text:
Author Archives: Jessica Cannon
Updated: Our Continued Response to COVID-19
PATH Foundation's fourth round of grants during the COVID-19 crisis contributes to a total of $2.3 million in response funding. On October 1, we announced $1,501,720 in Community Resilience Grants to 44 local organizations. Community Resilience Grants were a one-time grant cycle created to help ease the financial burden of increased expenses due to the […]
PATH announces $1.5 Million in New Community Resilience Grants
Warrenton, Va. — The PATH Foundation has announced more than $1.5 million in special Community Resilience Grants to 44 local organizations. The Community Resilience Grants were a one-time grant cycle created to help ease the financial burden of increased expenses due to the impact of COVID-19. The total granted in this round of giving was […]
Announcing $1.2M in Broadband, Program & Planning Grants
The PATH Foundation on Friday announced $1.2 million in new funding, including $750,000 for broadband initiatives in areas of Fauquier County that have limited or no access to high-speed internet. Of the $750,000 in broadband funding, $500,000 will support efforts launched in fiscal year 2020, which ended June 30. The remaining money will go toward […]
New Community Resilience Grant Cycle Opens July 6
The new, one-time grant cycle will help nonprofits facing financial difficulties from COVID-19 related circumstances. On June 29, the PATH Foundation announced a new Community Resilience grant cycle to support nonprofit organizations, with $1.5M in available funding. The grants were created in response to the difficulties many nonprofits are facing due to circumstances caused by […]
Third round of pandemic funding provides $206,500 in new COVID-19 relief
PATH Foundation's third round of grants during the COVID-19 crisis contributes to a total of $816,500 in response funding. May 10, 2020 — Warrenton, VA — On May 10, we announced $206,500 in new, immediate COVID-19 response funding to local organizations in our latest round of efforts to support our community through the crisis. This […]
$485,000 granted in Response to COVID-19
Our latest round of support to organizations during the coronavirus pandemic. The PATH Foundation has announced $485,000 in immediate grant funding to respond to COVID-19 in Fauquier and Rappahannock counties. This funding is in addition to the $125,000 granted on March 13 in anticipation of community needs, for a total of $610,000 to help our […]
How can we help a community struggling in a time of need?
Unprecedented times call for unprecedented actions, which we're seeing through previously unimagined cancellations, warnings and recommendations. In the past several days, we've been faced with finding solutions to ever-evolving problems. With the health and well being of our residents at stake, we're collaborating with partner organizations to help ease the increasing burdens people are facing. […]
PATH Foundation, VDOT and Fauquier County fund new trail, recreation opportunities
Warrenton, Va. — The PATH Foundation is partnering with the Virginia Department of Transportation (VDOT) and Fauquier County Government to fund the construction of a 1.3-mile, off-road pedestrian and bicycle trail connecting three subdivisions in Fauquier County. The PATH Foundation will contribute $528,332 towards the completion of the trail. VDOT has awarded $520,030 for the […]
Woods at Warrenton
PATH Foundation Announces $750,000 in Available Program and Planning Grants
Application window for Program and Planning Grants Opens Warrenton, Va. — The PATH Foundation has announced the opening of their annual Program and Planning Grants application window, with a total of $750,000 in funding available. Program and Planning Grants, given annually, provide up to $100,000 for organizations to spend on new or existing projects related […]
Program and Planning
Program and Planning Grants


text:
Talks & Conference Papers
Creative Writing & Classics
Gardens & Goddesses
Clodia and the Artist
The White Hind
Arachne and Athena
Xena - 20 Years On
Venus in Sussex
Nausikaa's Odyssey
Greek Mythology Comics Workshop
Greek Mythology Tour of Brighton
Conferences	/
Written by Zeus
Revisiting the Warrior Princess and her legacy: Xena 20 years on
18th and 19th June 2021
On 18th June 2021 it was twenty years since the controversial final episode of ground-breaking television series Xena: Warrior Princess, 'A Friend in Need Part II' aired. Although the series was first conceived as a spin-off from a series featuring a male hero, Hercules The Legendary Journeys, Xena's popularity surpassed that of Hercules, and ran for six seasons from 1995-2001. Xena paved the way for subsequent action heroines, from Buffy in Buffy the Vampire Slayer, who reached our television screens in 1997, to the re-envisioned Wonder Woman in the recent blockbuster films. The (initially subtextual) relationship between Xena and Gabrielle also influenced the depiction of LGBT relationships in popular culture. Xena garnered a large fan base and spawned an interest in the ancient world for a generation of young viewers.
This international virtual conference brought together scholars from across a range of disciplines to mark the 20th anniversary of the end of Xena: Warrior Princess and explored different aspects of the series and its impact. The organisers aim to publish selected papers following the conference.
Organisers: Amanda Potter (Open University) and Anise Strong (Western Michigan University)
Final programme with abstracts is available Here.
You can also download a PDF version Here
****************************************************************************************************************************************
If you would like to (re) watch some episodes help Amanda with her chapter for publication, 'How do you solve a problem like Callisto?' please watch 'Callisto' from season one and 'Return of Callisto' from season two, and complete a short survey:
https://www.surveymonkey.co.uk/r/2QG7NJ2
Amanda's debut novel, Nausikaa's Odyssey, is now available as an eBook on Amazon.
© 2022 21st Century Classics

redpajama_set_name: RedPajamaCommonCrawl


text:
package com.evolveum.midpoint.gui.api.model;

import org.apache.wicket.model.IModel;
import org.jetbrains.annotations.NotNull;

/**
 * @author mederly
 */
public interface NonEmptyModel<T> extends IModel<T> {

	@Override
	@NotNull
	T getObject();

	@Override
	void setObject(@NotNull T object);
}

redpajama_set_name: RedPajamaGithub


I was once again asked to photograph the opening night gala of the 2014 Toronto Japanese Film Festival, which was hosted last night at the Japanese Canadian Cultural Centre (JCCC). The JCCC is a really amazing place (the largest Japanese cultural centre/bunka kaikan in the world outside of Japan) that seems to be under appreciated because of the myth that it is horrendously far from downtown and terribly difficult to get to by transit. This is very much not the case, but apparently Torontonians are afraid of buses.
If you weren't there last night, here is what you missed. The full set of photos can be found in my TJFF 2014 Gallery.


text:
Now Your Kid Can Ride Like Rossi!
Eight-time world motorcycle champion Valentino Rossi has created a new line of two-wheel funsters – except these bikes have no motors.
Rossi teamed up with Italian company Cicli Adriatica to develop a line of children's bicycles – the VR46 bike collection. With this line of children's two-wheelers, you can now blame your child's skinned knee of the famed Fiat Yamaha rider!
The announcement of the VR46 bike collection came complete with a press release and a couple of photos of the bikes, complete with training wheels.
An excerpt from the press release after the jump.
"Valentino still remembers when, as a kid, took his first steps in the backyard of Dad Graziano, just riding an Adriatica bike. Since then, Rossi has continued to appreciate the two-wheeled world, though, of course, his interests have moved on the motorbike."
Not satisfied with churning out children's bikes, the VR46 line will apparently also include a racing bike and a mountain bike "from the highest level" to extend Rossi's "winning image."
The new bicycle line will be presented in Milan during EICMA. The show will start on Sept. 18 and run until Sept. 21.
By Lucas Cooney
See also: Yamaha Sketches out the Features on the New 2010 YZ450F [Video], 2010 Yamaha YZ450F Rumors [video], 2010 Yamaha YZ250F Preview [video].


Raymond L. Neubauer presents a view of nature that describes rising complexity in life in terms of increasing information content, first in genes and then in brains. The evolution of the nervous system expanded the capacity to store information with relatively open-ended programs, making learning possible. Portraying four species with high brain-to-body ratios--chimpanzees, elephants, ravens, and dolphins--Neubauer shows how each shares with humans the capability for complex communication, social relationships, flexible behavior, tool use, and powers of abstraction. He describes this constellation of qualities as an emergent self, arguing that humanity is not the only self-aware species and that human characteristics are embedded in the evolutionary process and are emerging in a variety of lineages on our planet. Neubauer ultimately shows that human culture is not a unique offshoot of a language-specialized primate, but an extension of a fundamental strategy that organisms have used since the beginning of life on earth to gather information and buffer themselves from environmental fluctuations.Neubauer also views these processes in a cosmic setting, detailing open thermodynamic systems that become more complex as the energy flowing through them increases. Similar processes of increasing complexity can be found in "self-organizing" structures in both living and non-living forms. Recent evidence from astronomy indicates that planet formation may be nearly as frequent as star formation. In February 2011, NASA announced that the Kepler space telescope had located fifty-four planets in the habitable zones around their stars. Life makes use of the elements most commonly seeded into space by burning and exploding stars, and the evolution of life and intelligence that occurred on our planet may be common across the universe.


The elective contribution limit for employees who participate in section 401(k), 403(b), or 457(b) plans increase from $17,000 to $17,500.
The limitation for the maximum annual addition for defined contribution plans increases from $50,000 to $51,000.
The annual compensation limit that can be considered under a plan increases from $250,000 to $255,000.
The limitation used in the definition of highly compensated employee remains unchanged at $115,000.
The dollar limitation concerning the definition of a key employee remains unchanged at $165,000.
The compensation amount regarding simplified employee pensions (SEPs) remains unchanged at $550.
The limitation regarding SIMPLE retirement accounts increases from $11,500 to $12,000.
The AGI limit for the saver's credit (also known as the retirement savings contributions credit) for low-and moderate-income workers is $59,000 for married couples filing jointly, up from $57,500 in 2011; $44,250 for heads of household, up from $43,125; and $29,500 for married individuals filing separately and for singles, up from $28,750.
The dollar amount for determining the deductible amount of an IRA contribution for taxpayers who are active participants filing a joint return or as a qualifying widow(er) is increased from $92,000 to $95,000. The applicable dollar amount for all other taxpayers (other than married taxpayers filing separate returns) is increased to $59,000. The applicable dollar amount under Section 219(g)(7)(A) for a taxpayer who is not an active participant but whose spouse is an active participant is increased from $173,000 to $178,000.
Fraser Stryker is a leader in employee benefits and tax law. Attorneys in the Firm's Employee Benefits and Taxation Practice Groups advise businesses, governments, and nonprofit/tax-exempt organizations on a wide variety of tax and employee benefits matters, transactions, and litigation. Fraser Stryker helps employers implement and maintain innovative and cost-effective employee benefit plans that attract and retain top talent. For more information, please contact Nicole R. Konen or Emily Wischnowski.


