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Dynamical properties of the single--hole $t$--$J$ model on a 32--site square lattice

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arXiv:cond-mat/9509087v1 15 Sep 1995Dynamicalpropertiesofthesingle–holet–Jmodelona32–sitesquarelattice

P.W.Leung∗PhysicsDepartment,HongKongUniversityofScienceandTechnology,ClearWaterBay,HongKong

R.J.Gooding†DepartmentofPhysicsandtheCentreforMaterialsScienceandEngineering,MassachusettsInstituteofTechnology,Cambridge,MA02139(February6,2008)

AbstractWepresentresultsofanexactdiagonalizationcalculationofthespectralfunctionA(k,ω)forasingleholedescribedbythet–Jmodelpropagatingona32–sitesquarecluster.Theminimumenergystateisfoundatacrystalmomentumk=(π2),consistentwiththeory,andourmeasureddisper-sionrelationagreeswellwiththatdeterminedusingtheself–consistentBornapproximation.Incontrasttosmallerclusterstudies,ourspectrashownoevidenceofstringresonances.WealsomakeaqualitativecomparisonofthevariationofthespectralweightinvariousregionsofthefirstBrillouinzonewithrecentARPESdata.

PACS:71.27.+a,74.25.Jb,75.10.Jm

TypesetusingREVTEX1Thet–Jmodelhasreceivedalotofattentioninrecentyears.Itisbelievedtobethesimpleststrong–couplingmodelofthelowenergyphysicsoftheanomalousmetallicstateofhigh–temperaturesuperconductors[1,2].TheHamiltonianofthemodelis

H=−t󰀁󰀅ij󰀆σ(˜c†iσ˜cjσ+H.c.)+J󰀁󰀅ij󰀆(Si·Sj−1

2,πThus,thispaperrepresentsamajoradvanceintheexact,unbiased,numericaltreatmentofanimportantstrong–couplingHamiltonian.Inorderforustocompletetheexactdiagonalizationonsuchalargelattice,weusetranslationandonereflectionsymmetrytoreducethetotalnumberofbasisstatestoabout150million.Atk=(3π4),noreflectionsymmetrycanbeusedandthetotalnumberofbasisstatesisabout300million.Tostudytheeffectoffinitesystemsizes,wewillsupplementourresultswithdataobtainedfromsmallersystems:theN=16(4×4)cluster,aswellasa24–site(√32)clusterthatincludesmanyoftheimportantwavevectors[7].Theelectronspectralfunctionisdefinedby

A(k,ω)=󰀁n|󰀊ψN−1n|˜ck,σ|ψN0󰀋|2δ(ω−EN0+EN−1n),(2)whereEN0andψN0arethegroundstateenergyandwavefunctionofthemodelathalffilling,respectively,andEN−1nandψN−1naretheenergyandwavefunctionofthentheigenstateofthesingle–holestate,respectively.A(k,ω)iscalculatedusingacontinuedfractionexpansion[8]with300iterationsandanartificialbroadeningfactorǫ=0.05.WeobtainA(k,ω)thatarewellconvergedusingthesequantities.Figure2(a)showsA(k,ω)atJ=0.3from(0,0)to(π,π).At(0,0),thespectrumhasaquasiparticlepeakatω∼1.34andabroadfeatureatlowerenergies.Askmovesawayfrom(0,0)alongthe(1,1)directiontowards(π,π),spectralweightshiftsfromthebroadfeaturetobothahigherenergyquasiparticlepeakandthelowenergytailofthespectrum.Thequasiparticlepeakincreasesinintensityandshiftstohigherenergies,reachingthevalencebandmaximumat(π2).Whenkgoesfurthertowards(π,π),thequasiparticlepeakmoves

tolowerenergiesanditsintensitydropssignificantly.Spectralweightmovetowardsthecentralpartofthespectrumagain,eventuallyleavingonlyaverysmallquasiparticlepeakatω=1.2313,andabroadlowenergystructure.Figure2(b)showsA(k,ω)atotherdistinctk.FromFig.2oneseesthatalongtheABZedgeA(k,ω)arequalitativelysimilar[9].Theyhavestrongquasiparticlepeakswhichdonotdispersemuch.Theintensityofthe

3quasiparticlepeakisthelargestat(π,0).Askmovesfrom(π,0)to(π2),theintensityof

thequasiparticlepeakdecreasesandisthesmallestat(π2)alongtheABZedge.Thiscanbemadequantitativebycalculatingthequasiparticleweight,whichisdefinedby

