Electronic Correlations and Beyond
- in Memory of Peter Fulde

Emergent dynamics of solids has been the focus of research in condensed-matter physics for the past decades. The interest is due to the fact that condensed-matter physics is expected to serve both technological application and answer fundamental questions: On one hand, it should provide guide lines for material design which implies that theory should give quantitative microscopic predictions. On the other hand, the large number of interacting degrees of freedom can lead to novel states of matter with unusual often enigmatic properties whose characterization require the development of new concepts and methods. The central goal of Peter Fulde’s research work was to develop a framework suitable for both tasks that in addition would allow to treat both weak and strong electronic correlations. He paid special attention to strongly correlated electron systems, in particular to heavy-fermion systems. In this talk, I will start with a historic overview. The main part will be devoted to materials with heavy fermions. Since their discovery almost half a century ago, heavy fermion systems have been a continuous source of surprising discoveries often challenging the theoretical understanding at the time. While heavy fermion systems in their “normal” state can be approximately described as Fermi liquids and therefore share common qualitative properties it is important to emphasize that there are different routes to heavy fermions. The focus of the present talk will be on intermetallic lanthanide (4f) and actinide (5f) compounds. I will present results on the recently discovered compound $CeRh_2As_2$ [1] that exhibits a complex phase diagram with rather unusual states at low temperatures. A prominent example is multi-phase superconductivity which seems to develop inside a normal state involving by itinerant multi-polar order [2]. The narrow quasiparticle bands arise from the Ce-4f degrees of freedom via the Kondo effect. We conjecture that the Kondo-induced quasi-quartet CEF ground state [3] in combination with pronounced nesting features of the Fermi surface [4,5] may give rise to ordered states involving multipolar degrees of freedom [6]. Consequences of topologically protected inter-band pairing in close analogy to UPt3 [7] will be discussed. References 1. S. Khim et al., Science 373, 1012 (2021) 2. D. Hafner et al., Phys. Rev. X 12, 011023 (2022) 3. Denise S. Christovam et al., Phys. Rev. Lett. 132, 046401 (2024) 4. Yi Wu et al, Chinese Physics Letters (2024) DOI 10.1088/0256-307x/41/9/097403 5. B. Chen et al., Phys. Rev. B 110, L041120 (2024) 6. P. Khanenko et al, arXiv:2409.11894 7. Z. Wang et al., Phys. Rev. B 96, 174511 (2017)

Altermagnetic materials are characterized by collinear magnetic order with a vanishing net magnetic moment, but nevertheless have a spin-splitting in their non-relativistic electronic band structure. From ab initio calculations we have identified around 60 altermagnetic materials. From a theoretical point of view several physical properties that render altermagnets different from canonical antiferro-, ferro- and ferri-magnets will be discussed. These include certain spin and heat transport features and piezomagnetic responses. By symmetry in principle also an anomalous Hall effect (AHE) may be allowed and we introduce an altermagnetic model in which the emergence of an AHE is driven by interactions. Quantum Monte Carlo simulations show that the system undergoes a finite temperature phase transition governed by a primary antiferromagnetic order parameter accompanied by a secondary altermagnetic one. The emergence of both orders turns the metallic state of the system, away from half-filling, into an altermagnet with zero net moment but finite AHE.

Fractionalization in condensed matter physics is a fascinating phenomenon where the collective excitations of a system exhibit properties distinct from those of its constituent particles. In this talk, I will begin by introducing a model proposed by Fulde et al., which allows for fractionalized charge excitations on frustrated lattices. The low energy physics of this model in the strong coupling limit is described by an emergent lattice gauge theory. I will then discuss how such emergent (2+1)D lattice gauge theories can be simulated on quantum computers, highlighting the potential for our understanding of fractionalized states of matter.

Several Heavy Fermion systems (notably, CeCu6-xAux ) show SDW instability and exhibit various NFL behaviors such as sub T-linear resistivity and log(T)-divergence in specific heat, hence called quantum critical phase (QCP). Furthermore, spin-fluctuation dynamics measured by neutron scattering exhibit a scaling behavior with a non-integer critical exponent of 0.75 (WF fixed point) in contrast to the Hertz-Millis theory. In this talk, I propose a possible way to cure the Hertz-Millis theory to generate a Non-Gaussian fixed point and Non-Fermi liquid behaviors.

