There you have it, I confess to my frequent imbalance. All summer I had tunnel vision for a specific creative project. Any spare time found was immediately invested in the process of fleshing out a series of artwork for a pressing exhibition deadline. I have been on a creative mission. I knew that in order to see this project to completion, I had to be focused and determined and maybe even a little obsessed, therefore some things had to temporarily take a back seat. Have no fear, I did not skirt my primary parental or spousal duties. I do admit the house was significantly neglected and physical exercise took a short hiatus. However, I managed to complete more artwork than expected for this series, and hang it all in time at its venue. I am pleased with the accomplishment.
before me in all its glory was the aftermath of my creative binging. It was a serious wreck. There was no way to work around it. I have had to take time to sort and reorganize. It is the part of being an artist that isn't enjoyable but is unavoidable for creative recovery. I have learned to appreciate these gaps in between my creative binges and realize that there is no way I can maintain creating at that level of adrenaline without eventually crashing or falling victim to fog brain. It is an imperative part of the artistic rhythm to take the opportunity to not only recover my workspace but also replenish my energy and emotional levels. If I don't make the effort to clean up and rest after such a season, I can't be released to move into the next. I am coming to recognize the responsibility that comes with being an artist. It isn't just about the discipline to create work. I also must learn to manage the margins in between well in order to recover from the expenditures demanded to create. Whether it is to implement physical exercise again, eat meals more regularly, or re-engage with friends and the world outside my door, I must make the intentional effort to spread out my attentions once again in a more balanced way and get myself back into shape.
As I clean up my workspace, I find I am also cleaning up my head space. Once the clutter is gone, I have room to gradually begin filling it again. It's a cycle I now embrace by being more aware of it. I clean my house in order to freely live and thrive in it, and my house needs cleaning regularly because I freely live and thrive in it. I clean my studio in order to do the same, to reclaim the space for the promise of yet more work and life to take place within it.


The first US Open Fat Bike Beach Championships are set to go off at the Blockade Runner Resort in Wrightsville Beach, NC on Saturday, March 14th, 2015.
What: A weekend of Fat Bike Riding and racing on Wrightsville Beach, NC. Three different divisions for the races, Beginner, Sport, Expert. Prizes for each age group in each division.
When: Friday March 13, 2015 check-in/packet pick-up 5-6pm. Saturday March 14 races 9:00am-1:00pm. Awards Party 1:30-3:00pm. Optional night ride 7:30-8:30pm on Wrightsville Beach. Sunday 9:00am optional pontoon ferry ride to uninhabited 8 mile island for a guided ride.
Where: Wrightsville Beach, North Carolina. Blockade Runner Resort.
Entry Costs: $42 (Includes race entry, Race T-Shirt, special rate for Sunday Island ride) (after 3-1-15 goes to $50).


text:
Q: What design pattern is better when I want to abstract interprocess communication? I have a class which communicates with other process with my socket library. But soon, some of my team members will release another socket library. So I have to replace socket communication module with it.
So I want to abstract the communication way with a design pattern. In such situation, what design pattern is better for me? My class shouldn't access my socket library directly.

A: Stop thinking sockets. Think interprocess communication API. Otherwise you will design an abstraction which is too low level.
What do you really want to do? Send application messages between processes? Send notifications?
Create a higher level abstraction instead. Something like:
public interface ICommunicationChannel
{
    ApplicationMessage Receive();
    void Send(ApplicationMessage);
    event EventHandler<FailureEventArgs> ChannelFaulted;
}

.. or, if your messages should be pushed from the other end point:
public interface ICommunicationChannel
{
    void Send(ApplicationMessage);
    event EventHandler<FailureEventArgs> ChannelFaulted;
    event EventHandler<MessageReceivedEventArgs> MessageReceived;
}

By doing it on that level you have a loose coupling with the communication layer and are free to design it using whatever technology you like.

A: I would suggest creating a ProcessCommuniactionInterface.
This interface should expose the basic operations and send() , receive(), callback() etc as per your need.
Right now it will have one implementation as say SocketLib1ProcessCommuniactionImpl .
Later on when any other Lib comes up , add one more implementation as SocketLib2ProcessCommuniactionImpl.
How you migrate would depend on your usecase:

*

*If the use case is to completely drop using 1st one and start using the second one , you could directly replace the first Implementation with the second one. (Most of the organizations don't go for this as it is risky and can lead to rollback)

*If the use case is to use on Lib for some flows and other Lib for other flow (This will most likely be case when some processes you communicate with are using Lib1 and others are using lib2) ,you should keep both implementations.  Here there are two cases:
    a. the flows are fixed when we have to use lib1 and when to use lib2 . Go for a static config file , and get the data from there and call the respective implemtation.
    b. we need the ability to dynamically use one of the libraries. Go for something like Strategy Pattern here.
Hope this helps:)

redpajama_set_name: RedPajamaStackExchange


Very little valid information exists about Solvation manufacturing. The company appears to be a sales broker or contract manufacturer of solar panel products, however specifications could not be verified. Solvation modules are listed as approved for use by the California Energy Commission. This listing implies that the modules meet a minimum standard for UL certification with a 10 year product warranty and 25 year performance guarantee.


text:
from heat.common import exception
from heat.common.i18n import _
from heat.engine import constraints
from heat.engine import properties
from heat.engine import resource
from heat.engine import support
from heat.engine import translation

class NovaQuota(resource.Resource):
    """A resource for creating nova quotas.

    Nova Quota is used to manage operational limits for projects. Currently,
    this resource can manage Nova's quotas for:

        - cores
        - fixed_ips
        - floating_ips
        - instances
        - injected_files
        - injected_file_content_bytes
        - injected_file_path_bytes
        - key_pairs
        - metadata_items
        - ram
        - security_groups
        - security_group_rules
        - server_groups
        - server_group_members

    Note that default nova security policy usage of this resource
    is limited to being used by administrators only. Administrators should be
    careful to create only one Nova Quota resource per project, otherwise
    it will be hard for them to manage the quota properly.
    """

    support_status = support.SupportStatus(version='8.0.0')

    default_client_name = 'nova'

    entity = 'quotas'

    required_service_extension = 'os-quota-sets'

    PROPERTIES = (
        PROJECT, CORES, FIXED_IPS, FLOATING_IPS, INSTANCES,
        INJECTED_FILES, INJECTED_FILE_CONTENT_BYTES, INJECTED_FILE_PATH_BYTES,
        KEYPAIRS, METADATA_ITEMS, RAM, SECURITY_GROUPS, SECURITY_GROUP_RULES,
        SERVER_GROUPS, SERVER_GROUP_MEMBERS
    ) = (
        'project', 'cores', 'fixed_ips', 'floating_ips', 'instances',
        'injected_files', 'injected_file_content_bytes',
        'injected_file_path_bytes', 'key_pairs', 'metadata_items', 'ram',
        'security_groups', 'security_group_rules', 'server_groups',
        'server_group_members'
    )

    properties_schema = {
        PROJECT: properties.Schema(
            properties.Schema.STRING,
            _('Name or id of the project to set the quota for.'),
            required=True,
            constraints=[
                constraints.CustomConstraint('keystone.project')
            ]
        ),
        CORES: properties.Schema(
            properties.Schema.INTEGER,
            _('Quota for the number of cores. '
              'Setting the value to -1 removes the limit.'),
            constraints=[
                constraints.Range(min=-1),
            ],
            update_allowed=True
        ),
        FIXED_IPS: properties.Schema(
            properties.Schema.INTEGER,
            _('Quota for the number of fixed IPs. '
              'Setting the value to -1 removes the limit.'),
            constraints=[
                constraints.Range(min=-1),
            ],
            update_allowed=True
        ),
        FLOATING_IPS: properties.Schema(
            properties.Schema.INTEGER,
            _('Quota for the number of floating IPs. '
              'Setting the value to -1 removes the limit.'),
            constraints=[
                constraints.Range(min=-1),
            ],
            update_allowed=True
        ),
        INSTANCES: properties.Schema(
            properties.Schema.INTEGER,
            _('Quota for the number of instances. '
              'Setting the value to -1 removes the limit.'),
            constraints=[
                constraints.Range(min=-1),
            ],
            update_allowed=True
        ),
        INJECTED_FILES: properties.Schema(
            properties.Schema.INTEGER,
            _('Quota for the number of injected files. '
              'Setting the value to -1 removes the limit.'),
            constraints=[
                constraints.Range(min=-1),
            ],
            update_allowed=True
        ),
        INJECTED_FILE_CONTENT_BYTES: properties.Schema(
            properties.Schema.INTEGER,
            _('Quota for the number of injected file content bytes. '
              'Setting the value to -1 removes the limit.'),
            constraints=[
                constraints.Range(min=-1),
            ],
            update_allowed=True
        ),
        INJECTED_FILE_PATH_BYTES: properties.Schema(
            properties.Schema.INTEGER,
            _('Quota for the number of injected file path bytes. '
              'Setting the value to -1 removes the limit.'),
            constraints=[
                constraints.Range(min=-1),
            ],
            update_allowed=True
        ),
        KEYPAIRS: properties.Schema(
            properties.Schema.INTEGER,
            _('Quota for the number of key pairs. '
              'Setting the value to -1 removes the limit.'),
            constraints=[
                constraints.Range(min=-1),
            ],
            update_allowed=True
        ),
        METADATA_ITEMS: properties.Schema(
            properties.Schema.INTEGER,
            _('Quota for the number of metadata items. '
              'Setting the value to -1 removes the limit.'),
            constraints=[
                constraints.Range(min=-1),
            ],
            update_allowed=True
        ),
        RAM: properties.Schema(
            properties.Schema.INTEGER,
            _('Quota for the amount of ram (in megabytes). '
              'Setting the value to -1 removes the limit.'),
            constraints=[
                constraints.Range(min=-1),
            ],
            update_allowed=True
        ),
        SECURITY_GROUPS: properties.Schema(
            properties.Schema.INTEGER,
            _('Quota for the number of security groups. '
              'Setting the value to -1 removes the limit.'),
            constraints=[
                constraints.Range(min=-1),
            ],
            update_allowed=True
        ),
        SECURITY_GROUP_RULES: properties.Schema(
            properties.Schema.INTEGER,
            _('Quota for the number of security group rules. '
              'Setting the value to -1 removes the limit.'),
            constraints=[
                constraints.Range(min=-1),
            ],
            update_allowed=True
        ),
        SERVER_GROUPS: properties.Schema(
            properties.Schema.INTEGER,
            _('Quota for the number of server groups. '
              'Setting the value to -1 removes the limit.'),
            constraints=[
                constraints.Range(min=-1),
            ],
            update_allowed=True
        ),
        SERVER_GROUP_MEMBERS: properties.Schema(
            properties.Schema.INTEGER,
            _('Quota for the number of server group members. '
              'Setting the value to -1 removes the limit.'),
            constraints=[
                constraints.Range(min=-1),
            ],
            update_allowed=True
        )
    }

    def translation_rules(self, props):
        return [
            translation.TranslationRule(
                props,
                translation.TranslationRule.RESOLVE,
                [self.PROJECT],
                client_plugin=self.client_plugin('keystone'),
                finder='get_project_id')
        ]

    def handle_create(self):
        self._set_quota()
        self.resource_id_set(self.physical_resource_name())

    def handle_update(self, json_snippet, tmpl_diff, prop_diff):
        self._set_quota(json_snippet.properties(self.properties_schema,
                                                self.context))

    def _set_quota(self, props=None):
        if props is None:
            props = self.properties

        kwargs = dict((k, v) for k, v in props.items()
                      if k != self.PROJECT and v is not None)
        self.client().quotas.update(props.get(self.PROJECT), **kwargs)

    def handle_delete(self):
        self.client().quotas.delete(self.properties[self.PROJECT])

    def validate(self):
        super(NovaQuota, self).validate()
        if sum(1 for p in self.properties.values() if p is not None) <= 1:
            raise exception.PropertyUnspecifiedError(
                *sorted(set(self.PROPERTIES) - {self.PROJECT}))

def resource_mapping():
    return {
        'OS::Nova::Quota': NovaQuota
    }

redpajama_set_name: RedPajamaGithub


Some 200 shops offering ladies fashions, accessories,cosmetics, and gourmet food.
Some 200 shops offering ladies fashions, accessories,cosmetics, and gourmet food. This convenient shopping area connects directly to JR Himeji Station. The rooftop plaza offers a splendid view of the World Heritage Site, Himeji Castle, and of Shinkansen trains pulling in and out. Restaurants have a full range of Japanese foods such as udon and donburi (rice bowls). Also find a drugstore and shops selling Himeji-only types of Japanese sweets. Stop by for a nice break when you come to visit Himeji Castle.
●Hours/<Main Building, Young-kan, Gochiso-kan> 10:00a.m.~9:00p.m.
◆Access/Short walk from JR Himeji station.


of our clients have worked with us on more than one assignment.
of our business is retained, tailored to suit our clients needs.
of the London residential property market have worked with GKR London.
The art of working exclusively.
I find Grant and the team professional, well driven and knowledgeable and having given them some tough assignments they continuously delivered."


