EFFECTS of ELECTRONIC CORRELATIONS on PHYSICAL PROPERTIES of UNCONVENTIONAL SUPERCONDUCTORS by SHINIBALI BHATTACHARYYA a DISSERT

EFFECTS OF ELECTRONIC CORRELATIONS ON PHYSICAL PROPERTIES OF UNCONVENTIONAL SUPERCONDUCTORS

By SHINIBALI BHATTACHARYYA

A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2020 © 2020 Shinibali Bhattacharyya I dedicate this to my parents - Ma and Babi. ACKNOWLEDGMENTS I sincerely thank Prof. Peter Hirschfeld for his seamless support, encouragement, motivation and guidance all throughout my journey as his graduate student. Not only has he mentored me to become a more independent researcher, he is also an extremely caring person who goes the extra mile to make his mentees more socially and culturally aware of the world at large. His wisdom, empathy and emotional support through some of the most challenging times during my doctoral studies have made so many hurdles much easier to overcome. His ability to identify individual potential of his mentees, and timely engagement to ensure optimal work output, has enabled me to successfully complete some difficult projects during my PhD. I am very grateful to Dr. Andreas Lindscheid, an ex-postdoctoral researcher in the Hirschfeld group, for teaching me all the relevant theoretical and computational basics of my research during the early stage of my doctoral studies. I thank Prof. Saurabh Maiti, Xiao Chen and Dr. Peayush Choubey for all the helpful discussions, insights and resources they shared with me whenever I reached out to them. I cannot exaggerate the continued help, support, insight and guidance that I receive from Prof. Andreas Kreisel on a weekly basis for my research work. His knowledge, readiness and eagerness to help others solve scientific problems is inspirational to me. I heartily thank Dr. Thomas Maier, Prof.Roser Maria Valentí, and Prof. Brian M. Andersen for giving me the precious opportunity to work at their prestigious host institutions for extended period of time, facilitating my visit to different countries, experiencing their cultures and people. I thank Dr. Chandan Settyfor choosing me to work in his project and, thus, giving me my first break in publication in a high impact journal. I deeply thank Dr. Kristofer Björnsson and Karim Zantout for all the in-depth physics discussions, hands-on practice and coding sessions that enabled me to tackle the most difficult project in my thesis. Besides, thanks to Daniel Steffensen andDr. Laura Fanfarillo, for their significant contribution in completion of my project. I thank Prof. Douglas Scalapino for sharing his ideas with Dr. Thomas Maier and me which turned into a successful physics investigation. I sincerely thank Prof. Dimitrii Maslov for teaching us

4 many-body physics and quantum condensed-matter theory, and for sharing his deep insights about various physics problems. I deeply thank Dr. Miguel Sulangi, Dr. Abhsihek Kumar, and Prof Pradeep Kumar for their helpful discussions from time to time. I acknowledge support in part through an appointment: (1) to the Oak Ridge National Laboratory ASTRO Program, sponsored by the U.S. Department of Energy and administered by the Oak Ridge Institute for Science and Education, (2) at the Goethe-Universität Frankfurt as a short-term research visiting scientist, sponsored by the Deutscher Akademischer Austauschdienst (DAAD), and (3) the University of Florida CLAS Dissertation Fellowship funded by the Charles Vincent and Heidi Cole McLaughlin Endowment, during various phases of my doctoral studies. I am extremely grateful for coming across some amazing people and intimate friends during this journey, as they have provided me strength and happiness during various phases in my life: Sandeep Upadhyay, Andrew Brooks, Matthias Hecker, Bettina and Michael Hecker, Eteri Svandize, Sananda Biswas. The colleagues who extended their warmth, welcomed me in foreign lands and made me feel integrated in new environments are: Panos Kotetes, Thomas Mertz, Alexander Razpopov, David Kaib, Caroline, Kira Ridl, Matthieu Barbier, Seher Karakazu, Anna Heffernan, Filipe Da Silva, and Geraldine Emig. Life would have been tougher without my friends from UF Physics: Yaqi Han, Prasanth Shyamsundar, Han-yu Chia, Kazi Ashraful Alam, Colin Barquist, and Rabindra Bhattarai. Thanks to other UF Physics colleagues: Prachi Sharma, Ammar Jahin, Hoda Akl, Mayar Shahin, Pancham Lal Gupta, Jasdeep Sidhu, for their company during various occasions. Thanks to my closest confidants: Indrani Ghosh and Churni Gupta for listening to my complaints, frustrations, excitements, and daydreams and for their unconditional friendship. Thanks to Bijurika Nandi, Sreeja Chowdhury, Srijata Samantha, and Ankita Sarkar for being very cordial and supportive cohabitants of our shared apartments. I sincerely thank Dipsikha Debnath, Divya Rajan, Sucharita Chakraborty and Subhadip Goswami, for helping and familiarising me with my transition from India to USA to start my PhD journey. I thank my undergrad friends: Eshita Samajpati, Tiyash Parira, Debadrita Jana, Shamreen Iram,

5 Riya Nandi, and Sucheta Majumder for providing me with familiar faces and company in this foreign land. Thanks to Doris and Jörg Hümbs, Ayan Bhattacharjee, Ayan Saha, Helen, Martin, Jeromé, Sara, Shashank, Tuya, Aditi Meshram, Agneya Loya, for their warmth and acquaintance in the most unlikely of circumstances, making my journey all the more cherishable. My Gainesville folks: Laura, Julia, Kristina, Anna, Joe, Basharat, Keith, Ladan, Victor, Sebi, Rocio, Simon, Ivelisse, Dragana, Bernard, Mei, Quentin, Jacob, I thank them for their presence in my life that gave me something to look forward to every week, outside of work. I earnestly thank Pam Marlin, Mary Remer, and Frau Anne Metz for handling our daily administrative tasks and making our lives much easier with less workload. I thank Mr. Charles Parks and Mrs. Debbie Parks for creating a very comforting and warm environment for all the graduate students in the UF Physics department, trying to make them feel at home, thousands of miles away from their original homeland. Thanks to my acquaintance, Dr. Anindya Bagchi, for being an encouraging mentor and showing his strong support for young women in their career journey. I cannot exaggerate the importance of privilege and the combination of fortunate and lucky events in my life which enabled me to reach where I am today. Lastly, I cannot thank my partner Mohammad Sayeb enough, for showing his unflinching support, strength, dedication and belief in me. I owe all of my little, as well as big achievements to my mom Sucheta Bhattacharyya and my dad Santi Ranjan Bhattacharyya. Their love, hard work, sacrifices and devotion in raising responsible adults have set the stage onwhichI stand today. I am thankful to have very supportive, generous, kind and loving brother Arkajyoti Bhattacharyya and cousin Srinwanti Chakrabarti. Thanks to my cousin-in-law and mentor Prof. Ayan Mukhopadhyay for his timely guidance in helping me make the right career decisions. I thank my aunt Sujata Chakrabarti and uncle Dr. Bikas Chakarabarti for supporting me with their generosity, care and kindness. I am thankful to my late grandmother Asha Roy and grandfather Birendra Kishore Roy for the many valuable life lessons they imparted upon us.

6 TABLE OF CONTENTS page ACKNOWLEDGMENTS ...... 4 LIST OF FIGURES ...... 9 ABSTRACT ...... 11

CHAPTER 1 INTRODUCTION AND BACKGROUND ...... 13 1.1 Superconductivity ...... 13 1.2 Fe-based Superconductors ...... 14 1.3 Earlier work highlighting the role of Electronic correlations ...... 17 1.4 Format of the Dissertation ...... 18 2 EFFECTS OF MOMENTUM-DEPENDENT QUASIPARTICLE RENORMALIZATION ON THE GAP STRUCTURE OF IRON-BASED SUPERCONDUCTORS ...... 19 2.1 Introduction ...... 19 2.2 Model ...... 23 2.3 Results ...... 26 2.3.1 Gap structure ...... 28 2.3.2 Analysis of the Gap structure in terms of Quasiparticle weights and the Pairing vertex ...... 30 2.3.3 Discussion: Tendency towards s± Pairing symmetry ...... 32 2.4 Summary ...... 33 3 NON-LOCAL CORRELATIONS IN IRON PNICTIDES AND CHALCOGENIDES . 34 3.1 Introduction ...... 34 3.2 Background: toy model illustrating Fermi-surface shrinkage ...... 38 3.3 Model and RPA scheme ...... 42 3.4 Results ...... 45 3.4.1 Electronic structure renormalization ...... 45 3.4.2 Effects of nearest-neighbor Coulomb repulsion ...... 46 3.4.3 Comparison of RPA, TPSC and FLEX results ...... 49 3.4.4 Scattering lifetime of Quasiparticles ...... 50 3.5 Summary ...... 53 4 TOPOLOGICAL ULTRANODAL PAIR STATES IN IRON-BASED SUPERCONDUCTORS ...... 55 4.1 Introduction ...... 55 4.2 Model and Physical observables ...... 58 4.2.1 Case of FeSe(S) ...... 61 4.2.2 Model details ...... 63

7 4.3 Results ...... 64 4.3.1 Spin-resolved spectral functions ...... 64 4.3.2 Bogoliubov Fermi surfaces, Specific heat and Tunneling conductance .... 64 4.3.3 Zeeman field ...... 68 4.3.4 Residual Specific heat in the presence of Zeeman field ...... 72 4.4 Summary ...... 73 5 BOGOLIUBOV QUASIPARTICLE INTERFERENCE IMAGING OF STRONTIUM RUTHENATE ...... 75 5.1 Introduction ...... 75 5.2 Model ...... 77 5.3 Results ...... 81 5.3.1 Homogeneous superconducting state ...... 81 5.3.2 Inhomogeneous superconducting state ...... 83 5.4 Summary ...... 87 6 CONCLUSIONS ...... 88

APPENDIX A SPIN DECOMPOSITION OF INTERACTION CHANNELS AND POLARIZATION BUBBLES ...... 91 A.1 Charge susceptibility ...... 91 A.2 Spin susceptibility ...... 94 A.3 Total RPA susceptibility ...... 97 A.4 Interaction matrices contributing to Self-energy ...... 99 B A SPIN-FLUCTUATION PAIRING STRENGTH FUNCTIONAL INCLUDING QUASIPARTICLE RENORMALIZATION ...... 101 C NUMERICAL DETAILS FOR SELF-ENERGY EVALUATION ...... 108 D RPA CALCULATION ON THE TOY TWO-BAND MODEL ...... 112 E COMPARISON OF QUASIPARTICLE WEIGHTS OBTAINED VIA RPA AND TPSC ...... 114 F MODEL DETAILS FOR ELECTRON-HOLE TOY BANDS FOR FESE(S) ..... 116 F.1 Derivation of equations for Specific heat evaluation ...... 118 F.2 Spectral function evaluation ...... 119 F.3 DOS and Tunneling conductance evaluation ...... 119 F.4 Zeeman coupling to external magnetic field ...... 120 G SPATIAL AND ORBITAL STRUCTURE OF THE GAP FUNCTION ...... 122 REFERENCES ...... 124 BIOGRAPHICAL SKETCH ...... 138

8 LIST OF FIGURES Figure page 1-1 Schematic diagram depicting phenomena of superconductivity ...... 13 1-2 Crystal structure and generic phase diagram of FeSC families ...... 15 1-3 Conventional Brillouin zones in FeSC ...... 16 1-4 Schematic diagram of gap symmetries in FeSCs ...... 16 2-1 DFT-predicted vs ARPES-fitted Fermi surface of Bulk FeSe and LiFeAs ...... 20 2-2 Experimental vs. theoretically predicted gap structure in Bulk FeSe and LiFeAs .. 22 2-3 LiFeAs 2-D FS and orbital weight ...... 26 2-4 Momentum-dependent quasiparticle weight ...... 28 2-5 Plot of gap structure around FS ...... 29 2-6 Map of singlet pairing vertex ...... 30 2-7 Gap functions showing change in gap symmetry ...... 32 3-1 Schematic plot showing generic Fermi surface shrinkage in FeSCs ...... 35 3-2 Schematic DFT-derived Fermi surfaces vs. ARPES intensity maps for LaFeAsO, LiFeAs, FeSe ...... 37 3-3 Schematic plot of electron and hole bands in 1-Fe zone of a generic FeSC ...... 40 3-4 Spectral function along high-symmetry path for real FeSC materials ...... 44 3-5 Effect of nearest neighbor Coulomb interactions on the Fermi surface ofspecific FeSCs ...... 48 3-6 Comparative plots for the real part of the orbitally-resolved static self-energy in an FeSC, computed along high-symmetry path via different evaluation schemes .. 49 3-7 Scattering lifetime of quasiparticles as a function of binding energy for LiFeAs ... 53

4-1 CV /T vs. T plot for FeSe1xSx from specific heat measurements ...... 56

4-2 Tunneling conductance data for FeSe1xSx from STM ...... 57 4-3 Schematic plot of the phase diagram of FeSe(S) ...... 62 4-4 Spin-resolved spectral function for FeSe(S) ...... 65 4-5 Bogoliubov Fermi surfaces, temperature dependence of the specific heat and tunneling conductance of FeSe(S) ...... 66 4-6 Spin-resolved spectral functions in presence of Zeeman field ...... 71

9 4-7 Residual specific heat in presence of Zeeman field ...... 72

5-1 Isosurface plots of wannier orbitals in Sr2RuO4 ...... 78

5-2 Schematic plot of cleave surface of Sr2RuO4 ...... 81

5-3 Band structure, Fermi surface and density of states of Sr2RuO4 ...... 82

5-4 Lattice DOS, continuum DOS and BQPI image for Sr2RuO4 ...... 84

5-5 Experimental results for BQPI image for Sr2RuO4 ...... 86 A-1 Bubble diagram for normal-state self-energy ...... 91 B-1 Frequency dependence of the effective pairing interaction ...... 104 D-1 Plot of the real part of the static self-energy as a function of momentum radius of integration ...... 113 E-1 Comparison of quasiparticle weight Z vs. Hubbard U values obtained via different evaluation schemes ...... 114

10 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy EFFECTS OF ELECTRONIC CORRELATIONS ON PHYSICAL PROPERTIES OF UNCONVENTIONAL SUPERCONDUCTORS By Shinibali Bhattacharyya December 2020 Chair: Peter J. Hirschfeld Major: Physics In this dissertation, we provide results of our investigation into effects of electronic correlation in unconventional superconductors, focusing primarily on Iron-based superconductors (FeSCs). The focus of this work has been to set up microscopic models within weak-coupling theory, combined with numerical techniques, for studying complex multi-orbital systems. In Chapter 2, we have investigated the influence of non-local correlations on the superconducting gap structure within spin-fluctuation pairing theory. Our microscopic model produces gap values in good agreement with Angle-resolved photoemission spectroscopy (ARPES) data available for a compound, LiFeAs, which until now has proven difficult to understand. It points towards a significant improvement over traditional weak-coupling theory, which could not reproduce certain important aspects of the gap structure. In Chapter 3, we have investigated electronic structure renormalization effects for FeSCs. Our results indicate that the less correlated systems LaFeAsO andLiFeAs show Fermi surface shrinkage in accordance with ARPES data, whereas the more strongly correlated system FeSe doesn’t show as drastic Fermi surface shrinkage as observed in experiments. We have proposed that long-range, next-nearest neighbor Coulomb interaction is playing a significant role in the FeSe system and can explain its strongly renormalized electronic dispersion. We have also included analysis of the momentum-dependent scattering lifetime in LiFeAs to address recent ARPES data. In Chapter 4, we show that multiband superconductors with dominant spin singlet, intraband pairing can undergo a transition to a state with topologically protected Bogolibuov Fermi surfaces. Our results show a strong

11 resemblance with data from specific heat measurements and Scanning tunneling microscopy (STM) measurements on FeSe(S) across its nematic transition. We extended our model to predict detailed experimental consequences of “ultranodal pair states” in the presence of a magnetic field. In Chapter 5, we study the effects of disorder in the superconducting state of strontium ruthenate (SRO). Using the T-matrix approach, we evaluate Fourier transform of the real-space continuum density of states to compare with STM conductance map and Bogoliubov quasiparticle interference (BQPI) images, providing information about the momentum-resolved superconducting gap structure. We aimed to address recent experimental data that claim d-wave order parameter to be the most consistent with their BQPI analysis for SRO gap structure.

12 CHAPTER 1 INTRODUCTION AND BACKGROUND The discoveries of high-temperature superconductivity (HTSC) in cuprates and the relatively newer class of FeSC materials, are widely acknowledged to be among the most significant breakthroughs in the field of condensed matter physics. Understanding the physical mechanism behind HTSC is a very challenging and exciting issue and crucial in contributing towards efficient design of future HTSC materials and their many potential applications. This chapter introduces the phenomenon of superconductivity and describes some of its characteristic signature in materials exhibiting this phenomenon, with emphasis on multiorbital Fe-based superconductors. This is followed by some background from experimental and theoretical works, performed on various materials, which have served as motivation for our current work on investigation of electronic correlation in superconducting materials. 1.1 Superconductivity

Superconductivity refers to a microscopic phenomenon that results in two important macroscopic signatures: (a) the disappearance of electrical resistance and (b) expulsion of magnetic field when a material is cooled below a certain critical temperatureT ( c) as shown in a Figure 1-1. Following the first discovery of superconductivity in mercury in 1911at

Figure 1-1. Schematic diagram depicting signatures of superconductivity: (a) Resistivity vanishing below TC, (b) Perfect diamagnetism below TC (Meissner effect)

TC = 4.2K, the list of newly discovered superconductors has grown dramatically. These

13 have been broadly categorized into conventional and unconventional superconductors. As per Bardeen-Cooper-Schrieffer (BCS) theory, a lattice (phonon)-mediated effective electron-electron attraction results in conventional superconductors, with the condensation of electrons in a coherent ground state of pairs with opposite momenta and spins. The assumption of a constant effective electron-electron interaction leads to an isotropic s-wave pair state in momentum space. All elemental superconductors and many other simple materials fall under this category. Superconductors in which electrons bind into Cooper pairs via the repulsive Coulomb interaction without significant aid from the ionic lattice, and the pair state isnota simple isotropic s-wave, are regarded as unconventional [1]. Cuprates and iron-based superconductors form two of the most interesting classes of unconventional superconductors because of their high critical temperatures, which goes beyond the paradigm of conventional BCS theory. 1.2 Fe-based Superconductors

The first high-temperature iron-based superconductor (FeSC) to be discovered was fluorine-doped LaFeAsO at TC = 26 K by Kamihara et al. [2] in 2008. The large and growing list of FeSCs includes various Fe pnictides and chalcogenides [3, 4]. The crystal structures of various families of FeSCs are shown in Fig. 1-2a. All the structures are layered and include square lattice planes of Fe with pnictogen and chalcogen atoms above and below the plane. The phase diagram of a typical FeSC is shown in Fig. 1-2b. The undoped parent compound is usually an antiferromagnet. The magnetic phase of the FeSC is often called the spin-density wave (SDW) state to stress that the magnetism arises from itinerant electrons. Superconductivity can be induced by either hole or electron doping. With decreasing temperature or by changing chemical/physical pressure, another ordered phase appears called the “nematic” phase which precedes the SDW phase, and has an electronic state that breaks the tetragonal rotational symmetry but preserves the spin rotational symmetry, time-reversal symmetry and translational symmetry.

14 Figure 1-2. (a) Crystal structures of various FeSC families. Picture adapted from [5]. (b) Schematic phase diagram of a typical FeSC in the parameter space of temperature and doping. Picture adapted from [4].

Crystal structure, Fermi surface topology and gap symmetry: Iron’s 6 valence electrons partially occupy five 3d orbitals. The electronic states near the Fermi surface (FS) of FeSC materials are heavily dominated by these d-orbitals, giving rise to Cooper pairing in different orbital channels, leading to multiorbital superconductivity. A common electronic feature of FeSCs is to have hole-like FS near the Γ-point in Brillouin zone (BZ) and electron-like near the zone boundary. The crystallographic unit cell contains two inequivalent

Fe positions (2-Fe Zone) and Fig. 1-3 depicts this graphically. For many FeSC, for the kz = 0 plane, there is a rigorous procedure of exact “unfolding” of bands to a reciprocal space corresponding to a single Fe atom[6, 7]. One can discuss much of the essential physics in this 1-Fe Zone representation which will be used throughout this document unless mentioned otherwise. The crucial feature that allows an unconventional Cooper pair state to take advantage of the repulsive Coulomb interaction is the fact that its superconducting gap function changes sign. Owing to the multiband nature of FeSCs, the superconductivity is believed to have −→ a gap function ∆( k ) with s± symmetry, i.e. a gap function which changes sign between electron and hole pockets but not necessarily within the same pocket [6]. The resonance peak in neutron scattering experiments serves as an evidence for sign changing gap symmtery [8]. Repulsive inter-pocket Coulomb interaction drives the s± superconductivity. Such mechanism

15 Figure 1-3. (a) General FeX (X = Pnictogen As or Chalcogen Se/Te) lattice indicating X=As above and below the Fe plane. Dashed green and solid blue squares indicate 1- and 2-Fe unit cells, respectively. (b) Schematic 2D Fermi surface in the 1-Fe BZ whose boundaries are indicated by a green dashed square. The arrow indicates folding wave vector QF. (c) Fermi sheets in the folded BZ whose boundaries are now shown by a solid blue square. Picture adapted from [6]. is believed to be enhanced due to spin fluctuations, because the magnetic ordering vector Q of the SDW state is the same as the nesting wave-vector between hole and electron pockets. Two other states have also been proposed for at least some FeSCs. One is a conventional s-wave state where the gap function has the same sign over the whole BZ [6]. Another one is dx2−y2 superconductivity, where the interpocket repulsion between two electron pockets become dominant over other interactions in the BZ [9]. Fig 1-4 depicts the different possible gap structures in FeSCs.

Figure 1-4. Schematic diagram of gap symmetry under discussion in the FeSC represented in the 2D 1-Fe BZ. Different colors stands for different signs of the gap. Picture adapted from [6].

16 1.3 Earlier work highlighting the role of Electronic correlations

Theoretically, correlated systems have been studied using two different approaches: first, different model Hamiltonians have been used to gain qualitative understanding of the underlying phenomena, studied either by using nonperturbative techniques such as quantum Monte Carlo, density matrix renormalization group and exact diagonalization or by approximate methods such as mean field theory, Random phase approximation (RPA) or Fluctuation-exchange approximation (FLEX). Second, real materials have been studied via electronic structure methods from ab-initio input, i.e., Density Functional Theory (DFT) combined with many-body techniques such as Dynamical Mean Field Theory, Cluster Perturbation Theory, to name a few. Ab-initio band structure evaluations (DFT) reproduce the Fermi surface topology with several hole- and electron-bands crossing the Fermi level; this has been a crucial input for the RPA approach to study FeSC. The agreement with experimentally observed Fermi surfaces is good in some materials, as in LaFePO and BaFe2As2[10], but highly mismatched in materials like bulk/monolayer FeSe [11], LiFeAs. In fact, the missing hole-pockets in monolayer FeSe poses a challenge to the usual spin-fluctuation-pairing theory which is largely based on assumptions of strong inter-pocket scattering at the Fermi level to explain superconductivity. It is believed that strong electronic correlations, not captured in ab-initio methods, result in the discrepancies observed in these correlated materials. The role of electronic correlations in FeSC has been investigated widely from different theoretical perspectives [12, 13]. However, there have been ongoing efforts to integrate correlation effects in the RPA approach in order to understand the physical properties and signatures of strongly-correlated FeSC [14]. A renormalized spin fluctuation theory, incorporating correlations at a phenomenological level, was used recently with great success to explain puzzling results on both the superconducting gap [15] and normal state electronic structure [16] obtained from quasiparticle interference detected via Scanning tunneling microscopy (STM). But this approach lacks a microscopic insight into the derivation of

17 quasiparticle weights Zorb from self-energy evaluations, or any other aspect of self-energy which is known to give rise to correlation effects in an interacting system. 1.4 Format of the Dissertation

My doctoral research has been primarily focused on setting up a microscopic model within weak-coupling theory for the multi-orbital, momentum-dependent self-energy and to study its effects on the physical properties of FeSC. I have worked on different materials, examining their normal and superconducting state signatures under the effect of non-local correlations. My various projects are related to each other only through their connection to weak-coupling theoretical approach in studying unconventional superconductors, and are otherwise fairly independent. Consequently, I present each of my projects in a separate chapter with dedicated introductory and concluding sections. In Chapter 2, we have provided a microscopic model to investigate the influence of non-local correlations on the superconducting gap structure via RPA within spin-fluctuation pairing theory. Our model produces gap values in good agreement with Angle-resolved photoemission spectroscopy (ARPES) data available for LiFeAs. In Chapter 3, we have investigated electronic structure renormalization effects for FeSCs from self-energy evaluation within spin-fluctuation theory and compared our results to ARPES measurements for various FeSCs. In Chapter 4, we have shown that multiband superconductors with dominant spin singlet, intraband pairing can undergo a transition to a state with topologically protected Bogolibuov Fermi surfaces. Our results show strong resemblance with specific heat measurements and STM measurements on FeSe(S) across its nematic transition. In Chapter 5, we have diversified our effort to study a different class of unconventional superconducting material: Sr2RuO4. Using the T-matrix approach, we study the effects of disorder in the superconducting state of Sr2RuO4. We evaluate continuum electronic density of states to compare with STM conductance map and Bogoliubov quasiparticle interference images, aiming to address recent experimental data. Further investigation is ongoing in this context.

