Understand before you leave what situations you are likely to encounter. For example, will you be invited out to dinner, or drinks, or karaoke bars? Should you travel armed with gifts for overseas colleagues? Make sure you know the protocol about who pays for dinner, what to wear and how to behave. Prepare for what unusual dishes may be presented to you – and have a feasible excuse ready if you think you won't want to try them. Learn about table manners in your destination, suitable topics of conversation and what to expect if you are invited to someone's home.
Don't put yourself under unnecessary pressure, for example, by cutting it fine to get to the airport, booking flights with tight connections or cramming too many meetings into one day in multiple locations. Look after yourself physically; this way, you will be more effective on the road. So use the hotel's pool, or research safe jogging routes before you leave home. Take home comforts with you – music, books, downloaded movies or box sets. Do whatever works for you to minimize jetlag.
Try not to get everyday travel stresses like crowds and airport delays get to you. Breathe, meditate, stop to think, smile. Or plug in your headphones and listen to music. Make time in even the most hectic of days to stop and think, and be peaceful.
A good travel manager can be a life-saver when things go wrong on the road and you're left with no wifi, missed flights and dwindling funds. Knowing someone is on hand to manage your itinerary is enormously reassuring. Similarly, support in the office is important; if you need quick decisions made, or access to someone senior in your organisation while you're away, make sure a communication system is set up, including out of hours.
Think about the impression you want to make if your overseas counterpart meets you from the flight – or how you will look if your luggage goes missing (which, if you travel enough, it will one day). Travel on hand luggage if you can and either way, pack spare clothes and wash kit in your carry on, as well as all important documentation. Learn about the dress code of the companies and the countries you will be visiting, which may be more conservative than the style you are used to at home.
Pack a universal adapter. Charge everything before you leave home. Use apps on your phone, so you can check in online, scan business cards, track expenses, order taxis and even track your health and diet. Back up your laptop and phone before you leave home.
Business travel can be lonely. Schedule times to chat via Skype or FaceTime with family. Use the time away to catch up with emails to friends. Stay active on social media so you feel a connection with home. Be sociable in your destination; accept invitations to dine out with colleagues (and make sure you have the authority to sign off entertaining bills if you reciprocate).
If you're on the road a lot, it's not difficult to reach more than the basic loyalty tiers with your favorite airline or hotel group. Even at the most basic level, for example, British Airways' silver grade, you get lounge access with food, drink and wifi; priority boarding; free seat selection; and the (usually unrealistic) hope of an upgrade. Loyalty with hotels means room upgrades and often, access to executive lounges for breakfast, or cocktail hour. Some credit cards come with perks like airport lounge access. Make sure you know before you leave what you're entitled to.
Yes, you may experience culture shock while on the road but you can also make new friends and broaden your professional network. Show an interest in people. Ask for introductions. Offer to buy a colleague lunch if they'll show you around. If you have downtime, make the most of it, even if it means doing something like booking a day trip somewhere, or getting last minute tickets to an event. By doing this, a business trip carries positive memories and becomes more personal.


I googled the name of the artist, Jean-Luc Moerman, and found out that he and i seem to have a lot in common. I mean, some of his work could have been made by me, or vice versa. Well, of course he is far more succesful, but i find some of the images that i have seen fro him look alike in an almost eery way.
The person who sent me the link thought that Jean-Luc Moerman might have got the idea to paint tattoos on magazine pictures from me. I have done this in 2005-2006 (here are some examples) and he started doing it a few years later. But when i look at some of his other works in which i recognise the same inspiration as i had in my drawings, it is the other way around. For example, i didn't start drawing abstract line drawings earlier than 2008, and that seems to be after or around the same timeJean-Luc Moerman started doing his.
I am curious to find out more about this man and his work. But most of all i wonder what he would think if he sees my work. This has never happened to me before, and i don't think it happens very often in general that you find somebody who is more or less doing the same things as you do.
By the way, because i began to realise that adding another few thousand abstract droodles to my oeuvre in 2011 does not seem like a wise way to spent my time, i have decided to try and do pick up writing letters again. I cannot show these here, but i can write you a letter if you would like to get one.


The wheel is the battle field where players put all their hope on. A slight mistake on the wheel rule can jeopardize all the victory and you could end up with nothing. From left, right, center and around, everything of the wheel needs to be understand the roulette wheel to enjoy betting by the player to ensure that you get maximum winnings. It is senseless if you call yourself a roulette player yet you have not mastered the rules and guidelines.
Let's see some of the secrets that are worth knowing. Put in mind that there are roulette wheels and Roulette tables which are all different.
They are the same like the same American wheel but these have a single zero. For people who would like to offset the variance, this is the best bet to consider at all times. The house edge happens to reduce with time and on the long run, players will have maximum amount of wins to compensate the loss. Actually, house edges are normally extracted from the zero point. The zero is also green in color here. This is what makes everything to be super fine at all times.
Learn the European based bets to be okay.
Some bets are tethered to the European wheel layout and these kind of bets are normally placed by the dealer on the casino. It starts after the player buys the chips and hands those chips to the dealer to make the bets in accordance with the European bets. For people who play the zero play, they cover numbers 12, 35, 3, 26,0,32 and 15. The bet is normally on four chips and the chip is on 0/3, second on 1 on 12-15 split, 1 on 26 straight up and another on 32-35 split. This is a good deal especially where there is more creativity on the player.
Another one is Voisini's the zero which means neighbors of zero. It covers a lot of numbers 22, 18, 29, 7, 28, 12, 35, 3, 26, 0, 32, 15, 19, 4, 21, 2 and 25. You normally have nine chips to bet on this kind of a bet and every chip has its own specific split of numbers which you must fall on to get rewarded. Two chips are normally on the trio while the others are spread in all the numbers.
There are other types of bets like tiers and neighbors but all you should understand is to ensure that you learn them and understand them perfectly. Numbers could be real confusing but if happen to get the perfect sequence of every bet, then you are okay to do your things perfectly. Just ensure that you get the best out of everything that you make. Preciseness is what is needed in Roulette so you need to ensure that you be as accurate as possible. Choose roulette wheels that do not offer asymmetrical rotations so that you can win money easily at all times.


text:
Oliver Hall
CCD Files Constitutional Challenge to Indiana's Restrictive Ballot Access Laws
March 17, 2022 - On behalf of a coalition of minor political parties, independent candidates and their voter-supporters, CCD today filed a case challenging the constitutionality of Indiana Election Code provisions that restrict access to the ballot for minor parties and independent candidates. The case alleges that Indiana denies voters the right to cast their votes effectively by guaranteeing ballot access to the two major parties -- Republicans and Democrats -- at taxpayer expense while imposing cost-prohibitive requirements on their potential competitors.
"Indiana's ballot access laws are among the most restrictive in the nation," said CCD founder and general counsel Oliver Hall. "No new minor party or independent candidate for statewide office has been able to qualify for Indiana's ballot in more than 20 years -- not since presidential independent Patrick Buchanan did in 2000."
Under Indiana law, major party nominees appear on the ballot automatically once they are selected by means of taxpayer-funded primary elections. Minor parties and statewide independents, by contrast, must submit nomination petitions signed by voters equal in number to 2 percent of the last vote for Secretary of State -- or 44,935 signatures in 2022. Independents for non-statewide offices must submit petitions signed by voters equal in number to two percent of the last vote for Secretary of State in their district.
But since there's no way to validate voters' signatures when they sign nomination petitions, minor parties and independents must collect approximately 50 percent more than the requirement, to ensure they have enough valid signatures. That means a statewide petition must have approximately 67,000 signatures to ensure a likelihood of success.
According to the Complaint, Indiana has not updated or improved the procedures by which minor parties and independents must demonstrate the requisite voter support in the 133 years since Indiana first began regulating ballot access: they still must collect signatures in person, by hand, on paper nomination petitions. This process is inherently laborious, time-consuming and, as the number of signatures required has increased, the cost of completing a statewide petition drive has skyrocketed.
"We intend to prove that statewide petition drives in Indiana now cost $465,000 - $565,000," Hall said. "That violates landmark Supreme Court decisions holding that states may not condition participation in their electoral processes on the ability of voters, candidates or parties to pay."
The Plaintiffs in the case are the Indiana Green Party, the Libertarian Party of Indiana, John Shearer, George Wolfe, David Wetterer, A.B. Brand, Evan McMahon, Mark Rutherford, Andrew Horning, Ken Tucker and Adam Muehlhausen.
The Plaintiffs assert claims under the First and Fourteenth Amendments for the violation of their right to cast their votes effectively, to speak and associate for political purposes, and to the equal protection of law. They seek a judgment declaring Indiana's statutory scheme unconstitutional and an injunction prohibiting the Defendant, Indiana Secretary of State Holli Sullivan, from enforcing the challenged provisions.
The case is captioned Indiana Green Party, et al. v. Sullivan, No. 1:2-cv-00518 (SDIN). It was filed in the Federal District Court for the Southern District of Indiana and has been assigned to Judge James R. Sweeney II.
CCD filed this case pro bono, with Advisory Board members Mark R. Brown and William P. Tedards, Jr. representing the Plaintiffs as co-counsel with Mr. Hall. To support CCD's work, please consider making a contribution. CCD is a 501(c)(3) non-profit, and all contributions are tax-deductible.
CCD Wins Major Case Striking Down Texas's Nomination Petition Procedures
In a huge win for Texas voters, on September 29, 2022 Federal District Judge Robert Pitman struck down Texas's nomination petition procedures as applied to independents and new or minor parties. Judge
CCD Defeats Democrats' Effort to Suppress Voter Choice in NC -- Court Orders Greens on Ballot!
August 5, 2022 (Raleigh, NC) -- United States District Judge James C. Dever III today issued an order directing the North Carolina State Board of Elections ("NCSBE") to certify the North Carolina Gree
Land of the Free?
Suppose you wanted to run for office, but you weren't exactly thrilled with the Republican or Democratic parties. Maybe, like Illinois emergency room doctor David Gill, you wanted to promote a single
©2021 by the Center for Competitive Democracy - all rights reserved.

redpajama_set_name: RedPajamaCommonCrawl


LeVar Burton's Reading Rainbow: First it was on your TV, then your iPad, and now (hopefully, with the help of Kickstarter funds) it'll be on the web. For free! Burton hopes that with $1 million, he can continue the legacy of the bookish show: "Reading Rainbow's digital collection already contains hundreds of books and video field trips … but with your help, we'll be able to make the Reading Rainbow library available on more of the devices modern kids use to consume content." He promises to provide the library to "over 1,500 schools," and if you give him $350, he'll follow you on Twitter (and you'll be doing a mitzvah and all of that, too).
Update: Not only did Burton and his Reading Rainbow team meet their $1 million goal, they've already surpassed it. With 34 days still left to go, the Reading Rainbow Kickstarter is at $1.6 mllion and counting. Someone better start doing their reward package Twitter shout-outs now.
(You can still!) help support Reading Rainbow on Kickstarter.