Standard Floor Level Cabinet Ovens from Grieve meet the requirements of National Fire Protection Association Standard 86, Industrial Risk Insurers, Factory Mutual and OSHA standards. For some applications, such as those involving flammable solvents or hazardous locations, the above organizations require additional safety devices.
Oven comes with Two (2) reinforced 100 lbs uniformly distributed load capacity nickel plated wire shelves. Has capacity for Nine (9) pair shelf support channels on 6″ centers; first shelf 8″ above factory floor.
Fully insulated 2″ thick floor reinforced for 500 lbs distributed loading.
Each features completely wired, top access UL listed control panel enclosing terminals for incoming power, temperature controllers, push buttons and pilot lights. Motor starter and heating element contactors electrically interlocked to shut off heaters if power to blower is interrupted and to permit opera- tion of blower without heat for cooling. Alloy wire heating elements. Exhaust outlet is 4″ diameter.
Control panel as detailed above and safety devices as listed on the front of this bulletin.
Automatic pre-ignition purge period and push button electric ignition contributes to ease of operation. Modulating gas burner is protected with electronic flame safety relay.
These electric or gas models come in two sizes. They feature full horizontal air flow with adjustable patented opposed louvers on full coverage supply and return duct work. Commonly used where work is shelf loaded and least restriction to free air passage is in the horizontal direction.
Size 36″ x 36″ x 60″ high provides 45 cubic feet of work space at temperatures to 350°F and 550°F respectively.
Size 48″ x 48″ x 60″ high provides 80 cubic feet of work space for larger parts and bigger loads.
· Additional Shelves, 100 lbs distributed load per shelf, 400 lbs maximum oven load; specify oven model when ordering.


Details: $5 off every Rise of the Eldrazi booster box with promo code ROE5Box.
Why are these codes still showing up???? EXPIRED!!
Details: Get 10% off supplies.
Details: Get 15% off RPGs.
Didn't work. Said can only be used 0 times per account.
Get 10% Off on Dice And Dice Bags.
Details: Get 10% off on dice and dice bags.
Today Through Sunday Get 10% Off All Fantasy Flight Lcgs With Promo Code Lcg10.
Details: Today through Sunday get 10% off all Fantasy Flight LCGs with Promo Code LCG10.
This Promo Code Ended on 03/29/2010.
10% Off Standard Magic Singles.
Details: 10% off Standard Magic singles.
"This Promo Code Ended on 04/16/2012"
Details: Get 10% off all Minis singles.
Get 10% Off All Minis.
Details: Get 10% off all minis.
Get 10% Off on All Minis Singles.
Details: Get 10% off on all Minis Singles.
All this does is send you to the Cool Stuff store webpage but no 86% off, no promo code, you place item in cart at $79.99 , and item will stay at $79.99. this Deal is no deal.
yes it did. all their games are 86% off retail.
Details: Get 10% off Yu-Gi-Oh Singles.
I applied the coupons, but I got the message: "Discounts do not stack"
Details: Get Free Shipping On All Booster Boxes.
Details: Get a Free Copy of Dragon punch ($15 value) with any purchase of a Level 99 Games product. Maximum 1 copy per order.
Get the top deals from 100s of retailers, including Cool Stuff, in the Best of RetailMeNot emails.


text:
Home Collections New York Hempstead
Hempstead Collections Lawyers
Mitchell A. Nathanson
Collections Lawyer Serving Hempstead, NY
Experienced & Determined Collections Law Representation! Competitive Fees!
Mitchell is often described by his colleagues and peers as a "lawyer's lawyer" and their "go to" lawyer for enforcing judgments against debtors, especially in New York State. However, Mitchell's expertise has resulted... Read More »
Mark Nathanson
New York Collections Attorney with 56 years of experience
Nassau County Bar Association and New York State Bar
Yale Pollack
Garden City, NY Collections Attorney
Touro College Jacob D. Fuchsberg Law CenterColgate University2nd Circuit, U.S. District Courts, Southern and Eastern Districts of New York, New Jersey and New YorkHerald Community Newspapers, Super Lawyers and Super LawyersAmerican Bar Association and New York State Bar Association
Claimed Lawyer ProfileLII GoldSocial Media
Brooklyn Law SchoolCity University of New York - City CollegeNew YorkNassau County Bar Association and New York State Bar
Michael A. Markowitz
Hewlett, NY Collections Attorney
Touro College Jacob D. Fuchsberg Law Center and Washington University School of LawNew York and 2nd CircuitNassau County Bar Association and Nassau County LegislatureNassau County Bar Association and Jewish Lawyers Association
Claimed Lawyer ProfileQ&ALII GoldSocial Media
Benjamin N. Cardozo School of LawYeshiva UniversityNew York, United States District Court, Eastern District of New York, United States District Court, Southern District of New York and United States District Court, Northern District of New YorkNew York State Bar
Richard Glenn Gertler
East Meadow, NY Collections Attorney
Brooklyn Law School1st Circuit, 2nd Circuit, U.S. Supreme Court and New YorkAmerican Bankruptcy Institute, National Association of Consumer Bankruptcy Attorneys...
Shane S. Hassin
Rockville Centre, NY Collections Attorney
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Bradley D. Schnur
Jericho, NY Collections Attorney
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Syosset, NY Collections Attorney
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Marilyn Jean Salzman
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Mr. Robert J. Nahoum Esq.
Bronx, NY Collections Attorney
Albany Law School of Union UniversityState University of New York at PlattsburghNew York, New York Bar, US District Court, Eastern District of New York, US District Court, Northern District of New York, US District Court, Southern District of New York, US District Court, Western District of New York, 3rd Circuit, Connecticut and New JerseyNew York State Bar Associationand National Association of Consumer AdvocatesFDCPA, FCRA Claims: Hidden Trustee Treasure?
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Golden Gate University and Golden Gate University School of LawRollins CollegeNew York, 2nd Circuit and New JerseyAvvoAssociation of the Bar of the City of New York, New York County Lawyers Association...
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Benjamin N. Cardozo School of LawHarvard University, Columbia University, City University of New York - Graduate School & University Center, New School University, Northwestern University and Laval University in MontrealConnecticut, New York, United States District Courts for the Southern and Eastern Districts of New York, 2nd Circuit and United States District Court for the District of ConnecticutNorthwestern UniversityNew York State Bar and Association of the Bar of the City of New YorkBusiness and Commercial Litigation--Fraudulent Conveyances, Web Jurisdiction over out-of-state and foreign Entities...
Mark S. Anderson
Baldwin, NY Collections Attorney
University of Baltimore School of LawUniversity of Wisconsin - MadisonUnited States District Court for the Southern & Eastern Districts of the State of New York and New YorkQueens County Bar Association and New York State Bar AssociationPublic Land Records: The Inevitable Tool for Fraud and the Need for Reform
Stephen Z. Starr
University of Southern California Gould School of LawColumbia University and Columbia UniversityCalifornia, District of Columbia, New Jersey and New YorkNew Jersey State Bar, American Bar Association...Corporate Crisis Managers and Disinterestedness Under the Bankruptcy Code and Debt Enforcement & Bankruptcy in Turkey
David Pierre Turchi
American University Washington College of LawNew York, Eastern District of New York, Southern District of New York and ConnecticutAmerican University Washington College of LawNew York City Bar Association and Westchester County Bar AssociationNew York Insurance Law
Bernard D'Orazio
St. John's University School of LawNew York University2nd Circuit, 3rd Circuit, U.S. District Court Eastern District of New York , U.S. District Court Northern District of New York , U.S. District Court Southern District of New York, U.S. Supreme Court and New YorkThe American Lawyer/ALM, The New York Law Journal Reader Rankings, Martindale-Hubbell and GlobalLawExpertsNew York State Bar, New York County Lawyers' Association..."Survey of N.Y. Practice" , 59 St. John's L. Rev. 436 , 1985
The LII Lawyer Directory contains lawyers who have claimed their profiles and are actively seeking clients. Find more Hempstead Collections Lawyers in the Justia Legal Services and Lawyers Directory which includes profiles of more than one million lawyers licensed to practice in the United States, in addition to profiles of legal aid, pro bono and legal service organizations.

redpajama_set_name: RedPajamaCommonCrawl


text:
Q: Python - Selenium: Find / Click YT-Like Button does anyone of you know how to find / click the YT-Like button in Python using selenium since it doesn't have a real Id, etc...
Thanks for the answers

A: you can select with CSS Selector like this:
if you want to like:
#watch8-sentiment-actions > span > span:nth-child(1) > button
if you want cancel like:
#watch8-sentiment-actions > span > span:nth-child(2) > button


Last week, President Trump launched an unprecedented assault on America's public lands when he ordered Interior Secretary Ryan Zinke to evaluate whether dozens of national monuments should be rescinded or reduced in size.
Trump was responding to pressure from Utah's Congressional delegation, which has long hated the Teddy Roosevelt-era Antiquities Act, the law that gives the president the authority to safeguard lands and waters with outstanding physical or cultural attributes. Some—though by no means all— Utahans are upset about President Obama's creation of the 1.35-million-acre Bears Ears National Monument, as well as President Clinton's 1996 designation of Grand Staircase Escalante National Monument.
The long-running campaign against the Antiquities Act comes with a lot of red-hot rhetoric.
along with how to text your message of defense and support for the national monument to Trump.
Also note that TWO DIFFERENT DEADLINES are established for public input–with Bears Ears getting only 15 days for comments while the rest get 60.
This is nothing compared to what lies ahead.


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

Understanding the effects of electron correlations is important not
only for comprehending the behavior of a wide class of materials. It
also allows one to get to fundamental principles of quantum
theory. Quantum dynamical effects caused by electron correlations are
still in the center of interest of condensed-matter
physicists. Although a lot of aspects of electron correlations have
been successfully explained, there are phenomena the picture of which
is incomplete or lacks substantial pieces. The main obstacle in
reaching a complete knowledge of the effects of electron correlations
is the difficulty of a reliable description of dynamical many-body
phenomena.

The major effort in grasping strong electron correlations is focused
on single-impurity Anderson model (SIAM).\cite{Anderson61} This model
was conceived to provide a simple picture of formation of local
magnetic moments. It appeared rather soon that formation of local
moments is connected with the Kondo effect.\cite{Kondo64} Since then
SIAM has been attracting standing attention of condensed-matter
theorists and experimentalists. The first wave of interest in SIAM was
crowned by finding an exact ground state of the model with a flat band
of conduction electrons for all interaction strengths.\cite{Tsvelik83}
The Kondo effect in SIAM was then explained and quantitatively
described. The exact expression for the Kondo asymptotics in SIAM has
become a hallmark and reference for dynamical effects of strong
electron correlations.

Impurity models suit for a description of quantum dots and
nano-particles\cite{Borda03,Choi05} but are unable to describe bulk,
macroscopic effects of electron correlations. It is, however,
generally believed that a local electron interaction in the
tight-binding scheme offers enough space for the description of most
of the dynamical effects of electron correlations. Success of the
dynamical mean-field theory (DMFT) of electron correlations has proved
this.\cite{Georges96} Dynamical mean-field approach, that is local
dynamical approximations, renewed interest in SIAM, since the basic
ingredient of DMFT is an impurity solver.  The latter essentially is
an impurity model with a self-consistent condition on the local
one-electron propagators. The exact thermodynamic solution of SIAM,
however, cannot be used, since it does not produce the necessary
dynamical renormalization of the energy spectrum. A new wave of
attempts to find approximate dynamical solutions of SIAM in the
strong-coupling regime arose.