"We'll leave the light on for you." So says the famous Motel 6 slogan. And I love Motel 6 – for the most part. Accommodations are predictable, inexpensive, and (usually) liveable. But my friends and I showed up at a hotel last Sunday night that not only didn't leave a light on for us, it left us out on the street with the front door locked.
The time was about 7:30 p.m. and we'd just pulled up – tired and ready to settle in – from a 3.5-hour drive from the Phoenix area. We were standing at the door to the historic 1902 Hotel San Ramon in teeny Bisbee, Arizona just 9 miles north of the Mexican international border. And there was nobody there to let us in.
I dialed the phone number written on a piece of paper that was taped to the glass window in the door. "Welcome to Verizon Wireless. Your call cannot be completed at this time." I tried again and got the same message. I looked at my road trip partners, Armando and Chris, with a bit of desperation. I already knew that vacancy was limited all over this teeny town of 5,000 people on a holiday weekend, and the chances of finding a back-up plan were slim. Would we end up driving 50 miles away to Benson to find a place to stay?
As a last ditch effort, I walked next door to the restaurant called Santiago's. "By chance do you know the keypad code to get into the hotel next door?" I asked the hostess. The girl disappeared behind a dividing wall and then returned with a woman who was already on her cell phone trying to come to our rescue. Luckily, the two were able to get in touch with the hotel proprietor and find out our 4-digit entry code.
Moments later, we were lugging our bags up a flight of stairs to room #6. The hotel, as it turns out, only has 6 rooms in total. Aside from the check-in experience, everything else about San Ramon was positively charming. Our room was unlocked when we found it, with 2 keys on the dresser. Silk rose petals were scattered on the floor, and 2 small Valentine's Day goody bags were on top with a few chocolates. Eat that, Motel 6!
It was just a little more than 3 years ago that I made a trip to Bisbee in my Legend. This latest experience was even more rich because I made it into an over-nighter as opposed to a rushed day-trip from the Phoenix area. After a delicious dinner at Santiago's, Armando and I explored the night life for a little bit while Chris decided to hit the sack.
Just steps away from the front door of San Ramon was the Stock Exchange Saloon. Its double doors and neon sign beckoned, so Armando and I went over to check it out. This building, dating back to 1905, once housed a stock brokerage firm called Duey and Overlock. According to the gal behind the counter who was born and raised in Bisbee, this was the only branch of the New York Stock Exchange outside of New York at the time. Today, the venue retains some of the original elements from those days including this board along the back wall under plexiglass. Armando and I cued up a few hits on the jukebox including "Walkin' on Sunshine" before calling it a night.
On Monday morning, I rolled open the wooden windows of the 2nd floor hotel room and admired the perfect weather outside. It was President's Day, and we departed the San Ramon and took the ILX to have breakfast at "Bisbee Breakast Club" (BBC) a mile and a half away in historic Lowell. If the streets of Bisbee were a snapshot of the 1880s, then Lowell moved us forward about 80 years to the 1960s. The entire main street was lined with classic cars, varying in condition from scrap heap to show-ready. We stepped inside and feasted on chiles rellenos.
The rest of our morning was filled with sights & sounds of the American Wild West. We stopped just briefly in Naco, Arizona at the Mexican border – but not too close – as none of us had our passports handy and we didn't want to chance an "accidental" international border crossing. We took a moment back in Bisbee to frame up the re-creation of a photo that I took 3 years ago with my Legend. (Thanks Chris & Armando for being my photographers!). Bisbee's strong heritage as a copper and gold mining town lives on and I will surely be going back for a tour of the mine soon.
I knew we had a 1:00 p.m. cave tour to get to back in Tucson, so by about 11:00 we said our farewells to B-town and headed out. Out last stop on the trip was to a large underground cave about 20 miles east of Tucson that I last visited 100,000 miles ago in September 2013. Our talkative tour guide stretched a 45-minute walking tour (covering roughly 1/2 mile and almost 400 steps) into about an hour and 15 minutes. I learned a few new things this time around, but I was surely glad to surface for daylight at the conclusion.
Thanks for joining for this weekend's adventure!
Other pics from this weekend follow.
Wouldn't be called Fountain Hills without an awesome fountain!
Loved this 1989 Porsche 911!
Bisbee is extremely hilly and the streets are narrow. It's no wonder the town wasn't built in squares.
Bisbee Breakfast Club awaits us!
How about that Chile Relleno?
Make sure you stop by Josh Clymer's blog, TSX Travels, where he's now on the Final Fifteen countdown to 500,000 miles on his 2005 Acura TSX.
Also, I highly recommend "Lasting Legend" body wash from Old Spice. For obvious reasons!
This entry was posted on February 16, 2016 at 4:56 pm and is filed under Arizona, ILX, NSX, Road Trip.	You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.
Oh, I know! Been there, done that! I've probably spent a combined 2 weeks in the backseat of my Legend coupe and it is NOT accommodating for an overnight stay 🙂 Hope you are doing well, Brad! Have a great weekend!
So Tyson when can we expect to see your little corner of Scottsdale take on cultural & historical significance and give Lowell a run for its money?


230 Guests visit Washington DC and New York on an incentive trip.
In the second week of June 2017, Incentive Action took 230 guests to the USA as part of an incentive reward. These guests had qualified over the course of 12 months through sales of our client's product.
Guests flew into Washington DC and spent two nights at the W Washington Hotel, overlooking the White House. Whilst in Washington DC, guests were treated to a city tour (including a visit to Capitol Hill) and a visit to the International Spy Museum. Guests also opted-in to various smaller tours including a bike tour, a Segway tour; a visit to George Washington's house at Mount Vernon; or a tour of Georgetown.
The highlight in Washington DC was the dinner at the National Air & Space Museum, where guests had the opportunity to explore the many displays as well as experience flight in the many simulators. American astronaut, Leroy Chiao, was the guest speaker at dinner, where he entertained guests with stories of his experience in space and aboard the International Space Station.
Guests were transferred to New York via the Amtrak and upon arrival, checked-in to the recently refurbished InterContinental Barclay. Three full days in New York saw the group partake in an Amazing Race challenge around Midtown and a city tour that included visits to SoHo, Chelsea, the Flatiron, 9/11 Memorial and High Line. Some guests also immersed themselves in smaller activities and tours such as scenic helicopter flights, an NYPD tour, biking in Central Park and a Gangs & Mobs walking tour.
The Farewell Party was held at The Foundry in Queens, situated right beside the Queensborough Bridge and accessed via a scenic Water Taxi tour down the East River, offering spectacular views of the Manhattan Skyline whilst enjoying some drinks.
Check out some of the amazing photos from this incredible trip.


text:
Prof Guoxiong Wu
WC1E 7JE
g.wu@ucl.ac.uk
https://sites.google.com/view/gx-wu
Professor of Computational Fluid Dynamics
Dept of Mechanical Engineering
Faculty of Engineering Science
After obtaining PhD at Department of Mechanical Engineering, UCL, Prof G.X. Wu worked in the same department as a researcher between 1987-1992, a lecturer between 1992-1997, a reader between 1997-2000, a professor since 2000. Prof Wu was a Royal Society Industry Fellow between 1997-1999 hosted by WS Atkins and has been a member of technical committee member of Lloyds Register.
Biomedical Engineering Research Group
Marine Research Group
Thermofluids Research Group
UCL Medical-Modelling Group
Professor Wu's research is primarily in modelling and simulation of fluid/structure interactions problems in the context of engineering applications, based on the finite element method, finite volume method, finite difference method, boundary element method, lattice Boltzmann method as well as many other methods. The principal applications of the work are (1) naval architecture, (2) coastal, offshore & deep-water engineering, (3) renewable energy, (4) polar engineering.
Professor Wu has been the chair of the joint research centre on deep-water challenges, involving UCL, Shanghai Jiao Tong University and Harbin Engineering University. The centre has trained many dozens of PhD and MSc students supervised by over a dozen professors, and has been supported by the Lloyd's Register Foundation which owns the commercial company Lloyd's Register (LR), one of leading classification societies in the world. The research covers a wide range of topics in naval architecture, ocean engineering and polar engineering. In addition to the cutting-edge research on some fundamental and challenging problems in engineering, the focus has been on applying the research to improve design, protect the environment, enhance safety and reduce cost. One of the applications was to optimisation of the structural design of a ship built at the end of 2016 by Jiangnan Shipyard (Group) Co. Ltd., based on the newest rules of the International Association of Classification Societies, whose members include LR.
The novel trimaran concept was pioneered at UCL in the 1990s. Professor Wu developed methods and codes for its simulation in collaboration with QinetiQ. His research with QinetiQ has also been used to improve the manoeuvrability and controllability of underwater vehicles at periscope depth.
Professor Wu has published extensively in JFM and Roy. Soc. Some of his publications in Physics of Fluids have been chosen as "Best Papers" of the journal or recommended as "Editors' Pick".
Many of novel methods developed by Professor Wu and the results obtained have made a major impact in industry. An example is the development of the commercial code Hydro-Star by Bureau Veritas. The author of that code stated (23rd Symposium on Naval Hydrodynamics) "…the author is greatly indebted to Professor Wu Guo Xiong … for providing comprehensive data … which have been critically useful in the development of computer codes."
The research output was also used in the development and verification of the computer code by the American Bureau of Shipping (JSR, Vol. 51).
Professor Wu has also been recognized as a pioneer in applying the finite element to nonlinear hydrodynamic problems, for example in work by authors from Denmark, Sweden as well as MIT in the USA (JCP, Vol. 318) who stated that "The use of FEM for fully nonlinear water waves was pioneered by Wu & Eatock Taylor (1994)."
More recently, Professor Wu has led a collaboration on polar engineering. Several novel methods and a commercial computer code have been developed to assess the hydrodynamic performance of a ship in the ocean covered by ice sheets (JFM, 2020, Vol.886; JCP, 2020, Vol.412), which have been used by shipyards.
The courses taught
1. Ship Hydrodynamics
2. Ship Dynamics
1986 Doctor of Philosophy University College London
1984 Master of Science University College London
1982 Bachelor of Science Shanghai Jiao Tong University


text:
\section{Introduction}
\label{sec:intro}

Strongly correlated electron systems with active orbital degrees of
freedom pose a variety of challenging problems. One of them is charge
propagation in systems with orbital order which has a whole plethora
of possible processes involving the ordered background~\cite{Zaa93}.
Some of them could support coherent hole propagation, but in general
the charge is expected to be confined when the spin-orbital order is
robust \cite{Woh09}. This problem simplifies tremendously when the
orbital degrees of freedom are quenched, as in high $T_c$
superconducting cuprates, and hole propagation in the antiferromagnetic
(AF) background may be studied for a two-dimensional (2D) square
lattice~\cite{Lee06} within the effective $t$-$J$ model or its
extensions~\cite{Toh94,*Bal95}. In these systems individual CuO$_2$
planes are separated from one another and it is sufficient to
investigate charge propagation within a single plane. A hole added to
a quantum antiferromagnet propagates then by processes which involve
quantum fluctuations in the spin background and this propagation is
controlled by a new energy scale set by the superexchange
$J$~\cite{Mar91}. In contrast, a hole is almost blocked in the absence
of quantum fluctuations for Ising exchange interactions --- then the
hopping in a square lattice is possible only by self-repairing
processes on plaquettes, as recognized by Trugman~\cite{Tru88}.

In this context the orbital models are of particular interest as they
offer several different scenarios as the exchange interactions are
always of lower symmetry than SU(2) and also the orbital flavor is not
conserved in some cases. They originate from the spin-orbital physics
\cite{Kug82,Fei97,*Fei98,Fei99,Kha00,Kha01,*Kha04,Ole05,Kha05,Rei05,
Cuo06,Cha08,Zaa09,Cor12,Brz12,*Brz13,*Brz11,*Cza15,Woh11,*Mar12,*Bis15,
*Plo16,*Woh13,*Che15,Kumar,Mil14,Ave15,*Hor11,*Ave13,Kat15,Brz15,*Brzjs},
and arise when spin order is ferromagnetic (FM) --- then the
superexchange reduces to orbital interactions only and spin-orbital
entanglement \cite{Ole12,*Ole15} is absent in the ground state
\cite{Ole06,You15,*You12,*Hor15}. Realistic 2D or three-dimensional
(3D) $e_g$ orbital models are also strongly frustrated, but
nevertheless alternating orbital (AO) order arises at finite
temperature and is induced by the strongest interaction~\cite{Ryn10},
while quantum effects are small \cite{vdB99}.
Doping of the $e_g$ orbital system suppresses
gradually orbital order \cite{Tan05,*Tan09} and gives a disordered
orbital liquid which plays a role in doped FM manganites~\cite{Fei05}.
It was argued that orbital non-conserving processes (see below) are
responsible for the onset of the disordered orbital liquid for larger
doping. Indeed, such a disordered orbital state supports the FM
metallic phase in doped manganites and was invoked to explain the
observed cubic symmetry in the 3D magnon dispersion for
La$_{1-x}$Pb$_x$MnO$_3$ \cite{Kil00,*Ole02}.

