Mott Transition in Strongly Correlated Materials: Many-Body Methods and Realistic Materials Simulations Tsung-Han Lee

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2017 Mott Transition in Strongly Correlated Materials: Many-Body Methods and Realistic Materials Simulations Tsung-Han Lee

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COLLEGE OF ARTS AND SCIENCES

MOTT TRANSITION IN STRONGLY CORRELATED MATERIALS: MANY-BODY

METHODS AND REALISTIC MATERIALS SIMULATIONS

TSUNG-HAN LEE

A Dissertation submitted to the Department of Physics in partial fulﬁllment of the requirements for the degree of Doctor of Philosophy

2017

Tsung-Han Lee defended this dissertation on July 12, 2017. The members of the supervisory committee were:

Vladimir Dobrosavljevic Professor Directing Dissertation

Naresh S. Dalal University Representative

Efstratios Manousakis Committee Member

Luis Balicas Committee Member

Jorge Piekarewicz Committee Member

The Graduate School has veriﬁed and approved the above-named committee members, and certiﬁes that the dissertation has been approved in accordance with university requirements.

ii To my beloved family.

iii ACKNOWLEDGMENTS

The time at Florida State University was certainly one of the best time in my life. I owe so much gratitude to the people who helped me and supported me during this period. I would like to express my appreciation to all of them. First of all, I would like to thank my advisor, Vladimir Dobrosavljevic, for all his support and encouragement. His deep knowledge and perspective in condensed matter physics always stimulated me to contemplate critically on the subjects that I have learned, and his bright ideas always inspired me the direction to tackle the problems. From him, I learned the attitude of being a rigorous scientist and a critical thinker. It was my pleasure to learn and work with him during my Ph.D. degree. I am also very grateful that he always supported me to study the problems that I am interested in, encouraged me to attend important summer schools and conferences, and helped me whenever I had any diﬃculties. During this process, I explored many interesting ﬁelds in condensed matter physics and broadened my horizon in material science. Besides my advisor, I would also like to thank my collaborators, Serge Florens, Nicola Lanata, Yongxin Yao, Jaksa Vucicevic, Darko Tanaskovic, Eduardo Miranda, and Vladan Stevanovic. From them, I learned so much invaluable knowledge and powerful techniques that will help me to proceed in my future career. I am especially thankful to Serge, Nicola, Yongxin, and Jaksa, who always gave me prompt responses regarding my technical and theoretical questions during our research. I would also like to express my gratitude to my committee members, Efstratios Manousakis, Luis Balicas, Jorge Piekarewicz, Simon Capstick, and Naresh S. Dalal, for spending their time to participate in my Ph.D. evaluation, and all their precious suggestions on my dissertation. My life as a Ph.D. student would not have been complete without colleagues and friends. I would like to thank my colleagues, Shao Tang, Hossein Javan Mard, and Samiyeh Mahmoudian. I enjoyed many interesting and inspiring discussions in our cubicle, which made our working environment very pleasant. I would also like to thank my friends, Shiuan-Fan Liou, Daniel Zuech, Mohammad Pouranvari, Wangwei Lan, and Niraj Aryal, for all the insightful discussions in our condensed matter study group. I am also happy to thank Yu-Che Chiu, Andrew Gallagher, Qiong Zhou, You Lai, Hiram Santiago, Nabin Rijal, Anish Bhardwaj, and Adewale Akinfaderin, for their

iv friendship. Furthermore, I would like to thank all my ﬁrends, whose names are too many to list, for accompanying me during this journey. Last but not least, I would like to thank my parents, Bong-Che Lee and Sau-Chin Ting, my wife, Ya-Huei Huang, and my brother, Hsien-Han Lee, for always being there to support me pursuing my dreams. I would like to dedicate this dissertation to them.

v TABLE OF CONTENTS

List of Tables ...... viii List of Figures ...... ix List of Symbols ...... xiii Abstract ...... xiv

1 Introduction 1

2 Models and Methods 3 2.1 Introduction ...... 3 2.2 First principle approach: Density functional theory ...... 4 2.3 Multiorbital Hubbard Model ...... 5 2.4 Dynamical mean field theory ...... 7 2.4.1 Mapping to Anderson impurity model ...... 8 2.4.2 Iterative Perturbation Theory (IPT) ...... 10 2.4.3 Slave-Rotor Non-crossing Approximation (NCA) ...... 11 2.4.4 Continuous Time Quantum Monte Carlo Method (CTQMC) ...... 13 2.5 Rotationally invariant slave-boson method (RISB) ...... 14 2.5.1 Rotationally invariant slave boson mapping ...... 15 2.5.2 Rotationally invariant slave boson Mean field theory ...... 16 2.6 The combination of first principle methods and many-body methods ...... 18 2.6.1 Correlated basis set ...... 19 2.6.2 LDA+RISB functional ...... 20

3 Unstable Domain Wall Solution in the Metal-Mott Insulator Coexistence Regime 22 3.1 Introduction ...... 22 3.2 Method ...... 24 3.2.1 Free energy and unstable solution ...... 24 3.2.2 Resistivity calculation and Sommerfeld approximation ...... 25 3.3 Result ...... 27 3.3.1 Along the ﬁrst order transition line ...... 27 3.3.2 Along the constant U trajectory ...... 29 3.4 Conclusion ...... 30

4 The Fate of Spinons at the Mott Point 33 4.1 Introduction ...... 33 4.2 Method ...... 35 4.3 Results ...... 37 4.3.1 Density of state and speciﬁc heat ...... 37 4.3.2 Phase diagram ...... 39

vi 4.3.3 Spinon orthogonality catastrophe ...... 41 4.4 Conclusion ...... 42

5 Emergent Bloch Excitations in Mott Matter 44 5.1 Introduction ...... 44 5.2 Ghost GA theory ...... 45 5.3 Results ...... 48 5.3.1 Benchmark: Energy, double occupancy, and quasiparticle weight ...... 48 5.3.2 Benchmark: Single-particle Green’s function ...... 50 5.4 Conclusion ...... 52

6 Structural Predictions and Polymorphism in Transition Metal Compounds 54 6.1 Introduction ...... 54 6.2 Method ...... 57 6.3 Results ...... 57 6.3.1 Predicted structures and unit cell volumes ...... 57 6.3.2 LDA ...... 58 6.3.3 LDA+RISB ...... 59 6.3.4 LDA+U ...... 61 6.3.5 Comparison: LDA+RISB and LDA+U ...... 63 6.4 Conclusion ...... 65

7 Nature of Am-O Bonding in Americian Chromate CsAm(CrO4)2 68 7.1 Introduction ...... 68 7.2 Method ...... 70 7.3 Electronic structure study ...... 71 7.3.1 LDA ...... 71 7.3.2 LDA+RISB ...... 72 7.4 Conclusion ...... 74

8 Conclusion and Outlook 75

Appendices A Parametrization of the Multiorbital Coulomb Interaction 78

B Analytical Continuation 80 B.1 Pade Approximation ...... 80 B.2 Maximum Entropy Method (MEM) ...... 81

C Total Energy and Speciﬁc Heat Calculation for Hubbard Heisenberg Model 83

References ...... 85 Biographical Sketch ...... 98

vii LIST OF TABLES

6.1 The smallest LDA+RISB quasiparticle weight, Z, among all the orbitals for all the materials at U = 13 eV and J = 0.9 eV at the experimental equilibrium volumes. Z = 0 corresponds to the Mott-insulating phase in Brinkman-Rice picture, and ﬁnite Z corresponds to the correlated metallic phase...... 61

6.2 The electronic phases predicted by LDA+U for all the materials and structures around the experimental equilibrium volumes. Here we set U = 13 eV and J = 0.9 eV. . . . . 63

6.3 Stable structures, electronic phases, and equilibrium volumes (in unit A˚3/f.u. ) predicted by LDA, LDA+U at U = 13 eV and J = 0.9 eV, and LDA+SB at U = 13 eV and J = 0.9 eV...... 67

7.1 Parameters of the Am-5f reduced density matrix computed by LDA+RISB assuming U=6 and J=0.7, see Eq. 7.1: Probability weights wi, corresponding quantum labels Ni (number of electrons) and Ji (total angular momentum)...... 74

viii LIST OF FIGURES

2.1 Schematic representation for the DMFT mapping: Mapping a lattice problem to an impurity coupled to a cavity ﬁeld, ∆(ω)...... 9

2.2 Schematic representation for the LDA+RISB algorithm...... 21

3.1 (a) Schematic representation for the near-ﬁeld microscopy measurement of V2O3 from McLeod et. al. [1]. (b) Near ﬁeld image of the metal Mott insulator coexistence regime by warming up the system from T = 164K to T = 184K. Blue areas correspond to the antiferromagnetic Mott insulating phase and red areas correspond to correlated metallic phase [1]...... 23

3.2 (a) The phase diagram of the half-fiiled Hubbard model, where Uc, Uc1, and Uc2 correspond to the first order transition line, the lower spinodal line, and the upper spinodal line, respectively. The trajectories of interest, in this project, are labeled by the blue solid arrow and the green dashed arrow, respectively. (b) The free energy functional along the first order transition line (FOTL), and (c) along a constant U

trajectory. Here Tc is the critical temperature, and TUc2 denotes the temperature at the spinodal line Uc2. l = 0 and l = 1 correspond to the insulator and metal solution, respectively...... 26

3.3 (a) The resistivity for metal (black circles), insulator (red squares), and unstable (green diamonds) solutions along the first order transition calculated from the continuous- time quantum Monte Carlo (CTQMC). The dashed lines are the resistivities calculated from the Sommerfeld approximation, Eq. 3.3. (b) Similar figure as in (a) calculated from the iterative perturbation theory (IPT). (c) The density of state (DOS) at the lowest temperature T = 0.1Tc (near Uc2(T = 0)). (d) The DOS near the critical temperature T = 0.75Tc (e) The quasiparticle weight, Z, for metal and unstable solution. (f) The scattering rate, Γ , for metal and unstable solution. The analytical continuation was done using the maximum entropy method (MEM) and the 5th order polynomial fitting (5th poly)...... 28

3.4 (a) The resistivity for metal (black circles), insulator (red squares), and unstable (green diamonds) solution along a constant U = 2.4 trajectory calculated from the continuous-time quantum Monte Carlo (CTQMC). The dashed lines are the resistivity calculated from the Sommerfeld approximation, Eq. 3.3. (b) Similar ﬁgure as in (a) calculated from the iterative perturbative theory (IPT) at U = 2.83. (c) The density of state (DOS) at low temperature T = 0.01. (d) The DOS near the spinadol line Uc2 at T = 0.019. (e) The quasiparticle weight, Z, for metal and unstable solution. (f) The scattering rate , Γ , for metal and unstable solution. The analytical continuation was done using maximum entropy method (MEM) and 5th order polynomial ﬁtting (5th poly)...... 31

ix 4.1 (a) Phase diagram for the κ-(BEDT-TTF)2Cu2(CN)3 where a spin liquid state is found in the Mott insulating phase from Kurosaki et. al. [2]. (b) Speciﬁc heat measurement from Yamashita et. al. for κ-(BEDT-TTF)2Cu2(CN)3 in the Mott insulating phase. −1 The ﬁnite intercept in CpT axis indicates a low-energy fermionic excitation in Mott insulating phase [3]...... 34

4.2 Proposed scheme to blend the static RVB approach with the DMFT, using an impurity solver based on emergent spinon degrees of freedom. The spinon self-energy Σf (iω), usually absent from a mean-field RVB treatment, is extracted from a spinon- based DMFT framework. Using the RVB self-consistent determination of the spinon bandwidth Jeff , this in turn allows to incorporate feedback effects due to scattering on charge excitations, which affect the gapless spin-excitations in the Mott state. . . . 35

4.3 Upper panel: electronic density of states across the Mott transition for T = 0.005D and J/D = 0.2, with increasing values of the Coulomb interaction U/D = 1, 2, 3. Lower panel: corresponding speciﬁc heat as a function of temperature. A striking linear in temperature contribution remains in CV at U/D = 3, while quasiparticles have disappeared from the density of states in the Mott phase...... 38

4.4 Phase diagram for J/D = 0.1 (upper panel) and J/D = 0.4 (lower panel) as a function of Coulomb strength U/D and temperature T/D. Continuous lines denote the metal- insulator boundaries, and dashed lines indicate the true first-order transition based on the free energy. Also, the region bounded by dots show the low-temperature onset of the spinon Fermi surface in the Mott insulator. The shaded region indicates metal that Jeff = 0 for the metallic state within the whole coexistence region. This demonstrates that spinons are only well defined quasiparticles in the low temperature Mott phase, and are never stable in the metal...... 40

4.5 Frequency-dependent spinon self-energy Σf (iωn) at temperature T = 0.005D for U/D = 1 (metal) and U/D = 3 (insulator). The strong enhancement in the metal is due to an x-ray edge eﬀect, which is a hallmark the spinon spectral density...... 42

5.1 Representation of a lattice including 2 ghost orbitals (α = 2, 3). The Hamiltonian of the system acts as 0 over the the auxiliary ghost degrees of freedom. In particular, the Hubbard interaction U acts only over the physical orbital α = 1...... 46

5.2 Evolution of total energy (upper panel), local double occupancy (middle panel) and QP weight (lower panel) as a function of the Hubbard interaction strength U for the single-band Hubbard model with semicircular DOS at half-ﬁlling. The Ghost-GA results are shown in comparison with the ordinary GA and with the DMFT+NRG results. The Ghost-GA boundaries of the coexistence region Uc1,Uc2 are indicated by vertical dotted lines. Inset: Integral of Ghost-GA local spectral weight over all frequencies (see discussion in main text)...... 49

5.3 Poles of the Ghost-GA energy-resolved Green’s function (bullets), see Eq. (5.7), in comparison with DMFT+NRG. The size of the bullets indicates the spectral weights

x of the corresponding poles. Metallic solution for U = 1, 2.5 and Mott solution for U = 3.5, 5...... 51

6.1 Four crystal structures, (a) NiAs-type, (b) rocksalt, (c) wurtzite, and (d) zincblende, studied in this project. NiAs-type and rocksalt have octahedral coordinate with hexagonal and cubic symmetry, respectively, and wurtzite and zincblende have tetrahedral coordinate with hexagonal and cubic symmetry, respectively...... 56

6.2 Predicted unit cell volumes and crystal structures. The colors, blue, green, red, and cyan, correspond to NiAs-type, rocksalt, wurtzite, and zincblende structures, respectively. The metallic and the Mott-insulating solution are labeld by ﬁlled and half-ﬁlled symbols, respectively. The circle, triangle, and diamond markers correspond to the experiment data, LDA, and LDA+RISB predictions, respectively. U and J is 13 eV and 0.9 eV, respectively, for LDA+RISB calculation...... 58

6.3 LDA energy-volume curves for (a) CoO, (b) CoS, (c) CoSe, (d) CrO, (e) MnO, and (f) FeO in the paramagnetic phase. The blue circle, green triangle, red square, and cyan diamond curves corresponds to NiAs-type, rocksalt, wurtzite, and zincblende structures, respectively. The experimental equilibrium volumes of each compound are indicated by the vertical colored solid lines, and the equilibrium volumes predicted by LDA are indicated by the colored dashed lines, where the colors correspond to the structures labeled in the legend...... 59

6.4 LDA+RISB energy-volume curves for (a) CoO, (b) CoS, (c) CoSe, (d) CrO, (e) MnO, and (f) FeO at U = 13 eV and J = 0.9 eV in the paramagnetic phase. The blue circle, green triangle, red square, and cyan diamond curves corresponds to NiAs-type, roscksalt, wurtzite, and zincblende structures, respectively. The solid and dashed curves correspond to the metallic and the Mott-insulating phase, respectively. The experimentally observed equilibrium volumes of each compound are indicated by the colored vertical solid lines, and the equilibrium volumes predicted by LDA+RISB are indicated by the colored dashed lines, where the colors correspond to the structures labeled in the legend...... 61

6.5 LDA+U energy-volume curves for (a) CoO, (b) CoS, (c) CoSe, (d) CrO, (e) MnO, and (f) FeO at U = 13 eV and J = 0.9 eV in the paramagnetic phase. The blue circle, green triangle, red square, and cyan diamond curves corresponds to NiAs- type, rocksalt, wurtzite, and zincblende structures, respectively. The experimental equilibrium volume of each compounds are indicated by the vertical colored solid lines, and the equilibrium volumes predicted by LDA+U are indicated by the colored dashed lines, where the colors correspond to the structures labeled in the legend. Note that the equilibrium volumes of CoS and CoSe are out side of the volume of interest. . 63

6.6 LDA+RISB energy diﬀerences relative to the ground state structure for all the selected structures as a function of interaction, U, and a ﬁxed exchange coupling, J = 0.9 eV. 64

xi 6.7 LDA+U energy diﬀerences relative to the ground state structure for all the selected structures in LDA+U as a function of interaction, U, and a ﬁxed exchange coupling, J = 0.9 eV...... 65

6.8 (a) LDA+RISB energy-volume curves for all the MnO polymorphs as a function of U with a ﬁxed J = 0.9 eV. The green, blue, red, and cyan curves correspond to rocksalt, NiAs-type, wurtzite, and zincblende structures, respectively. The solid and dashed line correspond to metallic and Mott-insulating phases. The black dashed line indicates the experimental equilibrium volume in the rocksalt structure. The rocksalt structures, evolves from a metastable state to a stable state after U 8 eV, and the wurtzite ≥ structure, predicted as a stable structure in LDA, becomes metastable. (b) LDA+U energy-volume curves for all the MnO polymorphs with the same parameters as in LDA+RISB. The experimentally stable rocksalt structures evolves from a metastable structure to a stable structure after U 6 eV, and the wurtzite structure predicted ≥ to be stable in LDA becomes metastable...... 66

7.1 Polyhedral representations of CsAm(CrO4)2. (a) View down the a axis showing the interlayer channels. (b) Single layer of one of the two-dimensional sheets. Am 3+ 2− is represented as dark red polyhedra, CrO4 as orange tetrahedra, and Cs + as tan spheres. (c) Coordination environment of Am3+ in CsAm(CrO4)2 with Am 3+ rep- 2− resented as dark red polyhedra, CrO4 as orange tetrahedra, and Cs+ as tan spheres...... 69

7.2 (a) Solid-state UV-Vis-NIR absorbance spectrum Vs. optical energy plot of CsAm(CrO4)2 showing a band gap of 1.65 eV...... 70 ∼

7.3 Local density approximation (LDA) density of state (DOS) for CsAm(CrO4)2. The energy is shifted with respect to the Fermi energy. The gap is around 0.6 eV...... 71

7.4 Local density approximation plus rotationally invariant slave-boson (LDA+RISB) density of state (DOS) for CsAm(CrO4)2. at U=6 eV and J=0.7 eV. The gap is around 2.2 eV...... 72

xii LIST OF SYMBOLS

The following short list of symbols are used throughout the document. The symbols represent quantities that I tried to use consistently.

EF Fermi energy E Energy V Volume T Temperature t Hopping amplitude U Coulomb interaction J Hund’s coupling D Half bandwidth Z Quasiparticle weight Γ Scattering rate ρ Resistivity CV Speciﬁc heat γ Sommerfeld coeﬃcient ω Real frequency ωn Ferminonic Matsubara frequency νn Bosonic Matsubara frequency G(ω) Green’s function A(ω) Spectral function Σ(ω) Selfenergy ∆(ω) Hybridization function

xiii ABSTRACT

Strongly correlated materials are a class of materials that cannot be properly described by the Den- sity Functional Theory (DFT), which is a single-particle approximation to the original many-body electronic Hamiltonian. These systems contain d or f orbital electrons, i.e., transition metals, actinides, and lanthanides compounds, for which the electron-electron interaction (correlation) effects are too strong to be described by the single-particle approximation of DFT. Therefore, complementary many-body methods have been developed, at the model Hamiltonians level, to describe these strong correlation effects. Dynamical Mean Field Theory (DMFT) and Rotationally Invariant Slave-Boson (RISB) approaches are two successful methods that can capture the correlation effects for a broad interaction strength. However, these many-body methods, as applied to model Hamilto- nians, treat the electronic structure of realistic materials in a phenomenological fashion, which only allow to describe their properties qualitatively. Consequently, the combination of DFT and many body methods, e.g., Local Density Approximation augmented by RISB and DMFT (LDA+RISB and LDA+DMFT), have been recently proposed to combine the advantages of both methods into a quantitative tool to analyze strongly correlated systems. In this dissertation, we studied the possible improvements of these approaches, and tested their accuracy on realistic materials. This dissertation is separated into two parts. In the first part, we studied the extension of DMFT and RISB in three directions. First, we extended DMFT framework to investigate the behavior of the domain wall structure in metal-Mott insulator coexistence regime by studying the unstable solution describing the domain wall. We found that this solution, differing qualitatively from both the metallic and the insulating solutions, displays an insulating-like behavior in resistivity while carrying a weak metallic character in its electronic structure. Second, we improved DMFT to describe a Mott insulator containing spin-propagating and chargeless fermionic excitations, spinons. We found the spinon Fermi-liquid, in the Mott insulating phase, is immiscible to the electron Fermi- liquid, in the metallic phase, due to the strong scattering between spinons in a metal. Third, we proposed a new approach within the slave-boson (Gutzwiller) framework that allows to describe both the low energy quasiparticle excitation and the high energy Hubbard excitation, which cannot be captured within the original slave-boson framework.

xiv In the second part, we applied LDA+RISB to realistic materials modeling. First, we tested the accuracy of LDA+RISB on predicting the structure of transition metal compounds, CrO, MnO, FeO, CoO, CoS, and CoSe. Our results display remarkable agreements with the experimental observations. Second, we applied LDA+RISB to analyze the nature of the Am-O chemical bonding in the CsAm(CrO4)2 crystal. Our results indicate the Am-O bonding has strongly covalent character, and they also address the importance of the correlation eﬀects to describe the experimentally observed electronic structure. In summary, we proposed three extensions within DMFT and RISB framework, which allow to investigate the domain wall structure in metal-Mott insulator coexistence regime, the metal-to- Mott-insulator transition with spinons excitation in the Mott-insulating phase, and the Hubbard excitation within RISB approach. Furthermore, we demonstrated that LDA+RISB is a reliable approximation to the strongly correlated materials by applying it to the transition metal compounds and the Americian chromate compounds.

xv CHAPTER 1

INTRODUCTION

Since Kohn and Sham developed Density Functional Theory (DFT) [4] in 1965, this powerful tool has been used a great deal within condensed matter physics and materials science. It serves as a reliable tool to predict the electronic structure of many materials. For examples, spin-polarized DFT accurately predicts the magnetic moments for Fe, Co, and Ni metals [5]. The modest extensions of DFT, such as DFT+U and GW, also show promising results on predicting the band gap for a wide range of semiconductors [6]. In addition, DFT is an indispensable tool for analyzing the electronic structure measurement based on Angle-Resolved Photoemission Spectroscopy (ARPES) and quantum oscillation techniques [7]. Besides the physics community, DFT is widely used in chemistry and in material science where they combine DFT with stochastic structural sampling to predict the stable crystal structures from the DFT formation enthalpy [8, 9]. The idea of DFT is to map the original many-body Hamiltonian of electrons, which is too complicated to solve, to a single-particle auxiliary problem, which can be calculated efficiently. This approximation faithfully predicts the electronic structure for various weakly or moderately correlated materials. However, the predictions become unsatisfactory for some materials containing transition metals, actinides, and lanthanides elements. This poor performance is now understood to be a result of strong electron-electron correlation effects, originating from the large on-site Coulomb interaction between the electrons in d or f shells in these materials. These kinds of systems are known as strongly correlated systems, displaying various intriguing emergent phenomena. For example, the Mott Metal-Insulator Transition (MIT) observed in transition metal oxides [10, 11], the strong mass enhancement found in heavy fermion compounds [12, 13], the high-temperature superconductivity discovered in cuprate and iron-pnictide materials[14], etc. In contrast to the weakly correlated materials, these phenomena cannot be described within the single-particle approximation of DFT. Therefore, complementary treatments are urgently needed. In more recent years, there have been significant developments in many-body methods. This methodology usually assumes a simple lattice model, e.g., a square, a cubic, or a hypercubic tight-

1 binding model, and the correlation eﬀects from Coulomb interaction, U, are added through a perturbative or a non-perturbative approach. The successful theories include Renormalization Group (RG), Slave-Boson (SB) approaches [15], Gutzwiller Approximation (GA) [16, 17, 18], and Dynamical Mean Field Theory (DMFT) [19]. Among these methods, the non-perturbative methods, such as DMFT, GA, and SB, provide a good description of the aforementioned strongly correlated phenomena by properly taking the low energy quasiparticle physics and the high energy atomic excitation into account. In contrary to DFT, these methods do not calculate the electronic structure from the ﬁrst principle. Consequently, they can only provide a qualitative description of the realistic materials. In the last decade, the incorporation of the realistic band structure produced by DFT within many-body methods, such as DFT+U, DFT+DMFT, and DFT+GA, have been developed intensively [20, 21, 22]. These techniques combine the power of DFT and many-body methods allowing to quantitatively investigate various phenomena in strongly correlated materials. For example, the origin of the high-spin to low-spin Mott transition in MnO [23], the volume collapse of plutonium

[24, 25], and the high-temperature superconductivity in iron pnictide and alkali-doped C60 [26, 27] have been resolved using the aforementioned methods. In this dissertation, I will focus on the extensions and the applications of these methods to various strongly correlated systems. The structure of this dissertation is as follows. In chapter 2, I will introduce the methods that we used in this dissertation, including DMFT, Iterative Perturbation Theory (IPT), Slave- Rotor Non-Crossing Approximation (NCA), Continuous-Time Quantum Monte Carlo (CTQMC), RISB and LDA+RISB. In chapter 3, I will present our research on the unstable solution in the metal-Mott insulator coexistence regime. In chapter 4, I will discuss our investigation on the role of spinon excitation in the Mott MIT. In chapter 5, I will introduce a new extension within the RISB framework that allows us to describe both the low-energy quasiparticle excitation and the high energy Hubbard excitation. In chapter 6, I will describe our investigation on the structural predictions for transition metal oxides, sulﬁdes, and selenides, using LDA+RISB approach. In chapter 7, I will present our study on the electronic structure of the Americium chromate compound,

CsAm(CrO4)2. Finally, in chapter 8, I will summarize the main ﬁndings in this dissertation.