Yttrium Iron Garnet is a much studied and widely used crystalline ferrimagnet but belongs to a whole series, when the non-magnetic yttrium ion is substituted for the other rare earth ions bearing a magnetic moment. While many details of the magnetic structurse were recognised long ago by Neel and co-workers, there has been a substantial renewal of interest in recent years. This is because of the potential applications in spintronics applications, exploiting the large spin-orbit coupling in the rare-earth and the strong coupling of the iron moments. New insights can de found by the use of spectroscopic methods, particularly spin-polarised neutron and X-ray beams, applied to crystals and film samples. Novel experimental techniques look at special dynamic properties at compensation temperatures, where either the net magnetisation or the net total angular moment vanish because of the different temperature dependence of the angular momenta of the rare earth ions and the iron moments.I will discuss new ideas in the dynamics of non-collinear magnetic structures that arise, with the interplay of the crystal field and exchange effects and the relation to measurements of the spin-Seebeck effects.

Electronic correlations can lead to highly entangled states of quantum matter in which electrons and spins are fractionalized into nonlocal novel elementary excitations. Quantum spin liquids (QSLs) in frustrated magnets are among such states and represent a timely field of search for manifestations of fractional quasiparticles. Here we will report on some of their fingerprints in transport and spectroscopy in Kitaev magnets, which are among the QSLs of great current interest, comprising a compass Ising model on the honeycomb lattice, allowing for fractionalization of spins into mobile Majorana matter and gauge flux visons.

In this talk, we report the progress of predicting four different phases (Haldane, Dimerized, Ferromagnetic, and critical phases) of 1-D spin-1 Bilinear-Biquadratic models by using unsupervised learning. The difficulty of detecting a second-order phase transition point is overcome by using three spin-spin correlation functions.

Coherent charge and heat transport through periodically driven nanodevices provide a platform for studying thermoelectric effects on the nanoscale. Here we study a junction composed of a quantum dot connected to two fermionic terminals by two weak links. An AC electric field induces time-dependent spin-orbit interaction in the weak links. We show that this setup supports DC charge and heat currents and that thermoelectric performance can be improved, as reflected by the effect of the spin-orbit coupling on the Seebeck coefficient and the electronic thermal conductance. Our analysis is based on the nonequilibrium Keldysh Green's function formalism in the time domain and reveals an interesting distribution of the power supply from the AC source among the various components of the device.

I will outline a set of results on the classical and quantum mechanics of fractons—particles defined by a consistent set of multipole conservation laws. I will demonstrate that the nonlinear, Machian dynamics of fractons are characterized by late-time attractors in position-velocity space, despite the absence of attractors in phase space as dictated by Liouville's theorem. These attractors violate ergodicity and lead to non-equilibrium steady states, which invariably break translational symmetry, even in spatial dimensions where the Hohenberg-Mermin-Wagner-Coleman theorem for equilibrium systems forbids such breaking. I will also discuss progress in quantizing this system.

The pseudogap phase in cuprate superconductors is one of the most intriguing challenges in the theory of high temperature superconductivity. In this talk we elaborate on the idea that pseudogap phenomena are mostly due to strong magnetic fluctuations. Using the two-dimensional Hubbard model to describe the electrons in the copper-oxygen planes of the cuprates, a gauge theory of fluctuating magnetic order is formulated and analyzed [1]. The theory is based on a fractionalization of electrons in fermionic chargons with a pseudospin degree of freedom and bosonic spinons, which leads to a SU(2) gauge redundancy. The chargons undergo Neel, spiral, or stripe order below a density dependent transition temperature $T^*$. Fluctuations of the spin orientation are described by a non-linear sigma model obtained from a gradient expansion of the spinon action. The spin stiffnesses are computed from a renormalization group improved random phase approximation. The spinon fluctuations prevent magnetic long-range order of the electrons at any finite temperature. The phase with magnetic chargon order exhibits many features characterizing the pseudogap regime in high-Tc cuprates: a strong reduction of charge carrier density, a spin gap, and Fermi arcs. A substantial fraction of the pseudogap regime exhibits electronic nematicity. [1] P. M. Bonetti and W. Metzner, Phys. Rev. B 106, 205152 (2022).