Concerning the hole propagation, it is important to realize that the
orbital models are more classical than spin ones, for instance the
orbital model for $t_{2g}$ orbitals has Ising superexchange and thus
realizes the paradigm introduced by Trugman~\cite{Tru88}. One expects
that in absence of orbital fluctuations holes should be immobile. Yet,
weak hole propagation becomes possible due to three-site processes for
$t_{2g}$ orbitals, which appear in doped systems in the same order
of the perturbation theory as the superexchange itself
\cite{Dag08,*Woh08,Wro10}. The three-site terms dominate also hole
propagation in the 2D compass model~\cite{Brz14,*Dag15} which is
related to the $e_g$ orbital model \cite{Cin10}. Finally, recently it
was argued that the concept of hole-localization in the absence of
quantum fluctuations does not to apply to three-band models
\cite{Ebr14,Ebr15}.

Perhaps the most complex situation arises in the $e_g$ orbital model
where both the orbital superexchange interactions~\cite{vdB99} and
the kinetic energy~\cite{Fei05} are radically different from their
counterparts in the spin $t$-$J$ model. The interactions are Ising-like
in a one-dimensional (1D) model~\cite{Dag04} and involve one directional
($3z^2-r^2$)-like orbital and one perpendicular to it ($x^2-y^2$)-like
orbital, both selected by the bond direction. Therefore, the 1D orbital
model is classical but quantum fluctuations increase with increasing
dimension: the 2D model and even more so the 3D model become quantum,
but still both have much weaker quantum fluctuations~\cite{vdB99} than
the SU(2) symmetric Heisenberg spin model.

In addition, the hopping in dimension higher than one does not
conserve the orbital flavor and includes both the intraorbital processes
with and interorbital ones without orbital conservation~\cite{Fei05}.
The orbital-flip processes allow an alternative mode of hole
propagation, where the orbital order of the ground state is not
disturbed at all.  As the hole thus needs neither spin-flip nor
three-site terms to move, one would expect to observe a quasiparticle
(QP) dispersion on a scale of the (interorbital) hopping $t$ rather
than on the scale of $J$. The additional presence of orbital-flipping
hoppings mediates interactions with the background that can further
renormalize the bandwidth, however the extremely small dispersion
found with the self-consistent Born approximation (SCBA)~\cite{Mar91}
comes as a surprise and seems to contradict this picture~\cite{vdB00}.

The purpose of this paper is to present a systematic study of hole
propagation in the 2D $e_g$ orbital model. A full understanding of
this orbital-polaron is a necessary ingredient -- together with the
full understanding of the spin-polaron -- before one can hope to
understand the complex behavior of spin-orbital polarons, \emph{i.e.},
of the QPs that form in systems with strong coupling to
both orbital and spin excitations \cite{Woh09}.

To study the polaron of the 2D $e_g$ orbital model, we go beyond the
simplest SCBA approach and also investigate the consequences of the
local real space constraints on the hole propagation. By comparing with
the variational approach, we establish that the SCBA, which includes
only rainbow diagrams, gives a surprisingly good qualitative picture.
However, we also find that its one weakness is that it gives
predominantly incoherent spectra with too small QP bandwidth. Indeed,
we find that the other approaches yield a fairly robust dispersion, which
is more easily reconciled with the underlying propagation mechanism.
We also develop and validate a variational approach to the problem
that can more easily be generalized to the full 3D model.

The paper is organized as follows: In Sec.~\ref{sec:2D} we introduce
the $e_g$ orbital Hamiltonian for a 2D $(a,b)$ plane and identify the
processes responsible for free hole propagation as well as interactions
which will have to be treated by some approximate method. In this paper
we employ several methods introduced in Sec.~\ref{sec:method}:
(i) variational approach in Sec.~\ref{sec:var},
(ii) self-consistent Born approximation (SCBA) in Sec.~\ref{sec:scba},
(iii) variational cluster approximation (VCA) in Sec.~\ref{sec:vca}, and
(iv) spectral moment approach in Sec.~\ref{sec:mom}.
We then discuss the convergence of the variational approach upon
increasing the variational space, and present and compare with one
another the numerical results obtained using these various methods in
Sec.~\ref{sec:results}. The summary and conclusions are given in
Sec.~\ref{sec:summa}. The paper is completed by Appendix~\ref{sec:2orb}
which elaborates on the details of the variational solution.

\section{The 2D orbital model}
\label{sec:2D}

KCuF$_{3}$ is a 3D perovskite material, with staggered FM $(a,b)$
planes along the $c$ cubic axis \cite{Bella}. Each $(a,b)$ plane has
AO order following the Goodenough-Kanamori rules \cite{Goode,Kan59}.
A carrier doped into this material will couple to both spin and
orbital excitations to realize its motion \cite{Woh09}. As already
mentioned, our current goal is to understand the orbital-polaron that
forms when the carrier is restricted to move within a single $(a,b)$
plane. The spin degree of freedom is then quenched and only orbital
interactions survive.

To address this situation we start with the on-site Coulomb
interactions for two degenerate $e_g$ orbitals $\{z,\bar{z}\}$ at Cu
sites; they read \cite{Ole83},
\begin{align}
  \label{intra}
    H_{\mathrm{int}}&= U_0 \sum_{i,\mu=z,\bar{z}} n_{i\mu,\uparrow}
    n_{i\mu,\downarrow}\nonumber\\
    &+\sum_{i} \left(U_0-\frac{5}{2}J_{\mathrm{H}}\right)n_{iz}n_{i\bar{z}}
    -2J_{\mathrm{H}}\sum_{i}\mathbf{S}_{iz}\cdot\mathbf{S}_{i\bar{z}}\\
    &+ J_{\mathrm{H}}\sum_{i} \left( d^\dagger_{iz,\uparrow}
      d^\dagger_{iz,\downarrow} d_{i\bar{z},\downarrow}^{}
      d_{i\bar{z},\uparrow}^{}+ d^\dagger_{i\bar{z},\uparrow}
      d^\dagger_{i\bar{z},\downarrow} d_{iz,\downarrow}^{}
      d_{iz,\uparrow}^{}\right).\nonumber
\end{align}
The basis $\{z,\bar{z}\}$ was chosen as consisting of one directional
$z$ orbital along the cubic $c$ axis, and orthogonal to it $\bar{z}$
orbital within the $(a,b)$ plane as follows:
\begin{equation}
  \label{basis}
  \begin{split}
    \ket{z}&\equiv \ket{3z^2-r^2}/\sqrt{6},\\
    \ket{\bar{z}}&\equiv \ket{x^2-y^2}/\sqrt{2}.
  \end{split}
\end{equation}
The interactions in Eq.~\eqref{intra} are parametrized by two parameters:
intraorbital Coulomb element $U_0$ and Hund's exchange $J_{\mathrm{H}}$.
These parameters decide about the energies of the ground and excited
states. In the present case of Cu$^{2+}$ ions with $d^9$ configuration,
$H_{\mathrm{int}}$ does not contribute to the ground states of a single hole
at each site.

Among the excited states at Cu$^{3+}$ ions with $d^8$ ($t_{2g}^6e_g^2$)
configurations, the high-spin $S=1$ states, as for instance
$|z\!\uparrow\bar{z}\!\uparrow\rangle$, have the lowest energy of $(U_0-3J_H)$
\cite{Ole05}. These states with two holes occupying two orthogonal
orbitals of $d^8$ ions arise by charge excitations on the bonds within
the $(a,b)$ planes of KCuF$_3$ and decide about the FM order. Thus, as
long as we limit ourselves to a single FM plane, we can consider
spinless $d^{\dagger}_{i\mu,\uparrow}\equiv d^{\dagger}_{i\mu}$ fermions
and replace Eq.~\eqref{intra} with
\begin{equation}
  \label{eq:hub}
  \mathcal{H}_{U} = U \sum_{i} n_{iz} n_{i\bar{z}},
\end{equation}
where the effective interaction between two holes with different orbital
flavors at the same site $i$ is,
\begin{equation}
  \label{eq:U}
  U \equiv U_0-3J_H.
\end{equation}
Thus, from now on the effective parameter $U$ stands for the
interaction $(U_0-3J_H)$ which is the only interaction left in the
subspace of high-spin (triplet) excited states \cite{vdB99}.

We supplement the above interaction term with the kinetic energy for
$e_g$ spinless holes on the full 3D lattice assuming FM order~\cite{Fei05}:
\begin{equation}
  \label{eq:3d}
  \mathcal{H}_{t} = -t\sum_{\alpha} \sum_{\mean{ij} \| \alpha}
  \left( d_{iz_{\alpha}}^{\dag} d_{jz_{\alpha}}^{} +\mathrm{H.c.}\right),
\end{equation}
which describes the hopping between nearest neighbor (NN) sites along
each bond $\mean{ij}$ that involves two $\sigma$-bonding
$\ket{3z_{\alpha}^{2}-r^{2}}$-type orbitals oriented along the bond
direction $\alpha=\{a,b,c\}$, with $z_c\equiv z$, \emph{etc.}
As the hopping $t$ follows from a two-step process via ligand F($2p$)
orbitals, the $\delta$-hopping between two
$\ket{x_{\alpha}^{2}-y_{\alpha}^{2}}$-type orbitals, which
are perpendicular to the direction $\alpha$, vanishes by symmetry. We
use again the same short-hand notation with $x_c\equiv x$, $y_c\equiv y$,
\emph{etc.} The above formulation, while concise, is not very practical due
to the orbital basis changing depending on the $\alpha$ direction.
By transforming the Hamiltonian~\eqref{eq:3d} to the standard orthogonal
orbital basis \eqref{basis} one obtains:
\begin{multline}
  \label{eq:trans3d}
  \mathcal{H}_{t} = -t \sum_{\mean{ij} \| c} d_{iz}^{\dag} d_{jz}^{} \\
  -\frac{t}{4}\sum_{\mean{ij} \| a} \left[(d_{iz}^{\dag} -\sqrt{3} d_{i\bar{z}}^{\dag})
  (d_{jz}^{} -\sqrt{3} d_{j\bar{z}}^{})+\mathrm{H.c.}\right] \\
  -\frac{t}{4}\sum_{\mean{ij} \| b} \left[(d_{iz}^{\dag} +\sqrt{3} d_{i\bar{z}}^{\dag})
  (d_{jz}^{} +\sqrt{3} d_{j\bar{z}}^{}) + \mathrm{H.c.}\right].
\end{multline}
The results presented below are in units of $t=1$.

Here we are interested both in the orbital Hubbard model for $e_g$
electrons/holes \cite{Fei05} given by Eqs.~\eqref{eq:hub} and
\eqref{eq:trans3d}, and in effective large-$U$ models, \emph{i.e.},
superexchange models derived by second-order perturbation expansion of
the Hamiltonian. Following a procedure analogous to the $t$-$J$ model
derivation \cite{Cha77}, one can find the exchange interaction, which
in this case consists of four terms expressed using singlet/triplet
projection operators in spin and orbital Hilbert spaces.

Although in KCuF$_3$ the spin interactions in $(a,b)$ planes are rather
weak, the ground state has $A$-type AF spin and $C$-type
AO order~\cite{Pao02,*Bella}, \emph{i.e.}, we
can view the system as consisting of strongly coupled $(a,b)$ planes
with AO order that are stacked with ferro-orbital (FO) order along the
$c$ axis. AF correlations between the layers strongly suppress
hole motion in $c$ direction, making it feasible to focus on the
in-plane orbital physics separately from the system's behavior along
the $c$-direction, at least as long as we are considering a
system near its ground state.