Picture prices do not include VAT.
Payment by Cheque is acceptable and works of art will be shipped anywhere in the world.
Gallery pictures can be ordered as commissions.
Works of art will be shipped framed.
Works of art will be shipped after a cheque is cleared.
After placing an order by email the work of art will be reserved for 30 days.
Shipping costs will include Insurance, VAT and postage.
Sizes and prices are given on each picture's individual webpage.
Gallery cards can be ordered as commissions.
Mounted in Aperture Cards, with Envelope and foil.
Prices are shown for each card, plus p&p and VAT.
Sizes shown are for the picture, which have about a 2cm white edge all around.
Printed Cards will be printed full page or with a 1cm border if required. Please specify.
Printed cards can be supplied with any of the designs on the Gallery page, if requested.


Very comfortable! 30'' Bar Stool (Set of 4) good design By Flash Furniture. 30'' Bar Stool (Set of 4) very well made, sleek and simple. Complete your living room furniture with a modern 30'' Bar Stool (Set of 4). Its cute sturdy, attractivce and it looks expensive and a best value for the money. 30'' Bar Stool (Set of 4) is one of the most cozy, cozy, beautiful look and exotic 30'' Bar Stool (Set of 4) especially for the price and made of excellent products. Great quality, easy to assemble, delivery on time and in perfect condition. 30'' Bar Stool (Set of 4) is good merchandise at fair prices and wonderful free shipping. Guarantee damaged claim by offering to send parts or to keep the item at a discounted price. Great buy would definitely recommend. Shop with our low-price guarantee and find great deals on 30'' Bar Stool (Set of 4) and more!. Reading the reviews helped you purchase.
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Typically now utilized in homes with a living room or open floor plan, they are usually large designs that seat lots of guests. In smaller areas, they works well for seats in an region which has a strange part or other room restriction. The ability to mix corner units, finish models and reclining sections according to space and person choice makes this sofa design super versatile. Sofa couches are also available in a wide variety of designs, from ultra modern or super luxe, to much more loved ones-friendly modern designs.
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Copyright © 30'' Bar Stool (Set of 4) By Flash Furniture in Patio Bar Furniture All right reserved.


text:
Truckee volleyball fall preview: Chemistry is key
Sports | September 9, 2004
Sierra Sun sports editor
Courtesy photo by Visual Sports
ALL |
Unity, continuity or team chemistry.
However you want to describe it, the Truckee varsity volleyball team is convinced they have it. More importantly, it's the significant difference between this year and last " when a Truckee team with plenty of talent brought plenty of drama and finished 4-10 in the Northern 3A.
"This (2004) team should qualify for state," said head coach Maggie Shane. "What really sets this team apart is their unity. Last year's team should have qualified for state, but we had personality problems that affected the team chemistry heavily. They were very talented, but they didn't want to work together."
It's one thing for a coach to mention it, but returning starter and senior Jesse Lamm was right in the thick of it.
"Last year, we definitely had a lot of separation with the seniors and juniors; a lot of authority went on," she said. "There was a lot of drama. (This year), the seniors and juniors really work well together, and we're pretty leveled out with our skills."
Despite the negative aura that surrounded the 2003 team, it still managed to find some cohesion when it mattered. Entering the regional tournament as the lowest seed in the Northern 3A (No. 6), Truckee defeated a strong No. 3 North Tahoe team led by heavy hitter Teal Ericsson.
After the strenuous match against their crosstown rivals, Truckee was wiped out physically. Without a break between matches, the Wolverines were ousted by Bishop Manogue, 3-0.
"We were playing really good that day, but North Tahoe went five games and every single game (was close)," said Lamm, Truckee's outside hitter. "We were just whipped."
Fittingly, Truckee opens the 2004 league season this afternoon (Sept. 10) at North Tahoe (4 p.m.) Last year, the Wolverines lost to the Lakers twice in the regular season, but then pulled off the upset in regionals. In 2002, it was the exact opposite scenario, Shane said.
Going into today's matchup, Truckee can rest a little easier knowing the 6-foot-plus Ericsson won't be on the other side of the net ready to strike.
"Last year they had their dominating hitter, Teal, so we don't have to face her," Lamm said. "They lost a lot of other players too, so I think we're a little more confident this year without Teal (there)."
Ericsson now plays for the University of Nevada, where she recently had 16 kills in a match against St. Mary's.
Shane has been involved in the Truckee volleyball program for 19 years, she said, and she has had two long stints as the varsity head coach. Her latest stint has covered 1997 until now. Going into this year, she is comfortable not relying on one or two girls.
"Right now, I probably wouldn't say that I have any one or two people that are stars, as opposed to just having a couple standouts that carry the team," she said. "Sometimes that's more important than just having one person (that excels above the rest of the team)."
Shane expects seniors Lamm and captain Alisa Garrison to lead the team, at least emotionally, on the court. Shane considers the defense the team's early season strength.
"We are working on our offense," she said. "Our defense is probably our strongest point right now; we're very good diggers and passers and very aggressive on the floor."
Senior captain and setter Kelly Murphy thinks the team's offseason practices also helped form stronger team chemistry.
"We've been playing club (volleyball) in the offseason, so we know how to play together," she said, referring to a club team that often traveled to Sacramento to play. "Our only weakness is we (tend to) play down to other teams' levels."
Coach Shane was satisfied with two big wins against 4A schools in a preseason tournament hosted by Lowry on Aug. 27-28. Truckee beat Elko (but also lost to them the next day) and Nampa, a 4A school out Boise, Idaho.
"Every game was very close, and they were very competitive," Shane said.
Last time Truckee went to the state championship tournament was 2001, Shane said. Southern Nevada's Moapa Valley has won 13 straight volleyball state titles, including eight in a row since it joined 3A competition in 1996. Those are numbers that exceed dominance.
"That school is (the community's) social life," Shane said. "There's not much else out there. All their teams do very well. We have a few more distractions here. They also have a club team down in Moapa that goes to Las Vegas to play."
Truckee's other captain, junior middle hitter Natalie Singer, suffered a mild fracture in her ankle during practice last week and is expected back before the season ends.
Day Date Opponent Site Time
Fri. Sept. 10 North Tahoe North Tahoe 4 p.m.
Tues. Sept. 14 Fernley Truckee 4 p.m.
Thurs. Sept. 16 Incline Incline 4 p.m.
Tues. Sept. 21 Sparks Truckee 4 p.m.
Fri. Sept. 24 Lowry Lowry 4 p.m.
Sat. Sept. 25 Spring Creek Spring Creek 10 a.m.
Tues. Sept. 28 Dayton Dayton 4 p.m.
Thurs. Sept. 30 Yerington Truckee 4 p.m.
Tues. Oct. 5 Fernley Fernley 4 p.m.
Thurs. Oct. 7 Incline Truckee 4 p.m.
Thurs. Oct. 14 Sparks Sparks 4 p.m.
Fri. Oct. 15 Lowry Truckee 4 p.m.
Sat. Oct. 16 Spring Creek Truckee 10 a.m.
Tues. Oct. 19 Yerington Yerington 4 p.m.
Thurs. Oct. 21 Dayton Truckee 4 p.m.
Thurs. Oct. 28 North Tahoe Truckee 4 p.m.
Sat. Nov. 6 Northern 3A regional tournament
Fri. Nov. 12 3A State Championship tournament
Lady Wolverines sweep season matchup with Greenwave
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Incline makes statement with rivalry win at North Tahoe
The gym at North Tahoe High School was packed Tuesday night for the rivalry game between the Lakers and league foe Incline. The matchup featured the top two teams in the Class 2A Northern League,...
VIDEO: 70th Warren Miller ski film screens Friday, Saturday by Resort at Squaw Creek
Year after year, skiers and snowboarders of all ages look forward to the coming of winter, and for many that =also means the latest Warren Miller ski film. On Friday and Saturday, the Resort at...