Zk=|󰀊ψN−1m|˜ckσ|ψN0󰀋|2

2,π

2,π2,πthe32–siteresults,exceptfordetailedvaluesoftheintensity.Weconcludethatthe24–sitelatticeislargeenoughtocapturetheessentialshapeofthespectralfunctionalongthe(1,1)direction,whilethe16–sitelatticeistoosmall,especiallywhenthelowerenergyfeaturesareconcerned.Inparticular,thewell–definedsecondarypeaksfoundat(π2)onthe16–sitelattice,whichwereinterpretedtoberelatedtothe“stringpicture”[12],arenotfoundinour32–sitesystem.Hencewefindnoevidencesupportingthestringpicture(atleastforthisvalueofJ).Further,thesingle–holeenergy,definedasEh=EN−10−EN0=−E(π2),

iscalculatedat0.1≤J≤0.8forthe32–sitesystem.FittingtotheformEh−J=a+bJνgivesa=−3.24,b=2.65,andν=0.72.Thisisconsistentwiththe16–siteresults[12],a=−3.17,b=2.83,andν=0.73,andalsowiththelargeclusterestimateoftheSCBAcalculations[11].However,νisnot2

2,π

2,π

2,π2,π

2,π

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关于矩阵Khatri-Rao乘积的若干矩阵不等式曾诚;何淦瞳【摘要】矩阵Khatri-Rao乘积作为一种特殊的矩阵乘积,被广泛地应用于控制理论、多元统计和动力学模型等领域的研究中.本文建立了一系列关于矩阵Khatri-Rao乘积的矩阵不等式,这些不等式包含或推广了相应的研究成果,在理论推导的过程中,采用的研究工具是矩阵Schur补和分块矩阵的性质.%It has been evident that the Khatri-Rao product has been widely used in the research of statistics,economic and dynamical systems as a special product form.The purpose of this paper is to present a family of matrix inequalities involving the Khatri-Rao product.Our theorems contain or extend some existing results.Schur complement and block matrices are a powerful tool.【期刊名称】《新疆大学学报（自然科学版）》【年(卷),期】2017(034)004【总页数】7页(P379-385)【关键词】Schur补;分块矩阵;Khatri-Rao乘积;矩阵不等式;Moore-penrose逆【作者】曾诚;何淦瞳【作者单位】贵州理工学院理学院,贵州贵阳550003;贵州大学数学与统计学院,贵州贵阳550025【正文语种】中文【中图分类】O151.210 引言在控制理论、多元统计和动力学模型等领域的研究中,矩阵的Hadamard乘积和Kronecker乘积扮演着重要的角色[1−7].当矩阵A,B是相同阶数的矩阵时,则A,B 间的Hadamard乘积A◦B是相应的Kronecker乘积A⊗B的主子阵[8−11].矩阵的Kronecker乘积最初起源于群论,物理上用来研究粒子理论,现在它常用来求解矩阵方程、分析灵敏度、以及线性方程组.通常情况下,可利用Vec算子来把矩阵方程AXB=C等价转化为¡BT⊗A¢V ecX=V ecC的形式,进而通过开展矩阵计算来求解矩阵方程.然而,对于含有分块为矩阵子块的未知矩阵X,却无法展开相应的矩阵计算,其原因在于该矩阵算子扰乱了系统的内部结构.因此,开展对矩阵块状Kronecker乘积的科学讨论是一个极其有价值的研究课题.关于矩阵块状Kronecker乘积主要有两种:即矩阵的Khatri-Rao乘积A∗B和矩阵Tracy-Singh乘积A∞B.Khatri和Rao[12]提出一类矩阵的块状Kronecker乘积,这类乘积本质上是矩阵Hadamard乘积和Kronecker乘积的推广,因而被称为矩阵的Khatri-Rao乘积;而Tracy-Singh乘积则是由Tracy和Singh[13]联合提出的.在文献[14]中,Wei和Zhang明确验证了矩阵的Kronecker乘积与矩阵的Tracy－Singh乘积之间可进行等价置换.而Rao等[15−17]则建立了关于Khatri-Rao乘积的一些矩阵等式.进一步,Horn和Mathias[18]把Hadamard乘积的Schur定理推广到Khatri-Rao乘积中,得到的结果是关于Khatri-Rao乘积的一些特征值不等式,奇异值不等式以及控制不等式.与此同时,Liu[19,20]对Khatri-Rao乘积和Tracy-Singh乘积建立了相应的渐近表达形式,描述了这两类特殊矩阵乘积之间的联系,并给出了关于这两类矩阵特殊乘积的若干矩阵不等式,同时可应用在多元统计分析中. 建立矩阵Khatri-Rao乘积的不等式,既可以作为矩阵Hadamard乘积的推广,同时也是为建立关于块状Kronecker乘积矩阵不等式的需要.尽管Khatri-Rao乘积可以看作是矩阵Hadamard乘积的推广,但与Hadamard乘积却有着本质的不同,其原因在于Khatri-Rao乘积具有不可交换性等特点,并且以块作为运算对象,导致乘积后的矩阵阶数迅速变大,同时其阶数仅仅依赖于矩阵的分块方式.另一方面,Hadamard乘积要求参与运算的矩阵必须具有相同的阶数,而Khatri-Rao乘积则体现了任意阶矩阵的任意分块方式,即只要求参与运算的矩阵在分块后的行块数和列块数分别保持相同即可,而对于矩阵的阶数,则可以不一致.因此,Khatri-Rao乘积不能看成Hadamard乘积的简单推广.文章首先将两个矩阵Khatri-Rao乘积和矩阵Tracy-Singh乘积之间的联系推广到有限个矩阵的情形,然后建立了一系列关于矩阵Khatri-Rao乘积的矩阵不等式.1 预备知识本文用Cn×m表示n×m阶复矩阵集;设A∈Cn×m,则A∗表示矩阵A的共轭转置,当A为方阵且可逆时,A−1¡A+¢表示矩阵A的逆（广义逆）;A>(≥)表示Hermite （半）正定矩阵;A>B表示A−B>0;In表示n阶单位矩阵.定义1[19]设n,m,p,q,ni,mj,pi,qj,(i=1···r,j=1···s)均为正整数,且满足再设分块矩阵A=(Aij)∈ Cn×m,B=(Bij)∈Cp×q,其中存在子矩阵Aij∈ Cni×mj,Bij∈Cpi×qj,定义矩阵的Khati-Rao乘积其中显然,若r=s=1时,则矩阵的Khati-Rao乘积就是矩阵的Kronecker乘积;若n=p,m=q,n1=···=nr=m1=···=ms=1,则矩阵的Khati-Rao乘积就是经典的Hadamard乘积.定义2[19]设Aij∈Cni×mj,BKl∈Cpi×qj分别为分块矩阵A∈Cn×m,B ∈Cp×q的位于(i,j),(k,l)处的子矩阵。

