Effective field theories for strongly correlated fermions - Insights from the functional renormalization group

'There are very few things that can be proved rigorously in condensed matter physics.' These famous words, brought to us by Nobel laureate Anthony James Leggett in 2003, summarize very well the challenging nature of problems researchers find themselves confronted with when entering the fascinating field of condensed matter physics. The former roots in the inherent many-body character of several quantum mechanical particles with modest to strong interactions between them: their individual properties might be easy to understand, while their collective behavior can be utterly complex. Strongly correlated electron systems, for example, exhibit several captivating phenomena such as superconductivity or spin-charge separation at temperatures far below the energy scale set by their mutual couplings. Moreover, the dimension of the respective Hilbert space grows exponentially, which impedes the exact diagonalization of their Hamiltonians in the thermodynamic limit. For this reason, renormalization group (RG) methods have become one of the most powerful tools of condensed matter research - scales are separated and dealt with iteratively by advancing an RG flow from the microscopic theory into the low-energy regime. In this thesis, we report on two complementary implementations of the functional renormalization group (fRG) for strongly correlated electrons. Functional RG is based on an exact hierarchy of coupled differential equations, which describe the evolution of one-particle irreducible vertices in terms of an infrared cutoff Lambda. To become amenable to numerical solutions, however, this hierarchy needs to be truncated. For sufficiently weak interactions, three-particle and higher-order vertices are irrelevant at the infrared fixed point, justifying their neglect. This one-loop approximation lays the foundation for the N-patch fRG scheme employed within the scope of this work. As an example, we study competing orders of spinless fermions on the triangular lattice, mapping out a rich phase diagram with several charge and pairing instabilities. In the strong-coupling limit, a cutting-edge implementation of the multiloop pseudofermion functional renormalization group (pffRG) for quantum spin systems at zero temperature is presented. Despite the lack of a kinetic term in the microscopic theory, we provide evidence for self-consistency of the method by demonstrating loop convergence of pseudofermion vertices, as well as robustness of susceptibility flows with respect to occupation number fluctuations around half-filling. Finally, an extension of pffRG to Hamiltonians with coupled spin and orbital degrees of freedom is discussed and results for exemplary model studies on strongly correlated electron systems are presented

Taming pseudo-fermion functional renormalization for quantum spins: Finite-temperatures and the Popov-Fedotov trick

The pseudo-fermion representation for $S=1/2$ quantum spins introduces unphysical states in the Hilbert space which can be projected out using the Popov-Fedotov trick. However, state-of-the-art implementation of the functional renormalization group method for pseudo-fermions have so far omitted the Popov-Fedotov projection. Instead, restrictions to zero temperature were made and absence of unphysical contributions to the ground-state was assumed. We question this belief by exact diagonalization of several small-system counterexamples where unphysical states do contribute to the ground state. We then introduce Popov-Fedotov projection to pseudo-fermion functional renormalization, enabling finite temperature computations with only minor technical modifications to the method. At large and intermediate temperatures, our results are perturbatively controlled and we confirm their accuracy in benchmark calculations. At lower temperatures, the accuracy degrades due to truncation errors in the hierarchy of flow equations. Interestingly, these problems cannot be alleviated by switching to the parquet approximation. We introduce the spin projection as a method-intrinsic quality check. We also show that finite temperature magnetic ordering transitions can be studied via finite-size scaling.Comment: 14 pages, 8 figures; minor clarifications, added reference

Spin-valley magnetism on the triangular moir\'e lattice with SU(4) breaking interactions

The discovery of correlated insulating states in moir\'e heterostructures has renewed the interest in strongly-coupled electron systems where spin and valley (or layer) degrees of freedom are intertwined. In the strong-coupling limit, such systems can be effectively described by SU(4) spin-valley models akin to Kugel-Khomskii models long studied in the context of spin-orbit coupled materials. However, typical moir\'e heterostructures also exhibit interactions that break the SU(4) symmetry down to SU(2)${}_{\mathrm{spin}}\otimes$U(1)${}_{\mathrm{valley}}$. Here we investigate the impact of such symmetry-breaking couplings on the magnetic phase diagram for triangular superlattices considering a filling of two electrons (or holes) per moir\'e unit cell. We explore a broad regime of couplings -- including XXZ anisotropies, Dzyaloshinskii-Moriya exchange and on-site Hund's couplings -- using semi-classical Monte Carlo simulations. We find a multitude of classically ordered phases, including (anti-)ferromagnetic, incommensurate, and stripe order, manifesting in different sectors of the spin-valley model's parameter space. Zooming in on the regimes where quantum fluctuations are likely to have an effect, we employ pseudo-fermion functional renormalization group (pf-FRG) calculations to resolve quantum disordered ground states such as spin-valley liquids, which we indeed find for certain parameter regimes. As a concrete example, we discuss the case of trilayer graphene aligned with hexagonal boron nitride using material-specific parameters.Comment: 20 pages, 16 figure

