 1 + e^{\mu_c - \beta |D(k)|},
\]
hence
\begin{equation}\label{eq:mean-occ-num}
  \langle n_c \rangle_\beta = \langle n_b \rangle_\beta \sim \frac 1{2\pi} \int_{-\pi}^\pi \mathrm{d}k\, \frac 1{1 + e^{\beta |D(k)|}}.
\end{equation}

In the low temperature limit $\beta \to \infty$, this integral is highly susceptible to the behavior of $D(k)$ around its zeros. Indeed, for $D(k) = k$ we find that
\[
  \langle n_{c/b} \rangle_{\beta} \sim \frac{\log 2}{\pi \beta} - \frac{\log [1 + e^{-\pi \beta}]}{\pi \beta}.
\]
The first term on the right hand side coincides with the continuum result, obtained by extending the range of integration in Eq.~(\ref{eq:mean-occ-num}) to infinity. The other term vanishes for $\beta \to \infty$, confirming that there is no fermion doubling. On the other hand, for $D(k) = \sin k$, we find
\[
  \langle n_{c/b} \rangle_{\beta \to \infty} \sim \frac{2\log 2}{\pi \beta},
\]
which is exactly twice as much as it should be. A comparative plot for general $\beta$ is given in Fig.~\ref{fig:fermion-density}.
\begin{figure}
\centering\includegraphics[width=80mm]{density}
\label{fig:fermion-density}
\caption{Mean fermion occupation number $\langle n \rangle_\beta$ at finite inverse temperature $\beta$, according to Eq.~(\ref{eq:mean-occ-num}). The naive lattice derivative (green) exhibits fermion doubling at low temperatures.}
\end{figure}

\section{Renormalization of the $XY$ and Ising quantum chain}
\label{IsingSection}

The Hamiltonian of the $XY$ model on a chain with $N$ sites and periodic boundary conditions is defined by
\begin{equation}
\label{xyhamiltonian}
H = -\sum_{n=0}^{N-1} \Bigl[ \frac{1+\gamma}2 \sigma^x_{n}\sigma^x_{n+1} +\frac{1-\gamma}2 \sigma^y_{n}\sigma^y_{n+1} + h \sigma^z_n\Bigr] ,
\end{equation}
where $\gamma$ is the anisotropy parameter and $h$ the transverse field. In the special case $\gamma=1$ one obtains the Hamiltonian of the Ising quantum chain.