2 CHAPTER 2

MODELS AND METHODS

2.1 Introduction

In condensed matter physics, the many-body Hamiltonian for electrons in solid reads,

2 2 2 2 2 Z e 1 e Z Z ′ e H = ∇i α + ∇α + α α , (2.1) − 2m − R r 2 r r ′ − 2M R R ′ α i ′ i i α ′ α α Xi Xi,α | − | Xi6=i | − | X α,αX | − | where ri is the coordinate of electron i, Rα is the coordinate of nuclei α, Zα is the atomic number for nuclei, m and M are the mass of electron and nuclei, and e is the charge of electron. Note that we set ¯h = 1 in this dissertation. In Born-Oppenheimer approximation, which assumes the electron moves much faster than nuclei, the electron and nuclei degrees of freedom are decoupled. Therefore, one can focus on the Hamiltonian for electrons,

2 2 2 i Zαe 1 e H = ∇ + = Te + Vext + Vee. (2.2) ′ − 2m − Rα ri 2 ′ ri ri Xi Xi,α | − | Xi6=i | − | When the interaction between electrons is set to zero, i.e., the last term of Eq. 2.2 is zero, the Hamiltonian becomes

2 Z e2 2 H = h(r )= [ ∇i α ]= [ ∇i v (r )]. (2.3) 0 i −2m − R r −2m − ext i α α i Xi Xi X | − | Xi The solution of Eq. 2.3 is a well-known Slater determinant,

φa1 (x1) φa1 (x2) ... φa1 (xN ) φa (x1) φa (x2) ... φa (xN ) 2 2 2 1 . ... Ψ(x ,x ,...,x )= , (2.4) 1 2 N √ . ... N! . ...

φ (x ) φ (x ) ... φ (x ) aN 1 aN 2 aN N

where φ (r) φ (r) = eikru (r) is the Bloch wavefunctions, and u (r) is a periodic function a ≡ nk nk nk related to the periodicity of the lattice. k is the wavevector conjugated to the real space coordinate,

3 ri, and the n is the band index for a speciﬁc wavevector. When the interaction between electrons becomes strong, the above non-interacting approximation is not suﬃcient to describe the system. Further treatments have to be applied to describe the interaction between electrons.

2.2 First principle approach: Density functional theory

One of the most popular approximations to the Hamiltonian in Eq. 2.2 is Density Functional Theory (DFT). The Hohenberg-Kohn theorem is the basis of this approach [28], which state the external potential, vext(ri), and the ground state electron density, n(ri), have a unique one to one correspondence. Therefore, the energy can be written into a functional of electron density n(ri),

E[n]= F [n]+ drvext(ri)n(ri)+ Enn, (2.5) Z where F [n]= T [n]+Eee[n] contains the kinetic energy term, T [n], and the electron-electron interaction, Eee[n], and Enn is the nucleus-nucleus interaction. The problem now becomes a minimization problem of an energy functional with respect to the electron density. Kohn and Sham propose that one can create an auxiliary non-interacting system with an external potential, vR(ri), and the corresponding ground state electron density is n(r )= ψ (r ) 2. With this auxiliary non-interacting i n | n i | system, the F [n] in Eq. 2.5 can be written into P

2 e n(ri)n(ri′ ) F [n]= T0[n]+ dri dri′ + Exc[n], (2.6) 2 r r ′ Z Z | i − i | where T0[n] is the kinetic energy of the auxiliary non-interacting system, and Exc[n] is the exchange- correlation functioanl. Now, we minimize the E[n] with respect to ψ with constraint <ψ ψ′ >= n n| n δn,n′ , we obtain the famous Kohn-Sham equation,

2 h(r )ψ (r )=[ ∇i + v (r )]ψ (r )= ǫ ψ (r ), (2.7) i n i 2m R i n i n n i where ǫn comes from the Lagrange multipliers for orthogonal constraint, and

2 2 Zαe e δExc[n] vR(ri)= + dri′ n(ri′ )+ . (2.8) − r R r r ′ δn α i α i i X | − | Z | − |

4 The exchange correlation functional, Exc[n], is unknown so the main diﬃculty in DFT is to determine a proper approximation to it. The popular approaches are the local density approximation (LDA), where the exchange correlation functional is approximated by

LDA 3 LDA Exc [n(ri)] = dri n(ri)ǫxc (n(ri)), (2.9) Z and the local spin-density approximation (LSDA), where the exchange correlation functional is approximated by

LSDA 3 LSDA Exc [n↑(ri),n↓(ri)] = dri n(ri)ǫxc (n↑(ri),n↓(ri)). (2.10) Z They are approximations derived from a homogeneous electron gas with density, n(ri), and spin density, nσ(ri). These approaches work amazingly well for a wide class of materials. In DFT algorithm, one only needs to provide the lattice potential, i.e., the position of the atoms, and the basis set to expand the wavefunction, ψn, e.g., linear augmented plane-wave (LAPW) or projector augmented-wave (PAW) methods [29]. Then, the electronic structure can be computed by solving the electron density, Kohn-Sham potential, and Kohn-Sham equation, iteratively. DFT has enormous success in studying the electronic structure of semiconductor and weakly correlated materials. However, the predictions of DFT become unsatisfactory when we apply it to strongly correlated systems. The failure of DFT is caused by the strong electron-electron correlation in these speciﬁc systems, which cannot be described by the single-particle picture in DFT. Therefore, other treatments to describe these special systems will be introduced in the following sections.

2.3 Multiorbital Hubbard Model

Beside the ﬁrst principle approach to Eq. 2.2, the model Hamiltonian approaches are also used intensively to describe the phenomena in condensed matter physics. The most famous models are the tight-binding model and the Hubbard model. These models are derived from the most general Hamiltonian, Eq. 2.2, with the assumption that the electron wavefunction are localized in space, which is true in most of the strongly correlated systems. Therefore, we can construct a local orbital basis called Wannier orbital basis [30] to downfold the most general Hamiltonian, Eq. 2.2, to the celebrated Hubbard model, which plays an important role in strongly correlated systems.

5 Starting from the second quantization form of Eq. 2.2,

2 2 2 † Zαe ′ † † ′ 1 e ′ H = drΨ (r)[ ∇ ]Ψ (r)+ drdr Ψ (r)Ψ ′ (r )[ ]Ψ ′ (r )Ψ (r), σ −2m − R r σ σ σ 2 r r′ σ σ σ α α ′ X Z X | − | Xσ,σ Z | − | (2.11) † where ψσ(r) (ψσ(r)) is the second quantization ﬁelds that annihilate (create) an electron with spin σ at coordinate r. Now, we introduce a localized Wannier orbital basis,

1 −ik.Ri wm(r Ri)= e φkm(r), (2.12) − √N kX∈BZ where Ri is the coordinate for lattice site i, k is the momentum label, m is the index for Wannier orbital, and N is the total number of sites. The electron ﬁelds can be expanded in the Wannier orbital basis, Ψσ(r) = wm(r Ri)dimσ, where dimσ annihlate an electron at site i, orbital m, i,m − and spin σ. Within thisP basis, Eq. 2.11 becomes

mm′ † 1 mm′nn′ † † H = tij dimσdjm′σ + Vijkl dimσdjm′σdkn′σdlnσ, (2.13) ′ 2 ′ ′ ′ ijmmX σ Xijkl mmXnn Xσσ where

2 mm′ ∗ t = drw (r R )[ ∇ + V (r)]w ′ (r R ), (2.14) ij m − i 2m m − j Z and

2 mm′nn′ ′ ∗ ∗ ′ e ′ V = drdr w (r R )w ′ (r R )[ ]w (r R )w ′ (r R ). (2.15) ijkl m − i m − j r r′ n − k n − l Z | − | Hubbard made a drastic assumption that the electron-electron interaction is completly local, mm′nn′ which means Vijkl = Vmm′nn′ δijδjkδkl, so the Hamiltonian reads,

mm′ † 1 † † H = tij dimσdjm′σ + Umm′nn′ dmσdm′σ′ dn′σ′ dnσ. (2.16) ′ 2 ′ ′ ′ ijmmX σ mmXnn Xσσ This Hamiltonian is called multiorbital Hubbard model. For the rotationally invariant interaction, which assumes the Coulomb interaction is completely atomic without the inﬂuence from the lattice

6 environment, the multiorbital Coulomb integral, Umm′nn′ , can be parametrized using the Slater- Condon parametrization as shown in Appendix. A. Although this approximation seems coarse, it actually works well to many strongly correlated systems. The reason is that the d and f orbitals, which are localized in space, are usually the dominant orbitals near the Fermi surface for most correlated materials. In addition, the screening eﬀect between electrons suppresses the long-range Coulomb interaction to a short-range Yukawa-like potential. Therefore, the local approximation to the electron-electron interaction is indeed quite good. Although the Hubbard model looks simple, it still cannot be solved exactly, even in the single orbital case, except in one and inﬁnite dimension. Seeking for the exact solution for the Hubbard model is, therefore, an ultimate goal in condensed matter physics. In this dissertation, we employed the Dynamical Mean Field Theory (DMFT) and Rotationally Invariant Slave-Boson (RISB) method, which are suitable to describe the Mott metal-insulator transition (MIT) and the mass enhancement in correlated materials, to study the Hubbard model.

2.4 Dynamical mean ﬁeld theory

Dynamical Mean Field Theory (DMFT) was ﬁrst devloped for solving the Hubbard model [19],

H = t (c† c + h.c.)+ U n n . (2.17) − ij iσ jσ i↑ i↓ σX Xi It has successfully reproduced the Mott MIT and the strong mass enhancement in correlated materials [19]. Similar to the Weiss mean field theory for the Ising model, DMFT is exact in infinite dimension, d , but is not an appropriate approximation to describe the quantum criticality in → ∞ low dimensional systems. In DMFT, the Hubbard model is mapped into an impurity coupled to an effective cavity field, which is an analogy to the Weiss field in the Ising model, as shown in Fig. 2.1. Then, the impurity problem and the cavity field is solved self-consistently with a set of DMFT equations. The limitation of DMFT is its inability to describe the spatial correlation. As a result, DMFT fails to describe the d-wave superconductivity in two-dimensional systems. This shortcoming can be improved by the cluster extensions of DMFT, such as cellular DMFT and Dynamical Cluster Approximation (DCA) [20]. In this dissertation, we focus on the Mott MIT. Therefore, DMFT is sufficient to describe our systems. In the following subsection, I will review the DMFT formalism and the methods for solving the impurity problem, i.e., Iterative Perturbation Theory

7 (IPT), Slave-Rotor Non-Crossing Approximation (NCA) and Continuous Time Quantum Monte Carlo (CTQMC).

2.4.1 Mapping to Anderson impurity model

The crucial step in DMFT is to map the Hubbard model to an Anderson Impurity Model (AIM). Starting from Eq. 2.17, the action for the Hubbard model is

† −S Z = DciσDciσe , (2.18) Z Yi where

β † ∂ † S = dτ[ ciσ(τ)( µ)ciσ(τ) tij(ciσ(τ)cjσ(τ)+ h.c.)+ U ni↑(τ)ni↓(τ)]. (2.19) 0 ∂τ − − Z Xi,σ i,j,σX Xi Then, we separate the action into three parts: the site with i = 0 (impurity), the sites with i = 0 6 (cavity), and the interaction between the site with i = 0 (impurity) and the sites with i = 0 6 (cavity). By integrating out the ﬁeld ci6=0,σ using the linked-cluster theorem [31], we obtain an eﬀective action

β β β S = dτ dτ ′ c† (τ)G−1(τ τ ′)c (τ ′)+ U dτn (τ)n (τ). (2.20) eff 0σ 0 − 0σ 0↑ 0↓ 0 0 σ 0 Z Z X Z The G−1(τ τ ′) in Eq. 2.20 is a Weiss ﬁeld consisted by a cavity Green’s function, 0 − G(0)(τ τ ′)=, ij − τ i6=0,σ j6=0

−1 ′ ∂ (o) ′ G (τ τ )=( µ)δ ′ + t t G (τ τ ). (2.21) 0 − ∂τ − τ,τ i0 0j ij − Xi,j (0) ′ 2 Note that for a Bethe lattice, the Fourier transform of ti0t0jGij (τ τ ) becomes t G0(iωn). The i,j − eﬀective action, Seff , describes an impurity coupled toP a cavity, as shown schematically in Fig. 2.1. From the Dyson equation, we can calculate the lattice Green’s function through

1 G(iωn)= . (2.22) G−1(iω ) Σ (iω ) 0 n − imp n The local Green’s function can be evaluated from

8 Figure 2.1: Schematic representation for the DMFT mapping: Mapping a lattice problem to an impurity coupled to a cavity ﬁeld, ∆(ω).

D (ǫ) G (iω )= dǫ 0 , (2.23) imp n iω + µ ǫ Σ (iω ) Z n − − imp n where D0(ǫ) is the non-interacting density of state. Eq. 2.20, 2.22, and 2.23 form a set of self- consistent equations, for which we can solve iteratively by giving an initial guess of the lattice

Green’s function, G(iωn), or selfenergy, Σ(iωn). The DMFT algorithm is shown below:

1. Guess an initial lattice Green’s function, G(iωn).

2 2. Calculate the cavity ﬁeld from ∆(iωn)= t G(iωn).

3. Use the impurity solvers, e.g. IPT, NCA, or CTQMC, to get the selfenergy, Σ(ω)=Σimp(ω), of Eq. 2.20.

4. Solve the local Green’s function, Gimp(iωn), using Eq. 2.23.

5. If the convergence criteria, Gk+1(iω ) Gk (iω ) < accuracy, is satisﬁed, stop the calcu- | imp n − imp n | lation, here k is the iteration index. Otherwise, go to step 2 and replace G(iωn) by the new

Gimp(iωn).

The most tricky part of this algorithm is solving the impurity model. Fortunately, the impurity model can be connected to the AIM, which can be solved with many exisiting methods. The AIM reads

9 † † † † † HAIM = ǫd dσdσ + ǫk,σck,σ + Vk(ck,σdσ + h.c.)+ Ud↑d↑d↓d↓. (2.24) σ X Xk,σ Xk,σ † † Here dσ and ckσ stand for the creation operators for the impurity and the conduction electrons, respectively, with spin σ and momentum k. ǫd and ǫkσ are the energy for the impurity and the conduction electrons. Vk is the hybridization between the impurity and the conduction electrons. U is the Coulomb energy for the electrons in impurity. By integrating out the conduction electron (0) ﬁelds, we obtain the same eﬀective action as in Eq. 2.20, but the ti0t0jGij (iωn) is replaced by i,j 2 Vk P . These two functions is called hybrdization function labeled by ∆(iωn). iωn−ǫk k P There are various methods that can solve the AIM, such as Exact Diagonalization(ED) [32], Iterative Perturbation Theory (IPT) [33], Slave-Rotor Non-Crossing Approximation (NCA) [34], and Continuous Time Monte Carlo (CTQMC)[35], etc. In this dissertation, we employed the IPT, slave-rotor NCA and CTQMC as our impurity solvers.

2.4.2 Iterative Perturbation Theory (IPT)

Iterative Perturbation Theory (IPT) is the most eﬃcient and easiest method to solve AIM approximately. It was brought out by Yoshida and Yamada in 1970 [36, 37, 38]. This method involves calculating a second order perturbation diagram of the eﬀective action, Eq. 2.20. The main equation we need to solve is

U β Σ(iω )= + U 2 dτeiωnτ Gˆ (τ)3. (2.25) n 2 0 Z0 This method can be eﬃciently implemented using the fast Fourier transformation between the imaginary time, τ, and the Matsubara frequency, ωn. Although IPT is a weak coupling approach, applying it in the DMFT self-consistent equations coincidentally generates reasonable solutions in the strong coupling limit, i.e., Mott insulator solutions. Therefore, IPT has the ability to smoothly interpolate the solutions from the weak coupling limit to the strong coupling limit, and succesfully produce a Mott transition at the critical point Uc. Although IPT has all these nice features, it cannot captures the exact Fermi-liquid behavior in metallic phase at low temperature. To study the precise behavior at low temperature, we need numerical exact methods, such as Continuous Time Quantum Monte Carlo (CTQMC) and Numerical Renormalization Group (NRG).

10 2.4.3 Slave-Rotor Non-crossing Approximation (NCA)

Slave-Rotor Non-Crossing Approximation (NCA) is a method to solve AIM [34]. It can capture the Kondo effect in AIM and gives quantitatively correct behavior for the energy scale larger than Kondo temperature, T = √2U∆exp[ πU/(8∆)]. However, for temperature lower than T , K − K the NCA type of approaches could not reproduce the exact local Fermi liquid behavior, i.e., the self-energy is ImΣ(ω) ω2. Instead, NCA approaches gives a non-Fermi liquid, ImΣ(ω) ωα, ∝ ∝ power-law behavior. This power law behavior is originating from the large spin, with N flavors, and large charge channel, with M flavors, approximation in NCA, which leads to an overscreening effect between the bath and impurity resulting to a non-Fermi liquid behavior at low energy. This approximation becomes a good approach when we study the multi-channel AIM. The idea of the slave-rotor method is based on the spin charge decomposition of the electron field. We can decouple the physical electron operator into a spinless and charge-carrying boson,

iθj † † iθj Xj = e , and a spin-carrying and chargeless fermion, fj,σ, such that dj,σ = fj,σe . Now, we apply the large N approximation to the spin ﬂavor, σ = 1, 2...N, by generalizing the AIM in Eq. 2.24 to SU(N) AIM,

U N H = ǫ d† d + ǫ c† + V (c† d + h.c.)+ [ d† d ]. (2.26) SU(N)AIM d σ σ k,σ k,σ k k,σ σ 2 σ σ − 2 σ σ X Xk,σ Xk,σ X † Within the slave-rotor framework, we can introduce an angular momentum operator, L = [fσf σ σ− 1 2 ], which conjugate to the rotor phase, θj. By applying the following Lagendre transformation,P S β dτ[ iL∂ θ + H + f †∂f], the action of the SU(N) AIM, Eq. 2.26, can be written into ≡ 0 − τ R

β (∂ θ(τ)+ ih)2 N S = dτ f †(τ)(∂τ + ǫ h)f (τ)+ τ + h σ d − σ 2U 2 0 σ Z X † −iθ(τ) + [ck,σ(τ)fσ(τ)e + H.c.]. (2.27) Xk,σ † 1 Here h is the constraint ﬁeld to enforce L = [fσf ]. By integrate out the bath fermion ﬁeld, σ σ − 2 † ck,σ(τ), we obtain the following form of the PSU(N) AIM

11 β (∂ θ(τ)+ ih)2 N S = dτ f †(τ)(∂τ + ǫ h)f (τ)+ τ + h σ d − σ 2U 2 Z0 σ β X β ′ + dτ dτ ′∆(τ τ ′) f †(τ)f (τ ′)eiθ(τ)−iθ(τ ), (2.28) − σ σ 0 0 σ Z Z X where ∆(τ τ ′) is the Fourier transform of the hybridization function, ∆(iω ). Now we apply the − n 2 large M approximation to the rotor ﬁeld, Xα, where α = 1,...,M, with the constraint Xα = M. α | | This approximation leads to the ﬁnal form of the action for the AIM in large N andPM limit,

β N h2 S = dτ f †(τ)(∂τ + ǫ h)f (τ)+ h M Mλ σ d − σ 2 − 2U − 0 σ Z X ∂ X (τ) 2 h + βdτ | τ α | + (X∗(τ)∂ X (τ) H.c.)+ λ X (τ) 2+ 2U 2U α τ α − | α | 0 α Z X β β 1 dτ dτ ′ ∆(τ τ ′) f †(τ)f (τ ′)X (τ)X∗(τ ′), (2.29) M − σ σ α α 0 0 σ,α Z Z X 2 here λ is the constraint field to enforce Xα = M. The saddle point equations of this action α | | can be obtained using the bilocal field methodP in Ref. [39] at the large N and the large M limits by fixing the = N/M ratio. The equations are N

G−1(iω )= iω ǫ + h Σ (iω ), (2.30) f n n − d − f n

ν2 2ihν G− 1(iν )= n + λ n Σ (iν ), (2.31) X n U − U − X n

Σ (τ)= ∆( τ)G (τ), (2.32) X −N − f

Σf (τ)=∆(τ)GX (τ), (2.33) where ωn and νn are the fermionic and the bosonic Matsubara frequency. Eq. 2.30-2.33 forms a set of self-consistent equations. One can start from an initial guess of the bosonic and the fermionic self-energy. Then, one can calculate the corresponding Green’s function from Eq. 2.30 and Eq. 2.31. The new Green’s function can be used to calculate the new self-energy using Eq. 2.32 and

12 Eq. 2.33. This loop is iterated until the self-energy and the Green’s function are converged to a certain criteria. The calculation can be eﬃciently implemented using the fast Fourier transformation between τ and ωn(νn) space.

