Recent experiments have observed strongly correlated physics in twisted bilayer graphene (TBG) at very small angles, along with nearly flat electron bands at certain fillings. A good starting point in understanding the physics is a continuum model (CM) proposed by Lopes dos Santos et al. [Phys. Rev. Lett. 99, 256802 (2007)] and Bistritzer et al. [Proc. Natl. Acad. Sci. USA 108, 12233 (2011)] for TBG at small twist angles, which successfully predicts the bandwidth reduction of the middle two bands of TBG near the first magic angle ${\theta }_0=1.{05}^{\circ }$. In this paper, we analyze the symmetries of the CM and investigate the low energy flat band structure in the entire moiré Brillouin zone near ${\theta }_0$. Instead of observing flat bands at only one “magic” angle, we notice that the bands remain almost flat within a small range around ${\theta }_0$, where multiple topological transitions occur. The topological transitions are caused by the creation and annihilation of Dirac points at either $\text{K},{\text{K}}^{\prime }$, or $\mathrm{\Gamma }$ points or along the high symmetry lines in the moiré Brillouin zone. We trace the evolution of the Dirac points, which are very sensitive to the twist angle, and find that there are several processes transporting Dirac points from $\mathrm{\Gamma }$ to K and ${\text{K}}^{\prime }$. At the $\mathrm{\Gamma }$ point, the lowest energy levels of the CM are doubly degenerate for some range of twisting angle around ${\theta }_0$, suggesting that the physics is not described by any two-band model. Based on this observation, we propose an effective six-band model (up to second order in quasimomentum) near the $\mathrm{\Gamma }$ point with the full symmetries of the CM, which we argue is the minimal model that explains the motion of the Dirac points around $\mathrm{\Gamma }$ as the twist angle is varied. By fitting the coefficients from the numerical results, we show that this six-band model captures the important physics over a wide range of angles near the first “magic” angle.

We present comprehensive measurements of the structural, magnetic, and electronic properties of layered van der Waals ferromagnet ${\mathrm{VI}}_3$ down to low temperatures. Despite belonging to a well-studied family of transition-metal trihalides, this material has received very little attention. We outline, from high-resolution powder x-ray diffraction measurements, a corrected room-temperature crystal structure to that previously proposed and uncover a structural transition at 79 K, also seen in the heat capacity. Magnetization measurements confirm ${\mathrm{VI}}_3$ to be a hard ferromagnet (9.1 kOe coercive field at 2 K) with a high degree of anisotropy, and the pressure dependence of the magnetic properties provide evidence for the two-dimensional nature of the magnetic order. Optical and electrical transport measurements show this material to be an insulator with an optical band gap of 0.67 eV—the previous theoretical predictions of $d$-band metallicity then lead us to believe ${\mathrm{VI}}_3$ to be a correlated Mott insulator. Our latest band-structure calculations support this picture and show good agreement with the experimental data. We suggest ${\mathrm{VI}}_3$ to host great potential in the thriving field of low-dimensional magnetism and functional materials, together with opportunities to study and make use of low-dimensional Mott physics.

The Weyl particle is the massless fermionic cousin of the photon. While no fundamental Weyl particles have been identified, they arise in condensed matter and metamaterial systems, where their spinor nature imposes topological constraints on low-energy dispersion and surface properties. Here we demonstrate a topological circuit with Weyl dispersion at low momentum, realizing a 3D lattice that behaves as a half-flux Hofstadter model in all principal planes. The circuit platform provides access to the complete complex-valued spin texture of all bulk and surface states, thereby revealing not only the presence of Weyl points and the Fermi arcs that connect their surface projections, but also the Berry curvature distribution through the Brillouin zone and the associated quantized chiral charge of the Weyl points. This work opens a path to exploration of interacting Weyl physics in superconducting circuits, as well as studies of how manifold topology impacts band topology in three dimensions.

