Condensed Matter Theory

Abstract:

Ruggedness—the prevalence of fitness peaks—and navigability—the existence of fitness-increasing paths to a target—are key factors affecting evolution on fitness landscapes. Here, we analyze these properties in landscapes that inherit biophysically grounded genotype–phenotype (GP) maps. By assuming a random phenotype-fitness assignment as a baseline, the structure of the GP maps is included without imposing further fitness correlations. We show analytically that the expected ruggedness can be predicted from two quantities: the sizes of neutral components (NCs)—mutationally connected genotype sets with the same phenotype—and their evolvabilities, defined as the number of distinct phenotypes among the NC’s mutational neighbors. Other features—such as robustness—influence ruggedness only indirectly via correlations with evolvability. Numerical results across diverse GP maps confirm that NC size and evolvability alone suffice to predict both the mean prevalence and heights of peaks. These calculations also provide new insights: Under random phenotype-fitness assignment, peaks arising from high-evolvability NCs have higher expected fitness than those from low-evolvability NCs. Thus, when evolvability correlates positively with NC size, the formation of large low-fitness peaks is impeded. We further derive an approximate scaling law for the minimal average evolvability required for navigability. Our framework applies broadly across GP maps, providing general insight into when and why fitness landscapes are expected to be rugged or navigable.

Abstract:

We use unrestricted Hartree-Fock, density matrix renormalization group, and variational projected entangled-pair state calculations to investigate the ground-state phase diagram of the triangular-lattice Hubbard model at “half doping” relative to single occupancy, i.e., at fillings of $(1\pm \frac{1}{2})$ electrons per site. The electron-doped case has a nested Fermi surface in the noninteracting limit, and hence a weak-coupling instability toward density-wave orders whose wave vectors are determined by Fermi-surface nesting conditions. We find that at moderate-to-strong interaction strengths, other spatially modulated orders arise, with wave vectors distinct from the nesting vectors. In particular, we identify a series of closely competing, itinerant long-wavelength magnetically ordered states, yielding to uniform ferromagnetic order at the largest interaction strengths. For half-hole doping and a similar range of interaction strengths, our data indicate that magnetic orders are most likely absent.

Abstract:

We review the theoretical modeling of moiré materials, focusing on various aspects of magic-angle twisted bilayer graphene (MA-TBG) viewed through the lens of Hartree–Fock mean-field theory. We first provide an elementary introduction to the continuum modeling of moiré bandstructures, and explain how interactions are incorporated to study correlated states. We then discuss how to implement mean-field simulations of ground state structure and collective excitations in this setting. With this background established, we rationalize the power of mean-field approximations in MA-TBG, by discussing the idealized ‘chiral-flat’ strong-coupling limit, in which ground states at electron densities commensurate with the moiré superlattice are exactly captured by mean-field ansätze. We then illustrate the phenomenological shortcomings of this limit, leading us naturally into a discussion of the intermediate-coupling incommensurate Kekulé spiral (IKS) order and its origins in ever-present heterostrain. IKS and its placement within an expanded Hartree–Fock manifold form our first ‘case study’. Our second case study involves time-dependence, and focuses on the collective modes of various broken-symmetry insulators in MA-TBG. As a third and final case study, we return to the strong-coupling picture, which can be stabilized by aligning MA-TBG to an hBN substrate. In this limit, we show how mean field theory can be adapted to the translationally non-invariant setting in order to quantitatively study the energetics of domain walls in orbital Chern insulating states. We close with a discussion of extensions and further applications. Used either as a standalone reference or alongside the accompanying open-source code, this review should enable readers with a basic knowledge of band theory and many-body physics to systematically build and analyze detailed models of generic moiré systems.

Abstract:

Interactions between active particles may be non-reciprocal, breaking action-reaction symmetry and leading to novel physics not observed in equilibrium systems. The non-reciprocal Cahn–Hilliard (NRCH) model is a phenomenological model that captures the large-scale effects of non-reciprocity in conserved, phase-separating systems. In this work, we explore the consequences of different variations of this model corresponding to different symmetries, inspired by the importance of symmetry in equilibrium universality classes. In particular, we contrast two models, one with a continuous SO(2) symmetry and one with a discrete C4 symmetry. We analyze the corresponding models by constructing three-dimensional linear stability diagrams. With this, we connect the models with their equilibrium limits, highlight the role of mean composition, and classify qualitatively different instabilities. We further demonstrate how non-reciprocity gives rise to out-of-equilibrium steady states with non-zero currents and present representative closed-form solutions that help us understand characteristic features of the models in different parts of the parameter space.