Multiloop functional renormalization group approach to quantum spin systems

Renormalization group methods are well-established tools for the (numerical) investigation of the low-energy properties of correlated quantum many-body systems, allowing to capture their scale-dependent nature. The functional renormalization group (FRG) allows to continuously evolve a microscopic model action to an effective low-energy action as a function of decreasing energy scales via an exact functional flow equation, which is then approximated by some truncation scheme to facilitate computation. Here, we report on our transcription of a recently developed multiloop truncation approach for electronic FRG calculations to the pseudo-fermion functional renormalization group (pf-FRG) for interacting quantum spin systems. We discuss in detail the conceptual intricacies of the flow equations generated by the multiloop truncation, as well as essential refinements to the integration scheme for the resulting integro-differential equations. To benchmark our approach we analyze antiferromagnetic Heisenberg models on the pyrochlore, simple cubic and face-centered cubic lattice, discussing the convergence of physical observables for higher-loop calculations and comparing with existing results where available. Combined, these methodological refinements systematically improve the pf-FRG approach to one of the numerical tools of choice when exploring frustrated quantum magnetism in higher spatial dimensions.Comment: 22 pages, 9 figure

MatsubaraFunctions.jl: An equilibrium Green's function library in the Julia programming language

The Matsubara Green's function formalism stands as a powerful technique for computing the thermodynamic characteristics of interacting quantum many-particle systems at finite temperatures. In this manuscript, our focus centers on introducing MatsubaraFunctions.jl, a Julia library that implements data structures for generalized n-point Green's functions on Matsubara frequency grids. The package's architecture prioritizes user-friendliness without compromising the development of efficient solvers for quantum field theories in equilibrium. Following a comprehensive introduction of the fundamental types, we delve into a thorough examination of key facets of the interface. This encompasses avenues for accessing Green's functions, techniques for extrapolation and interpolation, as well as the incorporation of symmetries and a variety of parallelization strategies. Examples of increasing complexity serve to demonstrate the practical utility of the library, supplemented by discussions on strategies for sidestepping impediments to optimal performance.Comment: 37 pages, 10 figure

TMDs as a platform for spin liquid physics: A strong coupling study of twisted bilayer WSe2_2

The advent of twisted moir\'e heterostructures as a playground for strongly correlated electron physics has led to a plethora of experimental and theoretical efforts seeking to unravel the nature of the emergent superconducting and insulating states. Amongst these layered compositions of two dimensional materials, transition metal dichalcogenides (TMDs) are by now appreciated as highly-tunable platforms to simulate reinforced electronic interactions in the presence of low-energy bands with almost negligible bandwidth. Here, we focus on the twisted homobilayer WSe$_2$ and the insulating phase at half-filling of the flat bands reported therein. More specifically, we explore the possibility of realizing quantum spin liquid (QSL) physics on the basis of a strong coupling description, including up to second nearest neighbor Heisenberg couplings $J_1$ and $J_2$, as well as Dzyaloshinskii-Moriya (DM) interactions. Mapping out the global phase diagram as a function of an out-of-plane displacement field, we indeed find evidence for putative QSL states, albeit only close to SU$(2)$ symmetric points. In the presence of finite DM couplings and XXZ anisotropy, long-range order is predominantly present, with a mix of both commensurate and incommensurate magnetic phases.Comment: 12 pages, 5 figures, supplemental material (3 pages, 1 figure