2.4.4 Continuous Time Quantum Monte Carlo Method (CTQMC)

Continuous Time Quantum Monte Carlo(CTQMC) is a numerical exact method to solve the AIM. It captures the precise physics for a broad parameters range. The only disadvantage of CTQMC is that it becomes ineﬃcient at low temperature. There are two types of CTQMC methods, weak coupling CTQMC and hybridization CTQMC. In this dissertation, we use the hybridization CTQMC to investigate our project [35, 40]. The formalism of CTQMC was developed by Wener, Gull, Rubtsov, and Millis. They separate the impurity Hamiltonian into three parts, H = Hloc + Hbath + Hhyb, which are

H = µ(n + n )+ Un n , (2.34) loc − ↑ ↓ ↑ ↓

† Hbath = ǫkckσckσ, (2.35) Xkσ and

σ † σ∗ † Hhyb = (Vk dσckσ + Vk ckσdσ), (2.36) Xkσ here the dσ and ckσ are the electron ﬁelds for impurity and bath, respectively. They are several ways to expand the partition function. One way is to expand it in the power of Hloc and averaging over

Hbath + Hhyb, which is called the weak-coupling approach (CT-AUX) or the interaction approach (CT-INT) [41, 42]. The other way is to expand it with respect to the hybridization term, H hyb ≡ † H Hd + Hd, and averaging over H = H + H , which is called hybridization approach 2 ≡ 2 2 1 loc bath (CT-HYB). The hybridization expansion in CT-HYB is shown below:

Z = Tr[exp( β(H + H ))] − 1 2 1 ′ ′ † † ′ ′ = ( )2 dτ ...dτ dτ ...dτ Tr[T e− R H1(τ)dτ Hd (τ )...Hd (τ )Hd(τ )...Hd(τ )]. (2.37) n! 1 n 1 n × τ 2 1 2 n 2 1 2 n n X Z

13 The partition function can be further reformulated to

σ σ′ σ σ′ −βHloc † σ σ′ † σ σ′ Z = Zbath dτ1 dτ1 ...dτn dτn Trd[e Tτ dσ(τ1 )dσ(τ1 )...dσ(τn )dσ(τn )] σ n Y X Z detM −1(τ σ...τ σ,τ σ′...τ σ′), (2.38) × σ 1 n 1 n

−βH where the term Trd[e loc ...] corresponds to the local average of the impurity electron fields, and −1 σ σ σ′ σ′ the term detMσ (τ1 ...τn ,τ1 ...τn ) corresponds to the average of the bath electron fields. The ma- −1 ′ trix Mσ (i,j)=∆σ(τi τj) is calculated by the Fourier transform of the hybridization function, σ 2 − |Vk | ∆σ(iωn) = . Note that this formalism does not have fermionic sign problem because the iωn−ǫk k −1 σ σ σ′ σ′ fermionic signP is absorbed into the bath determinant, detMσ (τ1 ...τn ,τ1 ...τn ). The Monte Carlo simulation can now be sampled on two continuous time line, τ σ, one for spin up and the other for spin down. For each proposed configuration, we calculate the ratio of the partition function between −βH the new and the old configurations from the local trace, Trd[e loc ...], and the bath determinant, −1 σ σ σ′ σ′ detMσ (τ1 ...τn ,τ1 ...τn ). Then, the metropolis algorithm is used to determine whether the con- new−1 old−1 figuration is accepted or not. Note that the ratio of the determinant, detMσ /detMσ , can be computed efficiently using the Sherman-Morrison algorithm, which is a standard technique in quantum Monte Carlo related methods. Since the calculation in CTQMC is calculated on the Matsubara frequency, the physical observables, e.g., Green’s function, has to be analytic continued to the real frequency. We use both the Pade approximation and the Maximum Entropy Method (MEM) to perform the analytic continuation. For detail, please refer to Appdendix. B.

2.5 Rotationally invariant slave-boson method (RISB)

Slave-Boson Mean Field (SBMF) methods can be viewed as a simplified version of DMFT [15], which is focused on the low energy physics near the Fermi level. Therefore, it is suitable to describe the low-energy quasiparticle parameters, such as quasiparticle weight, Z, corresponding to the inverse of the effective mass. It is equivalent to Gutzwillar approximation (GA), which is an exact approximation to the Gutzwiller variational problem in infinite dimension, at the saddle point level [15]. The dominant physics is the Mott metal-insulator transition in the Brinkman-Rice picture, in which the quasiparticle weight vanishes smoothly to zero at the Mott critical point, Uc. The disadvantage of this approximation is that it could not describe the high energy excitation, i.e., the

14 high energy Hubbard bands. Extending the SBMF methods to describe the high energy excitation is still an on-going research. In this dissertation, we proposed one of the methods called ghost GA that allows to describe the high energy excitation within the SBMF or equivalently GA framework in chapter 5. Rotationally Invariant Slave-Boson (RISB) method is one of the most eﬃcient methods to solve the multi-orbital Hubbard model as shown in Eq. 2.16. With the recent eﬀort from Lanata et. al [43, 25], this method can now be applied to multi-orbital systems with any kind of interactions. This breakthrough allows us to study complicate materials with d and f orbital electrons. Starting from the most general mult-orbital Hubbard model,

αβ † loc H = ǫk,ijckiαckjβ + H , (2.39) Xk ij=1X,..,na α=1X,..,Mi β=1X,..,Mj XRi where k is the momentum conjugate to the unit cell, R is the label for unit cell, i and j are the labels for atoms in unit cell, na is the number of atoms in unit cell, α and β are spin and orbital labels, and Mi is the number of spin and orbital for a given atom i. The local interaction can be represented in the local Fock state,

Hloc [Hloc] A, Ri B,Ri , (2.40) ≡ i AB| ih | XRi XAB The local Fock state is deﬁned by

† ν (A) † ν (A) A, Ri =[c ] 1 ... [c ] Mi 0 , (2.41) | i Ri1 RiMi | i M where νi(A) = 1 or 0 represents the occupancy of the local orbital, and A = 1,..., 2 i runs over all the conﬁguration space constructed by all the combinations of ν (A),..,ν (A) . { 1 Mi } 2.5.1 Rotationally invariant slave boson mapping

The RISB formalism introduces a new set of fermionic ﬁeld, fRia, called quasiparticle fermion.

For each quasiparticle fermion, we introduce a bosonic mode, ΦRiAn, providing a link between the physical, A, Ri , and quasiparticle, n,Ri , Fock states. Note that the two Fock states has the | i | i same particle number, N Mi ν (A)= N Mi ν (n). Since we enlarge the Hilbert space A ≡ a=1 a n ≡ a=1 a by introducing auxiliary bosonicP mode, we have toP apply the constraint to enforce our solution to

15 be in the physical Hilbert subspace. Therefore, we have the following constraints called Gutzwiller constraints,

K0 Φ† Φ Iˆ = 0, (2.42) Ri ≡ RiAn RiAn − XAn and

K f † f [F † F ] Φ† Φ = 0. (2.43) Riab ≡ Ria Rib − ia ib mn RiAn RiAm AnmX With this RISB mapping, the original Hamiltonian, Eq. 2.39, can be written into

αβ † loc † H = ǫk,ij ckiαckjβ + [Hi ]AB ΦRiAnΦRiBn, (2.44) n kijαβX RiABX X † † where cRiα RRiaα fRia with ≡ a P † † [Fiα]AB[Fia]nm † RRiaα = ΦRiAnΦRiBm. (2.45) nm NA(Mi NB) XAB X − p 2.5.2 Rotationally invariant slave boson Mean ﬁeld theory

The mean field approximation to the RISB formalism can be performed by introducing a mean field wavefunction, Ψ = Ψ φ , where Ψ is the Slater determinant corresponding to the | MF i | 0i⊗| i | 0i quasiparticle f and φ is the bosonic coherent state corresponding to the bosonic fields. Within Ria | i this approximation, the Gutzwiller constraint Eq. 2.42 and Eq. 2.43, become

Tr[φ†φ ] = 1 i, (2.46) i i ∀ and

[∆ ] Tr[φ†φ F † F ]= Ψ f † f Ψ ,i. (2.47) pi ab ≡ i i ia ib h 0| Ria Rib| 0i∀ The variational energy within this mean ﬁeld approximation becomes

1 1 Ψ H Ψ = Tr φ φ† Hloc + R ǫ R† Ψ f † f Ψ , (2.48) E ≡ h MF| | MFi i i i i k,ij j abh 0| kia kjb| 0i N i N kij ab X X X

16 where [R ] φ R φ is given by i aα ≡h | Riaα| i − 1 [R ] = Tr φ†F † φ F ∆ (1 [∆ ]) 2 . (2.49) i aα i iα i ib pi − pi ba Follow the detail derivation in [43, 25], the constraint minmization problem of this variational energy can be reformulated into the Lagrange function below,

L [ Φ ,Ec,R,R†,λ,D,D†,λc, ∆ ]= SB h | p 1 i(2m+1)πT 0+ lim T Trlog( † )e − T→0 i(2m + 1)π RǫkR λ + µ N Xk mX∈Z T − − + [ Φ Hemb[D ,D†; λc] Φ + Ec(1 Φ Φ )] h i| i i i i | ii i −h i| ii i X 1 c 2 [ ([λ ] +[λ ] )[∆ ] + ([D ] [R ] [∆ (1 ∆ )]ca + c.c.)] , (2.50) − i ab i ab pi ab i aα i cα pi − pi caα Xi Xab X where Hemb is called the embedding Hamiltonian describing a dimer system consisted by two atoms coupled to each other. The ﬁrst atom represents the physical impurity containing all the physical energy level and interactions. The second atom is an auxiliary non-interacting atom representing the environment. The Hemb reads

Hemb[D ,λc] Hloc[ cˆ† , cˆ ]+ ([D ] cˆ† fˆ + H.c.) + [λc] fˆ fˆ† . (2.51) i i i ≡ i { iα} { iα} i aα iα ia i ab ib ia aα X Xab By minimizing the function, Eq. 2.50, with respect to all the parameters, Φ ,Ec; R,R†,λ; D,D† h | c ,λ ; ∆p, we obtain the following saddle point equations that forms a root problem,

1 † [ Πif(RǫkR + λ)Πi]ba = [∆pi]ab (2.52) N k X 1 1 1 † † 2 [ Πi RǫkR f(RǫkR + λ)Πi]αa = [Di]cα[∆ip(1 ∆ip)]ac (2.53) Ri − N k c X 1 X ∂ 2 c p [∆pi(1 ∆pi)]cb[Di]bα[Ri]cα +c.c.+[l + l ]is = 0 (2.54) ∂dis − Xcbα Hemb[D ,λc] Φ = Ec Φ (2.55) i i i | ii i | ii 1 (1) † 2 [ ] Φ cˆ fˆ Φ [∆ (1 ∆ )]ca[R ] = 0 (2.56) Fi αa ≡h i| iα ia| ii− ip − ip i cα c X [ (2)] Φ fˆ fˆ† Ψ [∆ ] = 0 . (2.57) Fi ab ≡h i| ib ia| ii− pi ab

17 c c Here f(x) is the Fermi function, d, l, and l are the parameters for the matrices Di, λi, and λi parametrized in the Hermitian matrix basis. This set of equations can be solved using the quasi- Newton approximation by giving an initial guess of the R and the λ matrices. Then, the root solver can find the final R and λ matrices that satisfy Eq. 2.56 and 2.57. From the solved R and λ matrices, we can calculate the most important information in RISB, which is the quasiparticle weight for each orbital. The quasiparticle weight is defined by the eigenvalues of the matrix Z R†R, which corresponds to the mass renormalization for each orbital, ≡ m/m∗ = Z. The difference in quasiparticle weight for each orbital leads to many interesting phenomena, e.g., orbital differentiation and orbital-selective Mott transition [44, 45]. Also, from R and λ matrices, we can calculate the Green’s function and selfenergy defined by

1 G(k,ω)= , (2.58) iω ǫ + µ Σ(ω) − k − and

I R†R 1 1 Σ(ω)= ω − + λ . (2.59) − R†R R R† From them, we can further calculate the density of state and band dispersion that can be measured in ARPES and various experimental techniques.

2.6 The combination of ﬁrst principle methods and many-body methods

In the previous sections, we learned the advantages and the disadvantages of first principle simulations and many-body methods. It is interesting to notice that these two methods complement each other’s shortcoming. DFT does not have tunable parameters in the computation of band structure, but it cannot treat the correlation effects properly. On the other hand, the many-body methodology provides a good description of the correlation effects, but the non-interacting band structure to be modified by the many-body effects is usually generated from the tight-binding model with adjustable parameters. Therefore, it is intuitive to think that one can combine these two methods. Indeed, in the last decades, methods such as LDA+U, LDA+DMFT, and LDA+RISB have been intensively developed. They have been widely used to investigate the physics in strongly

18 correlated materials. In this section, I will introduce LDA+RISB, which was employed to study the transition metal compounds and the Americian compounds in this dissertation.

2.6.1 Correlated basis set

The first step of any LDA+X scheme is to project out the local correlated orbital from the Kohn-Sham basis in LDA. In order to do so, we need to choose a proper local basis set to describe the local correlated orbitals. The popular choices are maximum localized Wannier functions [30, 46], and the basis set proposed by Haule et. al [47]. Below, following the convention in Ref. [25], we define a set label P representing the local correlated orbitals, and a set label Q representing the uncorrelated orbitals. For generality, we define the local correlated basis set as χiπ, where π is the label for the local correlated orbitals, and i is the label for atoms. We can expand the electron field at coordinate r and spin σ in this basis set

X(r, σ)= χiπ(r, σ)ciπσ + XQ(r, σ)= XPi(r, σ)+ XQ(r, σ), (2.60) i,πX∈P Xi where the ﬁrst term represents the local correlated part of the electrons and the second term represents the uncorrelated part of the electrons. With Eq. 2.60, any one-body operator, A, can be separated into local correlated subspace, P , and uncorrelated subspace, Q,

A = drX†(r, σ)AX(r, σ) Aloc + Ahop, (2.61) ≡ i σ X Z Xi Aloc drX† (r, σ)AX (r, σ), (2.62) i ≡ Pi Pi σ X Z Ahop A Aloc. (2.63) ≡ − i Xi We can also use the local correlated basis set to project out the correlated Green’s function from the DFT results or to embed the selfenergy into the DFT calculations, which correspond to

′ ′ ∗ ′ giπ,π′ (ω)= drdr χiπ(r, σ)G(r, r )χiπ(r ,σ), (2.64) Z and

∗ ′ Σir,r′ (ω)= χiπ(r, σ)Σπ,π′ (ω)χiπ(r ,σ), (2.65) ′ Xπ,π

19 respectively. From the projected operators and the projected Green’s function, we can perform the many- body correction, e.g., orbital potential methods (+U), DMFT, or RISB, on top of the DFT band structure. Then, we can embed the self-energy, which contains the many-body correction, back to the Kohn-Sham basis to update the charge density for the next DFT iteration. This procedure is called DFT+X scheme, where X can be any many-body approximation.

2.6.2 LDA+RISB functional

It is straightforward do combine the LDA and the RISB functionals in Eq. 2.50 to the following LDA+RISB functional [25],

c † † c 0 KSH c † † c 0 Ω dc [ρ, J; Φ ,E ,R,R ,λ,D,D ,λ , ∆ ] Ω [J; Φ ,E ,R,R ,λ,D,D ,λ , ∆ ] V ;N h | p ≡ V dc;N h | p drJ(r)ρ(r)+ ELDA[ρ]+ E [ρ]+ E , (2.66) − Hxc ion ion−ion Z where J is the constraint ﬁeld to the electron density ρ(r). ΩKSH is the functional for the Kohn- Sham-Hubbard Hamiltonian,

ΩKSH [J; Φ ,Ec,R,R†,λ,D,D†,λc, ∆ ]= V dc;N h | p 1 + lim T Trlog( )ei(2m+1)πT 0 − T→0 i(2m + 1)π R[ 2,hop J hop]R† λ + µ N Xk mX∈Z T − ∇ − − + [ Φ Hemb[D ,D†; λc] Φ + Ec(1 Φ Φ )] h i| i i i i | ii i −h i| ii i X 1 c 2 [ ([λ ] +[λ ] )[∆ ] + ([D ] [R ] [∆ (1 ∆ )]ca + c.c.)] . (2.67) − i ab i ab pi ab i aα i cα pi − pi caα Xi Xab X The embedding Hamiltonian in LDA+RISB becomes

Hemb[D ,D†,λc] 2,loc[ cˆ† , cˆ ]+ J loc[ cˆ† , cˆ ]+ Hloc[ cˆ† , cˆ ] V dc c† c i i i i ≡ −∇i { iα} { iα} i { iα} { iα} i { iα} { iα} − i iα iα α X † ˆ c ˆ ˆ† + ([Di]aαcˆiαfia + H.c.) + [λi ]abfibfia. (2.68) aα X Xab Here we assume a linear double counting potential, Φdc V dcN loc. However, in our code, we used i ≡ i i the fully localized limit potential [48]. The optimization problem is summarized in Fig. 2.2. We

20 Initial electron density ρ0(r)

Construct J(r) and HHKS

Update electron density ρ(r) Solve HHKS with RISB (GA)

Total energy converged? No Yes Done

Figure 2.2: Schematic representation for the LDA+RISB algorithm.

start from an initial electron density, which is usually the electron density from bare LDA. Then, we calculate the constraint ﬁeld, J(r), and Kohn-Sham-Hubbard Hamiltonian, HKSH . After that, we solve the HKSH saddle point equations, Eq. 2.52-2.57, to get the new electron density and energy. If the electron charge and energy do not converge, we update the electron density for the next iteration. The loop is iterated until the charge and energy self-consistency are reached.

21 CHAPTER 3

UNSTABLE DOMAIN WALL SOLUTION IN THE METAL-MOTT INSULATOR COEXISTENCE REGIME

3.1 Introduction

A common feature of all ﬁrst-order transitions is domain formation in the phase coexistence regime, where two competing phases tend to form domains to minimize the internal energy. Between each pair of domains, there exist domain wall structures where their physical properties smoothly evolve from one phase to the other. Standard examples of domain formation are the ferromagnetic domains in transition metals, e.g., iron and nickel, below the Curie temperature. In strongly correlated materials, the domain formations have also been observed in the optical image in V2O3 and VO2 within the metal insulator coexistence regime [49, 50, 51, 1], where the domain wall structure displays an intermediate optical intensity between the metal (red region) and the insulator (blue region) domains as shown in Fig. 3.1. The free energy of these systems in the phase coexistence regime can be described by the Ginzburg-Landau free energy, which shows a double well structure with two minima corresponding to the stable and metastable phase, Fig. 3.2(b). The local maximum separating the stable and metastable phase is the unstable solution, which is known to correspond to the domain wall structure [52]. In metal-Mott insulator coexistence regime, the two local minima are the metal and Mott insulator solution, and the local maximum (unstable solution) is expected to describe the domain wall structure. To model the metal Mott-insulator transition (MIT), the fully frustrated Hubbard model turns out to be the minimum model to describe the system in the paramagnetic phase. The exact solution can be obtained by dynamical mean ﬁeld theory (DMFT) [19]. In the last decades, various exotic physics, such as the MIT phase diagram [19], resilient quasiparticle [53, 54], universal scaling behavior along the Widom line [55, 56], etc., have been discovered using DMFT. Despite the intense investigation, so far, these research were focused on the metal and the Mott insulator solution. The

22 Figure 3.1: (a) Schematic representation for the near-ﬁeld microscopy measurement of V2O3 from McLeod et. al. [1]. (b) Near ﬁeld image of the metal Mott insulator coexistence regime by warming up the system from T = 164K to T = 184K. Blue areas correspond to the antiferromagnetic Mott insulating phase and red areas correspond to correlated metallic phase [1].

previous studies on the unstable solution were restricted to the thermodynamics properties near the critical point [57, 58, 59]. The other investigations on the layer and inhomogeneous Hubbard model were only focused on the density of state (DOS) in the domain wall structure [60, 61]. Until now, there has not been any detailed investigation on the unstable solution’s transport and quasiparticle properties. Several questions come to mind. One, is the unstable solution more metallic or insulating? Two, how does the unstable solution differ from the metal and the Mott insulator? Producing the answer of these questions is the subject of this work. In this project, we explored the nature of the unstable solution using DMFT with Iterative Per- turbation Theory (IPT) and numerical exact Continuous Time Quantum Monte Carlo (CTQMC) solvers along two different trajectories, the first order transition line and a constant U trajectory, as shown in the phase diagram, Fig. 3.2 (a). In order to study the unstable solution, the multi- dimensional optimizer [58] was utilized to converge our solution to the local maximum of the free energy functional (the unstable solution). We also employed the free energy analysis [62, 63] to identify the first order transition line and the location of the unstable solution. Our results indicate that the unstable solution behaves like an incoherent metal with resistivity around and above the Mott limit, ρ ρ . Its quasiparticle peak remains finite in the entire ≥ Mott coexistence region. The resistivity for the unstable solution can be described by the Sommerfeld

23 quasiparticle approximation with two quasiparticle parameters, Z and Γ, where Z T 2 and Γ ∼ ∼ T −2 shows anomalous power law behavior diﬀer from the Fermi-liquid behavior.

3.2 Method

We studied the single band fully frustrated Hubbard model at half-ﬁlling

H = t (c† c + h.c.)+ U n n , (3.1) − iσ jσ i↑ i↓ σX Xi † where ciσ and ciσ are the electron creation and annihilation operators with spin σ at site i, niσ = † ciσciσ, t is the nearest neighbor hopping, and U is the on-site Coulomb interaction. The energy unit is set to the half band width, D = 2t. The Hubbard model can be mapped to the Anderson impurity model within the DMFT framework. To solve the impurity problem, we use the continuous time quantum Monte Carlo (CTQMC) and iterative perturbation theory (IPT) as impurity solvers. The analytical continuation was done using maximum entropy method (MEM), and the 5th order polynomial fitting for CTQMC. For IPT, we employed the Pade approximation. The DMFT phase diagram is shown in Fig. 3.2 (a). The system undergoes a second order transition at the critical temperature, Tc, along the Widom’s line indicated by the green dotted line above Tc. Below Tc, there is a metal-Mott insulator coexistence regime confined by the spinodal lines, Uc1(T ) and Uc2(T ), where the metal and the Mott insulator solutions coexist. The metal solution has a well-known Fermi-liquid behavior, where the resistivity and the scattering rate has a T 2 behavior. The Mott insulator solution shows a gap in the density of state, and its conductivity has an activated behavior. The first order transition line is indicated by the green dotted line below

Tc, where the metal and the Mott insulator solution’s free energy become the same. At T = 0, the ﬁrst order transition line merges with the spinodal line Uc2(T ). The metal-Mott insulator transition at Uc2(T = 0) along the T = 0 trajectory is known as a second order transition where the quasiparticle weight Z decreases smoothly to zero.