The Hamiltonian~(\ref{xyhamiltonian}) can be diagonalized following the standard methods by Lieb, Schulz, and Mattis~\cite{lieb1961two}. To this end one first rewrites the Hamiltonian in terms of the flip operators $\sigma_n^\pm = \frac12 (\sigma^x_n \pm \textrm{i} \sigma^y_n)$ by
\begin{equation}
H=h N \mathbbm 1 - \sum_{n=0}^{N-1} \Bigl[ \gamma\Bigl(\sigma^+_n\sigma^+_{n+1}+\sigma^-_n\sigma^-_{n+1}\Bigr) + \sigma^+_n\sigma^-_{n+1}+\sigma_n^-\sigma_{n+1}^++ + 2 h\sigma^+_n\sigma^-_n\Bigr].
\end{equation}
\paragraph{Jordan-Wigner transformation:}
The flip operators commute on different sites. In order to express the Hamiltonian in anticommuting operators, one performs a Jordan-Wigner transformation by introducing the operators
\begin{equation}
\psi_n:=(-\sigma^z_0)(-\sigma^z_1)\cdots(-\sigma^z_{n-1})\,\sigma^-_n\,,\qquad \psi_n^\dagger := [\psi_n]^\dagger.
\end{equation}
which obey the anticommutation relations $\{\psi_n,\psi_m\}=\{\psi^\dagger_n,\psi_m^\dagger\}=0, \;\; \{\psi_n,\psi_m^\dagger\}=\delta_{n,m}$. Rewriting the Hamiltonian in terms of these operator, the strings of $\sigma^z$-matrices drop out in all terms except for those between the last and the first site of the chain, where an operator $P=\prod_{n=0}^N (-\sigma^z_n)$ is encountered. Since this operator commutes with $H$ and has the eigenvalues $\pm 1$, the state space splits into two sectors. This means that the Hamiltonian can be written as
\begin{equation}
H=H_+\Pi_++H_-\Pi_-\,,
\end{equation}
where $\Pi_\pm=\frac12[\mathbbm 1 \pm P]$ are projectors on the respective sector. With $\psi=(\psi_0,\ldots,\psi_{N-1})$ and  $\psi^\dagger=(\psi^\dagger_0,\ldots,\psi^\dagger_{N-1})$ the sector Hamiltonians can be written in the bilinear form
\begin{equation}
\label{eq:xy-matrix}
  H_\pm = - \begin{bmatrix} \psi^{} \\ \psi^\dagger \end{bmatrix}^\dagger
  \begin{bmatrix} A^\pm & B^\pm \\ -B^\pm & -A^\pm \end{bmatrix}
  \begin{bmatrix} \psi^{} \\ \psi^\dagger \end{bmatrix}.
\end{equation}
where $A^\pm$ and $B^\pm$ are matrices of the form
\begin{equation}
  A^\pm = \frac 12
  \begin{bmatrix}
    2h &1 &  & &\mp 1 \\
    1 &2h &1 & &\\
    &\ddots &\ddots &\ddots &\\
    & &1 &2h &1 \\
    \mp 1 & & &1 &2h
  \end{bmatrix}
  \qquad \text{and} \qquad
  B^\pm = \frac{\gamma}2
  \begin{bmatrix}
    0 &1 &  & &\pm 1 \\
    -1 &0 &1 & &\\
    &\ddots &\ddots &\ddots &\\
    & &-1 &0 &1 \\
    \mp 1 & & &-1 &0
  \end{bmatrix}.
\end{equation}
\paragraph{Fourier transformation:}
As can be seen, the matrices $A^\pm$ and $B^\pm$ resemble a second and a first derivative on a lattice. It is therefore natural to Fourier-transform the Hamiltonian. Because of the corner matrix elements we expect two types of boundary conditions for the Fourier modes, namely, periodic b.c. in the negative sector and antiperiodic b.c. in the positive one. Assuming $N$ to be even we therefore introduce fermionic operators in momentum space
\begin{equation}
c_q = \frac{1}{\sqrt{N}} \sum_{n=0}^{N-1} e^{2\pi i n q/L } \,\psi_n
\end{equation}
and similarly $c^\dagger_q$, where the momentum index $q$ runs over the integers $-\frac N2,\ldots,+\frac N2-1$ in the negative sector and over half-integers $-\frac {N+1}2,\ldots,+\frac {N-1}2$ in the positive sector. The operators $c_q,c_q^\dagger$ still obey fermionic anticommutation relations. This allows the Hamiltonian to be rewritten as
\begin{equation}
\label{IsingC}
H_\pm\;=\; -{\sum_q}^\pm \Bigl( A^\pm_q \bigl(c^\dagger_q c_q-c_{-q}c^\dagger_{-q}\bigr) + B^\pm_q \bigl(c^\dagger_qc^\dagger_{-q}-c_{-q}c_q \bigr)\Bigr),
\end{equation}
where the sum $\sum^\pm_q$ runs over the ranges of $q$ specified above and where
\begin{equation}
A^\pm_q = h + \cos\frac{2 \pi q}{N}\,, \qquad B^\pm_q = i \gamma \sin\frac{2 \pi q}{N}
\end{equation}
are the eigenvalues of the two matrices $A^\pm$ and $B^\pm$ (with $\pm$ encoded in the range of $q$).
\paragraph{Decoupling of the edge modes:}
Now we would like to renormalize the Ising model by separating low and high momenta into two equal halves, as described in Sect.~\ref{sec:RGScheme}. In the positive sector, this is straight forward since the momentum index $q$ runs over $-\frac{N}{2}+\frac12,\ldots,-\frac32,-\frac12,\frac12,\frac32,\ldots,\frac{N}{2}-\frac12$, which splits naturally into an infrared part containing the modes $-\frac{N}{4}+\frac12,\ldots,\frac{N}{4}-\frac12$ and an ultraviolet part containing the other modes. However, in the negative sector, where $q$ runs from $-\frac N 2 $ to $\frac N 2-1$, the situation is more involved since the modes $q=\pm \frac N 4$ are exactly at the borderline between the IR and the UV sector. The corresponding contribution to the  Hamiltonian
\begin{equation}
-\begin{bmatrix} c_{N/4} \\ c_{-N/4} \\ c^\dagger_{N/4} \\ c^\dagger_{-N/4}  \end{bmatrix}^\dagger
\begin{bmatrix}
A^-_{N/4} &&& -B^-_{N/4} \\
& A^-_{N/4} & -B^-_{N/4} & \\
& B^-_{N/4} & -A^-_{N/4} & \\
B^-_{N/4} &&& -A^-_{N/4} \end{bmatrix}
\begin{bmatrix} c_{N/4} \\ c_{-N/4} \\ c^\dagger_{N/4} \\ c^\dagger_{-N/4}  \end{bmatrix}^\dagger
\end{equation}
couples the two edge modes via the off-diagonal elements. Therefore, we have to decouple the two edge modes in the negative sector \textit{before} the RG step is carried out. This can be done by means of a Bogoljubov transformation
\begin{equation}
c_{\pm N/4} \mapsto \tilde c_{\pm N/4} = \cos(\phi)\, c_{\pm N/4} \pm i \sin(\phi)\, c^\dagger_{\mp N/4}
\end{equation}
\textit{before} the momenta are separated. This transformation preserves the fermionic anticommutation relations and diagonalizes the contribution in the Hamiltonian by
\begin{equation}
\begin{bmatrix} c_{N/4} \\ c_{-N/4} \\ c^\dagger_{N/4} \\ c^\dagger_{-N/4}  \end{bmatrix}^\dagger
\begin{bmatrix}
\Lambda_{N/4} &&&  \\
& \Lambda_{N/4} && \\
&& -\Lambda_{N/4} & \\
&&& -\Lambda_{N/4} \end{bmatrix}
\begin{bmatrix} c_{N/4} \\ c_{-N/4} \\ c^\dagger_{N/4} \\ c^\dagger_{-N/4}  \end{bmatrix}^\dagger
\end{equation}
where $\Lambda_{N/2}=\sqrt{(A^-_{N/4})^2-(B^-_{N/4})^2}$ is the fermionic excitation energy of the edge modes. Now it is possible to separate the modes and to assign them to the IR and the UV sector.

