Effects of the spin-orbit coupling (SOC) and magnetic field on excitonic insulating (EI) states are investigated. We introduce the two-orbital Hubbard model with the crystalline field splitting, which is a minimal model for discussing the exciton condensation in strongly correlated electron systems, and analyze its effective Hamiltonian in the strong correlation limit by using the mean-field theory. In the absence of the SOC and magnetic field, the ground state changes from the nonmagnetic band-insulating state to the EI state by increasing the Hund coupling. In an applied magnetic field, the magnetic moment appears in the EI state, which is continuously connected to the forced ferromagnetic state. On the other hand, in the presence of the SOC, they are separated by a phase boundary. We find that the magnetic susceptibility is strongly enhanced in the EI phase near the boundary with a small SOC. This peculiar behavior is attributed to the low-energy fluctuation of the spin nematicity inherent in the high-spin local state stabilized by the Hund coupling. The present study not only reveals the impact of the SOC for the EI state but also sheds light on the role of quantum fluctuations of the spin nematicity for the EI state.
We present an $\textit{ab initio}$ study of the excitonic states of a prototypical high-temperature superconductor La$_2$CuO$_4$ and compare them to the isostructural single-layer nickelate La$_2$NiO$_4$. Key difference in the low-energy electronic structure leads to very different excitonic behavior. Excitons in La$_2$CuO$_4$ are delocalized and can freely move in the CuO$_2$ plane without disturbing the antiferromagnetic order. In contrast, in La$_2$NiO$_4$ we find the low-lying excitonic states to be extremely localized, producing a nearly flat dispersion. The theoretically obtained exciton…
Pezzini et al. reported an unconventional mass enhancement in topological nodal line semimetal ZrSiS (Nat. Phys. 14, 178 (2018), whose origin remains puzzling. In this material, strong short-range interactions might induce excitonic particle-hole pairs. Here we study the renormalization of fermion velocities and find that the mass enhancement in ZrSiS can be well understood if we suppose that ZrSiS is close to the quantum critical point between semimetal and excitonic insulator. Near this quantum critical point, the fermion velocities are considerably reduced by excitonic quantum fluctuation, …
We propose a mechanism of the spin Seebeck effect attributed to excitonic condensation in a nonmagnetic insulator. We analyze a half-filled two-orbital Hubbard model with a crystalline field splitting in the strong coupling limit. In this model, the competition between the crystalline field and electron correlations brings about an excitonic insulating state, where the two orbitals are spontaneously hybridized. Using the generalized spin-wave theory and Boltzmann transport equation, we find that a spin current generated by a thermal gradient is observed in the excitonic insulating state withou…
Optical responses in an excitonic insulating (EI) system with strong electron correlation are studied. We adopt the two-orbital Hubbard model with a finite energy difference between the two orbitals where the spin state degree of freedom exists. This model is analyzed by the variational cluster approach. In order to include the local electron correlation effect, the vertex correction is taken into account in the formulation of the optical conductivity spectra. We calculate a finite-temperature phase diagram, in which an EI phase appears between a low-spin band insulating state and a high-spin …
A sufficiently strong Coulomb interaction may open an excitonic fermion gap and thus drive a semimetal-insulator transition in graphene. In this paper, we study the Eliashberg theory of excitonic transition by coupling the fermion gap equation self-consistently to the equation of vacuum polarization function. Including the fermion gap into polarization function increases the effective strength of Coulomb interaction because it reduces the screening effects due to the collective particle-hole excitations. Although this procedure does not change the critical point, it leads to a significant enha…
We develop a theory of the excitonic phase recently proposed as the zero-field insulating state observed near charge neutrality in monolayer WTe$_2$. Using a Hartree-Fock approximation, we numerically identify two distinct gapped excitonic phases: a spin density wave state for weak non-zero interaction strength and spin spiral order at stronger interactions, separated by a narrow window of non-excitonic quantum spin Hall insulator. We introduce a simplified model capturing key features of the WTe$_2$ band structure, in which these phases appear as distinct valley ferromagnetic orders. We link …
A sufficiently strong long-range Coulomb interaction can induce excitonic pairing in gapless Dirac semimetals, which generates a finite gap and drives semimetal-insulator quantum phase transition. This phenomenon is in close analogy to dynamical chiral symmetry breaking in high energy physics. In most realistic Dirac semimetals, including suspended graphene, Coulomb interaction is too weak to open an excitonic gap. The Coulomb interaction plays a more important role at low energies in a two-dimensional semi-Dirac semimetal, in which the fermion spectrum is linear in one component of momenta an…