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Communications and Control Engineering For other titles published in this series,go to/series/61Series EditorsA.Isidori J.H.van Schuppen E.D.Sontag M.Thoma M.Krstic Published titles include:Stability and Stabilization of Infinite Dimensional Systems with ApplicationsZheng-Hua Luo,Bao-Zhu Guo and Omer Morgul Nonsmooth Mechanics(Second edition)Bernard BrogliatoNonlinear Control Systems IIAlberto IsidoriL2-Gain and Passivity Techniques in Nonlinear Control Arjan van der SchaftControl of Linear Systems with Regulation and Input ConstraintsAli Saberi,Anton A.Stoorvogel and Peddapullaiah SannutiRobust and H∞ControlBen M.ChenComputer Controlled SystemsEfim N.Rosenwasser and Bernhard mpeControl of Complex and Uncertain SystemsStanislav V.Emelyanov and Sergey K.Korovin Robust Control Design Using H∞MethodsIan R.Petersen,Valery A.Ugrinovski andAndrey V.SavkinModel Reduction for Control System DesignGoro Obinata and Brian D.O.AndersonControl Theory for Linear SystemsHarry L.Trentelman,Anton 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ChaosHuaguang Zhang,Derong Liu and Zhiliang Wang Stabilization of Navier–Stokes FlowsViorel BarbuDistributed Control of Multi-agent NetworksWei Ren and Yongcan CaoLars Grüne Jürgen Pannek Nonlinear Model Predictive Control Theory and AlgorithmsLars Grüne Mathematisches Institut Universität Bayreuth Bayreuth95440Germanylars.gruene@uni-bayreuth.de Jürgen Pannek Mathematisches Institut Universität BayreuthBayreuth95440Germanyjuergen.pannek@uni-bayreuth.deISSN0178-5354ISBN978-0-85729-500-2e-ISBN978-0-85729-501-9DOI10.1007/978-0-85729-501-9Springer London Dordrecht Heidelberg New YorkBritish Library Cataloguing in Publication DataA catalogue record for this book is available from the British LibraryLibrary of Congress Control Number:2011926502Mathematics Subject Classification(2010):93-02,92C10,93D15,49M37©Springer-Verlag London Limited2011Apart from any fair dealing for the purposes of research or private study,or criticism or review,as per-mitted under the Copyright,Designs and Patents Act1988,this publication may only be reproduced, stored or transmitted,in any form or by any means,with the prior permission in writing of the publish-ers,or in the case of reprographic reproduction in accordance with the terms of licenses issued by the Copyright Licensing Agency.Enquiries concerning reproduction outside those terms should be sent to the publishers.The use of registered names,trademarks,etc.,in this publication does not imply,even in the absence of a specific statement,that such names are exempt from the relevant laws and regulations and therefore free for general use.The publisher makes no representation,express or implied,with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made.Cover design:VTeX UAB,LithuaniaPrinted on acid-free paperSpringer is part of Springer Science+Business Media()For Brigitte,Florian and CarlaLGFor Sabina and AlinaJPPrefaceThe idea for this book grew out of a course given at a winter school of the In-ternational Doctoral Program“Identification,Optimization and Control with Ap-plications in Modern Technologies”in Schloss Thurnau in March2009.Initially, the main purpose of this course was to present results on stability and performance analysis of nonlinear model predictive control algorithms,which had at that time recently been obtained by ourselves and coauthors.However,we soon realized that both the course and even more the book would be inevitably incomplete without a comprehensive coverage of classical results in the area of nonlinear model pre-dictive control and without the discussion of important topics beyond stability and performance,like feasibility,robustness,and numerical methods.As a result,this book has become a mixture between a research monograph and an advanced textbook.On the one hand,the book presents original research results obtained by ourselves and coauthors during the