3.2.1 Free energy and unstable solution

We employed the Landau free energy functional method to study the structure of the free energy in the coexistence regime [62, 63]. In the DMFT formalism, we know the free energy is a functional of the Green’s function, G(iωn). As a result, the free energy functional can be

24 written into F [G]= F [G] t2T G2(iω ). The local minima of the free energy functional are imp − n n the physically stable or metastableP solutions. Inside the coexistence regime, we know the stable and the metastable solutions are the metal and the Mott insulator solutions. Consequently, the parameter space could be parametrized into G(l) = (1 l)G (iω)+ lG (iω) with a parameter − ins metal l. The change of the free energy along the parameterized space, l, can be calculated by an integral ∆F (l)= t2T l dl′e g(G(l′)), where e =(G G )/ G G and g = G (G) G. 0 l · l ins − met | ins − met| imp − In this project,R we focused on two trajectories, the first order line and a constant U trajectory, as shown in Fig. 3.2 (a). Fig. 3.2 (b) is the free energy functional along the first order line. The metal (l = 1) and the Mott insulator (l = 0) solutions have the same free energy, and the unstable solutions locate around l 0.5. Fig. 3.2(c) shows the free energy along a constant U ∼ trajectory. The unstable solution gradually shifts toward the Mott insulator solutions as we lower the temperature and finally merges with the Mott insulator solution at T = 0, which indicates the Mott insulator solution becomes unstable at T = 0. It is known the steepest descent fix point solver cannot converge the solution to the local maximum (unstable solution) [58]. Therefore, in our approach, we used the Broyden method as our fix point solver with an initial guess that is close to the local maximum of the free energy functional. This method can efficiently converge our solution to the unstable solution within several iterations.

3.2.2 Resistivity calculation and Sommerfeld approximation

To study the transport property, we calculated the temperature dependence of the resistivity along both the ﬁrst order transition line and the constant U trajectory. The resistivity was calculated by the Kubo formula [54]

∂f(ω) σ = σ dǫΦ(ω) dω( )A(ǫ,ω)2, (3.2) 0 − ∂ω Z Z where A(ǫ,ω)= (1/π)Im(ω+µ ǫ Σ(ω))−1 is the spectral function, Φ(ω) = Φ(0)[1 (ω/D)2]3/2 − − − − 2 corresponds to the velocity of the electron, and σ0 = 2πe /¯h is the conductance quantum. Σ(ω) is the selfenergy obtained by the maximum entropy method (MEM) for CTQMC, and the Pade approximation for IPT. The resistivity can be calculated by the inverse of the conductivity, ρ = 1/σ. 2 The Mott limit, ρMott =hD/e ¯ Φ(0), represents the scale in which the scattering process between

25 Trajectory 2: FOTL

Trajectoery 1: Constant U Tc

(a)

0e+00 0e+00

-2e-04 -1e-03 l) l) ( F( F ∆ ∆

T~T(U ) -4e-04 -2e-03 c2 T~Tc T~0.9T(U ) T~0.95Tc c2 T~0.9Tc T~0.8T(Uc2) T~0.85Tc T~0.7T(U ) T~0.8Tc c2 -0.4 0 0.4 0.8 1.2 0 0.2 0.4 0.6 0.8 1 1.2 l l (b) (c)

Figure 3.2: (a) The phase diagram of the half-fiiled Hubbard model, where Uc, Uc1, and Uc2 correspond to the first order transition line, the lower spinodal line, and the upper spinodal line, respectively. The trajectories of interest, in this project, are labeled by the blue solid arrow and the green dashed arrow, respectively. (b) The free energy functional along the first order transition line (FOTL), and (c) along a constant U trajectory. Here Tc is the critical temperature, and TUc2 denotes the temperature at the spinodal line Uc2. l = 0 and l = 1 correspond to the insulator and metal solution, respectively.

26 electrons becomes incoherent. Consequently, a bad metallic behavior can be found in the resistivity when ρ > ρMott. The resistivity, Eq. 3.2, can be simpliﬁed using the Sommerfeld approximation [53], where the lattice Green’s function is replaced by a Lorentzian, G(ω,ǫ) Z , with the quasiparticle ≃ ω−Zǫ+iΓQP weight Z = (1 ∂ReΣ(ω) )−1 and scattering rate Γ = ZImΣ(ω = 0). When Γ < T and − ∂ω ω=0 QP − QP T

σ 1 ZD tanh( ). (3.3) σMott ≈ Γ 2T

We found, in CTQMC, ΓQP < T and T

3.3 Result 3.3.1 Along the ﬁrst order transition line

To understand how the metal, the Mott insulator, and the unstable solutions behave around the

finite temperature critical point, Tc, and the zero temperature critical point, Uc2(T = 0), we studied the transport properties along the first order transition line where both sides of the transition line are terminated by these second order critical points. Fig. 3.3 (a) is the resistivities along the first order transition line calculated from CTQMC with the Kubo formula, Eq. 3.2, and the Sommerfeld approximation, 3.3, indicated by the dots and dashed lines, respectively. At critical temperature,

Tc, three solutions merge as expected, and the resistivity is around the order of the Mott limit,

ρMott. Below Tc, three solutions bifurcate to three trajectories. The unstable solution (green diamonds) has higher resistivity than the metal solution (black circles), and its resistivity remains around the order of the Mott limit, ρ ρ , to the lowest temperature. The resistivity of the ∼ Mott unstable solutions can be qualitatively described by the Sommerfeld approximation indicating the unstable solution is an incoherent metal. The IPT results also show similar behavior to CTQMC as shown in Fig. 3.3 (b).

In contrast to the ﬁnite temperature critical point, Tc, close to the zero temperature critical point, Uc2(T = 0), the resistivites of the three solutions do not merge together. This is due to the fact that the order parameter to describe the zero temperature phase transition is the quasiparticle weight, Z, not the Green’s function or the density of state (DOS), which directly relate to the

27 4 4 10 10 Metal Insulator Metal Unstable Insulator Metal Sommerfeld Unstable 2 Metal Sommerfeld Unstable Sommerfeld 2 10 10 Unstable Sommerfeld Mott Mott ρ/ρ ρ/ρ 0 0 10 10

-2 10 -2 0 0.2 0.4 0.6 0.8 1 1.2 10 0 0.2 0.4 0.6 0.8 1 1.2 T/T C T/Tc (a) CTQMC (b) IPT

2 2

Metal Metal T=0.75Tc Insulator T=0.1Tc Insulator 1.5 Unstable 1.5 Unstable ω) ω) 1 1 -ImG( -ImG(

0.5 0.5

0 0 -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 ω ω (c) CTQMC (d) CTQMC

0.2 2 Metal MEM Metal MEM Unstable MEM Unstable MEM Metal 5th poly Metal 5th poly 0.15 Unstable 5th poly 1.5 Unstable 5th poly Z 0.1 Γ 1

0.05 0.5

0 0 0 0.5 1 1.5 0 0.5 1 1.5 T/TC T/TC (e) CTQMC (f) CTQMC

Figure 3.3: (a) The resistivity for metal (black circles), insulator (red squares), and unstable (green diamonds) solutions along the first order transition calculated from the continuous-time quantum Monte Carlo (CTQMC). The dashed lines are the resistivities calculated from the Sommerfeld approximation, Eq. 3.3. (b) Similar figure as in (a) calculated from the iterative perturbation theory (IPT). (c) The density of state (DOS) at the lowest temperature T = 0.1Tc (near Uc2(T = 0)). (d) The DOS near the critical temperature T = 0.75Tc (e) The quasiparticle weight, Z, for metal and unstable solution. (f) The scattering rate, Γ , for metal and unstable solution. The analytical continuation was done using the maximum entropy method (MEM) and the 5th order polynomial fitting (5th poly). 28 conductivity through Eq. 3.2 and Eq. 3.3. Fig. 3.3 (c) and (d) shows the DOS around Uc2(T = 0) and around Tc, respectively. At the lowest temperature, Fig. 3.3 (c), we found both the metal (black line) and the unstable (green line) solutions have a narrow quasiparticle peak at the Fermi level.

The width of the quasiparticle peak for both solutions are expected to go to zero at Uc2(T = 0). Therefore, both the unstable and the metal solution’s quasiparticle weight, Z, can be smoothly extrapolated to zero when approaching Uc2(T = 0) as shown in Fig. 3.3 (e). This feature is consistent with the study of the Mott MIT at zero temperature with numerical renormalization group (NRG) [64]. Fig. 3.3 (e) shows the quasiparticle weight, Z, for the unstable (green diamonds) and the metal (black circles). The unstable solution has lower quasiparticle weight, Z, than the metal solution. Both solutions can be smoothly extrapolated to Z = 0 at low temperature as we expect for a second order transition. Fig. 3.3 (f) shows the scattering rate , Γ, for unstable (green diamonds) and the metal (black circles) solutions. The scattering rate for the unstable solutions are around the order of the half-bandwidth D = 1, which is the scale of the Mott limit for the scattering rate. With only two parameters, Z and Γ, the resistivity can be described by the Sommerfeld approximation,

Eq. 3.3 , qualitatively up to the critical temperature, Tc, as shown in the dashed lines in Fig. 3.3 (a) and (b). This is similar to the resilient quasiparticle behavior in Ref. [54].

3.3.2 Along the constant U trajectory

In order to understand the behavior of the unstable solution in the entire coexistence regime, we studied the resistivity along a constant U trajectory. From the free energy analysis, Fig. 3.2 (a), we know the unstable solution gradually shifts toward the Mott insulator solution and merges with the Mott insulator solution at T = 0. Therefore, it is expected that the unstable solution becomes more insulating at low temperature. Fig. 3.4 (a) shows the resistivities along the constant U trajectory, where U = 2.4 and U = 2.83 for CTQMC and IPT, respectively. The metal and the Mott insulator solutions has the Fermi-liquid and the activated behavior, respectively. On the other hand, for the unstable solution, we found the resistivity (green diamonds) increases as the temperature decreases. In addition, the resistivity of the unstable solutions can be two orders larger than the Mott limit, ρMott, at low temperature. However, in Fig. 3.4 (c), we found the unstable solution’s DOS at the lowest temperature still has a small quasiparticle peak at the Fermi level

29 indicating the unstable solution still carries metallic properties despite the insulating behavior in its resistivtiy. Fig. 3.4 (d) is the quasiparticle weight for Z for the metal (black circles) and the unstable (green diamonds) solutions. The metal has a normal Fermi-liquid behavior that Z saturates to a constant at low temperature. On the other hand, the unstable solution’s Z decreases with the decrease of temperature as a power law fashion, Z T 2, indicating the reduction of its quasiparticle ∼ peak at the Fermi level. Fig. 3.4 (e) shows the scattering rate for the metal and the unstable solutions. The metal solution’s scattering rate has a Γ T 2 Fermi-liquid behavior. In contrast, the ∼ unstable solution’s scattering rate increases as the temperature decreases as a function, Γ T −2. ∼ In addition, the scattering rate of the unstable solution excess the Mott limit, Γ > D = 1, at low temperature, again, indicating an incoherent scattering behavior. Although the unstable solution’s resistivities are several orders larger than the Mott limit, surprisingly, we found the Sommerfeld approximation, the dashed line in Fig. 3.4 (a), still captures its transport properties qualitatively to very low temperature.

3.4 Conclusion

In this project, we studied the behavior of the unstable, the metal, and the insulator solutions along the ﬁrst order transition line and a constant U trajectory using the DMFT with CTQMC and IPT impurity solvers. The unstable solution is expected to describe the domain wall structure in the metal-Mott insulator coexistence regime. Our results indicate the unstable solution resembles an incoherent metal, where the resistivity and the scattering rate are around or above the Mott limit, ρMott, in the entire coexistence regime. Along the ﬁrst order transition line, the unstable solution’s resistivity and scattering rate remain around the order of the Mott limit (ρ ρ and Γ D) representing its incoherent metallic ∼ Mott ∼ nature. In addition, we found both the metal and the unstable solution’s quasiparticle weight, Z, can be extrapolated to zero at Uc2(T = 0). The vanishing of Z leads to a sharp quasiparticle peak at the Fermi level for both solutions. Similar behavior has been reported by Bulla and Tong et. al [64, 57]. Along the constant U trajectory, the results are more striking. We found the unstable solution’s scattering rate and quasiparticle weight show a power law behavior, where Γ T −2 and ∼ Z T 2, resulting to an insulating behavior in its resistivity. Moreover, the scattering rate and the ∼

30 8 4 10 10 Metal Insulator 6 Metal 10 Unstable Insulator Metal Sommerfeld 2 Unstable Sommerfeld 10 Unstable 4 Metal Sommerfeld 10 Unstable Sommerfeld

Mott 2 Mott 10 ρ/ρ 0 ρ/ ρ 10 0 10

-2 10 -2 10 -4 0.01 10 0.01 T T (a) CTQMC (b) IPT

2 2

Metal Metal T=0.019 Insulator T=0.01 Insulator Unstable Unstable 1.5 1.5 ω) ω) 1 1 -ImG( -ImG(

0.5 0.5

0 0 -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 ω ω (c) CTQMC (d) CTQMC

2 10 Metal 5th poly 0.1 1 Metal MEM 10 Unstable MEM 2 Γ=569xT -2 0 Γ=0.00014xT 10 Metal 5th poly Z Metal MEM Γ Unstable MEM -1 2 10 Z=130.5xT

-2 10

0.01 -3 0.01 10 0.01 T T (e) CTQMC (f) CTQMC

Figure 3.4: (a) The resistivity for metal (black circles), insulator (red squares), and unstable (green diamonds) solution along a constant U = 2.4 trajectory calculated from the continuous- time quantum Monte Carlo (CTQMC). The dashed lines are the resistivity calculated from the Sommerfeld approximation, Eq. 3.3. (b) Similar ﬁgure as in (a) calculated from the iterative perturbative theory (IPT) at U = 2.83. (c) The density of state (DOS) at low temperature T = 0.01. (d) The DOS near the spinadol line Uc2 at T = 0.019. (e) The quasiparticle weight, Z, for metal and unstable solution. (f) The scattering rate , Γ , for metal and unstable solution. The analytical continuation was done using maximum entropy method (MEM) and 5th order polynomial ﬁtting (5th poly). 31 resistivity of the unstables solution can be two orders larger than the Mott limit at low temperature, which also indicate a strong incoherent character in the unstable solution. In the entire coexistence regime, we found the unstable solution always has a quasiparticle peak at the Fermi-level even at the lowest temperature. Consequently, the Sommerfeld quasiparticle approximation to the Kubo formula provides a good description to the resistivity. In conclusion, the unstable solution represents a new state of matter that carries strong incoherent metallic character. The physical properties of this matter are intermediate between the metal and the Mott insulator solution. This result can be connected to the optical image of VO2 and V2O3 in the metal insulator coexistence regime, in which the optical image of the domain wall structure has an intermediate optical scattering strength between the metallic and the insulating domains.

32 CHAPTER 4

THE FATE OF SPINONS AT THE MOTT POINT

4.1 Introduction

The physical nature of the Mott metal to insulator transition (MIT), a phenomenon generic to strongly correlated materials, still remains the subject of much controversy and debate. In contrast to conventional critical phenomena, the relevant degrees of freedom at the MIT cannot be easily identified using an appropriate order parameter of symmetry breaking principle, although different competing orders often do arise in its vicinity. As stressed in pioneering works by Mott and Anderson, however, a fundamentally different physical mechanism has to exist, because the Mott insulating state typically persists to temperatures much higher than any conventional order. A clear physical picture of how a sharp Mott transition can exist without any intervening order has emerged only recently, with the development of Dynamically Mean-Field Theory (DMFT) [19], which is formally exact in the limit of large coordination. Physically, it represents the limit of maximal frustration, therefore eliminating the precursors of any competing order, and retaining only purely local dynamical scattering processes. In contrast to alternative theoretical approaches describing dilute low-energy excitations, DMFT is most reliable at intermediate and high temperatures, where incoherent behavior prevails. The DMFT picture resulted in a finite temperature first-order boundary between the two phases, around which many fascinating phenomena organize themselves. Several of its most striking predictions were experimentally confirmed in a variety of systems, including organic charge-transfer salts of the κ-family, as well as various transition metal oxides. Here one can list the observation of strongly renormalized Fermi liquids [65], bad metal behavior, Ising universality near the Mott endpoint [66, 67], and even quantum critical scaling at higher temperatures [68, 69]. At lower temperatures, most Mott insulators still undergo magnetic ordering, but experimental studies of several frustrated organic materials have instead observed spin liquid behavior [2, 70, 3,

71, 72]. Two particular compounds have attracted a lot of attention, κ-(BEDT-TTF)2Cu2(CN)3 and EtMe3Sb[Pd(dmit)2]2. Most remarkably, thermodynamic measurement in the Mott insulating

33 (a) (b)

Figure 4.1: (a) Phase diagram for the κ-(BEDT-TTF)2Cu2(CN)3 where a spin liquid state is found in the Mott insulating phase from Kurosaki et. al. [2]. (b) Speciﬁc heat measurement from Yamashita et. al. for κ-(BEDT-TTF)2Cu2(CN)3 in the Mott insulating phase. The ﬁnite intercept −1 in CpT axis indicates a low-energy fermionic excitation in Mott insulating phase [3].

phase of these materials have revealed behavior normally expected for metals, including displaying linear in temperature specific heat and large thermal conductivity, indicating the presence of gapless magnetic excitations. In Fig. 4.1(a), we show the phase diagram for κ-(BEDT-TTF)2Cu2(CN)3. In the Mott insulating phase, the NMR measurement indicates there is no magnetic ordering down to the lowest temperature, 1K [2]. Fig. 4.1(b) shows the specific heat for the same materials [3]. The −1 intercept in the CpT axis indicates the low-energy excitation has a linear CV T fermionic behavior. Since the charge degrees of freedom is frozen in the insulating phase, the fermion contributing to the linear specific heat is not the normal electrons. These observations are easiest to rationalize using a time-honored idea, the resonating valence bond (RVB) theory [14], in which chargeless spin excitations embody a magnetic fluid with fermionic statistics. Despite this appealing picture, the RVB approach focuses on the zero temperature phases, and is at trouble in recovering the expected DMFT predictions upon warming up the system. One question that immediately arises concerns the compatibility of the two electronic

34 Jeﬀ

Dynamical Spinon RVB DMFT Σf (iω)

Figure 4.2: Proposed scheme to blend the static RVB approach with the DMFT, using an impurity solver based on emergent spinon degrees of freedom. The spinon self-energy Σf (iω), usually absent from a mean-field RVB treatment, is extracted from a spinon-based DMFT framework. Using the RVB self-consistent determination of the spinon bandwidth Jeff , this in turn allows to incorporate feedback effects due to scattering on charge excitations, which affect the gapless spin-excitations in the Mott state.

fluids, namely the Fermi liquid describing the correlated metallic state, and the magnetic fluid characterizing such a gapless Mott insulator. In this Letter, we address this question by proposing a consistent framework that succeeds in marrying the DMFT and the RVB approaches. This allows to preserve the known high temperature phenomenology of the DMFT, while introducing the strong thermodynamic signature of a gapless spin liquid state in the Mott phase. Akin to the impossibility of mixing oil and vinegar, we find that the Fermi liquid and the magnetic fluid are not miscible into each other in the transition region. While the insulating spin-liquid obviously cannot support mobile charge carriers, the Fermi liquid metal always shows strongly incoherent spinon excitations, due to an orthogonality catastrophe [73, 74, 75]. This observation has important consequences, because the two associated Fermi surfaces cannot be tuned into each other in a continuous way. Thus, the Mott localization into a gapless spin-liquid turns out to have first-order character even at zero temperature, similarly to the case of a spin-gapped insulator described by frozen short-range singlets. This result is in contrast with other scenarios based on the RVB picture only, in which the quasiparticles vanish continuously [76, 77, 78, 79], but is consistent with all available experiments.

4.2 Method

We start by describing our theoretical framework, that is sketched in Fig. 4.2. The main idea is to solve the DMFT equations by explicitly introducing spinon degrees of freedom, and to feedback the resulting spinon self-energy Σf (ω) into the RVB equations, thus aﬀecting the stability of the

35 spin liquid. To be concrete, we consider henceforth the half-ﬁlled Hubbard-Heisenberg Hamiltonian:

1 1 H = t d† d + U d† d d† d − iσ jσ i↑ i↑ − 2 i↓ i↓ − 2 hXi,jiσ Xi † ~τα,α′ † ~τσ,σ′ +J diα diα′ . djσ djσ′ . (4.1) " ′ 2 # " ′ 2 # Xhi,ji α,αX Xσ,σ Here t is the intersite hopping, U the local Coulomb interaction, J an explicit nearest neighbor exchange, and we have denoted by ~τ the set of Pauli matrices. The RVB picture results here from † a decomposition of the physical electron into a chargeless spin-carrying fermion fj,σ and a spinless iθ charge-carrying compact boson Xj = e j , using on each site j the so-called slave rotor [76, 80] de- † † † composition d = f eiθj , with i∂/∂θ = [f f 1 ]. This provides an effective Hamiltonian: j,σ j,σ j σ jσ jσ − 2 P 2 † i(θi−θj ) U ∂ Heff = fiσfjσ Jeff te 2 , (4.2) − − 2 ∂θi hXi,jiσ h i Xi with J = J f † f the RVB bond parameter. This mean-field Hamiltonian is consecu- eff − iσ jσ tively solved within the DMFT, by employing an impurity solver that is naturally based on the spinon/rotor decomposition [80], leading to the following respective Green’s functions in Matsubara domain:

G (iω )−1 = iω + µ Σ (iω ) ∆ (iω ), (4.3) f n n − f n − f n ν2 G (iν )−1 = n + λ Σ (iν ), (4.4) X n U − X n and self-energies in imaginary time:

Σf (τ) = ∆(τ)GX (τ), (4.5)

Σ (τ) = ∆(τ)G (τ), (4.6) X N f with = 3. The DMFT self-consistency equations account both for the physical electron and the N 2 2 spinon hybridization functions, which read for the Bethe lattice: ∆(τ)= t Gd(τ)= t Gf (τ)GX (τ) 2 and ∆f (τ)= Jeﬀ Gf (τ). The RVB equation can now be explicited on the Bethe lattice:

J Jeff Jeff = , (4.7) 2 2 2β zn zn 2 2 iωn + z + J + J X 2 n 4 eff eff q

36 with z = ω ImΣ (iω ) and β = 1/T the inverse temperature. We will set in what follows n n − f n t = 1/2, taking the electronic half-bandwith D = 2t = 1 as natural energy unit. The computation of the electronic speciﬁc heat results from the internal energy:

2 + U H = [ǫdG (k,iω )+ ǫf G (k,iω )]eiωn0 + D , (4.8) β k d n k f n 2 ↑↓ n,k X d f with ǫk / ǫk the electron and spinon dispersion relations, and Gd / Gf their respective Green’s functions. An important term to consider here is the double occupancy D↑↓, which is related to the local charge susceptibility by D↑↓ = (1/2)χc(τ = 0). This quantity can be expressed from either a f X spinon response χc or a rotor response χc :

f † 1 † 1 χ (τ) = [f (τ)f (τ) ][f ′ (0)f ′ (0) ] , c jσ jσ − 2 jσ jσ − 2 σ,σ′ X X ∂ ∂ χc (τ) = i (τ)i (0) . (4.9) ∂θj ∂θj

Both expressions are equivalent only provided the constraint is dealt strictly, but in a mean-field f −1 X −1 −1 treatment, one must use Nagaosa and Lee’s composition rule χc(iω)=[(χc ) +(χc ) ] [further technical details on the implementation of the specific heat calculation are given in Appendix.C][14]. The solution of the combined RVB and DMFT self-consistent scheme provides the physical density of states and specific heat curves shown in Fig. 4.3.