Magnetic frustration can lead to peculiar magnetic orderings that break a discrete symmetry of the lattice in addition to the fundamental magnetic symmetries (i.e., spin rotation invariance and time-reversal symmetry). In this work, we focus on frustrated quantum magnets and study the nature of the quantum phase transition between a paramagnet and a magnetically ordered state with broken threefold ($Z_3$) crystal rotation symmetry. We predict the transition to happen in two stages, giving rise to an intermediate nematic phase in which rotation symmetry is broken but the system remains magnetic…
Excitonic insulating system is studied from the viewpoints of the orbital physics in strongly correlated electron systems. An effective model Hamiltonian for low-energy electronic states is derived from the two-orbital Hubbard model with a finite energy difference corresponding to the crystalline field splitting. The effective model is represented by the spin operators and the pseudo-spin operators for the spin-state degrees of freedom. The ground state phase diagram is analyzed by the mean-field approximation. In addition to the low-spin state and high-spin state phases, two kinds of the exci…
The competing ground states of bilayer graphene are studied by applying renormalization group techniques to a bilayer honeycomb lattice with nearest neighbor hopping. In the absence of interactions, the Fermi surface of this model at half-filling consists of two nodal points with momenta $\mathbf{K}$, $\mathbf{K}'$, where the conduction band and valence band touch each other, yielding a semi-metal. Since near these two points the energy dispersion is quadratic with perfect particle-hole symmetry, excitonic instabilities are inevitable if inter-band interactions are present. Using a perturbativ…
We study the quantum criticality of the phase transition between the Dirac semimetal and the excitonic insulator in two dimensions. Even though the system has a semimetallic ground state, there are observable effects of excitonic pairing at finite temperatures and/or finite energies, provided that the system is in proximity to the excitonic insulating transition. To determine the quantum critical behavior, we consider three potentially important interactions, including the Yukawa coupling between Dirac fermions and the excitonic order parameter fluctuation, the long-range Coulomb interaction, …
The strong long-range Coulomb interaction between massless Dirac fermions in graphene can drive a semimetal-insulator transition. We show that this transition is strongly suppressed when the Coulomb interaction is screened by such effects as disorder, thermal fluctuation, doping, and finite volume. It is completely suppressed once the screening factor $μ$ is beyond a threshold $μ_{c}$ even for infinitely strong coupling. However, such transition is still possible if there is an additional strong contact four-fermion interaction. The differences between screened and contact interactions are als…
We consider the effective model of $H=-J_{1}\sum_{<i,j>}\mathbf{S}_{i}\cdot \mathbf{S}_{j}-J_{2}\sum_{<i,j>}(\mathbf{S}_{i}\cdot \mathbf{S}_{j}) ^{2}$, describing the Mott insulating phase with odd number of spin-1 bosons in optical lattices ($J_{2}>J_{1}>0$). In terms of an SU(3) boson representation, a valence bond mean field theory is developed. In 1D, a \textit{first-order} quantum phase transition from a spin singlet to a spin nematic phase with \textit{gapful} excitations is identified at $ J_{1}/J_{2}=0.19833$, while on a 2D square lattice a spin nematic ordered phase with gapless excit…
Artificial moiré superlattices in 2d van der Waals heterostructure is a new venue for realizing and controlling correlated electronic phenomena. Recently, twisted bilayer WSe$_2$ emerged as a new robust moiré system hosting a correlated insulator at moiré half-filling over a range of twist angle. In this work, we present a theory of this insulating state as an excitonic density wave due to intervalley electron-hole pairing. We show that exciton condensation is strongly enhanced by a van Hove singularity near the Fermi level. Our theory explains the remarkable sensitivity of the insulating gap …
Motivated by recent experiments, which give strong evidence for an excitonic insulating phase in $\rm TmSe_{0.45}Te_{0.55}$, we developed a scheme to quantitatively construct, for generic two-band models, the phase diagram of an excitonic insulator. As a first application of our approach, we calculated the phase diagram for an effective mass two-band model with long-range Coulomb interaction. The shielded potential approximation is used to derive a generalized gap equation controlling for positive (negative) energy gaps the transition from a semi-conducting (semi-metallic) phase to an insulati…
We consider insulating phases of cold spin-1 bosonic particles with antiferromagnetic interactions, such as Na-23, in optical lattices. We show that spin-exchange interactions give rise to several distinct phases, which differ in their spin correlations. In two- and three- dimensional lattices insulating phases with odd number of particles per site are always nematic. For insulating states with even number of particles per site there is always a spin-singlet phase and there may also be a first-order transition into the nematic phase. The nematic phase breaks spin rotational symmetry, but prese…
A recent paper on the insulating state of monolayer WTe2 reported the observation of large oscillations in the conductivity that are periodic in 1/B, resembling quantum oscillations in metals. This remarkable observation has inspired suggestions of exotic physics such as spin-charge separation. We show that a rather more conventional but still nontrivial explanation in terms of gap modulation may be possible in a model of excitonic insulator subject to a magnetic field.