lastfive years in a comprehensive and self contained way.On the other hand,the book also presents a number of results—both classical and more recent—of other authors.Furthermore,we have included a lot of background information from mathematical systems theory,op-timal control,numerical analysis and optimization to make the book accessible to graduate students—on PhD and Master level—from applied mathematics and con-trol engineering alike.Finally,via our web page we provide MATLAB and C++software for all examples in this book,which enables the reader to perform his or her own numerical experiments.For reading this book,we assume a basic familiarity with control systems,their state space representation as well as with concepts like feedback and stability as provided,e.g.,in undergraduate courses on control engineering or in courses on mathematical systems and control theory in an applied mathematics curriculum.However,no particular knowledge of nonlin-ear systems theory is assumed.Substantial parts of the systems theoretic chapters of the book have been used by us for a lecture on nonlinear model predictive con-trol for master students in applied mathematics and we believe that the book is well suited for this purpose.More advanced concepts like time varying formulations or peculiarities of sampled data systems can be easily skipped if only time invariant problems or discrete time systems shall be treated.viiviii PrefaceThe book centers around two main topics:systems theoretic properties of nonlin-ear model predictive control schemes on the one hand and numerical algorithms on the other hand;for a comprehensive description of the contents we refer to Sect.1.3.As such,the book is somewhat more theoretical than engineering or application ori-ented monographs on nonlinear model predictive control,which are furthermore often focused on linear methods.Within the nonlinear model predictive control literature,distinctive features of this book are the comprehensive treatment of schemes without stabilizing terminal constraints and the in depth discussion of performance issues via infinite horizon suboptimality estimates,both with and without stabilizing terminal constraints.The key for the analysis in the systems theoretic part of this book is a uniform way of interpreting both classes of schemes as relaxed versions of infinite horizon op-timal control problems.The relaxed dynamic programming framework developed in Chap.4is thus a cornerstone of this book,even though we do not use dynamic programming for actually solving nonlinear model predictive control problems;for this task we prefer direct optimization methods as described in the last chapter of this book,since they also allow for the numerical treatment of high dimensional systems.There are many people whom we have to thank for their help in one or the other way.For pleasant and fruitful collaboration within joint research projects and on joint papers—of which many have been used as the basis for this book—we are grateful to Frank Allgöwer,Nils Altmüller,Rolf Findeisen,Marcus von Lossow,Dragan Neši´c ,Anders Rantzer,Martin Seehafer,Paolo Varutti and Karl Worthmann.For enlightening talks,inspiring discussions,for organizing workshops and mini-symposia (and inviting us)and,last but not least,for pointing out valuable references to the literature we would like to thank David Angeli,Moritz Diehl,Knut Graichen,Peter Hokayem,Achim Ilchmann,Andreas Kugi,Daniel Limón,Jan Lunze,Lalo Magni,Manfred Morari,Davide Raimondo,Saša Rakovi´c ,Jörg Rambau,Jim Rawl-ings,Markus Reble,Oana Serea and Andy Teel,and we apologize to everyone who is missing in this list although he or she should have been mentioned.Without the proof reading of Nils Altmüller,Robert Baier,Thomas Jahn,Marcus von Lossow,Florian Müller and Karl Worthmann the book would contain even more typos and inaccuracies than it probably does—of course,the responsibility for all remaining errors lies entirely