We study the $t\text{−}V\text{−}{V}^{\prime }$ model in one dimension at half filling. It is known that for large enough $V$ fixed, as $V^{\prime }$ is varied, the system goes from a charge-density wave into a Luttinger liquid, then a bond-order phase, and then a second charge-density wave phase. We find that the Luttinger liquid state is further split into two, separating parts with distinct values of the many-body polarization Berry phase. Inside this phase, the variance of the polarization is infinite in the thermodynamic limit, meaning that even if the polarization differs, it would not be measurable. However, in the gapped phases on each side of the Luttinger liquid, the polarization takes a different measurable value, implying topological distinction. The key difference is that the large-$V^{\prime }$ phases are link-inversion symmetric, while the small-$V^{\prime }$ one is site-inversion symmetric. We show that the large-$V^{\prime }$ phase can be related to an $S=1$ spin chain, and exhibits many features of the Haldane phase. The lowest lying states of the entanglement spectrum display different degeneracies in the two cases, and we also find string order in the large-$V^{\prime }$ phase. We also study the system under open boundary conditions, and suggest that the number of defects is related to the topology.

We theoretically investigate the microscopic mechanism of conversion between the electron spin and the microscopic local rotation of atoms in crystals. In phonon modes with angular momenta, the atoms microscopically rotate around their equilibrium positions in crystals. In a simple toy model with phonons, we calculate the spin expectation value by using the adiabatic series expansion. We show that the time-averaged spin magnetization is generated by the microscopic local rotation of atoms via the spin-orbit interaction. On the other hand, in the system with a simple vibration of atoms, time-averaged spin magnetization becomes zero due to the time-reversal symmetry. Moreover, the magnitude of the time-averaged spin magnetization depends on the inverse of the difference of instantaneous eigenenergy, and we show that it becomes smaller in band insulators with a larger gap.

We present combined experimental and numerical work on light-matter interactions at nanometer length scales. We report numerical simulations of near-field infrared nanospectroscopy that consider detailed tip geometry and have no free parameters. Our results match published spectral shapes of amplitude and phase measurements even for strongly resonant surface phonon-polariton (SPhP) modes. They also verify published absolute scattering amplitudes. A unique, ultrabroadband light source enables near-field amplitude and phase acquisition into the far-infrared spectral range. This allowed us to discover a strong SPhP resonance in the polar dielectric $\mathrm{SrTi}{\mathrm{O}}_3$ (STO) at an $\sim 24\text{−}\mu \mathrm{m}$ wavelength of incident light.

We present a ${}^{35}\mathrm{Cl}$ NMR study for the spin $S=1/2$ perfect kagome antiferromagnet Ca-kapellasite $[{\mathrm{CaCu}}_3{\left(\mathrm{OH}\right)}_6{\mathrm{Cl}}_2·0.6{\mathrm{H}}_2\mathrm{O}]$ with a magnetic transition at $T^{*}=7.2\mathrm{K}$. The static magnetic structure in the ground state has been determined to be a chirality-ordered $Q=0$ state, which is selected by a finite Dzyaloshinskii-Moriya interaction. The low-energy magnetic excitations in the ordered state are investigated by a nuclear spin-lattice relaxation rate measurement. We detect a weakly temperature-dependent contribution in the magnetic fluctuations perpendicular to the kagome plane in addition to the dispersive spin-wave contribution in the kagome plane. The low-energy magnetic excitations from the coplanar spin structure are attributed to the zero mode originating from the flat band in this kagome antiferromagnet.

We propose a superfluid phase of “many-fracton system” in which charge and total dipole moments are conserved quantities. In this work, both microscopic model and long-wavelength effective theory are analyzed. We start with a second quantized microscopic model and formulate the coherent-state path-integral representation. With repulsive interactions and positive chemical potential, we calculate various properties of the resulting superfluid state and make comparison with a conventional superfluid. We deduce a highly nonlinear Euler-Lagrange equation as well as two Noether currents. We also formulate time-dependent Gross-Pitaevskii-type equations that govern hydrodynamical behaviors. We study the classical ground-state wave function, the associated off-diagonal long range order (ODLRO), supercurrents, critical current, and unconventional topological vortices. At length scale much larger than coherence length ${\xi }_{\mathrm{coh}}$, we derive the effective theory of our microscopic model. Based on the effective theory, we analyze gapless Goldstone modes and specific heat capacity at low temperatures as well as the fate of ODLRO against quantum fluctuations. Several future directions, e.g., numerical analysis of Gross-Pitaevskii equations, fermionic fractons, fractonic superconductors, and cold-atom experimental realization, are discussed.