18 CHAPTER 2 EFFECTS OF MOMENTUM-DEPENDENT QUASIPARTICLE RENORMALIZATION ON THE GAP STRUCTURE OF IRON-BASED SUPERCONDUCTORS ∗ 2.1 Introduction

Different theoretical perspectives have been employed to investigate the role of electronic correlation effects on the properties of iron-based superconductors10 (FeSC)[ , 12, 18–21]. As FeSC are multiorbital systems, the possibility of orbital selective physics implies that weak correlations may exist in the states of electrons of one orbital type while strong correlations may occur in others. Such behavior is believed to give rise to substantial differences in quasiparticle weights, interactions, magnetism, and orbital ordering; in addition, Cooper pairing itself can become orbital-selective [15, 22–24]. The conventional spin-fluctuation pairing model, proposed as the dominant source of Cooper pairing in FeSCs [25–27], can be modified by this orbital selective many-body mechanism that augments the gap anisotropy already present in such systems. The interband pair scattering between electron and hole pockets , facilitated by enhanced nesting [28], is believed to yield an s± pairing structure. Gap anisotropy results from the variation of the different d−orbital weights around any given Fermi surface (FS) sheet [29]. To the extent certain d orbitals are more incoherent than others due to correlations, pairing in these channels may be further suppressed, leading generally to enhanced gap anisotropy [30]. Across the various families of Fe-based superconductors, the degree of electronic correlation is known to vary considerably. Local-density approximation (LDA) + dynamical mean-field theory (DMFT) calculations have suggested that the 111 are considerably more correlated than, e.g., the well-studied 122 materials [10, 31, 32]. These works have also demonstrated that stronger interactions lead to a shrinkage of the inner- and expansion of outer-hole pockets but the size and shape of the electron pockets is maintained. LiFeAs is one of several FeSCs known to have a FS which is quite different from what is predicted from

∗ This chapter is a slightly modified version of17 [ ] published in Phys. Rev. B and has been reproduced here with permission of the copyright holders.

19 Density functional theory (DFT). ARPES measurements [33], [34] show that the Γ centered dxz/dyz hole pockets are considerably smaller than DFT predictions, but inclusion of local correlations within a DMFT scheme does not seem to solve this discrepancy. Recent work using non-local correlations within a Two-particle self-consistent (TPSC) approach [35] has shown that the FS can be significantly renormalized, resembling ARPES findings.

Figure 2-1. Schematic plot of DFT-predicted (dashed lines) and ARPES-fitted (solid lines) ′ ′ FS pockets at kz = π plane for (a) FeSe (bulk): α, ϵ and ϵ pockets. ϵ has not been detected in ARPES and γ has been found to be pushed below the Fermi level, hence omitted. (b) LiFeAs: α, β, β′ and γ pockets. Angle ϕ around the Fermi pockets are defined counterclockwise from the kx axis centered around each pocket, and will be referred to in the upcoming figures.

A phenomenological tight-binding band structure consistent with ARPES data [36], i.e., describing the correct spectral positions of the bands, was used in previous theoretical attempts [24, 36–39] to understand the ARPES-determined gap structure [33, 34, 40, 41]. A schematic plot of the DFT-predicted, as well as the ARPES-fitted FS in bulk FeSe and

LiFeAs have been shown in Fig.2-1, for the kz = π plane. Simultaneous pocket shrinkage for both hole and electron FS, compared to their DFT-predicted counterparts, have been observed across all FeSCs in experiments, and will be discussed in further details in Chapter 3. FeSe has a Z = (0, 0, π)-centered hole pocket α, and X′ = (π, 0, π)-centered electron

20 pocket ϵ (Fig.2-1a) which will be referred to in Fig.2-2 while discussing experimentally obtained gap structures. The Y ′ = (0, π, π)-centered ϵ′ electron pocket has not been observed spectroscopically in any experiments[15, 42], hence its ARPES-fitted schematic plot has been omitted (Fig.2-1a) and will not be discussed. The R = (π, π, π)-centered hole pocket γ is observed in ARPES to be pushed below the Fermi level, and this particular phenomena will be addressed in detail in Chapter 3. Fig.2-1b shows the schematic DFT-predicted and the ARPES-fitted FS for LiFeAs, with Z-centered hole pocket α, X′/Y ′-centered electron pocket β/β′ and R-centered hole pocket γ. Out of the two X′ − Y ′ electron pockets in LiFeAs, we will only discuss data for the β pocket as the other one is related by C4-symmetry. Some success was reflected in explaining some features of the gap structure, but not all significant details were reproduced properly; in particular, the large superconducting gaps on the inner hole pockets (α) were not recovered in the conventional spin fluctuation treatments (see Fig.2-2c). Subsequently, a good overall fit to experiment was claimed in Ref.[38] by including vertex corrections. A phenomenological approach assuming orbital selective renormalization of quasiparticle weights, primarily in the dxy channel, was used in Ref.[30], to test the proposition that the discrepancies in calculated gap structures arose from correlation effects neglected in conventional spin fluctuation theory (see Fig.2-2a and c for bulk FeSe and LiFeAs). The results of the dressed spin fluctuation approach, as depicted in2-2 Fig. b and d, showed good agreement with measured magnitudes and anisotropies of gaps from ARPES and Bogoliubov quasiparticle interference (BQPI) experiments for FeSe crystals (Fig.2-2b)[15, 42], LiFeAs (Fig.2-2d)[34, 40], and FeSe monolayers. While this approach lacked a microscopic framework to calculate the quasiparticle weights, it showed that quasi-particle renormalization effects were very important. Fluctuation Exchange Approximation (FLEX) calculations [43–45] have previously treated non-local correlation effects, but in this chapter, we explore a multi-orbital random-phase approximation (RPA) which is more simply related to the results of the phenomenological treatment [30]. The approach we discuss in here can be considered as a

21 Figure 2-2. Plot of the gap function around different FS pockets ϵ and α for FeSe (bulk): (a) the conventional spin-fluctuation calculation and (b) a calculation using the spin-fluctuation pairing in presence of quasiparticle weights. For direct comparison, the data from a Bogoliubov Quasiparticle interference analysis from Ref.[15] and a ARPES investigation [42] are displayed as well. Plot of the gap function around different FS pockets α, β and γ for LiFeAs: (c) the conventional spin-fluctuation calculation and (d) spin-fluctuation pairing in presence of quasiparticle weights. Data from a Bogoliubov Quasiparticle interference analysis from Ref.[40] and ARPES [34] are displayed as well. Picture adapted from Ref.[30].

FS restricted, one-loop FLEX calculation. It is numerically much less expensive and does not involve any ad-hoc restrictions as such. In this approach, both the paring and the single particle self-energy channels are approximated by the zero frequency RPA interactions (arising from spin and charge fluctuations) on the FS. In Sec.2.2, we introduce the five-orbital tight binding model that we will study along with the multi-orbital RPA effective interactions. Following this, the eigenvalue equation for the pairfield strength and the gap function along with the equation for the single particle

22 renormalization factor are discussed. In Sec.2.3, we demonstrate the results of solving these equations for LiFeAs, and discuss this in comparison to experiments Sec.2.3.1. In Sec.2.3.2 we analyze our results in terms of the quasi-particle renormalization weights and the pairing vertex, and in the following Sec.2.3.3 we discuss the tendency of iron-based systems towards s± pairing symmetry due to electronic correlations. Finally, we note the improvement obtained in predicting the pairing structure when the single particle self-energy is included. In Appendix B, we give the derivation of the dimensionless pairing strength functional and the resulting stationary eigenvalue equation for the pairing strength and gap function. 2.2 Model

We start with a five-orbital tight binding Hamiltonian H0 and include local interactions via Hubbard-Hund terms:

H =H0 + HI X X X X X tq † ′ = (t − µ δ δ )c c + U n ↑n ↓ + U n n ′ ij 0 ij tq itσ jqσ it it itσ iqσ (2–1) ijσ qt it i,t

′ ′ rotational invariance through the relations U = U − 2J and J = J . The kinetic energy H0 includes the chemical potential µ0 and is described by a tight-binding model spanned by five

Fe d orbitals dxy, dx2−y2 , dxz, dyz, d3z2−r2 . Here q and t are the orbital indices and i, j is the Fe-atom site. The spectral representation of the non-interacting Green’s function is:

X t q∗ 0 aµ(k)aµ (k) G (k, ωm) = (2–2) tq iω − E (k) µ m µ

q ⟨ | ⟩ | ⟩ where the matrix elements aµ(k) = q µk are spectral weights of the Bloch state µk with band index µ and wave vector k in the orbital basis and ωm = (2m + 1)πkBT are the fermionic Matsubara frequencies for a given temperature T . We will adopt Latin symbols

23 to denote orbital indices and Greek ones to denote band indices, throughout the rest of the discussion. The orbitally resolved noninteracting susceptibility is:

1 X 0 − 0 0 χpqst(q, Ωm) = Gtq(k, ωm)Gps(k + q, ωm + Ωm) Nkβ kωm X t q∗ p s∗ (2–3) 1 aµ(k)aµ (k)aν(k + q)aν (k + q) = − (f(Eµ(k)) − f(Eν(k + q))) Nk iΩm + Eµ(k) − Eν(k + q) kµν where Nk is the number of Fe lattice sites and β = 1/kBT is the inverse temperature,

Ωm = 2mπkBT is the bosonic Matsubara frequency. Within the RPA, the charge-fluctuation and spin-fluctuation parts of the RPA susceptibility are given by:

C 0 C −1 0 χ (q, Ωm) = [1 + χ (q, Ωm)U ] χ (q, Ωm) (2–4) S 0 S −1 0 χ (q, Ωm) = [1 − χ (q, Ωm)U ] χ (q, Ωm)

The interaction matrices U C and U S in orbital space have the following elements:

C S Upppp = U ,Upppp = U C ′ − S Uppss = 2U J,Uppss = J (2–5) C ′ S ′ Upssp = J ,Upssp = J C − ′ S ′ Upsps = 2J U ,Upsps = U .

Defining U SC = U S + U C , the particle-hole (N) and the singlet pairing particle-particle (A) interactions are: 3 3 1 1 S S S S C C C − C [Vpqst(q, Ωm)]N = U χ (q, Ωm)U + U + U χ (q, Ωm)U U 2 2 2 2 1 SC 0 SC − U χ (q, Ωm)U (2–6) 4 pqst 3 1 1 1 S S S S − C C C C [Vpqst(q, Ωm)]A = U χ (q, Ωm)U + U U χ (q, Ωm)U + U 2 2 2 2 pqst The relevant derivations for the above equation can be found in Appendix A. In the following, we assume that the dynamics of the interaction is cut off on a spin-fluctuation energy scale and we will restrict the treatment of the problem to Bloch states on the FS. We have assumed that the leading pairing instability occurs in the even-frequency channel.

24 Odd-frequency pairing, if it occurs, is expected to be associated with a reduced critical temperature [47]. Assuming that no bands cross each other in the vicinity of the Fermi level

( i.e. Ωm = Ω = 0 eV), each Fermi momentum point corresponds to a unique band quantum number, i.e. k ∈ µ, and k′ ∈ ν. The corresponding band representation for the interaction

′ vertices, describing scattering of particles between (kµ, kν) states at the Fermi level is X ′ q∗ s∗ ′ − ′ t p ′ [V (kµ, kν, 0)]N = Re aµ (kµ)aν (kν)[Vpqst(kµ kν, 0)]N aµ(kµ)aν(kν) pqst X (2–7) ′ p∗ t∗ − − ′ q ′ s − ′ [V (kµ, kν, 0)]A = Re aµ (kµ)aµ ( kµ)[Vpqst(kµ kν, 0)]A aν(kν)aν( kν) . pqst

The symmetrized singlet pairing vertex is then given by

1 [V (k , k′ , 0)] = [V (k , k′ , 0)] + [V (−k , k′ , 0)] . (2–8) µ ν A 2 µ ν A µ ν A

As discussed in Appendix B, the stationary solution of the pairing strength functional (Eq.B–22) is determined by the eigenvalue equation: I 1 [V (k , k′ , 0)] dd−1k′ − µ ν A ν ′ λg(kµ) = d ′ ′ g(kν). (2–9) (2π) ′ ∈ Z(k ) |v (k )| kν FS ν F ν with I d−1 ′ 1 ′ d kν Z(kµ) = 1 + d [V (kµ, kν, 0)]N ′ . (2–10) (2π) ′ ∈ |v (k )| kν FS F ν

′ Here, vF (kν) is the Fermi velocity of band ν and the integration is over its corresponding FS.

The eigenfunction g(kµ) (dimensions of energy) then determines the symmetry and structure of the leading pairing gap close to Tc. Traditional spin-fluctuation pairing calculations set

Z(kµ) to unity. Instead, we use the solution for the dimensionless Z(kµ) from Eq.2–10 in

Eq.2–9. Solving Eq.2–9 for the eigenfunction g(kµ) corresponding to the leading eigenvalue λ gives the modified gap function ∆(k ) = g(kµ) (in dimensions of energy). Note that we adopt µ Z(kµ) here the notation Z(k) consistent with Eliashberg’s convention, such that Z ≫ 1 corresponds to a highly incoherent quasiparticle. This is in contrast to the notation adopted in much of

25 the orbital-selectivity literature, where Z rather than Z−1 is the quasiparticle weight (see, e.g. Sprau et al. [15]). In the results presented here, we perform 2D calculations with a k-mesh on the order of 100 × 100 in the unfolded Brillouin Zone (BZ), and ≈ 500 total number of FS points. We set kBT = 0.01 eV for the rest of the calculations in this chapter. 2.3 Results

To study the effects of momentum-dependent correlations on the pairing structure, we have chosen the electronic dispersion relevant to LiFeAs as in Ref.[36]. LiFeAs is one of the few systems where good cleavage of its surfaces has led to reliable measurement of the gap functions via both STM and ARPES measurements [34, 40]. Certain Fermi sheets in LiFeAs are known to have significant kz-dispersion. However, to compare with experimental data of measured gap magnitudes, we only consider the kz = π plane for our analysis, where all the Z = (0, 0, π)-centered hole pockets are present. We stress that the 3D properties of the dispersion do not affect the conclusions of our results, since χ(q, 0) is nearly independent of qz [36].

Figure 2-3. (a) FS at kz = π plane for LiFeAs represented by its corresponding dominant orbital character as indicated in the color legend. (b) Orbital weight along the β1 electron pocket plotted as a function of angle ϕ shown in (a). (c) Diagonal orbital components of the static particle-particle effective interaction Vpppp(q, 0)A along high-symmetry path Z − R − X′ − Z − Y ′ with U and J as indicated. Plots as a function of the angle ϕ around the Fermi pockets are done with the angle measured counterclockwise from the kx axis.

26 As shown in Fig.2-3a, for the undoped material with a filling of n = 6 in the 1-Fe BZ, the FS include two small hole pockets α1 and α2 at the Z point with dominant dxz/dyz orbital characters, a large hole pocket γ at the R = (π, π, π) point of predominantly

′ dxy orbital character, and two electron pockets β1 (β2) situated at X = (π, 0, π)

′ ′ (Y = (0, π, π)) points of dyz/dxy (dxz/dxy) orbital characters. Since the Y -centered pocket is symmetry-related to the X′ pocket in this tetragonal system, it will not be

| t |2 discussed separately. In Fig.2-3b, we show the corresponding orbital weight aµ(k) for the β1 pocket as a function of the angle ϕ. We will relate this to Z(kµ) and the modulation in gap amplitude in the following analysis of our results.

Now, we will present our solutions to Eqs. (2–9-2–10) for Z(kµ) and the leading pairing eigenvectors (gap functions). We have evaluated the results for two sets of Hubbard-Hund parameters (U = 0.7 eV, J = 0.26 eV) and (U = 0.79 eV, J = 0.2 eV). These parameters are close to the standard parameters used in the literature employing the RPA approach to the pairing vertex. First, we will discuss the results for U = 0.7 eV, J = 0.26 eV and compare with experimental data. In Fig.2-3c, the diagonal components of the static particle-particle effective interaction [Vpppp(q, 0)]A evaluated from Eq. (2–6) is plotted along the high-symmetry path Z − R − X′ − Z − Y ′. The RPA susceptibility shows an enhanced incommensurate peak around q = π(1, 0.15) similar to the findings in Ref.[36]. The dxy orbital channel is the largest in magnitude as a consequence of favorable nesting condition

′ ′ between dxy-dominated parts of the electron pocket at X /Y and the hole pocket at R. In this 2D system under consideration, the phase space for scattering of quasiparticles between the dxz/dyz-dominated parts of the FS is restricted due to the unfavorable nesting conditions of the α hole pockets in connection to the electron pockets β. Thus, the magnitude of the dxz/dyz orbital channel for [Vpppp(q, 0)]A is much lower than the dxy channel. Both the static particle-particle effective interaction [Vpqst(q, 0)]A and particle-hole effective interaction

[Vpqst(q, 0)]N is dominated by the RPA spin-susceptibility contribution, and hence their underlying momentum structure is similar.

27 Figure 2-4. Z(ϕ) along Fermi pockets α2, β1 and γ evaluated from the static particle-hole effective interaction defined in2–10 Eq.( )

In Fig.2-4, we plot Z(kµ) evaluated from Eq. (2–10) as a function of the angle ϕ around the α2, β1 and γ Fermi pockets. Evidently, Z(ϕ) around the β1 pocket follows the orbital content of the dxy orbital; compare to Fig.2-3b for an angular plot of the orbital weight around the β1 pocket. This is expected from the current FS topology depending on two factors: (1) the interaction vertices are dominated by intraorbital processes, and (2) the dxy orbital with large weight at certain positions on the FS can take advantage of the strong − ′ ≈ peak in the susceptibility for scattering vectors kµ kν π(1, 0.15) roughly connecting ′ ◦ ◦ kµ and kν across Fermi pockets. For the β1 pocket, this happens around 0 , 180 . For the γ pocket, maxima in Z(ϕ) are seen at 0◦, 90◦, 180◦, 270◦. Due to the lower magnitude of the

[Vpppp(q, 0)]N vertex for the dxz/dyz orbitals, the magnitude of Z(ϕ) around the α2 pocket is smaller than around the β1 and γ pockets. 2.3.1 Gap structure

We show the results for the leading gap structure in Fig. 2-5a obtained from traditional

′ spin-fluctuation calculation with Z(kν) set to unity in Eq. (2–9)[36]. Fig. 2-5b is the most important result of this work, displaying the modified gap structure obtained by inclusion of momentum-dependent correlation effects Z(kµ) ≠ 1 via the linearized Eliashberg equations (2–9-2–10). For comparison of our calculations to experimental results, we have

28 Figure 2-5. Results for LiFeAs: (a) Angular plot of the s-wave gap function evaluated from traditional spin-fluctuation theory, around the Z-centered outer hole pocket α2, ′ X -centered electron pocket β1 and R-centered hole pocket γ. (b) The modified gap function obtained from the linearized Eliashberg equations (2–9-2–10) with quasiparticle renormalization Z(k) included. (c) Experimental results: Measured magnitudes of the gap from an ARPES experiment [34], symmetrized and displayed as diamonds and those from Bogoliubov QPI experiment [40] displayed as crosses. All calculations are done for U and J as indicated.

plotted C4-symmetrized ARPES data for the gap magnitude taken from Ref. [34] and BQPI data from Ref. [40] for the three Fermi pockets α2, β1, γ in Fig. 2-5c. ARPES sees only one band crossing at the Fermi level at Z on the kz = π plane with a large gap of the order of 6 meV. However, our current DFT-derived tight-binding structure always produces two α pockets. Hence, we will consider it to be roughly appropriate to speak of an average gap

± on the α pockets assigned to α2. In both Figs. 2-5a and 2-5b, we find an s -wave state with highly anisotropic but full gaps on the electron (negative gap) and hole (positive gap) pockets. With self-energy effects included, the gap functions undergo a remarkable change Fig. 2-5b relative to the traditional spin-fluctuation calculation Fig. 2-5a. First, we find a stronger tendency towards s± pairing symmetry, even for small values of J. We will discuss this in further detail in Section 2.3.3. Second, Z(kµ) induces a momentum-dependent modulation of the gap function on various pockets. Most importantly, in Fig. 2-5b we see that Z(kµ) enhances the magnitude of the nearly-isotropic gap on the small Z-centered hole pocket α2. This brings the results of spin-fluctuation theory much closer in line with experimental data, and corrects the crucial discrepancy in the calculation of Wang et al.

29 [36] relative to experiment. Although the angular positions of the gap maxima and minima on the γ pocket and the average gap magnitude is in agreement with experiment, we do not obtain a suppression of the anisotropy. We find weaker anisotropy of the gap function on the β1 pocket and an average gap magnitude comparable to experiment. However, the theoretical gap structure has 2 global maxima as opposed to 4 in the experimental data, and these maxima are located at positions where the experimental data has minima. In the next section we will analyze the results for the gap structure in terms of the quasiparticle renormalization factor Z(kµ) and the pairing vertex. 2.3.2 Analysis of the Gap structure in terms of Quasiparticle weights and the Pairing vertex

′ Figure 2-6. “Heatmap” plots of (a) the singlet pairing vertex [V (kµ, kν, 0)]A matrix resulting in the gap function plotted in Fig. 2-5a from the tight-binding model, (b) Matrix ′ ′ | ′ | elements of the quantity [V (kµ, kν, 0)]A dkν/ vF (kν) , (c) Matrix elements of the ′ ′ | ′ | ′ quantity [V (kµ, kν, 0)]A dkν/ vF (kν) /Z(kν). The value at any given point is proportional to the brightness of the color. The rows and the columns of the tiles ′ of (a)-(c) correspond to the Fermi points kµ and kν arranged in ascending order of their corresponding polar angle ϕ = 0◦ to 360◦ around each Fermi sheet. Band indices µ and ν represent the Fermi sheets α1, α2 at the Z, γ at the R, β1 at the ′ ′ X and β2 at the Y point. Unequal areas of the tiles are due to unequal number of Fermi points around each of the sheets.

We investigate the structure of the effective pair vertex from its graphical representation in Fig. 2-6a-2-6c. Each block demarcated by white lines in the image represents a matrix

′ (kµ, kν) consisting of values corresponding to the pairing vertex, with the Fermi points kµ ′ ◦ ◦ and kν arranged in ascending order of their corresponding polar angle ϕ = 0 to 360 around ′ each Fermi sheet (α1, α2, β1, β2, γ). By construction, as in Eq. (2–8), [V (kµ, kν, 0)]A is a

30 ←→ ′ symmetric matrix with respect to the interchange kµ kν as seen in Fig. 2-6a. Fig. 2-6b ′ ′ | ′ | shows matrix elements of the quantity [V (kµ, kν, 0)]A dkν/ vF (kν) . This is the matrix which is diagonalized in traditional spin-fluctuation theory to obtain the leading eigenstate. Itis ←→ ′ ′ not symmetric under the exchange kµ kν. Here, dkν is the length element of the Fermi ′ | ′ | sheet at the kν-th Fermi point. The factor 1/ vF (kν) acts as a momentum-dependent density of states for the corresponding band ν at the Fermi level. In the figures, the brightest set of blocks are the ones representing scattering processes among the pockets γ and β1 (β2). It is visually prominent that the dominant scattering processes occur for γ → β1 (β2). As discussed earlier, scattering between the dxy-dominated part of the Fermi pockets contributes to the primary pairing interaction leading to superconductivity. Although angular positions of 0◦, 90◦, 180◦, 270◦ on the γ pocket are favored to take advantage of the dxy-dominated scattering to either β1 or β2 pocket, positions

◦ ◦ ◦ ◦ of 45 , 135 , 225 , 315 take advantage of simultaneous scattering to β1 and β2 pockets. This renders gap maxima around 45◦, 135◦, 225◦, 315◦ on the γ pocket refer to Fig. 2-5a. It also

◦ ◦ explains why gap maxima occur at 0 , 180 on the β1 pocket due to favorable dxy scattering to the γ pocket. Due to restricted scattering between α2 and the other pockets, the overall gap magnitude is very low on this pocket. ′ ′ | ′ | ′ In Fig. 2-6c, we show matrix elements of the quantity [V (kµ, kν, 0)]A dkν/ vF (kν) /Z(kν).

The leading eigenfunction g(kµ) of this matrix gives the modified gap function as ∆(k ) = g(kµ) . Compared to Fig. 2-6b, the first noticeable difference is the enhanced µ Z(kµ) pairing (brighter spots) occurring between the α2 pocket and the β1 (β2) pockets. It appears as if the electronic correlations tend to lift the restriction on the phase space of scattering between the non-nested dxz/dyz-dominated parts of the α2 Fermi pocket. This is one of the

′ main effects of the momentum-dependent modulation of the pairing vertex dueto Z(kν). In addition, due to the smaller magnitude of Z(kα2 ) (refer to Fig. 2-4), the final gap function

∆(kα2 ) undergoes further enhancement in its magnitude as seen in Fig. 2-5b. We also find ◦ ◦ that the gap maxima shifts to 90 , 270 on the β1 pocket due to enhanced pairing to the α2

pocket via the dyz orbitals, combined with the effect of momentum modulation by Z(kβ1 ) on

31 the final gap function. The γ and β1 (β2) pocket pairing is not affected substantially due to

′ the Z(kν) modulation. Hence, ∆(kγ) does not undergo any drastic changes. 2.3.3 Discussion: Tendency towards s± Pairing symmetry

Figure 2-7. Angular plots of the gap function evaluated around the Fermi pockets as indicated by the color legend. The calculations show a change in the gap symmetry of the leading eigenfunction from (left) d-wave (obtained from traditional spin-fluctuation theory with Z(k) = 1) to (right) s-wave (obtained from linearized Eliashberg equations with Z(k) included). Both calculations are done for U and J as indicated.