We propose a general principle for constructing higher-order topological (HOT) phases. We argue that if a $D$-dimensional first-order or regular topological phase involves $m$ Hermitian matrices that anticommute with additional $p-1$ mutually anticommuting matrices, it is conceivable to realize an $n\mathrm{th}$-order HOT phase, where $n=1,...,p$, with appropriate combinations of discrete symmetry-breaking Wilsonian masses. An $n\mathrm{th}$-order HOT phase accommodates zero modes on a surface with codimension $n$. We exemplify these scenarios for prototypical three-dimensional gapless systems, such as a nodal-loop semimetal possessing SU(2) spin-rotational symmetry, and Dirac semimetals, transforming under (pseudo)spin-$\frac{1}{2}$ or 1 representations. The former system permits an unprecedented realization of a fourth-order phase, without any surface zero modes. Our construction can be generalized to HOT insulators and superconductors in any dimension and symmetry class.

We describe an approach for many-body calculations with a finite-temperature, grand canonical ensemble formalism using auxiliary-field quantum Monte Carlo (AFQMC) with a self-consistent constraint to control the sign problem. The usual AFQMC formalism of Blankenbecler, Scalapino, and Sugar suffers from the sign problem with most physical Hamiltonians, as is well known. Building on earlier ideas to constrain the paths in auxiliary-field space [Zhang, Phys. Rev. Lett. 83, 2777 (1999)] and incorporating recent developments in zero-temperature, canonical-ensemble methods, we discuss how a self-consistent constraint can be introduced in the finite-temperature, grand-canonical-ensemble framework. This together with several other algorithmic improvements discussed here leads to a more accurate, more efficient, and numerically more stable approach for finite-temperature calculations. We carry out a systematic benchmark study in the two-dimensional repulsive Hubbard model at $1/8$ doping. Temperatures as low as $T=1/80$ (in units of hopping) are reached. The finite-temperature method is exact at very high temperatures, and approaches the result of the zero-temperature constrained-path AFQMC as temperature is lowered. The benchmark shows that systematically accurate results are obtained for thermodynamic properties.

The Lieb-Schultz-Mattis theorem dictates that a trivial symmetric insulator in lattice models is prohibited if lattice translation symmetry and $U\left(1\right)$ charge conservation are both preserved. In this paper, we generalize the Lieb-Schultz-Mattis theorem to systems with higher-form symmetries, which act on extended objects of dimension $n>0$. The prototypical lattice system with higher-form symmetry is the pure Abelian lattice gauge theory whose action consists only of the field strength. We first construct the higher-form generalization of the Lieb-Schultz-Mattis theorem with a proof. We then apply it to the $U\left(1\right)$ lattice gauge theory description of the quantum dimer model on bipartite lattices. Finally, using the continuum field theory description in the vicinity of the Rokhsar-Kivelson point of the quantum dimer model, we diagnose and compute the mixed 't Hooft anomaly corresponding to the higher-form Lieb-Schultz-Mattis theorem.

Antiferromagnets are robust to external electric and magnetic fields, and hence are seemingly uncontrollable. Recent studies, however, realized the electrical manipulations of antiferromagnets by virtue of the antiferromagnetic Edelstein effect. We present a general symmetry analysis of electrically switchable antiferromagnets based on group-theoretical approaches. Furthermore, we identify a direct relation between switchable antiferromagnets and the ferrotoroidic order. The concept of ferrotoroidic order clarifies the unidirectional nature of switchable antiferromagnets and provides a criterion for the controllability of antiferromagnets. The scheme paves a way for perfect writing and reading of switchable antiferromagnets.

