UNIVERSITY OF CINCINNATI

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1 UNIVERSITY OF CINCINNATI Date: October 13, 2008 I, D.G. Sumith P. Doluweera hereby submit this work as part of the requirements Doctor of Philosophy for the degree of: in: Physics It is entitled: Effect of Weak Inhomogeneities in High Temperature Superconductivity This work and its defense approved by: Chair: Mark S. Jarrell F. Paul Esposito Michael Ma Leigh M. Smith Robert J. Endorf

2 Effect of Weak Inhomogeneities in High Temperature Superconductivity A dissertation submitted to the Division of Research and Advanced Studies of the University of Cincinnati in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY (Ph.D.) in the Department of Physics of the College of Arts and Sciences D.G. Sumith P. Doluweera M.S., University of Cincinnati, 2003 B.Sc., University of Ruhuna, Sri Lanka, 1994 Committee Chair: Prof. Mark Jarrell

3 Abstract We present results of three studies done using a dynamical cluster quantum Monte Carlo approximation. First, we investigate the d-wave superconducting transition temperature T c in the doped 2D repulsive Hubbard model with a weak inhomogeneity in hopping in the form of checkerboard pattern or a lattice of 2 2 plaquettes. Near neighbor hoppings within a plaquette is t and that of between the plaquettes is t. We investigate T c in the weak inhomogeneous limit 0.8t < t < 1.2t. We find inhomogeneity (t t) suppresses T c. The characteristic spin excitation energy (effective exchange energy) and the strength of d-wave pairing interaction decrease with decreasing T c. The latter observations suggest a strong correlation among effective exchange interaction, T c and the d-wave pairing interaction of the system. Second 1, we further find that enhancement of effective exchange interaction causes a slight increase in T c of a weakly disordered system with low impurity concentration, compared to the homogeneous system. Here the disorder is introduced to homogeneous repulsive 2D Hubbard model as a weak local potential disorder. Third, we present an improved maximum entropy method to analytically continue quantum Monte Carlo data with a severe sign problem. 1 A result from a collaborative study done with A. Kemper of Florida State University. i

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5 Preface This dissertation consists of three projects which I involved during my stay at the University of Cincinnati. First two projects deal with inhomogeneity and disorder of cuprate high temperature superconductors. The last project describes a modified maximum entropy method to analytically continue quantum Monte Carlo data with a severe sign problem. Prof. Mark Jarrell was the overall supervisor for me for all the projects. Except where reference is made to the work of others, this dissertation is the result of my own work and everything which is the outcome of work done in collaboration has been explicitly stated and cited as they are presented in the corresponding chapters. This dissertation has not been submitted in whole or in part for any degree or diploma at this or any other institute. D.G. Sumith P. Doluweera 2008 iii

6 To Chamini and Praveen

7 Acknowledgments I am thankful to the University of Cincinnati for providing me this opportunity to pursue my graduate studies and all the support provided. I wish to express my profound gratitude to my adviser Prof. Mark Jarrell for his financial support, encouragement, guidance, many stimulating discussions and wonderful opportunities provided to me during my graduate studies at UC. I am indebted to him since without his support, direction and advice this research has been impossible. I am highly inspired by his enthusiasm about the subject and the high degree of devotion to computational physics research. I am greatly thankful to him for his constructive criticisms and instructions and challenges he provided to me for better shaping my future. I am greatly indebted to Prof. Thomas Pruschke who informally served as my external adviser for his financial support, encouragement, stimulating discussions, guidance and the wholehearted hospitality which made my stay at Goettingen, Germany a memorable and wonderful experience. I am thankful to Dr. Alexandru Macridin for all the fruitful discussions we had over the years that broadened my understanding about the subject. Specially I want to express my gratitude to him for taking time to work with me, going through the hard work and made this work successful. I am grateful to my committee members Prof. Paul Esposito, Prof. Michael Ma, Prof. Leigh Smith and Prof. Robert Endorf for their kind advices and encouragements. Also I thank Prof. Robert Endorf for providing me an opportunity to shape up my teaching experience at UC. I am grateful to Prof. Alex Kegan and Prof. Philip Argyres for their role as graduate adviser. I acknowledge Prof. F. C. Zhang who initially served as a co-adviser before leaving UC. v

8 Effect of Weak Inhomogeneities in High Temperature Superconductivity I am very much grateful and indebted to Prof. Rohana Wijewardhana for his mentorship during my stay in Cincinnati as well as for providing this wonderful opportunity to me. I would like to acknowledge Dr. T. A. Maier and Dr. Juana Moreno for their collaborations with me. I thank former graduate students of Mark, Dr. Karan Aryanpour, Dr. M. A. Majid and Dr. Cyrill Slezak for their friendship, discussions, encouragements and time spent arguing and sharing their ideas with me. I also thank Dr. M. A. Majid for his collaboration with me on the charge density wave project. I sincerely acknowledge Alex Kemper who is a graduate student in the Florida State University for making a successful collaboration with me which is discussed in the third chapter of this dissertation. I am grateful to my friends Drs. Ajith and Theja for their wonderful friendship and love that started during our undergraduate studies in Sri Lanka and for the help I received to come and initially settle in Cincinnati. I wish to acknowledge all my friends including Sri Lankans at UC and Cincinnati who made a memorable wonderful living atmosphere and shared time with me in many ways. I thank Prof. Joseph Scanio, Elle Mengon, Donna Deutenberg, John Whitaker and Melody who helped me in numerous ways. I thank our group members for their friendship and discussions on the subject. My sincere thanks will go to my beloved father D. G. Peter whom I lost during my first year in Cincinnati and my loving mother Hema Gunasinghe for all the sacrifices they made for me to bring me up. I want to express my sincere thanks to my beloved mother-in-law Hilda Gurusinghe whom I lost during 2005 and my loving father-in-law George Gurusinghe for their extended help and encouragement toward my studies Finally I sincerely acknowledge and thank my loving wife Chamini and bright son Praveen who sacrificed many things in their lives due to my studies. Chamini had to sacrifice many privileges she enjoyed and yet had the courage to persuade me with never ending enthusiasm about my studies. Praveen lost a lot of play time with me both day and night during his early years and is still counting them. vi