We review our recent development of the Multiscale Functional Renormalization Group (MfRG) as an approach to the study of strongly correlated electronic materials in which both electron-electron (e-e) and electron-phonon (e-ph) interactions play important roles. Our MfRG method includes in a systematic manner the effects of the scattering processes involving electrons away from the Fermi surface and also permits proper inclusion of phonon retardation effects. After introducing the basic concepts, I will discuss in detail the application of this method to the 1D Extended Hubbard model and show that it correctly captures the subtle bond-order wave (BOW) phase in that model. We will then discuss additional applications of the MfRG method and show that it can be applied to multi-band models in higher dimensions, recovering previously known results and predicting novel behaviors that have been seen in experiment. Finally, we will discuss possible future applications of the MfRG approach.

We consider a PT-symmetric Fermi gas with an exceptional point, representing the critical point between PT-symmetric and symmetry broken phases. The low energy spectrum remains linear in momentum and is identical to that of a hermitian Fermi gas. The fermionic Green's function decays in a power law fashion for large distances, as expected from gapless excitations, albeit the exponent is reduced from -1 due to the quantum Zeno effect. In spite of the gapless nature of the excitations, the ground state entanglement entropy saturates to a finite value, independent of the subsystem size due to the non-hermitian correlation length intrinsic to the system. Attractive or repulsive interaction drives the system into the PT-symmetry broken regime or opens up a gap and protects PT-symmetry, respectively.

Starting with Peter Fulde's late works addressing the exponential-wall problem for large particle numbers, I will discuss loosely related approaches we have developed to approximately compute the smoothed density of states of certain many-body systems. Because of the vast growth of the many-body level density with excitation energy, its smoothed form is of central relevance for spectral and thermodynamic properties of interacting quantum systems. Our method provides predictions for excitation spectra that enable access to finite-temperature thermodynamics in large parameter ranges. Another application concerns many-body scattering through interacting media due to a fundamental relation between the smooth level density and the average dwell time.

Optically active polymers are intensively studied because of their optical properties, promising applications in light emitting and in energy harvesting. Still, there is no consensus on the very nature of the light emitting excitons. We develop an approach considering the polymers as generic one dimensional semiconductors with specific features of strongly correlated electronic systems. We demonstrate that optics of polymers can be understood within this combined frame of long range Coulomb interaction together with strong intra-monomer electronic correlations. Theory predicts different kinds of weakly and strongly bound excitons, in singlet and triplet states; all of them already show up in experiments. We shall mention also new polymers with a degenerate ground state which demonstrate solitons with expecting non-integer charge. These systems are promising in search of ferroelectricity in polymers.

The mean-field approximation provides a simple approach for many models of correlated particles on lattices. It replaces the original finite dimensional lattice with a generalized multidimensional version, in the infinite dimensional limit. The same approach sometimes works for zeta functions defined in terms of integers on lattices. We show how these zeta functions simplify in the infinite dimensional limit.

Experiments on the low temperature properties of multicomponent glasses revealed in 1998 evidence for a phase transition to a phase with strongly enhanced magnetosensitivity[1]. In the experimental paper [1] and in a theoretical investigation, initiated by Peter Fulde[2], it was suggested that this indicates a transition to a a coherent macroscopic quantum state of coupled tunneling systems. Here we review the status of the theory of quantum phase transitions in coupled tunneling systems, and whether the surprising transition to a phase with strongly enhanced magnetosensitivity discovered in the low temperature electric permittivity of multicomponent glasses can indeed be due to a quantum phase transition to a phase, where mesoscopically or even macroscopically large numbers of tunneling systems are coupled by the interaction and start tunneling in synchrony, yielding a strongly enhanced diamagnetic response as suggested in Ref. [2]. With the recent progress in understanding in long range interacting disordered quantum spin systems we aim to address this problem again, and review recent evidence for a Large Spin fixed point in such systems, which might provide the theoretical foundation for explaining the experimental results of Ref. [1], and thereby confirm the insightful intuition by Peter Fulde as revealed and developed in intensive discussions with the experimentalist Peter Strehlow and the author 25 years ago. [1] P. Strehlow, C. Enss, S. Hunklinger, Evidence for a phase transition in glasses at very low temperature: a macroscopic quantum state of tunneling systems?, Phys. Rev. Lett. 80, 5361 (1998). [2] S. Kettemann, P. Fulde, P. Strehlow, Correlated Persistent Tunneling Currents in Glasses, Phys. Rev. Lett. 83, 4325 (1999).