\paragraph{Renormalization and diagonalization:}
After separating the momenta, we diagonalize the Fourier-transformed Hamiltonian by Bogoljubov transformations in all non-diagonal excitation modes, considering the IR part as the new renormalized Hamiltonian. Following standard techniques, we arrive at the result that the renormalized Hamiltonian on a chain of $N$ sites, which has been obtained by $r$ renormalization steps from the original chain with $2^rN$ sites, can be written in the diagonal form
\begin{equation}
H_\pm^{(r)} = {\sum_q}^\pm \Lambda_q^{(r)} \eta_q^\dagger \eta_q
\end{equation}
with fermionic operators $\eta_q^\dagger,\eta_q $ and the excitation energies
\begin{equation}
\Lambda_q^{(r)} = \sqrt{\bigl(h+\cos\frac{2\pi q}{2^r N} \bigr)^2 -\gamma^2 \sin^2 \frac{2\pi q }{2^r N} }
\end{equation}
In the case of the \textbf{critical} Ising model ($h=-1, \gamma=1$) reduce to
\begin{equation}
\Lambda_q^{(r)} = \sqrt{2-2\cos\frac{2\pi q}{2^r N} }
\end{equation}
with the fixed point $\Lambda^*_q = \frac{2\pi q}{2^r N} $.

\paragraph{Conformal towers:}
\begin{figure}
\centering\includegraphics[width=95mm]{towers}
\vspace{-2mm}
\caption{Emergence of ``conformal towers'' in the spectrum of the quantum Ising model under renormalization. The figure shows the rescaled energy eigenvalues of a chain with $N$ obtained by $r$ renormalization steps from an ordinary Ising model with $2^r N$ sites at criticality. The points of the spectra are connected by straight lines.}
\label{fig:towers}
\end{figure}
The linearity of $\Lambda^*_q$ implies that the spectrum  of the critical Ising Hamiltonian at the RG fixed point consists of equally-spaced levels. It is well-known that the low-lying spectrum of the Ising model exhibits such an equally-spaced structure, so-called \textit{conformal towers}, in the thermodynamic limit $N \to \infty$, and that this structure reflects the underlying conformal symmetry (Verma module) on large scales. It is remarkable that in the present study these conformal towers'' can also be observed in the RG-limit of \textit{finite} systems. This is demonstrated in Fig.~\ref{fig:towers}, where we plotted the rescaled spectrum against the number of renormalization steps. As can be seen, the spectrum becomes equidistant for large $r$. This means that the Ising model with long-range interactions at the fixed point exhibits a spectrum with more degeneracies and thus with a higher degree of symmetry which is a signature of the full conformal symmetry in the thermodynamic limit.