Three-dimensional gapless semiconductors with quadratic band touching, such as HgTe, $α$-Sn, or Pr$_2$Ir$_2$O$_7$ are believed to display a non-Fermi-liquid ground state due to long-range electron-electron interaction. We argue that this state is inherently unstable towards spontaneous formation of a (topological) excitonic insulator. The instability can be parameterized by a critical fermion number $N_c$. For $N < N_c$ the rotational symmetry is spontaneously broken, the system develops a gap in the spectrum, and features a finite nematic order parameter. To leading order in the 1/N expansion…
Strong magnetic field induces at least two phase transitions in graphite beyond the quantum limit where many-body effects are expected. We report on a study using a state-of-the-art non-destructive magnet allowing to attain 90.5 T at 1.4 K, which reveals a new field-induced phase and evidence that the insulating state destroyed at 75 T is an excitonic condensate of electron-hole pairs. By monitoring the angle dependence of in-plane and out-of-plane magnetoresistance, we distinguish between the role of cyclotron and Zeeman energies in driving various phase transitions. We find that, with the no…
This is a commentary on two papers (PNAS 117 (51) 32244-32250 (2020) and arXiv:2011.06721), which observed a series of ordering transitions in a strongly correlated two-dimensional electron system confined to a AlAs quantum well. We summarize the main discoveries (electron nematicity, insulating ferromagnetism, etc.) of them and speculate on $T-r_s$ phase diagram based on theoretical considerations.
The stability of the excitonic condensation at low temperature driven by a coupling of electrons to vibrational degrees of freedom in semimetal two-dimensional electronic system is discussed. In the framework of the unrestricted Hartree-Fock approximation, we derive a set of equations to determine both the excitonic condensate order parameter and lattice displacement self-consistently. By lowering temperature we find out a semimetal-insulator transition in the system if the coupling is large enough. The insulating state typifies an excitonic condensation accompanied by a finite lattice distort…
We study the excitonic insulating (EI) phase in the two-band Hubbard models on the Penrose tiling. Performing the real-space mean-field calculations systematically, we obtain the ground state phase diagrams for the vertex and center models. We find that, in some regimes, the stable EI phase is induced by small interband interactions. We argue that this originates from the electron-hole pairing for the completely or nearly degenerate states, which are characteristic of the Penrose tiling. We also study spatial distribution of the order parameter, mapping it to the perpendicular space.