After the discovery of nematic topological superconductivity in ${\mathrm{Cu}}_x{\mathrm{Bi}}_2{\mathrm{Se}}_3$, carrier-doped topological insulators are established as a fertile ground for topological superconductors. The superconductor ${\mathrm{Cu}}_{1.5}{\left(\mathrm{PbSe}\right)}_5{\left({\mathrm{Bi}}_2{\mathrm{Se}}_3\right)}_6$ (CPSBS) contains ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ blocks as a constitutional unit, but its superconducting gap appears to have nodes [S. Sasaki et al., Phys. Rev. B 90, 220504 (2014)], which is in contrast to the fully opened gap in ${\mathrm{Cu}}_x{\mathrm{Bi}}_2{\mathrm{Se}}_3$ and the relation between the two superconductors remained an open question. Here, we report our observation of clear twofold symmetry in the in-plane magnetic field direction dependencies of the upper critical field and of the specific heat of CPSBS, where the direction of the maxima, which is different from that in ${\mathrm{Cu}}_x{\mathrm{Bi}}_2{\mathrm{Se}}_3$, indicates that the gap nodes are located in the mirror plane of the crystal lattice. This means that the topological nematic state with mirror-symmetry-protected nodes is realized in CPSBS.

Purification is a tool that allows to represent mixed quantum states as pure states on enlarged Hilbert spaces. A purification of a given state is not unique and its entanglement strongly depends on the particular choice made. Moreover, in one-dimensional systems, the amount of entanglement is linked to how efficiently the purified state can be represented using matrix-product states (MPS). We introduce an MPS based method that allows to find the minimally entangled representation by iteratively minimizing the second Rényi entropy. First, we consider the thermofield double purification and show that its entanglement can be strongly reduced especially at low temperatures. Second, we show that a slowdown of the entanglement growth following a quench of an infinite temperature state is possible.

The electrons forming a Cooper pair in a superconductor can be spatially separated preserving their spin entanglement by means of quantum dots coupled to both the superconductor and independent normal leads. We investigate the thermoelectric properties of such a Cooper pair splitter and demonstrate that cooling of a reservoir is an indication of nonlocal correlations induced by the entangled electron pairs. Moreover, we show that the device can be operated as a nonlocal thermoelectric heat engine. Both as a refrigerator and as a heat engine, the Cooper pair splitter reaches efficiencies close to the thermodynamic bounds. As such, our work introduces an experimentally accessible heat engine and a refrigerator driven by entangled electron pairs in which the role of quantum correlations can be tested.

We obtain multiple exact results on the entanglement of the exact excited states of nonintegrable models we introduced in Phys. Rev. B 98, 235155 (2018). We first discuss a general formalism to analytically compute the entanglement spectra of exact excited states using matrix product states and matrix product operators and illustrate the method by reproducing a general result on single-mode excitations. We then apply this technique to analytically obtain the entanglement spectra of the infinite tower of states of the spin-$S$ AKLT models in the zero and finite energy density limits. We show that in the zero energy density limit, the entanglement spectra of the tower of states are multiple shifted copies of the ground-state entanglement spectrum in the thermodynamic limit. We show that such a resemblance is destroyed at any nonzero energy density. Furthermore, the entanglement entropy $\mathcal{S}$ of the states of the tower that are in the bulk of the spectrum is subthermal $\mathcal{S}\propto logL$, as opposed to a volume law $\mathcal{S}\propto L$, thus indicating a violation of the strong eigenstate thermalization hypothesis (ETH). These states are examples of what are now called many-body scars. Finally, we analytically study the finite-size effects and symmetry-protected degeneracies in the entanglement spectra of the excited states, extending the existing theory.

We study a special class of topological phase transitions in two dimensions described by the inversion of bands with relative angular momentum higher than 1. A band inversion of this kind, which is protected by rotation symmetry, separates the trivial insulator from a Chern insulating phase with higher Chern number, and thus generalizes the quantum Hall transition described by a Dirac fermion. Higher angular momentum band inversions are of special interest, as the nonvanishing density of states at the transition can give rise to interesting many-body effects. Here we introduce a series of minimal lattice models which realize higher angular momentum band inversions. We then consider the effect of interactions, focusing on the possibility of electron-hole exciton condensation, which breaks rotational symmetry. An analysis of the excitonic insulator mean field theory further reveals that the ground state of the Chern insulating phase with higher Chern number has the structure of a multicomponent integer quantum Hall state. We conclude by generalizing the notion of higher angular momentum band inversions to the class of time-reversal invariant systems, following the scheme of Bernevig-Hughes-Zhang (BHZ). Such band inversions can be viewed as transitions to a topological insulator protected by rotation and inversion symmetry, and provide a promising venue for realizing correlated topological phases such as fractional topological insulators.