Presently, the most comprehensive quantitative approaches producing
dynamical properties of the impurity models are quantum Monte Carlo
(QMC)\cite{Fye88} with its continuous-time
extension,\cite{Rubtsov05,Werner06a} exact diagonalization (ED) with a
discretized bath,\cite{Caffarel94} and the numerical renormalization
group (NRG).\cite{Wilson75,Krishnamurthy80} The first two approaches
aim at a precise thermodynamics of the impurity problem, while the
last one best reproduces low-temperature and low-energy scales of
impurity models. The first two approaches work naturally in the
Matsubara formalism and need a numerical analytic continuation to
determine spectral properties, while NRG can be straightforwardly
extended to dynamical and spectral quantities.\cite{Costi94, Bulla98}
The numerical renormalization group represents now-a-days the most
accurate quantitative low-temperature and low-energy dynamical
solution of impurity problems.\cite{Hewson93}

Apart from pure numerical methods a series of analytically controlled
approximations have been proposed. An approximation based on
low-frequency renormalizations of Fermi-liquid parameters is able to
reproduce the Kondo scale in SIAM and fits for heavy Fermion
systems.\cite{Hewson93a} It is not, however, suitable for the
description of critical instabilities of the Fermi liquid
state. Strong-coupling expansions based on the infinite-interaction
model \cite{Keiter70,Pruschke89} reproduce as well the Kondo scale but
fail to reproduce the Fermi liquid regime in the weak-coupling limit.

Standard truncated weak-coupling expansions fail to reproduce
correctly the strong-coupling regime. An improvement can be reached if
the expansion is applied on vertex functions.  Expansions for
two-particle vertices are mostly based on multiple scatterings of
electrons on electrons (T-matrix, TMA) or electrons on holes (random
phase, RPA). The non-renormalized multiple electron-hole scatterings
lead to an unphysical pole that must be removed by a self-consistent
treatment. Standardly introduced one-electron self-consistency
(fluctuation-exchange approximation) removes the pole but fails to
reproduce the Kondo scale.\cite{Bickers89} A static one-electron
self-consistency of the Hartree type was introduced in an effective
approximate scheme, called local-moment approach.\cite{Logan98} The
expansion for the polarization operator (RPA vertex) uses one-electron
propagators with a broken spin symmetry in this approximation. The
spin symmetry of the equilibrium thermodynamic state is then recovered
via a construction of the self-energy calculated from the polarization
operator. In this way the RPA pole is not crossed and the
strong-coupling limit keeps the polarization operator in the critical
region of the RPA pole. The spectral function displays the correct
Kondo asymptotics of the quasiparticle peak. This theory, however,
contains an artificial symmetry breaking and an auxiliary fitting
parameter, the length of the local magnetic moment, in the
strong-coupling regime.

An alternative way of introducing a self-consistency at the
two-particle level was proposed by us.\cite{Janis07a,Janis08} We
suggested to use parquet equations constructed from multiple
electron-electron and electron-hole scatterings and use the
criticality of the RPA pole for a simplification leading to a
manageable theory. This approximation allows us to control
analytically the RPA pole and the genesis of the Kondo
asymptotics. The derived Kondo behavior reproduces universal features
of the exact Bethe-ansatz solution.

Real materials are more complex and cannot be described by elementary
impurity models. One has to extend the methods to multi-orbital
Hamiltonians. Not all the methods used as impurity solvers for
single-orbital models work also in the multi-orbital case.  The exact
solution can be extended to $SU(n)$ models only in the limit
$U\to\infty$.\cite{Ogievetski83} There are natural extensions of
QMC\cite{Bonca93,Kotliar06,Werner06b}, ED,\cite{Liebsch05,Inaba05} and
NRG\cite{Galpin05,Bulla08} methods to multiple-orbitals. All the
numerically exact approaches are, however, computationally rather
expensive for realistic atomic structures. There is hence still a need
for simpler, computationally less expensive semi-analytic approaches
that could qualitatively correctly describe the transition from the
weak to the strong coupling regimes in multi-orbital impurity and
lattice models.

Some of the existing analytically controlled approaches to strong
electron correlations have recently been extended to multi-orbital
models, mostly $SU(n)$ Anderson models. Various spectral and
thermodynamic quantities have recently been calculated within the
renormalized Fermi-liquid expansion,\cite{Nishikawa10} fluctuation
exchange,\cite{Drchal05} or local-moment
approach.\cite{Galpin09,Kauch09} The aim of this paper is to present a
generalization of a two-particle approach based on simplified parquet
equations to multi-orbital models and create a full-scale impurity
solver for realistic models of correlated electrons. Unlike the most
of the other multi-orbital approaches, we formulate the method so that
it can directly be applied not only to impurity models but also to
lattice models within the dynamical mean-field theory. The
approximation is formulated in real frequencies and hence the spectral
properties are directly available. The low and high frequency behavior
of the one-electron spectral function, its three-peak structure, is
qualitatively well reproduced and the method allows one to control
analytically the critical behavior of Bethe-Salpeter equations for the
two-particle vertex.

The paper is organized as follows. We introduce a general
multi-orbital perturbation expansion for two-particle vertices in
Sec.~\ref{sec:Renormalized_PT}. In Sec.~\ref{sec:Simplified_MPE} we
derive simplified parquet equations with electron-hole and
electron-electron multiple scatterings and show how the self-energy is
calculated from the full two-particle vertex. We present explicit
equations to be solved in a two-orbital case in
Sec.~\ref{sec:Two_orbital}.  The derived approximation is used to
obtain a numerical solution of a two-orbital model with Hund's
coupling and orbital splitting in Sec.~\ref{sec:Results}.  We also
compare our results with those from other approaches there.
Reliability and domains of applicability of the derived approximation
are discussed in the final Section~\ref{sec:Conclusions}.

\section{Perturbation expansion for multi-orbital models of correlated
  electrons}\label{sec:Renormalized_PT}

We start with a model Hamiltonian with multiple orbitals, where the
band structure is resolved and kinetic energy is diagonal in orbital
indices. The unperturbed Hamiltonian then reads
\begin{equation}\label{eq:H0}
  \widehat{H}_{0} = \sum_{\mathbf{k}}\sum_{\iota,\sigma}
  \epsilon_{\iota}(\mathbf{k})
  \tilde{c}^{\dagger}_{\iota\sigma}(\mathbf{k})
  \tilde{c}^{\phantom{\dagger}}_{\iota\sigma}(\mathbf{k})
\end{equation}
where $\mathbf{k}$, $\sigma$, and $\iota$ are momentum, spin and
orbital index of valence electrons.  The interaction multi-orbital
Hamiltonian can generally be represented in the direct space as
\begin{multline}\label{eq:HI}
  \widehat{H}_{I} = \frac 14
  \sum_{\substack{\mathbf{x}_{1},\mathbf{x}_{2}\\
      \sigma_{1},\sigma_{2}}}\sum_{\substack{\iota,
      \upsilon,\kappa,\lambda\\ \iota\le \upsilon, \kappa\\ \kappa \le
      \lambda}}
  U^{\lambda\iota,\sigma_{1}}_{\kappa\upsilon,\sigma_{2}}(\mathbf{x}_{1}
  - \mathbf{x}_{2} ) c^{\dagger}_{\iota\sigma_{1}}(\mathbf{x}_{1})
  c^{\dagger}_{\upsilon\sigma_{2}}(\mathbf{x}_{2}) \\ \times
  c^{\phantom{\dagger}}_{\kappa\sigma_{2}}(\mathbf{x}_{2})
  c^{\phantom{\dagger}}_{\lambda\sigma_{1}}(\mathbf{x}_{1})
\end{multline}
where all possible inter-orbital transitions are
included. Orbital-momentum conservation poses a restriction on the
selection of orbital indices of the interaction matrix being $\iota +
\upsilon = \kappa + \lambda$.  We use a tight-binding approximation
taking the coordinate $\mathbf{x}$ from the lattice space. We will
assume the interaction to be local in the lattice space, that is,
Hubbard-like. A graphical representation of the interaction matrix is
plotted in Fig.~\ref{fig:Int-general}, where we used latin indices for
all the dynamical degrees of freedom characterizing the electron
states. They are generally four-momenta or Matsubara frequencies in
local approximations (dynamical mean-field theory) together with
orbital indices. We use this abbreviated notation to simplify
summations over intermediate states in the multi-orbital perturbation
expansion.

\begin{figure}
  \includegraphics{U_ud_multorb.eps}
  \caption{General electron-electron interaction in the notation we
    use in this paper. Index $\sigma$ stands for spin, while $i,j,k,l$
    comprise all other dynamical degrees of freedom characterizing the
    electron state. Conservation law in the dynamical variables $i + j
    = k + l$ holds.}\label{fig:Int-general}
\end{figure}

It is necessary to classify all possible interconnections of
interaction vertices via fermion lines for building up the complete
multi-orbital perturbation expansion. It is not an easy task and we
will sum only selected classes of two-particle diagrams. We use the
graphical representation of the interaction vertex from
Fig.~\ref{fig:Int-general} and choose the upper part of it so that
spin (not necessarily orbital index) is conserved along the horizontal
fermion lines. We choose electron propagation from left to right and
hole propagation from right to left.  We have three possibilities how
to connect two interaction vertices by two fermionic lines. They
correspond to three two-particle scattering channels. \cite{Janis99b}

The first possibility to connect interaction vertices is via
propagation of two electrons, that is, by parallel fermionic lines,
cf. Fig.~\ref{fig:ee}. Mathematically this process can be represented
via a matrix multiplication
\begin{equation}\label{eq:EE-Channel}
  \left( UGG\star U\right)^{li,\sigma_{1}}_{kj,\sigma_{2}} = \sum_{i'} U^{li',\sigma_{1}}_{kj',\sigma_{2}} G_{i'\sigma_{1}}G_{j'\sigma_{2}} U^{i'i,\sigma_{1}}_{j'j,\sigma_{2}}
\end{equation}
where the latin indices stand for all dynamical variables of the
electron (momentum, frequency and orbital index). They obey a
conservation law $j'= k + l - i'$.

Second connection of interaction vertices is via propagation of an
electron-hole pair, that is, by antiparallel fermionic lines. Here we
must distinguish two options: electron-hole scatterings and
polarization bubbles. The former process is plotted in
Fig.~\ref{fig:eh} and is represented via another matrix multiplication
\begin{equation}\label{eq:EH-Channel}
  \left( UGG\bullet U\right)^{li,\sigma_{1}}_{kj,\sigma_{2}} =
  \sum_{i'} U^{li',\sigma_{1}}_{k'k,\sigma_{2}}
  G_{i'\sigma_{1}}G_{j'\sigma_{2}} U^{i'i,\sigma_{1}}_{jk',\sigma_{2}}
\end{equation}
where again the orbital-monentum conservation law $k'= i'+ k - l$
holds.