Engineering excitonic metal-insulator transitions in ultra-thin doped copper sulfides
Shanghai Synchrotron Radiation Facility, Shanghai Advanced Research Institute, Chinese Academy of Sciences（中国科学院上海高等研究院）；Hefei National Laboratory（合肥国家实验室）
Excitonic insulator (EI) was proposed in 60's as a distinct insulating state originating from pure electronic interaction, but its material realization has been elusive with extremely few material candidates and with only limited evidence such as anomalies in transport properties, band dispersions, or optical transitions. We investigate the real-space electronic states of the low temperature phase in Ta$_2$NiSe$_5$ with an atomic resolution to clearly identify the quasiparticle energy gap together with the strong electron-hole band renormalization using scanning tunneling microscopy (STM) and …
Slave rotor mean field (SRMF) theory is employed to study exciton formation in the extended Falicov-Kimball Model (EFKM). In this theory, charge and spin (or orbital) degrees are treated as independent degrees of freedom, coupled by a constraint. Using this formalism in single as well as cluster extension, we capture the effective many body scales beyond conventional mean-field theory. While the formation of exciton is favored by the interband hybridization $V$, it is strongly influenced by the on-site Coulomb interaction $U$. Beyond a critical hybridization, there is a condensation of exciton…
Magnetic field (H) effects on a correlated electron system with the spin-state degree of freedom are examined. The effective Hamiltonian derived from the two-orbital Hubbard model is analyzed by the mean-field approximation. Applying H to the low-spin (LS) phase induces the excitonic insulating phase, as well as the spin-state ordered phase where the LS and high-spin (HS) states are ordered alternately. In the case where H is applied to the HS phase, a reentrant transition for the HS phase appears. A rich variety of the phase diagrams are attributed to the spin-state degree of freedom and thei…
Two-dimensional multi-valley electronic systems in which the dispersion of individual pockets has low symmetry give rise to quantum Hall ferroelectric and nematic states in the presence of strong quantising magnetic fields. We investigate local signatures of these states arising near impurities that can be probed via Scanning Tunnelling Microscopy (STM) spectroscopy. For quantum Hall ferroelectrics, we demonstrate a direct relation between the dipole moment measured at impurity bound states and the ideal bulk dipole moment obtained from the modern theory of polarisation. We also study the many…
Motivated by the spate of recent experimental and theoretical interest in Mott insulating S=1 triangular lattice magnets, we consider a model S=1 Hamiltonian on a triangular lattice interacting with rotationally symmetric biquadratic interactions. We show that the partition function of this model can be expressed in terms of configurations of three colors of tightly-packed, closed loops with {\em non-negative} weights, which allows for efficient quantum Monte Carlo sampling on large lattices. We find the ground state has spin nematic order, i.e. it spontaneously breaks spin rotation symmetry b…
Magnetism in transition-metal compounds (TMCs) has traditionally been associated with spin degrees of freedom, because the orbital magnetic moments are typically largely quenched. On the other hand, magnetic order in 4f- and 5d-electron systems arises from spin and orbital moments that are rigidly tied together by the large intra-atomic spin-orbit coupling (SOC). Using inelastic neutron scattering on the archetypal 4d-electron Mott insulator Ca$_2$RuO$_4$, we report a novel form of excitonic magnetism in the intermediate-strength regime of the SOC. The magnetic order is characterized by ``soft…
Motivated by recent experimental processes, we systemically investigate strongly correlated spin-1 ultracold bosons trapped in a three-dimensional optical lattice in the presence of an external magnetic field. Based on a recently developed bosonic dynamical mean-field theory (BDMFT), we map out complete phase diagrams of the system for both antiferromagnetic and ferromagnetic interactions, where various phases are found as a result of the interplay of spin-dependent interaction and quadratic Zeeman energy. For antiferromagnetic interactions, the system demonstrates competing magnetic orders, i…
The electronic properties of excitonic insulators have been examined precisely in recent years. Pictures of exciton condensation may be applied to the spin-state transition observed in perovskite cobalt oxides. We examine the crystal-field and magnetic-field dependences of spatial spin structures on the basis of the density matrix renormalization group method using an effective model for the one-dimensional two-orbital Hubbard model in strong-coupling limit. We find an excitonic insulating (EI) phase and a spin-state ordering (SSO) phase in the intermediate region between low-spin and high-spi…
Graphite under high magnetic field exhibits consecutive metal-insulator (MI) transitions as well as re-entrant insulator-metal (IM) transition in the quasi-quantum limit at low temperature. In this paper, we identify the low-$T$ insulating phases as excitonic insulators with spin nematic orderings. We first point out that graphite under the relevant field regime is in the charge neutrality region, where electron and hole densities compensate each other. Based on this observation, we introduce interacting electron models with electron pocket(s) and hole pocket(s) and enumerate possible umklapp …