We all remember Peter's explanation of the correlation hole. I want to talk about the correlation hole that shows up in spatial patterns formed by correlated random walks [1]. I want to connect this to recent model calculations that help us to understand the spatial patterns formed by particles in turbulence [2]. I'll go rather quickly from $\hbar \neq 0$ to $\hbar = 0$. [1] Wilkinson & Mehlig, Phys. Rev. E 68 (2003) 040101 [2] Bec, Gustavsson & Mehlig, Annu. Rev. Fluid. Mech. 56 (2024) 189-213

Topological spin configurations in proximity to a superconductor have recently attracted great interest due to the potential application of the former in spintronics and as another platform for realizing nontrivial topological superconductors. Their application in these areas requires precise knowledge of the existing exchange fields and/or the stray fields, which are therefore essential for the study of these systems. In this talk we will analyse theoretically a hybrid heterostructure with magnetic textures including skyrmions inside a chiral ferromagnet interfaced by a thin superconducting film via an insulating barrier. We find that Pearl vortices (PV) are generated spontaneously in the superconductor within the skyrmion radius, while anti-Pearl vortices (PV) compensating the magnetic moment of the Pearl vortices are generated outside of the Sk radius, forming an energetically stable topological hybrid structure. Finally, we analyze the interplay of skyrmion and vortex lattices and their mutual feedback on each other. In particular, we argue that the size of the skyrmions will be greatly affected by the presence of the vortices offering another prospect of manipulating the skyrmionic size by the proximity to a superconductor [1]. Furthermore, we calculated the magnetic stray field and Meissner current in a superconductors S created by various nonhomogeneous magnetic texture in a F film incorporated in an S/F/S system[2] We also discuss the implications of the recent experimental results realizing these skyrmions-vortex interaction [3]. We analyse the collective modes (CM) in SFS heterostructure by taking into account the interplay between the de-magnetisation field, Hdem, and the field caused by the anisotropy of the ferromagnet Han, which was previously neglected. It turns out that the spectrum of composite collective modes, ω(k), has a qualitatively different form in the case of HdemHan. In the first case, the dependence ω(k) has the same form as in previous studies, whereas in the second case, the spectrum looks completely different. In particular, for moderate or weak anisotropy in ferromagnet the group velocity of collective modes demonstrates inflection point where the group velocity become infinite and is superluminal. Furthermore, this point separates purely real and complex-conjugate solutions for the collective modes and is also exception point. We show that the difference of the CMs spectra can be revealed by Fiske experiment by measuring the I−V characteristics in the presence of magnetic field and voltage[4]. [1] S. Dahir, A.F. Volkov, and I. Eremin, Interaction of Skyrmions and Pearl Vortices in Superconductor-Chiral Ferromagnet Heterostructures, Phys. Rev. Lett. 122, 097001 (2019). [2] Samme M. Dahir, Anatoly F. Volkov, and Ilya M. Eremin, Meissner currents induced by topological magnetic textures in hybrid superconductor/ferromagnet structures, Phys. Rev. B 102 , 014503 (2020) [3] A. P. Petrović, M. Raju, X. Y. Tee, A. Louat, I. Maggio-Aprile, R. M. Menezes, M. J. Wyszyński, N. K. Duong, M. Reznikov, Ch. Renner, M. V. Milošević, and C. Panagopoulos, Phys. Rev. Lett. 126, 11720513 (2021). [4] Pascal Derendorf, Anatoly F. Volkov, Ilya M. Eremin, arXiv: 2407.05457 (unpublished)