Remove deprecated interfaces more than one year old.
dbus: Module version bump for dbus-broker patch from Tom Gundersen.
Module version bumps for patches from David Sugar.
dhcp, logrotate: Module version bump.
mandb, spamassassin: Module version bumps.
gpg, pulseaudio, rpc: Module version bump.
dnsmasq, gnome, mon, mta, openoffice, pulseaudio, wm: Version bumps.
java, mozilla, mta, postfix: Module version bump.
portage: Fix usr_t map interface usage.
apache, portage: Module version bump.
dbus, policykit, wm: Module version bump.
alsa, mon, virt: Module version bump.
gpg, mozilla, rpc: Module version bump.
accountsd, policykit: Module version bump.
dbus, devicekit, modemmanager, networkmanager, virt: Module version bump.
cachefilesd, dbus, dirmngr, gnome, gpg, pulseaudio: Module version bump.
Replace deprecated mmap perm sets and pattern usage.
apache, dkim, monit: Module version bump.
Module version bump for usrmerge FC fixes from Jason Zaman.
Module version bump for cups patches from Guido Trentalancia.
Revert "cups/lpd: read permission for cupsd_var_run_t socket files"
Module version bump for cups revert.
Little misc patch from Russell Coker.
Module version bump for alsa and vnstatd fixes from cgzones.
samba: A few line moves.
Module version bump for samba patch from Russell Coker.
MTA fixes from Russell Coker.
Module version bump for dphysswapfile and mandb fixes from cgzones.
Module version bump for monit patch from cgzones.
networkmanager: adjust interface docs format.
Module version bump for gnome fix from Guido Trentalancia.
apache: Move blocks.  No rule changes.
Module version bump for patches from Russell Coker.
Module version bump for chronyd changes from Luis Ressel.
Module version bump for openoffice fix from Guido Trentalancia.
Module version bump for fixes from Guido Trentalancia.
Module version bump for misc patches from Guido Trentalancia.
gpg: Fix overspecified dependencies in gpg_agent_tmp_filetrans.
mozilla: Module version bump for patch from Luis Ressel.
rkhunter: Fix module version and move lines.
mozilla: Add fc for the files used by the firefox addon "vimperator"
Module version bump for patches from Jason Zaman.
Module version bump for Debian fprintd fc entry from Laurent Bigonville.
Module version bumps for mozilla and gpg patches from Luis Ressel.
rtkit, wm: Remove calls to nonexistant interfaces.
Module version bump for openoffice and wm patches from Guido Trentalancia.
Module version bump for wm patch from Guido Trentalancia.
Module version bump for capability2 fixes from Guido Trentalancia.
Module version bump for plymouth fix from Guido Trentalancia.
boinc: Update from Russell Coker.
Module version bump for mozilla update from Guido Trentalancia.
Module version bump for screen and exim changes from cgzones.
screen: Revert broken interface call.
Add additional comments in geoclue.
Module version bump for firewalld updates from Laurent Bigonville.
Module version bump for collectd update from Jason Zaman.
Module version bumps for user runtime fixes from Jason Zaman.
Boinc updates from Russell Coker.
rpcbind: Read /sys/devices/system/cpu/online from Russell Coker.
Module version bump for watchdog pidfile option from Russell Coker.
Module version bump for pulseaudio fc fix from Jason Zaman.
cpucontrol: revise cpucontrol_conf_t labeling, from Guido Trentalancia.
just a couple of files.
alsa: Add compatibility alias for alsa_etc_rw_t.
Module version bump for webalizer dead type removal from Sean Placchetti.
Module version bump for Evolution SSL fix from Guido Trentalancia.
evolution: Read user certs from Guido Trentalancia.
Module version bump for mozilla patch from Guido Trentalancia.
Module version bump for gnome patch from Guido Trentalancia.
Module version bump for evolution patch from Guido Trentalancia.
Module version bump for networkmanager fix from Naftuli Tzvi Kay.
webalizer: Rearrange a couple lines.
Module version bump for webalizer patch from Russell Coker.
Module version bump for changes to the redis module by Grant Ridder.
Module version bump for changes to the geoclue module by Nicolas Iooss.
Allow firewalld to create firewalld_var_run_t directory.
Module version bump for hadoop_admin() fix from Jazon Zaman.
Module version bump for fc typo in radius from Sven Vermeulen.
Module version bump for cron_admin interface from Jason Zaman.
Module version bump for vfio support for libvirt from Alexander Wetzel.
Module version bump for dbus systemd patch from Laurent Bigonville.
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text:
U.S. Navy war games to run on chicken fat
The U.S. military, in the largest ever government purchase of biofuel, bought a batch of fuel made from chicken fat and algae that will be burned by the Navy in war games next year, reports Reuters. The Defense Logistics Agency purchased 450,000 gallons of fuel from Dynamic Fuels LLC, a Louisiana-based venture owned by Tyson Foods Inc and Syntroleum Corp, which makes biofuel from used cooking oil, and from California-based Solazyme Inc., which makes fuel from algae.
"We are doing this for one simple reason, it makes us better war fighters," Navy Secretary Ray Maybus told reporters in a teleconference. "Our use of fossil fuels is a very real threat to our national security and to the U.S. Navy's ability to protect America and to project power overseas."
The military paid $12 million, or about $26 a gallon for the biofuel, far higher than the price of diesel fuel, which is currently under $4 a gallon. Maybus said the sky-high price was about half the amount the military paid for the fuels last year, and that costs should go down as the market develops.
Navy ships will use the animal fat oil while Navy aircraft will burn the algae fuel during a demonstration at a Rim of the Pacific Exercise, the world's largest war games, next summer off the coast of Hawaii. The fuel will be mixed in blends that are half biofuel and half traditional petroleum fuels that will cost about $15 a gallon. The biofuel is chemically identical to petroleum based fuels Maybus said.
KEYWORDS: biofuel fuel navy Tyson
NCC: China removes duties on U.S. Chicken but market remains closed to U.S. poultry and egg exports
China to extend anti-dumping penalties on U.S. chicken imports
Eagle's latest inline fat analysis and inspection systems help processors run leaner operations
NDC Technologies
NDC's InfraLab e-Series At-Line Meat Analyzer provides accurate measurements of fat, moisture and protein, and also collagen, in grab samples of meat taken from the process for quality assurance purposes. In addition, we work with our OEM partners who deliver NDC on-line fat measurements as part of a complete batch blending control system.
Eagle Product Inspection
Eagle Product Inspection is a pioneer in inline x-ray inspection and fat analysis systems, delivering robust technology for accurate bone & contaminant detection, fat measurement and weight verification to meat, poultry and seafood processors.


text:
\section{Introduction}

The spin-boson model describes a qubit (two-level system) coupled to a quantum
environment represented by a continuous bath of bosonic modes. It is a
paradigmatic model in many fields, ranging from quantum optics \cite{Scully
to condensed matter physics \cite{Leggett}  to open quantum systems \cit
{Breuer,weiss1}. The Hamiltonian is given by
\begin{equation}
H_{SB}=\frac{\Delta }{2}\sigma _{z}+\sum_{k}\omega _{k}a_{k}^{\dag
}a_{k}+\sum_{k}g_{k}\left( a_{k}^{\dag }+a_{k}\right) \sigma _{x},
\label{spinboson}
\end{equation
where $\sigma _{i}$ ($i=x,y,z$) is the Pauli matrices, $\Delta $ is the
qubit frequency, $a_{k}$ ($a_{k}^{\dag }$) is the bosonic annihilation
(creation) operator which can annihilate (create) a boson with frequency
\omega _{k}$, and  $g_{k}$ denotes the coupling strength between the qubit and
the bosonic bath, which is usually characterized by the power-law spectral
density $J(\omega )$,
\begin{equation}
J(\omega )=\pi \sum_{k}g_{k}^{2}\delta (\omega -\omega _{k})=2\pi \alpha
\omega _{c}^{1-s}\omega ^{s}\Theta (\omega _{c}-\omega ),
\end{equation
where $\alpha $ are a dimensionless coupling constant, $\omega _{c}$ is the
cutoff frequency, and $\Theta (\omega _{c}-\omega )$ is the Heaviside step
function. The bath exponent $s$ classifies the reservoir into super-Ohmic
\left( s>1\right) $, Ohmic $\left( s=1\right) $, and sub-Ohmic $\left(
s<1\right) $ types. In many theoretical studies, due to the
weak coupling strength in the real quantum optical and quantum dissipative
systems, the counterrotating terms involving higher excited states,
a_{k}^{\dag }\sigma _{+}$ and $a_{k}\sigma _{-}$, can be neglected, a condition which is the
so-called  rotating-wave approximation (RWA); thus the full Hamiltonian
\ref{spinboson}) can be reduced to the following RWA for
\begin{equation}
H_{SB}^{RWA}=\frac{\Delta }{2}\sigma _{z}+\sum_{k}\omega _{k}a_{k}^{\dag
}a_{k}+\sum_{k}g_{k}\left( a_{k}^{\dag }\sigma _{-}+a_{k}\sigma _{+}\right) .
\label{RWA}
\end{equation
It is generally believed that the RWA is a reasonably good approximation because
the counterrotating terms violate energy conservation, leading to virtual
processes, and thus are suppressed.

The total excitation number of the spin-boson model is $\hat{N
=\sum_{k}a_{k}^{\dag }a_{k}+\sigma _{+}\sigma _{-}$, where $\sigma _{\pm
}=\left( \sigma _{x}\pm i\sigma _{y}\right) /2$. With (without) the RWA, the
system possesses a $U(1)$ ($Z_{2}$) symmetry, which is characterized by the
action of the operator $\Re \left( \theta \right) =\exp \left( i\theta \hat{
}\right) $, where the arbitrary angle $\theta $ corresponds to $U(1)$
symmetry and the special value of $\theta =\pi \ $\  corresponds to $Z_{2}$ (parity)
symmetry. The parity operator $\hat{\Pi}=\Re \left( \pi \right) $ has two
eigenvalues \ $\pm 1$. \

With the advance of modern technology,  various qubit and oscillator
coupling systems can be engineered in many solid-state devices, such as
superconducting circuits \cite{Niemczyk,Yoshihara}, cold atoms \cite{Dimer},
and trapped ions \cite{Cirac}. Recently, the spin-boson model has been
realized by the ultrastrong coupling of a superconducting flux qubit to an
open one-dimensional (1D) transmission line ~\cite{Forn2}. The counterrotating terms can be
strongly suppressed in some proposed schemes \cit
{AnistropicDicke,Keeling,Fanheng}. In some systems, the anisotropy appears
quite naturally, because they are controlled by different input parameters~
\cite{Grimsmo1}. On the theoretical side, the anisotropic models where the
rotating and counterrotating terms are different have attracted
considerable attentions. Rich quantum phase transitions (QPTs) in the
anisotropic Dicke models including the Tavis-Cummings model \cite{TC}
without counterrotating terms have been reported \cite{AnistropicDicke} in
the thermodynamic limit, i.e., infinite qubits. It was found in~\cit
{JC_Plenio} that the Jaynes-Cummings (JC) model ~\cite{JC}, the quantum Rabi
model in the RWA, could also undergo the second-order QPT in the extreme
model parameter limit, $\Delta /\omega \rightarrow \infty $ where $\Delta $
and $\omega $ are the frequencies of qubit and cavity. It was demonstrated
that the ratio of frequencies $\Delta /\omega $ plays the same role as the
qubits number, and the second-order QPT can be identified by  finite
frequency scaling analysis.

Since both the Dicke and Rabi models in the RWA undergo the second-order
QPT, can the spin-boson model in the RWA  display the
second-order QPT? Are some extreme conditions for model parameters required
to realize the second-order QPT? As we know from the literature, the spin-boson Hamiltonian with the RWA is usually treated in
sub-space with  fixed total excitations $N$. In this way the second-order
QPT is absolutely excluded. The constrained condition for the given
sub-space should be relaxed to detect more physical phenomena.

In the sub-Ohmic spin-boson model without the RWA, the second-order QPT from the
delocalized phase, where spin has the equal probability in the two states, to
localized phase, in which  spin prefers to stay in one of the two states, has
been studied extensively \cite{Bulla,QMC,ED, Zhang,guo2012critical,Frenzel,CRduan}.
Many advanced numerical approaches in  quantum many-particle physics have
been applied and extended to this model, such as the numerical
renormalization group~\cite{Bulla}, quantum Monte Carlo simulations~\cit
{QMC}, sparse polynomial space approach~\cite{ED}, exact diagonalization in
terms of shift bosons \cite{Zhang}, and various matrix product state
approaches \cite{guo2012critical,Frenzel}. Some analytical approaches based
on the polaronic unitary transformation, also known as the Silbey-Harris ansatz
\cite{Silbey}, have been also developed for this model \cit
{zheng,Chin,mD1,multi_CS2,yzhao,Blunden,ZH,cao,HeShu1}. The single-coherent-states ansatz~\cite{Silbey} was improved by simply adding other coherent
states on an equal footing~\cite{mD1} and by superpositions of two
degenerate single coherent states~\cite{ZH}, which are generally termed as the
multi-coherent-states (MCS) ansatz. Actually, the MCS in the single-mode
model was proposed much earlier by Ren and Chen~\cite{tcs}.