9 Contents 1 Introduction Strongly Correlated Electronic Systems Hubbard Model Quantum Cluster Theories Dynamical Cluster Approximation (DCA) DCA/QMC Algorithm Quantum Monte Carlo Method (QMC) Unconventional Superconductivity Current Theories of High Temperature Superconductivity Inhomogeneities in high-t c Cuprates and Theory Outline of the Dissertation Other project collaborations d-wave superconductivity in the checkerboard Hubbard model An introduction to inhomogeneity and high-t c superconductivity Introduction Formalism

10 Effect of Weak Inhomogeneities in High Temperature Superconductivity 2.4 Results Discussion Conclusion Appendix Hamiltonian Green s Functions and Coarse Graining over the RBZ Weak Disorder Enhanced d-wave Superconductivity A brief introduction to disorder effects in superconductivity: motivation Introduction Formalism Results Discussion Analytic continuation of quantum Monte Carlo data with a sign problem An introduction to Maximum entropy method Entropic prior and prior probability Determination of α and the model selection Introduction MEM Formalism Data produced by QMC simulations with sign problem Sign problem in QMC simulations Sign problem and Maximum Entropy Method Likelihood function Formalism Discussion

11 Contents 4.7 Comparison of the spectra obtained with the two methods Conclusions Summary Summary Bibliography 89 3

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13 List of Figures 1.1 A schematic representation of the 2D one hole doped Mott insulator. When an adjacent site is vacant, electrons can hop into that site without on-site repulsion U. Where t is the near-neighbor hopping. This is equivalent to a hole moving in an antiferromagnetic background There are well established methods for the calculation of physical properties of materials in the free electron limit or when U <<< t and the other in the atomic limit or when U >>> t. The most interesting physics of cuprates is found in the intermediate region when U t and the model is difficult to solve Quantum cluster approaches, like the DMFA and DCA, map the infinite lattice problem onto a self-consistently embedded cluster problem. The figure and caption are from Ref. [61] One point vertex with entering (k 1, k 2 ) and leaving(k 3, k 4 ) momenta DCA/QMC algorithm [63] consists of three parts. 1) The self consistent loop, 2) Analysis code and 3) MEM analytic continuation code. See the text for description

14 Effect of Weak Inhomogeneities in High Temperature Superconductivity 1.6 Schematic T vs X phase diagram of cuprates obtained from experiments. Where X is the doping concentration. RHS: hole doped region, LHS: electron doped region. AF-Antiferromagnetism, T N -Néel temperature, T c -superconducting transition temperature, d SC -d-wave superconductivity, T*- crossover temperature, PG-pseudo-gap regime, SMstrange metal-non Fermi Liquid, E N -Nernst effect (below brown line), Yellow region between the AF and d SC is the spin glass region, FL-Fermi liquid A filtered STM image of local density of states of slightly over-doped Bi 2 Sr 2 CaCu 2 O 8+δ crystal [54]. The arrows indicate the direction of Cu- O bonds and the figure was taken from Ref. [67] Plaquette model. Here t is intra-cluster hopping and t is intercluster hopping (a)t c (t )/W for U/W = 1 and 10% doping. T c is suppressed for t t. (b) T c (t )/t for U = 6t and 8% doping. T c monotonically decreases with decreasing t. (c) T c /t for t = t and t = 0.88t as a function of doping δ when U = 8t. (d) Low energy density of states N(ω)t for t = t and t = 0.88t at T = 0.11t for 8% doping and U = 8t. Note the increase in low energy spectral weight for t = 0.88t (a) Normalized values of T c, V d and P d0 as a function of t at 8% doping, U = 6t and N c = 4. V d and P d0 are described in the text. (b) Néel temperature T N /t for different values of t at 2% doping for U = 8t