Recent advances in deep learning and neural networks have led to an increased interest in the application of generative models in statistical and condensed matter physics. In particular, restricted Boltzmann machines (RBMs) and variational autoencoders (VAEs) as specific classes of neural networks have been successfully applied in the context of physical feature extraction and representation learning. Despite these successes, however, there is only limited understanding of their representational properties and limitations. To better understand the representational characteristics of RBMs and VAEs, we study their ability to capture physical features of the Ising model at different temperatures. This approach allows us to quantitatively assess learned representations by comparing sample features with corresponding theoretical predictions. Our results suggest that the considered RBMs and convolutional VAEs are able to capture the temperature dependence of magnetization, energy, and spin-spin correlations. The samples generated by RBMs are more evenly distributed across temperature than those generated by VAEs. We also find that convolutional layers in VAEs are important to model spin correlations whereas RBMs achieve similar or even better performances without convolutional filters.

A system is non-Hermitian when it exchanges energy with its environment and nonreciprocal when it behaves differently upon the interchange of input and response. Within the field of metamaterial research on synthetic topological matter, the skin effect describes the conspiracy of non-Hermiticity and nonreciprocity to yield extensive anomalous localization of all eigenmodes in a (quasi) one-dimensional geometry. Here, we introduce the reciprocal skin effect, which occurs in non-Hermitian but reciprocal systems in two or more dimensions: Eigenmodes with opposite longitudinal momentum exhibit opposite transverse anomalous localization. We experimentally demonstrate the reciprocal skin effect in a passive RLC circuit, suggesting convenient alternative implementations in optical, acoustic, mechanical, and related platforms. Skin mode localization brings forth potential applications in directional and polarization detectors for electromagnetic waves.

Two-dimensional materials offer opportunities for unravelling unprecedented ordered states at the single-layer limit. Among such ordered states, the Mott phase is rarely explored. Here we study the Mott phase in van der Waals chromium (II) iodide ($\mathrm{Cr}{\mathrm{I}}_2$) films. High-quality $\mathrm{Cr}{\mathrm{I}}_2$ films with an atomically flat surface and macro size are grown on graphitized 6H-SiC(0001) substrate by molecular beam epitaxy. By in situ low-temperature scanning tunneling microscopy and spectroscopy, we reveal that the film has a band gap as large as $\sim 3.2\mathrm{eV}$, which is nearly thickness independent. Density functional plus dynamic mean-field theory calculations suggest that $\mathrm{Cr}{\mathrm{I}}_2$ films may be a strong Mott insulator with a ferromagnetically ordered ground state. The Mott phase is corroborated by the spectral band splitting and spectral weight transfer at charge dopants that is consistent with the extended Hubbard model. Our study provides a platform for studying correlated electron states at the single-layer limit.

The composite cosmological objects—Kibble-Lazarides-Shafi (KLS) walls bounded by strings and cosmic strings terminated by Nambu monopoles—could be produced during the phase transitions in the early universe. Recent experiments in superfluid ${}^3\mathrm{He}$ reproduced the formation of the KLS domain walls, which opened the new arena for the detailed study of those objects in a human controlled system with different characteristic lengths. These composite defects are formed by two successive symmetry breaking phase transitions. In the first transition the strings are formed, then in the second transition the string becomes the termination line of the KLS wall. In the same manner, in the first transition monopoles are formed, and then in the second transition these monopoles become the termination points of strings. Here we show that in the vicinity of the second transition the composite defects can be described by relative homotopy groups. This is because there are two well-separated length scales involved, which give rise to two different classes of the degenerate vacuum states, $R_1$ and $R_2$, and the composite objects correspond to the nontrivial elements of the group ${\pi }_n(R_1,R_2)$. We discuss this on example of the so-called polar distorted B phase, which is formed in the two-step phase transition in liquid ${}^3\mathrm{He}$ distorted by aerogel. In this system the string monopoles terminate spin vortices with an even winding number, while KLS string walls terminate on half-quantum vortices. In the presence of magnetic field, vortex skyrmions are formed, and the string monopole transforms to the nexus. We also discuss the integer-valued topological invariants of those objects. Our consideration can be applied to the composite defects in other condensed matter and cosmological systems.

We develop coupled-cluster theory for systems of electrons strongly coupled to photons, providing a promising theoretical tool in polaritonic chemistry with a perspective of application to all types of fermion-boson coupled systems. We show benchmark results for model molecular Hamiltonians coupled to cavity photons. By comparing to full configuration interaction results for various ground-state properties and optical spectra, we demonstrate that our method captures all key features present in the exact reference, including Rabi splittings and multiphoton processes. Furthermore, a path on how to incorporate our bosonic extension of coupled-cluster theory into existing quantum chemistry programs is given.