In this paper, we will extend the variational matrix product state (VMPS)
approach ~\cite{guo2012critical} to study the spin-boson model in the RWA
for all values of the bath exponents. The MCS variational approach and exact
diagonalization within truncated Hilbert space are also employed to provide
independent checks in different regimes. The paper is organized as follows.
In Sec. II, we briefly introduce the generalized spin-boson model. Some
methodologies including the VMPS, the MCS variational approaches, and exact
diagonalization in truncated Hilbert space are described. The rich phase
transitions revealed by the VMPS method are presented in Sec.  III, where
the MCS variational approaches and the exact diagonalization are also
applied to provide further evidence. The quantum criticality based on VMPS
studies is also analyzed. Finally, conclusions are drawn in Sec. IV.

\section{Generalized Model Hamiltonian and methodologies}

Based on Hamiltonian (\ref{spinboson}), the generalized spin-boson
Hamiltonian can be written as ($\hbar =1$)
\begin{widetext}
\begin{equation}
\hat{H_{SB}} =\frac{\Delta }{2}\sigma _{z}+\frac{\epsilon }{2}\sigma
_{x}+\sum_{k}\omega _{k}a_{k}^{\dag }a_{k}
+\frac{1+\lambda }{2}\sum_{k}g_{k}\left( a_{k}^{\dag }\sigma
_{-}+a_{k}\sigma _{+}\right) +\frac{1-\lambda }{2}\sum_{k}g_{k}\left(
a_{k}\sigma _{-}+a_{k}^{\dag }\sigma _{+}\right) ,  \label{Hamiltonian1}
\end{equation}
\end{widetext}
where $\epsilon $ is the energy bias applied in a two-level system ($\epsilon
=0$ except special statements) and $\frac{1+\lambda }{2}\ $and $\frac{1-\lambda
}{2}\ $ are the weights of the rotating-wave and counterrotating terms,
respectively. In this sense $\lambda $ is the anisotropy constant of this
model. Obviously, $\lambda =1$ ($\lambda =0$) corresponds to the spin-boson
model with (without) RWA.

For later use, Hamiltonian (\ref{Hamiltonian1}) can be rewritten as
\begin{eqnarray}
\hat{H} &=&\frac{\Delta }{2}\sigma _{z}+\frac{\epsilon }{2}\sigma
_{x}+\sum_{k}\omega _{k}a_{k}^{\dag }a_{k}+\frac{1}{2}\sum_{k}g_{k}\left(
a_{k}^{\dag }+a_{k}\right) \sigma _{x}  \notag \\
&&+\frac{\lambda }{2}\sum_{k}g_{k}\left( a_{k}-a_{k}^{\dag }\right) i\sigma
_{y}.  \label{Hamiltonian2}
\end{eqnarray
In the following, three methods are introduced to study this generalized
model.

\textsl{VMPS approach.-}. As is well known, the VMPS approach works efficiently
in one-dimensional chain models ~\cite{MPS_intro1,MPS_intro2}. To apply VMPS
in the spin-boson model, we therefore transform the model into a 1D chain
model. We first perform the logarithmic discretization of the spectral
density of the continuum bath~\cite{Bulla} with discretization parameter
\Lambda >1$; then, by using orthogonal polynomials $b_{n}^{\dag
}=\sum_{k}U_{nk}a_{k}^{\dag }$ ($b_{n}=\sum_{k}U_{nk}a_{k}$) as described in
Ref.~\cite{Plenio_Chin}, the spin-boson models can be mapped into the
representation of a one-dimensional semi-infinite chain with
nearest-neighbor interaction ~\cite{Friend}. Thus, Hamiltonian (\re
{Hamiltonian2}) can be written as:
\begin{eqnarray}
H_{\text{chain}} &=&\frac{\Delta }{2}\sigma _{z}+\frac{\epsilon }{2}\sigma
_{x}+\frac{c_{0}}{2}(b_{0}+b_{0}^{\dag })\sigma _{x}+\lambda \frac{c_{0}}{2
(b_{0}-b_{0}^{\dag })i\sigma _{y}  \notag \\
&&+\sum_{n=0}^{L-2}[\epsilon _{n}b_{n}^{\dag }b_{n}+t_{n}(b_{n}^{\dag
}b_{n+1}+b_{n+1}^{\dag }b_{n})],  \label{Hamitrans}
\end{eqnarray
where $b_{n}^{\dag }$($b_{n}$) is the creation (annihilation) operator for a new
set of boson modes in a transformed representation with $\epsilon _{n}$
describing frequency on chain site $n$, $t_{n}$ describing the
nearest-neighbor hopping parameter, and $c_{0}$ describing the effective
coupling strength between the spin and the new effective bath. All of the
parameters mentioned above, such as $t_{n},\epsilon _{n}$ and $c_{0}$, are
determined by the logarithmic discretization parameter $\Lambda $, the
cutoff frequency $\omega _{c}$, and a specific form of the spectral
density, which are expressed below{\
\begin{eqnarray*}
c_{0} &=&\sqrt{\int_{0}^{\omega _{c}}\frac{J\left( \omega \right)}{\pi}
d\omega} , \\
\epsilon _{n} &=&\xi _{s}\left( A_{n}+C_{n}\right) , \\
t_{n} &=&-\xi _{s}\left( \frac{N_{n+1}}{N_{n}}\right) A_{n},
\end{eqnarray*
where
\begin{eqnarray*}
\xi _{s} &=&\frac{s+1}{s+2}\frac{1-\Lambda ^{-\left( s+2\right) }}{1-\Lambda
^{-\left( s+1\right) }}\omega _{c}, \\
A_{n} &=&\Lambda ^{-j}\frac{\left( 1-\Lambda ^{-\left( j+1+s\right) }\right)
^{2}}{\left( 1-\Lambda ^{-\left( 2j+1+s\right) }\right) \left( 1-\Lambda
^{-\left( 2j+2+s\right) }\right) }, \\
C_{n} &=&\Lambda ^{-\left( j+s\right) }\frac{\left( 1-\Lambda ^{-j}\right)
^{2}}{\left( 1-\Lambda ^{-\left( 2j+s\right) }\right) \left( 1-\Lambda
^{-\left( 2j+1+s\right) }\right) }, \\
N_{n}^{2} &=&\frac{\Lambda ^{-n\left( 1+s\right) }\left( \Lambda
^{-1}:\Lambda ^{-1}\right) _{n}^{2}}{\left( \Lambda ^{-\left( s+1\right)
}:\Lambda ^{-1}\right) _{n}^{2}\left( 1-\Lambda ^{-\left( 2n+1+s\right)
}\right) },
\end{eqnarray*
with
\begin{equation*}
\left( a:q\right) _{n}=\left( 1-a\right) \left( 1-aq\right) ...\left(
1-aq^{n-1}\right)
\end{equation*
For details, one may refer to Ref.~\cite{Plenio_Chin}.

We now briefly introduce the VMPS approach~\cite{VMPS1,VMPS2,MPS_intro1}.
For the transformed spin-boson model of a 1D chain with $L$ sites, the ground-state wave function of Hamiltonian (\ref{Hamitrans}) can be depicted as
\begin{equation}
\left\vert \psi \right\rangle =\sum_{\{N_{n}\}=1}^{d_{n}}M\left[ N_{1}\right]
...M\left[ N_{L}\right] \left\vert N_{1},...,N_{L}\right\rangle ,
\label{MPSWF}
\end{equation
where $N_{n}$ is the physical dimension of each site $n$ with truncation
d_{n}$, and $D_{n}$ is the bond dimension for matrix $M$ with the open
boundary condition, bounding the maximal entanglement in each subspace. $M$
on each site is optimized through sweeping the 1D chain iteratively, where
accuracy of numerical results is determined by values of $d_{n}$ and $D_{n}
. In order to deal with the spin-boson model near the quantum critical region
effectively, we apply an optimized boson basis through an additional
isometric map with $d_{opt}\ll d_{n}$ like in Refs.~\cit
{guo2012critical,Friend}. In this way, we can effectively reduce the local boson
basis and improve the maximum  boson number in the quantum critical region.

We have recovered all the results in Ref. \cite{guo2012critical} and
confirmed the classical mean-field behavior for $s<1/2$ of the sub-Ohmic
spin-boson model. In the present paper, we will extend it to study the RWA
spin-boson model, mainly focusing on phase transitions in the sub-Ohmic baths,
namely $\lambda =1$, $0<s<1$. For the data presented below, we typically
choose the model parameters as $\Delta =0.1$, $\omega _{c}=1$, and $\epsilon =0
, if there are no special statements. Some parameters of orthogonal
polynomials transformation and VMPS are the same as those in Ref.~\cit
{guo2012critical}, e.g. \ the logarithmic discretization parameter $\Lambda
=2$, the length of the semi-infinite chain $L=50$, and optimized truncation numbers
d_{opt}=12$. In addition, we adjust the bond dimension to achieve better
convergence of the results. In this paper, we choose $D_{max}=20,40$ for
s=0.3,0.7$, respectively, which is sufficient to obtain the converged
results, as demonstrated in Appendix A in detail.

\textsl{Exact diagonalization in truncated Hilbert space.-}. It is well
known that the spin-boson model with the RWA possesses  $U(1)$ symmetry,
and the total excitation number $\hat{N}$ is conserved because $\left[ \hat{N},
\right] =0$. It has been reported in Ref. \cite{Tong} that the total excitation
number $N$\quad in the ground state of the RWA spin-boson model jumps from
0 $ to $1$ at a critical coupling strength. This instability can be also
called the first-order phase transition, because the first derivative of the
ground-state energy is discontinuous. Due to the possible sequence of
instabilities with the coupling strength, we can truncate the Hilbert space
up to a finite $N$ excitation number. To this end, we first separate the
Hilbert space into several subspaces with different excitation numbers
l=0,1,2...N$. The wave function in $l$-subspace $\left\vert \psi
_{l}\right\rangle $ can be written explicitly with $l$ excitations. E. g.
the wavefunctions for $l=0,1,2$ and $3$-subspace are listed in the following
\begin{eqnarray*}
\left\vert \psi _{0}\right\rangle &=&\left\vert 0\right\rangle \left\vert
\downarrow \right\rangle , \\
\left\vert \psi _{1}\right\rangle &=&c\left\vert 0\right\rangle \left\vert
\uparrow \right\rangle +\sum_{k}d_{k}{a_{k}}^{\dag }\left\vert
0\right\rangle \left\vert \downarrow \right\rangle , \\
\left\vert \psi _{2}\right\rangle &=&\sum_{k}e_{k}{a_{k}}^{\dag }\left\vert
0\right\rangle \left\vert \uparrow \right\rangle +\sum_{kk^{^{\prime
}}}f_{kk^{^{\prime }}}{a_{k}}^{\dag }{a_{k^{\prime }}}^{\dag }\left\vert
0\right\rangle \left\vert \downarrow \right\rangle , \\
\left\vert \psi _{3}\right\rangle &=&\sum_{kk^{^{\prime }}}p_{kk^{^{\prime
}}}{a_{k}}^{\dag }{a_{k^{\prime }}}^{\dag }\left\vert 0\right\rangle
\left\vert \uparrow \right\rangle \\
&&+\sum_{kk^{^{\prime }}k^{^{\prime \prime }}}q_{kk^{^{\prime }}k^{^{\prime
\prime }}}{a_{k}}^{\dag }{a_{k^{\prime }}}^{\dag }{a_{k^{\prime \prime }}
^{\dag }\left\vert 0\right\rangle \left\vert \downarrow \right\rangle .
\end{eqnarray*
Then the wavefunction in the truncated Hilbert space up to $N$ excitations
can be expressed a
\begin{equation}
\left\vert \psi \right\rangle ^{\leq N}=\sum_{l=0}^{N}\left\vert \psi
_{l}\right\rangle .  \label{ED}
\end{equation
For $N=0$, $\left\vert \psi \right\rangle ^{\leqslant 0}=\left\vert
0\right\rangle \left\vert \downarrow \right\rangle $, the ground-state
energy is $E_{0}=-\frac{\Delta }{2}$. However, it is very difficult to
obtain the analytical solution for $\left\vert \psi \right\rangle
^{\leqslant N}$ up to $N>0$ excitations, and numerically exact diagonalizations
are then required to obtain the converged  lowest energy. This approach is called
$N$ED below.