A polarization bubble, shielding the interaction, is represented via a
matrix multiplication
\begin{equation}\label{eq:V-Channel}
  \left( UGG\circ U\right)^{li,\sigma_{1}}_{kj,\sigma_{2}} =
  \sum_{\sigma'}\sum_{j'} U^{li,\sigma_{1}}_{k'j',\sigma'}
  G_{k'\sigma'}G_{j'\sigma'} U^{k'j',\sigma'}_{kj,\sigma_{2}}
\end{equation}
and is plotted in Fig.~\ref{fig:v}.  Notice that spin in this last
process is not conserved, but orbital-momentum conservation restricts
the intermediate indices of the dynamical variables to $k'= j'+ i -
l$.

\begin{figure}
  \includegraphics[width=8 cm]{channel_ee.eps}
  \caption{Connecting interactions via electron-electron
    propagation.}\label{fig:ee}
\end{figure}
\begin{figure}
  \includegraphics[width=8 cm]{channel_eh1.eps}
  \caption{Connecting interactions via electron-hole
    propagation.}\label{fig:eh}
\end{figure}
\begin{figure}
  \includegraphics[width=8 cm]{channel_eh2.eps}
  \caption{Connecting interactions via electron-hole triplet
    polarization bubbles (vertical channel).}\label{fig:v}
\end{figure}

We use these elementary scattering processes to construct the full
two-particle vertex $\Gamma$ from a two-particle perturbation
expansion. Once we have this vertex we construct the one-electron
self-energy as the fundamental quantity from which we derive all
thermodynamic properties. The self-energy is calculated from the
two-particle vertex via the Schwinger-Dyson equation, diagrammatically
represented in Fig.~\ref{fig:SDE}. Mathematically it reads
\begin{multline}\label{eq:SDE-general}
  \Sigma_{i\sigma} =
  \sum_{\sigma'}\sum_{j}U^{jj,\sigma'}_{ii,\sigma}G_{j\sigma'} -
  \sum_{j}U^{ji,\sigma}_{ij,\sigma} G_{j\sigma} \\
  - \sum_{\sigma'}\sum_{j',k'}
  U^{k'l',\sigma'}_{ij',\sigma}G_{j'\sigma} G_{k'\sigma'}
  G_{l',\sigma'} \Gamma^{l'k',\sigma'}_{j'i,\sigma} \ .
\end{multline}
Again the conservation law for intermediate variables $l'= i + k'- j'$
holds. The one-electron propagators in the perturbation expansion can
either be chosen as fully dynamically renormalized as suggested by
Baym and Kadanoff,\cite{Baym62} or only statically renormalized via
the Hartree approximation securing that the particle densities do not
change during the summation of diagrams. We discussed earlier that the
latter option is more appropriate in the strong-coupling regime, since
we can better control the two-particle singularity in the
electron-hole correlation function and consequently both low and high
energy scales are qualitatively better reproduced than in approximate
schemes with the fully renormalized propagators.\cite{Janis07a}

\begin{figure}
  \includegraphics[width=8 cm]{schwinger_dyson.eps}
  \caption{Schwinger-Dyson equation connecting the self-energy
    $\Sigma$ with vertex $\Gamma$. The first and the second term on
    the right-hand side of this equation are the Hartree and the Fock
    contribution.}\label{fig:SDE}
\end{figure}

\section{Simplified parquet equations: matrix
  formulation} \label{sec:Simplified_MPE}

The generic function for our perturbation expansion is the
two-particle vertex $\Gamma$. Its approximate expression together with
the Schwinger-Dyson equation \eqref{eq:SDE-general} define our
approximate scheme. We hence start with simple approximations on
vertex $\Gamma$. We resort our analysis only to local approximations,
that is, we stay within the dynamical mean-field theory with only
local fluctuations and Matsubara frequencies, orbital index and spin
as dynamical variables. We will treat these degrees of freedom
separately.

The simplest approximations for vertex $\Gamma$, beyond the bare
interaction, are multiple scatterings of the same form as defined by
elementary processes from the preceding section. Sums of all multiple
scatterings of the same type are solutions to Bethe-Salpeter
equations. They can be represented with the appropriate matrix
multiplications. For electron-hole and electron-electron multiple
scatterings we obtain
\begin{align}\label{eq:BS_bare-eh}
  \widehat{\Phi}(i\nu_{m}) = \widehat{U} - \widehat{U \chi}(i\nu_{m})
  \bullet \widehat{\Phi}(i\nu_{m})\ , \\ \label{eq:BS_bare-ee}
  \widehat{\Psi}(i\nu_{m}) = \widehat{U} - \widehat{U \psi}(i\nu_{m})
  \star \widehat{\Psi}(i\nu_{m}) \ ,
\end{align}
respectively. The kernels of these equations are composed from the
bare interaction and either the electron-hole bubble or the
two-electron propagator. The kernel in the electron-hole channel
explicitly reads
\begin{equation}\label{eq:Uchi}
  \left(U\chi\right)^{\lambda\iota,\sigma_{1}}_{\kappa\upsilon\sigma_{2}}
  = U^{\lambda\iota,\sigma_{1}}_{\kappa\upsilon,\sigma_{2}}
  \chi^{\iota\sigma_{1}}_{\kappa\sigma_{2}}(i\nu_{m}) \\
\end{equation}
with the electron-hole bubble
\begin{align} \label{eq:chi}
  \chi^{\iota\sigma_{1}}_{\kappa\sigma_{2}}(i\nu_{m}) &= \frac 1\beta
  \sum_{n}G_{\iota\sigma_{1}}(i\nu_{m} + i\omega_{n})
  G_{\kappa\sigma_{2}}(i\omega_{n}) \ .
\end{align}
Analogously we construct the kernel of the Bethe-Salpeter equation
\eqref{eq:BS_bare-ee} with multiple electron-electron scatterings. The
two-electron propagator then is
\begin{align}\label{eq:psi}
  \psi^{\iota\sigma_{1}}_{\upsilon\sigma_{2}}(i\nu^{\prime}_{m}) &=
  \frac 1\beta \sum_{n}G_{\iota\sigma_{1}}( i\nu^{\prime}_{m} -
  i\omega_{n}) G_{\upsilon\sigma_{2}}(i\omega_{n})\ .
\end{align}
We denote fermionic and bosonic Matsubara frequencies $\omega_{n}= (2n
+ 1)\pi T$ and $\nu_{m}=2m\pi T$, respectively (Boltzmann constant
$k_{B} =1$). Notice that the transfer frequency $\nu_{m}$, conserved
in electron-hole scatterings, is different from that conserved in the
electron-electron scattering channel, $\nu^{\prime}_{m}$. If the
incoming and outgoing frequencies of one electron are $\omega_{n}$ and
$\omega_{n'}$, respectively, we have $\nu^{\prime}_{m} = \nu_{m}+
\omega_{n} - \omega_{n'}$.

We do not take into account the third channel with triplet
polarization bubbles. There are two reasons why we can afford to do
this. First, the effects of the scatterings in the vertical channel
are similar to those from the electron-hole channel, at least
qualitatively. We hence do not lose track of any relevant effect in
the strong-coupling regime. Second, taking into account only two
scattering channels simplifies significantly approximate schemes for
the two-particle vertex.\cite{Janis09}

From the solutions of the Bethe-Salpeter equations for
$\widehat{\Phi}$ and $\widehat{\Psi}$ we obtain the full two-particle
vertex
\begin{multline}\label{eq:parguet-bare}
  \widehat{\Gamma}(i\nu_{m},i\nu_{m}+ i\omega_{n}- i\omega_{n'})\\ =
  \widehat{\Phi}(i\nu_{m}) + \widehat{\Psi}(i\nu_{m}+ i\omega_{n}-
  i\omega_{n'}) - \widehat{U}\ .
\end{multline}
Such a vertex defines the self-energy of the so-called FLEX
approximation if the one-electron propagators in the Bethe-Salpeter
equations \eqref{eq:BS_bare-eh} and \eqref{eq:BS_bare-ee} are fully
renormalized, that is, they contain the self-energy determined by the
approximate vertex $\Gamma$ from Eq.~\eqref{eq:parguet-bare}. Such an
approximation is known to be unreliable in the strong-coupling
regime.\cite{Janis99b}

Intermediate electron coupling is marked by proximity of a pole in the
electron-hole vertex $\Phi$.  If the transfer frequency $\nu=0$ then
we can define a small dimensionless scale
\begin{equation}
  a = 1+{\rm min}\ \text{Sp}\left[\widehat{U^{\sigma_{1}}_{\sigma_2} \chi^{\sigma_{1}}_{\sigma_2}}\right]
\end{equation}
approaching zero in the random-phase approximation with $\Gamma =
\Phi$, a solution of Eq.~\eqref{eq:BS_bare-eh}, when $U\to U_{c}$. We
used $\text{Sp}\left[ \widehat{A}\right]$ to denote the set of
eigenvalues of matrix $\widehat{A}$. This pole is unphysical and
indicates that the perturbation expansion for $U\ge U_{c}$ must be
renormalized. The critical interaction $U_{c}$ from RPA defines an
upper limit on the weak-coupling regime where perturbation theory with
selected classes of generic diagrams can safely be applied. We speak
about strong coupling if $U > U_{c}$, where only self-consistent
approximations may be reliable.

One, and most often used option how to suppress the unphysical pole,
or better to shift it to infinite interaction strength is to employ
one-particle dynamical self-consistency.  We use the self-energy
calculated from the Schwinger-Dyson equation \eqref{eq:SDE-general}
with vertex $\Gamma$ from Eq.~\eqref{eq:parguet-bare} to renormalize
the one-electron propagators in Bethe-Salpeter equations
\eqref{eq:BS_bare-eh} and \eqref{eq:BS_bare-ee}. This one-particle
self-consistency succeeds in shifting the unphysical pole to infinity,
but the Kondo regime defined as $a\to 0$ is not properly
reproduced. Moreover, FLEX-type approximations completely smear out
the large-frequency structures. There are no satellite Hubbard
bands.\cite{Janis99b}

A better systematic alternative how to move the RPA critical
interaction $U_{c}$ to infinity is to introduce a two-particle
self-consistency. The most straightforward way to do so is to use the
parquet approach. Its basic idea is to replace the bare interaction in
Bethe-Salpeter equations \eqref{eq:BS_bare-eh} and
\eqref{eq:BS_bare-ee} by irreducible vertex functions $\Lambda^{eh}$
and $\Lambda^{ee}$, respectively. In the exact theory then
$\Phi\left[\Lambda^{eh}\right] = \Psi\left[\Lambda^{ee}\right] =
\Gamma$ and equation~\eqref{eq:parguet-bare} transforms to $\Gamma + U
= \Lambda^{eh} + \Lambda^{ee}$. When we combine the two Bethe-Salpeter
equations with the parquet equation~\eqref{eq:parguet-bare} modified
as above we obtain a set of non-linear integral equations for the
irreducible vertices $\Lambda^{eh}\left[U, G\right],
\Lambda^{ee}\left[U, G\right]$ and finally also for the full vertex
$\Gamma\left[U, G\right]$. It is, however, impossible to solve the
parquet equations in the strong-coupling regime with a vanishing Kondo
scale $a\to 0$. The problem lies in that all the vertex functions,
solutions of the full parquet equations, generally depend on three
frequencies and there is presently no method, either analytic or
numerical, that would predict the full analytic structure of the
two-particle vertices.  We hence have to approximate the full parquet
equations.

We proposed a simplification of the parquet equations in the Kondo
regime of the single-impurity Anderson model.\cite{Janis07a,Janis08}
This simplification utilizes a partial separation of large and small
frequency fluctuations in the Kondo regime when the solution of the
Bethe-Salpeter equation in the electron-hole channel is almost
singular.  In the non-renormalized perturbation expansion it is the
RPA pole. The principal idea of our simplification is that we separate
large fluctuations diverging at the critical point from those
remaining finite. In this way we describe correctly universal
quantities connected with the critical point in the electron-hole
Bethe-Salpeter equation. We neglect all finite fluctuations and keep
only the relevant ones diverging at the critical point. We then
replace all finite frequency-dependent functions by constants and keep
only the frequency dependence in the variables controlling large
fluctuations. This simplification is kind of a mean-field (simplest)
approximation for the dynamics in the strong-coupling regime.