15 Figures 2.4 (a) Magnetic structure factor S(Q, ω) for U = 6t, δ = 0.08 and temperature T = 0.087t at Q = (0, π) for different values of t. The location of the peak is a measure of the characteristic spin excitation energy, here defined as J eff. Inset of (a) is a blow up to show the area around peak positions. T c (t ) and J eff (t, T = 0.087t) normalized to their values at t = t, as a function of t for (b) U = 6t, δ = 0.08, (c) U = 8t, δ = 0.08 and (d) U = 8t, δ = Note that both normalized T c (t ) and J eff (t ) monotonically increase with t The critical temperature T c as a function of impurity potential for N c = 16 and U = 4t, at impurity concentrations x = 3% and x = 6%. Error bars are calculated from the extrapolation of the pair-field susceptibility [89]. Inset: Blowup of the region of small impurity potential The critical temperature T c as a function of impurity concentration for N c = 16, and U = 4t, at impurity potentials V = t, V = 4t and V = 20t. Error bars are calculated from the extrapolation of the pair-field susceptibility [89]. The AG result is a fit to the critical concentration for V = 20t The dynamic spin susceptibility at Q = (0, π) for N c = 16, U = 4t and V = 4t, at impurity concentration x = 3%. The location of the peak, is a measure of the effective spin-wave exchange 2J eff [93]. Inset: Spin coupling constant J as a function of V Temperature times the difference susceptibility for N c = 16, and U = 4t, as a function of temperature for various impurity potentials and concentrations. The solid lines are guides to the eye

16 Effect of Weak Inhomogeneities in High Temperature Superconductivity 4.1 a) and b) histograms representing the distribution of Gs(β/16) and respectively sign s for 3000 measurements per bin. c) histogram representing the distribution of G(β/16) when a G point is obtained as a ratio of 3000 averaged Gs and 3000 averaged s points. d) histogram representing the distribution of G(β/2) when every G point is obtained by re-binning 30 initial G points obtained as a ratio of 600 averaged Gs and s points. The dashed lines represent the best Gaussian fit to the data One-particle spectra at K = (π, π/2) calculated with different amounts of data using a) the new method and b) the old method Two-particle spin susceptibility spectra A(ω) = χ (ω)/ω at K = (0, π/2) calculated for different amounts of data with a) the new method and b) the old method Imaginary part of the spin susceptibility χ (ω) = A(ω)ω at K = (0, π/2) calculated for different amount of data with a) the new method and b) the old method

17 Chapter 1 Introduction 1.1 Strongly Correlated Electronic Systems Strongly correlated electronic materials exhibit important and interesting phenomena such as Mott transition, magnetism, heavy fermion behavior, Kondo effect, high temperature superconductivity, charge ordering, and giant magneto resistance. Moreover, one also expects to see emerging new phenomena due to ever increasing complexity [26] of newly discovering materials. Increasing complexities and competing orders seen in these materials are not just because of the strong correlations alone, but due to comparable kinetic and potential energies of electrons in the material, which also render calculations of physical properties more difficult. Electrons or holes in those physics rich materials are correlated in the sense that their motion is strongly related to each others presence, both temporally and spatially. Thus, in addition to the basic intrinsic correlations among the electrons which are built into Fermi-Dirac statistics via Pauli exclusion principle, other specific correlations also play an important role in determining the properties of strongly correlated materials. For example, on-site Coulomb interaction between electrons or electron-phonon interaction between electrons and phonons 9

18 10 Effect of Weak Inhomogeneities in High Temperature Superconductivity may be essential for describing some of these phenomena. Therefore, it is reasonable to expect that collective effects of many particles or strong correlations among either different or same particles, as one of the key ingredients to have such phenomena in physics. Materials which can not be modeled or described using single particle approaches like Landau-Fermi liquid (FL) theory [17] or Band theory of solids generally refers under the category of strongly correlated electronic systems. FL theory assumes that even though interactions among the electrons are strong, physical quantities do not change qualitatively that from a non interacting Fermi gas. Thus, physical properties of some materials at or near zero temperature can be described independent of a model Hamiltonian, using a set of elementary excitations which represents collective behavior of all the particles in the system. These elementary excitations are known as quasi particles and have a finite life time depending on their distances from the Fermi surface. Quasi particles are treated as non interacting and their density of states form a delta function at the chemical potential. In a strongly correlated system, quasi particles are not well defined. Band theory of solids is constructed assuming that lattice has a periodic ionic potential which electron experiences and the Coulomb interaction between the electrons plays no role. According to the Band theory, a material is metallic if the highest energy band is partially filled. A famous example where band theory fails to capture physics of strongly correlated systems, even qualitatively is, insulating transition metal oxides (TMOs) [101]. Since TMOs have narrow electronic energy bands with valence electrons sitting on partially filled d-orbitals, Band theory predicts insulating TMOs as conductors. As early as in 1937, Mott and Pieierls [102] were the first to recognize that insulating behavior of TMOs is a result of the localization of otherwise itinerant d-electrons due