\textsl{MCS ansatz.-}. We also apply the MCS ansatz \cite{tcs,mD1,multi_CS2}
to the spin-boson model in the RWA. To facilitate the variational study and
visualize the symmetry breaking explicitly, we rotate the Hamiltonian (\re
{Hamiltonian2}) around the y axis by an angle $\pi /2$ with $\epsilon = 0$,
which gives
\begin{eqnarray}
H^{T} &=&-\frac{\Delta }{2}\sigma _{x}+\sum_{k}\omega _{k}a_{k}^{\dag }a_{k}
\frac{1}{2}\sum_{k}g_{k}\left( a_{k}^{\dag }+a_{k}\right) \sigma _{z}  \notag
\\
&&+\frac{\lambda }{2}\sum_{k}g_{k}\left( a_{k}-a_{k}^{\dag }\right) i\sigma
_{y}  \label{Hamiltonian3}
\end{eqnarray
The trial state $|\psi ^{T}\rangle $ is written in the basis of the spin-up
state $|\uparrow \rangle $ and spin-down state $|\downarrow \rangle $
\begin{equation}
|\psi ^{T}\rangle =\left(
\begin{array}{c}
\sum_{n=1}^{N_{c}}A_{n}\exp \left[ \sum_{k=1}^{L}f_{n,k}\left( a_{k}^{\dag
}-a_{k}\right) \right] |0\rangle \\
\sum_{n=1}^{N_{c}}B_{n}\exp \left[ \sum_{k=1}^{L}h_{n,k}\left( a_{k}^{\dag
}-a_{k}\right) \right] |0\rangl
\end{array
\right) ,  \label{VM_wave}
\end{equation
where $A_{n}$\ ($B_{n}$) are related to the occupation probabilities of the
spin-up (spin-down) state in the $n$th coherent state; $N_{c}$ and $L$ are
numbers of coherent states and total bosonic modes,  respectively; and $f_{n,k}$ (
h_{n,k}$) represents bosonic displacement of the $n$th coherent state and $k$
th bosonic mode. The symmetric MCS ansatz ($A_{n}=B_{n}$ and
f_{n,k}=-g_{n,k}$) can only be applied to the delocalized phase, so one can
easily detect the symmetry breaking.

The energy expectation value can be calculated as follows
\begin{equation}
E=\frac{\langle \psi ^{T}|H^{T}|\psi ^{T}\rangle }{\langle \psi ^{T}|\psi
^{T}\rangle },  \label{ehd}
\end{equation
where
\begin{eqnarray*}
\langle \psi ^{T}|H^{T}|\psi ^{T}\rangle
&=&\sum_{m,n}(A_{m}A_{n}F_{m,n}\alpha _{m,n} \\
&&+B_{m}B_{n}G_{m,n}\beta _{m,n}-\gamma _{mn}\Gamma _{m,n}A_{m}B_{n}), \\
\langle \psi ^{T}|\psi ^{T}\rangle &=&\sum_{m,n}\left(
A_{m}A_{n}F_{m,n}+B_{m}B_{n}G_{m,n}\right) ,
\end{eqnarray*
with
\begin{eqnarray*}
F_{m,n} &=&\exp \left[ -\frac{1}{2}\sum_{k}\left( f_{m,k}-f_{n,k}\right) ^{2
\right] , \\
G_{m,n} &=&\exp \left[ -\frac{1}{2}\sum_{k}\left( h_{m,k}-h_{n,k}\right) ^{2
\right] , \\
\Gamma _{m,n} &=&\exp \left[ -\frac{1}{2}\sum_{k}\left(
f_{m,k}-h_{n,k}\right) ^{2}\right] , \\
\alpha _{m,n} &=&\sum_{k}\left[ \omega _{k}f_{m,k}f_{n,k}+\frac{g_{k}}{2
\left( f_{m,k}+f_{n,k}\right) \right] , \\
\beta _{m,n} &=&\sum_{k}\left[ \omega _{k}h_{m,k}h_{n,k}-\frac{g_{k}}{2
\left( h_{m,k}+h_{n,k}\right) \right] , \\
\gamma _{m,n} &=&\left[ \Delta +\lambda \sum_{k}g_{k}\left(
f_{m,k}-h_{n,k}\right) \right] .
\end{eqnarray*}

Minimizing the energy expectation value with respect to variational parameters
gives the following self-consistent equations
\begin{equation*}
\frac{\partial E}{\partial A_{n}}=\frac{\partial E}{\partial B_{n}}=\frac
\partial E}{\partial f_{ij}}=\frac{\partial E}{\partial h_{ij}}=0
\end{equation*
which leads to
\begin{eqnarray}
&&\sum_{n}\left( 2A_{n}F_{i,n}\left( \alpha _{i,n}-E\right) -\Gamma
_{i,n}B_{n}\gamma _{i,n}\right) =0,  \label{aa} \\
&&\sum_{n}\left( 2B_{n}G_{i,n}\left( \beta _{i,n}-E\right) -\Gamma
_{n,i}A_{n}\gamma _{n,i}\right) =0,  \label{bb} \\
&&\sum_{n}\{-\Gamma _{i,n}B_{n}\left( h_{n,j}\gamma _{i,n}+\lambda
g_{j}\right)  \notag \\
&&+A_{n}F_{i,n}\left[ 2\left( \alpha _{i,n}+\omega _{j}-E\right)
f_{n,j}+g_{j}\right] \}=0,  \label{ff} \\
&&\sum_{n}\{-\Gamma _{n,i}A_{n}\left( f_{n,j}\gamma _{n,i}-\lambda
g_{j}\right)  \notag \\
&&+B_{n}G_{i,n}\left[ 2\left( \beta _{i,n}+\omega _{j}-E\right) h_{n,j}-g_{j
\right] \}=0.  \label{hh}
\end{eqnarray}

In practice, these parameters can be obtained by solving the coupled
equations self-consistently, which in turn give the ground-state energy and
wave function. The ground state with zero excitation $\left\vert
0\right\rangle \left\vert \downarrow \right\rangle $ to Hamiltonian (\re
{Hamiltonian2}) can be contained in the MCS wave function to Hamiltonian (\re
{Hamiltonian3}) by setting the constrained coefficients: $A_{n}=B_{n}$ and$\
f_{n,k}=h_{n,k}=0$\ for all $n$. However the state with nonzero total
excitations cannot be included in the \ MCS ansatz due to the finite number
of coherent states. It has \ been demonstrated that this wave function can
describe the localized phase of the spin-boson model \cite{Blunden}. The
number of the coherent states in the practical calculations in this paper is
$N_{c}=6$, which is sufficient to judge the existence of the second-order
QPT.

For the three approaches described above, discretization of the energy spectrum
of the continuum bath should be performed at the very beginning in the
practical calculations. The same logarithmic discretization is taken for
different approaches if comparison is made in the data presented below.

\section{Results and discussions}

We first describe our main results by VMPS method to the sub-Ohmic
\left(0<s<1\right )$ spin-boson model with the RWA described by Eq. (\re
{Hamiltonian2}) for $\lambda =1$. We observe a second-order QPT in the
spin-boson model under RWA, which was unnoticed in literature, to the best
of our knowledge. This surprising observation is further confirmed by the
MCS variational approach. The symmetry breaking was unambiguously found
above the critical point in this wave function based approach. The $N$ED
results are also given, which should be exact at  weak coupling and can
be regarded as a benchmark in this regime. Besides the second-order QPT, we
also find a few first-order QPTs before the critical point of the
second-order QPT for large bath exponent $s$. As $s$ decreases, the
first-order QPTs disappear successively, but the second-order phase
transition remains robust no matter how small $s$ is. We will discuss those
phenomena based on various numerical calculations in the following
subsections.

\subsection{Magnetization $\left\vert \left\langle \protect\sigma
_{x}\right\rangle \right\vert $}

\begin{figure}[tph]
\includegraphics[scale=0.4]{Sx_MPS_Ex.eps}
\caption{ (Color online) Magnetization $\left\vert \left\langle \protec
\sigma _{x}\right\rangle \right\vert $ as a function of $\protect\alpha $ in
the ground state for (a) $s=0.3$ and (b) $0.7$. Black lines with circles
denote the VMPS results and the red dashed lines denote the $N=3$-ED ones. $\protec
\lambda =1$, $\Delta =0.1$, $\protect\omega _{c}=1$, $\protect\epsilon =0$,
\Lambda =2$, $L=50$, $d_{opt}=12$, and $D=20,40$ for $s=0.3,0.7$,
respectively. }
\label{Magnetization}
\end{figure}
In terms of Hamiltonian (\ref{Hamiltonian2}), magnetization $\left\vert
\left\langle \sigma _{x}\right\rangle \right\vert $ can be regarded as the
order parameter in this model. In the second-order phase transition, due to
the symmetry breaking, the order parameter changes from zero to nonzero at
the critical points. In Fig.~\ref{Magnetization} we present the VMPS results
for $\left\vert \left\langle \sigma _{x}\right\rangle \right\vert $ as a
function of the coupling strength $\alpha $ for the spin-boson model in the
RWA (i.e.$\ \lambda =1$) for two typical values of $s=0.3$ and $0.7$.
Surprisingly, magnetization changes abruptly from zero to nonzero for both
cases. The critical points of the second-order QPTs are $\alpha
_{c}=0.016,0.685$ for $s=0.3,0.7\ $ respectively.