To find the exchange Hamiltonian for a single 2D plane we need to
project out the spin-singlet states from the full Hamiltonian.
After averaging over the in-plane spin-triplet states one arrives
at~\cite{vdB99}:
\begin{equation}
  \label{eq:Hj}
  \mathcal{H}_{J} = \frac{J}{2} \sum_{\mean{ij}} [
  T_{i}^{z}T_{j}^{z} + 3 T_{i}^{x} T_{j}^{x}
  \mp \sqrt{3} (T_{i}^{x}T_{j}^{z}+T_{i}^{z}T_{j}^{x}) ],
\end{equation}
where $J=t^2/U$, the upper/lower sign is for the $\mean{ij}$ bond
oriented along $a/b$ axis respectively, and $\{T_i^{\alpha}\}$
operators are the same as spin operators (1/2 times the respective
Pauli matrix), only acting in the orbital space in which $z/\bar{z}$
states correspond to spin up/down states.  Alternatively, one may
derive Eq.~\eqref{eq:Hj} from the Kugel-Khomskii model
\cite{Kug82,Fei97} by considering a FM system with superexchange
given by triplet charge excitations
$d_i^9d_j^9\leftrightarrow d_i^8d_j^{10}$.

\begin{figure}[t!]
  \centering
  \includegraphics[width=0.65\columnwidth]{model}
\caption{(Color online)
Schematic representation of the kinetic Hamiltonian $\mathcal{H}_t$
\eqref{eq:trans3d} and the ground state with AO order of the 2D orbital
model $\mathcal{H}_{J}$ \eqref{eq:Hj}. The
bow-shapes represent the $\ket{\pm}$ states which are highly directional
in the $(a,b)$ plane. Full orbitals represent a singly occupied state
(corresponding to the overall $d^9$ state) and the blue color represents
the orbitals occupied in the ground state. The dashed orbital contours
represent a doubly occupied central site (\emph{i.e.}, electron doping).
Blue contour refers to the orbital normally half-filled in the ground
state and the red one indicates the orbital that would have been wrongly
occupied if an electron were to be removed from it. The kinetic elements
of Hamiltonian~\eqref{eq:transfH} are a consequence of orbital symmetry
and overlap.}
\label{fig:model}
\end{figure}

For an FM plane in KCuF$_3$, or for the 2D compound K$_2$CuF$_4$, it has
been well established through a variety of analytical and numerical
treatments \cite{vdB99,Cin10,vdB04,Dag06} that the leading superexchange
term $3T^x_iT^x_j$ dominates Eq.~\eqref{eq:Hj} and stabilizes the AO
order shown schematically in Fig.~\ref{fig:model}. Indeed,
the orbital ground state has been found to be composed of alternating
eigenstates of $T^x$ operators, \emph{i.e.},
\begin{equation}
\ket{\pm}=\frac{1}{\sqrt{2}}(\ket{z}\pm\ket{\bar{z}}).
\label{eq:pm}
\end{equation}
Orbital excitations from such an ordered ground state have been studied
by several authors in the past
\cite{vdB99,Woh11,*Mar12,*Bis15,*Plo16,*Woh13,*Che15,Oka02,Ish05,For08,Nas13}.
Below we use them to investigate orbital polarons when a hole is added.

Quantum fluctuations around perfect N\'eel order involve one- and
two-orbiton (orbital excitation) processes and have consistently been
found to be small. As a further check, we carried out the SCBA with
and without these quantum fluctuations and have found the results to
be almost identical. Thus, while the other
two terms can be included in the treatment (they are automatically
included in the VCA) in the following we restrict the variational
approach to only the Ising-exchange term $ \mathcal{H}_{J} \approx
\frac{3}{2}J \sum_{\mean{ij}} T^x_iT^x_j$, for the sake of simplicity.

The kinetic Hamiltonian~\eqref{eq:trans3d}, after being restricted to
the $(a,b)$ plane, is transformed to the basis~\eqref{eq:pm}. Next,
the orbital degree of freedom is decoupled from the fermionic
operators by means of the slave boson  formalism~\cite{Mar91}:
\begin{equation}
  \label{eq:slave}
  d_{i0}^{\dag} = f_{i}^{\dag}, \quad d_{i1}^{\dag} = f_{i}^{\dag} a_{i}^{},
\end{equation}
where the $\{0,1\}$ indices on the left hand side indicate fermion
creation in the ground or excited state, respectively, \emph{i.e.},
$\ket{+}$ or $\ket{-}$~\eqref{eq:pm}, depending on the sublattice since
the orbital state is alternating. This transformation works under the
assumption that $f_{i}^{\dag}: d^9\to d^{10}$ and $f_{i}: d^{10}\to d^9$,
which is the consequence of disallowing doubly occupied states
in the Hilbert space.

After performing all the above transformations, we arrive at the final
form of the strong coupling Hamiltonian,
\begin{equation}
\mathcal{H}=\mathcal{H}_{J}+\mathcal{T}+\mathcal{V} ,
\end{equation}
where:
\begin{subequations}
  \label{eq:transfH}
  \begin{align}
    \label{eq:transfHJ}
    \mathcal{H}_{J} &= \frac{3}{8}J\sum_{\mean{ij}} (1 - \sigma_{i}^{z}\sigma_{j}^{z}),\\
    \label{eq:transfT}
    \mathcal{T} &= -\frac{t}{4}\sum_{\mean{ij}} (f_{i}^{\dag}f_{j}^{} + \mathrm{H.c.}) =
    \sum_{\mathbf{k}} \epsilon_{\mathbf{k}}^{} f_{\mathbf{k}}^{\dag} f_{\mathbf{k}}^{},\\
    \label{eq:transfV}
    \mathcal{V} &= -\frac{t}{4} \sum_{i,\delta} \left[
      (2+\sqrt{3} e^{i\pi_{y}\cdot\delta}
      e^{i \mathbf{Q}\cdot\mathbf{R}_{i}}) a_{i}^{\dag} +\right.\nonumber\\
    &\left.+(2-\sqrt{3} e^{i\pi_{y}\cdot\delta}
      e^{i \mathbf{Q}\cdot\mathbf{R}_{i}})
      a_{i+\delta}^{} + a_{i}^{\dag} a_{i+\delta}^{} \right]
    f_{i+\delta}^{\dag} f_{i}^{}.
  \end{align}
\end{subequations}
Here, $\epsilon_{\mathbf{k}}^{} = -t\gamma_{\mathbf{k}}^{}$ is the free
particle dispersion, with
$\gamma_{\mathbf{k}}^{}=(1/z)\sum_{\delta} e^{i\mathbf{k}\cdot\delta}$.
Also, for simplicity $\mathcal{H}_{J}$ and the corresponding ground
state have been transformed from AO order into the FO one, and the
leading Ising term has the exchange constant $J$ with an explicit
factor of $-3/8$. Note that we also subtracted the background energy
out of $ \mathcal{H}_{J}$. The free propagation $\mathcal{T}$
\eqref{eq:transfT} is the consequence of the kinetic energy terms
conserving the orbital flavor, while the interaction $\mathcal{V}$
\eqref{eq:transfV} arises by the same mechanism as the hole-magnon
coupling in the spin model \cite{Mar91}.
The phase factors serve to accommodate the model's dependence on
direction: $\pi_y=(0,\pi)$, $\mathbf{Q}=(\pi,\pi)$, and $\delta$ is
a vector variable pointing to the nearest neighbors of site~$i$.

\section{Methods}
\label{sec:method}

\subsection{Variational approximation (VA)}
\label{sec:var}
Starting from the Bloch state of a free electron,
\begin{equation}
\ket{\mathbf{k}}=f_{\mathbf{k}}^{\dag}\ket{0}=
\frac{1}{\sqrt{N}}\sum_{i}e^{i\mathbf{k}\cdot\mathbf{R}_i}f_{i}^{\dag}\ket{0},
\label{eq:kstate}
\end{equation}
we define the Green's function as
\begin{equation}
G(\mathbf{k},\omega)\equiv\bra{\mathbf{k}}\mathcal{G}(\omega)\ket{\mathbf{k}},
\end{equation}
where $\mathcal{G}(\omega)=(\omega+i\eta-\mathcal{H})^{-1}$  is the
resolvent. Here, $|0\rangle$ is the N\'eel AO ground-state of the
undoped 2D plane,  described above.

The variational approximation is based on a simplification of the
equations of motion which are obtained through repeated Dyson
expansions of the resolvent:
\begin{equation}
  \label{eq:dyson}
  \mathcal{G}(\omega) = \mathcal{G}_{0}(\omega)
  +\mathcal{G}(\omega)\mathcal{V}\mathcal{G}_{0}(\omega).
\end{equation}
Here,
$\mathcal{H}_{0} = \mathcal{T}+\mathcal{H}_{J}$ corresponds to
$\mathcal{G}_{0}(\omega)$ and $\mathcal{V}$ is given by Eq.~\eqref{eq:transfV}.

The rationale underlying the variational approach is that the creation
of every orbiton costs an energy proportional to $J$, so the bigger
the value of $J$ the less orbitons are likely to appear in the QP
cloud in the ground state. Therefore, one can restrict the Dyson
expansion in terms of the number of orbitons generated in the system,
and one can further decrease the Hilbert space by constraining the
orbitons to be located close to each other. This approach can of
course be tested by verifying the convergence of the result as a
function of the number of orbitons allowed in the system, \emph{i.e.},
by increasing the variational space. In this Section we show how this
approach works in the framework of a one orbiton approximation, when
only up to one orbiton is allowed to appear in the QP cloud. Then we
briefly discuss the generalization for the two-orbiton case, which is
further detailed in the Appendix. Note that we have implemented this
variational approach, and show results,  for  up to six orbitons in
the QP cloud. However, those equations of motion become too cumbersome
to write explicitly. We also note that similar variational schemes
have proved to be very accurate for the study of various models of
lattice polarons \cite{Ber06,Goo06,Ber07,Mar10} as well as spin
polarons \cite{Ber11,Tro13,Ebr14,Ebr15}.

Using the Dyson expansion and evaluating $\mathcal{V}\ket{\mathbf{k}}$
explicitly in real space, we arrive at the following expression:
\begin{equation}
  \label{eq:green}
  \begin{split}
&G(\mathbf{k},\omega) = \left[ 1-\frac{t}{2}\sum_{\delta}
F_{1}(\mathbf{k},\omega,\delta)+\right.\\
&\left.-\frac{\sqrt{3}t}{4}\sum_{\delta} \bar{F}_{1}(\mathbf{k},\omega,\delta)
e^{i\mathbf{\pi}_y\cdot\mathbf{\delta}} \right] G_{0}(\mathbf{k},\omega-4J'),
  \end{split}
\end{equation}
where
$G_0((\mathbf{k},\omega)=(\omega-\epsilon_{\mathbf{k}}+i\eta)^{-1}$ is
the free-electron Green's function, the shorthand notation $J'=3J/8$
has been introduced for convenience, and we define:
\begin{align}
  \label{eq:f1}
F_{1}(\mathbf{k},\omega,\delta) &=
\bra{\mathbf{k}} \mathcal{G}(\omega) \frac{1}{\sqrt{N}}
\sum_{i} e^{i\mathbf{k}\cdot\mathbf{R}_{i}}
f_{i+\delta}^{\dag} a_{i}^{\dag} \ket{0},\\
\label{eq:f1b}
\bar{F}_{1}(\mathbf{k},\omega,\delta) &= \bra{\mathbf{k}}
\mathcal{G}(\omega)
\frac{1}{\sqrt{N}} \sum_{i} e^{i(\mathbf{k+Q})\cdot\mathbf{R}_{i}}
f_{i+\delta}^{\dag} a_{i}^{\dag} \ket{0},
\end{align}
as the generalized Green's functions for the state with a single
orbital excitation above the ground state.