A close competitor of the leading s± state is the d-wave state, which has never been convincingly observed in any experiment in the context of FeSCs. In the traditional spin-fluctuation scenario, d and s wave solutions are nearly degenerate in LiFeAs [36], a consequence of its poor (π, 0) nesting properties [9],[31],[33]. In Fig. 2-7, we show the results for calculations with Hubbard-Hund parameters (U = 0.79 eV, J = 0.2 eV) which is a higher U and a lower J value than the previous calculation shown in Fig.2-5a-2-5b. With this particular set of U and J, the traditional spin-fluctuation theory yields a d-wave state as the leading pairing state (left panel in Fig. 2-7). We see a sign change in the gap structure around each pocket (α2, β1, γ) with a periodicity of π/2. For this case, the second leading eigen channel is the s-wave channel and the ratio of their eigen values is λd/λs = 1.05. In contrast, the inclusion of correlations effect via the linearized Eliashberg equations 2–9-2–10 stabilizes the s-wave state as the leading eigenfunction, and the d-wave state

32 becomes subleading. For this case, we get λd/λs = 0.97. This points towards quasiparticle renormalization effects playing an important role in stabilizing the s± pairing state in LiFeAs. 2.4 Summary

We have performed 2D calculations of the superconducting pairing state for the LiFeAs compound. It is one of the few multiband materials where ARPES experiments have measured the gap function on several different FS sheets and found significant gap anisotropy. We have provided a simple approximate theoretical framework for the calculation of the superconducting gap from spin fluctuation theory, including momentum-dependent self-energy effects at the single-particle level. This serves as a test of the proposal that a proper treatment of momentum dependent quasiparticle renormalization is the major missing ingredient in traditional spin fluctuation calculations of gap structures in FeSC. We have shown that this approach leads to robust s± pairing. In addition, it provides substantial improvement of the calculated gap structure on all the Fermi pockets compared to experiments, especially for the large gap on the inner hole pocket. We also discussed the role of self-energy effects in stabilizing the s± eigenstate. We conclude that within the framework of existing weak-coupling theories, the inclusion of self-energy effects is an important ingredient to understand the observed superconducting pairing structures in LiFeAs, and other FeSCs, which are dictated by the same underlying physical processes. Vertex corrections to capture the basic renormalization of the electronic structure and pairing vertex, as calculated in Ref.[38], do not appear to be necessary.

33 CHAPTER 3 NON-LOCAL CORRELATIONS IN IRON PNICTIDES AND CHALCOGENIDES ∗ 3.1 Introduction

Following the discovery of iron-based superconductors (FeSC), researchers realized that density functional theory (DFT) calculations of the electronic structure of these materials gave results that were qualitatively in agreement to ARPES measurements: the systems consisted of nearly compensated, quasi 2D hole and electron pockets centered at the high-symmetry points of the Brillouin zone (BZ) [49–52]. However, quantitative discrepancies remained: Fermi velocities and pocket sizes observed by de Haas-van Alphen (dHvA) and ARPES measurements were smaller as compared to those calculated by DFT [11, 31, 33, 53–57]. As already depicted before in Fig.2-1 in Chapter 2, here we show again in Fig. 3-1, a schematic plot of the band structure and Fermi surface (FS) of a generic FeSC, showing the discrepancy between DFT predictions and experimental findings. Earlier, the “pocket shrinkage” was explained in terms of a rather simple picture: in a toy multiband model for FeSC, simultaneous shrinkage of both electron and hole pockets (also called “red-blue shifts” [58]) were shown to arise from self-energy renormalizations induced by repulsive interband interactions [59, 60]. It is desirable to understand effects of correlations within a material-specific first principles method beyond DFT, as model calculations cannot distinguish easily among different materials. Early attempts by dynamical mean field theory (DFT+DMFT) to explain overall materials trends in Fe-based systems with local correlations (momentum-independent self-energy) saw some success [10]. It led to a picture where inter-orbital Hund’s exchange suppressed inter-orbital charge fluctuations and promoted strong orbital selective correlations, the so-called “Hund’s metal” state [12, 19, 61]. Subsequent DFT+DMFT calculations of LiFeAs were also reported as successful because

∗ This chapter is a slightly modified version of48 [ ] published in Phys. Rev. B and has been reproduced here with permission of the copyright holders.

34 Figure 3-1. Schematic plot showing (a) band structure along kx direction and (b) corresponding Fermi surface of a generic FeSC in the 1-Fe BZ. The dashed lines represent DFT-predicted Γ-centered hole (blue) and M-centered˜ electron (green) bands. The solid lines represent the renormalized bands, as observed in dHvA and ARPES experiments, with momentum-dependent energy shifts and shrunken pocket sizes.

some weak pocket shrinkage was observed for some specific kz values [31, 32, 62, 63], but did not account for the simultaneous shrinkage of both the electron and hole pockets. This is due to the fact that the DMFT-derived local self-energy at a given frequency has the same sign all over the Brillouin Zone, rather than having opposite signs at electron and hole pockets, that lead to simultaneous pocket shrinkage[59]. Within the framework of the fluctuation exchange approximation (FLEX), the first full-fledged numerical calculation that showed the correct trend of band renormalization effects in FeSC was performed in Ref.44 [ ]. However, ad hoc restrictions were imposed on the renormalization scheme as the extent of FS shrinkage reported in such calculations were uncontrollably large. The case of FeSe brought the discrepancies between DFT, DMFT and experimental low-energy band structures to stark relief. DFT calculations within the local density approximation (LDA) or the generalized approximation (GGA) show FS with two inner dxz/yz Γ-centered hole pockets surrounded by an outer dxy pocket, and two M-centered electron pockets of sizes similar to those found in DFT calculations for other FeSC. On the contrary, ARPES finds dramatically smaller pockets at 90K, just above the tetragonal-orthorhombic transition, with only a single hole pocket at Γ of dxz/yz character [64]. The dxy band appears to be pushed down by as much as 50 meV compared to

35 reported DFT calculations, such that it does not appear at the Fermi level at all, while it is proposed that the two dxz/yz bands can split due to spin-orbit coupling [65] or orthorhombic distortions [66]. Reported DFT+DMFT calculations do not reproduce the suppression of the dxy band below the Fermi level and also seem to fail in capturing the strong pocket shrinkage [67–69]. The FeSe system becomes strongly nematic without long range magnetism at lower temperatures, an effect that was not found in any first principles calculation until quite recently [70] with a combination of LDA+U and an enlarged unit cell scheme, whose generality is currently unclear. Recently, it has been proposed by Zantout et al. in Ref. [35] that a local approximation of correlations was not sufficient to account for band renormalizations in the Fe-pnictides. A significantly better description of the relative pocket shrinkage and deformations in LiFeAs band structure was obtained within a multi-orbital formalism of the Two-particle self-consistent scheme (TPSC) [71]. Certain local spin and charge sum rules, that are not accounted for in the more popular approximation schemes, are preserved in TPSC. The good agreement found with band renormalization and scattering lifetimes from ARPES experiments, left open the question of whether this success was due to some aspect of the particular approximation scheme employed, or to the general inclusion of non-local correlations. Thus, it seems necessary to compare different approximate approaches to calculating non-local self-energy to see the extent to which they agree with experimental evidences, and with each other. Understanding the origin of the apparent profoundly different effects of correlations in the two classes of iron pnictides (Pn) and chalcogenides (Ch) is also very important. In this chapter, we present calculations of the non-local electronic self-energy within the random phase approximation (RPA) and compare with TPSC calculations. For historical reasons, we also compare with FLEX calculations. Since our DFT-derived tight-binding parameters already account for important Hartree-Fock corrections, we ignore first-order Hartree-Fock contributions to the self-energy in contrast to the FLEX approach in Ref. [44]. To illustrate the difference between the Pn- and Ch-systems, we show a figure with schematic

36 depiction of DFT-derived Fermi surfaces for LaFeAsO (Fig.3-2a), LiFeAs (Fig.3-2c) and FeSe

(Fig.3-2e) in their tetragonal phase for a specific kz = 0 plane, and the corresponding ARPES intensity map of LaFeAsO (Fig.3-2b), LiFeAs (Fig.3-2d) and FeSe (Fig.3-2f) Fermi surfaces as obtained from experiments. The extent of simultaneous pocket shrinkage in Ch-system, compared to its DFT predictions, is more drastic than the Pn-systems.

Figure 3-2. Schematic DFT-derived Fermi surface of (a) LaFeAsO, (c) LiFeAs and (e) FeSe in their tetragonal phase in their 2-Fe Brillouin zone, and ARPES intensity map of the tetragonal Fermi surface of (b) LaFeAsO [52] and (d) LiFeAs [34] and (f) FeSe [57, 72]. Maximum pocket shrinkage is observed in FeSe compared to its DFT predictions. ARPES figures adapted from Ref.[34, 52, 72]

This chapter is organized as follows: first, we introduce in Section 3.2 the standard mechanism for Fermi pocket shrinkage in FeSC, as originally discussed by Ortenzi et al. [59]. We point out that the interplay between momentum-transfer and finite-energy scattering processes is somewhat subtle and needs to be considered carefully in the context of realistic models driven by spin-fluctuation interaction. In Section 3.3, we introduce the

37 multiorbital Hubbard-Hund Hamiltonian with which realistic calculations of the self-energy and renormalized band structure will be performed via RPA scheme. In Section 3.4.1, we present RPA results for electronic structure renormalization for LaFeAsO, LiFeAs and FeSe, which agrees very well with TPSC and FLEX evaluations. We show that in the case of LiFeAs and LaFeAsO (Pn), all the methods agree semi-quantitatively, giving rise to non-local band renormalizations and pocket shrinkage similar to ARPES findings. On the other hand for FeSe (Ch), while results of the three methods are still similar when starting with the same ab-initio model, they remain in dramatic disagreement with the experimental band structure. We show in Section 3.4.2 that one of the possible physical mechanisms that could explain drastic band renormalization for FeSe is significant nearest-neighbor Coulomb interactions. In Section 3.4.3, we explicitly benchmark the results for the static self-energy obtained from RPA, TPSC, and FLEX. We present in Section 3.4.4, our results for the quasiparticle lifetimes in LiFeAs at different points on its renormalized FS and compare with ARPES findings. Finally, Section 3.5 summarizes our conclusions. 3.2 Background: toy model illustrating Fermi-surface shrinkage

The phenomenon of Fermi pocket shrinkage, illustrated in Fig. 3-1, has been discussed in the context of a simple two-band model for FeSC by Ortenzi et al. in Ref. [59]. They analyzed the changes of the low-energy effective model induced by the coupling to collective modes, described within an Eliashberg framework via a self-energy function for each band. They showed that the multiband character of the electronic structure and the strong particle-hole asymmetry of the bands induce self-energy effects that lead to the shrinking of the FS. The FS of the interacting system is given by the pole of the renormalized Green’s

0 −1 −1 function G(k, ωm) = (G (k, ωm) − Σ(k, ωm)) , as obtained from Dyson’s equation,

0 where G (k, ωm) is the non-interacting Green’s function and Σ(k, ωm) the self-energy. Here

ωm = (2m + 1)πT are the fermionic Matsubara frequencies for a given temperature T . The renormalized dispersion arises from modification of the DFT dispersion caused bythe

′ real part of the static self-energy Σ (k, ω0) as T → 0. A momentum-dependent self-energy

38 ′ that has a negative (positive) value of Σ (k, ω0) around the hole (electron) pocket can result in simultaneous pocket shrinkage. Hence, for our purpose, we can investigate the sign of

′ Σ (k, ω0). Revisiting the shrinking mechanism discussed by Ortenzi et al., it is instructive to consider the result of lowest order perturbation theory for a two-band model as shown in Fig. 3-3, with a Γ = (0, 0) centered hole pocket and an M˜ = (π, 0) centered electron pocket. We assume purely two-dimensional parabolic bands with constant density of states, such that

| u| − | l | −1 l/u Nα = ( Eα Eα ) , where Eα are the extrema of α-band as shown in Fig. 3-3. Choice of l/u a constant density of states dictates the upper/lower energy cutoffs Eα of the α band. The one-loop fermionic self-energy mediated via an interaction Vαβ(q, Ωm) is generally given by

T X 0 − − Σα(k, ωm) = Vαβ(q, Ωm)Gβ(k q, ωm Ωm), (3–1) Nq q,Ωm where Ωm = 2mπT are the bosonic Matsubara frequencies and α, β are band indices. By approximating the full interaction with a momentum-independent repulsive interband interaction U(1 − δαβ)D(Ωm), mediated via a bosonic propagator D(Ωm) with a single

Einstein mode at ΩE, and solving Eq. (3–3) in the T → 0 limit, one arrives at the analytical result discussed in Ref. [59]:

Eu Σ′ (ω ) = −Ω UN ln β . (3–2) α 0 E β l Eβ

Notice that with electron and hole bands with the same masses, and in the particle-hole

| u| | l | ′ symmetric limit within each band where Eβ = Eβ , Eq. (3–2) leads to Σα(ω0) = 0. ′ However, for particle-hole asymmetric bands, Σα(ω0) is non-zero. It has negative value for a hole-like band since |Eu | > |El | and positive value for an electron band as |El | > |Eu|. This M˜ M˜ Γ Γ leads to simultaneous pocket shrinkage. We now illustrate how the pocket shrinkage mechanism proposed by Ortenzi et al. works within a slightly more complex model with a momentum-dependent interaction. Since the Fermi pockets in FeSC are small, zero-energy scattering processes at small momentum transfer can be associated with intraband interactions, and those at momenta close to the

39 antiferromagnetic instability wavevector (±π, 0) with interband interactions. Here we discuss which region in momentum space is actually associated with interband processes that lead to pocket shrinkage.

Figure 3-3. Schematic plot of Γ-centered hole band and M-centered˜ electron band in 1-Fe zone of a generic FeSC.

As shown in Fig. 3-3, in the case of finite-energy interband scattering of a quasiparticle from the Γ point with momentum transfer of π − qrad, one can end up at an energy level of ˜ − ˜ EM˜ (M qrad). Similarly, scattering from M can end at an energy level of EΓ(Γ + qrad). This means that for a given interaction with momentum transfer π − qrad, we need to perform the numerical integration over q in the sum in Eq. (3–1) using a patch of radius qrad centered around (±π, 0). By changing the area of integration we are effectively changing the range of energy considered, which allows us to rewrite the analytical result of Eq. (3–2) as a function of qrad,

E (M˜ − q ) (Σ′ ) = −Ω UN ln M˜ rad , (3–3) Γ an E M˜ El M˜ Eu ′ − Γ (ΣM˜ )an = ΩEUNΓln , (3–4) EΓ(Γ + qrad) where the subscript “an” stands for analytical. The sign of the self-energy renormalizations now crucially depends on the region of momenta used in the integration. A shrinking of the pockets is recovered as long as |E (M˜ − q )| > |El | and |E (Γ + q )| > |Eu|, while M˜ rad M˜ Γ rad Γ

40 the opposite result is obtained if one restricts the integration over a small region of q-space centered around (±π, 0)-fluctuation. We discuss this result in more detail in Appendix D, by performing numerical evaluations of Eq. (3–3) as a function of qrad with the interaction Vαβ(q, Ωm) treated in the usual RPA approach. The main conclusion of our analysis of the toy-model is that the shrinking mechanism proposed by Ortenzi et al. produces the correct trend for most band structures encountered in the FeSC. However, when attributing Fermi pocket shrinkage to interband scattering, though the interaction is sharply peaked around antiferromagnetic wavevector (±π, 0), one needs to account for the crucial role played by all other momentum transfer q-vectors participating in finite-energy scattering processes. Approximations that incorporate only large-q processes in an attempt to describe dominant interband scattering may fail to produce the correct pocket shrinkage, even if the interaction vertex is strongly peaked in q. Notice that the inclusion of the explicit momentum dependence of the effective interaction V (q, Ωm) in the above description of Fermi pocket shrinkage would lead to further renormalization, i.e., vertex corrections connected to the self-energy by standard Ward identities1 . We neglect such vertex corrections in the present work, though their effects could be important in the analysis of transport as shown, e.g., in Ref. [76]. It is also worth noticing that using a simplified renormalization group analysis for FeSC, one can generate non-local interactions that can eventually lead to shrinkage or expansion of Fermi pockets [77], the physical essence of which is similar to our findings.

1 In quantum field theory, symmetries and the associated conservation laws imply Ward identities, which are exact relations between different types of Green functions or vertex functions [73]. The constraints imposed by Ward identities can be very useful to devise accurate approximation schemes which do not violate conservation laws. For example, the Luttinger-Ward functional Φ[G] [74, 75] is a functional of the interacting Green’s function G and yields the self-energy Σ as a functional derivative Σ = δΦ/δG, whereas within the TPSC approach, one approximates the two-particle irreducible four-point vertex Γ = δ2Φ/δG2, to be static and momentum independent [71].

41 3.3 Model and RPA scheme

In the following sections, we perform numerical calculations for the realistic band structures of various FeSC. Based on the analysis of the simplified two-band model performed in section 3.2, we expect to find a shrinkage of the FS in systems with shallow, particle-hole asymmetric bands. However, the extent of this effect can be strongly modified by the number of hole/electron bands interacting via V (q, Ωm), the relative weight of the bands controlled by its density of states and the degree of particle-hole asymmetry of each band. The realistic calculations of renormalized band structure in FeSC begin with a five-orbital tight binding Hamiltonian H0 as already introduced in the previous Chapter

2, Sec.2.2. With the same definition as before, the kinetic energy H0 includes the chemical potential µ0 and is spanned in the basis of five Fe d orbitals [dxy, dx2−y2 , dxz, dyz, d3z2−r2 ]. The local interactions are included via Hubbard-Hund part HI where the interaction parameters U, U ′, J, J ′ are given in the notation of Kuroki et al. [46]. As mentioned before in Sec.2.2, U is the intra-orbital and U ′ the inter-orbital Coulomb repulsion terms, J is the Hund’s coupling term and J ′ the pair-hopping term. We consider cases which obey spin and orbital rotational invariance through the relations U ′ = U − 2J and J = J ′. Here q and t are the orbital indices and i is the Fe-atom site. The single-particle non-interacting Green’s function is given by

0 −1 G (k, ωm) = [iωm − H0(k)] . (3–5)

The orbitally resolved non-interacting susceptibility, the charge-fluctuation and spin-fluctuation parts of the RPA susceptibility and the interaction matrices U C and U S have been defined already in Sec.2.2. We note that our calculations are based on the local density approximation (LDA) ab-initio electronic band structure which already contains important Hartree-Fock corrections. The remaining leading order spin-fluctuation contributions to the self-energy within RPA is mediated through the particle-hole interaction 3 S S S 1 C C C 1 SC 0 SC Vpqst(q, Ωm) = U χ (q, Ωm)U + U χ (q, Ωm)U − U χ (q, Ωm)U (3–6) 2 2 4 pqst

42 where U SC = U S + U C . In the paramagnetic state, where the initial and the final spins for the normal self-energy are the same, the above interaction results from summing over all the possible triplet and singlet scattering channels [14]. Using Dyson’s equation, one obtains the interacting Green’s function

−1 0 −1 G(k, ωm) = G (k, ωm) − Σ(k, ωm), (3–7) where the self-energy in orbital basis is given by

1 X X 0 − − Σps(k, ωm) = Vpqst(q, Ωm)Gqt(k q, ωm Ωm). (3–8) βNq q,Ωm qt

To ensure particle number conservation, the chemical potential is reevaluated from the interacting Green’s function (details in Appendix C). As one approaches the antiferromagnetic instability point, close to a critical value of the interaction parameter U, the effective interaction can get strong enough to violate the simple perturbative RPAscheme for evaluating the one-loop electronic self-energy. Invoking FLEX (or TPSC) can bypass this issue by providing self-consistent solution for the self-energy, for weak-to-intermediate coupling range of the interaction parameter. However, similar qualitative results are obtained between RPA and the other methods, even close to the critical point of RPA scheme [78]. The scattering lifetime of quasiparticles at a specific momentum point k with energy − ′′ ′′ ω in orbital p is given by Zp(k)Σpp(k, ω) , where Σ refers to the imaginary part of the self-energy, and the quasiparticle weight Zp(k) is given by ′′ −1 ∂Σpp(k, ωm) Zp(k) = 1 − . (3–9) ∂ω + m iωm→0

Please note that the above definition of quasiparticle weight must not be confused with that in Chapter 2, which is defined as inverse of the above quantity under the Eliashberg approach. At sufficiently low temperatures, one can use the analytic properties of the

≈ − ′′ −1 Matsubara self-energy to approximate Zp(k) [1 Σpp(k, ω0)/ω0] , where ω0 = πT . We set T = 100K = 0.0086 eV for the rest of the calculations unless mentioned otherwise. We have used standard U and J values as used in the literature employing the RPA approach [79]. We

43 use the Padé approximation for numerical analytic continuation. Further numerical details are provided in Appendix C.

Figure 3-4. Spectral function A(k, ω) for interacting bands evaluated along high-symmetry path Γ − X − M − Γ for (a) LaFeAsO at U = 0.8 eV, (b) LiFeAs at U = 0.8 eV, and (c) FeSe at U = 0.89 eV in the 2-Fe BZ. The value of Hund’s coupling term was fixed at J = U/8. The black lines and the bright yellow curves denote the DFT and the renormalized bands, respectively. Contour plots of the DFT Fermi surface for (d) LaFeAsO, (e) LiFeAs, and (f) FeSe are represented by its corresponding dominant orbital character as indicated in the color legend on top. Spectral map of the renormalized Fermi surface for (g) LaFeAsO, (h) LiFeAs, and (i) FeSe. The momentum points marked as 1 − 5 on LiFeAs Fermi surface in (h) are used in evaluation of scattering lifetimes of quasiparticles later.

44 3.4 Results

3.4.1 Electronic structure renormalization

To present our results in a manner that is easy to compare to experimental data, we folded our spectral function from 1-Fe to 2-Fe BZ. As shown earlier in Fig. 3-1, the high-symmetry points in the 2-Fe BZ are Γ = (0, 0), X = (π, 0) and M = (π, π). In Fig. 3-4, P − 1 ′′ we show the spectral function A(k, ω) = π p Gpp(k, ω), obtained from the imaginary part of the retarded interacting Green’s function within the RPA scheme for LaFeAsO, LiFeAs and FeSe in their tetragonal phase as representatives of the pnictide and chalcogenide classes. For our DFT electronic dispersion, we have chosen the tight-binding parameters derived from DFT, relevant to LaFeAsO as in Ref. [79], LiFeAs as in Ref. [35], and FeSe as in Ref. [7]. Due to the quasi-2D nature of FeSC bands and to compare with available experimental data, our evaluations are restricted to the 2D plane at kz = 0. The obtained quasiparticle bands closely trace the position of the peak in the spectral density, as denoted by the bright yellow curves in the spectral function [Fig. 3-4a - LaFeAsO, 3-4b - LiFeAs, 3-4c - FeSe]. For easier comparison to the DFT bands, we have superimposed the DFT-derived band structure along high-symmetry path Γ − X − M − Γ with black lines in these figures. The corresponding DFT FSs with their respective orbital content are shown in Fig. 3-4d for LaFeAsO, 3-4e for LiFeAs, and 3-4f for FeSe. The RPA spectral maps for the renormalized FSs are provided in Fig. 3-4g for LaFeAsO, 3-4h for LiFeAs, and 3-4i for FeSe. All results shown in Fig. 3-4 evaluated within RPA are in agreement with findings from TPSC and FLEX (calculated and verified by S. Bhattacharyya and collaborators K. Zantout and K. Björnson). First of all, we notice the following universal trends across all the families of FeSC: the quasiparticle bands are strongly renormalized, resulting in noticeable changes in the FS, i.e., shrinkage around Γ and M points for both hole and electron pockets compared to their DFT counterparts, and momentum-dependent renormalization of the velocity of the bands under interaction. Upon comparison to experiment, we find that our renormalized FS for LaFeAsO [Fig. 3-4g] are similar to the ones observed in ARPES [52]. One should note that only a qualitative comparison can be made between ARPES intensity features

45 and our numerically evaluated spectral intensities. A quantitative comparison requires a more accurate knowledge of scattering matrix elements involved in the ARPES experiments

[80]. Out of the three Γ-centered hole pockets, the dxy pocket undergoes the most shrinkage while for the M-centered electron pockets, the dxz/yz parts shrink more than the dxy part. For LiFeAs, the renormalized FS in Fig. 3-4h agrees with ARPES findings from Ref.31 [ ,

33, 81, 82]. The Γ-centered dxy hole pocket does not show noticeable shrinkage but the inner-most dxz/yz hole pockets gets pushed down and show only residual spectral weight from the bands grazing the Fermi level. The renormalized FS for FeSe [Fig. 3-4i] also shows simultaneous shrinkage of both hole and electron pockets, however, it disagrees with ARPES data [11, 56, 57, 68, 72], where drastic shrinkage of all the Fermi pockets is reported, including vanishing of the dxy hole pocket from the Γ point. The most striking aspect of the band renormalization, i.e, the complete disappearance of the largest (dxy) hole pocket found in DFT calculations based on reported X-ray crystal structures, has not been addressed in the literature. As explained in Section 3.2 above, a momentum- and orbital-dependent self-energy that has a negative (positive) value of Σ′(k, ω = 0) around the hole (electron) pocket is a consequence of itinerant spin-fluctuations and can result in pocket shrinkage to various degrees in different orbitals. It should be noted that the above calculations are basedon DFT-derived tight-binding models without spin orbit coupling. Therefore, certain low-energy splitting have been neglected. In particular, in the case of LiFeAs, it is known that only a single band crosses the Fermi level at the Z point, and neither of the hole bands at the Γ point[36, 83]. Inclusion of spin-orbit coupling can account for this splitting and will allow for agreement with ARPES data only at somewhat higher values of the Coulomb interaction. 3.4.2 Effects of nearest-neighbor Coulomb repulsion

In this section we explore another physical mechanism that may be relevant for a quantitative understanding of band renormalizations. Longer-range Coulomb interactions have been previously shown to influence the pairing structure in FeSC84 [ ], and proposed as a candidate for the origin of d-wave nematic order in FeSe [85, 86]. These interactions can also

46 result in a drastically renormalized band structure [85–87]. In order to explore this effect, for the electronic structures of LiFeAs and FeSe considered here, we introduce the following nearest-neighbor (NN) Coulomb interaction X X X HV = V nitσnjqσ′ , (3–10) ⟨i,j⟩ t 0, and ⟨i, j⟩ indicates the summation over NN sites. To treat the effects of the NN Coulomb repulsion, we perform a Hartree-Fock mean-field decoupling and introduce the bond-order fields X tq −2V − ′ − ′ ⟨ † ⟩ Nkσ = [cos(kx kx) + cos(ky ky)] ck′qσck′tσ (3–11) Nk k′ Note that here we investigate only the effects of NN coupling on the bandstructure. It remains to be determined what are the effects of further next-nearest-neighbor interactions. In addition, we ignore Hartee-Fock terms arising from onsite interactions since these only introduce orbital-dependent onsite energy shifts that, we find for interaction parameters corresponding to Fig. 3-4, lead to small quantitative changes to the results discussed below. A quantitative exploration of the effects of a full set of neighboring Coulomb interaction parameters is left to a future study. By closely following the symmetry arguments in Refs.