However, the $N$ED within  excitation numbers up to $N$ shows that the
order parameter $\left\vert \left\langle \sigma _{x}\right\rangle
\right\vert $ remains zero for all coupling strengths. Note that the VMPS
has provided  convincing results for the full spin-boson model \cit
{guo2012critical}. $N$ED cannot yield  consistent results with the VMPS
ones, because the total excitation number may not conserve with $\alpha $
due to the unexpected symmetry breaking.

In the finite-size Dicke model with the RWA \cite{Buzek,Zhangyy}, the system
usually undergoes the first-order phase transition, i.e., sequence of
instabilities, among the phases within different conserved excitation
numbers as the coupling strength increases. In the limit $\Delta /\omega
\rightarrow \infty $, the second-order quantum phase transitions have been
observed in the quantum Rabi model with the RWA \cite{JC_Plenio}. It has been
also reported that the Dicke model under the RWA displays a second-order QPT
in the thermodynamic limit \cite{AnistropicDicke}. Contrary to the quantum
Rabi (Dicke) model in the RWA, the spin-boson model under the RWA could
undergo the second-order QPT even for a finite value of $\Delta /\omega $ (
one qubit).

{\ }

\subsection{Magnetization $\left\vert \left\langle \protect\sigma
_{z}\right\rangle \right\vert $ and the ground-state energy}

\begin{figure}[tph]
\centerline{\includegraphics[scale=0.4]{Sz_disE_MPS_Ex.eps}}
\caption{ (Color online) Magnetization $\left\vert \left\langle\protect\sigm
_{z}\right\rangle\right\vert$ and the ground-state energy difference (lower
panel) by the VMPS and $N=3$ED for $s=0.3$ (left panel) and $0.7$ (right
panel). $\protect\lambda =1$, $\Delta =0.1$, $\protect\omega _{c}=1$,
\protect\epsilon =0$, $\Lambda=2$, $L=50$, $d_{opt}=12$, and $D=20,40$ for
s=0.3,0.7 $ respectively. The inset in the right-upper panel shows the
enlarged view of the same plot where the kink can be clearly visible.}
\label{GS_quantity}
\end{figure}

{\ In the original spin-boson Hamiltonian, $\sigma _{z}$ describes a
tunneling two-level system ~\cite{zheng,Chin}. The magnetization along the
z-$direction }$\left\vert \left\langle \sigma _{z}\right\rangle \right\vert
{, simply the order parameter along z direction, is the renormalized factor
of the tunneling amplitude $\Delta $. In this subsection, we examine the
magnetization along the z direction as well as the ground-state energy by
both VMPS and $3$ED, which are exhibited in Fig.~\ref{GS_quantity}. }

Obviously, two observables begin to deviate only after the critical point
\alpha _{c}$. Especially we find that the ground-state energies by the VMPS
become lower than those by $3$ED after the critical points, indicating again
the invalidity of ED method at  strong coupling. Practically, we cannot
perform ED with a very large total excitation number due to the huge
Hilbert space. In this paper, our exact diagonalization is only performed up
to $\ N=3$. Of course, if one can really perform  true exact
diagonalization without the limitation of  total excitation numbers, the
true ground state could be also correctly described in the ED method.

{\ It is interesting to note from Figs.~\ref{Magnetization} and \re
{GS_quantity} that }$\left\vert {\left\langle \sigma _{x}\right\rangle
\right\vert ${\ and }$\left\vert {\left\langle \sigma _{z}\right\rangle
\right\vert ${\ exhibit different behaviors with the increase of the
coupling strength. We will discuss this issue based on  parity symmetry
breaking and  total excitation numbers in the  next subsection.}

\subsection{Parity symmetry breaking}

\begin{figure}[tph]
\centerline{\includegraphics[scale=0.4]{Pi_N_MPS_3Ex.eps}}
\caption{ (Color online) Parity (upper panel) and total excitations (lower
panel) in the ground-state by the three-excitation NED and the present VMPS
for $s=0.3$ and $0.7$. $\protect\lambda =1$, $\Delta =0.1$, $\protect\omega
_{c}=1$, $\protect\epsilon =0$, $\Lambda=2$, $L=50$, $d_{opt}=12$, and
D=20,40$ for $s=0.3,0.7$ respectively. }
\label{parity}
\end{figure}

As stated before, the spin-boson model with (without) the RWA possesses a
U(1)$ ($Z_{2}$) symmetry. $U(1)$ is a higher symmetry than $Z_{2}$, so in
the RWA spin-boson model, the system also has $Z_{2}$ symmetry, i.e.,
parity symmetry, like  in the full spin-boson model. In this section, we
study the behavior of the expectation value of the parity $\hat{\Pi}=\exp
\left( i\pi \hat{N}\right) $.

The finite order parameter above the critical points obtained by VMPS in the
previous subsection displays  spontaneous symmetry breaking, while parity
is generally just the criterion to determine whether symmetry is broken. The
upper panel in Fig.~\ref{parity} gives the expectation value of parity
\left\langle \hat{\Pi}\right\rangle \ $obtained by both methods for
s=0.3,0.7$. Before the critical points $\alpha_c$, both methods yield the
same results for the parity. It is interesting to find that in this regime,
the value of parity for all values of $s<1$ is either $1$ or $-1$, which
corresponds to even or odd parity respectively. It follows that the symmetry
is not broken in this regime. Nevertheless, the average parity $\left\langle
\hat{\Pi}\right\rangle $ becomes zero due to quantum fluctuations above the
critical points. It is not the eigenstate of the parity operator, indicating
 spontaneous parity symmetry breaking. Note also that the parity jumps
for a few times before the critical coupling for $s=0.7$, and remains
unchanged for $s=0.3$.

We also present the average value of the total excitation number
\left\langle \hat{N}\right\rangle $ in the ground-state as a function of the
coupling strength$\ \alpha $ in the lower panel of Fig.~\ref{parity}. Below
the critical points, both approaches give the same $\left\langle \hat{N
\right\rangle $, which jumps between different plateaus with different
integers, just like in the JC model~\cite{JC_Plenio}. For $s=0.3$,
\left\langle \hat{N}\right\rangle=0 $ remains until the second-order
critical point $\alpha_c$, while, for $s=0.7$, $\left\langle \hat{N
\right\rangle $ increases from $0$ to $2$ one by one before $\alpha _{c}$.
It follows that the lowest energies in different coupling regime belong to
the energy levels in different $\left\langle \hat{N}\right\rangle $
subspaces, leading to level crossing at some coupling strengths. Thus the
first derivative of the ground-state energy with respect to the coupling
strength must be discontinuous at these coupling strengths, so instability
of $\left\langle \hat{N}\right\rangle $ before $\alpha _{c}$ just
corresponds to the first-order phase transitions. In addition, the jump of
\left\langle \hat{N}\right\rangle $ can also account for the back and forth
of the parity between $1 $ and $-1$.

One can also note that the total excitation number$\ \left\langle \hat{N
\right\rangle $ by VMPS increases abruptly at the critical coupling $\alpha
_{c}$. The $N$ED approach can only describe the phase with excitation number
less than or equal to $N$. The total excitation number is not limited in
VMPS, so it in principle can describe all phases. The total excitation numbers are
not conserved in the ground state above $\alpha _{c}$, because of the
symmetry breaking. At the critical points, $\left\langle \hat{N
\right\rangle $ does not jump to a plateau with the finite larger integer,
different from the first-order phase transitions.

The instability of total excitation number $\left\langle \hat{N
\right\rangle $ at the  phase transitions can account for the rich behavior
of $\left\vert \langle \sigma _{z}\rangle \right\vert $ here. For any value
of $s$, {in} the weak-coupling regime, the corresponding ground state is the spin-down state with photonic vacuum for $\left\langle \hat{N}\right\rangle =0$,
i.e., $|\downarrow \rangle |0\rangle $, thus $\langle \sigma _{z}\rangle =-1
, as just demonstrated in the upper panel of {Fig. \ref{GS_quantity}} for
s=0.3,0.7$. Once $\left\langle \hat{N}\right\rangle \neq 0$ at the first
phase transition no matter whether it is of the first  or the second order,
the spin state in the ground state consists of both spin-up and spin-down
states, which is  drastically different from the spin state only including the
spin-down state at $\left\langle \hat{N}\right\rangle =0$, leading to the
jump of $\left\vert \langle \sigma _{z}\rangle \right\vert $ and
\left\vert \langle \sigma _{z}\rangle \right\vert \neq 1$, as exhibited in
Fig. 2. At the later phase transitions between two finite $\left\langle \hat
N}\right\rangle \neq 0$, the ground states do not change drastically
because both have spin-up and spin-down states, therefore $\left\vert
\langle \sigma _{z}\rangle \right\vert $ will exhibit kinks, as shown  in
Fig. 2 for $s=0.7$. The drop of $\left\vert \langle \sigma _{z}\rangle \right\vert $ to
zero is an artefact{\ due to} the plot using magnetization, which is the
absolute {\ value} of $\langle \sigma _{z}\rangle $ in our paper. In the
weak-coupling limit, $\langle \sigma _{z}\rangle $ is actually $-1$ as
stated above. In the strong-coupling limit, we always {\ find} $\langle
\sigma _{z}\rangle >0$ in the ground-state, so, {\ in between} $\langle
\sigma _{z}\rangle $ must cross zero, resulting in the drop of $\left\vert
\langle \sigma _{z}\rangle \right\vert $ to zero at some coupling that is
not at any phase transition point. $\left\vert
\langle \sigma _{x}\rangle \right\vert $ is the order parameter of the
second-order QPT, so it only becomes non-zero for couplings larger than the
critical one.

{\ }In short, we find  rich phase transitions in the sub-Ohmic spin-boson
model with the RWA. First, \ the second-order QPT occurs for any finite
model parameters, similar to its counterpart without the RWA. In both the Rabi
model and the Dicke model with the RWA, the second-order QPTs cannot happen for
finite ratio $\Delta /\omega $ \ or finite qubit number. Second, for larger
bath exponents, e.g., $s=$ $0.7$, both the first- and second-order QPTs occur
subsequently with the coupling strength. Several first-order phase
transitions before the critical points indicate a sequence of instabilities.
The first-order phase transition is absent for small bath exponent $s$, such
as for $s=0.3$.