Pseudo-fermion functional renormalization group for spin models

For decades, frustrated quantum magnets have been a seed for scientific progress and innovation in condensed matter. As much as the numerical tools for low-dimensional quantum magnetism have thrived and improved in recent years due to breakthroughs inspired by quantum information and quantum computation, higher-dimensional quantum magnetism can be considered as the final frontier, where strong quantum entanglement, multiple ordering channels, and manifold ways of paramagnetism culminate. At the same time, efforts in crystal synthesis have induced a significant increase in the number of tangible frustrated magnets which are generically three-dimensional in nature, creating an urgent need for quantitative theoretical modeling. We review the pseudo-fermion (PF) and pseudo-Majorana (PM) functional renormalization group (FRG) and their specific ability to address higher-dimensional frustrated quantum magnetism. First developed more than a decade ago, the PFFRG interprets a Heisenberg model Hamiltonian in terms of Abrikosov pseudofermions, which is then treated in a diagrammatic resummation scheme formulated as a renormalization group flow of $m$-particle pseudofermion vertices. The article reviews the state of the art of PFFRG and PMFRG and discusses their application to exemplary domains of frustrated magnetism, but most importantly, it makes the algorithmic and implementation details of these methods accessible to everyone. By thus lowering the entry barrier to their application, we hope that this review will contribute towards establishing PFFRG and PMFRG as the numerical methods for addressing frustrated quantum magnetism in higher spatial dimensions.Comment: Review article for Reports on Progress in Physics. Comments are welcome! (38+7) pages, (11+4) figures, 3 table

Pinch-points to half-moons and up in the stars: The kagome skymap

Pinch point singularities, associated with flat band magnetic excitations, are tell-tale signatures of Coulomb spin liquids. While their properties in the presence of quantum fluctuations have been widely studied, the fate of the complementary nonanalytic features—shaped as half moons and stars—arising from adjacent shallow dispersive bands has remained unexplored. Here, we address this question for the spin S=1/2 Heisenberg antiferromagnet on the kagome lattice with second and third neighbor couplings, which allows one to tune the classical ground state characterized by flat bands to one that is governed by shallow dispersive bands for intermediate coupling strengths. Employing the complementary strengths of variational Monte Carlo, pseudofermion functional renormalization group, and density-matrix renormalization group, we establish the quantum phase diagram of the model. The U(1) Dirac spin liquid ground state of the nearest-neighbor antiferromagnet remains remarkably robust till intermediate coupling strengths when it transitions into a pinwheel valence bond crystal displaying signatures of half moons in its structure factor. Our Letter thus identifies a microscopic setting that realizes one of the proximate orders of the Dirac spin liquid identified in a recent work [Song, Wang, Vishwanath, and He, Nat. Commun. 10, 4254 (2019)]. For larger couplings, we obtain a collinear magnetically ordered ground state characterized by starlike patterns

Moments and multiplets in moir\'e materials: A pseudo-fermion functional renormalization group for spin-valley models

The observation of strongly-correlated states in moir\'e systems has renewed the conceptual interest in magnetic systems with higher SU(4) spin symmetry, e.g. to describe Mott insulators where the local moments are coupled spin-valley degrees of freedom. Here, we discuss a numerical renormalization group scheme to explore the formation of spin-valley ordered and unconventional spin-valley liquid states at zero temperature based on a pseudo-fermion representation. Our generalization of the conventional pseudo-fermion functional renormalization group approach for $\mathfrak{su}$(2) spins is capable of treating diagonal and off-diagonal couplings of generic spin-valley exchange Hamiltonians in the self-conjugate representation of the $\mathfrak{su}$(4) algebra. To achieve proper numerical efficiency, we derive a number of symmetry constraints on the flow equations that significantly limit the number of ordinary differential equations to be solved. As an example system, we investigate a diagonal SU(2)$_{\textrm{spin}}$ $\otimes$ U(1)$_{\textrm{valley}}$ model on the triangular lattice which exhibits a rich phase diagram of spin and valley ordered phases.Comment: 20 pages, 7 figure

Cavity-induced quantum spin liquids

Quantum spin liquid states are realized in systems with frustrated magnetic interactions. Here, the authors show that tunable frustrated spin-spin interactions can be induced by coupling a quantum antiferromagnet to the quantized light of a driven optical cavity, giving rise to robust quantum spin liquid states. Quantum spin liquids provide paradigmatic examples of highly entangled quantum states of matter. Frustration is the key mechanism to favor spin liquids over more conventional magnetically ordered states. Here we propose to engineer frustration by exploiting the coupling of quantum magnets to the quantized light of an optical cavity. The interplay between the quantum fluctuations of the electro-magnetic field and the strongly correlated electrons results in a tunable long-range interaction between localized spins. This cavity-induced frustration robustly stabilizes spin liquid states, which occupy an extensive region in the phase diagram spanned by the range and strength of the tailored interaction. This occurs even in originally unfrustrated systems, as we showcase for the Heisenberg model on the square lattice