\section{Conclusions}

In this paper we have made  progress in the understanding of continuum quantum field theories on a lattice. Starting from the `bare' lattice action, we have defined an RG flow using infinitely extended wavelets, which ensures a perfect UV/IR decoupling. By construction, the fixed point of this flow exhibits a dispersion relation that coincides exactly with the continuum dispersion relation up to an energy scale given by the inverse lattice spacing. This implies that lattice effects are absent up to this energy scale. However, this comes at the price of introducing infinitely extended wavelets, which leads to highly non-local algebraically decaying interactions.

As applications, we have considered massive free fermions and bosons in $1+1$ dimensions, as well as the Ising quantum chain. We have demonstrated the usefulness of our method by calculating correlation functions and heat capacities. In particular, for massless free fermions we have shown how the fermion doubling problem is eliminated by the non-local nature of the wavelets.

In view of constructing holographic dual spaces,
we note that previous approaches involving  finite-range wavelets cannot be generalized directly to the infinite-range case: This is due to the fact that correlations functions between UV and IR modes vanish by construction in the approach taken here. In the finite range wavelet approaches, precisely these correlation functions were used for defining the AdS radius.
However, as demonstrated in Fig.~\ref{fig:towers}, our approach provides an exact renormalization of the Verma modules with the correct degeneracies up to a certain energy scale given by the inverse lattice spacing. This exact realization of the original conformal symmetry in a certain energy range is expected to be useful for a new holographic approach of constructing AdS/CFT duals using wavelets.
In this respect, a promising future direction is to consider the lattice version of coset minimal models \cite{Antonov:1995wv,Falceto:1992bf}. These are conjectured to be the field-theory duals of higher spin theories in the gravitational bulk \cite{Gaberdiel:2010pz,Gaberdiel:2012uj}.

\section{Acknowledgments}
IR acknowledges Perimeter Institute for hospitality during the final stage of this project.

redpajama_set_name: RedPajamaArXiv


text:
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    <div class="outer">...

And then use the following css:
    .outer:after {
        content:"";
        position:fixed;
        left:0;
        top:0;
        right:0;
        bottom:0;
        z-index:-1;
        display:block;
        background:url('bike-2756269_1920.jpg') no-repeat 80% 50%;
        background-size:cover;
    }
    .outer{
        position:fixed;
        left:0;
        top:0;
        right:0;
        bottom:0;
        max-height: 100%;
        overflow-y: scroll;
    }

This way you can use scroll and have the background in the after pseudo element - which is Not an elegant solution, unless you want to use it for some js parallax scrolling or similar.


text:
Q: Cost of iterating over cells in a UITableView Inside each UITableViewCell of my UITableView, I have a UIScrollView. The scroll view is setup so that when the user swipes right a menu will appear. This is similar to the behavior of the cells in the iPhone Twitter app. When a user swipes upon another cell I iterate over all visible cells to tell the UIScrollView to scroll back to the cell content (i.e. its initial position). The iteration is done in the scrollViewWillBeginDragging method with the following code:
   - (void)scrollViewWillBeginDragging:(UIScrollView *)scrollView {

        if( [scrollView tag] == 90 ) {

            NSLog(@"Dragging a scroll view inside a cell!");

            for (UITableViewCell *cell in self.tableView.visibleCells) {

                [(UICellContentScrollView *)[cell viewWithTag:90] scrollRectToVisible:CGRectMake(0.0f, 0.0f, 320.0f, [cell frame].size.height) animated:YES];

            }

        }

    }

In the method viewDidDisappear I iterate again over all cells to reset various things like so:
- (void)viewDidDisappear:(BOOL)animated {
    [super viewDidDisappear:animated];

    for( NSUInteger section = 0; section < [[self tableView] numberOfSections]; section++ ) {

        for( NSUInteger row = 0; row < [[self tableView] numberOfRowsInSection:section]; row++ ) {

            UITableViewCell *cell = [[self tableView] cellForRowAtIndexPath:[NSIndexPath indexPathForRow:row inSection:section]];
             // resetting cell here

        }

    }

}

My question is if I am (a) going about this the right way and (b) does anyone have any recommendations on a better solution considering the table view may be storing up 50 (no more than 100) items.