\subsection{Evidence for the second-order QPT by MCS variational studies}

\begin{figure}[tbp]
\centerline{\includegraphics[scale=0.4]{s_0d3_VMPS_VM_3Ex.eps}}
\caption{ (Color online) (a) The order parameter $\left\vert \left\langl
\protect\sigma_{x}\right\rangle\right\vert$.  (b) The total excitation number
\left\langle N \right\rangle$ as a function of the coupling strength within
VMPS, $3$ED, and MCS variational approaches. (c) The difference between the
VMPS (MCS) ground-state energy and that by $3$ED. $s = 0.3$, $\protec
\lambda =1$, $\Delta =0.1$, $\protect\omega _{c}=1$, $\protect\epsilon =0$,
\Lambda=2$, $L=20$, $d_{opt}=12$, $D=20$, $N_c=6$.}
\label{VM}
\end{figure}

Since this model at $s=0.3$ exhibits only one second-order QPT from the zero
excitation ground state, we can employ the MCS variational approach to
provide additional evidence, because the zero excitation can be realized and
the localized phase can be also described in the trial wave function (\re
{VM_wave}). In Fig.~\ref{VM}, we list results by  MCS, VMPS, and $N=3$ED
approaches for $s=0.3$. The logarithmic discretization parameter $\Lambda =2$
and $L=20$ bosonic modes are taken for all three approaches here. Note that
the number of bosonic modes here is smaller than those in other figures due
to the computational difficulties in the MCS approach, but it does not
influence the essential results at all. The MCS approach is used here to
account for the existence of the second-order QPT qualitatively, not for the
precise location of the critical points.

Before the critical points, the results by all three methods are the same.
Nonzero order parameter $|\left\langle \sigma _{x}\right\rangle |$ by MCS
approach appears above the critical points, providing   further convincing
evidence of the spontaneous symmetry breaking. The MCS ground-state energies
are lower than those by $N$ED method after the critical point, indicating
that the state with broken symmetry is more stable. The total excitation
number by MCS method increases suddenly above the critical point, because of
no limitation of total excitation number in the coherent state. All these
findings in the MCS variational study provide strong evidence of the
second-order QPT in the spin-boson model with RWA. As found recently by
Blunden-Codd et al.~\cite{Blunden} that a very accurate wave function can
be only obtained by at least $100$ coherent states,  by $N_{c}=6$
coherent states  the MCS results for the order parameter and energy still
slightly deviate from those by VMPS above the critical points.

\subsection{The critical exponent for magnetization}

\begin{figure}[tbp]
\centerline{\includegraphics[scale=0.4]{delta.eps}}
\caption{ (Color online) The log-log plot of the magnetization $\left\vert
\left\langle\protect\sigma_{x}\right\rangle\right\vert$ as a function of
bias $\protect\epsilon $ of the spin-boson model with (right) and without
(left) the RWA for $s=0.3$ (upper panel) and $s=0.7$ (lower panel). The
numerical results by VMPS are denoted by black circles, and the power-law
fitting curves are denoted  by the red dashed lines. All insets show the corresponding
linear plots. $\Delta =0.1$,$\protect\omega _{c}=1$, $\Lambda=2$, $L=50$,
d_{opt}=12$, and $D=20,40$ for $s=0.3,0.7$ respectively.}
\label{Fig6}
\end{figure}
Now we will study the nature of the second-order QPT in the sub-Ohmic
spin-boson model under RWA. The most important question is whether the RWA
changes the universality of the second-order QPT. The field relevant
order-parameter critical exponent $\delta $ can be determined through the
displayed power-law behavior $\left\langle \sigma _{x}\right\rangle \propto
\epsilon ^{1/\delta }$ at the critical coupling strength $\alpha =\alpha
_{c} $. Previously, various critical exponents in the full spin-boson model
have been calculated with different numerical approaches~\cite{QMC,
ED,Zhang, Chin, guo2012critical}. It is generally accepted that the exponent
$\delta $ takes the mean-field value $1/\delta =1/3$ for $s<1/2$, and the
non-classical one $1/\delta =\left( 1-s\right) /\left( 1+s\right) $ by the
exact hyperscaling for $s>1/2$ .

We present the magnetization by the VMPS method as a function of bias
\epsilon $ in a log-log plot for both $\lambda =1$ (RWA) and $\lambda =0$
(non-RWA) in Fig.~\ref{Fig6}. A very nice power-law behavior over three
decades is demonstrated in all cases. The results for the full model are
nearly the same as those in  Ref. \cite{guo2012critical}. It is
interesting to find that $1/\delta $ is around $1/3$ for $s<1/2$ for both
RWA and non-RWA cases. Very surprisingly, even for $s=0.7$ in the RWA model,
$1/\delta $ is still close to that in the full model within the statistical
errors, also indicating  hyperscaling. Our results suggest that
counterrotating terms would have no effect on critical exponent $\delta $
in the spin-boson model.

\subsection{Extensions to the Ohmic bath.}

\begin{figure}[tbp]
\centerline{\includegraphics[scale=0.4]{s_1_full.eps}}
\caption{ (Color online) (a) The magnetization $\left\vert \left\langl
\protect\sigma_{z}\right\rangle\right\vert$, (b) parity $\left\langle \hat
\Pi}\right\rangle$, and (c)  total excitation number $\left\langle N
\right\rangle$ as a function of the coupling strength within VMPS and $3$ED
approaches for the Ohmic bath. $s = 1$, $\protect\lambda =1$, $\Delta =0.1$,
$\protect\omega _{c}=1$, $\protect\epsilon =0$, $\Lambda=2$, $L=50$,
d_{opt}=12$, and $D=20$. }
\label{Ohmic}
\end{figure}

Now we turn to the Ohmic spin-boson model under the RWA. By both VMPS and
 $3$ED methods, the above observables are also calculated. In Fig.~\re
{Ohmic}, we collect magnetization $\left\langle \sigma _{z}\right\rangle $,
parity $\left\langle \hat{\Pi}\right\rangle $, and  total excitations
\left\langle N\right\rangle $ as a function of the coupling strength $\alpha
$ in the range $(0,2)$. During this wide regime, one can find that the model
undergoes a few first-order QPTs with the increment $1$ of the total
excitation number. The two approaches almost give the same results. It
follows that in the Ohmic bath, the $3$ED approach is suited to the wide
coupling regime where no second-order QPT occurs. This result is obviously
different from that in Ref. \cite{Tong} where only $N=1$, i.e., single
excitation, is considered. We have also studied the super-Ohmic bath $s=3/2
, and find that there is only one first-order QPT in the regime $\alpha
=[0,2]$ (not shown here).

\section{Conclusion}

In this paper, we study the spin-boson model in the RWA by the VMPS method,
MCS variational ansatz, and exact diagonalizations within the truncated
Hilbert space. Surprisingly, we find the second-order QPT in the RWA model
for any bath exponent $s<1$ for the first time, to the best of our
knowledge. A rich picture for the quantum phase transitions is observed.
Besides the second-order phase transition, the first-order phase transition
also appears in the same model, which could only vanish for small bath
exponents. The coexistence of both first- and second-order QPTs in the same
model has never been observed in  other spin-boson-like models, such as
the quantum Rabi and Dicke models in the RWA.

Within the statistical error, for all values of the bath exponents $s<1$,
the critical exponent $\delta$ is found to be nearly the same as those in
the full model. It is then suggested that the counterrotating terms would
have almost no effect on critical exponent $\delta$ in the critical regime.
The analytical argument about the quantum-to-classical mapping ~\cit
{map,berry,Kirchner} in the spin-boson model in the RWA would be helpful to
account for this robust nature uniquely from the rotating-wave terms.

\textbf{ACKNOWLEDGEMENTS}  We acknowledge useful discussions with Stefan Kirchner. This work is supported by the National Science
Foundation of China (Nos. 11834005, 11674285), the National Key Research and
Development Program of China (No. 2017YFA0303002),

$^{*}$ Email:qhchen@zju.edu.cn

\setcounter{figure}{0}

\renewcommand{\thefigure}{A\arabic{figure}}

\begin{appendix}

\section{Convergence of results by the variational matrix product state
approach}

We provide  evidence for the full convergence of our VMPS results here.

\begin{figure}[tbp]
\centerline{\includegraphics[scale=0.4]{s0.3_0.7_lbd1_Sx_converge.eps}}
\caption{ (Color online) The plot of the magnetization $\left\vert
\left\langle\protect\sigma_{x}\right\rangle\right\vert$ as a function of
coupling strength $\alpha$ with bond dimension (a) D=12, 20, 30 for $s=0.3$ and (b)
D = 12, 24, 40$ for $s=0.7$. Other parameters: $\protect\lambda =1$,
$\Delta =0.1$, $\protect\omega _{c}=1$, $\protect\epsilon = 0$,
$\Lambda = 2$, $L = 50$, $d_{opt} = 12$.}
\label{FigA4}
\end{figure}

The most important point of our paper is whether the second-order QPT occurs
in the spin-boson model in the RWA. To demonstrate this point convincingly,
we have  checked the magnetization as a function of the coupling
strength with increasing bond dimension (D). The results are shown in
Fig.~\ref{FigA4}. One can see that the magnetization is converged even for the smallest one $D=12$, indicating that the second-order QPT in our model is  reliable.

\begin{figure}[tbp]
\centerline{\includegraphics[scale=0.4]{s0.3_0.7_lbd1_D_converge.eps}}
\caption{ (Color online) Convergence check for the VMPS parameter $D$ in the critical regime.
The panels shows the log-log plot of the magnetization $\left\vert
\left\langle\protect\sigma_{x}\right\rangle\right\vert$ as a function of
bias $\epsilon$ with bond dimension (a) $D=12, 20, 30$ for s=0.3 and (b)D = 20, 40, 60
for $s=0.7$. Other parameters: $\protect\lambda =1$, $\Delta =0.1$, $\protec
\omega _{c}=1$, $\Lambda = 2$, $L = 50$, $d_{opt} = 12$.}
\label{FigA1}
\end{figure}

\begin{figure}[tbp]
\centerline{\includegraphics[scale=0.4]{s0.3_0.7_lbd1_L_d_converge.eps}}
\caption{ (Color online) Convergence check for the VMPS parameter $\Lambda$ and $d_{opt}$ in the critical regime.
The panels show log-log plots of the magnetization $\left\vert
\left\langle\protect\sigma_{x}\right\rangle\right\vert$ as a function of
bias $\epsilon$ with (a)discrete logarithms $\Lambda = 1.5, L = 80$; $\Lambda = 2, L = 50$; and
$\Lambda = 3, L = 30$ for $s = 0.3$ with $D = 20$, $d_{opt} = 12$; and (b) optimized
physical dimension $d_{opt} = 8,12,16$ for $s = 0.7$ with $\Lambda = 2, L = 50$,
$D = 40$. Other parameters: $\protect\lambda =1$, $\Delta =0.1$, $\protec
\omega _{c}=1$.}
\label{FigA2}
\end{figure}

The field relevant
order-parameter critical exponent $\delta $ should be determined precisely to study the universality class of the second-order QPT in this model. In doing so, we should  check its
convergence for the VMPS parameters carefully. Figures~\ref{FigA1} and ~\ref{FigA2} exhibit
the nice  power-law curves  with the same fitting exponent  for the selected bond dimension ($D$), optimized physical dimension ($d$),
and discrete logarithm ($\Lambda $) for the same values of the bath exponent $s$.  It is clearly demonstrated that we have obtained a convincing critical exponent  of the second-order QPT in this paper.

\begin{figure}[tbp]
\centerline{\includegraphics[scale=0.4]{s0.3_0.7_lbd1_Entropy_converge.eps}}
\caption{ (Color online) The plot of the entanglement entropy as a function
of $1/D$ (D is bond dimension) with the fixed coupling strength $\protec
\alpha = 0.1,1$ for $s=0.3, 0.7$ respectively. Other
parameters: $\protect\lambda =1$, $\Delta =0.1$, $\protect\omega _{c}=1$,
\protect\epsilon = 0$, $\Lambda = 2, L = 50$, $d_{opt} = 12$ .}
\label{FigA5}
\end{figure}

Finally we turn to the convergence of the entanglement as a function of the
inverse of the bond dimension 1/D. In the spin-boson model, the entanglement
entropy can be rewritten as \cite{Chin}
\begin{equation}
E=-P_{+}\log _{2}P_{+}-P_{-}\log _{2}P_{-}
\end{equation}
where $P_{\pm }=\left( 1\pm \sqrt{\left\langle \sigma _{x}\right\rangle
^{2}+\left\langle \sigma _{y}\right\rangle ^{2}+\left\langle \sigma
_{z}\right\rangle ^{2}}\right) /2$, which describes the correlation between
the spin and the bosonic bath.