The temperature-dependent intrinsic stacking fault Gibbs energy is computed based on highly converged density-functional-theory (DFT) calculations for the three prototype face-centered cubic metals Al, Cu, and Ni. All relevant temperature-dependent contributions are considered including electronic, vibrational, magnetic, and explicit anharmonic Gibbs energy contributions as well as coupling terms employing state-of-the-art statistical sampling techniques. Particular emphasis is put on a careful comparison of different theoretical concepts to derive the stacking fault energy such as the axial-next-nearest-neighbor-Ising (ANNNI) model or the vacuum-slab approach. Our theoretical results are compared with an extensive set of previous theoretical and experimental data. Large uncertainties in the experimental data highlight the necessity of complementary parameter-free calculations. Specifically, the temperature dependence is experimentally unknown and poorly described by thermodynamic databases. Whereas calphad derived data shows an increase of the stacking fault energy with temperature for two of the systems (Cu and Ni), our results predict a decrease for all studied systems. For Ni, the temperature induced change is in fact so strong that in the temperature interval relevant for super-alloy applications the stacking fault energy falls below one third of the low temperature value. Such large changes clearly call for a revision of the stacking fault energy when modeling or designing alloys based on such elements.

A violation of the Wiedemann-Franz law in a metal can be quantified by comparing the Lorentz ratio, $L=\kappa \rho /T$, where $\kappa$ is the thermal conductivity and $\rho$ is the electrical resistivity, with the universal Sommerfeld constant, $L_0=({\pi }^2/3){(k_B/e)}^2$. We obtain the Lorentz ratio of a clean compensated metal with intercarrier interaction as the dominant scattering mechanism by solving exactly the system of coupled integral Boltzmann equations. The Lorentz ratio is shown to assume a particular simple form in the forward-scattering limit: $L/L_0=\overline{{\mathrm{\Theta }}^2}/2$, where $\mathrm{\Theta }$ is the scattering angle. In this limit, $L/L_0$ can be arbitrarily small. We also show how the same result can be obtained without the benefit of an exact solution. We discuss how a strong downward violation of the Wiedemann-Franz law in a type-II Weyl semimetal ${\mathrm{WP}}_2$ can be explained within our model.

We report on direct observation of interface superconductivity in single-unit-cell ${\mathrm{SnSe}}_2$ films grown on graphitized SiC(0001) substrate. The tunneling spectrum in the superconducting state reveals a rather conventional character with a fully gapped order parameter. The occurrence of superconductivity is further confirmed by the observation of vortices under external magnetic field. Through interface engineering, we unravel the mechanism of superconductivity that originates from a two-dimensional electron gas formed at the interface of ${\mathrm{SnSe}}_2$ and graphene. Our finding opens up novel strategies to hunt for and understand interface superconductivity based on van der Waals heterostructures.

We consider two-dimensional Fermi systems with quadratic band touching and $C_3$ symmetry, as realizable in Bernal-stacked honeycomb bilayers. Within a renormalization-group analysis, we demonstrate the existence of a quantum critical point at a finite value of the density-density interactions, separating a semimetallic disordered phase at weak coupling from a gapped ordered phase at strong coupling. The latter may be characterized by, for instance, antiferromagnetic, quantum anomalous Hall, or charge density wave order. In the semimetallic phase, each point of quadratic band touching splits into four Dirac cones as a consequence of the nontrivial interaction-induced self-energy correction, which we compute to the two-loop order. We show that the quantum critical point is in the $(2+1)$-dimensional Gross-Neveu universality class characterized by emergent Lorentz invariance and a dynamic critical exponent $z=1$. At finite temperatures $T>0$, we hence conjecture a crossover between $z=2$ at intermediate $T$ and $z=1$ at low $T$, and we construct the resulting nontrivial phase diagram as function of coupling strength and temperature.