A: Check out the NSNotification documentation. You could register all of your UITableViewCell objects to receive a notification you could call something like "cellWasSwiped" or "needToResetCells" or whatever. Then whenever you want to reset the cells you just post the notification. All of your UITableViewCell objects that are registered to receive it will get the notification and can then call whatever method you need.


South African recording artiste Kay juices out a fresh hit after much time in the studio and he has tagged this Lala featuring rising star acts Bhizer, Tonic Jazz, Sbopho & Manqonqo who all shook up the track. . Download Kay Lala Ft Bhizer, Tonic Jazz, Sbopho & Manqonqo below and enjoy with pleasure and delight and also remember to share.


No matter how big or small you are, whether you have budget or not – you need to be monitoring changes in Active Directory.
The Windows Security Log and Active Directory auditing faithfully log a cryptic and noisy trail of security significant changes made anywhere in Active Directory. But there's no good way to report or visualize this data.
to give you easy but powerful visualization of security signicant changes in Active Directory with the power of Splunk Free while staying under the Splunk's 500mb per day limit.
Filter out the noise and collect only those events from each domain controller's security log that represent significant security change. This is usually less than 0.1% of the log. We do this with a managed security log filter that comes with Supercharger for Windows Event Colllection which is free for use with manange domain controllers.
Collect these change events from each domain controller to one central server. No agents to install. No remote collection. No credentials or holes to open in Windows Firewall - WEC is zero touch. And when you install a domain controller, it automatically begins sending events. That's the beauty of native Windows Event Collection managed by Supercharger.
Once WEC delivers these events to Splunk, our LOGbinder for Splunk App leverages our deep experience with the Windows Security Log to refine the raw and cryptic logs into easy to understand dashboards that help you zero in on the important changes in AD, who made them and when.
Too often Splunk Apps are lopsided. Apps are either written by someone who knows Splunk but is not an expert in the particular logs the app pertains to. Or the author is a subject matter expert in the logs being analyzed but isn't a Splunk expert. This leads to predictable results.
Our Splunk App for LOGbinder is a collaboration between security log SME Randy Franklin Smith and talented Splunk engineers that uses advance features of Splunk to do more than just filter raw events.
Knowledgeable Splunk engineers may wonder if the app depends on "index time" parsing. And the answer is "no". We do it the right way with search time fields extractions. So the app works on old events already indexed and more importantly doesn't impact indexing. We tried to do this right. Plus, we made sure this app works whether you consume events directly from the Security log on each computer or via Windows Event Collection (which is what we recommend with the help of Supercharger).


The 340B Coalition would like to thank our exhibitors and sponsors for their generous support! Below is a list of current exhibitors for this event. Please note that additional exhibitors will be added as they sign up, so please check back for updates! Don't forget to visit them at their booths during the event to learn more about how they can help you. We also encourage you to use the mobile app (coming soon) to help you target your visits by sorting exhibitors by functions to find the companies offering the services you need.
Companies, don't miss out on this opportunity to meet with hundreds of pharmacists and other clinicians, hospital and health center executives, federal, state and local government officials, GPOs, wholesalers, drug company decision makers and other leading health care professionals in attendance at this conference. Join attendees for plenary sessions, roundtables, and the largest exhibitor forum for 340B networking.
We look forward to having your company be a part of the public conversation on how 340B serves patients, supports safety-net providers, and benefits the community. If you have any questions, please contact Lee-Anne Gabrielli at (202) 552-5856 or lee-anne.gabrielli@340bhealth.org.