We know from the analysis of multiple electron-electron and
electron-hole scatterings that only the latter can cause a singularity
in the vertex function. It means that the irreducible vertex in the
electron-hole channel $\Lambda^{eh}$ remains finite. We hence can
replace it by an effective interaction that we denote
$\overline{U}$. The renormalized Bethe-Salpeter equation in the
electron-hole channel is RPA-like with the following replacement
\begin{equation}\label{eq:Effective-interaction}
  \widehat{U} \longrightarrow \widehat{\overline{U}}, \qquad
  \widehat{\Phi}(i\nu_{m}) \longrightarrow\
  \widehat{\overline{\Phi}}(i\nu_{m})\ .
\end{equation}
The effective interaction $\overline{U}$ will be determined from the
Bethe-Salpeter equation in the electron-electron channel by using the
parquet self-consistency. Within the scope of our approximation the
Bethe-Salpeter equation in the electron-electron scattering channel
holds only in an averaged form smearing out finite frequency
differences.  We proposed in Ref.~\onlinecite{Janis08} a decoupling of
the Bethe-Salpeter equation in the electron-electron scattering
channel so that an integral equation is converted to an algebraic,
solvable one. In this approximate simplification an averaged integral
kernel is assumed to diagonalize the integral equation. Such a
decoupling is not uniquely defined and should be chosen so that the
effective interaction is a real self-adjoint (positive definite)
matrix.  An equation for the effective interaction derived from a
decoupling with the right-averaged integral kernel used in the
single-orbital case reads
\begin{equation}\label{eq:EI-right}
  \widehat{\overline{U}\psi} = \widehat{U\psi} -
  \widehat{L}\star\left[\widehat{1} +
    \widehat{L}\star\right]^{-1}\widehat{L}
\end{equation}
where we denoted
\begin{multline}\label{eq:L-function}
  L^{\lambda\iota,\sigma_{1}}_{\kappa\upsilon,\sigma_{2}}(i\omega_{n})
  \\ = \frac 1{\beta} \sum_{n'}
  \Lambda^{\lambda\iota,\sigma_{1}}_{\kappa\upsilon,\sigma_2}(i\nu_{-n-n'})
  G_{\iota\sigma_1}(i\omega_{n'}) G_{\upsilon\sigma_2}(i\omega_{-n'})\
  .
\end{multline}
This one-sided averaging in the multi-orbital case can lead to a
matrix of the effective interaction that is not self-adjoint. To avoid
this spurious behavior we introduce also a left-averaged integral
kernel in the electron-electron Bethe-Salpeter equation. We then
define another equation for the effective interaction
\begin{equation}\label{eq:EI-left}
  \widehat{\psi \overline{U}} =
  \ \widehat{\psi U}   - \widehat{\mathcal{L}}\star\left[\widehat{1} +
    \widehat{\mathcal{L}}\star\right]^{-1}\widehat{\mathcal{L}} \ ,
\end{equation}
with a left-averaged integral kernel
\begin{multline}\label{eq:calL-function}
  {\mathcal{L}}^{\lambda\iota,\sigma_{1}}_{\kappa\upsilon,\sigma_{2}}(i\omega_{n})
  \\ = \frac 1{\beta} \sum_{n'} G_{\lambda\sigma_1}(i\omega_{n'})
  G_{\kappa\sigma_2}(i\omega_{-n'})
  \Lambda^{\lambda\iota,\sigma_{1}}_{\kappa\upsilon,\sigma_2}(i\nu_{-n-n'})
  \ .
\end{multline}
We demand that the matrix of the effective interaction
$\widehat{\overline{U}}$ be self-adjoint. It is always the case if we
use a sum of equations \eqref{eq:EI-right} and \eqref{eq:EI-left} to
determine the effective interaction.  That is, we choose an equation
symmetric with respect to right and left averaging of the integral
kernel of the Bethe-Salpeter equation in the nonsingular
electron-electron channel.

The dominant contribution to the integral kernel in the
electron-electron scattering channel comes from the singular part of
the vertex from the electron-hole scattering channel, being
$\widehat{\Lambda} = \widehat{\Phi} - \widehat{U}$. Since the averaged
integral kernels $L$ and $\mathcal{L}$ depend on a fermionic Matsubara
frequency, we assume that Eqs.~\eqref{eq:EI-right}, \eqref{eq:EI-left}
are valid only for the lowest frequency, being zero at zero
temperature. At finite temperatures we use the lowest or a few low
frequencies, symmetric around zero, and add all equations for these
frequencies. The effective interaction is then determined form an
average over Eq.~\eqref{eq:EI-right} and Eq.~\eqref{eq:EI-left} for a
few lowest-lying fermionic frequencies. The low-temperature
asymptotics is in this way well reproduced. The higher the temperature
the more Matsubara frequencies we must take into account to maintain
accuracy of the zero-temperature solution. We know, however, that
thermal fluctuation suppress the Kondo resonance and smear out the RPA
singularity in the electron-hole correlation function.\cite{Janis08}

Equations \eqref{eq:BS_bare-eh},
\eqref{eq:EI-right}-\eqref{eq:calL-function} are our simplified
parquet equations determining the two-particle vertices
$\overline{\Phi}$ and $\overline{U}$. The full two-particle vertex
$\Gamma$ is determined from equation~\eqref{eq:parguet-bare}. Within
our approximation it reduces to the renormalized RPA vertex
$\overline{\Phi}$.  We use it to determine the self-energy. The
respective Schwinger-Dyson equation reads
\begin{widetext}
  \begin{multline}\label{eq:SDE-SPE}
    \Sigma_{\iota\sigma}(i\omega_{n}) = \frac 1\beta
    \sum_{m}\sum_{\sigma'\upsilon}U^{\upsilon\upsilon,\sigma'}_{\iota\iota,\sigma}G_{\upsilon\sigma'}(i\omega_{n
      + m}) - \frac 1\beta
    \sum_{m}\sum_{\upsilon}U^{\upsilon\iota,\sigma}_{\iota\upsilon,\sigma}
    G_{\upsilon\sigma}(i\omega_{n + m})  \\
    - \frac 1{\beta^{2}}
    \sum_{n',m}\sum_{\sigma'}\sum_{\iota',\kappa'}
    U^{\kappa'\lambda',\sigma'}_{\iota\iota',\sigma}G_{\iota'\sigma}(i\omega_{n+m})
    G_{\kappa'\sigma'}(i\omega_{n'})
    G_{\lambda',\sigma'}(i\omega_{n'-m})
    \overline{\Phi}^{\lambda'\kappa',\sigma'}_{\iota'\iota,\sigma}(i\nu_{m})\
    ,
  \end{multline}
\end{widetext}
where conservation of the orbital index $\lambda'= \iota + \kappa'-
\upsilon'$ holds.

The approximation is almost complete. The last thing we have to do is
to specify the one-electron propagators in the parquet equations for
the two-particle vertices. We know that the full one-particle
self-consistency where the one-electron propagators are renormalized
by the self-energy from Eq.~\eqref{eq:SDE-SPE} does not lead to the
Kondo scale as $U\to\infty$ and also smears out the satellite Hubbard
bands.\cite{Janis08} We hence use the Hartree one-electron propagators
that in the dynamical mean-field theory is
\begin{multline}\label{eq:1GF-general}
  G_{\iota\sigma}(i\omega_{n}) \\ = \int\frac {d\epsilon
    \rho_{\iota}(\epsilon)} {i\omega_{n} + \mu_{\iota} + \sigma h -
    \epsilon - \frac 12 \sum_{\lambda,\sigma'}
    U^{\lambda\lambda,\sigma'}_{\iota\iota,\sigma}n_{\lambda\sigma'}}
\end{multline}
and for the impurity Anderson model
\begin{multline}\label{eq:1GF-SIAM}
  G_{\iota\sigma}(i\omega_{n}) \\ = \frac 1{i\omega_{n} + \mu_{\iota}
    + \sigma h - \Delta(i\omega_{n}) - \frac 12 \sum_{\lambda,\sigma'}
    U^{\lambda\lambda,\sigma'}_{\iota\iota,\sigma}n_{\lambda\sigma'}}
  \ .
\end{multline}
We denoted $\mu_{\iota} = \mu - E_{\iota}$, where $E_{\iota}$ is the
center of the $\iota$'s orbital and $\mu$ is the chemical potential
fixing the total charge density. The Hartree propagators are
statically self-consistent, that is, the particle densities
$n_{\lambda \sigma}$ are determined from the fully renormalized
propagator
\begin{equation}\label{eq:1P-density}
  n_{\lambda\sigma} = \frac 1\beta
  \sum_{n}G_{\lambda\sigma}\left(i\omega_{n} -
    \Delta\Sigma_{\lambda\sigma}(i\omega_{n})\right)\ .
\end{equation}
We denoted $\Delta\Sigma$ the dynamical part of the self-energy, a
correction to the Hartree term. The chemical potential is then
adjusted so that the total particle density is fixed. The approximate
scheme based on the simplified parquet equations is now complete. We
demonstrate explicitly applicability of this approximation on a
two-orbital model in the next sections.

\section{Two-orbital model}\label{sec:Two_orbital}

\begin{widetext}

  \begin{figure}
    \includegraphics{int_U.eps}
    \caption{Different elementary scattering processes in the
      two-orbital model. Each of these processes is renormalized
      independently.}\label{fig:Mult_int}
  \end{figure}

\end{widetext}

We use a general two-orbital interacting Hamiltonian comprising also
Hund's exchange coupling\cite{Imada98}
\begin{multline}\label{eq:Hamiltonian-2O}
  H_I = U \sum_\iota n_{\iota\uparrow} n_{\iota\downarrow} +
  \sum_{\sigma\sigma'} \left[U_{1}\eta_{\sigma\sigma'} +
    U_{2}\delta_{\sigma\sigma'}\right] n_{1\sigma} n_{2\sigma'}\\
  -
  \left(J_{1}c^{\dagger}_{2\uparrow}c^{\dagger}_{1\downarrow}c^{\phantom{\dagger}}_{2\downarrow}c^{\phantom{\dagger}}_{1\uparrow}
    +
    J_{2}c^{\dagger}_{2\uparrow}c^{\dagger}_{2\downarrow}c^{\phantom{\dagger}}_{1\downarrow}
    c^{\phantom{\dagger}}_{1\uparrow}+H.c.\right)
\end{multline}
where $\eta_{\sigma\sigma'} + \delta_{\sigma\sigma'} =1$. A graphical
representation of single terms contributing to this two-orbital
Hamiltonian is presented in Fig.~\ref{fig:Mult_int}. Each input
parameter of this Hamiltonian will be renormalized independently.

Only one of the bare interacting terms denoted $U_{2}$ is spin
triplet.  This term does not mix with the other scattering processes
in the Hamiltonian from Eq.~\eqref{eq:Hamiltonian-2O}. The simplified
parquet equations for this term are identical with those from the
single-orbital case.  The effective interaction
$\overline{U}^{\sigma}_{2}$ is
\begin{equation}\label{eq:U2-bar}
  \overline{U}^{\sigma}_{2} = U_{2} -
  \frac{\left(L^{11,\sigma}_{22,\sigma}\right)^2}{\left(1+L^{11,\sigma}_{22,\sigma}\right)\psi^{1\sigma}_{2\sigma}}
\end{equation}
with the averaged integral kernel from the Bethe-Salpeter equation in
the electron-electron channel
\begin{multline}\label{eq:L_upup}
  L^{11,\sigma}_{22,\sigma}(i\omega_n) \\ = \frac 1{\beta}
  \sum_{n'}G_{1\sigma}(i\omega_{n'}) G_{2\sigma}(i\omega_{-n'})
  \Lambda^{11,\sigma}_{22,\sigma}(i\nu_{-n -n'})\ .
\end{multline}
The dynamical part of the (reducible) vertex from the electron-hole
scattering channel is
\begin{equation}\label{eq:Lambda_upup}
  \Lambda^{11,\sigma}_{22,\sigma}(i\nu_{m}) = -
  \frac{\left(\overline{U}^{\sigma}_{2}\right)^{2}
    \chi^{1\sigma}_{2\sigma}(i\nu_{m})}{1+\overline{U}_{2}
    \chi^{1\sigma}_{2\sigma}(i\nu_{m})}\ .
\end{equation}

The other terms in the interaction Hamiltonian from
Eq.~\eqref{eq:Hamiltonian-2O} mix in the parquet equations and must be
treated simultaneously. Since only specific inter-orbital processes
are allowed we can simplify the representation of the interaction term
from the preceding section and reduce super-matrices (tensors) with
four indices to regular matrices with two indices.  A relation between
matrix indices and super-indices can be chosen as follows
$1\equiv(1,1)$, $2\equiv(2,2)$, $3\equiv(1,2)$, $4\equiv(2,1)$. If we
go over from super-matrices to matrices we have to distinguish two
representations corresponding to two matrix multiplications used in
constructing the Bethe-Salpeter equations. The interaction matrix
suitable for summation of multiple electron-electron scatterings is
\begin{equation}

  \widehat{U}^{ee} =\left(
    \begin{array}{cccc}
      U & J_2 & 0& 0 \\
      J_2 & U & 0 &0 \\
      0 & 0 & U_1 & J_1\\
      0 & 0 & J_1 & U_1
    \end{array}
  \right)\ .

\end{equation}
The interaction matrix for the electron-hole multiple scatterings
interchanges the two Hund couplings $J_{1}$ and $J_{2}$. Hence we have
\begin{equation}

  \widehat{U}^{eh} = \left(
    \begin{array}{cccc}
      U & J_1 & 0 & 0 \\
      J_1 & U & 0 & 0 \\
      0 & 0 & U_1 & J_2\\
      0 & 0 & J_2 & U_1
    \end{array}
  \right)\ .