We show that spontaneous time-reversal-symmetry (TRS) breaking can naturally arise from the interplay between pair density wave (PDW) ordering at multiple momenta and nesting of Fermi surfaces (FS). Concretely, we consider the PDW superconductivity on a hexagonal lattice with nested FS at 3/4 electron filling, which is related to a recently discovered superconductor CsV3Sb5. Because of nesting of the FS, each momentum on the FS has at least two counterparts on the FS to form finite momentum Cooper pairs, resulting in a TRS and inversion broken PDW state with stable Bogoliubov Fermi pockets. Possible implications to the Kagome superconductor CsV3Sb5 and future experiments will be discussed.

Early proposals for creating Majorana zero modes involve spin triplet superconductivity while almost all known superconductors belong to the spin singlet class with a few possible exceptions. As a result, various ingenious heterostructures of superconducting nano-wires have been proposed to generate the required p-wave pairing component. In this presentation, I will review the recent findings of two-component superconductivity with a dominant p-wave part in CoSi2/TiSi2 heterostructures In particular, I will review the effect of interface-induced Rashba spin orbit coupling on the conductance of a three terminal ``T" shape superconducting device and present our calculations for the differential conductance for this device within the quasi-classical formalism that includes the mixing of triplet-singlet pairing due to the Rashba spin orbit coupling.

The low energy crystalline electric field states and heavy electron bands in strongly correlated 4f- and 5f electron compounds exhibit possibilities of broken symmetry states and collective excitations at low temperatures as well as distinct interaction effects with the lattice. Characteristic examples such as charge ordering, multipolar order and coexistence of magnetism and unconventional superconductivity is presented, focusing on the Ce- Yb- and U- based compounds, including the most prominent heavy fermmion materials. Emphasis is also placed on the methods to investigate order parameter symmetries and the associated collective excitations.

It will describe how the diffusion process is used in generative AI model and a new physics-based AI model.

Wavefunctions can tell everything. The computation of many-electron wavefunctions from first principles in solid-state context is one of the long-term projects set up by Peter Fulde. A few illustrative examples are reviewed in this contribution, ranging from the calculation of quasiparticle bands in either weakly [1] or strongly [2] correlated electronic systems to the nature of ground-state wavefunctions across group-5 mixed-valent 'breathing' pyrochlores [3] and in alkaline-earth hexaborides [4]. [1] L Hozoi, U Birkenheuer, P Fulde, A Mitrushchenkov, and H Stoll, Ab initio wavefunction-based methods for excited states in solids: correlation corrections to the band structure of ionic oxides, Phys. Rev. B 76, 085109 (2007). [2] L Hozoi, M S Laad, and P Fulde, Fermiology of cuprates from first principles: from small pockets to the Luttinger Fermi surface, Phys. Rev. B 78, 165107 (2008). [3] T Petersen, P Bhattacharyya, U K Roessler, and L Hozoi, Resonating holes vs molecular spin-orbit coupled states in group-5 lacunar spinels, Nat. Commun. 14, 5218 (2023). [4] T Petersen, U K Roessler, and L Hozoi, Quantum chemical insights into hexaboride electronic structures: correlations within the boron p-orbital subsystem, Commun. Phys. 5, 214 (2022).

Peter Fulde's early work on the so-called "local ansatz" for a wavefunction in extended systems provided the basis for the application of correlation methods, developed mainly in Theoretical Chemistry for finite systems, to extended systems namely solids. Within this contributions I will focus on the method of increments and its application to different kind of solids, from insulators to metals, from van der Waals bound rare gas crystals to metallic binding in mercury.

We study interacting fermions on a finite chain in a Wannier-Stark linear potential using the density matrix renormalization group numerical technique. For the generalized extended Hubbard model, we find that the ground state exhibits a staircase of (quasi) plateaus in the average local site density along the chain, decreasing from being doubly filled to empty as the potential increases. These “plateaus” represent locked-in commensurate phases of charge density waves together with band and Mott insulators. These phases are separated by incompressible regions with incommensurate fillings, as observed from dynamical properties. We suggest that experimental variations of the slope of the potential and the range of the repulsive interactions will produce such a coexistence of phases. We also study the effect of correlations on the localisation length of an extended spinless model with nearest neighbour repulsion and compare with exact analytical results.