4.3 Results 4.3.1 Density of state and speciﬁc heat

The electronic density of states shows at low temperature the expected behavior: the quasiparticle peak narrows down for increasing values of U, with strong spectral weight transfer towards Hubbard bands located at U/2. This situation persists upon a discontinuous disappearance of the ± quasiparticle peak, leading to the formation of an insulating state with a large Mott gap (shown for U = 3). The formation of heavy quasiparticles can equally be witnessed in the speciﬁc heat

(bottom panel in Fig. 4.3), with a strong enhancement of the γ = CV /T coefficient at the lowest temperature. However, in strong contrast with the usual DMFT predictions, we find the persistence of a finite γ coefficient in the Mott phase, instead of the usually observed activated behavior for a high entropy paramagnetic insulator. This result is in agreement with the thermodynamic measurements made in several organic materials showing spin liquid behavior. Fitting from Fig. 4.3

37 2.0 U =1 U =2 1.5 ) U =3 ω ( d 1.0 ImG

− 0.5

0.0 3 2 1 0 1 2 3 ω

U =1 0.6 U =2 U =3

) 0.4 T ( v C 0.2

0.0 0.0 0.1 0.2 0.3 0.4 0.5 T

Figure 4.3: Upper panel: electronic density of states across the Mott transition for T = 0.005D and J/D = 0.2, with increasing values of the Coulomb interaction U/D = 1, 2, 3. Lower panel: corresponding speciﬁc heat as a function of temperature. A striking linear in temperature contribution remains in CV at U/D = 3, while quasiparticles have disappeared from the density of states in the Mott phase.

38 2 the low-temperature linear slope of CV in the Mott insulating phase, we ﬁnd γ = 27kB/D and 2 γ = 14kB/D for J/D = 0.2 and J/D = 0.4 respectively. Taking the half-bandwidth in the range D = 200 meV, from recent estimates on various organic systems [81, 82], we have approximately γ = 80 mJK−2mol−1 and γ = 40 mJK−2mol−1 for J/D = 0.2 and J/D = 0.4 (these correspond to typical values of the exchange constant in organics [83]). Our predictions are somewhat larger, but of the right magnitude, with the experimental value γ = 20 mJK−2mol−1 measured both for the

EtMe3Sb[Pd(dmit)2]2 [71] and κ-(BEDT-TTF)2Cu2(CN)3 compounds [3]. We now demonstrate that this thermodynamic effect has a strong influence of the metal-insulator phase diagram. The main reason is the quenching of the entropy of the Mott insulator by the presence of a finite exchange interaction J, which typically bends the transition lines towards a stabilization of the metal upon heating. Indeed, the entropic contribution to free energy of the insulator is strongly diminished by exchange, leading to a reentrance of the first order transition lines [84, 85, 86]. We show in Fig. 4.4 two phase diagrams, for J/D = 0.1 and J/D = 0.4 respectively. The continuous lines denote the metal to insulator boundaries, Uc1(T ) and Uc2(T ) at which the insulator and metallic solution disappear respectively. For small J (upper panel), only the Uc1 line is bent, while the whole transition region is affected for large J (lower panel), with a slight increase of the maximum critical temperature Tc due to the coupling to spinon degrees of freedom. Most importantly, the critical domain moves towards strongly reduced values of the Coulomb strength U at increasing J, so that the Brinkman-Rice transition, associated to a diverging γ coefficient, is strongly pre-empted by spin fluctuations. This readily explains the small values of γ that were found in our calculations.

4.3.2 Phase diagram

We then consider the fate of the gapless insulating spin liquid. The phase diagrams of Fig. 4.4 also show as dots the onset of the spinon Fermi surface, namely the low-temperature domain where the effective spinon bandwidth J = 0. We find that this domain corresponds precisely to the eff 6 region of existence of the Mott insulator (only at low temperature for small J, and at temperatures up to the critical Tc of the terminal Mott endpoint for large enough J). Said otherwise, the high entropy local moment insulator is always unstable to the formation of a low entropy spin liquid, even in the coexistence region of the MIT where the Mott gap is strongly reduced. Since the gapless spin-liquid state penetrates fully the part of the phase diagram where metal and insulator coexist,

39 0.03 Uc1

Uc2

0.02 Uc spinons T Tc

0.01 met Jeff =0

0.00 1.8 2.0 2.2 2.4 2.6 2.8 3.0 U

0.05

Uc1 0.04 Uc2

0.03 Uc

T T spinons 0.02 c J met =0 0.01 eff

0.00 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 U

Figure 4.4: Phase diagram for J/D = 0.1 (upper panel) and J/D = 0.4 (lower panel) as a function of Coulomb strength U/D and temperature T/D. Continuous lines denote the metal-insulator boundaries, and dashed lines indicate the true first-order transition based on the free energy. Also, the region bounded by dots show the low-temperature onset of the spinon Fermi surface in the metal Mott insulator. The shaded region indicates that Jeff = 0 for the metallic state within the whole coexistence region. This demonstrates that spinons are only well defined quasiparticles in the low temperature Mott phase, and are never stable in the metal.

40 one can wonder whether the metallic state is capable of hosting non-trivial spinon excitation. We find however, for all our simulations, that the spinon bandwidth always vanishes in the metal, metal namely Jeff = 0. Since the critical value Uc2(T = 0) for the loss of the metal steadily decreases with increasing J, this shows that the Mott transition becomes intrinsically first order at zero temperature, in contrast to the case J = 0 (standard DMFT), where the quasiparticle weight continuously vanishes.

4.3.3 Spinon orthogonality catastrophe

We finally show that spinons are dramatically repelled from the Fermi liquid because of the generic occurence of an orthogonality catastrophe in the spinon self-energy Σf (ω) for a Fermi liquid state. The reason is deeply rooted in the spinon/rotor decomposition, and its associated constraint Q = i∂/∂θ [f † f 1 ] = 0. While the spectral function of the physical electron j − σ jσ jσ − 2 † † iθj djσ = fjσe connects excitationsP within the physical subspace Q = 0 only, the spinon density of states corresponds to processes that link the physical subspace Q = 0 to an unphysical subspace Q = 1 containing one extra auxiliary slave-particle. Thus, the spinon density of states involves matrix elements of the type Φ(1) f † Φ(0) 2, where Φ(Q) are wavefunctions living in different | | jσ| | | Q-subspace, which thus experience in a metal a singular x-ray edge. Due to this mechanism, the spinon self-energy acquires an anomalous frequency dependence Σ (ω) ωα at low-energy f ∝ (typically α< 1/2). This crucial property is well obeyed by construction in our impurity solver [80], and can be shown from an exact numerical solution of quantum impurity problems [73, 74], as well as from 1/N fluctuations [87] around the condensed slave boson mean-field theory (upon which the static RVB picture is based). The calculated spinon self-energy is shown in Fig. 4.5 for the metallic and insulating phases. The anomalously large spinon scattering rate is clearly seen for U/D = 1, while a regular behavior is found for U/D = 3. Spinon are thus very incoherent in the metallic state, insomuch as to fully destroys their Fermi surface, as we discuss now. The spinon orthogonality catastrophe is indeed the key physical mechanism leading to the instability of the spinon Fermi surface in the whole metallic phase. The absence of spinons in the metal metal, Jeff = 0, can be formulated as a simple inequality from the linearized version of the RVB equation (4.7): 1 1 1 > 2 . (4.10) J 2β [ωn ImΣf (iωn)] Xiωn −

41 0.14

0.12 U =1

) 0.10 U =3 n

iω 0.08

Σ( 0.06

Im 0.04 − 0.02

0.00 0.0 0.5 1.0 1.5 2.0 2.5 3.0 ωn

Figure 4.5: Frequency-dependent spinon self-energy Σf (iωn) at temperature T = 0.005D for U/D = 1 (metal) and U/D = 3 (insulator). The strong enhancement in the metal is due to an x-ray edge eﬀect, which is a hallmark the spinon spectral density.

Using the anomalous spinon self-energy Σ (iω) = iC ω αSign(ω) at low energy, and evaluating f − f | | the Matsubara sum at low temperatures, we find the inequality for stability of the a true Fermi liquid: 1 ∞ dω 1 > (4.11) J π [ω + C ωα]2 Z0 f Since α < 1/2, the infrared divergence in the integral is cut off and the inequality is fulfilled for small J. In contrast, the Mott insulator is always unstable to the spin-liquid because the integral diverges for Σf = 0. These analytic arguments are in complete agreement with our numerical findings.

4.4 Conclusion

In conclusion, we have examined the role of gapless spin excitations, characterizing frustrated Mott insulators, in the vicinity of the Mott metal-insulator transition at half-filling. Our result indicate that the spin-liquid arises simultaneously with Mott gap opening, suggesting that local magnetic moments generically tend to form a zero-entropy gapless state in absence of magnetic ordering. However, a spinon Fermi surface cannot be stabilized upon closing of the Mott gap, due to an orthogonality catastrophe we identified with emerging charge fluctuations. This mechanism leads to the conclusion that the Mott transition associated with the loss of quasiparticles at Uc2

42 is inherently of first-order type, even at zero temperature and for a zero entropy Mott insulator. From this perspective, we advocate a scenario where behavior consistent with quantum criticality (QC) has purely local character and emerges only at high temperatures, above the metal-insulator coexistence region, consistent with experiments. Our result indicate, thus, that spin liquid correlations do not play a significant role within this high temperature QC region, which marks the closing of the Mott gap. The physical picture we propose is dramatically different from the perspective provided from alternative theories [76, 77, 78, 79], which postulate the dominance of spin-liquid excitation in the entire QC regime surrounding the Mott point.

43 CHAPTER 5

EMERGENT BLOCH EXCITATIONS IN MOTT MATTER

5.1 Introduction

The physical nature of the excited states in strongly interacting quantum systems has long been a subject of much controversy and debate. Deeper understanding was achieved by Landau, more than half a century ago [88], who realized that in systems of fermions the Pauli principle provides a spectacular simplification. He showed that many properties of Fermi systems can be understood in terms of weakly interacting quasiparticles (QP), allowing a precise and detailed description of strongly correlated matter. Modern experiments provide for even more direct evidence of such QP excitations, for example from using angle-resolved photoemission spectroscopy (ARPES) [7] or scanning-tunneling microscopy (STM) methods [89]. The Fermi liquid paradigm, however, describes only the low energy excitations. At higher energies, the physical properties are often dominated by incoherent processes, which do not conform to the Landau picture. The task to provide a simple and robust theoretical description of such incoherent excitations has therefore emerged as a central challenge of contemporary physics. An intriguing apparent paradox is most evident around the Mott point. Here, ARPES and STM experiments provide often clear evidence of additional well-defined high energy excitations (Hubbard bands) which, while being fairly incoherent, still display relatively well defined Bloch character with pronounced momentum dispersion, see, e.g., Ref. [90]. As a matter of fact, it is often difficult to experimentally even distinguish the Hubbard bands found in Mott insulators from ordinary Bloch bands found at high energy in conventional band insulators. While such behavior can be already numerically reproduced by some modern many-body approximations [19, 91], a simple conceptual picture for the apparent Bloch character of such high energy charge excitations is not still available. In particular, variational methods such as the Gutzwiller Approximation (GA) [18] — which are often able to reproduce the numerical results in a much simpler semi-analytical fashion — generally

44 capture only the low-lying QP features on the metallic side, but cannot provide a description of charge excitations around the Mott point and in the insulating regime. The goal of this Letter is to write an appropriate variational wave function able to capture both the (low energy) QP bands and the (high energy) Hubbard bands, within the same theoretical framework. A particularly interesting fact emerging from our theory is that many important features of both types of excitations are encoded in the bare density of states (DOS) of the uncorrelated system and a few renormalization parameters — in a similar fashion as for the QP excitations in Landau theory of Fermi liquids. This is accomplished, similarly as in many other theories for many-body systems, see, e.g., Refs. [92, 93, 94], by enlarging the Hilbert space by introducing auxiliary ”ghost” degrees of freedom. In particular, this construction sheds light on the physical origin of the ”hidden” Bloch character of the Hubbard bands mentioned above. Our calculations of the single-band Hubbard model, which are benchmarked against the Dynamical Mean Field Theory (DMFT) [19, 91] solution, show that the new wave function quantitatively captures not only the dispersion of the QP but also the Hubbard bands. Furthermore, our theory enables us to describe the Mott transition and the coexistence region between the metallic and the Mott-insulator phases.

5.2 Ghost GA theory

For simplicity, our theory will be formulated here for the single-band Hubbard model

ˆ † H = tRR′ cRσcR′σ + U nˆR↑nˆR↓ (5.1) ′ σ XRR X XRσ at half-ﬁlling. The generalization to arbitrary multi-orbital Hubbard Hamiltonians is straightforward. In order to construct the Ghost-GA theory we are going to embed the physical Hamiltonian of the system [Eq. (5.1)] within an extended Hilbert space obtained by introducing auxiliary Fermionic ”ghost” degrees of freedom not coupled with the physical orbitals, see Fig. 5.1. Let us represent Hˆ within the extended Hilbert space mentioned above as follows:

ˆ ˜αβ † H = tRR′ cRασcR′βσ + U nˆR1↑nˆR1↓ ′ XRR Xαβσ XR αβ † = ǫ˜k ckασckβσ + U nˆR1↑nˆR1↓ , (5.2) Xk Xαβσ XR

45 Figure 5.1: Representation of a lattice including 2 ghost orbitals (α = 2, 3). The Hamiltonian of the system acts as 0 over the the auxiliary ghost degrees of freedom. In particular, the Hubbard interaction U acts only over the physical orbital α = 1.

˜11 11 where tRR′ = tRR′ are the physical hopping parameters,ǫ ˜k = ǫk are the eigenvalues of the ﬁrst αβ αβ term of Hˆ , t˜ ′ =ǫ ˜ = 0 (α,β) = (1, 1) and σ is the spin. RR k ∀ 6 Our theory consists in applying the ordinary multi-orbital GA theory [95, 96, 97, 98, 99] to Eq (5.2). In other words, the expectation value of Hˆ is optimized variationally with respect to a Gutzwiller wave function represented as Ψ = ˆ Ψ , where Ψ is the most general Slater | Gi PG| 0i | 0i determinant, ˆ = ˆ , and ˆ acts over all of the local degrees of freedom labeled by R — PG R PR PR including the ghostQ orbitals α > 1. The variational wave function is restricted by the following conditions:

Ψ ˆ† ˆ Ψ = Ψ Ψ (5.3) h 0| PRPR | 0i h 0| 0i Ψ ˆ† ˆ c† c Ψ = Ψ c† c Ψ , (5.4) h 0| PRPR Rασ Rβσ | 0i h 0| Rασ Rβσ | 0i which are commonly called ”Gutzwiller constraints”. Furthermore, the so called ”Gutzwiller Ap- proximation” [18] — which is exact in the limit of inﬁnite dimensions (where DMFT is exact) — is employed. The minimization of the variational energy will be performed by employing the algorithms derived in Ref. [100]. The basis of our theory is that extending the Hilbert space by introducing the ghost orbitals does not aﬀect the physical Hubbard Hamiltonian Hˆ , as all of its terms involving ghost orbitals are multiplied by 0, see Eq. (5.2). The advantage of enlarging the Hilbert space arises exclusively

46 from the fact that the corresponding Ghost-GA multi-orbital variational space is substantially more rich with respect to the original GA variational space (where ˆ acts only on the physical degrees PR of freedom). In this respect, our scheme presents analogies with the solution of the Aﬄeck, Lieb, Kennedy and Tasak (AKLT) model [92] and with the theory of matrix product states (MPS) [93, 94], which are also constructions obtained involving virtual bipartite entanglement and projections. As shown in previous works, see, e.g., Refs. [97], the variational energy minimum of Hˆ is realized by a wave function Ψ = ˆ Ψ where Ψ is the ground state of a quadratic multi | Gi PG| 0i | 0i band Hamiltonian represented as

Hˆ = ˜ǫ˜ ˜† + λ˜ f † f qp R kR ab kaσ kbσ Xk Xabσ ∗ † = ǫ˜kn ψknσψknσ , (5.5) Xknσ where f are related to c by a proper unitary transformation [95, 97], the matrices ˜ and kaσ kaσ R ˜ ∗ ˆ λ are determined variationally andǫ ˜kn are the eigenvalues of Hqp. The states represented as Ψp = ˆ ψ† Ψ and Ψh = ˆ ψ Ψ , where ψ† are the eigen-operators of Hˆ , | Gknσi PG knσ| 0i | Gknσi PG knσ| 0i kaσ qp represent excited states of Hˆ [101, 102]. The energy-resolved Green’s function of the physical degrees of freedom (α = 1) can be evaluated in terms of the excitations mentioned above and represented as

G(ǫ ,ω)= ω ǫ˜ Σ(˜ ω) −1 = ω ǫ Σ(ω) −1, (5.6) k − k − 11 − k − where Σ(˜ ω) = ω I ˜† ˜ ˜† ˜ −1 + ˜−1λ˜ ˜†−1 and the subscript ”11” indicates that we are − − R R R R R R interested only in the physical component α = β = 1 of the Green’s function. We point out that, since Eq. (5.6) involves a matrix inversion, the Ghost-GA approximation Σ(ω) to the physical self- energy is generally a non-linear function [103, 104] — while, by construction, the self-energy is linear in the ordinary GA theory. Note also that, since

† 1 G(ǫk,ω)= ˜ ˜ , (5.7) ˜ ˜† ˜ "R ω ǫ˜k + λ R# − R R 11 ∗ ˆ the poles of G(ǫk,ω) coincide with the eigenvaluesǫ ˜ka of Hqp, see Eq. (5.5).

47 5.3 Results 5.3.1 Benchmark: Energy, double occupancy, and quasiparticle weight

Below we apply our approach to the Hubbard Hamiltonian [Eq. (5.2)] at half-filling assuming a semicircular DOS, which corresponds, e.g., to the Bethe lattice in the limit of infinite connectivity, where DMFT is exact [19]. The half-bandwidth D will be used as the unit of energy. The extended Ghost-GA scheme will be applied following the procedure of Ref. [100], utilizing up to 2 ghost orbitals. In Fig. 5.2 is shown the evolution as a function of the Hubbard interaction strength U of the Ghost-GA total energy, the local double occupancy and the QP weight z. Our results are shown in comparison with the ordinary GA theory and with DMFT in combination with Numerical Renor- malization Group (NRG). In particular, we employed the ”NRG Ljubljana” impurity solver [105]. The agreement between Ghost-GA and DMFT is quantitatively remarkable. In particular, the Ghost-GA theory enables us to account for the coexistence region of the Mott and metallic phases, which is not captured by the ordinary GA theory. The values of the boundaries of the coexistence region U 2, U 2.79 are in good agreement with the DMFT results available c1 ≃ c2 ≃ in the literature [106, 107, 108, 109], i.e., U 2.39, U 2.94. The Ghost-GA value of U , c1 ≃ c2 ≃ c2 which is the actual Mott transition point at T = 0, is particularly accurate. The method also gives a reasonable value for the very small energy scale characterizing the coexistence region, which we can estimate as T E (U ) E (U ) 0.02, consistently with both DMFT and c ≃ ins c1 − met c1 ≃ experiments [110, 111]. We point out also that, as shown in the second panel of Fig. 5.2, the Ghost- GA approach captures the charge fluctuations in the Mott phase, while this is approximated by the simple atomic limit (which has zero double occupancy) within the Brinkman-Rice scenario [112]. Interestingly, while at least 2 ghost orbitals are necessary to obtain the data illustrated above for the Metallic solution, 1 ghost orbital is sufficient to obtain our results concerning the Mott phase. Increasing further the number of ghost orbitals does not lead to any appreciable difference. As we are going to show, this is connected with the fact that the electronic structures of the Mott and the metallic phases are topologically distinct.

48 Pwih lwrpnl safnto fteHbaditrcinstrengt interaction Hubbard occupanc the double of local function panel), a (upper as panel) energy (lower total weight of QP Evolution 5.2: Figure snwt h riayG n ihteDF+R eut.TeGhost-G The results. DMFT+NRG the with and re region GA Ghost-GA coexistence ordinary The the half-ﬁlling. with at DOS ison semicircular with model Hubbard pcrlwih vralfeunis(e icsini antext main in discussion (see frequencies all over weight spectral U c 1 Etot U , −0.4 −0.3 −0.2 −0.1

z 0.05 0.15 0.25 0.2 0.4 0.6 0.8 0.1 0.2 c 2 0 1 0 0 r niae yvria otdlns ne:Itga fGotG l Ghost-GA of Integral Inset: lines. dotted vertical by indicated are 5 4 3 2 1 0 49 U insulator DMFT insulator DMFT metal GA ghost with 1 insulator GAmetal ghost with 2 std.GA c1 U U 0.9 1 c2 4 2 0 z+h ). h ut r hw ncompar- in shown are sults h U mdl ae)and panel) (middle y onaiso the of boundaries A o h single-band the for ocal 5.3.2 Benchmark: Single-particle Green’s function

Let us now analyze the Ghost-GA single-particle Green’s function G(ǫ,ω), see Eq. (5.7). In Fig. 5.3 is shown the Ghost-GA energy-resolved spectral function A(ǫ,ω)= 1 ImG(ǫ,ω) in com- − π parison with DMFT [113]. Although the broadening of the bands (scattering rate), is not captured by our approximation (as it is not captured by the ordinary GA), the positions and the weights of the poles of the Ghost-GA spectral function encode most of the DMFT features, not only at low energies (QP excitations), but also at high energies (Hubbard bands). In order to analyze how the spectral properties of the system emerge within the Ghost-GA theory, it is particularly convenient to express the QP Hamiltonian [Eq. (5.5)] in a gauge where λ˜ is diagonal. In the metallic phase, an explicit Ghost-GA calculation obtained employing 2 ghost orbitals shows that the matrices ˜ and λ˜ are represented as follows: R

λ˜ = lδ (δ δ ) (5.8) ij ij 2i − 3i ˜ = δ √zδ + √h (δ + δ )/√2 , (5.9) Rij j1 i1 i2 i3 where δij is the Kronecker delta, and l, z and h are real positive numbers determined numerically as in Ref. [100]. The corresponding self-energy, see Eq. (5.6), is given by:

ω 1 z 2 Σ(ω)= 1 = − ω + o(ω ) . (5.10) 1+ 2 2 2 − z z− ω −l +2hω ω2−l2

Thus, the variational parameter z of Eq. (5.9) represents the QP weight, whose behavior was displayed in the third panel of Fig. 5.2. Note that the overall spectral weight dω dǫ ρ(ǫ) A(ǫ,ω), where ρ(ǫ) is the semicircular DOS, is not z as in the ordinary GA theory, butR it is zR+h =[ ˜† ˜] , R R 11 which is almost equal to 1 for all values of U (see the inset of the third panel in Fig. 5.2). The additional spectral contribution h, which is not present in the ordinary GA approximation, enables the Ghost-GA theory to account for the Hubbard bands. In the Mott phase, an explicit Ghost-GA calculation obtained employing 1 ghost orbital shows that the matrices ˜ and λ˜ are represented as follows: R

λ˜ = lδ (δ δ ) (5.11) ij ij 1i − 2i ˜ = δ √h (δ + δ )/√2 , (5.12) Rij j1 i1 i2

50 1 0.5 U=1.0 0 −0.5 −1 1 0.5 U=2.5 0 −0.5 −1 k 1 0.5 U=3.5 0 −0.5 −1 1 0.5 U=5.0 0 −0.5

ε −1 −4 −2 0 2 4 ω

Figure 5.3: Poles of the Ghost-GA energy-resolved Green’s function (bullets), see Eq. (5.7), in comparison with DMFT+NRG. The size of the bullets indicates the spectral weights of the corresponding poles. Metallic solution for U = 1, 2.5 and Mott solution for U = 3.5, 5.