text:
Posted inBites
Cult favorite Dumpling Time will open its first East Bay location in Berkeley
Tanya Holland's new restaurant opens in Oakland; Mama returns with a tipless model.
by Eve Batey June 23, 2021, 10:21 a.m. August 4, 2022
Dumpling Time is bringing its vast menu of Asian dumplings to Berkeley. Credit: Dumpling Time
One of the Bay Area's hottest dumpling shops is headed to Fourth Street
Big news for XLB fans: Dumpling Time, the passionately beloved restaurant from the Omakase Restaurant Group (SF's Niku Steak House and Udon Time, among others), is preparing to open its first East Bay location in Berkeley's buzzy Fourth Street dining district.
Look, almost all dumplings are good, but Dumpling Time's are exceptional: Diners rave about its Chinese/Japanese giant soup dumplings; its sticky and succulent pan-fried, gyoza-style dumplings; and its eminently Instagrammable colored dumpling skins. But until now, folks who wanted a taste of owner Kash Feng's expansive menu had to travel to San Francisco (or Japan) to quell their cravings.
That's set to change in early 2022, a Dumpling Time spokesperson tells Nosh. That's when Dumpling Time expects to open its first spot on this side of the Bay, to an 1800-square-foot space at 1795 Fourth St. in Berkeley.
According to Feng, it was customer demand that spurred the opening. "East Bay residents have expressed great interest in our offerings," he told Nosh. "We're eager to finally make our cuisine more readily available to the community here."
When Berkeley's Dumpling Time opens in the "first quarter of 2022" (according to its spokesperson), expect a 480-square-foot outdoor dining area in addition to its expansive dining room. Will a gigantic soup dumpling taste better if enjoyed al fresco? Beats me, but I can't wait to find out. Dumpling Time, 1795 Fourth St. (near Delaware Street), Berkeley
Mama's Oakland dining room will reopen on June 25. Credit: Emma K. Morris/Mama
Mama will reopen with a new menu — and a no tip policy — on June 25
After a long hiatus and reshuffle in ownership, Oakland's Italian-inspired restaurant Mama will reopen to patrons on June 25. Opened on Grand Avenue by Bay Grape's Stevie Stacionis and Josiah Baldivino in the summer of 2019, the popular spot struggled during the pandemic, eventually shutting its doors. Last month, it announced that it would reopen as part of a new partnership with Hi Neighbor Hospitality Group, the San Francisco restaurant company behind lauded spots like Trestle, Corridor and the Vault.
Since then, Hi Neighbor's Jason Halverson has revamped the menu, tweaking the restaurant's fixed-price dinner plan, but keeping it just under $35 per person. The current lineup includes offerings like a porcini risotto with kale gremolata and a ricotta gnudi with pork shoulder sugo; add-ons include a whole Mediterranean sea bass for an additional $39 or a $44 veal chop.
Also new is a mandatory service charge of 20% per order: according to Mama, the fee is "a progressive decision … enabling us to take better care of our entire staff by providing full benefits, vacation, and professional development resources." That means that when you get your check, there won't be an optional line for the tip, the math will be done for you, a likely relief to all of us who lose the ability to calculate percentages after a cocktail or two. Reservations for indoor and outdoor seating are going fast, and can be made via Resy. Mama Oakland, 388 Grand Ave. (between Perkins Street and Staten Avenue), Oakland
Oakland chef Tanya Holland's newest restaurant has just opened inside the Oakland Museum of California. Photo: Smeeta Mahanti
Tanya Holland's newest restaurant has opened inside the Oakland Museum of California
Town Fare, Brown Sugar Kitchen founder Tanya Holland's anticipated new restaurant, opened last weekend — nearly 10 months after its landlord, the Oakland Museum of California (OMCA), expected it to.
The restaurant was first announced in February 2020, with a plan to open the spot — billed then as "California soul food, featuring high-quality yet approachable menu items for breakfast, lunch, and select dinners" — last August. As with all things, the pandemic snarled that plan, and Holland turned her attention to non-restaurant projects like podcasting, telling Nosh last July that her hope is to pivot away from the kitchen and into a career as a public figure in food.
It's unclear how much Holland will be involved in the day-to-day operations at Town Fare, which currently offers "a limited menu of to-go picnic boxes, a la carte dishes and drinks," the SF Chronicle reports. A look at its current menu shows a blackened shrimp skewer dish, a selection of toasts, and a fried chicken salad, for example. To enjoy those dishes, one must pay to enter the museum, the OMCA says, as "orders are only available to OMCA visitors with a ticket reservation." Those tickets range from $7-16 and can be purchased here, and Town Fare preorders (which the museum recommends) can be placed here. Town Fare at The Oakland Museum of California, 1000 Oak St. (near 10th Street), Oakland
Chef Tanya Holland is ready to become the next household name in food
What you can order at these East Bay ghost and collaborative kitchens
Oakland wins big in Bib Gourmand picks for good food at moderate prices
Tagged: Berkeley food and drinks, Fourth Street Berkeley, New East Bay restaurants, Oakland food and drinks
Eve Batey
eve@berkeleyside.org
Eve Batey has worked as a reporter and editor since 2004, including as the co-founder of SFist, as a deputy managing editor of the SF Chronicle and as the editor of Eater San Francisco. More by Eve Batey