[85–87], we decompose the bond-order fields into C4-symmetry-preserving and -breaking

tq tq terms, denoted Nkσ,br and Nkσ,sb, respectively. The additional subscripts of the fields refer to band renormalization (br) and symmetry-breaking (sb). For our current study, we assume no spontaneous nematic transition to occur, and therefore neglect the C4-symmetry-breaking

tq terms. This leaves us with the task of calculating Nkσ,br self-consistently, while keeping the electron density n fixed. As conjectured in Ref.[88], the cRPA values for U, J in FeSe are higher than other Pn compounds, say, LiFeAs. However, we will use the same specific interaction potential strength, strong enough to show substantial reconstruction of the Fermi surface in both Pn and Ch systems. The resulting FSs for LiFeAs and FeSe are depicted in Fig. 3-5 for different interaction potential strengths V as used in the standard literature [87]. It is evident

47 Figure 3-5. The effects of NN Coulomb interactions on the Fermi surfaces of LiFeAs and FeSe. (a,c,e) Fermi surface for LiFeAs with the interaction potential strength V = 0, 0.30, 0.46 eV, respectively. (b,d,f) Same as in (a,c,e), but for FeSe. All figures were obtained using T = 0.01 eV and a filling of n = 6. Calculations done by: D. Steffensen. that the FSs are significantly modified even for rather small V , and that the two systems are affected quite differently by the NN Coulomb repulsion. These observations areseen specifically for V = 0.30 eV [Fig. 3-5(c) and 3-5(e)], where the dxz/yz-dominated Γ-pockets of the LiFeAs band get completely removed from the Fermi level, whereas for the FeSe band in Fig. 3-5(d) and 3-5(f), it is the dxy-dominated Γ-pocket that gets removed for V = 0.46 eV while the dxz/yz-dominated pockets prevail. The origin of this difference between the two materials can be traced to the initial band structures [see Fig. 3-4b and 3-4c]. Specifically, the dxy-dominated DFT hole band at Γ reaches significantly larger energies for LiFeAs

48 (≈ 250 meV) than for FeSe (≈ 100 meV), thus simply making it more robust towards renormalization effects. 3.4.3 Comparison of RPA, TPSC and FLEX results

′ Figure 3-6. Real part of the orbitally-resolved static self-energy Σ (k, ω0) computed along high-symmetry path Γ − M˜ − Γ˜ − Γ in the 1-Fe BZ of LiFeAs via (a) RPA, (b) TPSC, and (c) FLEX evaluation scheme. The orbital character and its corresponding color choice is same as before as marked in the color legend at the bottom of Fig (b). The solid black line demarcates the value of the new chemical potential µ. The RPA and FLEX evaluations were carried out at U = 0.8 eV and J = U/8. The TPSC parameters used are the same as in Ref. [35]. RPA, TPSC and FLEX calculations done by S. Bhattacharyya, K. Zantout and K. Björnson, respectively.

In order to provide a quantitative basis of comparison of the self-energy obtained via different methodologies mentioned in this chapter, we focus on the real partofthe ′ ˜ orbitally-resolved static self-energy Σ (k, ω0) computed along high-symmetry path Γ − M − ˜ ˜ Γ − Γ in the 1-Fe BZ of LiFeAs. Γ hosts the small dxz/yz-dominated hole pockets. Γ refers to the (π, π) point in the 1-Fe BZ which gets folded back to the Γ point in the 2-Fe BZ ˜ representation, and hosts the large dxy-dominated hole pocket. M hosts the dyz/xy-dominated electron pocket which gets folded and represented by the M-point in the 2-Fe BZ. In Fig. 3-6, ′ ˜ ˜ we plot Σ (k, ω0) along Γ − M − Γ − Γ obtained via RPA (3-6a), TPSC (3-6b) and FLEX evaluations (3-6c). We have demarcated the new chemical potential µ by the solid black line to illustrate

′ the relative sign contrast of Σ (k, ω0) between different momentum points. First, we notice

′ the qualitative similarities in Σ (k, ω0) obtained via the three methods. The dxz/yz component at the Γ point lies below the new chemical potential µ resulting in a relative negative value

49 ˜ of the self-energy, whereas the dyz/xy component at the M point lying above µ, results in a relative positive value. As discussed before in Section 3.2, due to the above behavior of the self-energy, the resulting renormalized Fermi surface shows pocket shrinkage for both the inner dxz/yz hole and dyz/xy electron pockets. Note that the outer dxy hole pocket centered ˜ ′ around Γ doesn’t show much shrinkage, as is also evident from its Σ (k, ω0) value which is grazing the level of the new chemical potential. It is worth noting that the self-energy components displayed in Fig. 3-6 for TPSC are larger than the magnitudes shown for RPA and FLEX. This is due to the fact that TPSC is able to access higher values of the Coulomb interaction U/J parameters, comparable to those obtained from realistic constrained RPA calculations [35]. In contrast, the interaction parameters used in the weak-coupling RPA/FLEX approach are restricted to smaller magnitudes away from the antiferromagnetic instability point. This apparent discrepancy is understood as an effective downward renormalization of the U that leads to the RPA susceptibility χC/S evaluation [89]. Despite the large quantitative difference between the self-energy values obtained via TPSC and RPA, we still have qualitatively similar Fermi level features. This is due to the fact that some of the self-energy shifts are undone by the shift of the new chemical potential, which is also much larger in TPSC than in RPA. In Appendix E, we have presented a detailed comparison of quasiparticle weights obtained via RPA and TPSC. 3.4.4 Scattering lifetime of Quasiparticles

In this section, we will address ARPES findings for momentum-dependent scattering lifetime of quasiparticles in LiFeAs for momentum points 1 − 5 along Γ − M direction as marked in Fig. 3-4h. This particular high-symmetry direction includes points on the Fermi surface with orbital weight arising purely from a single specific orbital. This makes it easier to link the behavior at any specific Fermi momentum point to its corresponding orbital. Within Fermi liquid theory, the imaginary part of the retarded self-energy in 2D obeys the Fermi liquid relation within logarithmic accuracy with respect to the binding energy ω near

50 the Fermi level at temperature T [90]:    2 2 C1ω log |ω/Λ| if ω/T >> 1 − ′′ Σ (kF, ω; T ) =  (3–12)  2 2 C2T log |T/Λ| if ω/T << 1. where Λ ∼ EF is some upper energy cutoff, C1 and C2 are proportionality constants. To satisfy the first-Matsubara rule [90], the above two asymptotic form of the self-energy equation can be combined into one as: ω ω2 + T 2 −Σ′′(ω, T ) = f (ω2 + π2T 2)log (3–13) T Λ2 ω C /π2 − C where, f = C + 2 1 T 1 1 + ω/T

This self-energy vanishes at ω = iπT satisfying the first-Matsubara rule. One can recover the asymptotic form of the self-energy in the limits, ω/T << 1 and ω/T << 1, to arrive at the Eqs. 3–12 from 3–13. The scattering lifetime of quasiparticles nearby the Fermi level is given

′′ by −Z(kF)Σ (kF, ω) as discussed earlier in Section 3.3. Previous TPSC calculations [35] have found Fermi-liquid and non-Fermi liquid-like behavior of quasiparticles over intermediate T -ranges. In ARPES measurements, the quasiparticle lifetimes are typically compared to the dominant quadratic-ω Fermi liquid term in Eq.3–13 as ℏ/τ = γ (ℏω)2 + (πT )2 (3–14)

In Ref. [81], Brouet et al. claimed that the Fermi liquid behavior of scattering lifetime depends on the orbital character rather than its position on the Fermi surface (hole or electron pocket). They found that momentum points [3 and 4] dominated by dxy orbital on both hole and electron pockets show strong Fermi liquid behavior with a large γ, unlike small γ value for dxz/yz orbital [points 2 and 5]. Brouet et al. explained that for a simple metal, the renormalized bandwidth ZW would set a low-energy scale for the coherent part of the spectrum Ecoh to observe Fermi liquid behavior, i.e., Ecoh ≪ ZW and γ ∼ 1/ZW [91].

In case of ZW being smaller for dxz/yz than for dxy, it may not be possible to establish

51 its Fermi liquid regime within current experimental precision. In an independent ARPES study Ref. [82], Fink et al. claimed that the scattering lifetime shows strong momentum dependence which can be attributed to nesting conditions and availability of phase-space for interband scattering. They found linear-ω behavior for dyz/xy orbitals at momentum points 1,

3 (hole) and 5 (electron) while quadratic behavior for dxz/xy orbitals at points 2 (hole) and 4 (electron). The discrepancy between these two experimental works remains an open question. We performed two sets of RPA calculations for Coulomb interaction parameters U =

0.85Ucrit and U = 0.95Ucrit, with J = U/8 and Ucrit = 1.19 eV being the critical value causing antiferromagnetic instability at T = 100K. With the larger U = 0.95Ucrit, the shrinkage of the FS pushes the inner hole pockets below the Fermi level by almost 15 meV. As mentioned earlier in Section 3.4.1, since the observed spin-orbit splitting at Γ in LiFeAs is of the same order of magnitude, we expect that inclusion of spin-orbit coupling will give rise to a single band grazing the Fermi level, as seen in ARPES data. In Fig.3-7, we show the scattering lifetime as a function of frequency for the above two values of U. In panel b, we do not show data for momentum points 1 and 2 since in our calculation without spin-orbit coupling, both the inner-hole pockets are pushed below the Fermi level. We find that all our numerical data obtained at different momentum points can be fit to the Fermi liquid behavior as inEq.3–13, with fitting coefficients C1,C2, Λ.

We find that the dxy orbitals become more correlated than dxz/yz as U is increased, i.e.

Z(dxy) decreases by a factor of 2.7 compared to 1.7 for Z(dyz). The relative magnitudes for

′′ −ZΣ (0) changes upon increasing U, specifically the dyz electron pocket (point 5) acquires

′′ a larger −ZΣ (0) than dxy hole and electron pockets (points 3 and 4). This hierarchy of the −ZΣ′′(0) values agree with the ARPES findings in Ref.81 [ ]. Also, for larger U we find that

2 the fitting coefficient C1 for ω Fermi-liquid behavior is larger for momentum points 3 and 4

(dxy orbital content) than for 5 (dyz content), which is also consistent with Ref. [81] claims. Independent of the values of U, the analysis of the RPA scattering lifetime yields Fermi liquid behavior compatible with some experimental data[81]. The analysis close to criticality also shows enhancement of orbital differentiation in qualitative agreement with this data.

52 Figure 3-7. Scattering lifetime of quasiparticles as a function of binding energy ω evaluated within RPA for (a) U = 0.85Ucrit, (b) U = 0.95Ucrit, J = U/8 and T = 100K, at momentum points 1 − 5 as denoted in Fig. 3-4b for LiFeAs. Solid lines are fit to Fermi liquid behavior as in Eq. 3–13. Colors refer to particular momentum points as marked in the plots, and are labelled in descending order of their −ZΣ′′(0) value from top to bottom.

3.5 Summary

We have performed 2D calculations for the three prototypical FeSC: LaFeAsO, LiFeAs, and FeSe and evaluated their momentum-dependent dynamic self-energy and the corresponding electronic structure renormalization effects. We showed that the correct sign of the self-energy that causes Fermi surface shrinkage is determined by the repulsive interband finite-energy scattering processes that are sensitively dependent on the upper/lower edge of the band structure. Our results point towards a universal trend across the families of FeSC, i.e., Fermi surface shrinkage of both hole and electron pockets arising due to non-local, orbitally selective self-energy renormalizations within both RPA, TPSC and FLEX schemes, which treat itinerant spin-fluctuations arising from local Coulomb interactions. Theories advocating for local correlation effects, like DFT+DMFT19 [ ], cannot explain the phenomena of simultaneous shrinkage of both hole and electron pockets. We found that the renormalized Fermi surfaces for Fe-Pn-systems LaFeAsO and LiFeAs agree well with ARPES findings,

53 whereas the Fe-Ch-system FeSe does not yield the drastic Fermi surface shrinkage observed in experiments. We proposed that the band structure renormalization of the Fe-Ch-system is significantly affected by the nearest-neighbor Coulomb interaction. In this context, our calculations show desirable results for FeSe. We showed that the momentum-dependent scattering lifetime of quasiparticles in LiFeAs exhibits Fermi liquid behavior. We have also included detailed discussions about ARPES findings for the same. We conclude that understanding the observed electronic structure of the moderately correlated Fe-Pn-systems requires the important ingredient of including non-local, orbitally resolved self-energy effects within the framework of existing spin-fluctuation theories. However, the more strongly correlated Fe-Ch-systems might be dictated by other underlying physical mechanisms, for example, non-local Coulomb interactions. Further investigation devoted to a detailed study of the band renormalization in bulk FeSe is desirable in this context.

54 CHAPTER 4 TOPOLOGICAL ULTRANODAL PAIR STATES IN IRON-BASED SUPERCONDUCTORS ∗ 4.1 Introduction

The standard s± paradigm for superconductivity in iron-based superconductors is based on the simple notion that repulsive interband interactions would force a sign change of the order parameter between hole and electron pockets, separated by a near-nesting wave vector at which the magnetic susceptibility is peaked [26, 28]. This approach proved rather successful for the iron pnictide superconductors with doping near 6 electrons/Fe, but was questioned in the case of “end-point” iron based systems where either electron or hole pockets disappear. The latter case includes several FeSe intercalates, including the FeSe/SrTiO3 monolayer system which has the highest Tc of the iron-based class. For these cases, where the standard s± scenario apparently breaks down, a number of exotic alternatives for pairing have been proposed, some of which involve interorbital pair states. Normally such states are energetically disfavored, since they generically force interband pairing of states k and −k, which must occur off the Fermi level and therefore lose the Cooper logarithm that drives robust pairing. However recent works [94, 95] have shown that infinitesimal spin-orbit coupling can induce a Cooper log, such that novel interorbital pair states may be expected to occur in special circumstances. Spin-orbit coupling has been proposed to provide the principal source of hybridization between two electron pockets in FeSe monolayers and intercalates, interpolating between d-wave and so-called bonding-antibonding s± states [6, 96, 97]. At the same time, there has been an ongoing discussion about the possibility of time-reversal symmetry breaking (TRSB) states in the Fe-based materials. These can mix two degenerate representations like s and d at a degeneracy point, leading to a relative

∗ This chapter is a slightly modified version of92 [ ] and [93] published in Nature Communi- cations and Phys. Rev. B, respectively, and has been reproduced here with permission of the copyright holders.

55 π/2 phase shift as in, e.g. s + id pairing [98, 99], or occur in more complicated fashion with arbitrary phase shifts among bands in a system with at least 3 bands [100, 101]. Recently, it has been claimed that muSR experiments confirm a predicted TRSB state in highly hole-doped BaFe2As2[102]. The detection of TRSB in bulk FeSe and the sulfur doped compound was reported by another group[103]. Time reversal symmetry can also be broken in the presence of a pseudo-magnetic field arising from interband pairing104 [ , 105]. It has been pointed out that such terms can be generated by spin-orbit coupling in iron based systems [94, 95].

Figure 4-1. The electronic component of the specific heat divided by temperature,CV /T , vs. T in FeSe1xSx for (A) x = 0, (B) x = 0.08, (C) x = 0.13, and (D) x = 0.20. Dotted horizontal lines indicate the normal state values of CV /T . Picture adapted from Ref.[106].

Since both spin-orbit effects and time reversal symmetry breaking can individually lead to deviations from the s± paradigm, it is interesting to investigate their interplay, particularly in the context of unusual phenomena observed in the Fe chalcogenides. Recently, a rather unexpected and unusual form of the T,H-dependent specific heat was observed just after the nematic state disappeared with S doping in the Fe(Se,S) system[106]. The authors of this work were convinced primarily by the change in the magnetic field dependence that the gap structure was changing abruptly at the nematic transition; however, another unusual aspect of the data was the large (O(N state)) value of the apparent residual T → 0 Sommerfeld coefficient (see Fig.4-1C and D). In a clean superconductor, even with line nodes,

γs = C/T → 0 as T → 0, and while disorder can lead to a nonzero value, STM measurements

56 on these samples suggest that they are very clean, inconsistent with γs/γN of O(1)[107]. Both specific heat and STM[107] analyses suggest that the form of the gap function or at least the density of low-energy quasiparticle excitations is varying rapidly near the nematic critical point. We also present the STM data [107] in Fig.4-2 obtained from a more closely spaced sequence of S-dopings in FeSe1xSx. This analysis was able to establish that a transition took place between x = 0.13 and x = 0.17, corresponding roughly to the nematic transition. At this transition, the coherence peak position in the conductance spectrum red-shifted weakly, while the zero bias conductance became nonzero, growing with increased S-doping.

Figure 4-2. Averaged tunneling spectra of FeSe1xSx showing the SC gaps. Each curve corresponding to a given x is shifted vertically by 5 nS for clarity. Picture adapted from Ref.[107].

In this chapter, we propose an explanation for the temperature dependence of the specific heat and the form of the STM conductance spectrum1 inFeSe −xSx, based upon a prescribed evolution of a superconducting order parameter in the presence of spin orbit coupling that leads to a state with Bogoliubov Fermi surface (BFS), topologically protected patches of finite area where zero-energy excitations exist in the superconducting104 state[ ]. The only absolute restriction on the forms of pair wavefunction components arise from the Pauli principle. In a one band system, only even parity spin singlet pairing or odd parity spin triplet pairing are allowed in the absence of spin-orbit coupling. In a multiband system, on the other hand, the additional degrees of freedom allow for novel pairing structures, including odd parity-spin singlet and even parity-spin triplet (both band singlet) components.

57 Note the pairing problem is often discussed in an orbital instead of band basis. In order to demonstrate the emergence of a BFS, we consider a three-pocket model appropriate for iron-based superconductors with a charge and parity (CP ) symmetric pairing structure. The latter contains interband, TRS and TRSB, spin-triplet-band singlet components. In addition, the pairing structure also contains an intra-band spin singlet-band triplet component with, generally, both isotropic and anisotropic momentum dependences. As a function of the ratio of inter-and intra-orbital pairing, our model exhibits a topological phase transition from a fully gapped s-wave superconductor (Z2 trivial) to a gapless system with extended BFSs

(Z2 non-trivial). Such a transition is absent when time reversal symmetry is preserved; it is broken here because the interband pairing gives rise to a pseudo-magnetic field. The topologically non-trivial phase discussed here retains characteristics of both the superconductor and the normal Fermi liquid, e.g., the ratio CV /T has the usual discontinuity at Tc within mean field theory, but saturates to a non-zero constant as T → 0 due to the finite density of zero-energy excitations on the104 BFS[ ]. We demonstrate this dichotomy for the specific case of FeSe1−xSx by considering a minimal model for the gaps on the two hole and two electron pockets, qualitatively consistent with the gap structures proposed on the basis of thermodynamic, transport[106] and STM measurements[107]. Within this phenomenology, doping is assumed to tune the anisotropy of the electron pocket gap, driving the topological transition. This simple model can explain essentially all qualitative features observed in FeSe1−xSx across the nematic transition. 4.2 Model and Physical observables

In the following, we discuss the role of charge-conjugation (C), parity (P ) and time reversal (T ) symmetries which can all be defined on the momentum space Hamiltonian as

−CH(−k)C−1 = H(k) (4–1)

PH(−k)P −1 = H(k) (4–2)

TH(−k)T −1 = H(k), (4–3)

58 respectively. While P is a unitary operator represented by a matrix UP , T is anti-unitary and can be denoted as T = UT K where UT is a unitary matrix and K complex conjugation

(similar notation for C with unitary matrix UC ). Before we narrow down our discussions to specific forms of the gap, we begin by highlighting our result for a generic Hamiltonian ofthe form X 1 † Hˆ = Hˆ + Hˆ = Ψ (H (k) + H (k)) Ψ , 0 ∆ 2 k 0 ∆ k k † † † † where the Nambu operator is defined in the basis Ψk = cki↑, cki↓, c−ki↑, c−ki↓ and ckiσ is the electron creation operator in pocket i with spin σ. H0(k) and H∆(k) are the normal and pairing terms in the Hamiltonian written in momentum space. In the basis chosen above where the orbitals/bands transform trivially under time reversal, the corresponding unitary matrices for CPT symmetries would take the form

UP = π0 ⊗ τ0 ⊗ σ0,

UT = π0 ⊗ τ0 ⊗ iσy,

UC = πx ⊗ τ0 ⊗ σ0 (4–4)

for a two-pocket model. Here πi, τi, σi are Pauli matrices in particle-hole, band (pocket) and spin space, respectively. P ˆ † We choose the normal state part of the Hamiltonian as H0 = kiσ ϵi(k)ckiσckiσ written in the band basis. For the superconducting part, we choose a gap Hamiltonian having both spin singlet and triplet along with TRS and TRSB components. These terms are written as (i and j are band/pocket indices and i ≠ j) X 1 † † † † Hˆ = ∆ c c + c c + h.c. − (i ↔ j) ∆ 2 0 ki↑ −kj↑ ki↓ −kj↓ i

59 Here ∆0 and ∆i(k) are the TRS components of the pairing and δ is the degree of TRS breaking; ∆i(k)(∆0) appears as an intraband (interband) spin-singlet (spin-triplet) term. It can be verified that the pairing Hamiltonian above, along with the band dispersion, is CP symmetric. Therefore, following the arguments of Agterberg et al. [104], one can similarity transform the Hamiltonian to a basis where it is completely anti-symmetric; hence, the Pfaffian of such a system is well defined (the Pfaffian of any skew-symmetric matrixisdefined as the square root of its determinant). ˆ ˆ Substituting the pairing structure H∆ in the total Hamiltonian H and evaluating the determinant, one finds that the Pfaffian is givenby 4 2 2 2 2 − 2 2 − Pf(∆0, ∆i, δ, ϵi) =∆0 + ∆1(k)∆2(k) + δ + ϵ1(k)ϵ2(k) 2∆0 δ ϵ1(k)ϵ2(k) 2 2 2 2 2 − 2 + ∆1(k) ϵ2(k) + ∆2(k) ϵ1(k) + 2(δ ∆0)∆1(k)∆2(k) (4–6)

We next determine the condition for the existence of a BFS by checking for a change in sign of Pfaffian. The function Pf(∆0, ∆i, δ, ϵi) acquires arbitrarily large positive values for arbitrarily large (in magnitude) dispersions |ϵi(k)|. To determine if the Pfaffian turns negative, we minimize Pf(∆0, ∆i, δ, ϵi) with respect to ϵi (ϵ1(k) ≠ ϵ2(k)). For a non-zero δ and ∆0, the minimum value depends on the relative sign of ∆1(k) and ∆2(k) and is given

   2 | || | − 2 δ ( ∆1(k) ∆2(k) ∆0) if ∆1(k)∆2(k) > 0 Min[Pf] = (4–7)   2 | || | − 2 ∆0 ( ∆1(k) ∆2(k) δ ) if ∆1(k)∆2(k) < 0.

Given that the Pfaffian is large and positive for momenta k corresponding to energies far from the Fermi level, it changes sign only if the minimum is negative. Looking at Eq. (4–7) for the case when the order parameters on both the bands have the same sign

(∆1(k)∆2(k) > 0), this can be achieved if the following two conditions are fulfilled. First, there must be a nonzero time-reversal symmetry breaking component, |δ| > 0. Second, the

2 term in parentheses has to become negative, leading to the condition |∆0| > |∆1(k)||∆2(k)|, i.e. the magnitude square of the inter-band pairing exceeds that of the product of the

60 intra-band pairings on the two pockets. On the other hand, when the order parameters on the two bands have opposite signs as is typical in FeSC, the roles of δ and ∆0 are switched, as can be deduced from Eq. (4–6) for the Pfaffian. In the numerical calculations to follow, we have set δ = ∆0 so that the results are independent of the relative sign of the two gaps. With the above conditions satisfied, a BFS emerges from a fully gapped superconducting phase giving rise to a topological phase transition at a critical combination of the order parameters.

At low temperatures far away from Tc, the low energy excitations close to the BFS resemble that of a normal metal yielding a finite residual specific heat. Hence, the system represents a unique example where one can have non-zero superconducting order parameter coexisting with a fully gapless Fermi surface. In contrast to the example of gapless superconductivity in disordered superconductors[108], this phase can occur in a completely clean system, and the BFS does not coincide in momentum space with the normal metal Fermi surface. Such a state is characterized by a conventional specific heat jump at the superconducting critical temperature, but a non-zero Fermi level density of states and Sommerfeld coefficient CV /T as T → 0. We therefore adopt the name ultranodal superconducting state to indicate that the phase space for Bogoliubov quasiparticle excitations is larger than in either the point or line nodal case familiar from studies of unconventional superconductivity. 4.2.1 Case of FeSe(S)

As a concrete example to highlight the physics discussed above, we consider a simplified three-pocket model (i = X,Y, Γ) in two dimensions to capture the essential electronic structure of iron-based superconductors, and in particular the FeSe1−xSx system that has been noted to display anomalous thermodynamics for doping beyond the nematic transition[106]. We set the band dispersions to be quadratic, centered around Γ and X/Y points; the quantities that influence the existence and location of the BFSs are thegap functions. Choosing an intra-pocket pairing ansatz of the form ∆j(k) = ∆ja(k) + ∆j, where ∆ja(k) (∆j) is the anisotropic (isotropic) component on pocket j = X,Y, Γ, we decrease with doping the isotropic components on each pocket for a fixed inter-pocket pairing ∆0 and anisotropic intra-band component ∆ja, as sketched roughly in Fig. 4-3.