Spherical cap shaped liquid crystal drops have potential for use in tunable multifocal optical lenses. To achieve this, one needs to understand the liquid crystal director structure in various external fields. We present detailed experimental and theoretical studies of the director structure in magnetic fields of less than 1 mm diameter nematic liquid crystal drops on glass surfaces ($x\text{−}y$ plane), forming spherical cap shape with average molecular directions initially perpendicular to the boundaries. We show that at sufficiently high magnetic fields a Néel wall-type metastable inversion wall forms in the middle of the drop and then moves outward when the field is applied in $x$ direction. Applying fields above a critical angle between the plate of the spherical cap and the magnetic field, a uniform director structure forms. Drops with uniform director structure can be used as tunable optical lenses, where the focal length can be controlled by light polarization, viewing angle, magnetic, or electric fields. We also propose a theory that explains the texture variation in small magnetic fields and accounts for the formation and motion of the inversion wall. The theory explains the physical origin of the movement and gives the right magnitude of the speed at the first 10–15 min of the motion.

We show that a one-dimensional topological superconductor can be realized in carbon nanotubes, using a relatively small magnetic field. Our analysis relies on the intrinsic curvature-enhanced spin-orbit coupling of the nanotubes, as well as on the orbital effect of a magnetic flux threaded through the nanotube. Tuning experimental parameters, we show that a half-metallic state may be induced in the nanotube. Coupling the system to an Ising superconductor, with an appreciable spin-triplet component, can then drive the nanotube into a topological superconducting phase. The proposed scheme is investigated by means of real-space tight-binding simulations, accompanied by an effective continuum low-energy theory, which allows us to gain some insight on the roles of different terms in the Hamiltonian. We calculate the topological phase diagram and ascertain the existence of localized Majorana zero modes near the edges. Moreover, we find that in the absence of a magnetic field, a regime exists where sufficiently strong interactions drive the system into a time-reversal-invariant topological superconducting phase.

Recently there have been a number of experiments on ${\mathrm{Ce}}_2{\mathrm{Zr}}_2{\mathrm{O}}_7$ and ${\mathrm{Ce}}_2{\mathrm{Sn}}_2{\mathrm{O}}_7$, suggesting that these materials host a three-dimensional quantum spin liquid with emergent photons and fractionalized spinon excitations. However, the interpretation of the data to determine the precise nature of the quantum spin liquids is still under debate. The Kramers doublet in the ${\mathrm{Ce}}^{3+}$ local moment offers unusual pseudospin degrees of freedom as the $x$ and $z$ components transform as a dipole and the $y$ component as an octupole. This leads to a variety of possible quantum spin liquid (or quantum spin ice) phases on the pyrochlore lattice of ${\mathrm{Ce}}^{3+}$ moments. In this work, we theoretically propose that magnetostriction would be able to distinguish the dipolar (D-QSI) and octupolar (O-QSI) quantum spin ice, where the dipolar or octupolar components possess the respective spin ice correlations. We show that the magnetostriction in various configurations can be used as a selection rule to differentiate not only D-QSI and O-QSI but also a number of competing broken-symmetry states.

We investigate the ground state properties of the spin $S=1$ Kitaev honeycomb model under a magnetic field based on the density matrix renormalization group (DMRG) calculation. With the time-reversal symmetry breaking due to the magnetic field, a gapped Kitaev spin liquid is identified for both ferromagnetic (FM) and antiferromagnetic (AFM) Kitaev couplings. The topological nature of such a Kitaev spin liquid is manifested by a nearly quantized Wilson loop, degeneracy in the entanglement spectra, and the existence of edge modes. While the FM Kitaev spin liquid is destroyed by a weaker magnetic field $H_{*}^{\text{FM}}$, the AFM one demonstrates a robustness up to an order of magnitude larger critical field $H_{*}^{\text{AFM}}$. Moreover, an intermediate nonmagnetic phase appears only for the AFM case at larger fields, $H_{*}^{\text{AFM}}<H<H_{**}^{\text{AFM}}$, before the transition to a high-field polarized paramagnet. The stability of the Kitaev spin liquid against the Heisenberg interactions is also examined. Our findings may further inspire the investigation of recently proposed $S=1$ Kitaev materials.

Employing time-resolved photoelectron spectroscopy we analyze the relaxation dynamics of hot electrons in the charge density wave/Mott material $1T\text{−}{\mathrm{TaS}}_2$. At 1.2 eV above the Fermi level we observe a hot electron lifetime of $12\pm 5$ fs in the metallic state and of $60\pm 10$ fs in the broken symmetry ground state—a direct consequence of the reduced phase space for electron-electron scattering determined by the Mott gap. Boltzmann equation calculations which account for the interaction of hot electrons in a Bloch band with a doublon-holon excitation in the Mott state provide insight into the unoccupied electronic structure in the correlated state.