redpajama_set_name: RedPajamaCommonCrawl


(Bangkok, 15 September 2018) – The Asian Forum for Human Rights and Development (FORUM-ASIA), with the participation of the Asian Network for Free Elections (ANFREL), conducted a fact-finding mission in Dhaka, Bangladesh from 7-10 September 2018. In light of the testimonies and data gathered during the mission, we express our utmost concern about the deterioration of human rights and the lack of a conducive environment for free and fair elections in the country.
The following issues indicate how human rights, fundamental freedoms and democratic space are under unprecedented attack in Bangladesh.
The excessive use of force against peaceful protesters calling for a reform of the quota system for civil service jobs and road safety in the country was a blatant violations of Bangladeshi people's right to peaceful assembly and express dissent.
More than 100 people, majority students, have been arrested between 29 July and 15 August as at least 52 cases were filed in different police stations. 5,000 unnamed people were accused for causing violence during the road safety protests.
The right to freedom of expression is systematically curtailed by the Information Communication and Technology (ICT) Act, in particular by its vague Section 57.
The recent unlawful arrest of well-known photographer and social activist Shahidul Alam under Section 57 showcases the lack of free safe space for expressing opinions or criticisms towards the Government both online and offline.
As of 31 August 2018, a total of 128 persons have been arrested under the ICT Act. In August 2018 alone, 23 persons have been arrested for spreading alleged false and confusing information on Facebook and other social media during the safe road movement protests.
The replacement of the ICT Act with a Digital Security Bill 2018 poses additional concerns, due to the broader nature of its provisions and disproportionate sentences.
367 cases of extrajudicial killings have been recorded from January to August 2018, mainly committed by the Police and the Rapid Action Battalion (RAB). From 01 January 2009 to 31 August 2018, a total of 1822 persons have been extra-judicially killed. 228 persons have been extra-judicially killed in the context of a war on drugs started in May this year.
As of 31 August 2018, 27 persons disappeared allegedly by state agencies, while 442 persons are reported disappeared since 2009. Among them there also high profile civil servants, like former Bangladeshi diplomat Maroof Zaman, missing since 4 December 2017.
The combination of judicial harassment, extrajudicial killings and enforced disappearances is nurturing a climate of fear in the country. As a result, civil society organisations, human rights defenders, journalists and members of the opposition are particularly under threat.
By blocking access to financial resources, legal harassments, surveillance and intimidations by different agencies, civic space is increasingly shrinking. Intimidations such as the one against SHUJAN's Secretary Badiul Alam Majumdar, whose house was attacked on 4 August 2018, exemplify the climate of fear that human rights defenders deal with in Bangladesh.
The integrity of democratic institutions is cumulatively compromised, as we recorded a widespread deficit trust towards their independence and fairness. The case of Chief Justice Surendra Kumar Sinha, reportedly forced to leave the country in October 2017, casts shadow on the independency of the judiciary and the functioning of democracy in the country.
Serious concerns were raised during the mission by civil society and media about the independence of the Election Commission and the existence of a conducive environment for free and fair elections.
Hand over power to a neutral caretaker government mandated to hold free and fair elections within 90 days from assuming power.
Support civil society's efforts in publicly raising concerns and awareness about human rights violations in the country, in particular crackdown on protests, extra-judicially killings and enforced disappearances.
Continue to regularly address and highlight human rights concerns through international mechanisms such as Universal Periodic Review and treaty bodies, and support calls for a conducive environment for free and fair elections.
Office of the High Commissioner for Human Rights, and relevant UN Special Procedure mandate holders should closely monitor and report to the UN Human Rights Council on the deteriorating human rights situation in Bangladesh, in particular, in the context of the upcoming General Elections.
We believe that South Asian countries, and India in particular, have a strategic role to play. We call upon them to support the promotion and protection of human rights and democracy in Bangladesh.
 Under this section the publishing of any material in electronic format that 'causes the deterioration of law and order, prejudices the image of the state or person; or causes to hurt religious beliefs' can be punished with minimum of 7 up to a maximum of 14 years imprisonment, without any possibility to get bail.
 Shahidul was arrested hours after posting on Facebook a video of the ongoing protests in Dhaka calling for road safety, and criticizing the Government and the police's handling of the protests in an interview with Al Jazeera.
 Data provided by Odhikar as of 31 August 2018.