In order to find these new propagators, we use the Dyson expansion
again to generate their equations of motion. In the one-orbiton
variational scheme we include only the
part of $\mathcal{V}$ which does not create new orbitons.
However, at this point one has to impose additional constraints on the
movement of the electron, namely: ($i$) the electron and the orbiton
cannot both occupy the same site, and ($ii$) when the electron and
orbiton are far apart the energy increase is by $12J'$ (4 broken bonds
accompanied by 4 excited bonds), however if they are on adjacent sites
the energy increase is only by $10J'$ (one excited bond less). The
above constraints can be taken into account by adding the following
terms to the $\mathcal{H}_{0}$ Hamiltonian, in
order to cancel the corresponding processes:
\begin{equation}
  \label{eq:vi}
  \mathcal{V}_{1} = \frac{t}{4} \sum_{\epsilon}
  (f_{i}^{\dag} f_{i+\epsilon}^{} + \mathrm{H.c.})
  -2J'\sum_{\epsilon} n_{i+\epsilon},
\end{equation}
where $i$ is the location of the orbiton and $n_{i+\epsilon}$ is
the electron number operator for the sites NN to site $i$. The above
expression modifies the free part of the Hamiltonian, which means
now instead of
$\mathcal{G}_{0}(\omega)$ one has to use $\mathcal{G}_{1}(\omega)$
(corresponding to $\mathcal{H}_{0}+\mathcal{V}_{1}$) in the Dyson
expansion of the generalized Green's function:
\begin{equation}
\mathcal{G}(\omega) = [1+\mathcal{G}(\omega)\mathcal{V}]
\mathcal{G}_{1}(\omega).
\end{equation}
However, $\mathcal{G}_{1}(\omega)$ is no longer diagonal in Fourier
space, and so its propagator needs to be solved first. This can again
be done using the Dyson equation,
\begin{equation}
\mathcal{G}_{1}(\omega) = [1+\mathcal{G}_{1}(\omega)\mathcal{V}_{1}]
\mathcal{G}_{0}(\omega),
\end{equation}
which leads to the following set of equations of motion:
\begin{equation}
  \label{eq:gi}
  \begin{split}
&G_{1}(n,i\!+\!\delta,\omega) = G_{0}(n,i\!+\!\delta,\omega-12J')
+\!\sum_{\epsilon} G_{1}(n,i\!+\!\epsilon,\omega)\\
    &\times\left[\frac{t}{4}G_{0}(i,i\!+\!\delta,\omega-12J')
    -2J'G_{0}(i\!+\!\epsilon,i\!+\!\delta,\omega-12J')\right],
  \end{split}
\end{equation}
where $G_{1}(n,i\!+\!\delta,\omega)=
\langle 0|f_n\mathcal{G}_1(\omega) f^{\dagger}_{i+\delta}|0\rangle$
describes the propagation of an
electron in real space, in the presence of the forbidden site $i$ (due
to the orbiton located there) from site  $\ket{i+\delta}$ to any site
$n\ne i$. However, as we show below, the only  propagators needed are
for NN sites $n=i+\gamma$. This leads to a reduction of
Eq.~\eqref{eq:gi} to a $4\times4$ matrix equation for the set of
functions $G_{1}(i\!+\!\gamma,i\!+\!\delta,\omega)$, describing
propagators exclusively around the orbiton excitation:
\begin{equation}
  \label{eq:gimatrix}
\mathbb{G}_{1}^{\gamma\delta} = \mathbb{G}_{0}^{\gamma\epsilon}
\left[\mathbb{I}^{\delta\epsilon} -\tfrac{t}{4}\mathbb{G}_{0}^{\delta 0}
+2J'\mathbb{G}_{0}^{\delta\epsilon}\right]^{-1}.
\end{equation}

What makes this method of including the site occupation constraints
particularly appealing, in spite of its complicated nature, is the
fact that the real space Green's function depends only
on the distance between sites:
\begin{equation}
  \label{eq:G0}
  G_{0}(m,n,\omega) = \frac{1}{\pi^{2}} \int_{0}^{\pi}d^{2}\mathbf{k}\,
  G_{0}(\mathbf{k},\omega)\,e^{i\mathbf{k}\cdot(\mathbf{R}_m-\mathbf{R}_n)},
\end{equation}
and not the specific location of the $\{\ket{m},\ket{n}\}$ states.
Moreover, it can be shown (e.g. see the approach introduced by Morita
\cite{Mor71a,*Mor71}) that for 2D lattice Green's functions, the
contributions $G_{0}(m,n,\omega)$ can be expressed exactly in terms of
the complete elliptic integrals $K(\kappa),E(\kappa)$, where
$\kappa=t/\omega$. Therefore, this method in principle allows for exact
analytical calculation of both the free as well as the constrained
non-interacting real space Green's functions.

Going back to the calculation of the $F_{1}$ functions, one has
to use the Dyson equation as explained above, which leads to:
\begin{widetext}
  \begin{align}
    \label{eq:f1s}
    F_{1}(\mathbf{k},\omega,\delta) &= -\frac{t}{4}\sum_{\gamma} \left[
      2G(\mathbf{k},\omega)+\sqrt{3}\bar{G}(\mathbf{k},\omega)e^{i\pi_{y}\cdot\gamma}
      + F_{1}(\mathbf{k},\omega,-\gamma)e^{-i\mathbf{k}\cdot\mathbf{\gamma}}\right]
    G_{1}(i+\gamma,i+\delta,\omega),\\
    \label{eq:f1bs}
    \bar{F}_{1}(\mathbf{k},\omega,\delta) &= -\frac{t}{4}\sum_{\gamma}
    \left[2\bar{G}(\mathbf{k},\omega)+\sqrt{3}G(\mathbf{k},\omega)
      e^{i\mathbf{\pi}_y\cdot\mathbf{\gamma}} -\bar{F}_{1}(\mathbf{k},\omega,-\gamma)
      e^{-i\mathbf{k}\cdot\mathbf{\gamma}}\right] G_{1}(i+\gamma,i+\delta,\omega),
\end{align}
where
\begin{equation}
    \label{eq:gb}
\bar{G}(\mathbf{k},\omega) = \bra{\mathbf{k}}\mathcal{G}(\omega)\ket{\mathbf{k}+\mathbf{Q}} =
-\frac{t}{4}\sum_{\delta} \left[2\bar{F}_{1}(\mathbf{k},\omega,\delta)
+\sqrt{3}F_{1}(\mathbf{k},\omega,\delta)e^{i\mathbf{\pi}_y\cdot\mathbf{\delta}}\right]
G_{0}(\mathbf{k}+\mathbf{Q},\omega-4J').
\end{equation}
\end{widetext}
Altogether, Eqs.~\eqref{eq:green},~\eqref{eq:gimatrix}, and
Eqs.~\eqref{eq:f1s} to~\eqref{eq:gb} form a set of 14 coupled equations.
Solving this system numerically yields the desired Green's function
$G(\mathbf{k},\omega)$ for an electron moving in an AO system with at
most a single orbiton.

Following a procedure similar to the one outlined above, one can
perform a multi-orbiton approximation. For example, for a two-orbiton
variational space, one needs to perform
Dyson expansion three times, disallowing the three-orbiton states in
the last expansion. This leads to the creation of a series of
$F_{2}(\mathbf{k},\omega,\epsilon,\gamma)$ generalized Green's
functions. Here $\{\epsilon,\gamma\}$ are unit vectors setting the
path of the moving charge, leaving behind a trail of two orbitons,
described by
$\sum_i e^{i\mathbf{k}\cdot\mathbf{R}_i} f_{i+\epsilon+\gamma}^{\dag}a_{i+\epsilon}^{\dag}a_{i}^{\dag}\ket{0}$.
The set of equations is now closed  by tying
$F_{2}$ only to other $F_{2}$ functions and
$F_{1}$ functions defined before. This system of equations can
be easily solved numerically, similarly to the previous case.

For two-orbiton states there are, of course, two
no-double-occupancy constraints, and therefore now one has to
calculate new constrained ``free'' Green's functions, denoted
$G_{2}(m,n,\omega)$. Their matrix equation turns out to be very
similar to the one-orbiton case~\eqref{eq:gimatrix}, where all the
summations over nearest neighbor vectors have to be replaced by
summation over all the sites adjacent to the string of orbitons
present in the system.

A more detailed discussion of this two-orbiton solution is found in
the Appendix. The generalization to more orbitons
follows along similar lines. While in principle the relative distance
between the orbitons can be arbitrarily large, we expect that the
configurations with the highest weight in the QP cloud are those with
orbitons located in close vicinity to each other. We therefore impose
the relatively liberal restriction that any two orbitons have to be
within a distance not greater than the total number of orbitons in the
current state. This allows for boson clouds that are spatially
constrained, yet are not limited to unbroken strings.

\subsection{Self-consistent Born approximation (SCBA)}
\label{sec:scba}

Alternatively, the model can be solved by means of the SCBA, a method
well established in polaron research~\cite{Mar91,Ram98,vdB00}, which
has been proven to be highly accurate for spin-polarons in the $t$-$J$
model \cite{DMC,Konrad} but very poor for lattice polarons at strong
couplings, \emph{e.g.} in the Holstein model \cite{Ber06,Goo06}. It is
therefore not \emph{a priori} clear how accurate it is for orbiton
polarons. SCBA has been discussed extensively in the literature so here
we will limit ourselves to stating its form for the specifics of our
model.

SCBA is a self-consistent method based on the calculation of the
self-energy $\Sigma(\mathbf{k},\omega)$, assuming that all crossing
diagrams are negligibly small. In order to use it, we first need to
transform the Hamiltonian into its momentum representation using a
discrete Fourier transform. Furthermore,
$\mathcal{H}_J$ has to be simplified to linear orbital wave (LOW) order
by representing it with a quadratic form in bosonic orbiton operators
$\{a^{\dag},a\}$. Orbital fluctuations can be included at little extra
cost only requiring an additional step of the Bogoliubov transformation.

Our Hamiltonian transformed to Fourier space takes the form:
\begin{align}
  \label{eq:fouH}
\mathcal{V} &= \frac{-t}{\sqrt{N}} \sum_{\mathbf{k}\mathbf{q}}
\left[f_{\mathbf{k}}^{\dag} f_{\mathbf{k}-\mathbf{q}}^{}
\left( M_{\mathbf{k}\mathbf{q}}^{} \alpha_{\mathbf{q}}^{}
+ N_{\mathbf{k}\mathbf{q}}^{} \alpha_{\mathbf{q}+\mathbf{Q}}^{}
\right) + \mathrm{H.c.}\right],\\
\mathcal{H}_{J} &= \sum_{\mathbf{q}} \omega_{\mathbf{q}}^{}
\alpha_{\mathbf{q}}^{\dag}\alpha_{\mathbf{q}}^{},
\end{align}
where $\alpha_{\mathbf{q}}^{(\dag)}$ is the Bogoliubov-transformed
orbiton operator, $\{u_{\mathbf{q}},v_{\mathbf{q}}\}$ are the Bogoliubov
coefficients,
$\eta_{\mathbf{q}}=\gamma_{\mathbf{q}+\mathbf{\pi}_y}$, and
\begin{equation}
\omega_{\mathbf{q}}=3J\sqrt{1+\gamma_{\mathbf{q}}/3}\approx 3J
\end{equation}
is the orbiton energy (with and without fluctuations, respectively).
The carrier-orbiton interaction gives two vertex functions,
\begin{align}
M_{\mathbf{k}\mathbf{q}}&=2\,\left(u_{\mathbf{q}}
\gamma_{\mathbf{k}-\mathbf{q}}+v_{\mathbf{q}}\gamma_{\mathbf{k}}\right), \\
N_{\mathbf{k}\mathbf{q}}&=\sqrt{3}\,\left(u_{\mathbf{q}+\mathbf{Q}}
\eta_{\mathbf{k}-\mathbf{q}}+v_{\mathbf{q}+\mathbf{Q}}\eta_{\mathbf{k}}\right).
\end{align}
Having written the Hamiltonian in
this form, we can calculate the self-energy from:
\begin{multline}
  \label{eq:scbaself}
\Sigma(\mathbf{k},\omega) = \frac{t^{2}}{N}\sum_{\mathbf{q}}
\left[M_{\mathbf{k}\mathbf{q}} G(\mathbf{k}-\mathbf{q},\omega
  -\omega_{\mathbf{q}}) M_{\mathbf{k}\mathbf{q}}\right.\\
\left.+N_{\mathbf{k}\mathbf{q}} G(\mathbf{k}-\mathbf{q},\omega
-\omega_{\mathbf{q}+\mathbf{Q}}) N_{\mathbf{k}\mathbf{q}}\right],
\end{multline}
where $G$ is the SCBA Green's function, which is \emph{a priori}
unknown. Therefore, in the self-consistent calculation we start with
the free Green's function $G_{0}(\mathbf{k},\omega)=1/\omega$. Having
calculated the first order approximation of the self-energy, we can
find the next order approximation of the Green's function with
\begin{equation}
  \label{eq:scbagreen}
G(\mathbf{k},\omega) = \left[G_{0}^{-1}(\mathbf{k},\omega)
-\Sigma(\mathbf{k},\omega)\right]^{-1}.
\end{equation}
This solution can then be plugged back into Eq.~\eqref{eq:scbaself}
to obtain a better approximation of the self-energy, and so on, until
one reaches a stable result. This procedure usually converges very
fast, often finding a stable solution after as few as one or two steps.