19 1. Introduction 11 to strong on-site Coulomb repulsion. Therefore insulators of this kind are known as Mott insulators and the relevant metal to insulator transition is called Mott transition. Mott insulators are antiferromagnetic in nature, meaning that neighboring spins are antiferromagnetically aligned. Their insulating behavior is due to many-body effects. The copper-oxygen based ceramic compound La 2 CuO 4 is another strongly correlated system and a Mott insulator. Excitingly enough, when appropriately doped with Ba, the Mott insulator becomes superconducting at low temperatures, but at a significantly higher temperature (35K in the original discovery) than known conventional superconductors at that time. It was the discovery made in 1986 by Alex Müller and Georg Bednorz [19] and initiated the field of high temperature superconductivity (high- T c superconductivity). They won the Nobel prize in the following year. The family of such high temperature superconducting compounds which contain copper-oxide planes are commonly known as cuprates. Oxygen atoms located in between the Cu-O planes are called apical oxygen and believed to play a major role [139]. Cuprates are perhaps the most peculiar strongly correlated electronic system known today and further discussed in Sec Hubbard Model In 1963 J. Hubbard [55] in his seminal paper, Electron correlations in narrow energy bands, proposed a simple many body Hamiltonian which includes kinetic energy (or band width) and on-site screened Coulomb interaction U of the system to describe TMOs. 1 During the same year M. C. Gutzwiller [41] and J. Kanamori [65] almost simultaneously presented the same Hamiltonian in different papers. However, later the Hamiltonian became famous as the Hubbard model. Hubbard model and its variants 1 Hubbard concluded that the insulating behavior of TMOs is due to strong correlations and apparently did not know about the previous work by Mott and Pieierls [1].

20 12 Effect of Weak Inhomogeneities in High Temperature Superconductivity like the t-j model are now among the extensively studied models in condensed matter theory with respect to strongly correlated electronic lattice systems in addition to the Periodic Anderson Model and Kondo model. Despite the simplicity and numerous studies on the Hubbard model, still there is no at least nearly a complete solution to the model except in one [77] and the infinite [34,58] dimensions. In this dissertation we shall use the Hubbard model in order to study two very interesting problems in strongly correlated lattice electronic systems as described in Sec The model received a sparkling attention by condensed matter theorists soon after the discovery of high temperature superconductivity since Anderson [8] as well as Zhang and Rice [139] proposed one band two dimensional (2D) Hubbard model as a suitable candidate to study cuprates. Now we know that the model is rich enough to capture the most of the physics of cuprates [115]; for example, antiferromagnetism, d-wave superconductivity, pseudo-gap, waterfall structures and stripes. We basically focus on some issues related to d-wave superconductivity as introduced in brief in Secs.1.6 and 1.7. In this dissertation, we shall model cuprate superconductors using the Hubbard model. The one-band homogeneous Hubbard model is given in Eq H = t i,j σ(c iσ c jσ + c jσ c iσ) + U i n i n i, (1.1) where c iσ (c iσ) creates (destroys) an electron at site i with spin σ, n iσ = c iσ c iσ and i, j represents the nearest neighbors. t is the nearest neighbor hopping and U is the on-site Coulomb repulsion between the electrons, which is site independent. When U < 0 the model is referred as negative U Hubbard model. The model is particle-hole symmetric at half filling only if the lattice is bipartite.

21 1. Introduction 13 Figure 1.1: A schematic representation of the 2D one hole doped Mott insulator. When an adjacent site is vacant, electrons can hop into that site without on-site repulsion U. Where t is the near-neighbor hopping. This is equivalent to a hole moving in an antiferromagnetic background. Hubbard model describes the effect of two opposite tendencies. The kinetic energy or the electron hopping tends to delocalize the electrons into Bloch or itinerant states and favors metallic behavior. The on-site Coulomb repulsion or the potential energy tends to localize the electron and could favor insulating states with magnetic behavior. At half filling and U greater than some critical value t c, the conduction band splits into two sub-bands. They are called upper and lower Hubbard bands and the chemical potential falls in between the sub-bands resulting an insulator. It should be pointed out that the very existence of the upper Hubbard band depends on the electron filling or the site occupation since it is a many body effect. Inclusion of different interactions with magnitudes which are comparable or larger than the kinetic energy in models of strongly correlated systems renders the problem difficult or impossible to solve using conventional techniques like perturbation theory, due to the absence of a small parameter. However, there are well established methods

22 14 Effect of Weak Inhomogeneities in High Temperature Superconductivity to calculate physical properties of materials in the limit of U << t like Band theory and in the limit when U >>> t or very strongly correlated limit, like the atomic solution. Those methods are associated with constructing wave functions of the system. They employ different bases for constructing wave functions. In the limit U << t, delocalized plane waves or Bloch waves form the appropriate basis for constructing wave functions. In contrast, for very strongly correlated systems, the Wannier orbitals form the appropriate basis where localized electrons sit. It is because of this problem of complementarity, i.e. having two different bases for the solutions at those limits; one can not obtain an interpolated solution between them 2, which includes the interested region where most of new phenomena occur such as competing orders in cuprates. This necessitates the use of approximation techniques and numerical simulations for solving such models. Methods like finite size numerical simulations (FSS) have been vastly employed for studying those models on simple lattices and they have contributed a great deal of knowledge to the subject. Disadvantages of FSS are the finite size effects in calculated bulk properties of the system due to small lattice sizes used for simulations. Therefore it is important to look for methods that can be used to simulate correlated systems in the thermodynamic or large N limit. 1.3 Quantum Cluster Theories Metzner and Vollhardt [98] introduced Dynamical Mean Field Theory (DMFT) for strongly correlated systems, simplifying the calculations, showing that in the limit of infinite dimension, inter-site correlations can be neglected while retaining local dynamical correlations. In other words, in the limit of infinite dimension electron self energy looses its momentum dependence, but retains imaginary frequency. In 1992, 2 Conversations with Mark Jarrell.