51 where l and h are real positive numbers determined numerically as in Ref. [100]. Note that h = [ ˜† ˜] 1 (see the inset of the third panel in Fig. 5.2). The corresponding self-energy, see R R 11 ≃ Eq. (5.6), is given by: 1 h l2 1 Σ(ω)= − ω + . (5.13) − h h ω The pole of the self energy at ω = 0, which is the source of the Mott gap, is captured by the Ghost-GA theory. The analysis above clarifies also why, by construction, within the Ghost-GA approximation the self-energy can develop poles, see Eqs. (5.10) and (5.13), but can not capture branch-cut singularities on the real axis. It is important to point out that the Hilbert space extension that we have introduced in this work has been essential in order to capture the effect of the electron correlations on the topology of the excitations — such as the change of the number of bands at the Mott transition (between 3 bands in the metallic phase and 2 bands in the Mott phase). In fact, without extending the Hilbert space, the ordinary GA theory enables only to renormalize and shift the band structure with respect to the uncorrelated limit U = 0, without affecting its qualitative topological structure. On the other hand, extending the Hilbert space enables us to relax this constraint, as G(ǫ,ω), see Eq. (5.7), is variationally allowed to have any number of distinct poles equal or smaller to the corresponding total (physical and ghost) number of orbitals. It is for this reasons that only 1 ghost (2 orbitals) is sufficient to describe the Mott phase of the single-band Hubbard model, while at least 2 ghosts (3 orbitals) are necessary in order to describe its metallic phase — whose spectra includes the QP excitations and the 2 Hubbard bands. A remarkable aspect of this construction is that, within the Ghost-GA theory, the information concerning the spectral function — including the Hubbard bands — is entirely encoded in only 3 parameters (z,h,l) in the Metallic phase, and in 2 parameters (h,l) in the Mott phase, see Eqs. (5.8), (5.9), (5.11), (5.12).

5.4 Conclusion

We derived a uniﬁed theoretical picture for excitations in Mott systems based on a generalization of the GA, which captures not only the low-energy QP excitations, but also the Hubbard bands. The essential idea consists in extending the Hilbert space of the system by introducing auxiliary ”ghost” orbitals. This construction enables us to express analytically many important features of both

52 types of excitations in terms of the bare DOS of the uncorrelated system and a few renormalization parameters, in a similar fashion as for the QP excitations in Landau theory of Fermi liquids. In particular, this idea provides us with a conceptual picture which assigns naturally a Bloch character to the Hubbard bands even in Mott insulators. In this respect, we note that our theory presents a few suggestive analogies with the interesting (but rather speculative) idea of ”hidden Fermi liquid” previously introduced by P. W. Anderson [114] within the context of the BCS wave function (for superconductors) and the Laughlin’s Jastrow wave function (for the Fractional Hall Effect). In fact, they both propose a descriptions of non-Fermi liquid states related to ordinary Fermi liquids residing in unphysical Hilbert spaces, see, e.g., Ref. [115]. From the computational perspective, our Ghost-GA theory constitutes a very promising tool for ab-initio calculations in combination with Density Functional Theory (DFT) [116, 117, 99, 118, 97], as it is substantially more accurate with respect to the ordinary GA approximation, without much additional computational cost. In fact, within the numerical scheme described in Refs. [100, 97], our theory results in solving iteratively a finite impurity model with 2ν 1 bath sites, where ν is the total number of orbitals − (physical and ghost). Since there exist numerous available techniques enabling to solve efficiently this auxiliary problem, see, e.g., Refs. [119, 120, 121, 122, 123], this opens an exciting avenue for realistic modeling of many challenging materials, including predictions of ARPES spectra for complex orbitally-selective Mott insulators. Furthermore, since the Ghost-GA theory is based on the multi-orbital GA [100, 97], it can be straightforwardly generalized to finite temperatures [124, 125, 126] and to non-equilibrium problems [127, 128]. For the same reason, the Ghost-GA theory can be reformulated [129, 95, 97] in terms of the rotationally invariant slave boson (RISB) theory [130, 131], whose exact operatorial foundation recently derived in Ref. [100] constitutes a starting point to calculate further corrections [132]. It would be also interesting to apply the ghost-orbital Hilbert space extension in combination with the Variational Monte Carlo method [133] or the generalization of the GA to finite dimensions of Ref. [134], which might lead to a more accurate description of strongly correlated electron systems even beyond the DMFT approximation.

53 CHAPTER 6

STRUCTURAL PREDICTIONS AND POLYMORPHISM IN TRANSITION METAL COMPOUNDS

6.1 Introduction

Predicting ground-state structure of periodic systems became an active area of research with the advent and efficient numerical implementation of total energy methods, density functional theory (DFT) in particular. Various approaches to exploring potential energy surface of solids have been proposed [9, 135] including simulated annealing [136, 137], evolutionary algorithms [138, 139, 140], metadynamics [141], minima hopping [142], random structure search [143], methods based on data mining and machine learning [144], structure prototyping [145, 8, 146], etc. Alongside, an increased activity in exploring the space of low energy polymorphs in searching for new functional materials [147, 148] and advancing our understanding of the phenomenon of polymorphism itself and realizability of metastable structures [146, 149] can be observed. The common trait of all of the above-mentioned methods is their critical dependence of the accurate representation of the potential energy surface, i.e., on the correct electronic Hamiltonian that defines the potential energy of the ions within the adiabatic approximation. Given the usual method of choice, typically based on DFT, it is not a surprise that most of the structure prediction works try to avoid transition metals and their compounds. This is because of the inability of DFT and related approaches to correctly describe effects of strong electronic correlations present in these systems, e.g., the strong mass enhancement in heavy fermion compounds and the Mott insulator metal transition (IMT) in transition metal oxides [13, 10]. One manifestation of the problems that can arise due to the effects that go beyond DFT is the inaccurate dependence of the total energy on crystal structure leading to wrong energy ordering of different polymorphs. This can even cause incorrect predictions of the stable structure. For example, the tendency to erroneous predictions of the lowest energy structure of compounds that contain transition metal atoms has recently been recognized for MnO, which according to DFT and DFT+U has lower

54 energy in the tetrahedrally coordinated zincblende structure in antiferromagnetic phase contrary to experimentally known rocksalt ground state [150, 151], and for MnO2 where DFT, DFT+U and hybrid HSE functionals favor α phase instead of the experimentally known β ground state structure [152]. In the case of MnO, the improved description of electronic correlations within the random phase approximation [150] or fixed node diffusion Monte Carlo [151] provides correct energy ordering of different structures and correct prediction of the rocksalt ground-state structure. Despite the success of DFT+U and RPA treatments in the aforementioned magnetic ordered phases, in general, most of the transition metal oxides are paramagnetic Mott insulators at the ambient condition and show a Mott IMT by applying pressure [153, 154]. For example, MnO, FeO, and CoO are paramagnetic Mott insulators at the ambient condition and experience an antiferromagnetic transition below the Neel temperatures, 116K, 198K, and 291K, respectively. It is known DFT+U and RPA related methods fail to produce paramagnetic insulating solutions in FeO and CoO. Therefore, more accurate and efficient methods to treat the correlation effect are required in the field of structure prediction. In this work, we investigate a general preference of DFT based methods (LDA) toward tetrahedral coordination of transition metals in their oxide compounds (manganese oxide included). Namely, we find that, in DFT, the transition metal oxides such as MnO, CrO, FeO, and CoO in their rocksalt structure, which consists of edge-sharing coordination octahedra, are higher in energy than in their wurtzite or zincblende structures within which the transition metal atoms are tetrahedrally coordinated. This is in contradiction with the rocksalt structure being the experimentally known the stable structure for all these systems in the ambient condition. To elucidate the physical reasons behind this preference and investigate further the role of electronic correlations on the total energy difference between different crystal structures in transition metal compounds, we employ the local density approximation (LDA) plus rotationally-invariant slave-boson (RISB) approach as the treatment for the correlation effect in LDA. This state of art method is capable of describing the large mass enhancement and the Mott IMT due to the strong local Coulomb interaction between electrons [21, 155]. It also has been shown to have qualitatively agreements with sophisticated LDA plus dynamical mean field theory (LDA+DMFT) approach [20] but is computationally more efficient for performing large scale calculations. LDA+U, which is one of the most popular methods to treat correlation effects, is also used to compare the performance of LDA+RISB.

55 Figure 6.1: Four crystal structures, (a) NiAs-type, (b) rocksalt, (c) wurtzite, and (d) zincblende, studied in this project. NiAs-type and rocksalt have octahedral coordinate with hexagonal and cubic symmetry, respectively, and wurtzite and zincblende have tetrahedral coordinate with hexagonal and cubic symmetry, respectively.

To benchmark our approach, we studied six transition metal binary oxides and chalcogenides including CrO, FeO, MnO, CoO, CoS, and CoSe, in four diﬀerent crystal structures: rocksalt, NiAs-type, zincblende, and wurtzite, as shown in Fig. 6.1. This set of crystal structures is chosen to help analyzing the interplay between the local coordination versus the long range order as they combine octahedral coordination with cubic symmetry (rocksalt), octahedral coordination with hexagonal symmetry (NiAs-type), tetrahedral coordination with cubic symmetry (zincblende) and tetrahedral coordination with hexagonal symmetry (wurtzite). Another observation which follows from the available experimental data is that, while transition metal monoxides favor the rocksalt structure, as the anion changes down along the group-16 of the periodic table, from O to S, Se and Te, the ground state structure in some systems changes to hexagonal symmetry and/or tetrahedral coordination. This is the case for CoS and CoSe which adopt the NiAs-type structure rather than the rocksalt. To elucidate the importance of strong correlation eﬀects, we constrain our system in the paramagnetic phase, corresponding to the high temperature phase for these materials, where DFT related methods fail to produce the experimentally observed structures in these transition metal oxides. Our results show that LDA+RISB correctly captures the experimentally observed crystal and electronic structures for all six transition metal binary oxides and chalcogenides. It also reproduces the Mott IMT in MnO, FeO, and CoO, as found in experiments. Moreover, LDA+RISB predicts accurate equilibrium volumes of all the oxides and chalcogenides compared to the data from inorganic crystal structure database (ICSD).

56 6.2 Method

The electronic structure calculation was performed using the density functional theory within the local density approximation (LDA) and LDA+U as implemented in the WIEN2k package [156]. The LDA plus rotationally invariant slave boson (LDA+RISB) calculation were performed using the CyGutz package interfaced with the LDA in WIEN2k as describe in Chapter. 2. The calculation was carried out using 50000 k-points and RKmax = 8 for all the calculations. The total energy and charge convergence criteria were set to 10−5 Ry and 10−3 electrons, respectively. For the double counting functional, we used the fully-localized limit (FFL) in LDA+RISB and the self-interaction correction (SIC) in LDA+U. For the Coulomb interaction parametrization in LDA+RISB, we use the rotationally invariant Coulomb interaction within the Slater-Condon parametrization as described in Appendix. A. For all the cases, we only consider the crystal field (CF) effect and ignore the spin-orbit coupling (SOC) effect because the energy scale for SOC is much smaller than the CF in 3d transition metal compounds.

6.3 Results 6.3.1 Predicted structures and unit cell volumes

To demonstrate the importance of the electronic correlation, we show the predicted unit cell volumes and structures from LDA, LDA+RISB and the experiment data in Fig. 6.2. The experimentally observed and the predicted structures are color coded by blue, green, red, cyan, for NiAs-type, rocksalt, wurtzite, zincblende, respectively, and the filled and half-filled symbols correspond to the the metallic and the Mott-insulating phase, respectively. As shown in the figure, without the correction of the correlation effects, LDA (triangles) fails to predict the ground state structures for all the oxides, and the predicted equilibrium volumes of CoS and CoSe are inaccurate compared to the experimental data (circles). LDA also predicts the MnO, FeO, and CoO as metal instead of the experimentally known Mott insulator. After the inclusion of electronic correlation effects by using LDA+RISB, the LDA+RISB data (diamonds) faithfully capture the correct experimental structures and unit cell volumes for all the selected materials, CrO, MnO, FeO, CoO, CoS, and CoSe. This result indicates that LDA+RISB is a reliable method to treat the correlation effects in transition metal compounds. In the following section, we will compare the difference between

57 LDA+RISB and LDA+U, the most popular method to add electronic correlation, and show that the LDA+RISB is the most proper method to treat correlated materials.

32 )

3 NiAs-type Metal

Å 30 Rocksalt Insulator Wurtzite 28 Zincblende 26 Experiment LDA 24 LDA+RISB

Predicted unit cell volumes ( volumes unitPredicted cell 16 CrO MnO FeO CoO CoS CoSe Materials

Figure 6.2: Predicted unit cell volumes and crystal structures. The colors, blue, green, red, and cyan, correspond to NiAs-type, rocksalt, wurtzite, and zincblende structures, respectively. The metallic and the Mott-insulating solution are labeld by ﬁlled and half-ﬁlled symbols, respectively. The circle, triangle, and diamond markers correspond to the experiment data, LDA, and LDA+RISB predictions, respectively. U and J is 13 eV and 0.9 eV, respectively, for LDA+RISB calculation.

6.3.2 LDA

We first show our LDA energy-volume plot in Fig. 6.3. The black vertical solid lines in each figure indicate the experimental equilibrium volumes of each material. The colored vertical dashed lines indicate the equilibrium volumes predicted by LDA, and the colors correspond to the predicted ground state structures. As shown in the figure, LDA fails to predict the experimentally stable structures for all the oxides which are known as in the rocksalt structure (green triangles) in experimental observation. Instead, LDA predicts the wurtzite structures (red squares) of CoO [Fig. 6.3 (a)], CrO [Fig. 6.3 (d)], and CoO [Fig. 6.3 (e)], and zincblende structure (cyan diamonds) of FeO [Fig. 6.3 (f)] are favored in energy. In addition, LDA predicts all the oxides are metals in contrary to the experimentally known Mott insulator in MnO, FeO, and CoO. The inability of

58 2.5 CoO LDA 2.0 CoS LDA 3.0 CoSe LDA V V (a) LDA Exp (b) VLDA VExp (c)VLDA VExp 2.5 2.0 1.5 2.0 1.5 1.0 1.5 1.0 1.0

E(eV/f.u) 0.5 NiAs LDA 0.5 0.5 Rocksalt LDA 0.0 0.0 0.0 Wurtzite LDA 15 16 17 18 19 20 21 22 23 22 24 26 28 30 32 24 26 28 30 32 34 36 38 Zincblende LDA CrO LDA MnO LDA FeO LDA 2.5 3.0 2.0 (d) VLDA VExp (e)VLDA VExp (f) VLDA VExp 2.5 2.0 1.5 2.0 1.5 1.5 1.0 1.0 1.0

E(eV/f.u) 0.5 0.5 0.5 0.0 0.0 0.0 15 16 17 18 19 20 21 16 18 20 22 24 15 16 17 18 19 20 21 V(Å 3/f.u)

Figure 6.3: LDA energy-volume curves for (a) CoO, (b) CoS, (c) CoSe, (d) CrO, (e) MnO, and (f) FeO in the paramagnetic phase. The blue circle, green triangle, red square, and cyan diamond curves corresponds to NiAs-type, rocksalt, wurtzite, and zincblende structures, respectively. The experimental equilibrium volumes of each compound are indicated by the vertical colored solid lines, and the equilibrium volumes predicted by LDA are indicated by the colored dashed lines, where the colors correspond to the structures labeled in the legend.

LDA to describe the electronic correlation effect, i.e., band renormalization and Mott transition, resulting inaccurate energy ordering for all the oxides. Despite the unsatisfactory performance on the oxides, LDA captures the correct NiAs-type structures (blue circles) for CoSe [Fig. 6.3 (b)] and CoS [Fig. 6.3(c)]. This is because the electronic correlation effects, in these systems, are relatively weak compared to the oxides, and cannot drive the system to the Mott-insulating phase. Therefore, the single-particle description of LDA still works for these materials. However, the equilibrium volumes predicted from LDA, for CoS and CoSe, are quite far from the experiment data, as shown in the figure. Later, we will show the equilibrium volume can be improved by taking the electronic correlation into account.

6.3.3 LDA+RISB

We now discuss how the electronic correlation effects modify the energy ordering among different polymorphs in LDA+RISB method. Fig. 6.4 shows the energy ordering among different structures within LDA+RISB approach at U = 13 eV and J = 0.9 eV. An additional feature in Fig. 6.4 is the Mott IMT as indicated by the crossing between the solid and dashed lines representing

59 the metallic and the Mott-insulating phase, respectively. Within this approach, we recovered the experimentally observed Mott-insulating solution in rocksalt structure (green triangles dashed line) for CoO [Fig. 6.4 (a)], MnO [Fig. 6.4 (e)], and FeO [Fig. 6.4 (f)] in the paramagnetic phase with the equilibrium volume indicated by the green vertical dashed lines. However, for CrO [Fig. 6.4 (d)], LDA+RISB predicts its ground state is a metal in rocksalt structure (green triangle solid line). It is also interesting to notice that the NiAs-type structures (blue circles) experience a Mott transition at large volume, indicated by the crossing of the blue solid and the blue dashed lines, in CoO, MnO, FeO. In our calculation, the Mott transition appears only in the octahedral crystal field structures, i.e., rocksalt and NiAs-type, in MnO, FeO, and CoO. For CoS [Fig. 6.4 (b)] and CoSe [Fig. 6.4 (c)], the stable crystal and electronic structures remain in NiAs-type structure (blue circles) and metallic phase, as in LDA. However, the equilibrium volumes are significantly improved by including the electronic correlation effects, as indicated by the vertical lines in Fig. 6.4 (b) and (c). Fig. 6.4 demonstrates the reliability of the LDA+RISB approach as it captures the experimentally observed crystal and electronic structures for all the compounds. This success is because the LDA+RISB can capture the most important physics in strongly correlated material, i.e., the renormalization of the band width and the Mott IMT due to the strong local Coulomb interaction between electrons. These effects, which can not be systematically described in LDA and LDA+U, are faithfully reproduced in LDA+RISB. Table. 6.1 shows the smallest quasiparticle weight, Z, among all the orbitals for all the compounds at its equilibrium volume. Z can be directly related to the inverse of the effective mass, Z = m/m∗, or the renormalized bandwidth. The quasiparticle weight for MnO, FeO, and CoO in rocksalt and NiAs-type structure are zero, Z = 0, at the equilibrium volume, indicating the systems are in the Mott-insulating phase in Brinkman-Rice picture. For all the other metallic compounds, we find significant band renormalization effects, Z < 1, which is responsible for the improvement of the predicted equilibrium volumes within LDA+RISB approach.

60 Table 6.1: The smallest LDA+RISB quasiparticle weight, Z, among all the orbitals for all the materials at U = 13 eV and J = 0.9 eV at the experimental equilibrium volumes. Z = 0 corresponds to the Mott-insulating phase in Brinkman-Rice picture, and ﬁnite Z corresponds to the correlated metallic phase.

NiAs-type rocksalt wurtzite zincblende CoO 0.0 0.0 0.65 0.67 CoS 0.67 0.69 0.77 0.78 CoSe 0.64 0.65 0.77 0.75 CrO 0.62 0.34 0.68 0.64 MnO 0.0 0.0 0.51 0.49 FeO 0.0 0.0 0.62 0.65

3.0 CoO LDA+RISB U=13 J=0.9 CoS LDA+RISB U=13 J=0.9 CoSe LDA+RISB U=13 J=0.9 (a) V V 1.4 (b) 1.4 (c) Exp SB VExp VSB VExp VSB 2.5 1.2 1.2 2.0 1.0 1.0 0.8 0.8 1.5 0.6 0.6 1.0 0.4 0.4 E(eV/f.u) NiAs Metal 0.5 0.2 0.2 NiAs Mott Ins. 0.0 0.0 0.0 Rocksalt Metal 15 16 17 18 19 20 21 22 22 24 26 28 30 32 24 26 28 30 32 34 36 38 Rocksalt Mott Ins. CrO LDA+RISB U=13 J=0.9 MnO LDA+RISB U=13 J=0.9 FeO LDA+RISB U=13 J=0.9 2.0 5 2.0 Wurtzite Metal (d)V V (e) V V (f) V V EXP SB Exp SB Exp SB Zincblende 1.5 4 1.5 3 1.0 1.0 2

E(eV/f.u) 0.5 0.5 1

0.0 0 0.0 17 18 19 20 21 22 23 15 16 17 18 19 20 21 22 23 15 16 17 18 19 20 21 V(Å 3/f.u)

Figure 6.4: LDA+RISB energy-volume curves for (a) CoO, (b) CoS, (c) CoSe, (d) CrO, (e) MnO, and (f) FeO at U = 13 eV and J = 0.9 eV in the paramagnetic phase. The blue circle, green triangle, red square, and cyan diamond curves corresponds to NiAs-type, roscksalt, wurtzite, and zincblende structures, respectively. The solid and dashed curves correspond to the metallic and the Mott-insulating phase, respectively. The experimentally observed equilibrium volumes of each compound are indicated by the colored vertical solid lines, and the equilibrium volumes predicted by LDA+RISB are indicated by the colored dashed lines, where the colors correspond to the structures labeled in the legend.