As already stated, the SCBA only sums non-crossing (rainbow)
diagrams. It therefore ignores contributions from processes where the
electron absorbs the orbitons in a different order than their reversed
creation order. It also fails to impose the constraints of at most one
orbiton at one site, and of not allowing the electron and an orbiton on
the same site. As such, SCBA is expected to be less accurate than the
variational method which implements these local constraints exactly and
also includes the crossed diagrams (in its multi-orbiton flavors).

\subsection{Variational cluster approximation (VCA)}
\label{sec:vca}

We complement the approaches discussed above with a numerical treatment
based on the self-energy functional approach, the VCA. In this scheme,
the one-particle spectral density of a large system (e.g. in the
thermodynamic limit) is obtained by approximating its self-energy by
the self-energy of a small cluster~\cite{Pot03}. We do this here
numerically using the Lanczos algorithm to solve the Hamiltonian on
eight-, ten- and twelve-site clusters. Electronic correlations and
quantum fluctuations on short length scales within the cluster are
thus taken into account.

Long-range order is also included on a mean-field level via an
optimization of the grand potential with respect to the order parameter
for orbital order~\cite{Dah04}. The grand potential is in turn again
obtained from:
(i) the grand potential of the small cluster, and
(ii) the Green's function of the large system.
The wave function of the ordered orbital can likewise be optimized;
this procedure reproduces the expected result that the $\ket{\pm}$
orbital combinations~\eqref{eq:pm} alternate in a checkerboard pattern,
as depicted in Fig.~\ref{fig:model}. The ``optimal'' state
corresponding to a stationary point of the grand potential can then be
used to evaluate the one-particle spectral density for comparison with
the other approaches.

Unfortunately, the self-energy approach is only valid for interactions
that are contained purely within the directly solved cluster. In our
case, this implies that we cannot address the $t$-$J$ Hamiltonian
comprised of Eqs.~\eqref{eq:3d} and~\eqref{eq:Hj}, but instead have to
work on the itinerant Hubbard variant given by Eqs.~\eqref{eq:3d} and
\eqref{eq:hub}. In order to meaningfully compare the orbital Hubbard
model to the $t$-$J$-model results, we then have to restrict ourselves
to the regime of large $U/t$, \emph{i.e.}, small $J/t$. The physics of
this regime, however, can be expected to be well described if the
impact of quantum fluctuations can be considered short-range enough to
be captured by the directly solved cluster. As we find that results of
clusters with eight, ten, and twelve sites agree, we assume that
finite-size effects are not too severe.

\subsection{Spectral moments}
\label{sec:mom}

In order to gauge the accuracy of SCBA and the variational approach,
it is useful to calculate their numerical spectral moments and compare
them against an analytical calculation. The spectral moments are defined
as
\begin{equation}
\label{eq:moment}
M_{n}(\mathbf{k})= \int\limits_{-\infty}^{\infty}
A(\mathbf{k},\omega)\,\omega^n\,d\omega,
\end{equation}
where
\begin{equation}
\label{eq:ak}
A(\mathbf{k},\omega)=-\frac{1}{\pi}\,\Im[G(\mathbf{k},\omega)],
\end{equation}
is the normalized spectral function. Eq.~\eqref{eq:moment} is useful
for numerical integration of a calculated spectral function
$A(\mathbf{k},\omega)$. On the other hand, the analytical expression for
a spectral moment can be calculated directly from the model Hamiltonian,
using the algebraic formula~\cite{Nol72}:
\begin{equation}
  \label{eq:moment2}
  M_{n}(\mathbf{k}) =
  \bra{0} \big\{
  \underbrace{[[[f_{\mathbf{k}}^{},H],H],\ldots]}_{n-p},
  \underbrace{[\ldots,[H,[H,f_{\mathbf{k}}^{\dag}]]]}_{p}
  \big\} \ket{0},
\end{equation}
where the value of $0\le p\le n$ is arbitrary, \emph{i.e.}, it can be
chosen in the most convenient way for the problem at hand. Obviously,
$M_{0}(\mathbf{k})=1$ is just the spectral function normalization,
which is consistent with the integral of~\eqref{eq:ak}. Several higher
order moments were calculated explicitly:
\begin{align}
  \label{eq:mom}
  M_{1}(\mathbf{k}) &= \epsilon_{\mathbf{k}} +4J',\\
  M_{2}(\mathbf{k}) &=(\epsilon_{\mathbf{k}} +4J')^{2} +\tfrac{7}{4}t^{2},\\
  M_{3}(\mathbf{k}) &=(\epsilon_{\mathbf{k}} +4J')^{3}
  +\tfrac{57}{16}t^{2}\epsilon_{\mathbf{k}} +\tfrac{63}{2}t^{2}J'.
\end{align}
$M_{4}(\mathbf{k})$ was also calculated and is shown below, however its
expression is too long to be written here explicitly.

\section{Results and discussion}
\label{sec:results}

\subsection{Convergence of the variational approach}
\label{sec:convergence}

We start by analyzing the convergence of the spectral weight predicted
by the variational approach, with the increase of the variational
space (maximum number $n$ of orbitons allowed). Note that we always plot
$\tanh[A(\mathbf{k},\omega)]$ to highlight the low amplitude part of
the spectra. This transformation has the advantage that the relative
amplitudes of values smaller than 1 are nearly unaffected
(since $\tanh(x)\approx x-x^{3}/3+\ldots$ is nearly linear),
while the big values are all treated uniformly, because of the upper
limit of 1. Therefore, density maps presented herein indicate actual
maxima in black, while the incoherent parts of the spectral function
are shown in shades of gray. The dashed red lines serve as reference,
and mark the location of the free electron band at
$\omega=\epsilon_{\mathbf{k}}+4J'$.

\begin{figure}[t!]
  \centering
  \includegraphics[width=\columnwidth]{spectral-J0p5}
\caption{(Color online) Momentum dependence of the spectral function
for the variational calculation with a maximum of $n=1$ to $n=5$
orbitons. The 6$^{\mathrm{th}}$ panel shows the corresponding SCBA results.
Parameters: $t=1$, $J=0.5$.
The dashed red line indicates the free electron dispersion.
The spectral function is plotted on a nonlinear $\tanh$-scale to
emphasize the low-amplitude incoherent part of the spectrum.
The high-symmetry points are: $\Gamma=(0,0)$, $X=(\pi,0)$,
$Y=(0,\pi)$, $S=(\pi/2,\pi/2)$, and $M=(\pi,\pi)$.}
\label{fig:VA05}
\end{figure}

Fig.~\ref{fig:VA05}
shows $A(\mathbf{k},\omega)$ along high symmetry cuts in the Brillouin
zone for $n=1$ to $n=5$; the SCBA results are shown in the sixth panel.
Note that we cannot show meaningful VCA results for this large $J/t$
value, for reasons explained above.
The spectrum consists of a low-energy QP band which has most of the
weight, and a broad but low-weight continuum at higher energies.
Focusing first on the QP band, we see that with increasing $n$ it
moves to lower energies and its bandwidth decreases. This is standard
polaronic behavior: a bigger cloud results in a more stable but also
heavier polaron. However, we also see that the QP band is already
converged for $n\approx 2$, in other words the QP binds very few
orbitons in its cloud.

This is expected because for this relatively large $J$, orbiton
excitations are expensive. Moreover, the variational approach properly
implements the constraint of allowing only up to one orbiton per site.
Thus, a cloud with many orbitons would be spread out over many sites.
However, the electron can only interact with orbitons on its NN sites,
so orbitons that are far from it cost (orbital) exchange energy but do
not contribute to a lowering of total energy through carrier-orbiton
interactions. As such, their presence is energetically unfavorable.

Another way to think about it is in terms of an ``effective coupling''.
For lattice polarons, this is defined as $\lambda \sim g^2/(\Omega D)$,
where $g$ is the (suitably averaged) electron-phonon coupling strength,
$\Omega$ is the optical phonon energy and $D$ is the free-electron
bandwidth. For small $\lambda$ the polaron cloud has very few phonons
and the QP properties are hardly changed compared to those of the free
carrier. For large $\lambda$, on the other hand, a so-called
``small polaron'' forms. It has a large boson cloud, with many phonons,
and consequently is a heavy QP.

\begin{figure}[t!]
  \centering
  \includegraphics[width=\columnwidth]{spectral-J0p1}
\caption{(Color online) Same as in Fig.~\ref{fig:VA05} for $J/t=0.1$.
The variational result for $n=6$ is shown in Fig.~\ref{fig:spectral}.
The high-symmetry points are as in Fig.~\ref{fig:VA05}.}
\label{fig:VA01}
\end{figure}

For our problem, $g \sim t, D \sim t, \Omega \sim J$; as a result a
large $J/t$ implies a small $\lambda$ and \emph{vice versa}. The results of
Fig.~\ref{fig:VA05} are, therefore, consistent with this polaron
paradigm of having a small cloud and weakly renormalized QP at small
$\lambda$. Moreover, we also see fairly good agreement with SCBA,
which is also known to be true for small $\lambda$ Holstein lattice
polarons \cite{Ber06,Goo06}. This is easy to understand: for a cloud
consisting of very few bosons, the probability that the carrier
absorbs the last emitted boson (which is the only process allowed in
SCBA) as opposed to any other boson, is of order 1. It is only for
clouds with many bosons that this probability becomes very small. In
this latter case, SCBA is expected to become very inaccurate (as, indeed,
is the case for large $\lambda$ Holstein polarons \cite{Ber06,Goo06})
unless there is some other physical reason to strongly favor the
absorption of the last emitted boson.