We theoretically investigate a folded bilayer graphene structure as an experimentally realizable platform to produce the one-dimensional topological zero-line modes. We demonstrate that the folded bilayer graphene under an external gate potential enables tunable topologically conducting channels to be formed in the folded region, and that a perpendicular magnetic field can be used to enhance the conducting when external impurities are present. We also show experimentally that our proposed folded bilayer graphene structure can be fabricated in a controllable manner. Our proposed system greatly simplifies the technical difficulty in the original proposal by considering a planar bilayer graphene (i.e., precisely manipulating the alignment between vertical and lateral gates on bilayer graphene), laying out a strategy in designing practical low-power electronics by utilizing the gate-induced topological conducting channels.

We compute the phase diagram of twisted bilayer graphene near the magic angle where the occurrence of flat bands enhances the effects of electron-electron interactions and thus unleashes strongly correlated phenomena. Most importantly, we find a crossover between $d+\mathit{id}$ superconductivity and antiferromagnetic insulating behavior near half filling of the lowest electron band when the temperature is increased. This is consistent with recent experiments. Our results are obtained using unbiased many-body renormalization group techniques combined with a mean-field analysis of the effective couplings. We provide a qualitative understanding by considering the competition between Fermi-surface nesting and van Hove singularities.

We present a comprehensive theory for Bogoliubov Fermi surfaces in inversion-symmetric superconductors which break time-reversal symmetry. A requirement for such a gap structure is that the electrons posses internal degrees of freedom apart from the spin (e.g., orbital or sublattice indices), which permits a nontrivial internal structure of the Cooper pairs. In a pairing state that breaks time-reversal symmetry, the Cooper pairs are generically polarized in the internal degrees of freedom, in analogy to spin-polarized pairing in a nonunitary triplet superconductor. We show that this polarization can be quantified in terms of the time-reversal-odd part of the gap product, i.e., the matrix product of the pairing potential with its Hermitian conjugate. This product is essential for the appearance of Bogoliubov Fermi surfaces and their topological protection by a ${\mathbb{Z}}_2$ invariant. After studying the appearance of Bogoliubov Fermi surfaces in a generic two-band model, we proceed to examine two specific cases: a cubic material with a $j=3/2$ total-angular-momentum degree of freedom and a hexagonal material with distinct orbital and spin degrees of freedom. For these model systems, we show that the polarized pairing generates a magnetization of the low-energy states. We additionally calculate the surface spectra associated with these pairing states and demonstrate that the Bogoliubov Fermi surfaces are characterized by additional topological indices. Finally, we discuss the extension of phenomenological theories of superconductors to include Bogoliubov Fermi surfaces, and identify the time-reversal-odd polarization of the Cooper pairs as a composite order parameter, which is intertwined with superconductivity.

It is known that iron selenide superconductors exhibit unique characteristics distinct from iron pnictides, especially in the electron-doped region. However, a comprehensive study of continuous carrier doping and the corresponding crystal structures of FeSe is still lacking, mainly due to the difficulties in controlling the carrier density in bulk materials. Here we report the successful synthesis of a new family of bulk $\mathrm{L}{\mathrm{i}}_x{\left({\mathrm{C}}_3{\mathrm{N}}_2{\mathrm{H}}_{10}\right)}_{0.37}\mathrm{FeSe}$, which features a continuous superconducting dome harboring Lifshitz transition within the wide range of $0.06\le x\le 0.68$. We demonstrate that with electron doping, the anion height of FeSe layers deviates linearly away from the optimized values of pnictides and pressurized FeSe. This feature leads to a new superconducting zone with unique doping dependence of the electronic structures and strong orbital-selective electronic correlation. Optimal superconductivity is achieved when the $\mathrm{Fe}3d{t}_{2g}$ orbitals have almost the same intermediate electronic correlation strength, with moderate mass enhancement between $3\sim 4$ in the two separate superconducting zones. Our results shed new light on achieving unified mechanism of superconductivity in iron-based superconductors.