6.3.4 LDA+U

Fig. 6.5 shows the LDA+U results at U = 13 eV and J = 0.9 eV in the paramagnetic phase. For CoO [Fig. 6.5 (a)], LDA+U predicts the stable structure is an insulator in zincblende structure

61 instead of the experimentally observed rocksalt structure. For the experimentally observed rocksalt structure, LDA+U predicts it as a metal. The NiAs-type and the wurtzite structures of CoO are predicted as insulators. For CoS [Fig. 6.5 (b)] and CoSe [Fig. 6.5 (c)], LDA+U predicts the stable structures are metals in zincblende structures with very large equilibrium volumes instead of the experimentally observed NiAs-type structures. The rocksalt and the wurtzite structures of CoS and CoSe are predicted as metals. For CrO [Fig. 6.5 (d)], LDA+U obtains the experimentally observed rocksalt structure but the equilibrium volume is too large compared to experiment. Besides the wurtzite structure of CrO, for which LDA+U predicts an insulating solution, all other structures of CrO are predicted as metals. In MnO [Fig. 6.5 (e)], we obtained similar results as in the LDA+RISB predictions. LDA+U predicts the experimentally observed Mott-insulating solution in rocksalt structure. Besides the NiAs-type structure of MnO, for which LDA+U obtains a Mott- insulating solution, the wurtzite and the zincblende of MnO are predicted as metal in LDA+U. For FeO [Fig. 6.5 (f)], LDA+U predicts the stable structure is in the metallic NiAs-type structure instead of the experimentally known Mott-insulating rocksalt structure. Moreover, in the rocksalt phase, LDA+U predicts it as a metal instead of a Mott insulator. The wurtzite and the zincblende structure of FeO are predicted as metals. From the above results, we conclude LDA+U performs unsatisfactorily in all these transition metal compounds in the paramagnetic phase. It predicts incorrect stable crystal structures in CoO, CoS, CoSe, and FeO. In addition, it fails to predict the experimentally known Mott-insulating rocksalt structures in CoO and FeO. In MnO, LDA+U obtained identical electronic phases as in LDA+RISB. However, the Mott IMT is absent in the LDA+U pictures. Table. 6.2 summarized the electronic phases obtained from LDA+U among all the structures. It is interesting to notice that MnO has identical prediction in the electronic phases compared to the prediction in LDA+RISB, as shown in Table. 6.1. This might be the reason that the LDA+U has fairly good prediction only in MnO and works poorly in all the other compounds.

62 Table 6.2: The electronic phases predicted by LDA+U for all the materials and structures around the experimental equilibrium volumes. Here we set U = 13 eV and J = 0.9 eV.

NiAs-type rocksalt wurtzite zincblende CoO insulator metal insulator insulator CoS metal metal metal metal CoSe metal metal metal metal CrO metal metal metal metal MnO insulator insulator metal metal FeO metal metal metal metal

CoO LDA+U U=13 J=0.9 5 CoS LDA+U U=13 J=0.9 5 CoSe LDA+U U=13 J=0.9 (a) (b) (c) 4 VExp V+U VExp VExp 4 4 3 3 3

2 2 2

E(eV/f.u) 1 1 1 NiAs Rocksalt 0 0 0 Wurtzite 15 16 17 18 19 20 21 22 23 22 24 26 28 30 32 26 28 30 32 34 36 38 Zincblende 2.5 CrO LDA+U U=13 J=0.9 5 MnO LDA+U U=13 J=0.9 4.0 FeO LDA+U U=13 J=0.9 (d) (e) V V (f) +U Exp 3.5 VExp V+U 2.0 4 VExp V+U 3.0 1.5 3 2.5 2.0 1.0 2 1.5

E(eV/f.u) 1.0 0.5 1 0.5 0.0 0 0.0 17 18 19 20 21 22 23 16 18 20 22 24 15 16 17 18 19 20 21 V(Å 3/f.u)

Figure 6.5: LDA+U energy-volume curves for (a) CoO, (b) CoS, (c) CoSe, (d) CrO, (e) MnO, and (f) FeO at U = 13 eV and J = 0.9 eV in the paramagnetic phase. The blue circle, green triangle, red square, and cyan diamond curves corresponds to NiAs-type, rocksalt, wurtzite, and zincblende structures, respectively. The experimental equilibrium volume of each compounds are indicated by the vertical colored solid lines, and the equilibrium volumes predicted by LDA+U are indicated by the colored dashed lines, where the colors correspond to the structures labeled in the legend. Note that the equilibrium volumes of CoS and CoSe are out side of the volume of interest.

6.3.5 Comparison: LDA+RISB and LDA+U

To elucidate how the correlation eﬀects modify the energy surface in LDA+RISB and LDA+U, we show their energy-volume curves as a function of interaction, U, with a ﬁxed exchange interaction, J = 0.9 eV, in Fig. 6.8 (a) and (b). We select the MnO as an example, where both LDA+RISB and LDA+U predict the experimentally observed stable structure in the ambient condition by in-

63 cluding the interaction, U. In the ambient condition, MnO is a Mott insulator in rocksalt structure 3 (green curves) with equilibrium volume, V0 = 21.97 A˚ /f.u., indicated by the black dashed line. However, LDA, corresponding to the curves at U = J = 0 in Fig. 6.8 (a) and (b), predicts the metallic wurtzite structure (red curve) has the lowest energy among all the structures. Within the LDA+RISB framework, as shown in Fig. 6.8 (a), the stable structure evolves from the wurtzite structure (red curve), at U = J = 0 eV, to the rocksalt structure (green curves), at U 8 eV and ≥ J = 0.9 eV. For U 6 eV, the rocksalt (green curves) and the NiAs-type (blue curves) structures ≥ experience a Mott IMT at large volume, where the energy-volume curves shows a kink at the transition, as indicated by a dot between the Mott insulator (dashed lines) and the metallic (solid lines) solutions. Remarkably, after the Mott transition, the equilibrium volume is significantly improved to V = 21.41 A˚3/f.u. for U 8 eV. In addition, the energy gap between the tetrahedral coordi- 0 ≥ nate structures, zincblende (cyan curves) and wurtzite (red curves), and the octahedral structures, rocksalt (green curves) and NiAs-type (blue curves), increases significantly with increasing U. For comparison, we show the LDA+U results for MnO in Fig. 6.8 (b). Although LDA+U predicts an experimentally stable rocksalt structure (green curves), it can not capture the physics of Mott IMT. Moreover, the energy gap between the octahedral and the tetrahedral coordinated structures does not have significant change as we increase the interaction, U.

0.8 CoO 0.6 CoS 0.6 CoSe (a) (b) (c) 0.7 0.5 0.5 0.6 0.5 0.4 0.4

(eV) 0.4 0.3 0.3

GS 0.3 0.2 0.2 0.2 0.1 0.1 E-E 0.1 0.0 0.0 0.0 NiAs Meatl 0 6 8 10 13 0 6 8 10 13 0 6 8 10 13 NiAs Mott U (eV) Rocksalt Metal Rocksalt Mott 1.2 CrO 2.5 MnO 1.2 FeO (d) (e) (f) Wurtzite Metal 1.0 1.0 2.0 Zincblende Metal 0.8 0.8 1.5

(eV) 0.6 0.6

GS 0.4 1.0 0.4 0.2 0.5 0.2 E-E 0.0 0.0 0.0 0 6 8 10 13 0 6 8 10 13 0 6 8 10 13 U (eV)

Figure 6.6: LDA+RISB energy diﬀerences relative to the ground state structure for all the selected structures as a function of interaction, U, and a ﬁxed exchange coupling, J = 0.9 eV.

64 2.5 CoO 2.5 CoS 2.0 CoSe (a) (b) (c) 2.0 2.0 1.5 1.5 1.5

(eV) 1.0

GS 1.0 1.0 0.5

E-E 0.5 0.5 NiAs 0.0 0.0 0.0 0 6 8 10 13 0 6 8 10 13 0 6 8 10 13 Rocksalt U (eV) Wurtzite Zincblende 2.5 CrO 1.6 MnO FeO (d) (e) (f) 1.4 1.5 2.0 1.2 1.0 1.5 1.0

(eV) 0.8

GS 1.0 0.6 0.4 0.5 0.5 E-E 0.2 0.0 0.0 0.0 0 6 8 10 13 0 6 8 10 13 0 6 8 10 13 U (eV)

Figure 6.7: LDA+U energy diﬀerences relative to the ground state structure for all the selected structures in LDA+U as a function of interaction, U, and a ﬁxed exchange coupling, J = 0.9 eV.

To check further how the interaction, U, influence the energy ordering among all the polymorphs, we plot the energy differences relative to the ground state structure for all the structures and materials in Fig. 6.6 for LDA+RISB, and in Fig. 6.7 for LDA+U. As shown in Fig. 6.7, LDA+U fails to obtain the experimentally stable rocksalt structures in CoO and FeO for all the selected U. However, in LDA+RISB, the experimentally observed rocksalt structures for all the oxides can be obtained with large enough interaction, where U = 8 eV for MnO, U = 10 eV for CrO, and U = 13 eV for CoO and FeO are sufficient to have a stable Mott-insulating rocksalt solution. Finally, we benchmark our LDA, LDA+U, and LDA+RISB results against the experimental observation in Table. 6.3. The LDA only obtains the experimentally stable structures and phases for CoS and CoSe. In LDA+U, only the experimental structure of MnO is correctly predicted. In contrary to the unsatisfactory performance in LDA and in LDA+U, our LDA+RISB approach captures all the experimentally stable structure and electronic phases correctly. The equilibrium volumes are also improved significantly, within 4% error compared to the experimental data, after including the electronic correlation effect.

6.4 Conclusion

In summary, we tested the reliability of LDA, LDA+U, and LDA+RISB as a structure prediction tools on CrO, MnO, FeO, CoO, CoS, and CoSe in NiAs-type, rocksalt, wurtzite, and zincblende

65 (a) Rocksalt Metal NiAs Metal Wurtzite Metal Zincblende Metal Rocksalt Mott 5 NiAs Mott 3 V0=21.97Å /f.u. 4

2 E(eV/f.u.)

14 0 16 U=0 J=0 18 U=6 J=0.9 V( U=8 J=0.9 Å 3/f.u.) 20 U=10 J=0.9 22 U=11 J=0.9 U=13 J=0.9

(b)

Rocksalt NiAs 8 Wurtzite 7 Zincblende 3 6 V0=21.97Å /f.u. 5 4

3 E(eV/f.u.) 2 1

14 0 16 U=0 J=0 18 U=6 J=0.9 V( U=8 J=0.9 Å 3/f.u.) 20 U=10 J=0.9 22 U=11 J=0.9 U=13 J=0.9

Figure 6.8: (a) LDA+RISB energy-volume curves for all the MnO polymorphs as a function of U with a ﬁxed J = 0.9 eV. The green, blue, red, and cyan curves correspond to rocksalt, NiAs- type, wurtzite, and zincblende structures, respectively. The solid and dashed line correspond to metallic and Mott-insulating phases. The black dashed line indicates the experimental equilibrium volume in the rocksalt structure. The rocksalt structures, evolves from a metastable state to a stable state after U 8 eV, and the wurtzite structure, predicted as a stable structure in LDA, ≥ becomes metastable. (b) LDA+U energy-volume curves for all the MnO polymorphs with the same parameters as in LDA+RISB. The experimentally stable rocksalt structures evolves from a metastable structure to a stable structure after U 6 eV, and the wurtzite structure predicted to ≥ be stable in LDA becomes metastable.

66 Table 6.3: Stable structures, electronic phases, and equilibrium volumes (in unit A˚3/f.u. ) predicted by LDA, LDA+U at U = 13 eV and J = 0.9 eV, and LDA+SB at U = 13 eV and J = 0.9 eV.

LDA+U LDA+SB experiment structure phase V0 structure phase V0 structure phase V0 CoO Zincblende Insulator 23.5 Rocksalt Insulator 20.48 Rocksalt Insulator 19.35 CoS Zincblende Metal >35 NiAs Metal 26.25 NiAs Metal 25.77 CoSe Zincblende Metal >40 NiAs Metal 30.85 NiAs Metal 30.27 CrO Rocksalt Metal 20.74 Rocksalt Metal 18.27 Rocksalt N/A 18 MnO Rocksalt Insulator 21.8 Rocksalt Insulator 22.06 Rocksalt Insulator 21.97 FeO NiAs Insulator 20.2 Rocksalt Insulator 21.23 Rocksalt Insulator 20.35

structures within the paramagnetic phase, corresponding to the room temperature phase for these materials. Our results show that the LDA+RISB is a superior method, compared to LDA and LDA+U, by demonstrating its remarkable consistency with the known experimental observations. The crucial ingredient for the success of LDA+RISB is its ability to capture the Mott transition and the band renormalization effects, which is absent in LDA and LDA+U. With the inclusion of these electronic correlation effects, LDA+RISB predicted the experimentally known rocksalt Mott-insulating phase for MnO, FeO, and CoO compounds in the ambient condition, and the equilibrium volumes of CoS and CoSe are significantly improved. In addition, we found LDA+RISB and LDA+U have qualitatively different behavior when turning up the interaction, U. The gap between the octahedral (NiAs-type and rocksalt) and the tetrahedral (zincblend and wurtzite) coordinated structures increase drastically with U in LDA+RISB. However, in LDA+U, the gap does not have significant change. These promising results show the potential of LDA+RISB as a tool for structural prediction for correlated materials. It is also important to note that the computational speed of our LDA+RISB implementation is in the same order as LDA+U. With this accuracy and efficiency, LDA+RISB opens a new avenue for structural prediction and material by design in the field of strongly correlated electronic systems.

67 CHAPTER 7

NATURE OF AM-O BONDING IN AMERICIAN CHROMATE CsAm(CrO4)2

7.1 Introduction

2− The chromate moiety (CrO4 ) can be found in various minerals, where the characteristic color is usually red, orange, or yellow. One of the examples is the lead chromate crocoite, PbCrO4, which was discovered in 18th century and displays orang-yellow color. In the chromate compounds containing f-electrons, it is widely believed that the f-electrons do not participate in chemical bonding with the other atoms. Since the absorption rate of the f-f transition is weak due to the selection rule in an atomic environment, ∆l = 1, these chromate compounds are expected to have ± orange-yellow color originating from the charge-transfer gap, 2-3 eV, of the chromate alone. This is, indeed, true for all the lanthanide compounds, CsLn(CrO4)2, studied by Arico et. al. However, the actinide compound, CsAm(CrO4)2, measured by the same group displays anomalous dark red color, and the UV-vis-NIR absorption spectrum shows a smaller band gap of 1.65 eV, as shown in Fig. 7.2. This discrepancy indicates that, contrary to the other lanthanide compounds, the Am(III) may participates to the chemical bonding with chromates. The bonding between Am(III) and chromates leads to diﬀerent behaviors in the electronic structure, such as a smaller band gap in dark red color. Therefore, the nature of the Am-O bonding in CsAm(CrO4)2 requires detailed experimental and theoretical investigations. Below, I will ﬁrst review the experimental result from Arico et. al. Then, I will present our theoretical investigation.

The Am(CrO4)2 crystallizes in the triclinic space group P1.¯ It forms a two-dimensional layered structure that incorporates Cs+ cations in the interlayer channels similar to the early and mid- 2− lanthanide structures, but there are two crystallographically-unique chromium centers in the CrO4 moieties present that alter the layer composition. The layers extend through the ab plane and are composed of edge-sharing chains of Am(III) polyhedra connected by alternating corner- and edge- 2− 2− sharing CrO4 tetrahedra (Cr1) and strictly corner-sharing CrO4 tetrahedra (Cr2), as shown in Fig. 7.1 (a) and (b). The Am(III) polyhedra are eight-coordinate adopting a bicapped trigonal

68 Figure 7.1: Polyhedral representations of CsAm(CrO4)2. (a) View down the a axis showing the interlayer channels. (b) Single layer of one of the two-dimensional sheets. Am 3+ is represented 2− as dark red polyhedra, CrO4 as orange tetrahedra, and Cs + as tan spheres. (c) Coordination 2− environment of Am3+ in CsAm(CrO4)2 with Am 3+ represented as dark red polyhedra, CrO4 as orange tetrahedra, and Cs+ as tan spheres.

prism geometry. The coordination environments are composed of eight oxygen atoms; seven CrO42− tetrahedra donate to Am(III), which can be seen in Fig. 7.1 (c). Average Am-O bond lengths are 2.438 A˚.

In sharp contrast to the observed spectra of the lanthanide compounds, that CsAm(CrO4)2 crystals feature a very strong absorption extending through the visible spectrum, 390 nm - 700 nm (1.7 eV - 3.1 eV), to around 720 nm (1.72 eV), which can be found in Fig. 7.2. This is obvious in the very dark red color of the crystals, so dark that they actually appear black to the naked eye. Upon inspection of well-known UV-Vis-NIR spectra of americium compounds in solution (such as solutions of Am(III) in perchlorate or carbonate) the 7F 5 L and 7F 7 F transitions 0 → 6 0 → 6 routinely appear around 2.47 eV and 1.53 eV, respectively. The sharp peak at 2.47 eV cannot be conﬁrmed in this experiment due to the strong, extensive charge transfer absorption which obstructs that wavelength region. A triplet is observed in the NIR with local maxima occurring at 1.49 eV, 1.52 eV, and 1.57 eV. In the aforementioned solution-state spectra, the feature that appears around 1.52 eV is typically broad and weakly absorbing. In that same region of these solid-state spectra, a very strongly absorbing triplet is resolved; this indicates the complexities of the local environment

69 Figure 7.2: (a) Solid-state UV-Vis-NIR absorbance spectrum Vs. optical energy plot of CsAm(CrO ) showing a band gap of 1.65 eV. 4 2 ∼

around the trivalent americium centers in these compounds. Such strong absorption across the visible range in these compounds is an indication of semiconducting behavior, which is analyzed in the absorbance vs. optical energy plot in Fig.7.2. In order to disentangle the nature of the Am-O bonding and the anomalous band gap in

CsAm(CrO4)2, we investigate the electronic structure of this compound using local density approximation plus rotationally invariant slave-boson method (LDA+RISB). Our results shows the strong correlation eﬀect in the Am(III) f-electrons, which enhance the bare LDA band gap in the J=5/2 and 7/2 sector, is the key igredient to capture the correct band gap in this material. In addition, our result indicates the f electrons in Am(III) entangles strongly with rest of the system. Therefore, the f electrons in Am(III) has strong convalent character that participate intensively in the chemical bonding. The compound is a non-magnetic strongly renormalized band insulator with Pauli like behavior in magnetic susceptibility which is consistent with the experimental measurement.

7.2 Method

We employ the LDA+RISB method [99]. As in Refs. [21, 155], we utilize the DFT code WIEN2K [156] and employ the Local Density Approximation (LDA) and the standard fully-

70 50 Total Am f Cr d 40 O p

20 DOS(States/eV)

0 -8 -6 -4 -2 0 2 4 6 8 Energy (eV)

Figure 7.3: Local density approximation (LDA) density of state (DOS) for CsAm(CrO4)2. The energy is shifted with respect to the Fermi energy. The gap is around 0.6 eV.

localized limit form for the double-counting functional. Consistently with previous works[21], we assume that the Hunds coupling constant is J=0.7eV and the value of the screened Coulomb interaction strength is U=6.0 eV. Since our experiments have been all performed above the Neel temperature of the system, in our simulations we have assumed from the onset a paramagnetic wavefunction, i.e., a solution which does not break spontaneously the symmetry of the system. For simplicity, the approximation of averaging over the crystal ﬁeld splittings (CFS) of the Am-5f electrons was employed, see the supplemental material of Ref. [155] for a detailed description of the CFS averaging procedure.

7.3 Electronic structure study 7.3.1 LDA

In this subsection, we discuss the LDA band structure for this compound. The LDA density of state (DOS) is shown in Fig.7.3. The Fermi level is shifted to 0, and the f electron from Am has the dominant weight near the Fermi level. The gap is around 0.6 eV, which is due to the spin-orbit splitting between the J = 5/2 and the J = 7/2 sectors. The d electron from Cr also has salient

71 Figure 7.4: Local density approximation plus rotationally invariant slave-boson (LDA+RISB) density of state (DOS) for CsAm(CrO4)2. at U=6 eV and J=0.7 eV. The gap is around 2.2 eV.

weight near Fermi surface implying the signiﬁcant hrbridization between the f and the d electrons from Am and Cr. The occupancy for 5/2 sector is 5.68 and 7/2 is 0.26. The nominal valence for f electron is 6. Clearly, the band gap, 0.6 eV, is not comparable to the experimental observed band gap, 1.65 eV. Therefore, other important physics, which is absent in LDA, should be taken into ∼ account. In the next subsection, we employed the LDA+RISB to treat the strongly correlation eﬀect, which plays an important role in the strongly localized f-electron compounds.

7.3.2 LDA+RISB

In this subsection, we discuss our LDA+RISB results. Let us investigate the band gap of the system, which is 1.6 eV according to our absorption experiments. While bare LDA predicts a band gap of only 0.6 eV, within the LDA+RISB method, which takes into account more accurately the strong Am-5f electron correlations, the band gap is about 2 eV (for the U, J considered) as shown in Fig. 7.4, which is in much better agreement with the absorption experiments. Such a large enhancement of the band gap with respect to LDA constitutes an unequivocal evidence of the importance of the strong electron correlations in this material.

72 In order to analyze the electronic structure of the system and characterize the role of the strong f-electron correlations in this material, here we consider the local reduced density matrix of the

Am-5f electrons, ρf , which is formally obtained from the LDA+RISB ground state wavefunction of the solid by tracing out all degrees of freedom except the f shell of one of the Am atoms in the crystal. For later convenience, we represent ρf as:

ρf = wiri (7.1) X where ri = Pi/T r[Pi], Pi are projectors over the eigenspaces Vi of ρf , and the probability weights wi are sorted in descending order. Note that, since we made the approximation of averaging over the CFS, each one of the eigenspaces Vi has well defined electron occupation Ni and total angular momentum Ji. However, the orbital angular momentum L and the spin angular momentum S are not good quantum numbers, as the spin orbit coupling (SOC) between the J=5/2 and J=7/2 Am-5f electrons is very large in this material (∆ 1.25 eVs). Within our calculations, the average SOC ∼ number of f electrons per Am atom is n = Tr[ρ N] 6.02. As shown in Table 1, the Am-5f f f ∼ electronic structure is dominated by a singlet with N=6 electrons and total angular momentum 2 2 J=0, which carries a probability weight w0 = 0.88. Interestingly, we find that Tr[r0S ]= Tr[r0L = 2.25 (2.25+1)]. This indicates that the reason why the Am-5f electronic structure is dominated by a × non-degenerate ground state is the SOC, which is sufficiently strong to contaminate considerably the spin and orbital angular-momentum quantum numbers and, at the same time, to lift substantially the corresponding degeneracy in favor a J=0 singlet, in agreement with the third Hund’s rule. We note also that the probability weight arising from different multiplets besides the dominant J 2 = 0 singlet discussed above is more than 10 %, which is non-negligible. This fact is reveled also by the value of the Am-5f entanglement entropy Sf = Tr[ρf ln[ρf ]], see Refs. [157, 158], which is 0.72 according to our calculations, while it would be 0 if was a pure state. This observation provides us with a clear indication of the fact that the Am-5f electrons are entangled with the rest of the lattice, i.e., their contribution to the f-bonding is not purely ionic, but has also a non-negligible covalent component. In order to quantify the importance of the covalency effects relative to the Am-5f electrons, i.e., their covalent energy contribution to the bonding of the system, here we compare the total energy of the system with the energy that the system would have if the Am-5f shell hosted exactly 6 electrons

73 Table 7.1: Parameters of the Am-5f reduced density matrix computed by LDA+RISB assuming U=6 and J=0.7, see Eq. 7.1: Probability weights wi, corresponding quantum labels Ni (number of electrons) and Ji (total angular momentum).

wi 0.88 0.06 0.04 0.002 Ni 6 7 5 6 Ji 0 3.5 2.5 6

entirely isolated from the rest of the system (i.e., if Sf was exactly 0). According to our calculations, such a state would have an energy ∆E 1.85 eV higher with respect to the ground state of cov ∼ the system. Based our analysis we deduce that the electron correlations are very strong in this material. However, the electron correlations do not lead to the formation of a local moment, i.e., this material does not qualify as a Mott insulator, but as a strongly renormalized band insulator. This conclusion is consistent with the experimental evidence that the magnetic susceptibility displays Pauli-like behavior within the whole range of temperatures explored experimentally.