Regarding the higher energy continuum, we see that it keeps on
changing up to higher $n$ values, although it converges for $n\approx 5$
(this is why we do not show here the $n=6$ result). This is also expected,
because configurations with many orbitons are energetically expensive
and  contribute primarily to the higher-energy excited states. To
properly describe these high-energy states, therefore, one needs to
consider bigger variational spaces. The agreement between the
variational approach and SCBA is much poorer for this higher energy
continuum; we expect the latter to be less accurate here for reasons
discussed in the previous paragraph.

\begin{figure}[t!]
  \centering
  \includegraphics[width=\columnwidth]{spectral-J0p05}
\caption{(Color online) Same as in Fig.~\ref{fig:VA05} for $J/t=0.05$.
The variational result for $n=6$ is shown in Fig.~\ref{fig:spectral}.
The high-symmetry points are as in Fig.~\ref{fig:VA05}.}
\label{fig:VA005}
\end{figure}

\begin{figure}[b!]
  \centering
  \includegraphics[width=\columnwidth]{converge}
\caption{(Color online) Ground state energy $E(\mathbf{k})$ and
spectral weight $Z(\mathbf{k})$ calculated numerically for $J=0.1$
(see Fig.~\ref{fig:VA01}) and a maximum of $n=1$ to $n=6$ orbitons
from top to bottom as indicated by arrows.
The high-symmetry points are as in Fig.~\ref{fig:VA05}.}
  \label{fig:converge}
\end{figure}

We have verified that increasing $J/t$ more, \emph{i.e.}, moving toward
even smaller effective couplings, further confirms all the inferences
we have made so far. Clearly, the more interesting question is what
happens in the opposite limit, when the effective coupling increases.
This is shown in Figs.~\ref{fig:VA01} and \ref{fig:VA005} for $J/t=0.1$
and 0.05, respectively. As expected, as we move towards
stronger effective coupling, the QP band reaches convergence more
slowly (at $n\approx 5$ in these cases), moves to even lower energies
and becomes narrower.
For a more direct verification of this see Fig.~\ref{fig:converge}
showing the evolution of QP energy $E(\mathbf{k})$ and spectral weight
$Z(\mathbf{k})$ for increasing number of orbitons at $J=0.1$. One can
clearly see the monotonic convergence of the ground state results, as
should indeed be the case for the variational method. Both of the curves
are practically indistinguishable from one another for the cases $n=5$
and $n=6$, indicating that the result is practically converged for
$n\approx 5$.

The higher energy continuum is also more structured, but it is probably
not yet fully converged. On the
low-energy side, it exhibits ``copies'' of the QP band. The agreement
with SCBA for the QP band is quite good for $J/t=0.1$, but for
$J/t=0.05$ SCBA predicts a much heavier polaron than the variational
approximation (see also Sec. \ref{sec:compare} below).
This is likely due to the fact that SCBA does not
enforce the constraint of allowing only up to one orbiton per site. As
a result, at strong couplings it is likely to overestimate the number
of orbitons in the QP cloud, and therefore predict a heavier QP.

\subsection{Comparison with VCA}
\label{sec:compare}

\begin{figure}[b!]
  \centering
  \includegraphics[width=\columnwidth]{spectral-new}
\caption{(Color online) Comparison of the spectral weights predicted by
SCBA (left), VCA (center) and the variational approach with $n=6$, for
$J/t=0.05$ (top panels) and 0.1 (bottom panels).
The high-symmetry points are as in Fig.~\ref{fig:VA05}.}
\label{fig:spectral}
\end{figure}

Figure~\ref{fig:spectral} presents the comparison of the spectral
function for $J/t=0.05$ and 0.1, for SCBA, VCA and the variational
approach with $n=6$ orbitons. On the first glance, the QP band in the
VCA solution looks more similar to the variational result, but as shown
below the SCBA gives also quite satisfactory results for the QPs.
Although all the solutions exhibit a similar shape, with a lot of
spectral weight around the minimum at the $\Gamma=(0,0)$ point and a
substantial drop in weight with additional
flattening of the spectrum around the $M=(\pi,\pi)$ point, those
features are much more pronounced in the SCBA than in the other two
methods.

Above the QP band, both SCBA and the variational approach show the
emergence of a ladder structure of excited states dissolving quickly
into a broad continuum. Such a ``shake-off'' band also emerges below
the continuum for VCA for $J/t=0.05$, although its weight is rather
small. Thus, it seems reasonable to expect that such a feature does
emerge in the strong coupling limit, although the variational method
and SCBA may overestimate it.

\begin{figure}[t!]
  \centering
  \includegraphics[width=\columnwidth]{weight}
\caption{(Color online)
Spectral weight $Z(\mathbf{k})$ (top) and ground-state energy
$E(\mathbf{k})$ (bottom) for $J=0.05$
(left) and 0.10 (right) as obtained in the $n=6$ orbiton variational
approach (6-orbiton), in VCA and in SCBA.
The high-symmetry points are as in Fig.~\ref{fig:VA05}.}
\label{fig:weight}
\end{figure}

At even higher energies, SCBA predicts a very featureless continuum,
unlike VCA and the present variational method, which show a more
structured one. However, since neither SCBA nor the variational method
aim to be highly accurate at such high energies, such comparisons are
of limited use. It should also be noted that the apparent dense peak
structure obfuscating the VCA results is not physical but just a
finite-size effect which is the usual byproduct of this method.

Returning to the low-energy polaron, we show its QP spectral weights
$Z(\mathbf{k})$ and ground state energies $E(\mathbf{k})$ in
Fig.~\ref{fig:weight}, obtained by least squares fitting of a
Lorentzian curve to the first peak of the spectrum. As can be seen,
while SCBA and the variational result are in good agreement as to the
spectral weight in both of the presented cases, SCBA always predicts
a heavier QP, especially for the smaller value of $J$,
where it clearly overestimates the binding energy due to not including
the particle constraints, as already discussed. On the other hand, one
can see that the VCA results are in disagreement with both other
results. This can be attributed partly to those calculations being
based on a different model, and so can be affected by a ground state
energy shift, and partly due to the systematic error of the method as
demonstrated by the dense peak structure, which affects the quality of
the peak fit. In any case, the VCA results are in agreement with other
results only qualitatively, although they are usually more similar to
the variational results.

\begin{figure}[t!]
  \centering
  \includegraphics[width=\columnwidth]{moments}
\caption{(Color online)
First four spectral moments for the calculations performed
in the one-orbiton and two-orbiton variational approximation, and SCBA
as compared to the analytical solution. The high-symmetry points as in
Fig.~\ref{fig:VA05}.}
\label{fig:moments}
\end{figure}

As a final way to compare SCBA and the variational approach in the
strong coupling limit, Fig.~\ref{fig:moments} shows the predicted
first four spectral moments compared to their analytical expressions.
While the agreement is reasonable for both methods up to
$M_3(\mathbf{k})$, the variational approach is clearly more accurate.
Note however, that while agreement with sum rules is obviously
desirable, nevertheless one has to be careful when assessing how
meaningful such an agreement really is \cite{Goo06}.

\subsection{Significance  of inter-orbital hopping}
\label{sec:hopping}

After discussing the similarities and differences between the SCBA
and the variational approach, let us briefly focus on the relative
importance of the $t$ and $J$ energy scales for the QP propagation.
In the strong coupling limit, Fig.~\ref{fig:weight} shows that the
QP bandwidth changes with $r=J/t$, but this dependence is rather weak
for VCA and the variational approach. The bandwidth is reduced from
the free value expected for unhindered inter-orbital hopping, but it
is clearly not proportional to $J$.

\begin{figure}[t!]
  \includegraphics[width=\columnwidth]{VCA}
\caption{
VCA-spectra for AO state at $U=10t$ (corresponding to $J=0.10$) and:
    (a) the $\ket{z}$ orbital,
    (b) the $\ket{\bar{z}}$ orbital,
    (c) the orbital $\ket{+}$, as well as
    (d) the trace over the orbital basis (basis invariant),
    \emph{i.e.}, the normalized sum of (a) and (b).
The high-symmetry points as in Fig.~\ref{fig:VA05}.
\label{fig:AO}}
\end{figure}

We can further substantiate the importance of inter-orbital hopping by
looking at orbitally resolved spectra. To do so, here we show the hole
spectra projected onto the original $\ket{z}$ and $\ket{\bar{z}}$
orbitals. These are given by
\begin{align}
&A_{\alpha\alpha}(\vec{k},\omega) =  \\
&-\frac{1}{\pi} \lim_{\delta\to 0}\Im \left[
  \langle 0| c^{\dagger}_{\vec{k}\alpha}
  \frac{1}{\omega-(\mathcal{H}_t +
  \mathcal{H}_U-E_0)+i\delta}
  c^{}_{\vec{k}\alpha}|0\rangle \right], \nonumber
\end{align}
for $\alpha=z,\bar{z}$ referring to the $3z^2-r^2$ and $x^2-y^2$
orbitals \eqref{basis}. $\mathcal{H}_t + \mathcal{H}_U$ is the
`Hubbard-like' Hamiltonian comprised of Eqs.~(\ref{eq:trans3d}) and
(\ref{eq:hub}), whose half-filled ground state is denoted by
$|0\rangle$ and has energy $E_0$. Spectra for $z$ and $\bar{z}$ are
shown in Figs.~\ref{fig:AO}(a) and~\ref{fig:AO}(b), respectively.

The first notable feature is that the dispersion does not reflect the
doubled unit cell of the AO ground state, as might \emph{a priori} be
expected. This doubling of the unit cell does manifest itself in the
spectrum calculated for the orbital $\ket{\alpha}=\ket{+}$, see
Fig.~\ref{fig:AO}(c), and in the trace shown in Fig.~\ref{fig:AO}(d).
The second relevant aspect is that the dispersion is ``opposite'' for
the $\ket{z}$ and $\ket{\bar{z}}$ orbitals, \emph{i.e.}, the minimum is
either at the $\Gamma$ point (for $\ket{z}$) or at the $M$ point
(for $\ket{\bar{z}}$).

Such an orbital-specific lack of any signature of the unit-cell size
is not expected in a propagation mechanism driven by three-site terms
or orbital flips, and doubling of the unit cell is in fact observed in
the $t_{2g}$ or SU(2) symmetric $t$-$J$ models. However, this feature
arises naturally for a dispersion supported mainly by inter-orbital NN
hopping. The AO ordered ground state consist of alternating $\ket{+}$
and $\ket{-}$ orbitals occupying sublattices $a$ and $b$ respectively.
The interorbital hopping $t_{+-}=t/4$ allows a $\ket{+}$ electron on
sublattice $a$ to move to the $\ket{-}$ orbital of a previously empty
NN site on sublattice $b$ -- the hole has thus moved between NN sites
without disturbing the AO background. The $\ket{z}$ orbital has
positive overlap $+\tfrac{1}{\sqrt{2}}$ with both $\ket{\pm}$ orbitals
and consequently a minimum develops at $\Gamma$ like for a free
particle, see Fig.~\ref{fig:AO}(a). The $\ket{\bar{z}}$ orbital, on the
other hand, has an overlap $\pm\tfrac{1}{\sqrt{2}}$ alternating from
site to site. It translates into the momentum shift moving the minimum
to $M$, as observed in Fig.~\ref{fig:AO}(b).

We have thus identified interorbital hopping as the main driving force
of hole propagation in the $e_g$ model. While the bandwidth is reduced
from its free value through interaction with the AO background, the
SCBA overestimates this effect. This may in turn assign too strong a
role to orbital fluctuations and consequently obscure the origin of the
QP propagation. Finally, we note that an orbital-selective ARPES study
might fail to see the doubling of the unit cell of an orbitally ordered
state, depending on the wave function of the ordered orbital.

\section{Conclusions}
\label{sec:summa}

By analogy with work done for other polaron models, here we
constructed a method to calculate the Green's function for an
orbiton-polaron in a variational space including configurations with
up to six orbitons. We then compared these results with two different
and well established methods, SCBA and VCA. We found that our
variational Green's function converges fast, especially for the
low-energy QP, and that this important feature agrees well with the
corresponding feature obtained in VCA in the strong effective coupling
limit. In contrast, the quasiparticle predicted by SCBA has a
noticeably smaller bandwidth, although general quantitative trends of
the spectra are similar for all three methods.

The higher accuracy of the variational approach can be understood by
noting that its equations of motions allow orbitons to be absorbed
in any order and it consequently includes crossed diagrams, which
are neglected by the SCBA. Moreover, it also implements the local
constraints ignored by SCBA.
In fact, at first sight it is perhaps surprising that the agreement
with SCBA remains as good as it is in the strong effective coupling
limit. This is in contrast to what happens for lattice polarons but
in agreement with what is observed for spin polarons. The common link
with the latter is that like magnons, orbitons are also restricted to
at most one per site. As a result, the cloud is spread over a larger
area and the carrier is more likely to absorb one of the last emitted
orbitons, since they are close to its location (note that at energies
characteristic for the polaron band, which lie well below the
free-particle band, the free carrier cannot propagate). Thus, even
though there is no analogue of the total spin conservation, which
leads to the vanishing of many crossed diagrams and thus improves the
validity of SCBA for spin models, uncrossed diagrams still have higher
probabilities than crossed ones in the present problem. By contrast,
in the small polaron limit of a lattice polaron, the number $n$ of
phonons on the same site can be quite large.
Because they are indistinguishable, the carrier is equally likely to
absorb them in any of $n!$ possible orders,  and SCBA, which only
allows them to be absorbed in the order inverse to that of their
creation, underestimates the contribution of such processes by $1/n!$.