61 Figure 4-3. Schematic plot of the phase diagram and intra-pocket order parameters used on the electron and hole bands in different phases across the transition. Light blue and purple colors refer to assumed opposite signs of order parameters on hole and electron pockets, which are, however, qualitatively irrelevant for the conclusions discussed here. Dashed Fermi surface in x = 0 case refers to pocket that has not been observed spectroscopically. Data from Refs. [106, 107, 109].

This is in accordance with the data and conclusions in Ref. [106] for FeSe1−xSx where, as a function of sulfur doping, all the pockets were posited to become nodal or close to nodal across the nematic quantum critical point. The anisotropic component on each pocket,

2 − 2 ∆ja(k) = ∆ja(kx ky), with k measured from the center of the pocket, is chosen to yield

C2-symmetric gap structures in the nematic superconducting phase consistent with both ARPES [42] and STM[110]. Note the gap structures discussed here are simply plausible guesses respecting the symmetry of the various phases and agreeing qualitatively with the evolution suggested by experiment; as yet, there is no microscopic theory of the ultranodal state.

62 4.2.2 Model details

The calculations were performed by taking simple parabolic dispersion for the electronic structure,

4α ϵ (k) = − k2 + E Γ π2 + h − i 4α kx π 2 ky 2 ϵ (k) = + ) − E− X π2 1 + ϵ 1 − ϵ h i 4α kx 2 ky − π 2 ϵ (k) = + − E− (4–8) Y π2 1 − ϵ 1 + ϵ with the parameters α = 2 and E+ = 0.6 , E− = 0.6, ε = 0.2 and additionally inserting symmetry related electron bands having band minima at (0, −π) and (−π, 0). In the numerical implementation, arbitrary energy units are chosen which then are assigned to the real units by fixing the critical temperature Tc and the position of the coherence peaks

∆ΓA, see Fig. 4-5(e,f). The order parameters are explicitly given by

4∆ Γa 2 − 2 ∆Γ(k) = ∆Γ + 2 (kx ky) π h i 4∆ k − π k ∆ (k) = ∆ + Xa − x 2 + y 2 X X π2 1 + ϵ 1 − ϵ h i 4∆ k k − π ∆ (k) = ∆ + Y a x 2 − y 2 (4–9) Y Y π2 1 − ϵ 1 + ϵ and the corresponding order parameters on the symmetry related electron bands. We use for the anisotropic gap components ∆Γa = 0.1 , and ∆Xa = ∆Y a = 0.4. The isotropic gap components are denoted as [∆Γ, ∆X , ∆Y ] and are assumed to decrease continuously as a function of sulfur doping, and we define sets of parameters A-D with0 A:[ .40, 0.35, 0.35], B:[0.35, 0.27, 0.35], C:[0.16, 0.20, 0.25], D:[0.07, 0.07, 0.07]. The inter-band gap component is chosen to be ∆0 = 0.4 and the time reversal broken component δ = ∆0. These parameters were adopted to describe a situation where the gap in Fe(Se,S) evolves from a highly anisotropic, nematic state with nodes along one axis of each Fermi surface pocket, to an even more anisotropic state with four nodes on the Γ pocket as the tetragonal phase is reached, consistent with experiment[106, 107, 111, 112]. Further details about the model is elaborated in Appendix F.

63 4.3 Results

4.3.1 Spin-resolved spectral functions

We begin by evaluating the spin dependent intensities as measured by ARPES.

Diagonalizing the Hamiltonian, we obtain the eigenenergies Eµ(k) of the Bogoliubov

th jσ quasiparticles in µ band and a unitary transformation with the matrix elements aµ (k) such that the spin-resolved spectral function reads

1 1 X η |a1σ(k)|2 A (k, ω) = − Tr Im (Gσ (k, ω)) ≃ µ . (4–10) σ π 11 π η2 + (ω − E (k))2 µ µ

σ where G11(k, ω) refers to the diagonal Gor’kov Green’s function, σ is the spin index and η is an artificial broadening parameter used for numerical evaluations. InFig.4-4, we show the spin-resolved spectral function along the high-symmetry path X − Γ − Y for down-spin component and up-spin components for cases A-D. Case A (Fig.4-4a and 4-4e) shows that the system will be fully gapped without any residual BFSs. With evolution in sulfur doping content of the system, as mimicked in the transition from case A to D, the spectral map reveals a Fermi level crossing of the bands as can be seen in Fig.4-4b-4-4d, 4-4b-4-4d. The BFSs become larger as more momentum space points satisfy the Pfaffian sign change condition. Such features can be easily detected in ARPES measurements. With spin-resolved ARPES, it is possible to probe into different momentum sections of the same bandas depicted in Fig.4-4. The calculation of the spectral function was carried out on a momentum path of size 1200 points in each segment of X − Γ and Γ − Y , and on a frequency grid of 1500 points. The artificial broadening η was set to 0.008. 4.3.2 Bogoliubov Fermi surfaces, Specific heat and Tunneling conductance

To obtain specific heat, one can start from calculating the entropy S of the free Fermi gas of Bogoliubov quasiparticles, in terms of the spectrum Ek of the Hamiltonian H(k) and

−1 use CV = T dS/dT to obtain 2 X −∂n(E ) E2 ∂E 2 X n(E (k))n(−E (k)) ∂E (k) k k − k µ µ 2 − µ CV = Ek = 2 Eµ(k) TEµ(k) Nk ∂Ek T ∂T Nk T ∂T k µ,k (4–11)

64 Figure 4-4. Spin-resolved spectral function Aσ(k, ω) for the “realistic” model described in the text, appropriate for Fe(Se,S), evaluated along high-symmetry path X − Γ − Y at temperature T/Tc = 0.02 for the example cases (a) A, (b) B, (c) C, and (d)D (with the energy axis normalized to hole pocket anisotropic gap maximum ∆ΓA). The arrow pointing downwards refers to spin-down component σ =↓. (e-h) Same as in (a-d) but for spin-up component σ =↑. where n(x) = 1/(exp(x) + 1) is the Fermi function. The temperature dependence of the order parameter is assumed to follow a mean field behavior with dimensionless function p d(T ) = tanh(1.76 Tc/T − 1), where we have set Tc = 0.15 which is chosen to yield

2∆ΓA/Tc ≈ 7 i.e. larger than the BCS value and very close to the ratio discussed for FeSe[111]. We calculate the differential conductance dI(V )/dV as a function of external bias voltage V and external field. For this, we first calculate the density of states by

1 X ρ(E) = Aσ(k,E) (4–12) Nk k,σ and then perform a convolution with the derivative of the Fermi function to obtain the differential conductance[113] Z dI(V ) EU ρ(E)e(E−eV )/T ∝ dE , (4–13) (E−eV )/T 2 dV −EL (1 + e )

dI(V ) where the solution of ρ(E) is used to evaluate dV . The integration was performed from

−2π ≤ kx,y ≤ 2π. The upper and lower boundaries of the integral, EU ,EL, were set to extend

65 the plotted range over several scales of the temperature, such that the derivative of the Fermi function outside that window is numerically zero. The calculation of DOS was carried out on a momentum grid of size 800 × 800 points. The energy grid was spaced by 0.0015 and artificial broadening η was set to 0.0004.

(a) (c) (e) A C N 10 A B C 5 D

0 0 0.5 1 1.5

(b) (d) (f) B D N A B C D 1.5

0.5

0 -1 0 1

Figure 4-5. Transition into the Ultranodal state. (a-d) Normal state Fermi surface (red contour) and Bogoliubov Fermi surface in superconducting state (blue/green patches) for different values of the isotropic gap parameters on each pocket. Note that while results are plotted over a putative 1st Brillouin zone, the model is actually continuous. Note the C2 symmetry of the nodes for larger isotropic gaps, consistent with ARPES. (e) Temperature dependence of the specific heat CV /T for the sets of gap components on each pocket (A-D). (f) Tunneling conductance dI/dV normalized to normal state value vs STM bias eV , normalized to hole pocket intraband gap ∆ΓA evaluated at temperature T = 0.07Tc. Curves are calculated by convolving density of states ρ(E) of 3-pocket model with Fermi function derivative[113]. The sets of gap values A-D span the nematic transition, with decreasing isotropic gap component ∆j, between gapped/near nodal state (A) and ultranodal states (B-D). Normal state conductance (red) is also given for reference.

66 As argued before, when the isotropic intraband component becomes sufficiently small in the presence of interband pairing and spin orbit coupling, the Pfaffian changes sign, and the spectrum of low-energy excitations is altered dramatically. As shown in Fig. 4-5(a-d), as the ratio of intraband isotropic to anisotropic components decreases with doping, the system develops a BFS, which grows and becomes more C4 symmetric as nematic order disappears. We emphasize that it is impossible within the present model framework to associate a unique set of gaps with a particular doping. However, one can plot the specific heat, which allows a rough comparison with experiment. Fig. 4-5(e) shows a plot of CV /T as a function of temperature for the same sets (A-D) of isotropic gap components on the individual pockets. CV /T goes to zero as T → 0 for larger isotropic components (A), but saturates at a nonzero value for smaller ones (B-D), reflecting the ultranodal state. Hence a system such as FeSe1−xSx can exhibit properties of both a superconductor and a normal Fermi liquid. Close to critical values of the isotropic gap (see Fig. 4-5 (b)), the BFS shrinks continuously to a point.

This behavior is reminiscent of experiments on FeSe1−xSx[106]: as one dopes through the nematic transition, the specific heat ratio ∆CV /CN at the transition is observed to fall, while at the same time the residual Sommerfeld coefficient in the superconducting state increases. Thermal conductivity[106] exhibits a similar low temperature evolution. The abruptness of the transition is not entirely clear from the thermodynamic data, since the samples are spaced relatively far apart in doping. However, recent STM data analysis obtained from a more closely spaced sequence of S-doping in FeSe1xSx [107], as discussed before in Sec.4.1 (see Fig.4-2), was able to establish that a nematic transition took place between x = 0.13 and x = 0.17. At this transition, the coherence peak position in the conductance spectrum red-shifted weakly, while the zero bias conductance became nonzero, growing with increased S-doping. Precisely this behavior is seen in the superconducting density of states across the topological transition into the ultranodal state, as shown in Fig. 4-5(f). As expected, with the development of the ultranodal state, the zero-energy value of the DOS increases due to the presence of BFSs of increasing size. In addition, the weight in the coherence peaks

67 is suppressed and their position weakly red shifted in the ultranodal phase, resulting in a gap-filling rather than a gap-closing phenonmenon very similar to experiment. 4.3.3 Zeeman field

We now study the effect of a weak Zeeman field on BFSs close to the topological transition. To demonstrate the underlying physics, we consider a simple toy model that includes the Zeeman field (which is a realistic model for an in-plane field in a 2Dsystem, without any coupling to the orbital motion of the electrons). We choose this model to be of the form X 1 † Hˆ = Hˆ + Hˆ + Hˆ j = Ψ H (k) + H (k) + Hj (k) Ψ . (4–14) 0 ∆ Z 2 k 0 ∆ Z k k Here j = x, y, z is the direction of the magnetic field and the Zeeman term reads explicitly P ˆ j σσ¯ † j HZ = kσσ¯ hjσj ckσckσ¯. In the expanded particle-hole basis, it is written as HZ(k) = hj (πz ⊗ τ0 ⊗ σj). For the pairing, we work with two special cases: sign-change (+−) and no-sign-change (++) pairing on the two pockets. These pairing terms are written as

++ ⊗ ⊗ ⊗ H∆ (k) = ∆(k)(τ0 iσy) + ∆0(iτy σ0) + δ(iτy σz) +− ⊗ ⊗ ⊗ H∆ (k) = ∆(k)(τz iσy) + ∆0(iτy σ0) + δ(iτy σz) with a real ∆(k) and ∆(k) = ∆(−k). The ∆(k) term defines the pairing amplitude fora spin-singlet intra-pocket pair with the same order parameter magnitudes on both pockets.

When the magnetic field is in-plane (h∥, j = x, y), the relevant Pfaffian minima with respect to the band energies are    2 2 − 2 − 2 4δ ∆(k) ∆0 h∥ ++ Min{Pf(k)} = (4–15)   2 2 − 2 2 2 − 4∆0∆(k) 4δ ∆0 + h∥ + .

Hence an in-plane field always pushes the system into the topologically non-trivial state for both phase distributions as long as δ ≠ 0. Moreover, and as one should anticipate, this conclusion is independent of the direction of in-plane field. For the case when the field is out-of-plane (h⊥, j = z), the total Pfaffian is written as a sum of two terms: one quadratic in

68 the field and another linear. It takes the form

{ } { 2 } − Pf(k)± δ, ∆(k), ∆0, h⊥, ϵi = Pf(k)2,± δ, ∆(k), ∆0, h⊥, ϵi h⊥δ∆0 (ϵ1 + ϵ2) (4–16)

± where the first term, Pf(k)2,±, is quadratic in the field and denotes the sign-change and no-sign-change cases respectively. For a given set of bands with dispersion ϵi, the relative signs of the field h⊥ and the TRSB component δ determines the sign of the linear term. While for generic field strengths both terms are important in determining the existence of BFSs, the linear term dominates the physics at small fields. In this limit, one can ignore the field dependence of Pf(k)2,± and we obtain

{ } ≃ { } − Pf(k)± δ, ∆(k), ∆0, h⊥, ϵi Pf(k)2,± δ, ∆(k), ∆0, 0, ϵi h⊥δ∆0 (ϵ1 + ϵ2) . (4–17)

Close to the topological critical point, we know that the first term { } ≃ Pf(k)2,± δ, ∆(k), ∆0, 0, ϵi 0. Hence whether the Pfaffian changes sign in this regime is completely determined by the linear term in h⊥, i.e., the relative signs of h⊥ and δ. If the Pfaffian has a certain sign for a given direction of the weak field,it must change sign when the direction of the field is flipped. Therefore, if there exists no BFS for a certain direction of the field, h⊥, one must emerge for −h⊥. This statement is independent of the details of the band structure ϵi, provided one has purely electron or hole like pockets and the field has little effect on the internal electronic structure of the material, and therefore, forms adistinct signature of topological phase transition. The above signatures are expected to show up in the specific heat and tunneling DOS close to the topological critical point. In the case where one has mixed electron and hole pockets as is the case with FeSe, the situation is less unambiguous. As is evident from the last term in Eq. 4–17, the relative sign between the two pockets with respect to the Fermi level becomes important. In such a scenario, the electron and hole pockets satisfy the Pfaffian sign change condition separately for opposite direction of the field. Hence, close to the topological transition, BFSs formonly on the hole pockets for one direction of the field and on the electron pockets for the opposite direction. Nonetheless, a key of signature of BFSs would manifest itself in the asymmetry of

69 the residual specific heat and tunneling conductance spectra with respect to flipping thefield direction, as the hole and electron pockets generally have different density of states at the Fermi energy (see the next sub-section on specific heat). We now present the spectral map of the system evaluated under the presence of a magnetic field close to the topological transition. Choosing the same model as discussed in

Section 4.3.1, we select the isotropic gap components as ∆Γ = 0.23, ∆X = 0.28, ∆Y = 0.33, inter-band gap component to be ∆0 = 0.3 and time-reversal broken component δ = ∆0.

Anisotropic gap components are ∆Γa = 0.1 , and ∆Xa = ∆Y a = 0.3. This choice of parameters is made such that the system is very close to the transition into the topological state. Exactly at the transition, the BFS reemerges upon application of an infinitesimal magnetic field as seen in Eq.4–16. We use three values of magnetic field h⊥ to generate BFS as seen in Fig.4-6a h⊥ = −0.03, 4-6d h⊥ = 0 and 4-6g h⊥ = +0.03. Note that the sign of the magnetic field (±h⊥) is chosen with respect to the sign of the inter-band gap component

∆0. The red lines denote the normal-state FS contour under the same parameter values for magnetic field. Notice that the spin-degeneracy is lifted under the presence of the magnetic field, giving rise to very closely-spaced concentric Fermi pockets in the normal state. Case 4-6d shows no BFS under zero magnetic field close to the topological transition. We recover ultranodal BFS states on the electron pockets in case 4-6a (blue) with negative magnetic field and on the hole pockets in case 4-6g (pink) with positive field.

The corresponding spectral function for h⊥ = 0 case (Fig.4-6e and 4-6f) shows that the system will be fully gapped without any residual Bogoliubov surfaces. With the application of −h⊥, the residual BFS appears along Γ − X direction which is shown in the Fermi level crossing of the spectral map in Fig.4-6b and 4-6c. Upon flipping the direction of the magnetic field to +h⊥, the Fermi level crossing shifts towards the Γ − Y direction as shown in Fig.4-6h and 4-6i. Since the effects described here are small but observable, it is worth stating clearly that the best chance for a “smoking gun” experiment where the BFS is induced by an external probe requires operation very close to the transition point.

70 Figure 4-6. (a) Normal state Fermi surface (red contour) and Bogoliubov Fermi surface in superconducting state (blue/pink patches) for magnetic field h⊥ = −0.03, (b) Spin-resolved spectral function Aσ(k, ω) evaluated along high-symmetry path X − Γ − Y at temperature T/Tc = 0.02 (with the energy axis normalized to hole pocket anisotropic gap maximum ∆ΓA). The arrow pointing downwards refers to spin-down component σ =↓. (c) Same as in (b) but for spin-up component σ =↑. (d-f) Same as in (a-c) but for magnetic field h⊥ = 0. (g-i) Same as in (a-c) but for magnetic field h⊥ = +0.03. Note that while results are plotted over a putative 1st Brillouin zone, the model is actually continuous. Note that the sign of the magnetic field (±h⊥) is chosen with respect to the sign of the inter-band gap component ∆0.

71 A B C D

Figure 4-7. Residual specific heat CV /T at zero temperature, normalized to the value in the normal state. For fields in plane, h∥ the CV /T is an even function of the field (a), while for out of plane field, h⊥, there are also contributions odd in the field (b). The explicit behavior depends on the details of the order parameter (and the band structure); here we present results for a model for Fe(Se,S) with parameters for the cases A-D as given in the text. The inset shows the expected zero field behavior as function of temperature. Calculations for the plots above were done by A. Kreisel.

4.3.4 Residual Specific heat in the presence of Zeeman field | Results of (CV /T ) T →0 as a function of fields in plane h∥ and out of plane h⊥ are shown in Fig.4-7. The numerical evaluations are carried out at a low temperature of T/Tc = 0.0007. While for in plane fields, the specific heat is an even function of the field4-7 (seeFig. (a)), it also acquires a dependence on odd powers of the field h⊥ in Fig.4-7 (b); finite values | of (CV /T ) T →0 are signatures of the ultranodal state. Starting from the case A, where no BFS exists, it is possible to tune into the topological state by fields in any direction, while the field in z direction is more effective. In the parameter set B it is not possible to | decrease (CV /T ) T →0 with fields in plane; in contrast the curve for h⊥ has a finite slope at zero field such that leaving the topological state might be possible. Note that qualitatively similar behavior of CV /T is expected at any temperature T ≤ Tc; this quantity will be an even function of the in plane fields, but acquire odd powers of h⊥. We have not calculated the corrections to the specific heat due to the low energy states in the vortex phaseof

72 the superconductor (e.g. Volovik contribution from extended states[114], or Caroli-de Gennes-Matricon states in the core[115]). However, we stress that these will always increase the value of CV /T and are independent of the direction of the field; they therefore do not change the conclusion that in the state with BFS, CV /T acquires dependence on h⊥ of odd powers. A momentum grid of 1200 × 1200 was used to calculate the specific heat to yield convergence at the lowest temperature. 4.4 Summary

It is important to reemphasize the distinction between the two ways in which pairing terms can break TRS. The time reversal operator is T = UT K; hence, TRSB can occur either because the order parameter acquires an imaginary component (type 1) or/and because the spin sector in the pairing term does not transform properly under the unitary part of the time reversal operator (type 2). From our analysis, it is clear that a pure intra-pocket type 1 TRSB cannot yield a BFS since the Pfaffian preserves sign (irrespective of the number of input bands). Therefore, one necessarily requires a non-zero (even infinitesimally small, as is evident from Eq. (4–15)) type 2 TRSB, δ2 > 0, to achieve a non-trivial phase when the sign of the order parameter is the same on the two pockets. Finally, we note that combined thermodynamic and ARPES data, together with our analysis, point to deep minima on at least one of the Γ hole pockets coinciding with those on the electron pockets at the same momentum direction near the nematic quantum critical point. The general condition for the Pfaffian to change sign requires that the product of two intraband gaps be smaller than the TRS interband gap (TRSB component δ) when two pockets have the same (opposite) sign of the order parameter. Since interband gaps and TRSB breaking components are generically small, this requires that nodal or deep minima align such that the product of intraband hole and electron gaps ∆e(k)∆h(k) is also small. Fig. 4-3 shows how such a coincidence of nodal structures along X and Y might occur in the tetragonal phase. We pointed out that intrapocket nodal structures are essential to allow for the formation of the ultranodal state without the need for large interband interactions,

73 typically required to overcome the missing Cooper logarithm due to pairing states away from the Fermi level. In summary, we explored the possible existence of an ultranodal superconductor: a state of matter uniquely characterized by topologically protected extended zero-energy surfaces (BFSs) in multi-band superconductors with spin-1/2 pairs such as iron-based systems. This phenomenon occurs in the presence of a spin-orbit coupling-induced triplet pairing component, interband pairing, and type-2 broken time reversal symmetry. We derived conditions for the existence of such surfaces and studied the behavior of the specific heat, the Sommerfeld ratio CV /T , and the density of states close to and away from the transition. We argued that the topologically non-trivial phase retains thermodynamic and electronic properties of both the superconducting state and the normal Fermi liquid. Finally, we argued that our results have direct, immediate experimental relevance by examining recent evidence for an excess of low energy quasiparticle states in the clean iron-based superconductor FeSe1−xSx near its nematic transition. We showed that theoretical specific heat and conductance spectra agreed remarkably well with experiment across the transition, and concluded that an ultranodal state can exist in this system. Our further analysis of the effect of a weak Zeeman field on the electronic and thermodynamic properties of BFSs close to the topological critical point reveals a distinguishing feature of BFSs: the dependence of observable such as the zero temperature residual specific heat on the sign of the out-of-plane external field. Generic features arising from the spin-resolved spectral functions can be verified using spin-polarized ARPES. More direct probes of BFSs such as ARPES and quantum oscillations could help paint a fuller picture of this rapidly developing story and pave the way towards a deeper understanding of the ultranodal state.

74 CHAPTER 5 BOGOLIUBOV QUASIPARTICLE INTERFERENCE IMAGING OF STRONTIUM RUTHENATE ∗ 5.1 Introduction

Sr2RuO4 is considered to be a strong candidate for being a correlated topological superconductor, leading to intense research interest recently. The determination of the structure and symmetry of the superconducting gap ∆(k), which encodes its pairing mechanism and is one of the critical inputs in the analysis of topological properties of

Sr2RuO4, has resulted in investigation over decades [116–119], but new perspectives have been highlighted recently. Earlier observations claimed p + ip symmetry of the gap order

17 in Sr2RuO4; O Knight shift [120] and spin-polarized neutron scattering [121] reported no decrease in spin susceptibility below TC, and muon spin rotation [122] and Kerr effect [123] implied TRS breaking [124]. However, recent studies provides contradicting evidence. The temperature dependence of the London penetration depth [125], linear temperature-dependence of electronic specific heat capacity at lowest temperature126 [ ], and field-oriented specific heat measurements [127] have hinted at the existence of nodes or deep minima in ∆(k). Recent thermal conductivity measurements also implied that the nodes/minima are oriented parallel to the crystal c axis [128]. A substantial drop in the

17 Knight shift below TC has also been reported from in-plane O NMR study [129], which is naturally consistent with spin-singlet symmetry of the gap structure [130]. Also, time-reversal symmetry (TRS) is shown to be preserved using Josephson tunnel junctions in current-field inversion experiments [131]. All these findings have lead to a detailed revaluation ofthe

Sr2RuO4 superconductivity theory [132–136].

Correct representation of the electron-electron interactions in Sr2RuO4 is a challenging topic. Besides pervasive onsite and intersite Coulomb U interactions, Hund’s coupling J between the Ru d orbitals give rise to orbital-selective phenomena. The dxy dominant bands

∗ This chapter is part of an ongoing investigation, with a peer-reviewed publication in sight.