\end{equation}
Each matrix representation is chosen so that the scattering processes
in the respective channel are simple matrix multiplications.

The matrix integral kernel for the Bethe-Salpeter equation in the
electron-hole channel is
\begin{equation}
  \widehat{K} \equiv \widehat{\overline{U} \chi} =
  \left(
    \begin{array}{cccc}
      \overline{U}_{01} \chi^{1}_{1} & \overline{J}_1 \chi^{2}_{2} & 0 & 0 \\
      \overline{J}_1 \chi^{1}_{1} & \overline{U}_{02} \chi^{2}_{2} & 0  & 0 \\
      0 & 0 & \overline{U}_1\chi^{1}_{2}  & \overline{J}_2 \chi^{2}_{1}\\
      0 & 0 & \overline{J}_2 \chi^{1}_{2} & \overline{U}_1 \chi^{2}_{1}
    \end{array}
  \right),
\end{equation}
while the right-averaged kernel in the Bethe-Salpeter equation in the
electron-electron channel reads
\begin{equation}
  \widehat{L}^{ee} =
  \left(
    \begin{array}{cccc}
      L^{11}_{11} &L^{12}_{12}  & 0 & 0 \\[1pt]
      L^{21}_{21} & L^{22}_{22} & 0 & 0\\
      0 & 0 & L^{11}_{22} & L^{12}_{21}\\[1pt]
      0 & 0 &  L^{21}_{12} & L^{22}_{11}
    \end{array}
  \right)\ ,
\end{equation}
with
\begin{multline}
  L^{\lambda\iota,\sigma_{1}}_{\kappa\upsilon,\sigma_{2}}(i\omega_n)\\
  = \frac 1{\beta} \sum_{n'}
  \Lambda^{\lambda\iota,\sigma_{1}}_{\kappa\upsilon,\sigma_{2}}(i\nu_{-n
    -n'}) G_{\iota\sigma_{1}}(i\omega_{n'})
  G_{\upsilon\sigma_{2}}(i\omega_{-n'})
\end{multline}
and $\widehat{\Lambda} = \widehat{\overline{\Phi}} -
\widehat{\overline{U}}$. Analogously we construct the left-averaged
kernel $\mathcal{L}$, if necessary. The matrix form of the simplified
parquet equations then is in the electron-hole channel
\begin{align}\label{eq:SPE_2O-EH}
  \widehat{\overline{\Phi}}& = \left[\widehat{1}+
    \widehat{K}\bullet\right]^{-1} \widehat{\overline{U}}^{eh}\
\end{align}
and in the electron-electron channel with right and left-averaged
integral kernels
\begin{multline}\label{eq:SPE_2O-EE}
  \widehat{\overline{U}\psi} + \widehat{\psi\overline{U}} =
  \widehat{U\psi}
  - \widehat{L}^{ee}\star\left[\widehat{1} + \widehat{L}^{ee}\star\right]^{-1}\widehat{L}^{ee} \\
  +\ \widehat{\psi U} -
  \widehat{\mathcal{L}}^{ee}\star\left[\widehat{1} +
    \widehat{\mathcal{L}}^{ee}\star\right]^{-1}\widehat{\mathcal{L}}^{ee}
  \ .
\end{multline}

All matrices appearing in the parquet equations~\eqref{eq:SPE_2O-EH}
and \eqref{eq:SPE_2O-EE} are block diagonal. We denote
\begin{align}
  \overline{\Phi} = \left(
    \begin{array}{cc}
      \overline{\Phi}_{\rm upp}  &  0\\
      0 &  \overline{\Phi}_{\rm low}
    \end{array}
  \right) \ , & \quad \overline{U} = \left(
    \begin{array}{cc}
      \overline{U}_{\rm upp}  & 0 \\
      0   &  \overline{U}_{\rm low}
    \end{array}
  \right)
\end{align}
and solve the parquet equations for each block separately. The
solution in the electron-hole scattering channel has the following
representation
\begin{widetext}
  \begin{align}
    \overline{\Phi}_{\rm upp} &= \frac{1}{1^{\phantom{\big(}}+
      \overline{U}_{01}\chi^{1}_{1} + \overline{U}_{02}\chi^{2}_{2} +
      \left(\overline{U}_{01}\overline{U}_{02} -
        \overline{J}_{1}^{2}\right)\chi^{1}_{1} \chi^{2}_{2}
    }
    \left(
      \begin{array}{cc}
        \overline{U}_{01}  + \left( \overline{U}_{01}\overline{U}_{02}
          - \overline{J}_{1}^{2}\right)\chi^{2}_{2} \quad&
        \overline{J}_{1} \\
        \overline{J}_{1}\quad & \overline{U}_{02} + \left(
          \overline{U}_{01}\overline{U}_{02} -
          \overline{J}_{1}^{2}\right) \chi^{1}_{1}
      \end{array}
    \right)\
    \label{Phi_EH_uppeL^{ee}_block_Ubar}
  \end{align}
  and
  \begin{align}
    \overline{\Phi}_{\rm low} &= \frac{1}{1^{\phantom{\big(}}+
      \overline{U}_{1}\left(\chi^{1}_{2} + \chi^{2}_{1}\right) +
      \left(\overline{U}_{1}^{2} -
        \overline{J}_{2}^{2}\right)\chi^{1}_{2} \chi^{2}_{1}
    }
    \left(
      \begin{array}{cc}
        \overline{U}_{1}  + \left( \overline{U}_{1}^{2} - \overline{J}_{2}^{2}\right)\chi^{2}_{1} \quad& \overline{J}_{2} \\
        \overline{J}_{2}\quad & \overline{U}_{1} + \left( \overline{U}_{1}^{2} - \overline{J}_{2}^{2}\right) \chi^{1}_{2}
      \end{array}
    \right)\ .
    \label{Phi_EH_loweL^{ee}_block_Ubar}
  \end{align}
  Notice that the Hubbard interaction $U$ splits into two renormalized
  values $\overline{U}_{01}$ and $\overline{U}_{02}$ if the bubbles in
  different orbitals are different, $\chi^{1}_{1} \neq
  \chi^{2}_{2}$. Note that we suppressed the frequency index at the
  electron-hole bubble leading to a frequency dependence of vertex
  $\widehat{\overline{\Phi}}$.

  We now choose the singular part of the renormalized electron-hole
  vertex $\widehat{\Lambda} = \widehat{\overline{\Phi}} -
  \widehat{\overline{U}}$ and use it to obtain a solution of the
  Bethe-Salpeter equation in the electron-electron channel with an
  averaged (right, left) kernel that determines the matrix of the
  effective (renormalized) interaction. Its upper matrix block reads
  \begin{multline}\label{eq:EI-O2-upp}
    \widehat{\overline{U}_{upp}\psi} \equiv \left(
      \begin{array}{cc}
        \overline{U}_{01} \psi^{1}_{1} & \overline{J}_{2}\psi^{2} _{2}\\
        \overline{J}_{2}\psi^{1}_{1} & \overline{U}_{02} \psi^{2}_{2}
      \end{array}
    \right) = \left(
      \begin{array}{cc}
        U  \psi^{1}_{1} & J_2  \psi^{2}_{2}\\[1pt]
        J_2  \psi^{1}_{1} & U  \psi^{2}_{2}
      \end{array}
    \right)  - \frac{1}{(1 + L^{ee}_{11})(1 + L^{ee}_{22}) - L^{ee}_{12} L^{ee}_{21}} \\
    \times \left(
      \begin{array}{cc}
        \left(1 + L^{ee}_{22}\right)\left( L^{ee}_{11}\right)^{2} +
        L^{ee}_{12} L^{ee}_{21} \left( 1 - L^{ee}_{11}\right) \quad &
        L^{ee}_{12}\left( L^{ee}_{11} + L^{ee}_{22} +  L^{ee}_{11} L^{ee}_{22} - L^{ee}_{12} L^{ee}_{21}\right)
        \\[1pt]
        L^{ee}_{21}\left( L^{ee}_{11} + L^{ee}_{22} + L^{ee}_{11} L^{ee}_{22} - L^{ee}_{12} L^{ee}_{21}\right)    \quad &
        \left(1 + L^{ee}_{11}\right) \left(L^{ee}_{22}\right)^{2} + L^{ee}_{21} L^{ee}_{12} \left( 1 - L^{ee}_{22}\right)
      \end{array}
    \right) \ .

  \end{multline}
  The lower block analogously is
  \begin{multline}\label{eq:EI-O2-low}
    \widehat{\overline{U}_{low}\psi} \equiv \left(
      \begin{array}{cc}
        \overline{U}_{1} \psi^{1}_{2} & \overline{J}_{1}\psi^{2}_{1}\\
        \overline{J}_{1}\psi^{1}_{2} & \overline{U}_{1} \psi^{2}_{1}
      \end{array}
    \right) = \left(
      \begin{array}{cc}
        U_{1}  \psi^{1}_{2} & J_1  \psi^{2}_{1}\\[1pt]
        J_1  \psi^{1}_{2} & U_{1}  \psi^{2}_{1}
      \end{array}
    \right)  - \frac{1}{(1 + L^{ee}_{33})(1 + L^{ee}_{44}) - L^{ee}_{34} L^{ee}_{43}} \\
    \times \left(
      \begin{array}{cc}
        \left(1 + L^{ee}_{44}\right)\left( L^{ee}_{33}\right)^{2} +
        L^{ee}_{34} L^{ee}_{43} \left( 1 - L^{ee}_{33}\right) \quad &
        L^{ee}_{34}\left( L^{ee}_{33} + L^{ee}_{44} +  L^{ee}_{33} L^{ee}_{44} - L^{ee}_{34} L^{ee}_{43}\right)
        \\[1pt]
        L^{ee}_{43}\left( L^{ee}_{33} + L^{ee}_{44} + L^{ee}_{33} L^{ee}_{44} - L^{ee}_{34} L^{ee}_{43}\right)    \quad &
        \left(1 + L^{ee}_{33}\right) \left(L^{ee}_{44}\right)^{2} + L^{ee}_{43} L^{ee}_{34} \left( 1 - L^{ee}_{44}\right)
      \end{array}
    \right) \ .
  \end{multline}
\end{widetext}
The electron-electron bubble $\psi$ and vertices $L^{ee}$ are taken
here (zero temperature) only at zero frequency.  If $\psi^{1}_{1} \neq
\psi^{2}_{2}$ we must symmetrize Eqs.~\eqref{eq:EI-O2-upp} and
\eqref{eq:EI-O2-low} by adding the Bethe-Salpeter equation with the
left averaged kernel $\mathcal{L}$ so that to keep the matrix of the
effective interaction self-adjoint.

We use the electron-hole dynamical vertex $\widehat{\overline{\Phi}}$
resulting from the simplified parquet equations and calculate the
self-energy correction to the Hartree term from Schwinger-Dyson
equation \eqref{eq:SDE-SPE}. It is in the super-matrix representation
\begin{multline}\label{eq:SDE-two_orbitals}
  \Delta \Sigma_{\iota\sigma}(i\omega_{n}) = - \frac 1\beta\sum_{m}
  \sum_{\lambda,\lambda'}\sum_{\iota',\sigma'} U^{\iota
    \iota',\sigma}_{\lambda\lambda',\sigma'}\chi^{\iota',\sigma}_{\lambda',\sigma'}
  \overline{\Phi}^{\iota'\iota,\sigma}_{\lambda'\lambda,\sigma'}(i\nu_{m})\\
  \times G_{\lambda\sigma'}(i\omega_{n} + i\nu_{m})\ .
\end{multline}
The dynamical part of the self-energy $\Delta\Sigma$ is then used to
determine the particle densities from Eq.~\eqref{eq:1P-density} so
that the static one-electron self-consistency is fulfilled.