An interesting follow-up question is whether SCBA is still expected to
be qualitatively accurate (even though quantitatively less accurate
than the variational method) for the combined spin-orbiton polarons
\cite{Woh09}, given that it performs reasonably well for either species
of bosons. We expect this to not be the case, because there is no
reason why diagrams with crossed magnon and orbiton lines should be
small compared to the non-crossed ones included in SCBA:
magnons and orbitons can be located on the same site and there is no
conservation law restricting the order in which they can be absorbed.

Moreover, SCBA does not lend itself easily to schemes with multiple
bosonic flavors, which would be the case for the spin-orbital problem,
while the variational method proposed here has already proved highly
accurate for spin-polaron problems (as well as lattice polaron
problems), and can be easily extended to such multi-boson models.
It is thus a promising candidate for investigating carrier propagation
in the full 3D $A$-AF/$C$-AO order in KCuF$_{3}$ which needs to include
both orbiton and spin degrees of freedom, similar to LaMnO$_3$
\cite{Bal01}. Such studies are under way at present.

\begin{acknowledgments}
We thank Krzysztof Wohlfeld for insightful discussions.
We kindly acknowledge support by UBC Stewart Blusson Quantum Matter
Institute,
by Natural Sciences and Engineering Research Council of Canada (NSERC),
by Narodowe Centrum Nauki (NCN, National Science Center)
under Project No.~2012/04/A/ST3/00331, and
by the German Research Foundation (DFG) under grant No. DA 1235.
\end{acknowledgments}

redpajama_set_name: RedPajamaArXiv


Please enter your registration/license plate number (or wording) & colour or paint code in the boxes above. Double check to ensure you have entered this correctly, as refunds can not be made.
PLEASE NOTE: These prints are despatched direct to you from our supplier - please allow 7-10 days for delivery.
Choose Post or AirMail as the shipping method if the print is not being combined with any other items you are ordering.
Order in your colour with your plate number!
Available to order - Old unit required first.


Greg Greger, the oldest member of the PSK at 89, hosted the Tri-Cities Knap-in at Howard Amon Park over the Memorial Day weekend. While not a large turnout by PSK standards (probably because another PSK Knap-in, Illahee Flats started at the same time), it was nevertheless a great time!
Having a Knap-in at the Howard Amon Park in Richland, WA turned out to be a great ideal. Richland, WA, along with Pasco and Kennewick make up the 'Tri-Cities', three small cities that are located at the confluence of the Columbia, Yakima and Snake Rivers. These cities grew up around the Hanford Nuclear Reservation back in the 1940/50s when Hanford was building nuclear reactors and processing uranium into plutonium. The Howard Amon Park in Richland, WA is located on the banks of the Columbia River just north of where the two other rivers enter the Columbia. The weather was great, lots of public access and plenty of room for knapping.
We were so busy knapping and demonstrating knapping to the public that we didn't get much of an opportunity to take a lot of photos. In fact much of our time was spent teaching youngsters to knap - along with half dozen or so adults! Lots of people stopped by to watch and ask questions and kept us busy the entire three days!
And some 'oldsters' asa well!
Douglas Macomber, who spent some time in Saudia Arabia working in the Oil Industry, stopped by to show us some of the artifacts he collected while there.


Little quantitative information is available on the mixing patterns of children in school environments. Describing and understanding contacts between children at school would help quantify the transmission opportunities of respiratory infections and identify situations within schools where the risk of transmission is higher. We report on measurements carried out in a French school (6–12 years children), where we collected data on the time-resolved face-to-face proximity of children and teachers using a proximity-sensing infrastructure based on radio frequency identification devices.
Data on face-to-face interactions were collected on Thursday, October 1st and Friday, October 2nd 2009. We recorded 77,602 contact events between 242 individuals (232 children and 10 teachers). In this setting, each child has on average 323 contacts per day with 47 other children, leading to an average daily interaction time of 176 minutes. Most contacts are brief, but long contacts are also observed. Contacts occur mostly within each class, and each child spends on average three times more time in contact with classmates than with children of other classes. We describe the temporal evolution of the contact network and the trajectories followed by the children in the school, which constrain the contact patterns. We determine an exposure matrix aimed at informing mathematical models. This matrix exhibits a class and age structure which is very different from the homogeneous mixing hypothesis.
We report on important properties of the contact patterns between school children that are relevant for modeling the propagation of diseases and for evaluating control measures. We discuss public health implications related to the management of schools in case of epidemics and pandemics. Our results can help define a prioritization of control measures based on preventive measures, case isolation, classes and school closures, that could reduce the disruption to education during epidemics.
The following movie was provided as supplementary information.


text:
AFI Top 100
Religious Music & Entertainment
Confess your okayness Fletch
January 20, 2023 by Coyle Neal
Confess your okayness Fletch January 20, 2023 Coyle Neal
Image: IMDB
Obviously a movie like Confess, Fletch can't be watched outside of the context of the book or the earlier movies. Fortunately, I've never read the book and haven't seen the Chevy Chase movies in more than 20 years (which is probably a good thing, since I have mildly fond memories of the old movies and I suspect that fondness wouldn't remain if I were to go back and watch them again). Which means I kind-of get to watch this movie on its own with only a vague knowledge of the canon it's adding to.
And, as far as that goes, it was a fine movie. Not a great movie, not a terrible movie, just an okay one. [Probably some spoilers from here on out.] The plot is kind-of simple: Fletch comes into his Air BnB and finds a dead body. He calls the police and promptly becomes the prime suspect in an apparently art-related crime. His wealthy heiress girlfriend, her Countess gold-digging step mother, the homeowner and his ex-wife, local art dealer/Harvard adjunct Dale Cooper Ronald Horan, and weirdo neighbor Eve are all somehow involved and possibly suspects.
As far as it goes, the mystery is pretty well done. While there aren't any real curveballs, they do leave it somewhat up in the air as to who the murderer is and what the motive was until the big reveal. Likewise the humor mostly lands well–it helps that the film is exceptionally well cast and the actors all do the best they can with the script they've got. The weak part here is the writing. And here I'll have to guess a bit, but based on what I've heard about the book the main character in the novel is a very… 1970's… man. That is, he sleeps with anything that moves, isn't afraid of substances, has no respect for anything, and still solves the crime. That combination of things obviously isn't going to play as well in the post-Me Too era. This isn't a bad thing other than it affects the attempt to turn a 1970s book into a 2020s movie. The characteristics of a 1970s lead haven't really been replaced by anything, and the plot and humor, while good, aren't quite strong enough to make up the difference for a lack of strong charcter.
From a Christian point of view, Confess Fletch is a film that represents the transitional time film is in. As a demographic that is for all intents and purposes outside of what's going on in Hollywood these days (and that's not automatically a bad thing), we should be keeping an eye on the culture as it tries to feel its way towards a coherent metanarrative. We need to celebrate and enjoy the times when the culture manages to find something true, mourn when they embrace falsehood, and admit when something is just an okay and average representation of what's going on in film. Confess Fletch is just such a representation.
Dr. Coyle Neal is co-host of the City of Man Podcast an Amazon Associate (which is linked in this blog), and an Associate Professor of Political Science at Southwest Baptist University in Bolivar, MO
Common Grace, 3.27
January 24, 2023 Common Grace, 3.28
Select a Category Action AFI Top 100 Animation Best Picture Winner Biography Biopic Books Childrens Christian Classics Classics Comedy Crime Documentary Drama Family Fantasy Fantasy Films Foreign Historical Historical Fiction History Horror Horror IMDB Top 250 Indie Memoir Modern Novel Musical Mystery Non-Fiction Oscars Philosophy Poetry Popular Literature Reading and Writing Science Fiction Science Fiction Sports Thriller TV Show Uncategorized Western Westerns Young Adult
Select a Month January 2023 December 2022 November 2022 October 2022 September 2022 August 2022 July 2022 June 2022 May 2022 April 2022 March 2022 February 2022 January 2022 December 2021 November 2021 October 2021 September 2021 August 2021 July 2021 June 2021 May 2021 April 2021 March 2021 February 2021 January 2021 December 2020 November 2020 October 2020 September 2020 August 2020 July 2020 June 2020 May 2020 April 2020 March 2020 February 2020 January 2020 December 2019 November 2019 October 2019 September 2019 August 2019 July 2019 June 2019 May 2019 April 2019 March 2019 February 2019 January 2019 December 2018 November 2018 October 2018 September 2018 August 2018 July 2018 June 2018 May 2018 April 2018 March 2018 February 2018 January 2018 December 2017 November 2017 October 2017 September 2017 June 2017 January 2017 December 2016 October 2016 September 2016 August 2016 July 2016 June 2016 May 2016 April 2016 March 2016 February 2016 January 2016 December 2015 November 2015 October 2015 September 2015 August 2015 July 2015 June 2015 May 2015 April 2015 March 2015 February 2015 January 2015 December 2014 November 2014 October 2014 September 2014 August 2014 July 2014 June 2014 May 2014 April 2014 March 2014 February 2014 January 2014 December 2013 November 2013 October 2013 September 2013 August 2013 July 2013 June 2013 May 2013 April 2013 March 2013 February 2013 January 2013 December 2012 November 2012 October 2012 September 2012 August 2012 July 2012
Related posts from Schaeffer's Ghost
Reaching for a better Reacher
Schaeffer's Ghost
The Bored Housewives' Detective Agency
Potent Story Lacks Punch in The Innocents
A Vertiginous Horror
TRENDING AT PATHEOS Entertainment
Review of "Everything Everywhere All at Once" directed by Daniel Scheinert
Those 20 Best Hymns That You All Know...
We all love listening to hymns. They are the songs we grew...
'Yellowstone' Actor Neal McDonough Talks New Film 'Left...
The Antichrist ascends to power in the latest installment of the popular...
Zach Williams Set to Debut New Songs During...
This weekend, GRAMMY Award-winning singer-songwriter Zach Williams will headline the 25th Rock...
Minority Report and Judgment
Like many of the films we discuss here, Steven Spielberg's 2002 film,...
Kuyper on the corruption of the relationship between church and state
view all Entertainment blogs


For the second time this season, the NASCAR Camping World Truck Series (NCWTS) will embark on new territory - making racing history once again - this time north of the border at Canadian Tire Motorsports Park (CTMP) in Bowmanville, Canada. The 2.459-mile track marks the series' first visit to a road course in over 13 years - the last time the series raced on anything other than an oval was June 24, 2000 at Watkins Glen (N.Y.) International.
Although Coulter has only experienced the 10-turn course through simulation racing and videos - much like a majority of the 30 truck field - the driver of the Monster Energy Toyota Tundra does have an advantage on the field. Since 2009, the Miami Springs, Fla. native has made three starts on a road course in the ARCA Racing Series, including two laps led en route to one top-five and two top-10 finishes. The 23-year-old hopes his knowledge of turning left and right will pay dividends in Sunday's inaugural event.
Heading into this weekend's Truck Series event at CTMP, Coulter is solidly inside the top 10 in points - a position he hasn't been in since April - after finishing a season-high second at Kansas Speedway in Kansas City. In a field mixed with a few road course ringers and mostly road course rookies, the No. 18 Kyle Busch Motorsports (KBM) driver hopes to claw into the competition by capitalizing on his road course racing experience to make "Monster" size strides in the points.