75 are more correlated than the dxz/yz ones [137, 138] and spin-orbit coupling is also believed to play a strong role. Contemporary theories [132–136, 139, 140] have considered various combinations of Coulomb and Hund’s interactions to obtain ∆(k) predictions, focusing on the symmetry of the predominant ∆(k). Weak coupling analyses have found that the phase space of the order parameter, as a function of spin-orbit coupling strength λ and J/U ratio, exhibit B1g(dx2−y2 ) symmetry and A1u(helical p) symmetry and even more complex triplet orders [132–134]. However, many of these states are not consistent with experimental inferences. Direct measurement of the superconducting energy gap ∆(k) has not been possible so far because its energy scales are very low (|∆| ≤ 350 µeV), which requires ultra-low temperatures and energy resolution to distinguish between anisotropic k-space features. However, Bogoliubov quasiparticle interference (BQPI) imaging is a powerful technique capable of high-precision measurement of multiband ∆(k) [15, 141, 142]. When a highly anisotropic ∆(k) opens on a given band, Bogoliubov quasiparticles |k(ω)⟩ exist in the

min max energy range ∆k < ω < ∆k . Within this range, interference of impurity-scattered quasiparticles produces characteristic real space modulations in the density of states ρ(r, ω). The Bogoliubov quasiparticle dispersion E(k) = ω then exhibits closed constant energy contours surrounding FS k-points where minima in ∆(k) occur. These k-space locations can then be determined because ρ(r, ω) modulations occur at the set of wavevectors q(ω) connecting them. These q(ω) are identified from maxima in ρ(q, ω), the Fourier transform of ρ(r, ω). The BQPI technique has recently been implemented to analyze ∆(k) in Sr2RuO4 [143]. This analysis, combined with observations from recent work [128, 129, 131–134], has lead to a claim of dx2−y2 gap symmetry for Sr2RuO4. Here, we will briefly discuss the mechanism by which STM obtains BQPI images. STM is a widely used tool for the real space analysis of surfaces [113] and can be used in several modes. Constant height mode is used to measure the sample density of states (DOS) as a function of energy. The tunneling current (I) is measured by sweeping the bias voltage (V ) keeping the STM tip-sample separation fixed. The tunneling conductance dI/dV can

76 be shown to be proportional to the sample DOS [113]. STM can provide the most local picture of the impurity states in superconductors, and can measure lattice DOS with atomic resolution which gives a measure of ρ(r, ω). The Fourier transform of this quantity gives a BQPI map. This chapter is organized as follows: in Section 5.2, we introduce the multiorbital Hubbard Hamiltonian with the pairing and impurity terms, and describe it using the Bogoliubov-de Gennes (BdG) equations. We write down the equations for computing the lattice DOS and continuum DOS [144, 145], in the presence of impurities using the T-matrix approach. In Section 5.3, we explain the BQPI findings from Ref.[143] and then we present our results for lattice DOS, continuum DOS and BQPI for Sr2RuO4 using two different gap order parameters with the following gap symmetry: triplet p-state and singlet d-state. Finally, we present our conclusions in Section 5.4 and elucidate our efforts directed towards further investigation in this context. 5.2 Model

Here we will lay out the theoretical framework on how the surface of a material is imaged by STM, as described in Ref.[144] and [145]. Starting from first-principle derived electronic structure, followed by projecting onto a Wannier basis preserving all local symmetries, we get the resulting multi-orbital tight-binding Hamiltonian of the electrons on the lattice: X X qt † † ′ − H0 = tRR′ cRqσcR tσ µ0 cRqσcRqσ (5–1) RR′qtσ Rqσ

† Here, cRqσ (cRqσ) is the creation (annihilation) operator for an electron in the unit cell R, qt orbital q with spin σ. tRR′ is the amplitude for hopping from unit cell R, orbital q to the unit cell R′, orbital t. DFT studies show that only Ru d-orbitals contribute to the states around the Fermi energy. Thus, the tight-binding model includes three d-orbitals, dxz, dyz, dxy in the unit cell. The ab-initio approach also yields a set of Wannier orbitals (at the surface of a slab of the sample as considered in the DFT calculation), as shown in Fig.5-1. Individual Wannier functions exhibit significant weight on nearby atoms.

77 Figure 5-1. Isosurface plots of (a) dxz, (b) dyz and (c) dxy Ru Wannier orbitals in Sr2RuO4. Red and blue colors indicate the opposite phases of the wave function. Source for ab-initio data: T. Berlijn.

The full Hamiltonian has three terms, namely the kinetic energy term H0 as given above,

BCS like interaction term HBCS and single-impurity term Himp.

H = H0 + HBCS + Himp (5–2) X qt † † − ′ HBCS = VRR′ cRq↑cR′t↓cR t↓cRq↑ ′ XRR ,qt qt † Himp = VimpcR⋆qσcR⋆tσ R⋆qtσ

qt ′ ⋆ VRR′ is the effective attraction between unit cell R, orbital q and unit cell R , orbital t. R is qt the impurity site and Vimp is the (on-site only) non-magnetic impurity potential. Mean-field treatment: According to the BCS mean-field assumption, the pair creation † † operator cRq↑cR′t↓ has a finite expectation value and its fluctuation around the average is small. Thus we can project the quartic operator in Hamiltonian given by Eq.5–2 onto the ‘Cooper channel’. The resulting mean-field Hamiltonian is the following: X X qt † † ′ − HMF = tRR′ cRqσcR tσ µ0 cRqσcRqσ RR′qtσ Rqσ X h i X − qt † † qt † ∆RR′ cRq↑cR′t↓ + h.c. + VimpcR⋆qσcR⋆tσ (5–3) RR′qt R⋆qtσ

78 qt qt ⟨ ′ ⟩ where the pairing field ∆ is given by ∆RR′ = VRR′ cR t↓cRq↑ . The following equations will be discussed in momentum space, i.e., Fourier transformed from the real space unless mentioned othecnrwise. Bogoliubov-de Gennes (BdG) equations: Upon applying a Bogoliubov transformation and working in the momentum space, the superconducting gap is obtained from the self-consistent solution of the BdG equation. For this, the full BdG Hamiltonian is given by:    Hˆ (k) ∆(ˆ k)  Hˆ =   (5–4) ∆ˆ †(k) −Hˆ T (−k) written in the Nambu spinor basis Ψ†(k) = (Φ†(k), ΦT (−k)) with 12 degrees of freedom

† † † † † † † (orbital and spin). Here, Φ (k) = (cxz,↑(k), cxz,↓(k), cyz,↑(k), cyz,↓(k), cxy,↑(k), cxy,↓(k)). ˆ Upon diagonalizing H on a k-grid, we obtain the eigenvalues {±Eµ(k)} and the unitary transformation matrix Uˆ(k) that diagonalizes Hˆ , a 12 × 12 matrix indicated by the underscore. The orbitally resolved-gap structure ∆ˆ qt(k) is also expressed in this Nambu basis. The orbitally-resolved homogeneous DOS in the superconducting state is ρt(ω) = P − 1 ˆ π Im k Gtt(k, ω) where Gˆ(k, ω) = (ω − Hˆ (k) + iδ)−1 (5–5) is the Green’s function for the BdG Hamiltonian. The real-space bare lattice Green’s function is obtained from the Fourier transform X 0 ′ 0 Gˆ (R, R′, ω) = e−ik.(R−R )Gˆ(k, ω) = Gˆ (R − R′, ω) (5–6) k Impurity states and T-matrix approach: With intra-orbital, on-site impurity

qq potential terms Vimp in Himp of Eq.5–2, we can construct the lattice Green’s function in the presence of the impurity via the T-matrix approach:

0 0 0 Gˆ(R, R′, ω) = Gˆ (R − R′, ω) + Gˆ (R, 0, ω)Tˆ(ω)Gˆ (0, −R′, ω). (5–7)

79 The T-matrix is given by ˆ − ˆ ˆ −1 ˆ T (ω) = [I V impG(ω)] V imp, (5–8) P ˆ ˆ with the local Green’s function G(ω) = k G(k, ω). It is assumed that the corrections to the nearest-neighbor hopping and potentials are generally significantly smaller, so we do not include them here. The local lattice DOS in the presence of impurity is

1 ρ(R, ω) = − Im Tr’Gˆ(R, R, ω) (5–9) π with Tr’ being the orbital trace over the normal part of the Nambu Green’s function. Wannier functions to calculate tunneling current: For a given bias voltage V , the differential tunneling conductance in an STM experiment is given by:

dI 4πe (r, eV ) = |M|2ρ (0)ρ(r, eV ) (5–10) dV ℏ tip where r = (x, y, z) denote the coordinates of the tip, ρ(r, eV ) is the continuum LDOS

2 (cDOS), ρtip(0) is the DOS of the tip, and |M| is the square of the matrix element for the tunneling barrier. The cDOS can be calculated by

1 ρ(r, ω) = − ImG11(r, r, ω), (5–11) π

11 where G (r, r, ω) = G↑(r, r, ω) is the normal part of the Nambu continuum Green’s function given by   ′ ′  G↑(r, r , ω) F (r, r , ω)  G(r, r′, ω) =   (5–12) ∗ ′ ′ F (r, r , ω) −G↓(r, r , −ω) defined in a basis described by the field operators ψσ(r). These are related to the lattice operators cRqσ through the Wannier functions matrix elements wRq(r) as P ψσ(r) = Rq cRqσwRq(r). Employing the Wannier basis transformation[144, 145], we can obtain continuum Green’s function as: X ′ ˆ ′ ′ G(r, r , ω) = Gqt(R, R , ω)wRq(r)wR′t(r ) (5–13) R,R′,q,t

80 and thereafter, evaluate the conductance from Eq.5–10. The cDOS ρ(r, ω) involves local (R = R′) as well as non-local (R ≠ R′) Green’s function contributions. Owing to this, the spectral and spatial properties of cDOS are both qualitatively and quantitatively different from that of lattice DOS. 5.3 Results

Sr2RuO4 crystal structure is composed of alternating layers of SrO and RuO2 planes. The cleaving of the sample in ultrahigh vacuum at low temperatures is considered to reveal atomically flat SrO cleaved surface [143]. A schematic plot for the exposed SrO surface of the sample, as probed by an STM tip, is shown in Fig.5-2.

Figure 5-2. Schematic plot for the exposed SrO cleave surface of Sr2RuO4 sample, as probed by the STM tip (Sr atoms = pink squares, O atoms = blue circles). Ru atoms lie vertically below the O atoms. Depicted over 11 × 11 unit cells here, the R = (0, 0) center is marked with a blue star and will be considered as the position for the single on-site impurity potential in the upcoming analysis.

5.3.1 Homogeneous superconducting state

Fig.5-3(a) and (b) shows the band structure and the Fermi surface (FS) for 2D Sr2RuO4, respectively, with the corresponding dominant orbital contribution. To be consistent with experimental measurements, we renormalized the band structure by an overall factor of 1/5

81 compared to the DFT values. There are three FS pockets: hybridisation between the two quasi-1D bands that originate from Ru dxz (purple) and dyz (yellow) orbitals lead to the electron-like β band around Γ point and hole-like α band surrounding the M point; and the Ru dxy (blue) orbital generates the electron-like quasi-2D γ band surrounding the Γ point. For the current work, we will restrict ourselves to 2D evaluations without any loss of information. Fig.5-3(c) shows the orbitally-resolved homogeneous DOS ρt(ω) in the normal

Figure 5-3. (a) Band structure of Sr2RuO4 along high-symmetry path Γ-X-M-Γ, and (b) Fermi surface of Sr2RuO4 showing the corresponding dominant orbital contribution as indicated by the color legend; (c) Orbitally-resolved DOS in the normal state of Sr2RuO4; and DOS in the superconducting state with (d) d-wave order parameter (e) p-wave order parameter as described in the main text. Yellow lines in panel (c)-(e) depicts degenerate DOS contributions from orbitals dxz and dyz.

state, with degenerate contributions from dxz and dyz orbitals (depicted in yellow). The flat dxy band near the Fermi level at X-point contributes to a large singular-like feature in the normal state DOS. Next, we introduce a gap symmetry of our choice to analyze different possible situations:

82 • Case I: gap order parameter with dx2−y2 symmetry; purely intra-orbital spin-singlet

| ↑↓⟩ − | ↓↑⟩ qq ∆0 − ( ) components ∆ (k) = 2 (cos(kx) cos(ky)), with ∆0 = 3.5 meV.

• Case II: gap order parameter with p + ip symmetry used from Ref. [146]; purely

intra-orbital spin-triplet (| ↑↓⟩ + | ↓↑⟩) components: ! X qq qq qq ∆ (k) = ∆0 ∆xjgxj(k) + i∆yjgyj(k) j=1,2,3

gx1(k) = sin(kx)

gx2(k) = sin(kx) cos(ky)

gx3(k) = sin(3kx) (5–14)

yz,yz xz,xz xz,xz yz,yz xy,xy xy,xy with gyj(kx, ky) = gxj(ky, kx), and ∆yj = ∆xj , ∆yj = ∆xj = 0, ∆xj = ∆yj ∀ xz,xz xz,xz xz,xz xy,xy xy,xy xy,xy j. Moreover, (∆x1 , ∆x2 , ∆x3 ) = (0, 0.2, 1) and (∆x1 , ∆x2 , ∆x3 ) =

(0.18, 0.15, −0.3).

• Case III: We also intend to evaluate BQPI patterns arising due to gap paramaters

predicted via “realistic” spin-fluctuation pairing calculations in 132Ref.[ ]. Fig.5-3(d) and (e) shows the orbitally-resolved homogeneous DOS in the superconducting state for the choice of order parameters described in Case I and II, respectively. We can clearly identify the coherence peaks at ∆0 = ±3.5 meV in Fig.5-3(d) and the valley-like DOS features in both Fig.5-3(d) and (e). We have used 500 × 500 k-grid and a broadening of 0.1 meV for all the calculations discussed here. 5.3.2 Inhomogeneous superconducting state

To consider scattering from a single point-like impurity, we introduce an impurity substituting one of the Ru atoms in the center of the finite 11 × 11 lattice expanse in our calculations (see Fig.5-2). We consider a non-magnetic impurity scatterer that is

qq purely intra-orbital, and diagonal in the spin space, i.e. (Vimp)σσ. With a suitable choice of

Vimp = −0.1 eV used in Eq.5–7 and 5–8, the impurity produces bound states within the

83 energy range of the superconducting gap for a resonance frequency. The following results will be discussed for the resonance frequency ω = 1.1 meV for Case I, and ω = 0.32 meV for Case

II, obtained corresponding to the choice of Vimp = −0.1 eV. Upon subsequent evaluation of the lattice Green’s function Gˆ(R, R′, ω), the lattice DOS is obtained from Eq.5–9 and is shown in Fig.5-4(a) for Case I (d-wave) and Fig.5-4(b) for Case II (p-wave). Each small square represents a Ru site and the LDOS at these sites are indicated by the color scale. Typically, a clear enhancement of LDOS is observed in the vicinity of the impurity atoms, indicating the presence of the bound impurity states. The intensity maxima is seen to occur at the impurity site R = (0, 0), with some tails on the nearest-neighbor (NN) sites. Next, Fig.5-4(c) and (d) show the Wannier function-modified

Figure 5-4. For gap order parameter of Case I (singlet d-state): (a) Plot of lattice DOS over 11 × 11 unit cells, (c) Wannier function-modified continuum DOS and (e) BQPI map showing prominent q-features at resonance frequency ω = 1.1 meV for Vimp = −0.1 eV. (b)-(d)-(f) Same as in (a)-(c)-(e) but for gap order parameter of Case II: triplet p-state at resonance frequency ω = 0.32 meV. Refer to Sec.5.3.1 for details about Case I and II. continuum LDOS pattern of the bound states for Case I and II, respectively, obtained from

84 Eq.5–11 and 5–13. The spatial cDOS is calculated on the xy-plane at a specific z-height above the topmost SrO layer of the sample (here, z = 5.7Å is used). In both Cases I and II, cDOS is suppressed at the impurity site R = (0, 0). The bound state has maxima at the next-NN sites, just above the Sr atoms. Before we discuss about the BQPI images Fig.5-4(e) and (f) obtained from Fourier transformation of the real-space maps Fig.5-4(c) and (d), respectively, we will briefly present the BQPI results from the experimental work from Ref.[143]. As pointed out in earlier

BQPI studies [147] of normal state Sr2RuO4, the dxy dominated γ band remains virtually undetectable [143], probably due to the small wave-function overlap of the dxy surface wave functions of the sample with the STM tip. The dxz/yz dominated α, β bands are detected from the normal-state scattering interference wave-vectors, and subsequently, yield prominent signatures in the superconducting state. The pictures in Fig.5-5 are adapted from Ref.[143]. Fig.5-5(a) shows a schematic plot of k-space regions with significant quasiparticle density of states in the superconducting state of Sr2RuO4, under the assumption of a dx2−y2 gap order parameter with nodes along (±1, ±1) on α, β bands. The major scattering wave-vectors qj

(j = 1, 2, 3, 4, 5), connecting these k-regions are also shown. q1, q2, q3 scatters quasiparticles on β band due to its gap minima/nodes, whereas q4, q5 connects gap minima/nodes regions in the α band. Detecting BQPI intensities at these specific q vectors serves as a direct evidence for a superconducting gap structure with gap minima/nodes along (±1, ±1) on the

α, β bands of Sr2RuO4. Fig.5-5(b) shows the measured BQPI pattern at energy ω = 100 µeV, well within the maximum gap value (350 µeV), with red/blue circles highlighting the qj features as mentioned above. Low |q| features in the experiments are believed to be due to long-range disorder/drift in real space rather than any scattering interference, and hence, are ignored.

Our BQPI evaluations presented in Fig.5-4(e) (for Case I with dx2−y2 order parameter) does show some similar key |q| features as in Fig.5-5(b), unlike the results of Case II: helical p order parameter, Fig.5-4(f). But further analysis is required before making any conclusive statements. Future investigations in this context will involve:

85 Figure 5-5. (a) Schematic plot showing regions of significant quasiparticle density for α, β bands when gapped with a d-state order parameter with gap minima/nodes along (±1, ±1). Major scattering vectors (qj; j = 1 − 5) are shown in red and blue, corresponding to the β and α band, respectively. (b) Measured BQPI pattern at a given energy (within the superconducting gap) with circles highlighting features arising from qj vectors depicted in panel (a). Pictures adapted from Ref.[143].

• Realistic approximation of the gap order parameter to be defined only on the Fermi

surface and extrapolated to the rest of the Brillouin zone using a Gaussian cutoff (refer

to Appendix.G for further details). This will eventually lead to the representation of

the superconducting pairing term in orbital basis in real-space, the same basis in which

the tight-binding hopping elements are represented.

• Evaluating BQPI signature for different gap order parameters calculated in132 Ref.[ ]

and for different impurity potentials Vimp.

• A careful scrutiny of the other significant q features in the BQPI pattern from

Ref.[143], and analysis of other experimental evidences in favor of triplet/singlet

superconducting state of Sr2RuO4.

86 5.4 Summary

We have performed 2D calculations of the real-space lattice density of states and

Wannier function-modified continuum electronic density of states2 forSr RuO4. Using the T-matrix approach for analyzing scattering effects from a single impurity point, wehave provided real-space tunneling conductance map and its Fourier transformed BQPI image to compare with STM measurements [143]. We have shown our evaluations for two possible gap order parameters: Case I with dx2−y2 symmetry and Case II with helical p symmetry. We explained the meaning of the BQPI pattern as exhibited in Ref.[143] and showed our effort in connecting this to our theoretical predictions. Although some similarities in q features are seen between Case I BQPI image and that of Ref.[143], further investigation in this topic is ongoing to make more robust conclusive statements. We then mentioned the steps being taken for current and future investigations in this context.

87 CHAPTER 6 CONCLUSIONS In this dissertation, we provided results of our investigation into effects of electronic correlation in unconventional superconductors, focusing primarily on FeSCs. We set up microscopic models within weak-coupling theory combined with numerical techniques to study complex multi-orbital systems. In Chapter 2, we discussed the influence of momentum-dependent correlations on the superconducting gap structure in iron-based superconductors. Within the weak coupling approach including self-energy effects at the one-loop spin-fluctuation level, we constructed a dimensionless pairing strength functional which includes the effects of quasiparticle renormalization. The stationary solution of this equation determines the gap function at

Tc. The resulting equations represent the simplest generalization of spin fluctuation pairing theory to include the effects of an anisotropic quasiparticle weight. We obtained good agreement with experimentally observed anisotropic gap structures in LiFeAs, indicating that the inclusion of quasiparticle renormalization effects in the existing weak-coupling theories can account for the observed anomalies in the gap structure of Fe-based superconductors. In Chapter 3, we discussed the effects of non-local correlations on electronic properties in the normal state of FeSCs. Deviations of low-energy electronic structure of iron-based superconductors from density functional theory predictions have been parametrized in terms of band- and orbital-dependent mass renormalizations and energy shifts. The former have typically been described in terms of a local self-energy within the framework of dynamical mean field theory, while the latter appears to require non-local effects dueto interband scattering. By calculating the renormalized bandstructure in both random phase approximation (RPA) and the Two-particle self-consistent (TPSC) approximation, we showed that correlations in pnictide systems like LaFeAsO and LiFeAs can be described rather well by a non-local self-energy. In particular, Fermi pocket shrinkage as seen in experiment occurs due to repulsive interband finite-energy scattering. For the canonical iron chalcogenide system FeSe in its bulk tetragonal phase, the situation is however more complex since even

88 including momentum-dependent band renormalizations cannot explain experimental findings. We proposed that the nearest-neighbor Coulomb interaction may play more important role in band-structure renormalization in FeSe. We further compared our evaluations of non-local quasiparticle scattering lifetime within RPA and TPSC with experimental data for LiFeAs. In Chapter 4, we discussed about topologically protected ultranodal pair states in FeSCs. Bogoliubov Fermi surfaces (BFSs) are contours of zero-energy excitations that exist in the superconducting state. In this chapter, we showed that multiband superconductors with dominant spin singlet, intraband pairing of spin-1/2 electrons can undergo a transition to a state with BFSs if spin-orbit coupling, interband pairing and time reversal symmetry breaking are also present. These latter effects may be small, but drive the transition to the topological state for appropriate nodal structure of the intra-band pair. Such a state should display nonzero zero-bias density of states and corresponding residual Sommerfeld coefficient as for a disordered nodal superconductor, but occurring even in the purecase. We presented a model appropriate for iron-based superconductors where the topological transition associated with creation of a BFSs can be studied. The model gives results that strongly resemble experiments on FeSe1−xSx across the nematic transition, where this ultranodal behavior may already have been observed. We further studied the effect of a weak Zeeman field close to the topological transition and proposed distinguishing features ofBFSs using residual specific heat. We also presented additional signatures of BFSs in spin-polarized spectral weights. In Chapter 5, we illustrated our ongoing effort towards determination of the momentum-dependence of the superconducting gap structure to compare with Bogoliubov quasiparticle interference (BQPI) imaging data for Sr2RuO4. This material is a strong candidate for being a correlated topological superconductor, thus, generating intense research interests in the recent years. The energy scales of the gap order parameter in Sr2RuO4 are very low, hence, its momentum structure has proven difficult to measure using existing experimental probes. However, recent work [143] employing a BQPI imaging technique capable of high-precision measurement of multiband gap structure, has provided data

89 claiming dx2−y2 gap symmetry to be the most consistent with the analysis for Sr2RuO4. We presented our theoretical results for continuum electronic density of states and its

Fourier-transformed BQPI map in the superconducting state of Sr2RuO4, using two different gap order parameters with helical p and dx2−y2 symmetry, to compare with STM images in Ref.[143]. Further investigation is ongoing in this context to provide more conclusive scientific statements.

90 APPENDIX A SPIN DECOMPOSITION OF INTERACTION CHANNELS AND POLARIZATION BUBBLES

Figure A-1. Orbital (a) and spin (b) structure of the RPA bubble diagram for the self-energy in the multiorbital system Σps(k, ωm). Interaction lines contain four orbital ˆ indices, U = Uwzst. The shaded bubble denotes the RPA susceptibility χ(q) = χ(q, Ωm). Incoming and outgoing indices s and p carry the same spin σ. χ1,2,3 are different susceptibility channels and σ = −σ.

Here, we will show the derivation of the equations leading to evaluation of the interaction channel contributing towards the normal state self-energy. The self-energy in orbital basis as shown in Fig.A-1 is given by

1 X X 0 − − Σps(k, ωm) = Vpqst(q, Ωm)Gqt(k q, ωm Ωm). (A–1) βNq q,Ωm qt

A.1 Charge susceptibility

For spin indices α, β, etc. Greek characters and orbital indices v, u, etc. Latin characters, the creation/annihilation operators can be expressed as row/column matrix as c† (k, τ) = † † , where α ≠ β. The charge density operator and the v cvα(k, τ) cvβ(k, τ) charge susceptibility are: X † nvu(q, τ) = cvα(k + q, τ)δαβcuβ(k, τ) k,αβ     X 1 0 cu↑(k, τ) = † †     cv↑(k + q, τ) cv↓(k + q, τ) k 0 1 cu↓(k, τ)

91 Z 1 β χC (q, iΩ ) = dτeiΩmτ ⟨T n (q, τ)n (−q, 0)⟩ vuwz m 2 τ vu zw Z0 β X 1 † † ′ ′ iΩmτ − = dτe Tτ cvα(k + q, τ)δαβcuβ(k, τ) czγ(k q, 0)δγδcwδ(k , 0) 2 0 ′ Z kk ,αβγδ β X 1 † † ′ ′ = dτeiΩmτ T c (k + q, τ)c (k, τ) c (k − q, 0)c (k , 0) 2 τ vα uα zγ wγ 0 kk′,αγ (A–2)

Figure A-2. (a) Bare single particle Green’s function G0. (b) Charge polarization bubble C χvuwz for interaction free system. (c) And the same, in the presence of interaction, which can now connect bubbles of different spin composition. U denotes the renormalized interaction from series summation under RPA.