23 1. Introduction 15 Figure 1.2: There are well established methods for the calculation of physical properties of materials in the free electron limit or when U <<< t and the other in the atomic limit or when U >>> t. The most interesting physics of cuprates is found in the intermediate region when U t and the model is difficult to solve. Jarrell [58] and Georges and Kotlier [34] successfully applied DMFT to solve Hubbard model in the infinite dimension. DMFT becomes exact only in the infinite dimension. Therefore its applications in any finite dimension are called Dynamical Mean Field Approximation (DMFA). DMFA has been very successful in studies of strongly correlated systems [70, 132]. However, to address some problems in strongly correlated systems, DMFA had to be modified to incorporate inter-site correlations and the resulting approximation had to satisfy the causality. Cluster extensions to DMFA, which satisfy causality requirement, such as Dynamical Cluster Approximation (DCA) [47,48,60,91], Cellular Dynamical Mean field theory (CDMFT) [69] and Cluster Perturbation Theory (CPT) were introduced in order to account for the inter-site or non local correlations. Now it is customary to call these approaches commonly as Quantum cluster theories [90]. In DCA, the infinite size periodic lattice problem is mapped onto a self-consistently embedded cluster problem (Fig. 1.3). When the cluster size is one, DCA recovers the DMFA and as the cluster size increases DCA systematically incorporates non local correlations into the problem. Therefore even with small cluster sizes, DCA has been very successful for addressing problems where short range correlations are the most important; for example, short-

24 16 Effect of Weak Inhomogeneities in High Temperature Superconductivity Figure 1.3: Quantum cluster approaches, like the DMFA and DCA, map the infinite lattice problem onto a self-consistently embedded cluster problem. The figure and caption are from Ref. [61]. range antiferromagnetic correlations in high-t c superconductivity. In DCA, the simplified finite size periodic cluster problem is solved using a quantum cluster solver (see Fig.1.3). Quantum cluster solvers could be based on either diagrammatic perturbation theories or non perturbative methods. Perturbation techniques may involve second order perturbation theory, fluctuation exchange approximation or non-crossing approximation. Non-perturbation techniques are the most important of these and may be quantum Monte Carlo (QMC), exact diagonalization or Wilson renormalization group method. In this thesis work we will be using DCA with QMC cluster solver for calculations and they are briefed in Secs. 1.4 and 1.5 respectively. A complete review about quantum cluster theories is given by T. A. Maier et al. in ref. [90]. 1.4 Dynamical Cluster Approximation (DCA) As mentioned before [47, 48, 60, 63, 91], the motivation for DCA is systematically incorporating non-local spacial correlation into DMFA. Non-local spacial correlations are included by partially restoring the momentum conservation which is neglected in DMFA [35,109]. Consider the one-point vertex shown in the Fig Momentum conservation at

25 1. Introduction 17 Figure 1.4: One point vertex with entering (k 1, k 2 ) and leaving(k 3, k 4 ) momenta. the vertex is characterized by the Laue function. If k 1 and k 2 (k 3 and k 4 ) are the entering and (leaving) momenta of the Green s functions at each vertex, conservation of momentum requires, (k 1,k 2,k 3,k 4 ) = r exp [ir (k 1 + k 2 k 3 k 4 )] (1.2) = Nδ k1 +k 2,k 3 +k 4, up to reciprocal lattice vector. DMFA assumes (k 1,k 2,k 3,k 4 ) 1 at finite dimensions [35,63,109] and simplifies the application of momentum conservation at each vertex. This is equivalent to freely summing over all momenta k of the propagators at each vertex. Therefore the summation yields a coarse graining transformation that leads to a local self energy and Green s functions which only depend upon Matusbara frequency ω n = (2n + 1)πT at temperature T. DCA restores the momentum dependence of the propagators by restricting the coarse graining to individual cells of equal size within the first Brillouin zone (FBZ). To form the DCA cluster problem, FBZ is divided into N c = L D cells with size k = 2π/L instead of a single cell used in DMFA. Where L is the linear cluster size. Let K be the momenta at the center of a cell. As in the DMFA, propagators are coarse grained

26 18 Effect of Weak Inhomogeneities in High Temperature Superconductivity in k, but locally, within an each cell (Eq. 1.3). In this way propagators retain their momenta due to the momentum conservation between the cells. This is explained via the Laue function for the DCA shown in Eq: 1.4 and ensures the inclusion of non-local correlations up to the linear cluster size L. By increasing the cluster size N c, DCA interpolates solution between the DMFA (N c = 1) and the exact solution. Explicit details about the microscopic derivation of DCA is discussed in Ref. [63]. Ḡ(K,iω n ) = N G(K + k;iω n ) (1.3) N c k DCA (k 1,k 2,k 3,k 4 ) = N c δ M(k1 )+M(k 2 ),M(k 3 )+M(k 4 ) (1.4) DCA assumes that local and irreducible quantities on the cluster is equal to the local and irreducible quantities of the lattice. Thus the irreducible cluster self energy Σ c (K, ω) is equal to the lattice self energy Σ(K, ω) DCA/QMC Algorithm The Fig. 1.5 shows a sketch of the DCA/QMC algorithm [63]. First part of the algorithm self consistently calculates the cluster Green s functions and self energy by employing a QMC method. Second part or the Analysis code uses the converged cluster Green s functions and the cluster self energy to calculate lattice quantities. Lattice quantities include lattice coarse grained Green s function, kinetic and potential energies, charge and spin susceptibilities, momentum distribution function, leading Eigenvalues of particle-hole and particle-particle channels and approximated d-wave pairing interaction. Third part of the algorithm uses Maximum Entropy method (MEM) to analytic continue imaginary time single and two particle QMC Green s functions data to real frequencies. Single particle density of states, angle resolved photo-emission spec-