7.4 Conclusion

In summary, we calculated the electronic structure of CsAm(CrO4)2 by employing LDA+RISB, and reproduced the band gap observed in the optical absorption experiment. We found a significant difference between the band gaps produced by LDA+RISB and LDA, demonstrated the importance of the many-body effects in this system. Moreover, our theoretical analysis indicates that this material is not a Mott insulator, but a strongly renormalized band insulator, in which the f electrons in Am participate strongly in the chemical bonding. The formation of a local moment is prevented by the strong SOC of the system, which favors a singlet state with total angular momentum J=0. Therefore, the magnetic susceptibility shows a Pauli like behavior at low temperatures, which is consistent with the experimental results.

74 CHAPTER 8

CONCLUSION AND OUTLOOK

In this dissertation, we studied the Mott metal-insulator transition (MIT) in various type of systems, which include the domain wall structure in metal-Mott insulator coexistence regime, the Mott MIT in the presence of spinon excitations, the structure prediction for the transition metal oxides, sulfides, and selenides, and the electronic structure of the Americian compounds. We applied both the many-body techniques, Dynamical Mean Field Theory (DMFT) and Rotationally Invarianty Slave-Boson (RISB) methods, and the first principle simulations, local density approximation (LDA), LDA+U, and LDA+RISB, as our tools to investigate these problems. The main findings of our investigations are summarized in the following paragraphs. In chapter 3, we employed the DMFT and the Broyden method to investigate the transport and the quasiparticle properties of the unstable solution in Mott metal-insulator coexistence regime. Physically, this solution is expected to describe the properties of the domain wall separating the metallic and the Mott-insulating regions in a spatially inhomogeneous system. We found that, within our theory, the unstable solution represents a new phase of matter, differing qualitatively from the conventional metal and insulator, displaying an incoherent weakly insulating-like behavior down to the lowest temperatures. Still, the unstable solution has a well-defined quasiparticle peak at the Fermi level in the entire coexistence regime, and the resistivity can be described qualitatively by the Sommerfeld quasiparticle approximation. In chapter 4, we blended the static resonanting-valence-bond theory of a gapless insulating quantum spin liquid carrying a Fermi surface with the DMFT using an impurity solver based on emergent spinon degrees of freedom. This idea allows to reproduce the salient thermodynamic features recently observed in the organic compounds κ-(BEDT-TTF)2Cu2(CN)3 and EtMe3Sb[Pd(dmit)2]2, providing a consistent picture of the Mott transition towards a gapless low entropy insulator. In the Fermi liquid metal, the strong local scattering of the magnetic degrees of freedom occurs due to a generic orthogonality catastrophe in the spinons density of states, destroying discontinuously the spinon Fermi surface at the insulator-to-metal phase boundary.

75 In chapter 5, we developed a unified theoretical picture for excitations in Mott systems, por- traying both the heavy quasiparticle excitations and the Hubbard bands as features of an emergent Fermi liquid state formed in an extended Hilbert space, which is non-perturbatively connected to the physical system. This observation sheds light on the fact that even the incoherent excitations in strongly correlated matter often display a well defined Bloch character, with pronounced momentum dispersion. Furthermore, it indicates that the Mott point can be viewed as a topological transition, where the number of distinct dispersing bands displays a sudden change at the critical point. Our results, obtained from an appropriate variational principle, display also remarkable quantitative accuracy. This opens an exciting avenue for fast realistic modeling of strongly correlated materials. In chapter 6, we studied six common transition metal binary compounds, CrO, MnO, FeO, CoO, CoS, and CoSe, and demonstrated the critical role of the strong electronic correlation effects on the total energy differences between different crystal structures. By explicitly including the effects of strong correlations in the total energy calculations, at the LDA plus rotationally-invariant slave- boson (RISB) level of theory, we are able to reproduce the known ground state structures for all 6 compounds. The significance of this result is in that it deepens our understanding of the role that strong correlations play in determining the crystal structure of transition metal compounds, paves the way for predicting and studying metastability and polymorphism in these systems, and finally, opens the door for extending materials by design ideas and the large-scale (high-throughput) calculations to strongly correlated systems. In chapter 7, we investigated the electronic structure of the Americium chromate compounds,

CsAm(CrO4)2, displaying anomalous dark red color corresponding to a small band gap, 1.65 eV, as observed in the optical absorption measurement. Our calculation indicates that the strong correlation eﬀect in the Am-5f electrons, which enhance the spin-orbit gap in the J=5/2 and the J=7/2 sectors, is important to capture the observed band gap. Our calculation further shows the Am-5f electrons are strongly entangled with the rest of the systems, indicating a pronounced covalent character in the Am-O bonding. In addition, the atomic structure analysis shows that the strong spin-orbit coupling (SOC) prevent the Am-5f electron to form local moments. Consequently, the magnetic susceptibility has a Pauli like behavior, which is consistent with the experimental measurement.

76 From the above results, we conclude that DMFT, RSIB, and their combinations with density functional theory (DFT) serve as reliable tools to explain various complicated phenomena in strongly correlated systems. They can remedy the inability of DFT related methods to describe the strong correlation effects, and provide a good description of correlated materials. However, there still remain many challenges for these approaches. For example, since the local approximation in DMFT and RISB does not have the ability to describe the spatial correlations, which are important in one and two dimensional systems, the cluster and the diagrammatic extensions have to be applied. Finding an unbiased and efficient methods to systematically include these spatial correlations are still an active field of research. Also, since the popular methods to study strongly correlated materials, such as DFT+U, DFT+DMFT, and DFT+RISB, all treat the Coulomb interaction, U, and Hund’s coupling, J, as fitting parameters to the experimental data, seeking for a more objective tool to avoid these tunable parameters is still under intense investigation. Therefore, we still need to make many efforts to develop a systematic tool that can describe the important physics in strongly correlated systems in an unbiased fashion.

77 APPENDIX A

PARAMETRIZATION OF THE MULTIORBITAL COULOMB INTERACTION

The multiorbital Coulomb interaction can be approximated by a rotationally invariant form, which is exact for an atomic environment. It is also a good approximation to the d and f electrons, because their wavefunction are localized in real space. Within this approximation, the Coulomb interaction can be parameterized in the following fashion, called Slater-Condon parametrization. Starting from the most general form of the interaction,

† † Umm′nn′ cmσcm′σ′ cnσ′ cn′σ, (A.1) ′ ′ ′ mmXnn Xσ,σ and

′ ∗ ∗ ′ 1 ′ U ′ ′ = drdr w (r R)w ′ (r R)[ ]w (r R)w ′ (r R). (A.2) mm nn m − m − r r′ n − n − Z | − | 1 We can approximate our basis by an atomic basis, wm(r,θ,φ)= Rl(r)Ylm(θ,φ), and expand |r−r′| in this basis,

∞ k k 1 r< 1 ∗ ′ ′ = 4π k+1 Ykq(r)Ykq(r ). (A.3) r r r> 2k + 1 | − | Xk=0 X−k Within this approximation, we have the Coulomb interaction in the following form,

k k ∗ U ′ ′ = F Y Y Y ′ Y ′ Y Y , (A.4) mm nn h 2m| kq| 2n ih 2m | kq| 2ni Xk qX=−k where F k is the Slater integral,

4π rk F k = r2dr r′2dr′ < R2(r)R2(r′). (A.5) 2k + 1 rk+1 l l Z Z >

78 0 The F term corresponds to the Coulomb interaction, U. The exchange term involving Umnmn corresponds to the Hund’s interaction J. For d electron, the parametrization are U = F 0 and J = 1/14(F 2 + F 4).

79 APPENDIX B

ANALYTICAL CONTINUATION

The IPT, NCA and CTQMC that we utilized in this dissertation were calculated in the Matsubra space, iωn. As a result, our Green’s function and selfenergy were a function of Matsubara frequency,

G(iωn) and Σ(iωn). The physical observables, such as density of states, conductivity, susceptibility, etc, are a function of real frequency, G(ω) and Σ(ω). Therefore, we need to analytical continuate the Matsubara functions to real fequency, ω, to extract the physical information. For closed-form functions, the analytical continuation can be transformed by directly substi- tuting the imaginary frequency to real frequency plus a inﬁnitesimal small imaginary number, iω ω + iη. However, the results we obtained from IPT, NCA, and CTQMC were a set of data n → on Matsubara space. Thus, we need numerical analytical continuation algorithms to transform the data to real frequency. The Pade approximation [159, 160] is one of the most widely used methods for the data generated by the methods without statistical error, such as IPT and NCA. However, for CTQMC, the statistical error (small random ﬂuctuations in the data) becomes a problem in Pade approximation. In this case, the Maximum entropy method (MEM) [161] has to be utilzed for the analytical continuation.

B.1 Pade Approximation

When the impurity solver does not have statistical error, e.g. IPT and NCA, the Pade approximation is a reliable method to perform the analytical continuation. Pade approximation is a m j Pj=0 aj x fractional polynomial ﬁtting to a rational function, R(x)= n k , with ﬁtting parameters ai 1+Pk=1 bkx and bi, which can be calculate by a recursive manner [160]. Our Pade method is based on one of its application to Green’s function, H. J. Vidberg and J. W. Serene’s [159] formalism, as described below.

Given a set of data representing a Green’s function on Matsubara frequency, ui, where i = 1...N is the indices of Matsubara frequency points, we can ﬁt the Green’s function by a continuous fraction,

80 a G(z)= 1 , a2(z−z1) 1+ a3(z−z2) 1+ a (z−z ) 1+.... n−1 n−2 1+an(z−zn−1) where the coeﬃcient an can be calculate by a recursive relation,

ai = gi(zi), g1(zi)= ui(i = 1,...N), and

g (z ) g (z) g = p−1 p−1 − p−1 , (p 2). p (z z )g (z) ≥ − p−1 p−1 The Green’s function can, then, be calculated by

AN GN (z)= . BN

Here N represent the order of Pade approximation. AN and BN can be calculated by the following recursive relation,

A (z)= A (z)+(z z )a A (z) i+1 i − i i+1 i−1 and

B (z)= B (z)+(z z )a B (z), i+1 i − i i+1 i−1 with the initial condition A0 = 0, A1 = a1, B0 = B1 = 1. After obtaining the GN (z), we could set the Matsubara frequency, z, to real frequency, ω, to calculate the real frequency Gree’s function, G(ω).

B.2 Maximum Entropy Method (MEM)

When the QMC method is used, the Green’s function has statistical noise leading to uncontrol- lable ﬂucutations in the Pade approximation. Therefore, the maximum entropy method (MEM) needs to be utilized to obtain a reasonable real frequency Green’s function.

81 For a general analytical continuation problem, we need to ﬁnd a spectral function, A(ω) = 1 ImG(ω), for a given Green’s function G(τ), such that they satisfy the following equation, − π

∞ e−ωτ ∞ G(τ)= dω −βω A(ω)= dωK(τ,ω)A(ω) Kl,jAj = Gl. −∞ 1+ e −∞ ≃ Z Z Xl Here l and j are the indeces for the discretized form of the τ and ω, respectively. The direct inversion of this equation can not be implemented effectively, because the e−ωτ term leads to an exponentially small Kernel, K(τ,ω), at large frequency. Consequently, due to the limitation of machines precision, there exist many A(ω) at large frequency that can yield the same G(τ), resulting an ill-defined inversion problem. The MEM can be utilized here to overcome this difficulty. The idea of MEM is that we can treat the spectral function, A(ω), as a statistical observable, and we need to find the most probable A(ω) for a given G(τ). The probability for this problem has been shown to be [161]

2 P (A G, m, α) eαS−χ /2 | ∼ with the entropy, S = dω[A(ω) m(ω) A(ω)ln( A(ω) )]. Here, m(ω) is the default model for our − − m(ω) G −P K Aj spectral function (usuallyR is a constant density of state) and χ2 = ( l j l,j )2 is the guassian σl l distribution describing the deviation of the observation, A(ω), fromP the given G(τ). α can be think as a paramter describing the competition between the entropy and the gaussian distribution. σl is a adjusting parameter that control the width of the guassian distribution. The main task here is to ﬁnd the A(α) that maximize the function, αS χ2/2, for a given α. Here, we use the Bryan’s − algorithm, as implemented in A. Sandvik’s code, to solve the spectral function by averaging the spectal function over α

∞ = dαP (α G, m)A(α). | Z0 Here, P (α G, m) = dAP (A, α G, m). After obtained the spectral function, A(ω), we used the | | Hilbert transformationR to calculate the full Green’s function, G(ω).

82 APPENDIX C

TOTAL ENERGY AND SPECIFIC HEAT CALCULATION FOR HUBBARD HEISENBERG MODEL

Starting from eﬀective Hamiltonian of the main text (adding the constant mean ﬁeld contribution, last term in the equation below),

2 2 † i(θi−θj ) U ∂ Jeﬀ H = fiσfjσ[Jeﬀ te ] 2 + | | . (C.1) − − 2 ∂θi J hXi,jiσ Xi hXi,jiσ the internal energy per site (with Ns sites) is calculated by standard Green’s function methods [19]

2 H + + T iωn0 T iωn0 U 1 Jeﬀ Eint = = [ǫkGdσ(k,iωn)]e + [ǫkGfσ(k,iωn)]e + D↑↓ + | | . Ns Ns Ns 2 Ns J n,k,σ n,k,σ X X hXi,jiσ (C.2) Here, the ﬁrst and second term correspond to the kinetic energy for electrons and spinons, that can be computed using a spectral decomposition:

T + + [ǫ G (k,iω )]eiωn0 = 2T eiωn0 dǫ ǫρ (ǫ)G (ǫ,iω ), (C.3) N k d/f,σ n d/f d/f,σ n s n n,k,σX X Z with the semi-circular density of states of the Bethe lattice for the electronic density of states, ρ (ǫ) = 1 1 [ǫ/(2t)]2. The spinons also follow a semi-circular density of states, ρ (ǫ) = d πt − f 1 2 1 [ǫ/p(2Jeff )] , that involves a different bandwidth 4Jeff associated to the spinon dispersion. πJeff − 1 Herep the lattice electron and spinon Green’s functions are given by G (ǫ,iωn)= , d/f iωn−ǫ−Σd/f (iωn) using the fact that the DMFT self-energies Σd/f (iωn) are purely local (but frequency dependent). The third term in Eq. (C.2) is associated to the Coulomb interaction, and is expressed as a function of the double occupancy, D↑↓, which is related to the dynamical charge susceptibility by

D↑↓ = (1/2)χc(τ = 0). The charge susceptibility can be computed in principle either from spinon

83 f X response χc or from rotor response χc [80],

f † 1 † 1 χ (τ) = [f (τ)f (τ) ][f ′ (0)f ′ (0) ] = 2G (τ)G (τ), c jσ jσ − 2 jσ jσ − 2 f f σ,σ′ X X ∂ ∂ 2 2 2 χc (τ) = i (τ)i (0) = 2 GX (τ)[∂τ GX (τ)+ Uδ(τ)] [∂τ GX (τ)] . (C.4) ∂θj ∂θj U { − }

Both quantities should be equal in absence of approximation, but they do diﬀer at the saddle point f −1 X −1 −1 level. For this reason, we use Nagaosa and Lee composition rule, χc(iω)=[(χc ) +(χc ) ] [14], which allows to recover the correct behavior of the physical charge response both in the Fermi liquid and in the Mott state.

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97 BIOGRAPHICAL SKETCH

Tsung-Han Lee

Education

Florida State University, USA Theoretical Condensed Matter Ph.D. 2017 Advisor: Prof. Vladimir Dobrosavljevic

National Tsing Hua University, Taiwan Theoretical Condensed Matter M.S. 2010 Advisor: Prof. Chung-Hou Chung, Prof. Po-Chung, Chen

National Chung Cheng University, Taiwan Physics B.S. 2008

Professional Experience

2012-present Research Assistant, Florida State University and National High Magnetic Field Laboratory

Investigating the energy ordering among diﬀerent polymorphic structures for 3d transition • metal oxides, sulﬁdes, and selenides with the combination of density functional theory and rotationally invariant slave-boson (equivalently Gutzwiller approximation) method (DFT+RISB).

Collaborating with Professor Thomas E Albrecht-Schmitt’s group to study the electronic • structure of f-electron compounds, CsAm(CrO4)2, CsSm(CrO4)2 , and Eu(CrO4)OH.

Developing non-equilibrium method for ghost RISB (Gutzwiller) theory. • Developed a uniﬁed algorithm to describe both the low and high energy (Hubbard band) • excitation by introducing a ghost orbital in the RISB (Gutzwiller) framework.

Incorporated the resonating valence bond theory (RVB) and dynamical mean ﬁeld theory • (DMFT) to investigate the spinon’s inﬂuence on Metal Mott insulator transition within slave-rotor framework.

Employed DMFT with iterative perturbative theory (IPT) and continuous time quantum • Monte Carlo (CTQMC) solvers to study the unstable domain wall solution in Metal Mott insulator coexisting region.

98 Collaborated with Professor Martin Dressel’s group to investigate the universality of metal- • insulator transition phase diagrams for organic compounds, κ-(BEDT-TTF)2Ag2(CN)3, ′ κ-(BEDT-TTF)2Cu2(CN)3, and β -EtMe3Sb[Pd(dmit)2]2.

2011-2012 Research Assistant, National Chiao Tung University

Developed the non-equilibrium Non-crossing approximation (NCA) code to study the • universal scaling of the nonlinear transport properties in pseudo-gap two channel Anderson model.

2008-2010 Research Assistant, National Chiao Tung University

Incorporated the Fano resonance physics and two stage Kondo screening process to study • the tunneling density of state in side-coupled double quantum dot system.

Honors and Fellowship

Dirac-Hellman Award, Florida State University. • Dean Doctoral Scholarship, Florida State University. •

Publications

”Fate of Spinons at the Mott Point” Tsung-Han Lee, Serge Florens, and Vladimir • Dobrosavljevic, Phys. Rev. Lett. 117, 136601(2016)

”Universal scaling of nonlinear conductance in the two-channel pseudogap Anderson model: • Application for gate-tunned Kondo eﬀect in magnetically doped graphene” Tsung- Han Lee, Kenneth Yi-Jie Zhang, Chung-Hou Chung, Stefan Kirchner, Phys. Rev. B 88 085431(2013).

”Tunable Fano-Kondo resonance in side-coupled double quantum dot systems” Chung- Hou • Chung and Tsung-Han Lee, Phys. Rev. B 82, 085325(2010).

Submitted Publications

”Emergent Bloch Excitations in Mott Matter” Nicola Lanata, Tsung-Han Lee, Yong-Xin • Yao, and Vladimir Dobrosavljevic, arXiv:1701.05444 submitted to PRL(2017).

”The Widom Line in Pristine Mott insulators: Dynamical Properties of Quantum Spin • Liquids” A. Pustogow, M. Bories, E. Zhukova, B. Gorshunov, M. Pinteri, S. Tomi, J.A. Schlueter, A. Lhle, R. Hbner, T. Hiramatsu, Y. Yoshida, G. Saito, R. Kato, S. Fratini, T.- H. Lee, V. Dobrosavljevic, and M. Dressel, submitted to Science Advances.

99 Manuscripts In Preparation

”Do Cluster Models Capture the Electronic Properties of Heavy Element Materials with • Extended Structures? An Examination of Structure-Property Relationships in CsM(CrO4)2 (M = La - Sm; Am)” Alexandra A. Arico, Tsung-Han Lee, Yongxin Yao, Xiaoyu Deng, Shane S. Galley, Julia S. Storbeck, Vlad Dobrosavljevic, Thomas E. Albrecht-Schmitt, Nikolas Kaltsoyannis, Nicola Lanata

”Structure predictions and polymorphism in transition metal compounds: An LDA • +Rotationally-invariant slave-boson study” Tsung-Han Lee, Nicola Lanata, Yongxin Yao, Vladan Stevanovic, and Vladimir Dobrosavljevic.

”Unstable Domain-Wall Solution in the Metal-Mott Insulator Coexisting Regime” Tsung- • Han Lee, Vladimir Dobrosavljevic, Jaksa Vucicevic, Darko Tanaskovic, Eduardo Miranda.

Oral Presentations

”Polymorphism in transition metal compounds: An LDA+Rotationally-invariant slave- • boson study” Tsung-Han Lee, Nicola Lanata, Yongxin Yao, Vladan Stevanovic, Vladimir Dobrosavljevic, APS March Meeting (2017)

”Inﬂuence of spinons ﬂuctuations near the spin liquid Mott transition” Tsung-Han Lee, • Serge Florens, Vladimir Dobrosavljevic, APS March Meeting (2016).

”Unstable Domain-Wall Solution in the Metal-Mott Insulator Coexisting Regime” Tsung- • Han Lee, Vladimir Dobrosavljevic, Jaksa Vucicevic, Darko Tanaskovic, Eduardo Miranda, APS March Meeting (2015).

Posters

”Fate of Spinons at the Mott Point” Tsung-Han Lee, Serge Florens, and Vladimir • Dobrosavljevic, Julich Autumn School on Correlated Electrons (2016).

”Fate of Spinons at the Mott Point” Tsung-Han Lee, Serge Florens, and Vladimir • Dobrosavljevic, Gordon Research Conference (2016).

”Tunable Fano-Kondo resonance in side-coupled double quantum dot systems.” Tsung-Han • Lee, Chung-Hou Chung, Annual Meeting of the Physical Society of ROC Taiwan (2010).

100 Conference Attendance

March 2017: American Physical Society March Meeting, New Orleans, LA, USA. • September 2016: Julich Autumn School on Correlated Electrons: Quantum Materials: • Experiments and Theory, Julich, Germany.

August 2016: Gordon Research Conference: Conductivity and Magnetism in Molecular • Materials, South Hadley, MA, USA.

March 2016: American Physical Society March Meeting, Baltimore, MD, USA. • July 2015: SPICE-Workshop on Bad Metal Behavior in Mott Systems, Mainz, Germany. • March 2015: American Physical Society March Meeting, San Antonio, TX, USA. • September 2014: Julich Autumn School on Correlated Electrons: DMFT at 25: inﬁnite • dimensions, Julich, Germany.

May 2014: International Summer School on Computational Methods for Quantum • Materials, Sherbrooke, Canada.

March 2012: Physical Society of ROC annual meeting, Chiayi, Taiwan. • March 2010: Physical Society of ROC annual meeting, Tainan, Taiwan. •

101