0 † We define bare single particle Green’s function as Gvwσ(k, τ) = Tτ cvσ(k, τ)cwσ(k, 0) , which means a particle with momentum k and spin σ is destroyed in orbital w at time 0, while the particle with same momentum and spin is created in orbital v at time τ (Refer to Fig.A-2a). This imposes the condition that a single-particle Green’s function conserves momentum and doesn’t flip spin of the particle at any time. The Fourier transformed R 0 β 0 iωmτ Green’s function in frequency domain: Gvwσ(k, iωm) = 0 dτGvwσ(k, τ)e ; vice versa:

92 X 0 1 0 −iωmτ Gvwσ(k, τ) = β Gvwσ(k, iωm)e . Thus, following Eq. A–2 imposing momentum ωn conservation and applying Wick’s theorem to group fermionic operators, we write:

Z β X 1 † † χC (q, iΩ ) = − dτeiΩmτ T c (k + q, τ)c (k + q, 0) T c (k, 0)c (k, τ) vuwz m 2 τ vα wγ τ zγ uα 0 k,αγ 1 X X = − G0 (k + q, iω + iΩ )G0 (k, iω ) (A–3) 2β vαwγ n m zγuα n ωn k,αγ

The above derivation carried out for an interaction free system, will require spin conservation to be imposed for single-particle Green’s function (Fig.A-2b), which means α = γ, hence:

1 X X χC (q, iΩ ) = − G0 (k + q, iω + iΩ )G0 (k, iω ) vuwz m 2β vwσ n m zuσ n ω k,σ n (A–4) 1 X X = − G0 (k + q, iω + iΩ )G0 (k, iω ) β vw↑ n m zu↑ n ωn k where the sum over spins σ yields a factor of 2 that cancels out the pre-factor. This comes

0 0 0 0 from spin symmetry where we get Gvw↑Gzu↑ = Gvw↓Gzu↓. In the case of interaction, the charge bubbles can be connected through Interaction lines, as shown in the picture. Thus, we see that the interaction lines can connect bubbles of opposite spins (Fig.A-2c), and the effective charge bubble can then be written to have:

1 X X χC (q, iΩ ) = − G (k + q, iω + iΩ )G (k, iω ) vuwz m 2β vσwσ n m zσuσ n (A–5) ωn k,σ where we denote G to be the effective Green’s function to differentiate it from thebare single-particle Green’s function. The σ denotes the opposite spin of σ. And we get rid of the pre-factor half by considering sum-over different spins, following from spin symmetry

Gv↓w↑Gz↑u↓ = Gv↑w↓Gz↓u↑. Thus, in presence of interaction, the total charge susceptibility (Fig.A-3) after spin decomposition turns out to be:

1 X X χC (q, iΩ ) = − G0 (k + q, iω + iΩ )G0 (k, iω ) vuwz m β vw↑ n m zu↑ n ωn k (A–6) 1 X X − G ↑ ↓(k + q, iω + iΩ )G ↓ ↑(k, iω ) β v w n m z u n ωn k

93 C Figure A-3. Total charge susceptibility χvuwz with its corresponding decomposition from the bare bubble and the interaction bubble.

A.2 Spin susceptibility

The spin operator is defined as:

1X S (q, τ) = c† (k + q, τ)σ c (k, τ) vu 2 vα αβ uβ (A–7) k,αβ

Here the vector of Pauli matrices is denoted as σ, which is given by:       0 1 0 −i 1 0  σ = (σx, σy, σz), σx =   , σy =   , σz =   , (A–8) 1 0 i 0 0 −1

The spin susceptibility calculated from the Matsubara spin-spin correlation function is: Z 2 β χS (q, iΩ ) = dτeiΩmτ ⟨T S (q, τ).S (−q, 0)⟩ vuwz m 3 τ vu zw Z0 β X 1 † † ′ ′ iΩmτ i − i = dτe Tτ cvα(k + q, τ)σαβcuβ(k, τ) czγ(k q, 0)σγδcwδ(k , 0) 6 0 ′ Z kk ,i,αβγδ 1 β X = dτeiΩmτ ⟨T 6 τ 0 kk′,α≠ β × † † ′ − ′ [2cvα(k + q, τ)cuβ(k, τ)czβ(k q, 0)cwα(k , 0) † † ′ − ′ + cvα(k + q, τ)cuα(k, τ)czα(k q, 0)cwα(k , 0) − † † ′ − ′ ⟩ cvα(k + q, τ)cuα(k, τ)czβ(k q, 0)cwβ(k , 0)] (A–9)

We denote the first term in Eq.A–9 as Transverse Spin susceptibility which comes from the

(SxSx + SySy) component: Z β X D E − 1 † † ′ ′ χ+ (q, iΩ ) = dτeiΩmτ T c (k + q, τ)c (k, τ)c (k − q, 0)c (k , 0) vuwz m 2 τ vα uβ zβ wα 0 kk′,α≠ β (A–10)

94 and rest of the term coming from SzSz component as: Z β X 1 † † ′ ′ χzz (q, iΩ ) = dτeiΩmτ ⟨T [c (k + q, τ)c (k, τ)c (k − q, 0)c (k , 0) vuwz m 2 τ vα uα zα wα 0 kk′,α≠ β − † † ′ − ′ ⟩ cvα(k + q, τ)cuα(k, τ)czβ(k q, 0)cwβ(k , 0)] (A–11)

Thus, as shown in Fig.A-4a,

2 1 χS (q, iΩ ) = χ+− (q, iΩ ) + χzz (q, iΩ ) (A–12) vuwz m 3 vuwz m 3 vuwz m

S +− Figure A-4. (a) Total spin susceptibility χvuwz. (b) Transverse spin susceptibility χvuwz. (c) ZZ Longitudinal spin susceptibility χvuwz.

95 Applying momentum conservation and Wick’s theorem as we have seen before in the case of charge susceptibility, we rewrite for the transverse spin susceptibility (see Fig.A-4b): Z β X D E − 1 † † χ+ (q, iΩ ) = − dτeiΩmτ T c (k + q, τ)c (k + q, 0) T c (k, 0)c (k, τ) vuwz m 2 τ vα wα τ zβ uβ 0 k,α≠ β 1 X X = − G0 (k + q, iω + iΩ )G0 (k, iω ) 2β vαwα n m zβuβ n ωn k,α≠ β 1 X X = − G0 (k + q, iω + iΩ )G0 (k, iω ) (A–13) β vw↑ n m zu↓ n ωn k

For longitudinal susceptibility (see Fig.A-4c): Z β X 1 † † zz − iΩmτ χvuwz(q, iΩm) = dτe Tτ cvα(k + q, τ)cwα(k + q, 0) Tτ czα(k, 0)cuα(k, τ) 2 0 Z k,α β X D E 1 † † + dτeiΩmτ T c (k + q, τ)c (k + q, 0) T c (k, 0)c (k, τ) 2 τ vα wβ τ zβ uα 0 k,α≠ β 1 X X = − G0 (k + q, iω + iΩ )G0 (k, iω ) β vw↑ n m zu↑ n ωn k 1 X X + G ↑ ↓(k + q, iω + iΩ )G ↓ ↑(k, iω ) (A–14) β v w n m z u n ωn k When there are no interactions present, as discussed earlier, spin conservation for single particle Green’s function won’t allow for occurrence of Gvσwσ processes. Hence X zz − 1 0 for non-interacting cases, one can have: χvuwz(q, iΩm) = β Gvw↑(k + q, iωn + k,ωn 0 +− iΩm)Gzu↑(k, iωn). This is also quantitatively equal to χvuwz(q, iΩm) and thus, also from Eq.A–4, we write:

2 1 χS (q, iΩ ) = χ+− (q, iΩ ) + χzz (q, iΩ ) = χzz (q, iΩ ) = χ+− (q, iΩ ) vuwz m 3 vuwz m 3 vuwz m vuwz m vuwz m 1 X (A–15) = − G0 (k + q, iω + iΩ )G0 (k, iω ) = χC (q, iΩ ) β vw n m zu n vuwz m k,ωn

S Also, spin-rotation invariance of a paramagnetic system guarantees χvuwz(q, iΩm) =

zz +− χvuwz(q, iΩm) = χvuwz(q, iΩm) even in the presence of interactions, at all times. As shown in

96 Fig.A-5, this is equivalent to the bare susceptibility of the system and is given by:

1 X aw(k + q)av∗(k + q)au(k)az∗(k) 0 − µ µ ν ν − χvuwz(q, iΩm) = [f(Eµ(k + q)) f(Eν(k))] (A–16) Nk iΩm + Eµ(k + q) − Eν(k) k,µν

0 Figure A-5. Bare susceptibility χvuwz for a spin-rotation invariant system for non-interacting case.

A.3 Total RPA susceptibility

From Eq.A–6 and Eq.A–12 we define the total susceptibility as:

T ot χvuwz(q, iΩm) 1 = χC (q, iΩ ) + χS (q, iΩ ) 2 vuwz m vuwz m 1 1 1 = χC (q, iΩ ) + χ+− (q, iΩ ) + χzz (q, iΩ ) 2 vuwz m 3 vuwz m 6 vuwz m 1 X = − [G0 (k + q, iω + iΩ )G0 (k, iω ) + G (k + q, iω + iΩ )G (k, iω )] 4β vwσ n m zuσ n vσwσ n m zσuσ n k,ωnσ 1 X 1 X − G0 (k + q, iω + iΩ )G0 (k, iω ) − G0 (k + q, iω + iΩ )G0 (k, iω ) 6β vwσ n m zuσ n 12β vwσ n m zuσ n k,ωnσ k,ωnσ 1 X + G (k + q, iω + iΩ )G (k, iω ) 12β vσwσ n m zσuσ n k,ωnσ

1 2 3 = Aχvuwz(q, iΩm) + Bχvuwz(q, iΩm) + Cχvuwz(q, iΩm) (A–17)

where A, B, C are pre-factors accompanying χ1,2,3 respectively and for the definition of χ1,2,3, we look at specific spin channels to obtain:

• σv = σw = σz = σu: 1 1 X 2 Aχ1 (q, iΩ ) = − − G0 (k + q, iω + iΩ )G0 (k, iω ) = χ1 (q, iΩ ) vuwz m 4β 12β vwσ n m zuσ n 3 vuwz m k,ωnσ (A–18)

97 X 2 1 −1 0 0 where A = 3 and χvuwz(q, iΩm) = 2β Gvwσ(k + q, iωn + iΩm)Gzuσ(k, iωn). k,ωn,σ

• σv ≠ σw = σz ≠ σu: 1 1 X 1 Bχ2 (q, iΩ ) = − + G (k + q, iω + iΩ )G (k, iω ) = χ2 (q, iΩ ) vuwz m 4β 12β vσwσ n m zσuσ n 3 vuwz m k,ωnσ (A–19) X 1 2 −1 where B = 3 and χvuwz(q, iΩm) = 2β Gvσwσ(k + q, iωn + iΩm)Gzσuσ(k, iωn). k,ωnσ

• σv = σw ≠ σz = σu: 1 X 1 Cχ3 (q, iΩ ) = − G0 (k + q, iω + iΩ )G0 (k, iω ) = χ3 (q, iΩ ) vuwz m 6β vwσ n m zuσ n 3 vuwz m k,ωnσ (A–20) X 1 3 −1 0 0 where C = 3 and χvuwz(q, iΩm) = 2β Gvwσ(k + q, iωn + iΩm)Gzuσ(k, iωn). k,ωnσ

1 Figure A-6. (a) First Triplet channel given by χvuwz. (b) Second Triplet channel given by 2 3 χvuwz. (c) Singlet channel given by χvuwz.

98 From these above definitions, to express χC/+−/zz in terms of χ1,2,3, we rewrite:

C 1 2 χvuwz(q, iΩm) = χvuwz(q, iΩm) + χvuwz(q, iΩm) zz 1 − 2 χvuwz(q, iΩm) = χvuwz(q, iΩm) χvuwz(q, iΩm) (A–21)

+− 3 χvuwz(q, iΩm) = χvuwz(q, iΩm) and using χS = χ+− = χzz, we can invert the above equations to express χ1,2,3 in terms of χC/S:

1 χ1 (q, iΩ ) = χC (q, iΩ ) + χS (q, iΩ ) vuwz m 2 vuwz m vuwz m 1 χ2 (q, iΩ ) = χC (q, iΩ ) − χS (q, iΩ ) vuwz m 2 vuwz m vuwz m 3 S χvuwz(q, iΩm) = χvuwz(q, iΩm) (A–22)

So the total susceptibility stands out to be: T ot 1 C S χvuwz(q, iΩm) = χvuwz(q, iΩm) + χvuwz(q, iΩm) 2 (A–23) 2 1 1 = χ1 (q, iΩ ) + χ2 (q, iΩ ) + χ3 (q, iΩ ) 3 vuwz m 3 vuwz m 3 vuwz m

T ot 0 In the absence of interactions, this gives χvuwz(q, iΩm) = χvuwz(q, iΩm). A.4 Interaction matrices contributing to Self-energy

Figure A-7. Diagram for interaction channel Uvuwz with incoming and outgoing spin and orbital indices labelled on each leg.

The orbital part of interactions in each channel U 1,U 2,U 3 are:

• σv = σw = σz = σu:

1 1 ′ − 1 1 − ′ Uaaaa = 0,Ubbaa = U J, Uabab = 0,Ubaab = J U (A–24)

• σv ≠ σw = σz ≠ σu:

2 2 ′ 2 ′ 2 Uaaaa = U, Ubbaa = U ,Uabab = J ,Ubaab = J (A–25)

99 • σv = σw ≠ σz = σu:

3 − 3 − 3 − ′ 3 − ′ Uaaaa = U, Ubbaa = J, Uabab = J ,Ubaab = U (A–26) where U, U ′, J, J ′ terms correspond to the notations of Kuroki et al. Expressing U 1,2,3 in terms of U C/S gives:

1 1 C S 2 1 C − S 3 S (A–27) U = 2 (U + U ),U = 2 (U U ),U = U

One can see by inspection that channels (1) and (2) couple to χ1,2, and channel (3) couples

3 to χ . Thus, the interaction V (q, Ωm) at any given (q, Ωm) will contain the following matrix structure:

V = U 1χ1U 1 + U 2χ1U 2 + U 1χ2U 2 + U 2χ2U 1 + U 3χ3U 3 (A–28)

Using Eq.A–22 and A–27 in Eq.A–28 above, we can rewrite V in terms of U C/S and χC/S as:

3 1 V = U SχSU S + U C χC U C (A–29) 2 2

However, the second-order bubbles get double-counted from the spin and charge channels in this process, and can be taken care of by a simple subtraction:

3 1 1 V = U SχSU S + U C χC U C − U SC χ0U SC (A–30) 2 2 4 where U SC = U S + U C . This expression has been used for normal-state effective interaction in Chapter 2 and 3.

100 APPENDIX B A SPIN-FLUCTUATION PAIRING STRENGTH FUNCTIONAL INCLUDING QUASIPARTICLE RENORMALIZATION In the multiband Nambu basis given by:   ↑  cα,k  † † Ψαk =   , Ψ = − ↓ † αk cα,k↑ , cα, k cα,−k↓ the band index α ranges from 1 to 5 in FeSCs equal to the number of Fe 3d orbitals. We X † express the non-interacting Hamiltonian as H0 = εα(k)Ψαkτ3Ψαk where εα(k) is the αk dispersion in the α band in the normal state of the system. The Pauli matrices are denoted conventionally by τi, i = 0, 1, 2, 3. The Hamiltonian H0, the full renormalized Green’s function G(k, ωm), the non-interacting Green’s function G0(k, ωm), and the self-energy

Σ(k, ωm) are all 10 × 10 matrices in this multiband basis for every set of (k, ωm). The

−1 non-interacting band Green’s function Gα0(k, ωm) = (iωm − εα(k)) gives the full matrix as − −1 G0(k, ωm) = (iωmτ0 εα(k)τ3) . The particle-hole and the particle-particle Green’s function are defined, respectively, as: Z β † − iωmτ ⟨ ⟩ Gα(k, ωm) = dτe Tτ cα,kσ(τ)cα,kσ(0) Z 0 β (B–1) iωmτ Fα(k, ωm) = dτe ⟨Tτ cα,k↑(τ)cα,−k↓(0)⟩ 0 and the corresponding full Green’s function:  

Gα(k, ωm) Fα(k, ωm)  G(k, ωm) =   ∗ − − − Fα(k, ωm) Gα( k, ωm)

The full self-energy is:  

Σα(k, ωm)Φα(k, ωm)  Σ(k, ωm) =   (B–2) ∗ − − − Φα(k, ωm) Σα( k, ωm)

101 where the particle-hole and particle-particle components are given by: X 1 ′ ′ Σα(k, ωm) = VαβN (k − k , ωm − ωn)Gβ(k , ωn) βNk′ n,β,k′ X (B–3) 1 ′ ′ Φα(k, ωm) = VαβA(k − k , ωm − ωn)Fβ(k , ωn) βNk′ n,β,k′

′ The particle-hole interaction between band α and β is denoted by VαβN (k − k , ωm − ωn)

′ whereas the particle-particle interaction is denoted by VαβA(k − k , ωm − ωn). From Dyson’s equation, we have: −1 −1 − G (k, ωm) = G0 (k, ωm) Σ(k, ωm) (B–4)

Using the Pauli matrices, Σ can be written as:

Σ(k, ωm) = iωm[1 − Zα(k)]τ0 + Xα(k, ωm)τ3 + Φα(k, ωm)τ1 + Φα(k, ωm)τ2 (B–5)

with yet unknown and independent functions Zα(k),Xα(k, ωm), Φα(k, ωm), Φα(k, ωm). From − 2 − 2 Dyson’s Eq.B–4 and the Pauli matrix identity (a0τ0 +⃗a.⃗τ)(a0τ0 ⃗a.⃗τ) = (a0 ⃗a )τ0, one finds: G(k, ωm) = iωmZα(k)τ0 + (εα(k) + Xα(k, ωm))τ3 + Φα(k, ωm)τ1 + Φα(k, ωm)τ2 /Dα(k, ωm) (B–6)

−1 2 2 2 with Dα(k, ωm) = detG (k, ωm) = (iωmZα(k)) − (εα(k) + Xα(k, ωm)) − Φα(k, ωm) −

2 Φα(k, ωm) . Using Eq.B–3 and B–5, we obtain the following self-consistent equations for the four unknown functions:

X ′ − 1 − ′ − iωnZβ(k ) iωm(1 Zα(k)) = VαβN (k k , ωm ωn) ′ βNk′ Dβ(k , ωn) n,β,k′ X ′ ′ 1 − ′ − εβ(k ) + Xβ(k , ωn) Xα(k, ωm) = VαβN (k k , ωm ωn) ′ βNk′ Dβ(k , ωn) n,β,k′ X ′ (B–7) 1 − ′ − Φβ(k , ωn) Φα(k, ωm) = VαβA(k k , ωm ωn) ′ βNk′ Dβ(k , ωn) n,β,k′ X ′ 1 − ′ − Φβ(k , ωn) Φα(k, ωm) = VαβA(k k , ωm ωn) ′ βNk′ Dβ(k , ωn) n,β,k′

102 With the modified dispersion given by the poles of the Green’s function, wedefine: s 2 2 2 (εα(k) + Xα(k, ωn)) Φα(k, ωn) + Φα(k, ωn) Eα(k, ωn) = 2 + 2 (B–8) Zα(k) Zα(k)

The normal state corresponds to a solution Φ = Φ = 0. Z is the quasiparticle renormalization factor, and X describes shifts in the electron energies. The superconducting state is characterized by a non-zero Φ or Φ. From Eq.B–8 the gap function given by:

Φα(k, ωn) − iΦα(k, ωn) ∆α(k, ωn) = (B–9) Zα(k) describes the energy gap in the quasiparticle spectrum. Φ(k, ωn) and Φ(k, ωn) obey the same equations and are expected to have the same functional form up to a common phase factor. This phase factor becomes important in the description of Josephson junctions, but is irrelevant for the thermodynamic properties of a homogeneous superconductor. In the following, we choose the simple gauge Φ(k, ω ) = 0, rendering ∆ (k, ω ) = Φα(k,ωn) . We also n α n Zα(k) 2 2 rewrite Dα(k, ωn) = (iωnZα(k)) − (Eα(k, ωn)Zα(k)) . For simplifying the self-consistent equations further, we will ignore Xα(k, ωn) considering energy shifts to the dispersion being much smaller and negligible compared to the value of εα(k) itself. This leaves us with the following two equations to solve: X − 1 − ′ − iωn iωm(1 Zα(k)) = VαβN (k k , ωm ωn) ′ 2 ′ 2 (B–10a) βNk′ Zβ(k ) ((iωn) − Eβ(k , ωn) ) n,β,k′ X ′ 1 − ′ − Φβ(k , ωn) Φα(k, ωm) = VαβA(k k , ωm ωn) 2 ′ 2 ′ 2 (B–10b) βNk′ Z (k ) ((iωn) − Eβ(k , ωn) ) n,β,k′ β

We will simplify the equations further, under assumptions at T = TC , to obtain solutions without implementing numerically expensive self-consistency loops. The interaction

′ vertices VαβN/A(k − k , ωm − ωn) is expressed in terms of the spin fluctuation contribution S − ′ − C − ′ − χαβ(k k , ωm ωn) and the charge fluctuation contribution χαβ(k k , ωm ωn).

Let’s denote (ωm − ωn) by Ωmn. Fig.B-1 shows the Matsubara frequency dependence of these two terms and of the interaction vertices for a momentum transfer of (π, 0.15π) (incommensurate antiferromagnetic wave-vector) connecting the electron and hole pocket at

103 ′ Figure B-1. The frequency dependence of the effective interaction V (k − k , ωm − ωn) a typical interaction strength for our multiorbital pnictide system. Here, the spin fluctuation contribution is dominant and falls off on a frequency scale that is small compared withthe bandwidth. Thus, the gap equation is dominated by important k and k′ values restricted by this frequency cutoff to remain near Fermi Surface. Without band crossing in the vicinityof the Fermi level, each FS point corresponds to a unique band quantum number, i.e. k ∈ µ, and k′ ∈ ν. Henceforth, we transfer the band index as a subscript of momentum index k. − ′ For Ωmn = 0 in the Matsubara axis, the real quantity VA/N (kµ kν, Ωmn = 0) translates − ′ to the real part of the Retarded quantity Re VA/N (kµ kν, Ω = 0) . We assume the form of the interaction to have the following explicit dependence on the Matsubara frequency: V (k − k′ , Ω = 0) Re V (k − k′ , Ω = 0) − ′ A/N µ ν mn A/N µ ν VA/N (kµ kν, Ωmn) = = (B–11) 1 + W |Ωmn| 1 + W |ωm − ωn|

∝ −1 where W ξC , ξC being the spin-fluctuation energy cut-off. We show the band representation of the effective interaction vertices from orbital space2–7 inEq. in the main text. In Fig.B-1, we also show the numerically fitted curve (in black) to the formof

Eq.B–11. It shows good fit for low frequency region |ωm − ωn| < ξC which makes the largest contribution to the frequency sum in Eq.B–10a and B–10b. As for the external frequency, we

104 solve the Eliashberg equations for low energy physics at the Fermi level at ωm = ω0 = πkBTC

(lowest Fermionic Matsubara frequency). This makes |ωm − ωn| = |ω0 − ωn| = |2nπkBTC |. For simplicity, we assume that the gap function is frequency independent Φµ(k, ωn) = Φ(kµ). First, we apply the above assumptions in d-dimensions to Eq.B–10a: I Z ′ X∞ ∞ 1 [V (kµ, kν, 0)]N d−1 ′ ωn ′ 1 ω0(1−Z(kµ)) = − (d k )∥ dε ′ d | ′ | ν | | kν 2 ′ 2 β(2π) k′ ∈FS vF (kν) 1 + W 2nπkBTC −∞ ω + ε ′ ν n=−∞ n kν (B–12)

′ [V (kµ, kν, 0)]N is expressed in Eq.2–7. The momentum summation over full BZ has been reduced to a transverse and longitudinal integration over the Fermi surface values of the ′ ′ ′ ≈ quantities involved in the equation. At TC , we have ε(kν) >> Φ(kν), hence E(kν, ωn) ′ ε(kν ) ′ ′ = ε ′ . Since the effective interaction is peaked mostly for scattering vectors connecting Z(kν ) kν ′ states (kµ, kν) on the FS, one can identify that the full BZ summation can be replaced by integration over small shells wrapped around the FS. Within the infinitesimal volume of each cubes surrounding individual FS points, we assume that there is only longitudinal variation

′ ′ ′ of [V (kµ, kν, 0)]N , Z(kν) and Φ(kν) parallel to the FS, and no variation in the transverse ′ ′ direction. Hence, they are constant within this infinitesimal cube at any FS point k . ε ′ ν kν is constant along the FS and only varies in the transverse direction. The integration yields 2 tan−1 (∞) = π and together with the factor 1/β in the RHS it cancels the factor |ωn| |ωn|

ω0 = πkBTC = π/β in the LHS, while ωn/|ωn| yields a sign (ωn) in the RHS. Eq.B–12 is rewritten as: I d−1 ′ X∞ 1 ′ (d kν)∥ sign (ωn) Z(kµ) = 1 + d [V (kµ, kν, 0)]N ′ (B–13) (2π) ′ ∈ |vF (k )| 1 + W |2nπkBTC | kν FS ν n=−∞

The Matsubara sum just survives for n = 0 and is cancelled out for all positive (+n) and negative (−n) pairs due to the oddness of sign (ωn). Thus, with sign (ω0) = +1, we arrive at the following equation that needs to be numerically solved for Z(kµ): I d−1 ′ 1 ′ (d kν)∥ Z(kµ) = 1 + d [V (kµ, kν, 0)]N ′ (B–14) (2π) ′ ∈ |v (k )| kν FS F ν

105 Returning to Eq.B–10b, we proceed similarly as above: I ′ ′ d−1 ′ X∞ π [V (k , k , 0)] Φ(k ) (d k )∥ 1 1 − µ ν A ν ν Φ(kµ) = d ′ ′ . (B–15) β(2π) ′ ∈ Z(k ) |vF (k )| 1 + W |2nπkBTC | |ωn| kν FS ν ν n=−∞

We can perform the Matsubara sum. Rewriting in terms of the dimensionless parameter

A = 1/(2πW kBTC ) :

∞ ∞ ∞ ∞ π X 1 1 X A 1 X A 1 X A 1 . = . = . + . β 1 + |n|/A |ω | A + |n| |2n + 1| A + 1 + n 2n + 1 A + n 2n + 1 n=−∞ n n=−∞ n=0 n=0 (B–16) where we shifted n to (n − 1) in the first infinite summation term. For low critical temperatures, A >> 1, and we can absorb A + 1 ≈ A which gives:

∞ ∞ π X 1 1 X A 1 . = 2 . (B–17) β 1 + |n|/A |ω | A + n 2n + 1 n=−∞ n n=0

We refer to identities of the Digamma function Ψ(z), one of which is:

∞ X z − 1 Ψ(z) = −γ + (B–18) (2n + 1)(z + n) n=0

γ is the Euler-Macheroni constant. For z >> 1, we have z − 1 ≈ z and Ψ(z) ≈ log(z). Thus, rewriting Eq.B–17:

∞ π X 1 1 . ≈ 2 (γ + log(A)) = 2log (eγA) (B–19) β 1 + |n|/A |ω | n=−∞ n

This makes the linearized gap equation from Eq.B–15 take the following form: I γ ′ d−1 ′ 2log (e A) [V (k , k , 0)] (d k )∥ − µ ν A ν ′ Φ(kµ) = d ′ ′ Φ(kν) (B–20) (2π) ′ ∈ Z(k ) |v (k )| kν FS ν F ν

The gap equation above turns out to be an eigen value equation which can now be solved

106 Defining λ as:

H H d−1 ′ d−1 ′ 1 (d kµ)∥ [V (kµ,kν ,0)]A (d kν )∥ ′ 2d ′ ∈ ∈ | | Φ(kµ) ′ | ′ | Φ(kν) − (2π) kν FS kµ FS vF (kµ) Z(kν ) vF (kν ) λ = H d−1 (B–22) (d kµ)∥ 2 d Φ (kµ) kµ∈FS (2π) |vF (kµ)| and dropping the constant log term from our numerical solution, we get the following eigen value equation from Eq.B–20 that needs to be numerically evaluated:

X d−1 ′ 1 1 (d k )∥ λg(k ) = − [V (k , k′ , 0)] ν g(k′ ) (B–23) µ d ′ µ ν A | ′ | ν (2π) ′ ∈ Z(kν) vF (kν) kν FS

The final modified gap structure due to Z(k ) is given by ∆(k ) = g(kµ) . µ µ Z(kµ)

107 APPENDIX C NUMERICAL DETAILS FOR SELF-ENERGY EVALUATION Here, we will outline few generic relations that are practical, saves memory and speeds up the actual numerical calculations. For a paramagnetic normal state that is time-reversal invariant, the Green’s function obeys the relation:

∗ Gps(k, ωm) = Gsp(k, −ωm) (C–1)

0 ∗ Following this, the non-interacting susceptibility obeys the relation χpqst(q, Ωm) = 0 − χstpq(q, Ωm). The interaction matrix elements as mentioned in Eq. 2–5 are symmetric under the interchange Upqst = Ustpq, which allows for the spin and charge RPA susceptibilities and the effective interaction to obey the same symmetry relation like the non-interacting susceptibility:

∗ Vpqst(q, Ωm) = Vstpq(q, −Ωm) (C–2)

These relations help in restricting our calculations to only one-half of the Matsubara plane while the other half can be symmetry related. With the temperature set to T = 100K, we used 400 Matsubara frequencies in the upper half plane for all our calculations. We performed 2D calculations with a k-mesh of 40 × 40 in the unfolded 1-Fe BZ. The orbitally resolved non-interacting susceptibility is evaluated using the Lindhard expression:

1 X 0 − 0 0 χpqst(q, Ωm) = Gtq(k, ωm)Gps(k + q, ωm + Ωm) Nkβ kωm X t q∗ p s∗ (C–3) 1 aµ(k)aµ (k)aν(k + q)aν (k + q) = − [f(Eµ(k)) − f(Eν(k + q))] Nk iΩm + Eµ(k) − Eν(k + q) kµν where f(ϵ) = 1/(eβϵ + 1) is the Fermi-Dirac distribution function. To evaluate the self-energy as in Eq. 3–8, we used circular convolution theorem with Fast Fourier transform along the momentum space.