27 1. Introduction 19 Figure 1.5: DCA/QMC algorithm [63] consists of three parts. 1) The self consistent loop, 2) Analysis code and 3) MEM analytic continuation code. See the text for description. tra (ARPES), dynamic spin and charge susceptibilities are examples for the quantities calculated in the MEM part. DCA self consistent loop is initialized by calculating Ḡ using an initial guess for the self energy Σ either by second order perturbation theory or zero. 2) The host Green s function G, which serves as the bare Green s function to the QMC cluster solver is calculated. Introduction of G is required to avoid over-counting the diagrams [61,63]. At this point the host Green s function is Fourier transformed from momentum-frequency variables to space-imaginary time variables to be suitable for QMC calculations. 3) QMC measurements are made for various types of cluster Green s functions. 4) New Σ is extracted using G and the cluster Green s functions. Then Σ is Fourier transformed back to momentum-frequency variables. 5) The new self energy is used to initiate a new iteration 3. 3 Technical details about the algorithm such as high frequency conditioning, use of point group symmetries to reduce errors and QMC code optimizations are discussed in Refs. [47,48,60,63,91].

28 20 Effect of Weak Inhomogeneities in High Temperature Superconductivity 1.5 Quantum Monte Carlo Method (QMC) Quantum Monte Carlo methods can be used to solve the cluster problem in quantum cluster theories. QMC method is a systematically exact way of calculating Green s functions in imaginary time τ as an alternative to perturbation methods. There are several QMC methods available for this purpose. Studies reported in this thesis were done using Hirsch-Fye QMC method [51]. The essential idea of the Hirsch-Fye QMC method is to recast the interacting electron problem in Eq. 1.1 into an imaginary time dependent non-interacting problem which can be solved without any approximation. The derivation of Hirsch-Fye method in relation to DCA, which uses Suzuki-Trotter decomposition and the Hirsch-Hubbard- Stratonovisch transformation is given in Ref. [63]. There the only error comes from the discretitation of imaginary time and is proportional to τ 2. The non-interacting electrons obtained in this way interact with space and imaginary time dependent Isinglike fields which are coupled to the Z component of the non-interacting electrons. A particular field configuration corresponds to a snap shot of the evolution of the Green s function over space and imaginary time and bears a probability to be in that configuration. QMC algorithm proposes a change from a filed configuration to another by flipping an Ising-like spin at a particular space and imaginary time point. If the flip is accepted according to either Metropolis [97] or heat bath algorithm [63], the Green function is updated. QMC measurements are made when the system reaches the equilibrium. The main disadvantage of the method is the QMC sign problem and is discussed in detail in chapter 4.

29 1. Introduction Unconventional Superconductivity Superconductivity that can not be explained using the BCS [16] theory (non S-wave superconductivity) falls under the category of unconventional superconductivity. Unconventional superconductivity observed in high-t c cuprates and some organic compounds and might have the same origin of superconductivity. The highest critical temperature at ambient pressure observed in cuprates is 135K [75, 121] (one report indicates 200K cuprate compound which does not form stoichiometrically [2]). In organic compounds present highest T c is around 12K under pressure [75]. Undoped cuprates are antiferromagnetic Mott insulators and superconductivity emerges upon either electron or hole doping and as the temperature lowers under ambient pressure. In the case of organic superconductors pressure substitutes doping, and the material remains half filled while it undergoes a transition from Mott insulating phase to superconductivity. Since high-t c superconductivity emerges proximity to Mott insulator we are inclined to view high-t c superconductors as doped Mott insulators [76,106]. In this study we focus on high-t c cuprates. Literature regarding the high-t c superconductivity is enormous and perhaps the most studied phenomena in the condensed matter physics. It may be best to introduce cuprates is via the commonly accepted features of the phase diagram. These features are introduced without references since they are commonly discussed elsewhere. Fig.1.6 shows the typical temperature vs doping (T-x) schematic phase diagram of high-t c cuprates. It is experimentally obtained and does not depend on any particular high-t c theory. One can immediately notice the asymmetry of electron and hole doped sides. The reason for this is still unknown. We concentrate on the hole doped side where inhomogeneous structures are observed experimentally. Upon doping up to 5% and at low temperatures cuprates still exhibit antiferromagnetic Mott insulating phase as the parent compound does. It has a very unconventional and mysterious normal