Biological sensory systems exhibit remarkable power efficiency, largely due to their capacity to store and recall learned solutions. Emulating this efficiency in artificial sensory systems necessitates adaptability and analog functionality. One promising approach involves computing based on the complex dynamics of interacting oscillators, particularly when these oscillators serve as the building blocks of artificial intelligence. This talk explores how the Coulomb blockade effect in single molecule transistors can form the foundation of the smallest conceivable oscillators. We will discuss a potential pathway from utilizing Coulomb blockade to developing oscillatory neural networks capable of executing complex tasks such as image recognition and tackling computationally intensive problems like vertex coloring.

In 1964, Fulde and Ferrell as well as Larkin and Ovchinnikov independently predicted the existence of unconventional, nonuniform superconductivity above the Pauli paramagnetic limit. There is now robust evidence that quasi-two-dimensional organic superconductors realize such so-called FFLO states at high magnetic fields. High-resolution specific-heat and magnetization measurements show a clear upturn of the upper critical field towards lowest temperatures and the existence of an additional high-field low-temperature superconducting state that appears only in a very narrow angular region close to in-plane field orientation. Besides these thermodynamic results, NMR studies give strong microscopic evidence for the realization of the FFLO state. A one-dimensional sinusoidal order-parameter modulation can favorably explain the NMR spectra in this high-field low-temperature phase. All these results clearly mark the existence of FFLO states in organic superconductors and motivate further detailed studies of this unconventional superconductivity.

“Nonreciprocity” means “not going the same way backward as forward”. The well-known is the rectification by using PN junctions in electronics where electrons flow in one direction but not the other. The spin Hall effect and its inverse are the interconversion mechanism between electron current and spin current. It has been shown that the interconversion is reciprocal and described by the Onsager reciprocal relation [1]. On the contrary, many of the spin transport phenomena are nonreciprocal. Here, the nonreciprocity in spin transport is discussed together with various examples. The key is that the spin current is a flow of spin angular momentum, in contrast to the electric current. A flow of electrons can have the orbital angular momentum , which is called “vorticity”, and may be interconverted with spin current [2]. However, since the vorticity of electron flow is highly nonlinear, the conservation mechanism, i.e., the spin-vorticity coupling, is also nonlinear and, in general, nonreciprocal[1]. As a result, we find a variety of nonreciprocal phenomena in spin transport. The nonreciprocity of surface acoustic waves in magnetic films [3], the magnetic skyrmion generation and annihilation by electric current in magnetic films with notch structure [4], and the spin current generation in the graded materials [5] will be discussed. [1] T. Kimura, Y. Otani, T. Sato, S. Takahashi and S. Maekawa: Phys. Rev. Lett. 98, 156601 (2007), [2] M.Matsuo, E.Saitoh and S.Maekawa, Chapter 25 in “Spin Current” ed. S.Maekawa et al. (Oxford University Press, 2017). [3] M. Xu, K. Yamamoto, J. Puebla, K. Baungaert, B. Rana, K. Miura, H. Takahashi, D. Grundler, A. Maekawa and Y. Otani: Sci. Adv. 6, eabb1724 (2020), [4] J.Fujimoto, W.Koshibae, M.Matsuo and S.Maekawa, Phys. Rev. B103, L220404 (2021). [5] G.Okano, M.Matsuo, Y.Ohnuma, S.Maekawa, and Y.Nozaki, Phys. Rev. Lett. 122, 217701 (2019).

Recent discovery of high temperature superconductivity La3Ni2O7 under high pressure has attracted a lot of attention. In this talk, I shall present our theory for electronic structure of the compound and address the similarity and difference of this newly discovered bilayer nickelate superconductor in comparison with high temperature superconducting cuprates and with infinite layer nickelate superconductor Nd_(1-x)Sr_xNiO2. I shall propose that La3Ni2O7 is a self-doped molecule Mott insulator, where molecule stands for the strongly coupled nickel ions on the top and bottom layers.