In Fig.~\ref{FigA5}, we plot the entanglement entropy as a function of 1/D
for s=0.3 (upper) and s=0.7 (lower) at fixed coupling strength. It is
clearly shown that the entropy  saturates even before  $D=20$ for both cases, providing
further strong evidence for the convergence of our results for $D \ge 20$, i.e.,
the value selected in our calculation. Thus we believe that our
algorithm has converged to the global minimum, the ground state is the
correct one, and the results given here should be reliable.

\end{appendix}

redpajama_set_name: RedPajamaArXiv


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I was so excited to try these when I saw the recipe. I didn't have kielbasa, so I used Aidells Chicken Apple Sausage. They turned out perfect! Great flavor and another one to add to the mix of party foods.
Cut sausage into slices (about 1/3" thick).
In a medium skillet, bring the pomegranate juice and ketchup to a boil. Boil for 2-3 minutes or until it becomes syrupy.
Add the sausage slices to the sauce and cook over high heat. Turn the slices occasionally, until the mixture reduces almost completely and coats the sausage.
Serve with toothpicks or with crackers.
I tried these for the Super Bowl and they were delicious! I can't wait to make them again. I even stocked up on frozen wings last week when they were on sale. If you don't have a Trader Joe's near you, there are quite a few brands that have a Carolina Gold BBQ sauce that you can find at your grocery store.
Put the frozen chicken wings in crockpot.
Add Franks hot sauce. Stir the chicken wings in the sauce.
Add Carolina Gold sauce and stir.
Once they are done, remove chicken wings and place in a baking dish. Brush some Carolina Gold Sauce on them and broil until nice and crispy. Remove and brush more sauce on them and then put back in the oven for a few minutes.
This one is a favorite of my youngest daughter. She requests them every time we have an event that I'm making snacks for. So I stock up on Lil Smokies whenever they are on sale and freeze them.
Add all the ingredients to crockpot.
Stir together and cook on low for 1 hour.
I love appetizers! And I love any opportunity to make them. These meatballs are super easy and great for a crowd. I usually use grape jelly, but I have also used apple jelly, apricot jelly, and peach jelly. All turn out great.
Cook for 3-4 hours on high (7-8 hours on low).
I first discovered these little bites of heaven years ago at a baby shower. I mean you can't go wrong with a sandwich toasted in buttery oniony goodness! They are also great for a quick, easy dinner.
These packages of Hawaiian Rolls are often on the "soon to be expired rack" at our grocery store, so whenever I see them, I stock up to make these sandwiches. FYI, the rolls and sandwiches both freeze really well.
I also use whatever lunch meat (ham or turkey) and cheese (Swiss, Colby Jack, Cheddar, American) that we have lying around. Or a combination of them. They have all turned out great!
I also usually leave out the poppy seeds just because I don't keep them on hand. Mix and match the ingredients and make these your own.
It's time for Super Bowl snacks! I love snack foods and appetizers. And when I make them, it usually means that we are having a party with family and friends!
I discovered this recipe last year and made it for NYE. My first time making it, I forgot to drain the Rotel. It still tasted great, but it was pretty runny. I also hate the chunks in Rotel, so I blend it up before adding it.
Serve this warm with tortilla chips. It's not pretty, but it tastes amazing!
In a medium sized bowl, add corn, black beans, rotel, cream cheese, chili powder, garlic powder, cumin, cayenne, & 1/2 C shredded cheese. Stir together. I use my KitchenAid.
Potatoes are my FAV! Seriously, my favorite food. I was raised on a potato farm and potatoes are in my blood.
Love me a baked potato. Loaded with butter. And cheese. And bacon. And… You get the idea. But I don't like the texture when they are cooked in the microwave and I'm not patient enough to wait for them to cook in the oven.
So when I found out I could cook them in the Instant Pot, I was sold! And I made them that night. To my wonderment, they were soft and fluffy. No hard spots, no cooking for an hour, and they were cooked all the way through. Finally!
Potatoes (just don't fill above the line)1 C waterInstructions Put water in Instant Pot. Add trivet.
Cut potatoes into similar sizes and add to Instant Pot.
Allow pressure to naturally release for 10 minutes, then do a quick release.
What started as a blog for foot zoning is turning into a place for me to share more of what I love. Trying new recipes, DIY, finding amazing deals, and so much more.


text:
Home of the 10-time World of Outlaws Sprint Car Champion...
DRIVER: DONNY SCHATZ
He has nearly 300 World of Outlaws Sprint Car victories to his name. He's a 10-time Knoxville Nationals Champion and a 10-time World Of Outlaws Sprint Car Series Champion. He will go down as one of the very best to ever strap into a Sprint Car. Meet Donny Schatz...
TEAM: TONY STEWART RACING
Tony Stewart Racing (TSR) is the title-winning World of Outlaws Sprint Car Series team owned by three-time Monster Energy NASCAR Cup Series champion Tony Stewart. TSR has captured 23 short-track racing titles (14 in USAC and nine WoO) since its inception in 2001.
MORE ON TSR
LATE MODEL RACING
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LIVE ON DIRTVISION
Helmet Design Submission
DONNY SCHATZ
2021 Paint Scheme Revealed January 20, 2021 READ MORE Schatz set to open 2021 season in Late Model at Wild West Shootout January 9, 2021 READ MORE GALLERY: Last Call at Charlotte November 10, 2020 READ MORE GALLERY: Late Model at Charlotte November 10, 2020 READ MORE Schatz kicks off Last Call with Late Model doubleheader November 4, 2020 READ MORE
HOMEJohnathan Bateman2020-10-13T18:43:07-04:00
2021 Paint Scheme Revealed
The 2021 Tony Stewart Racing No. 15 is set to make its maiden voyage this weekend at the Wild Wing Shootout at Arizona Speedway in San Tan Valley, Arizona.
Mike Spieker2021-01-20T10:22:58-05:00January 20th, 2021|
Schatz set to open 2021 season in Late Model at Wild West Shootout
FARGO, N.D. (January 9, 2021) – Ten-time World of Outlaws Sprint Car champion, Donny Schatz, will be kicking off his 2021 season tonight at the Wild West Shootout at Arizona Speedway near Phoenix, Ariz. in his Petro Stopping Center / Fastlane Car Wash No. 15 Late Model. This will mark Schatz's
Mike Spieker2021-01-09T11:47:23-05:00January 9th, 2021|
Schatz kicks off Last Call with Late Model doubleheader
CHARLOTTE, N.C. (November 4, 2020) – Donny Schatz is set to make his sixth and seventh appearances with the World of Outlaws Morton Buildings Late Models tonight, Nov. 4th and Thursday, Nov. 5th at the Dirt Track at Charlotte. The two-night Late Model event precedes the season finale for the World
Mike Spieker2020-11-04T13:25:11-05:00November 4th, 2020|
Schatz races to runner-up finish at Kokomo Speedway
Mike Spieker2020-10-26T22:20:00-04:00October 26th, 2020|
FARGO, N.D. (October 26, 2020) – Donny Schatz picked up his 23rd top five finish of the 2020 season last Saturday night at Kokomo Speedway in Indiana. The driver of the Tony Stewart / Curb-Agajanian Racing No. 15 was strong all night, picking up second place finishes in his qualifying group
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Career World of Outlaws Feature Starts
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This day was beautiful, with sunshine which created good thermalling conditions. I flew with Richard again and we found the thermals strong, wide and right up to cloud base at over 5500 ft. We were up for almost two hours again and first flew over the east ridge but then, I took us over to the west ridge and south to the Lincoln Gap. We encountered other gliders in the same thermals and had good close looks at them.
The sky would periodically over-develop then open up again. But as the day went on, the over-development stayed around and thermal were harder to find. We went back to the east ridge and after losing a bit of height, decided to end the flight. I flew for a bit more and found a good thermal that took us back to cloud base but had to make tight, steep turns to stay in the best lift. Those circles are evident on the track of our flight due east of the airport. That was the last climb of the flight.


How do I add GA to a Canvas Page? I've tried putting in the code via the standard way, but it just strips out the code. I'd like the ability to track usage on a public available free site on canvas's site.
This will be short since I have no personal experience using Google Analytics except as a user being tracked.
Now, the question becomes how does that help? Canvas is collecting the Google Analytics but I don't know if they share that with you in any way other than through the Analytics pages. Some of it is starting to show up in Canvas Data, but I'm not sure if that will be sufficient for your needs or you need to interact with Google Analytics directly.
I do remember going to a presentation at a previous InstructureCon (2013 or 2014) where a guy talked about using Google Analytics. At the time, it was my first introduction to what Google Analytics was and I was both blown away and freaked out at the same time.
What was the "standard way" you tried? Was it an iframe? If not that could help, follow steps 3 and 4 on this guide Embedding a webpage in an assignment (iframes) to embed the analytics page into the rich content editor.
It was just copy/paste the GA Code into the page via the HTML editor.
Is there a non-iframe method or is that the only method?
The other option would be to use the redirect tool add a menu item to send the students to the Google Analytics page, but this does not embed it, rather it sends the students away from Canvas.
Also important to note that Google Analytics does NOT count any page views using the mobile Apps, just Web Browsers.
Victor Yee, I wanted to make sure you saw Jeremy Perkins new resource, How to Set Up Google Analytics for Canvas, and his upcoming CanvasLIVE session, CanvasCon USF: Using Google Analytics to Understand Canvas Usage.