30 22 Effect of Weak Inhomogeneities in High Temperature Superconductivity Figure 1.6: Schematic T vs X phase diagram of cuprates obtained from experiments. Where X is the doping concentration. RHS: hole doped region, LHS: electron doped region. AF-Antiferromagnetism, T N -Néel temperature, T c -superconducting transition temperature, d SC -d-wave superconductivity, T*- crossover temperature, PG-pseudo-gap regime, SM-strange metal-non Fermi Liquid, E N -Nernst effect (below brown line), Yellow region between the AF and d SC is the spin glass region, FL-Fermi liquid. state regime called strange metal (SM), so called because it does not follow standard FL behavior. A pseudo-gap (PG) regime exists where the normal state shows a gap in some wave vectors even before superconducting. Origin of the pseudo-gap is not completely understood yet, but may be coming from pre-formed pairs according one lines of thought. The pseudo-gap regime is more peculiar since it shows inhomogeneous structures like stripes and checkerboard patterns on the nanometer scale. The superconducting dome starts at about 5% hole doping (x) and the highest T c occurs at optimal doping about x = 15%. Superconductivity vanishes around 25% doping. Experiments find that the symmetry of the superconducting gap is d-wave and the gap increases towards half filling with decreasing super-fluid density. The dome region above and below the optimal doping are called over-doped and under-doped regions

31 1. Introduction 23 respectively. At or around 1 doping an anomaly or a suppression of T 8 c occurs in LSCO and YBCO (YBa 2 Cu 3 O y ) compounds where static charge order form. Experiments have observed a Nernst (EN) region just above the superconducting dome near optimal and under doped regions, where they find a vortex state. A FL regime is present above 25% doping and at low temperatures. It is important to note some of the differences and similarities between conventional BCS and high-t c superconductivity from a theoretical point of view. Conventional or BCS superconductors are S-wave superconductors and they emerge from an weak coupling instability to the Fermi surface of a good metal [23]. Here the good in the sense that the normal state of the metal is a Fermi liquid. In contrast, high- T c superconductivity emerges as an instability to a non-fermi liquid state and the superconducting state exhibits d-wave symmetry [68,114]. Coherence length ζ of high- T c superconductors is short and phase fluctuations are strong compare to those of conventional superconductors. Another important difference is the way phase coherence occurs. In the conventional case, formation of Cooper pairs and the phase coherence occur simultaneously at T c. In high-t c superconductors, pairs may be formed well before the phase coherence temperature T c [23,30] and define two distinct energy scales for pairing and T c. Apart from the d-wave order parameter symmetry and nodal quasi particles in high-t c superconductors, other properties of the superconducting state are similar in both cases. Independent of the type of superconductivity, super current is carried by Cooper paired electrons or charge 2e bosons. While electron-phonon interaction is responsible for the pairing in conventional case [16], it is still an open question for the high-t c superconductivity.

32 24 Effect of Weak Inhomogeneities in High Temperature Superconductivity Current Theories of High Temperature Superconductivity Understanding the high temperature superconductivity is one of the great challenges in condensed matter physics. Even after two decades of extensive search for an explanation of the underlying high-t c pairing mechanism, the scientific community is still in debate. Given the ability to reproduce most of the experimental observations of cuprates [115] at least qualitatively or semi quantitatively, it is reasonable to believe that either one band Hubbard model or it s variants like t-j model in its homogeneous (or inhomogeneous) version contain the essential physics of high-t c superconductors. Apparently there are five schools of thoughts on the table for a while. Following Philip Anderson s original RVB idea or similar concepts some claim exchange interaction is responsible for pairing and the pairing interaction is non-retarded [9 11,76]. The SO5 theory S. Zhang [140] introduced a theory with a five dimensional order parameter (3 for space and 2 for superconductivity) to tackle the problem of antiferromagnetism and superconductivity. According to D. J Scalapino and others [56, 99, 116] it is the spin fluctuation mechanism with retarded pairing interaction. S. K. Kivelson and collaborators [12, 23] take a different direction and propose a spin gapped microscopic mechanism in which inhomogeneities of the system are primarily responsible for establishing high-t c superconductivity. Those four scenarios one way or another share the paradigm of magnetic origin of high-t c superconductivity which arise as a result of strong repulsion between electrons. The fifth school, following the conventional wisdom, argue that some form of electron phonon coupling is still responsible for pairing in high-t c cuprates [103].

33 1. Introduction Inhomogeneities in high-t c Cuprates and Theory Jan Zaanen and Olle Gunnarsson predicted the existence of inhomogeneous structures that could occur in the normal state of cuprates using the Hubbard model with a very large U [138]. Later experiments confirmed the presence of quasi 1-D stripes and 2-D checkerboard patterns in under doped LSCO, BSCO (Bi 2 Sr 2 CaCu 2 O 8+δ ) and YBCO [53,54,67,100,120,128]. As mentioned before these inhomogeneities start to develop above the superconducting T C. Fig.1.7 shows checkerboard charge ordered state as observed by scanning tunneling microscopy [54]. Stripes observed through neutron scattering experiments can readily be regarded as a bulk property while 2D-charge modulations observed through STM are in question. Presently there is no evidence that these ordering or inhomogeneities are essential for superconductivity. However static charge order seems to suppress superconductivity [23] in cuprates. On the other hand one may speculate that, if exists, dynamic charge order might enhance or compete with the superconductivity in cuprates. Therefore, based on experimental observations and numerous other calculations that suggest isolated stripes have strong pairing correlations [15, 25, 104, 105], it was suggested that stripes or charge inhomogeneities are essential for superconductivity in cuprates [12]. In this dissertation we shall attempt to explore the role of weak inhomogeneity in high temperature superconductivity. We model cuprates using the Hubbard model and use dynamical cluster approximation with Quantum Monte Carlo cluster solver to simulate the problem.