\section{Results}\label{sec:Results}

We analyze a general two-band Hubbard model within the dynamical
mean-field theory. We set the model parameters $U_{1} = U - 2J$,
$U_{2} = U - 3J$ and $J_{1}=J_{2}=J$, which corresponds to a
spherically symmetric situation.\cite{Imada98} We choose the Hund
exchange so that all three electron interactions remain positive.  We
use the one-electron propagator from Eq.~\eqref{eq:1GF-general} with
the semi-elliptic density of states, $\rho_{1}(\epsilon) =
\rho_{2}(\epsilon) =2\pi^{-1}\sqrt{1 - \epsilon^{2}}$ the bandwidth of
which is $w = 2$.  We investigate only the spin symmetric case where
the one-electron propagators are spin-independent. The critical
interaction in the random phase approximation, separating the weak and
strong coupling regimes, then is $U_{c} =3\pi/8\approx 1.18$. It does
not depend on the number of orbitals if the Hund coupling $J=0$. We
perform all calculations at zero temperature.

We first investigate the simplest situation of the charge-symmetric
model at half filling, that is $n=2$, with no orbital splitting
$\Delta = |E_{2} - E_{1}|/2=0$ and with vanishing Hund coupling,
$J=0$.  This situation mimics the single-orbital model. The emergence
of the Kondo scale is plotted in Fig.~\ref{fig:var_U}. The
quasiparticle peak starts to develop from the bare density of states
for interactions around the critical one from RPA, $U_{c}\approx
1.18$. The satellite Hubbard bands are well formed in the
strong-coupling regime $U>U_{c}$ and lie slightly beyond the atomic
values $\pm U/2$.  The Kondo scale remains exponentially small but
non-zero even for large interaction strengths. The Kondo asymptotics
for the semi-elliptic density of states is $a\propto \exp\{-
4U\rho_{0}/3\}$, where $\rho_{0}$ is the density of states at the
Fermi energy.  This result is in discrepancy with general confidence
that the Hubbard model at half-filling undergoes in the paramagnetic
phase a Mott-Hubbard metal insulator transition at a finite
interaction strength.  The existence of a metal-insulator transition
in DMFT is concluded from extrapolations of low-temperature
Monte-Carlo\cite{Georges96} or NRG\cite{Bulla01,Bulla08}
data. Although there is no doubt about the existence of an insulating
solution for very large interaction strengths matching the atomic
limit, numerical methods cannot presently exclude the existence of a
metallic solution with a very narrow (below numerical resolution)
quasiparticle peak. The existence of a sharp metal-insulator
transition has not yet been rigorously proved and there is at present
neither a consistent solution at the critical point of a transition
nor there is an analytic formula for the vanishing width of the
quasiparticle peak. If there is a metal-insulator transition, the
metallic side of the transition cannot be a Fermi liquid. The
low-frequency asymptotics of the electron-hole correlation function in
Fermi liquid is due to sum rules of order $\omega^{-1}$.  Such a
behavior is then incompatible with Bethe-Salpeter equations that have
to be fulfilled self-consistently in the electron-hole and
electron-electron scattering channels by an exact solution. Only
integrable singularities can emerge in an exact solution and Fermi
liquid hence does not allow for a divergence in the electron-hole
correlation function expected at the Mott-Hubbard metal-insulator
transition. In this respect the simplified parquet approximation is
consistent with the exact two-particle behavior.

Nonexistence of a sharp metal-insulator transition in the simplified
parquet equations is not caused by approximations made in this
approach but rather it is a consequence of a Fermi-liquid solution
fulfilling self-consistently Bethe-Salpeter equations simultaneously
in the electron-hole and electron-electron scattering channels. There
is an insulating paramagnetic state in parquet-based theories for $U >
U_{c}$ (critical interaction of the RPA pole) being a superposition of
Hartree polarized solutions, but it coexists with the metallic one in
the strong-coupling regime and has a higher total energy. A
metal-insulator transition in the Fermi liquid can either be connected
with a breaking of a global symmetry and/or can be noncritical as we
discuss later for orbital splitting in a crystal field.
\begin{figure}
  \includegraphics[width=8.5cm]
  {var_U_HF.eps}
  \caption{(Color online) Emergence of the Kondo scale in the
    two-orbital, charge, spin and and orbital symmetric model without
    Hund exchange (one electron per orbital). Semi-elliptic density of
    states.  }\label{fig:var_U}
\end{figure}

We compared the spectral function calculated from the simplified
parquet equations with other diagrammatic approximations based on
summations of Feynman diagrams for multiple two-particle
scatterings. We chose a value of the interaction strength $U=3$ so
that to be in the strong-coupling regime where renormalizations of the
perturbation expansion are necessary. We used the spectral function
for the self-consistent T-matrix, iterated perturbation theory in
second order (IPT) and for T-matrix with IPT (topological)
self-consistency, that is, the local renormalized propagator of the
Hubbard model $\mathcal{G}$ is defined as $\mathcal{G}^{-1}(\omega) =
G^{-1}(\omega - \Sigma(\omega)) + \Sigma(\omega)$.  The results of the
comparison are plotted in Fig.~\ref{fig:U3}.  We can see that except
for the fully self-consistent T-matrix, the chosen three
approximations display the expected three-peak structure. They differ,
however, in the way the satellite bands are attached to the central
quasiparticle peak. The simplified parquet approximation keeps a small
finite density of states between the central and satellite bands. It
means that this approximation allows for an energy transfer between
the central and the satellite bands. Approximations based on the IPT
self-consistency develop a quasi-gap between the low and high energy
states and no energy exchange occurs. The satellite bands are less
spread in the parquet approximation than in the IPT-based ones. The
detailed shape of the central quasi-particle peak in these
approximations is magnified in Fig.~\ref{fig:U3_detail}.  The most
narrow is the central peak for IPT suggesting vicinity of a
metal-insulator transition with zero width of the quasiparticle
peak. The parquet-based approximations, on the other hand, slow down
the tendency toward a metal-insulator transition. They actually screen
it, since there may be no critical metal-insulator transition in these
theories as discussed above.

\begin{figure}
  \includegraphics[width=8.5cm]
  {U3.eps}
  \caption{(Color online) Spectral function for the symmetric
    situation calculated in various diagrammatic approximations:
    simplified parquets, self-consistent T-matrix (SCTMA), iterated
    perturbation theory (IPT), and T-matrix with IPT
    propagators. }\label{fig:U3}
\end{figure}
\begin{figure}
  \includegraphics[width=8.5cm]
  {U3detail.eps}
  \caption{(Color online) Detailed structure of the quasiparticle peak
    for the approximations from
    Fig.~\ref{fig:U3}.}\label{fig:U3_detail}
\end{figure}

The simplified parquet equations show best agreement with the exact
solution of the single-impurity Anderson model at half filling, where
the density of states at the Fermi surface is fixed. The same holds
also for other fillings at intermediate and strong coupling calculated
within the dynamical mean-field approximation. Apart from the
charge-symmetric situation the simplified parquet equations reproduce
qualitatively well the low and high frequency features of the spectral
function, although the width of the quasiparticle peak and the Kondo
temperature derived from it do not follow as closely the exact
solution for the flat-band model as for the half-filled
case.\cite{Janis08} The spectral function of a strong-coupling
solution changes when receding from half filling so that the central
quasiparticle peak broadens and slightly moves below the Fermi
surface.  The lower satellite band moves toward the Fermi energy. The
lower band and the central peak eventually merge, the quasiparticle
peak is absorbed by the satellite one, when the electron-hole
asymmetry becomes prominent. This feature seems to be universal for
all theories and is demonstrated in our approach in
Fig.~\ref{fig:var_fill}.
\begin{figure}
  \includegraphics[width=8.5cm]
  {var_fill.eps}
  \caption{(Color online) Spectral function in charge asymmetric
    situations for various filings $n$, averaged number of electrons
    per orbital.}\label{fig:var_fill}
\end{figure}

We hitherto have shown the results for the two-orbital model without
Hund exchange and orbital splitting, that is for $J=0$ and $\Delta =
0$. Such a situation is close to the single-orbital model as we
demonstrated.  The most important change in the model with nonzero
Hund coupling is a splitting of the interaction strength, that is,
$U_{1}= U - 2J$ and $U_{2}= U - 3J$. It means that the interaction
strength controlling the approach to the critical point in the
electron-hole channel is the Hubbard $U$ between different spins on
the same orbital. The Hund exchange decreases the critical interaction
at which the electron-hole vertex diverges in the bare perturbation
theory. It would then narrow the width of the Kondo resonant peak at a
fixed renormalized Coulomb repulsion $\overline{U}$. In the real
situation we, however, do not fix the renormalized but rather the bare
Coulomb repulsion.  We find that though larger Hund exchanges lead to
higher values of the averaged kernels $L^{ee}$, as we can see from
Eqs.~\eqref{eq:EI-O2-upp} and \eqref{eq:EI-O2-low}, they
simultaneously cause decrease in the renormalized repulsions
$\overline{U}_{\alpha}$ at a fixed bare Coulomb interaction.
Consequently, the weight of the central quasiparticle peak in the
self-energy decreases. We plotted the impact of the Hund coupling on
the spectral function in Fig.~\ref{fig:var_J}.  We observe that the
width of the Kondo peak shrinks but only on a narrower frequency
interval near the Fermi energy.  The Hund coupling at a fixed bare
Coulomb repulsion then leads to filling of the states between the
central peak and the satellite bands. Further, the satellite bands are
positioned closer to the central one and become less pronounced with
the increasing Hund exchange.
\begin{figure}
  \includegraphics[width=8.5cm]
  {var_J_U3.eps}
  \caption{(Color online) Effect of the Hund exchange coupling on the
    spectral function.}\label{fig:var_J}
\end{figure}

The most interesting on the two-orbital model are changes in the
spectral function due to a broken orbital symmetry induced by a
crystal field. That is, when the one-electron propagators are
different on different orbitals. We achieve this here by splitting the
centers of the two bands. We set $E_{1}= - E_{2}= \Delta$, but keep
the same band structure, $\rho_{1}(\epsilon) =
\rho_{2}(\epsilon)$. The effect of the orbital splitting depends on
whether we have a finite or infinite bandwidth of macroscopically
occupied energies.  Orbital splitting leads in the former case to a
metal-insulator transition unlike the latter one. The genesis of a
metal-insulator transition in the model with the semi-elliptic density
of states is plotted in Fig.~\ref{fig:MIT-orbital}. We chose a
moderate interaction strength, $U = 1.5$, so that to be able to follow
the process more closely. When we increase the orbital splitting
parameter $\Delta$ there is almost no observable change at the Fermi
energy with a pronounced quasiparticle peak. It slightly splits into
two but stays at the Fermi energy.  More apparent is the change caused
by orbital splitting in the bulk of the energy band where the states
are rearranged so that the orbital polarization is enhanced. Such a
response on the band splitting effectively pushes the system out from
the critical region of the RPA pole.  With the increasing band
splitting the dynamical renormalization of the band structure, apart
from the vicinity of the Fermi energy, becomes less important and the
solution approaches the Hartree one. The process continues up to the
point when one of the orbital saturates and the other is emptied. The
insulating solution is then that of the Hartree approximation with no
dynamical correlations. Changes in the quasiparticle peak, split and a
broadening, are observed only close to the transition. They vanish
practically at the transition point. There are, however, no singular
two-particle functions at the transition point, the metal-insulator
transition is hence noncritical. It is worth noting that the Kondo
quasiparticle peaks exist in a pronounced way only in the
strong-coupling regime, cf. Fig.~\ref{fig:MIT-orbital_weak}.

\begin{figure}
  \includegraphics[width=13cm]
  {kondo_decay_U1_5.eps}
  \caption{(Color online) Orbital splitting and a metal-insulator
    transition in the half-filled case for the semi-elliptic density
    of states and $U=1.5$. }\label{fig:MIT-orbital}
\end{figure}

\begin{figure}
  \includegraphics[width=13cm]
  {kondo_decay_U0_5.eps}
  \caption{(Color online) Orbital splitting and a metal-insulator
    transition in weak coupling, $U=0.5$.}\label{fig:MIT-orbital_weak}
\end{figure}