Using a controlled weak-coupling renormalization group approach, we establish the mechanism of unconventional superconductivity in the vicinity of spin or charge ordered excitonic states for the case of electrons on the Bernal stacked bilayer honeycomb lattice. With one electron per site this system exhibits nearly parabolically touching conduction and valence bands. Such a state is unstable towards a spontaneous symmetry breaking, and repulsive interactions favor excitonic order, such as a charge nematic and/or a layer antiferromagnet. We find that upon adding charge carriers to the system, t…
In contrast to bulk FeSe, which exhibits nematic order and low temperature superconductivity, atomic layers of FeSe reverse the situation, having high temperature superconductivity appearing alongside a suppression of nematic order. To investigate this phenomenon, we study a minimal electronic model of FeSe, with interactions that enhance nematic fluctuations. This model is sign problem free, and is simulated using determinant quantum Monte Carlo (DQMC). We developed a DQMC algorithm with parallel tempering, which proves to be an efficient source of global updates and allows us to access the r…
We investigate the half-filled two-orbital Hubbard model with the crystalline electric field using dynamical mean-field theory combined with the continuous-time quantum Monte Carlo simulations. We systematically study how the interplay of the intra- and interorbital Coulomb interations together with the Hund coupling realizes the diagonal and off-diagonal ordered states. It is found that the antiferroorbital ordered state is realized in the Hubbard model, in addition to the antiferromagnetically ordered and excitonic states. The competition between the antiferroorbital ordered and excitonic st…
Tuning many-body electronic phases by an external handle is of both fundamental and practical importance in condensed matter science. The tunability mirrors the underlying interactions, and gigantic electric, optical and magnetic responses to minute external stimuli can be anticipated in the critical region of phase change. The excitonic insulator is one of the exotic phases of interacting electrons, produced by the Coulomb attraction between a small and equal number of electrons and holes, leading to the spontaneous formation of exciton pairs in narrow-gap semiconductors/semimetals. The layer…
We directly trace the near- and mid-infrared transmission change of a VO$_2$ thin film during an ultrafast insulator-to-metal transition triggered by high-field multi-terahertz transients. Non-thermal switching into a metastable metallic state is governed solely by the amplitude of the applied terahertz field. In contrast to resonant excitation below the threshold fluence, no signatures of excitonic self-trapping are observed. Our findings are consistent with the generation of spatially separated charge pairs and a cooperative transition into a delocalized metallic state by THz field-induced t…
Correlated Metals, Metamagnetism, and Orbital Nematic Order in moiré Materials with Neural Quantum States
ETH Zürich（苏黎世联邦理工学院）；Institute of Theoretical Physics, Chinese Academy of Sciences（中国科学院理论物理研究所）；University of Chinese Academy of Sciences（中国科学院大学）；École Polytechnique Fédérale de Lausanne (EPFL)（洛桑联邦理工学院）
We derive from a microscopic model the effective theory of nematic order in a system with a spontaneous quantum anomalous Hall effect in two dimensions. Starting with a model of two-component fermions (a spinor field) with a quadratic band crossing and short range four-fermion marginally relevant interactions we use a 1/N expansion and bosonization methods to derive the effective field theory for the hydrodynamic modes associated with the conserved currents and with the local fluctuations of the nematic order parameter. We focus on the vicinity of the quantum phase transition from the isotropi…
Graphene-based moiré systems have attracted considerable interest in recent years as they display a remarkable variety of correlated phenomena. Besides insulating and superconducting phases in the vicinity of integer fillings of the moiré unit cell, there is growing evidence for electronic nematic order both in twisted bilayer graphene and twisted double-bilayer graphene (tDBG), as signaled by the spontaneous breaking of the threefold rotational symmetry of the moiré superlattices. Here, we combine symmetry-based analysis with a microscopic continuum model to investigate the structure of the n…
On the basis of the multi-orbital dynamical mean field theory, a three-orbital Hubbard model with a relativistic spin-orbit coupling (SOC) is studied at five electrons per site. The numerical calculations are performed by employing the continuous-time quantum Monte Carlo (CTQMC) method based on the strong coupling expansion. We find that appropriately choosing bases, i.e., the maximally spin-orbit-entangled bases, drastically improve the sign problem in the CTQMC calculations, which enables us to treat exactly the full Hund's coupling and pair hopping terms. This improvement is also essential …
University of Southern California（南加州大学）；Tamkang University（淡江大学）；The University of British Columbia（不列颠哥伦比亚大学）；The University of Texas at Austin（德克萨斯大学奥斯汀分校）；University of Washington（华盛顿大学）；University of California at Berkeley（加利福尼亚大学伯克利分校）；Lawrence Berkeley National Laboratory（劳伦斯伯克利国家实验室）