34 26 Effect of Weak Inhomogeneities in High Temperature Superconductivity Figure 1.7: A filtered STM image of local density of states of slightly over-doped Bi 2 Sr 2 CaCu 2 O 8+δ crystal [54]. The arrows indicate the direction of Cu-O bonds and the figure was taken from Ref. [67]. 1.8 Outline of the Dissertation This dissertation is a collection of results of three projects I participated during my graduate studies at the University of Cincinnati. My contributions are described in detail in each of the corresponding chapters due to the highly collaborative nature of these projects. The organization of the dissertation is as follows. In Chapter two we use the checkerboard Hubbard model to investigate the effect of weak inhomogeneity on d-wave superconductivity. Using a four site cluster DCA/QMC simulation, we show that a weak inhomogeneity in hopping suppresses the d-wave superconducting T c. We show that the effective exchange interaction of the system is proportional to T c. In the third Chapter, we, in collaboration with A. Kemper from the Florida state University investigate the effect of local disorder on d-wave superconductivity. Confirming the observations made in Chapter three, we show that the d-wave T c of the weakly disordered system increases with increasing effective exchange energy compare

35 1. Introduction 27 to the homogeneous system. In Chapter four we propose a method to analytically continue QMC data with a severe sign problem by improving the existing maximum entropy method. Finally, in Chapter five we conclude the final outcome of the three studies. 1.9 Other project collaborations In addition to the three projects mentioned before, I involved in the following three projects and their results have not reported in this dissertation. In a collaboration with A. Macridin, M. Jarrell and Th. Pruschke, I developed a dynamical cluster quantum Monte Carlo approximation for the two chain Hubbard model including corresponding maximum entropy codes. I also developed DCA codes for a system of coupled Hubbard ladders where inter-ladder coupling is treated in the mean field level. I collaborated mainly with M.A. Majid for a study about ferromagnetism in the periodic Anderson Model and the preprint Charge Density Wave Driven Ferromagnetism in the Periodic Anderson Model is published in condensed matter archives [92]. I also made another collaboration with M. Jarrell, R. Scaletter and his group on a project to study the physics of Hubbard Multi-layers. In this project I modified relevant maximum entropy codes to analytically continue QMC data to calculate density of states for different layers. This is an on-going project and is expected to be completed soon.

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37 Chapter 2 d-wave superconductivity in the checkerboard Hubbard model 2.1 An introduction to inhomogeneity and high-t c superconductivity The normal states of conventional or BCS superconductors are metals and do not show different kinds of ordering which compete or co-exist with superconductivity. In contrast, high-t c cuprates are antiferomagnetic doped-mott insulators and behave as bad metals [23]. They show different kinds of orders as discussed in sec.1.7. Energies of these ordering states and the d-wave superconducting state seem to be comparable. Addressing physics of these bad metals is hard and there is no complete theoretical understanding of how different orders such as superconductivity, antiferromagnetism and charge ordering prefer one over the other or the co-existence. Carlsen et al. [23] argue that since the electron density of strongly correlated systems are much more prone to inhomogeneity than in normal metals, mesoscale intermediate structures are important in determining physics of high-t c superconductors. An important question 29

38 30 Effect of Weak Inhomogeneities in High Temperature Superconductivity naturally arise is, are these mesoscale structures essential for the pairing mechanism in high-t c superconductivity? Along this line of thoughts S.K Kivelson and co-workers have contributed a great deal of theoretical work to the subject and we refer to excellent article by Carlsen et al. [23] for a complete description. Numerical studies on ladders and square lattices using Hubbard and t-j models provide different results about the ground state of these models, perhaps due to the limitations or biases of the methods employed. However it is well known that doped isolated Hubbard and t-j ladders exshibit d-wave like superconducting correlations [15, 25, 104, 105]. In addition, DCA studies done on isolated Hubbard ladders also show diverging d-wave like superconducting correlations 1. These superconducting correlations are due to the strong on-site Coulomb repulsion and correlations are strong along the transverse direction to the ladder. Thus one could imagine that a system of weakly coupled ladders, in which ladders primarily exhibit d-wave like superconducting correlations might provide a mechanism for superconductivity. Based on the fact that these isolated ladders possess a large spin gap, Emery et al. suggested Spin gap proximity effect as a mechanism of superconductivity and referred it as magnetic analog of the usual superconducting proximity effect [31]. According to them one expects a d-wave superconducting ground state for a system of hole doped coupled ladders(i.e.2d) since hole pairs can hop between the ladders due to the Joshepson coupling between the ladders. Where the long range order with a superconducting phase coherence establishes in the ladder system as a dimensional crossover from quasi 1D to 2D. Instead, density matrix renormalization group (DMRG) calculations reported formation of stripes on doped 6-leg Hubbard ladders [135] for values of U/t ranging from 6 to 20 and their results have been verified by George Hager et al. [42]. However, other numerical calculations [46,88,126] on Hubbard and t-j models report the existence of 1 Unpublished studies by Doluweera et al.