We further studied the effect of the Hund exchange on the orbital
metal-insulator transition. We set the parameters of the model so that
we could compare our results with the existing Monte-Carlo simulations
of Ref.~\onlinecite{Werner07}.  We chose three values of the bare
orbital splitting $2\Delta = 0.2, 0.4, 1.0$. For each value of the
band splitting we fixed four ratios of the Hund coupling $J/U = 0,
0.05, 0.1, 0.2$ and plotted the filling (per spin) of the upper
orbital $n_{1}$ as a function of the bare interaction at half filling
$n_{1} + n_{2}= 1$ in Fig.~\ref{fig:fill_U_J}.  We observe that small
Hund couplings do not prevent the system from falling into the
orbitally polarized state as demonstrated in
Fig.~\ref{fig:MIT-orbital}. For large $\Delta$ the occupation of the
upper orbital decreases practically monotonically with the increasing
interaction. For smaller splittings the filling $n_{1}$ increases and
may get to the Kondo regime at intermediate interaction strengths
where both orbitals are almost equally occupied ($n_{1}\approx n_{2}
\approx 0.5$).  Unlike the Monte-Carlo data, we do not reach, however,
the Mott insulating phase from the Fermi-liquid solution. If $J/U <
0.2$ the system does not stay in the Kondo regime for ever. Strong
electron repulsion increases orbital polarization due to the Hartree
shift and the system eventually goes over into a polarized insulator
$n_{1}=0$. For ratios $J/U>0.2$ the Hartree term starts to act against
the crystal field and diminishes orbital polarization. The filling
$n_{1}$ then asymptotically approaches half filling.  Apart from
nonexistence of the Mott transition these effects of the Hund coupling
on the occupation of orbitals in a crystal field are described by the
simplified parquet equations in a good agreement with the Monte-Carlo
simulations. It is worth mentioning that the Kondo resonance for $J/U
< 0.2$ survives to a certain value of the interaction strength beyond
which it is rather abruptly suppressed, similarly to the scenario from
Fig.~\ref{fig:MIT-orbital}. Vanishing of the Kondo resonance is
followed by a sharp saturation of orbital polarization. The numerical
solution then becomes unstable, since the metallic phase in the
extreme Kondo regime and the orbitally polarized insulating solution
coexist with almost the same energies.

\begin{figure}
  \includegraphics[width=8.7cm]
  {fill_U_J_D.eps}
  \caption{(Color online) Filling per spin of the upper orbital as a
    function of the bare interaction strength $U$. Different curves
    for a given split $2\Delta$ correspond to different $J$ values
    from bottom to top $J/U=0, 0.05, 0.1, 0.2$. A metal-insulator
    transition ($n_{1}=0$) takes place for values of $J/U < 0.2$.  The
    Kondo regime, vicinity of the Mott-Hubbard transition
    ($n_{1}\approx 1/2$) is reached only for larger values of
    $J/U$. }\label{fig:fill_U_J}
\end{figure}

\begin{figure}
  \includegraphics[width=13cm]
  {kondo_decay_lorentz.eps}
  \caption{(Color online) Orbital splitting in the single-impurity
    Anderson model for $U=5$. No metal-insulator transition takes
    place. }\label{fig:MIT-orbital_Lorentz}
\end{figure}

The scenario of orbital splitting changes if the bare energy bands are
infinite. When an orbital symmetry breaking is switched on, the system
with infinite band-widths behaves for small splittings analogously to
the model with finite energy bands. The changes take place in the
bulk, away from the Fermi energy so that the orbital splitting is
enhanced, cf. Fig.~\ref{fig:MIT-orbital_Lorentz}. Similarly there are
almost no observable changes on the quasiparticle peak. The system is
slowly pushed away from the critical region of the RPA transition by
increasing the orbital splitting and approaches orbital
saturation. But due to the infinite band width, there is no transition
to the Hartree insulator and the density of states at the Fermi energy
remains nonzero for all interaction strengths. The Kondo peaks do not
survive to infinite interaction.  A sudden macroscopic cleft breaks in
between the central quasiparticle peaks of different orbitals at a
finite interaction strength. The split Kondo peaks are then soon
absorbed by the bulk energy bands. The solution approaches a
weak-coupling (Hartree-Fock) state with almost separated energy bands
centered around energies $\pm (\Delta + Um_{o}/2)$, where $m_{o}$ is
the orbital moment.

\section{Discussion and conclusions}\label{sec:Conclusions}

Dynamical effects of strong electron correlations in impurity models
manifest themselves in a three-peak structure of the spectral function
with a narrow central quasiparticle peak near the Fermi energy and two
satellite bands centered around eigenvalues of the local part of the
interacting Hamiltonian. There is no exact solution proving such a
picture, but a number of approximate numerical and analytic-numerical
solutions confirm this behavior of the spectral function. There are a
number of such approximate schemes for the simplest single-orbital
impurity and DMFT models. Extensions of approximations showing the
expected three-peak spectral function in strong coupling to
multi-orbital models face a considerable increase of complexity in the
strong-coupling regime. Monte-Carlo simulations as well as the
numerical renormalization group applied on multi-orbital models need
to resort to additional approximations to reach sufficient accuracy of
dynamical and spectral properties.

Most of the existing approximations on dynamical properties of
strongly-correlated electron systems make directly accessible only
one-electron properties. Only few of them, even in single-orbital
models, are able to approach two-particle quantities and in
particular, to control two-particle divergencies that develop in
Bethe-Salpeter equations.  We extended in this paper an approximation
based on parquet equations controlling the RPA singularity in the
electron-hole scattering channel to a general multi-orbital model of
correlated electrons. Approximations used to simplify complexity of
the full parquet approach resulting from a separation of large and
small dynamical fluctuations in the critical region of the RPA pole
allow us to reproduce the expected three peak structure of the
spectral function of SIAM in the Kondo regime. An extension of this
approximation from single to multi-orbital models proved to be
manageable with only a moderate increase in numerical requirements. An
$l$-orbital model is in the most general case described by
$4l^{2}\times 4l^{2}$ matrices and the numerical expenses increase
polynomially with the number of orbitals. In real situations
symmetries and conservation laws further reduce complexity of the
simplified parquet equations.

The impurity solver for the dynamical mean-field theory we built up
within the parquet approximation suits best for describing the
dynamical and spectral properties of the metallic (Fermi liquid) phase
from weak up to moderately strong electron correlations. No
self-consistency is needed in the weak-coupling regime below the
critical interaction strength $U_{c}$ at which the RPA vertex
diverges. The parquet self-consistency is indispensable in
intermediate ($U\approx U_{c}$) and strong ($U > U_{c}$) coupling. The
numerical procedure for the solution of the simplified parquet
equations gets unstable for large interaction strengths. We were able
to reach a stable metallic solution up to $U \approx 5 U_{c}$. Unlike
other approaches such NRG, QMC or IPT this approximation does not
allow for a \textit{critical} metal-insulator transition without a
symmetry breaking in Fermi liquid. This is a general feature of all
impurity solutions where Bethe-Salpeter equations in the electron-hole
and electron-electron channels should be obeyed simultaneously in a
self-consistent manner. Parquet equations are hence unable to describe
a Mott insulator without a symmetry breaking. The insulating solution
obtained within the local parquet approach is a superposition of
fluctuations-free Hartree saturated solutions and the transition to
such a state is noncritical.

The simplified parquet equations presented in this paper are not
quantitatively as accurate as numerically exact solutions, but they
offer a numerically manageable scheme with an analytic control of
two-particle singularities in Bethe-Salpeter equations.  The
self-energy and the spectral function calculated within this
approximation obey Fermi-liquid rules in the metallic phase and
produce the typical three-peak structure in the density of states in
the strong-coupling regime. The width of the central quasiparticle
peak is exponentially small for large interaction strengths. This
approximation may not reproduce the exact asymptotics of the Kondo
scale and temperature in the single-impurity model as well as
e.~g. the local moment approach, but unlike the latter it does not
show any unphysical behavior and does not use auxiliary fitting
parameters. The approximation based on the simplified parquet
equations is formulated in such a way that it can directly be applied
not only to impurity models but to lattice models within the dynamical
mean-field approach as an impurity solver.

The impurity solver based on the simplified parquet equations is a
semi-analytic theory for local spectral properties formulated in real
frequencies with a full control of analytic properties of the
approximation. From this reason the approximation best suits to the
calculation of spectral properties and the dynamical self-energy
directly accessible from a perturbation theory. Its unique feature is
that it offers analytic control of two-particle singularities in the
Bethe-Salpeter equations. The simplified parquet approximation is well
suited to problems involving two-particle dynamical, and in particular
critical, functions. The approximation is less suited to global
thermodynamic quantities being averages of dynamical functions over
frequencies. An extension to finite temperatures does not represent a
problem and the effective interactions can be calculated also at
nonzero temperatures as discussed in this paper. The quantitative
reliability of the simplified parquet equations decreases with the
reduced influence of the critical behavior of the vertex function in
the vicinity of the RPA pole (Kondo regime). Only in the critical
region of a pole in the two-particle vertex we can rely on a
separation of large and small dynamical fluctuations, that is, a
decoupling of singular and non-singular functions.

To our knowledge, the simplified parquet equations is the only scheme
that uses a two-particle self-consistency to control the Kondo
asymptotics and criticality of the RPA pole. There are attempts to use
parquet equations for renormalizations of the perturbation expansion,
but they are based on the full one- and two-particle
self-consistency.\cite{Bickers04,Yang09} It is, however, important to
realize that the full dynamical one-particle self-consistency impedes
control of two-particle divergencies and the Kondo scale in the
spectral function is no longer correctly reproduced. Moreover, the
full dynamical one-electron self-consistency ultimately leads to
smearing of the satellite bands.\cite{Janis07a} It is hence of
importance that the parquet equations are solved with the Hartree
propagators so that to reproduce the three-peak spectral function with
the correctly scaled central quasiparticle peak.

The extension of the simplified parquet equations was used in this
paper as an impurity solver for the two-band Hubbard model with Hund
coupling and orbital splitting. We showed emergence of the Kondo
quasiparticle peak with the increasing interaction strength at half
filling. The quasiparticle peak survives only close to the
electron-hole symmetry where multiple electron-hole scatterings are
dominant. When deviations from this symmetry are significant the
quasiparticle peak is absorbed by one of the satellite bands. We also
demonstrated how a metal-insulator transition induced by a crystal
field takes place in a model with finite energy bands. The parquet
equations do not allow for a critical transition in the local Fermi
liquid. We showed that the orbitally induced metal-insulator
transition is noncritical, but with a Kondo behavior almost up to the
transition point, if the interaction is sufficiently strong.

Last but not least, the approximate impurity solver we built up on
parquet equations uses the formalism of Green functions and can
conveniently be combined with ab-initio computational schemes to
assess correlation effects in materials with a nontrivial atomic
structure. It should be suitable in particular for metallic systems
where we expect Kondo-like behavior or strong valence fluctuations.

\acknowledgments \noindent We thank V. Drchal and A. B. Shick for
valuable discussions and helpful comments. This research was carried
out within project AV0Z10100520 of the Academy of Sciences of the
Czech Republic and supported in part by Grant No.  202/07/J047 of the
Czech Science Foundation.

redpajama_set_name: RedPajamaArXiv


text:
Kang Sun-woo (; born 2 June 1978) is a South Korean politician.

She earned her Bachelor's in English education and master's degree in consumer studies and human development from Ewha Womans University. She received her doctorate degree in human development and family studies from University of Wisconsin–Madison where she worked as an instructor from 2010 to 2012. She continued to teach as an assistant professor at South Dakota State University from 2012 to 2016.

In 2016 she joined the Democratic Party and ran for 2016 general election as the number 29 on the list for proportional representation. From 2016 to 2017 she worked as a standing deputy spokesperson of her party. She worked at then-candidate Moon Jae-in's presidential campaign in 2017. She also participated in various policy advisory instruments of Moon government such as National Education Council, National Unification Advisory Council and Presidential Committee for Balanced National Development.

In 2020 she earned the party nomination for Gangseo district for the 21st general election defeating the incumbent parliamentarian Geum Tae-sub by a significant margin. In the election, she defeated the former parliamentarian and diplomat Ku Sang-chan from the main opposition party.

In August 2020, the newly elected leader of her party, Lee Nak-yon, appointed her as one of three standing spokesperson of the party.

Electoral history

References

1978 births
Living people
People from Daegu
Ewha Womans University alumni
University of Wisconsin–Madison School of Education alumni
South Dakota State University faculty
South Korean women academics
21st-century South Korean women politicians
21st-century South Korean politicians
Members of the National Assembly (South Korea)
Minjoo Party of Korea politicians
Female members of the National Assembly (South Korea)

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