108 To ensure particle number conservation, one has to evaluate a new chemical potential µ X such that the electron density is a given n = 2 f(Eν(k)). Here, Eν(k) is the eigenvalue of νk the unperturbed Hamiltonian H0(k). With the interacting Green’s function, we can evaluate n with the following equation:

X 2 X − 0 n = 2 f(Eν(k)) + Gpp(k, ωm) Gpp(k, ωm) (C–4) βNk νk p,k,ωm

In order to produce the plots in Fig. 3-4 with high resolution, we have interpolated our Matsubara self-energy data at each frequency point to a k-mesh of 250 × 250 points, followed by the evaluation of the corresponding renormalized Green’s function. We folded our spectral function from 1-Fe to 2-Fe BZ to present our results in a manner that is easily comparable to experimental data. Padé Approximation: Once we have evaluated a quantity in the Matsubara plane, Padé approximation for analytic continuation can be used to evaluate its corresponding value in the Retarded plane, under two specific circumstances:

1. when the function to be continued is not given analytically (otherwise, one can use iωm → ω + iδ)

2. when the function is given without statistical errors (for example, Quantum Monte Carlo Green’s function uses Maximum Entropy Method) There is one specific drawback of the Padé approximation: it is a polynomial representation that has limited precision for functions which are hard to approximate by a polynomial. It can show the wiggles that are typical for polynomial interpolations. The algorithm for calculating the Padé approximant was written by Vidberg and Serene [148]. Given the values um of a function at N complex points zm(m = 1, ..., N), we define the continued fraction

a1 CN (z) = − (C–5) 1 + (z z1)a2 1+...(z−zN−1)aN

109 Here, z can be the retarded frequency ω, and zm = iωm. The coefficients am are to be determined so that

CN (zm) = um, m = 1, ..., N (C–6)

The coefficients are then given by the recursion

am = gm(zm), g1(zm) = um, m = 1, ..., N

gp−1(zp−1) − gp−1(z) gp(z) = , p ≥ 2 (C–7) (z − zp−1)gp−1(z)

This requires the following calculation:

g1(z1) = a1 = u1, g1(z2) = u2, g1(z3) = u3, g1(z4) = u4,...

g1(z1) − g1(z) a1 − g1(z2) a1 − u2 g2(z) = , g2(z2) = a2 = = , (z − z1)g1(z) (z2 − z1)g1(z2) (z2 − z1)u2

a1 − g1(z3) a1 − u3 a1 − u4 g2(z3) = = , g2(z4) = ,... (z3 − z1)g1(z3) (z3 − z1)u3 (z4 − z1)u4 g2(z2) − g2(z) a2 − g2(z3) a2 − g2(z4) g3(z) = , g3(z3) = a3 = , g3(z4) = ,... (z − z2)g2(z) (z3 − z2)g2(z3) (z4 − z2)g2(z4) g3(z3) − g3(z) a3 − g3(z4) g4(z) = , g4(z4) = a4 = ,... (z − z3)g3(z) (z4 − z3)g3(z4)

Thus, the following triangular matrix pm,j has to be calculated:

j = 1 j = 2 j = 3 j = 4 ...

m = 1 a1 = u1 u2 u3 u4 ...

m = 2 a2 g2(z3) g2(z4) ... (C–8) m = 3 a3 g3(z4) ...

m = 4 a4 ......

This can be done as:

pm−1,m−1 − pm−1,j p1,j = uj, j = 1, ..., N and pm,j = , j = 2, ..., N and m = 2, ..., j (C–9) (zj − zm−1)pm−1,j

110 The diagonal elements of this matrix then contains the coefficients am = pm,m , which are used to evaluate the continued fraction Eq.C–5 to obtain the analytically continued value of the given quantity.

111 APPENDIX D RPA CALCULATION ON THE TOY TWO-BAND MODEL In Section 3.2, we illustrated the pocket shrinking mechanism proposed by Ortenzi et al. in Ref. [59] within a toy two-band model for FeSC with two-dimensional (2D) parabolic bands. Here, we analyze the case of a momentum-dependent spin-fluctuation interaction and verify that, while it seems natural to associate scattering process at (±π, 0) with interband interactions, in order to recover the shrinking of the pockets one has to consider the region in momentum space large enough to accommodate scattering processes that connect sufficiently high energy states in an electron (hole) band to those near the top (bottom) of the hole (electron) band. The hole and electron band dispersion are:

− 2 2 EΓ(k) = γ(kx + ky) + µ (D–1) − 2 2 − EM˜ (k) = γ (kx π) + ky µ (D–2)

With γ = 1.5, and µ = 1, we fix the units of energy in terms of µ for our current analysis. We ignore the presence of the M-centered electron pocket since it doesn’t add any further insight to our analysis. The non-interacting susceptibility in band-basis (α, β) is:

T X 0 − 0 0 χαβ(q, Ωm) = Gα(k, ωm)Gβ(k + q, ωm + Ωm) (D–3) Nk k,ωm

C/S and the charge- and spin-fluctuation parts of the RPA susceptibility is χ (q, Ωm) =

0 −1 0 [1 ± χ (q, Ωm)U] χ (q, Ωm). The corresponding particle-hole interaction is: 3 2 S 1 2 C 2 0 Vαβ(q, Ωm) = U χ (q, Ωm) + U χ (q, Ωm) − U χ (q, Ωm) (D–4) 2 2 αβ

A bare Coulomb interaction parameter of U = 8µ and T = 0.05µ was used for the numerical evaluation. We provide both analytical predictions using Eq. 3–3-3–4 and numerical results performed using Vαβ(q, Ωm) from Eq. D–4 in Eq. 3–1 for the static self-energy, as a function

′ ′ of qrad. In Fig. D-1, we show that both the analytical (Σα)an and numerical (Σα)num results

112 ′ Figure D-1. Plot of the real part of the static self-energy Σα(ω0) (in units of energy) at Γ and ˜ M point as a function of varying qrad. Both numerical results and analytical predictions have been included. Inset: The shaded region in red depicts the area of q-integration as in Eq. 3–1 within the radius qrad, centered around the (±π, 0) antiferromagnetic wavevector.

agree qualitatively, changing sign at a specific qrad. It eventually yields a desirable value for FS shrinkage, in the limit where qrad encompasses the full BZ integration. While the zero of the analytical functions is found exactly at the particle-hole symmetry point (see Eq. 3–3-3–4), the change of sign for the numerical self-energies occurs at a slightly different momenta due to the effect of the momentum dependence of Vαβ(q, Ωm). The agreement between analytical predictions and numerical results is a strong verification of our correct understanding of the interplay between momentum transfer and finite-energy scattering, and its role in FS shrinkage. With our findings, we re-emphasize that finite energy scattering processes that are sensitively dependent on the upper/lower edge of the band structure are also important in determining the correct sign of the self-energy that causes FS shrinkage.

113 APPENDIX E COMPARISON OF QUASIPARTICLE WEIGHTS OBTAINED VIA RPA AND TPSC

Figure E-1. (a) Renormalized Fermi surface of LiFeAs evaluated via RPA at U = 0.8 eV and J = U/8. (b) Plot of the evolution of quasiparticle weight Z with increasing U value evaluated via RPA for momentum points 3 and 5 as marked in figure (a). (c) Same as in (b) but for TPSC calculation using Hubbard-Kanamori interaction matrices.

In this section, we will compare quasiparticle weights Z(k) obtained via different methods and analyze its evolution with respect to the Hubbard interaction. The quasiparticle weight, strictly speaking, is a quantity defined only at the Fermi level. This means only

Zdxz/yz and Zdxy can be analyzed for the orbitals carrying weight on the Fermi surface of FeSCs, leaving out Z and Z . dx2−y2 d3z2−r2 The comparison of Z values obtained from RPA and TPSC is not straightforward because the parameters corresponding to U, J, etc. have slightly different meanings and must be interpreted as renormalized quantities. For example, it is well-known that the RPA produces a rather good approximation to the structure of the exact (Quantum Monte Carlo) magnetic susceptibility of the Hubbard model in momentum and frequency space, but requires a strong downward renormalization of U to do so[89]. RPA and TPSC approximations therefore work at different scales of U, rendering qualitatively similar but quantitatively different values of various quantities. To be more consistent with comparing results, at the same scale of U, the TPSC evaluation presented in this section has been obtained using Hubbard-Kanamori form of the interaction matrices as used in RPA with J

114 set to U/8, unlike the main text where cRPA values of the interaction parameters were used to compare with Ref. [35]. In Fig. E-1b and E-1c, we have shown the evolution of Z as a function of U for LiFeAs from RPA and TPSC calculations, respectively. As already discussed in Section 3.4.4, upon increasing U value, the two inner hole pockets of LiFeAs are pushed below the Fermi level making momentum points 1 and 2 (see Fig. E-1a) vanish. Hence, data for these points are not shown. The UC = 1.19 eV marked in Fig. E-1b signifies the critical U value causing antiferromagnetic instability in RPA calculation. On the other hand, TPSC calculations are not inhibited by the same UC at such low values of U. One can see the expected trend of

Z decreasing as U increases. For all the cases, we see that Zdxy < Zdxz/yz , but the TPSC Z values are much larger than the RPA Z values. Point 4 has dxy orbital content which is the same as point 3, hence not repeated in the plot.

115 APPENDIX F MODEL DETAILS FOR ELECTRON-HOLE TOY BANDS FOR FESE(S) In order to depict realistic model for FeSe(S), the calculations were performed by taking simple parabolic dispersion for the electronic structure in continuum space with inversion symmetry preserved. Each of the pockets were chosen to have a quadratic dispersion,

2 ϵΓ(k) = −αk + E+ ! 2 2 kx − π ky ϵ (k) = α + − E− +X 1 + ε 1 − ε ! 2 2 kx + π ky ϵ− (k) = α + − E− X 1 + ε 1 − ε (F–1) ! 2 2 kx ky − π ϵ (k) = α + − E− +Y 1 − ε 1 + ε ! 2 2 kx ky + π ϵ− (k) = α + − E− Y 1 − ε 1 + ε

2 with the parameters α = 8/π and E+ = 0.6 , E− = 0.6, and ε = 0.2 in arbitrary units. The continuum space limit kx = ky = −π : π mimics the First Brillouin Zone of an FeSC.

The energy is fixed in units of (∆Γ)A. The order parameters are explicitly given by

2 − 2 ∆1(k) = ∆Γ + ∆Γa(kx ky) ! k − π 2 k 2 ∆ (k) = ∆ + ∆ − x + y 2 X Xa 1 + ε 1 − ε ! k + π 2 k 2 ∆ (k) = ∆ + ∆ − x + y 3 X Xa 1 + ε 1 − ε (F–2) ! k 2 k − π 2 ∆ (k) = ∆ + ∆ + x − y 4 Y Y a 1 − ε 1 + ε ! k 2 k + π 2 ∆ (k) = ∆ + ∆ + x − y 5 Y Y a 1 − ε 1 + ε

2 The parameter values for the anisotropic components are ∆Γa = 0.4/π , ∆Xa = ∆Y a = 0.16/π2. The set of isotropic values which were tuned for mimicking Fermi Surface evolution in FeSe1−xSx from zero doping to higher doping (A → D), are given as [∆Γ, ∆X , ∆Y ] in the

116 set: A = 0.40, 0.35, 0.35 ,B = 0.35, 0.27, 0.35 C = 0.16, 0.20, 0.25 ,D = 0.07, 0.07, 0.07

As observed later from the STM spectrum evaluation, the largest peak of the STM from set A occurs at (∆Γ)A = 0.5, hence all energy scales in the plots have been normalized to represent energy in units of (∆Γ)A.

In Nambu basis, the normal state Hamiltonian HN is given by this 10 × 10 matrix:   ϵΓ(k) 0 0 0 0 0 0 0 0 0       0 ϵΓ(k) 0 0 0 0 0 0 0 0       0 0 ϵ (k) 0 0 0 0 0 0 0   +X       0 0 0 ϵ+X (k) 0 0 0 0 0 0       0 0 0 0 ϵ−X (k) 0 0 0 0 0    HN (k) =    0 0 0 0 0 ϵ− (k) 0 0 0 0   X       0 0 0 0 0 0 ϵ+Y (k) 0 0 0       0 0 0 0 0 0 0 ϵ+Y (k) 0 0       0 0 0 0 0 0 0 0 ϵ− (k) 0   Y 

0 0 0 0 0 0 0 0 0 ϵ−Y (k)

117 The gap Hamiltonian H∆ with only inter-band spin triplet component (∆0)ΓX =

(∆0)ΓY = ∆0 = 0.4 is given by:    0 ∆1(k) 2∆0 0 2∆0 0 2∆0 0 2∆0 0      −∆1(k) 0 0 0 0 0 0 0 0 0       −2∆ 0 0 ∆ (k) 0 0 0 0 0 0   0 2     −   0 0 ∆2(k) 0 0 0 0 0 0 0       −2∆0 0 0 0 0 ∆3(k) 0 0 0 0    H∆(k) =    0 0 0 0 −∆ (k) 0 0 0 0 0   3     −   2∆0 0 0 0 0 0 0 ∆4(k) 0 0       0 0 0 0 0 0 −∆4(k) 0 0 0       −2∆ 0 0 0 0 0 0 0 0 ∆ (k)  0 5 

0 0 0 0 0 0 0 0 −∆5(k) 0

Temperature dependence of the gap was introduced by a phenomenological BCS-like functional such that H∆(k) was multiplied by ∆(T ) = BCS(f(T )) and gives

H∆(k,T ) = H∆(k) × ∆(T ). The full BdG Hamiltonian H(k,T ) is:  

 HN (k) H∆(k,T )  H(k,T ) =   † − T − H∆(k,T ) HN ( k)

Diagonalizing the above H(k,T ) gives the full BdG dispersion E(k,T ), which is then used for the following calculations. F.1 Derivation of equations for Specific heat evaluation

We have the equation for specific heat as derivative of the entropy of the system:

1 dS −2 X ∂n(E ) 1 ∂β −1 C (T ) = = E k,T where, β = , = V T dT T 2 k,T ∂β T ∂T T 2 k We have

1 ∂n(E ) −e(Ek,T /T ) ∂n(E ) −E ∂n(E ) n(E ) = , k,T = , k,T = k,T k,T k,T (E /T ) (E /T ) 2 e k,T + 1 ∂Ek,T T (e k,T + 1) ∂T T ∂Ek,T

118 From chain rule, we get the following: ∂n(E ) ∂n(E ) ∂T ∂n(E ) ∂n(E ) ∂n(E ) ∂E k,T = k,T = −T 2 k,T = −T 2 k,T + k,T k,T ∂β ∂T ∂β ∂T ∂T ∂Ek,T ∂T ∂n(E ) −E ∂E = −T 2 k,T k,T + k,T ∂Ek,T T ∂T

Putting this back for evaluating CV , we get: − X X − E2 2 ∂n(Ek,T ) ∂n(Ek,T ) k,T − ∂Ek,T CV (T ) = 2 Ek,T = 2 Ek,T T ∂β ∂Ek,T T ∂T k k

−∂n(E) A numerically efficient way to calculate ∂E is by rewriting it in terms of n(E) as following:

1 1 e(E/T ) n(E)n(−E) = = = (e(E/T ) + 1)(e(−E/T ) + 1) 2 + e(E/T ) + e(−E/T ) 2e(E/T ) + e(2E/T ) + 1 which means we can have

e(Ek,T /T ) ∂n(E ) n(E )n(−E ) = = −T k,T , k,T k,T (E /T ) 2 (e k,T + 1) ∂Ek,T (F–3) X n(E )n(−E ) ∂E C (T ) = 2 k,T k,T E2 − TE k,T V T 2 k,T k,T ∂T k

F.2 Spectral function evaluation

The spectral function is given by: X X | j |2 1 1 aµ(k) A(k, ω) = − Im (Gjj(k, ω)) = − Im π π ω + iη − Eµ(k) j∈particles j∈particles,µ

X | j |2 1 η aµ(k) A(k, ω) = 2 2 (F–4) π η + (ω − Eµ(k)) j∈particles,µ

F.3 DOS and Tunneling conductance evaluation

In a multiband system, we have the density of states given by:

X X | j |2 1 1 η aµ(k) ρ(E) = A(k,E) = 2 2 (F–5) Nk πNk η + (E − Eµ(k)) k j∈particles,µ,k

119 For the dI/dV spectrum as observed in STM, we evaluate the following. We know that:

Z ∞ 4πe 2 It→s(V ) = − |M| ρS(E)ρt(E − eV )[n(E − eV ) − n(E)] dE ℏ −∞

Let’s rewrite the equation for I(V ) such that we only consider the crucial response

ρS(E) = ρ(E) from the sample. The tip ρt(E − eV ) can be absorbed as a constant for low temperatures and for constant DOS for normal material of the tip for energy scales close to Fermi level (eV << EF ). For a grid of Energy bias E ranging between −EL to EU close to the Fermi level, we can rewrite: Z EU I(V ) = C ρ(E)[n(E − eV ) − n(E)] dE −EL Z Z (F–6) dI(V ) EU d eC EU ρ(E)e(E−eV )/T = C ρ(E) [n(E − eV )] dE = dE (E−eV )/T 2 dV −EL dV T −EL (1 + e )

F.4 Zeeman coupling to external magnetic field

We used the same parabolic dispersion for the non-interacting normal state as before. But in order to mimic realistic electronic structure of FeSe system further, we made the hole bands heavier than the electron bands by doubling its band mass:

k2 ϵ (k) = −α + E (F–7) Γ 2 +

2 0.6 with the parameters α = 8/π and E+ = 2 . The corresponding superconducting order parameter of the hole band is given by

k2 − k2 ∆ (k) = ∆ + ∆ x y (F–8) 1 Γ Γa 2

With all other definitions remaining same as in the previous model, the parameter values

2 2 for the anisotropic components are ∆Γa = 0.4/π , ∆Xa = ∆Y a = 0.12/π . We only choose to study the system at the critical topological transition point between vanishing of BFSs and appearance of fully-gapped superconducting order. The corresponding set of critical isotropic values is ∆Γ = 0.23 ,∆X = 0.28, ∆Y = 0.33. The inter-band spin triplet component is set to ∆0 = 0.3.

120 The hope is that changing the direction (+z to −z and vice versa) of the external magnetic field makes the BFSs reappear/disappear on hole/electron pockets. In the presence of weak external magnetic field, the perturbation to the normal state Hamiltonian isgiven by the Zeeman coupling term: σ.h, where σ is the Pauli matrix in spin basis and h is the external magnetic field. In Nambu basis, the Zeeman perturbation term(σZ hZ ) for magnetic field along z-direction yields:   +h 0 0 0 0 0 0 0 0 0       0 −h 0 0 0 0 0 0 0 0       0 0 +h 0 0 0 0 0 0 0         0 0 0 −h 0 0 0 0 0 0       0 0 0 0 +h 0 0 0 0 0    Hh =    0 0 0 0 0 −h 0 0 0 0         0 0 0 0 0 0 +h 0 0 0       0 0 0 0 0 0 0 −h 0 0       0 0 0 0 0 0 0 0 +h 0    0 0 0 0 0 0 0 0 0 −h

The full normal state Hamiltonian is now given by: HN (k) + Hh. Diagonalizing the full H(k,T ) gives the full BdG dispersion E(k,T ), which is then used for evaluating various response quantities as described before.

121 APPENDIX G SPATIAL AND ORBITAL STRUCTURE OF THE GAP FUNCTION The gap function ∆(k) encodes the spatial and orbital structure information of the

Cooper pairs. The gap function has its largest magnitude ∆µ(kF) on the µ-th Fermi band and falls off away from the FS, a behavior which can be parametrized in terms of a Gaussian cutoff: X 2 (−(|k−kF|/λC ) ) ∆µ(k) = ∆µ(kF)exp (G–1)

kF∈µ where λC = 2π/ξ0 with ξ0 taken to be of order 3 times the lattice vector a. This provides a local picture of the internal orbital structure of a pair which continues out to a radius set by the coherence length ξ0. While extrapolating the gap from the FS over the full BZ, one should keep in mind the translational invariance condition for the BZ, i.e, to obtain ∆µ(k)

st values in the 1 BZ (−π : π), contributions of ∆µ(kF) values from the neighboring BZs (−2π : 2π) should also be taken into account. The following derivation has been adapted from Ref.[79]. The gap operator ∆ˆ can be written in the BdG representation as X ˆ 1 ∆ = ∆ν(k)γν↑(k)γν↓(−k) (G–2) Nk ν,k where γνσ(k) is the destruction operator for an electron in the ν-th band with wavevector

q k and spin σ. Using the band-orbital unitary transformation matrix elements aν(k), we can write: X q γνσ(k) = aν(k)cqσ(k) (G–3) q where cqσ(k) is the destruction operator for an electron in q-th orbital. Thus, we rewrite the gap operator in terms of the orbital operators as X ˆ 1 ∆ = ∆qt(R1 − R2)cq↑(R1)ct↓(R2) (G–4) Nk R1,R2

122 with

1 X q t − −ik.R ∆qt(R) = ∆ν(k)aν(k)aν( k)e (G–5) Nk ν,k where R = R1 − R2. The amplitude ∆qt(R) encodes the internal spatial and orbital structures of the pair. Another simple Fourier transform back to the momentum space can X ik.R yield the corresponding orbitally-resolved pairing structure ∆qt(k) = ∆qt(R)e . R

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137 BIOGRAPHICAL SKETCH Shinibali Bhattacharyya was born in Burdwan, an old city in the state of West Bengal, India. She grew up in the suburbs of Kolkata, in a little town called Barasat, where she pursued her preliminary education. She attended the same high school for 14 years: starting from kindergarten all through her 12th grade. There, she consistently excelled in her curriculum and fuelled a naive dream of becoming an astronaut one day. Her parents always sustained her curiosity and hunger for success. They knew that she would need to emerge out of the shell of her tiny hometown and venture out in the larger world to fulfill her dreams. Shinibali went on to pursue her 3-year BSc degree in physics from University of Calcutta, followed by a 2-year MSc degree from Indian Institute of Technology (IIT) Bombay. Majoring in physics exposed her to a variety of other interesting sub-fields and eventually, she found her calling in theoretical condensed matter physics. She joined the University of Florida in fall 2014 for earning her doctorate in the same. With 6 years of toil, ample luck and fortunate moments, and under the heaps of multiple sacrifices and gains, she turned out to be the first doctorate in her family, bringing unbounded joy and pride to her parents. She received her Ph.D. from the University of Florida in December, 2020. Shinibali knows this is only the beginning of her uphill climb, and the best of her achievements are yet to come.

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