with us and we appreciate all comments on errors,typos,miss-ing references and the like.Beyond proof reading,we are grateful to Thomas Jahn for his help with writing the software supporting this book and to Karl Worthmann for his contributions to many results in Chaps.6and 7,most importantly the proof of Proposition 6.17.Finally,we would like to thank Oliver Jackson and Charlotte Cross from Springer-Verlag for their excellent rs Grüne Jürgen PannekBayreuth,Germany April 2011Contents1Introduction (1)1.1What Is Nonlinear Model Predictive Control? (1)1.2Where Did NMPC Come from? (3)1.3How Is This Book Organized? (5)1.4What Is Not Covered in This Book? (9)References (10)2Discrete Time and Sampled Data Systems (13)2.1Discrete Time Systems (13)2.2Sampled Data Systems (16)2.3Stability of Discrete Time Systems (28)2.4Stability of Sampled Data Systems (35)2.5Notes and Extensions (39)2.6Problems (39)References (41)3Nonlinear Model Predictive Control (43)3.1The Basic NMPC Algorithm (43)3.2Constraints (45)3.3Variants of the Basic NMPC Algorithms (50)3.4The Dynamic Programming Principle (56)3.5Notes and Extensions (62)3.6Problems (64)References (65)4Infinite Horizon Optimal Control (67)4.1Definition and Well Posedness of the Problem (67)4.2The Dynamic Programming Principle (70)4.3Relaxed Dynamic Programming (75)4.4Notes and Extensions (81)4.5Problems (83)References (84)ix5Stability and Suboptimality Using Stabilizing Constraints (87)5.1The Relaxed Dynamic Programming Approach (87)5.2Equilibrium Endpoint Constraint (88)5.3Lyapunov Function Terminal Cost (95)5.4Suboptimality and Inverse Optimality (101)5.5Notes and Extensions (109)5.6Problems (110)References (112)6Stability and Suboptimality Without Stabilizing Constraints (113)6.1Setting and Preliminaries (113)6.2Asymptotic Controllability with Respect to (116)6.3Implications of the Controllability Assumption (119)6.4Computation ofα (121)6.5Main Stability and Performance Results (125)6.6Design of Good Running Costs (133)6.7Semiglobal and Practical Asymptotic Stability (142)6.8Proof of Proposition6.17 (150)6.9Notes and Extensions (159)6.10Problems (161)References (162)7Variants and Extensions (165)7.1Mixed Constrained–Unconstrained Schemes (165)7.2Unconstrained NMPC with Terminal Weights (168)7.3Nonpositive Definite Running Cost (170)7.4Multistep NMPC-Feedback Laws (174)7.5Fast Sampling (176)7.6Compensation of Computation Times (180)7.7Online Measurement ofα (183)7.8Adaptive Optimization Horizon (191)7.9Nonoptimal NMPC (198)7.10Beyond Stabilization and Tracking (207)References (209)8Feasibility and Robustness (211)8.1The Feasibility Problem (211)8.2Feasibility of Unconstrained NMPC Using Exit Sets (214)8.3Feasibility of Unconstrained NMPC Using Stability (217)8.4Comparing Terminal Constrained vs.Unconstrained NMPC (222)8.5Robustness:Basic Definition and Concepts (225)8.6Robustness Without State Constraints (227)8.7Examples for Nonrobustness Under State Constraints (232)8.8Robustness with State Constraints via Robust-optimal Feasibility.2378.9Robustness with State Constraints via Continuity of V N (241)8.10Notes and Extensions (246)8.11Problems (249)References (249)9Numerical Discretization (251)9.1Basic Solution Methods (251)9.2Convergence Theory (256)9.3Adaptive Step Size Control (260)9.4Using the Methods Within the NMPC Algorithms (264)9.5Numerical Approximation Errors and Stability (266)9.6Notes and Extensions (269)9.7Problems (271)References (272)10Numerical Optimal Control of Nonlinear Systems (275)10.1Discretization of the NMPC Problem (275)10.2Unconstrained Optimization (288)10.3Constrained Optimization (292)10.4Implementation Issues in NMPC (315)10.5Warm Start of the NMPC Optimization (324)10.6Nonoptimal NMPC (331)10.7Notes and Extensions (335)10.8Problems (337)References (337)Appendix NMPC Software Supporting This Book (341)A.1The MATLAB NMPC Routine (341)A.2Additional MATLAB and MAPLE Routines (343)A.3The C++NMPC Software (345)Glossary (347)Index (353)。