Fracton order is an intriguing new type of order which shares many common features with topological order, such as topology-dependent ground-state degeneracies, and excitations with mutual statistics. However, it also has several distinctive geometrical aspects, such as excitations with restricted mobility, which naturally lead to effective descriptions in terms of higher-rank gauge fields. In this paper, we investigate possible effective field theories for three-dimensional fracton order, by presenting a general philosophy whereby topological-like actions for such higher-rank gauge fields can be constructed. Our approach draws inspiration from Chern-Simons and BF theories in $2+1$ dimensions, and imposes constraints binding higher-rank gauge charge to higher-rank gauge flux. We show that the resulting fractonic Chern-Simons and BF theories reproduce many of the interesting features of their familiar two-dimensional cousins. We analyze one example of the resulting fractonic Chern-Simons theory in detail, and show that upon quantization it realizes a gapped fracton order with quasiparticle excitations that are mobile only along a subset of one-dimensional lines, and display a form of fractional self-statistics. The ground-state degeneracy of this theory is both topology and geometry dependent, scaling exponentially with the linear system size when the model is placed on a three-dimensional torus. By studying the resulting quantum theory on the lattice, we show that it describes a ${\mathbb{Z}}_s$ generalization of the Chamon code.

The theoretical model of the short-range interacting Luttinger liquid predicts a power-law scaling of the density of states and the momentum distribution function around the Fermi surface, which can be readily tested through tunneling experiments. However, some physical systems have long-range interaction, most notably the Coulomb interaction, leading to significantly different behaviors from the short-range interacting system. In this paper, we revisit the tunneling theory for the one-dimensional electrons interacting via the long-range Coulomb force. We show that, even though in a small dynamic range of temperature and bias voltage the tunneling conductance may appear to have a power-law decay similar to short-range interacting systems, the effective exponent is scale dependent and slowly increases with decreasing energy. This factor may lead to the sample-to-sample variation in the measured tunneling exponents. We also discuss the crossover to a free Fermi gas at high energy and the effect of the finite size. Our work demonstrates that experimental tunneling measurements in one-dimensional electron systems should be interpreted with great caution when the system is a Coulomb Luttinger liquid.

We demonstrate the equivalence between two distinct Luttinger liquid impurity problems. The first concerns a one-dimensional topological superconductor coupled at one end to the ends of two single-channel Luttinger liquids. The second concerns a point contact in the quantum spin Hall effect, where four helical Luttinger liquids meet at a point. Both of these problems have been studied previously and exhibit several stable phases, depending on the Luttinger parameter $K$, that can be characterized in terms of simple conformally invariant boundary conditions describing perfect normal (or Andreev) transmission or reflection. In addition, both problems exhibit critical points that are described by “intermediate” fixed points similar to those found in earlier studies of an impurity in a Luttinger liquid with spin. Though these two models have different symmetries and numbers of modes, we show they are equivalent and are related by a duality transformation, and we show that the nontrivial intermediate critical points are the same. In the noninteracting limit, $K=1$, the duality involves two distinct free fermion representations that are related by a nonlocal transformation that derives from the triality of $\text{SO}\left(8\right)$. Using the explicit translation between the two theories, we translate results from one problem to the other and vice versa. This allows us to make predictions about the topological superconductor–Luttinger liquid junction, including predictions about the global behavior of the critical conductance $G^{*}\left(K\right)$, as well predictions for the critical exponents and universal crossover scaling functions. In this paper, we introduce both models from scratch, using a common notation that facilitates their comparison, and we discuss in detail the dualities that relate them, along with their free fermion limits. We close with a discussion of open problems and future directions.

We consider a junction between two semi-infinite narrow ribbons of a two-dimensional (2D) topological insulator (TI) and calculate the electron current through the interface carried by one-dimensional (1D) edge states formed in the bulk band gap. In addition to an abrupt interface between ribbons, we also consider a configuration where one or two rectangular islands of finite length are sandwiched between these ribbons. We demonstrate that it is feasible to create “current-on” and “current-off” states by modifying the energy dispersion of 1D interface edge states and the resonant levels localized at the boundary of the island regions.

In this Rapid Communication, we develop a systematic weak localization and antilocalization theory fully considering the anisotropy and Berry phase of the system, and apply it to various phases of few-layer black phosphorus (BP), which has a highly anisotropic electronic structure with an electronic gap size tunable even to a negative value. The derivation of a Cooperon ansatz for the Bethe-Salpeter equation in a general anisotropic system is presented, revealing the existence of various quantum interference effects in different phases of few-layer BP, including a crossover from weak localization to antilocalization. We also predict that the magnetoconductivity at the semi-Dirac transition point will exhibit a nontrivial power-law dependence on the magnetic field, while following the conventional logarithmic field dependence of two-dimensional systems in the insulator and Dirac semimetal phases. Notably, the ratio between the magnetoconductivity and Boltzmann conductivity turns out to be independent of the direction, even in strongly anisotropic systems. Finally, we discuss the tunability of the quantum corrections of few-layer BP in terms of the symmetry class of the system.