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Equilibrium probes have been widely used in various noisy quantum metrology schemes. However, such an equilibrium-probe-based metrology scenario severely suffers from the low-temperature-error divergence problem in the weak-coupling regime. To circumvent this limit, we propose a strategy to eliminate the error-divergence problem by utilizing the strong coupling effects, which can be captured by the reaction-coordinate mapping. The strong couplings induce a noncanonical equilibrium state and greatly enhance the metrology performance. It is found that our metrology precision behaves as a polynomial-type scaling relation, which suggests the reduction of temperature can be used as a resource to improve the metrology performance. Our result is sharply contrary to that of the weakcoupling case, in which the metrology precision exponentially decays as the temperature decreases. Paving a way to realize a high-precision noisy quantum metrology at low temperatures, our result reveals the importance of the non-Markovianity in quantum technologies.

A key feature of the newly discovered altermagnet is that its spin degeneracy is lifted, although it has an antiferromagnetic order and zero net magnetization. In this work, we investigate a frustrated spin-1/2 $J_1$-$J_2$-$\delta$ Heisenberg model on the square lattice by the tensor network methodin combination with the linear spin-wave theory, with our focus on both the magnon excitations and longitudinal excitations.For a small $J_2$ and a finite range of $\delta$ we demonstrate that such a model hosts an altermagnetic ground state. Its magnon spectrum is split into two branches and the largest splitting occurs at $\left(\pm\pi/2, \pm\pi/2\right)$ in the Brillouin zone. The magnitudes of splitting in the two magnon modes are equal with respect to the case of $\delta=0$. Dynamical spin structure factors show that the low-energy peak in the longitudinal spectral weight around $(\pi/2, \pi/2)$ is also split, and thus the relative positions of the magnon modes and longitudinal modes in energy may change in the presence of a finite $\delta$. These findings demonstrate that the altermagnets harbor more complex quantum dynamics than the conventional collinear antiferromagnets., Comment: The splitting in the low-energy longitudinal spectral peak is explained

Novel collinear magnet, the altermagnet (AM) with spin-splitting energy band and zero net magnetization have attracted great interest due to its potential spintronic applications. Here, we demonstrate AM-like phases in a microscopic Kondo lattice model, widely used for heavy fermion compounds. With the framework of fermionic parton mean-field theory, we find the $d$-wave AM state can coexist with the intrinsic Kondo screening effect in such itinerant-local electron system if an alternating next-nearest-neighbor-hopping (NNNH) is included. Such alternating NNNH take nonmagnetic atoms, neglected in usual antiferromagnetism study, into account when encountering real-life candidate AM materials. The AM-like states are characterized by their spin-splitting quasiparticle bands, Fermi surface, spin-resolved distribution function and conductivity. It is suggested that the magnetic quantum oscillation and charge transport measurement can detect those AM-like phases. We hope the present work may be useful for exploring AM-like phases in $f$-electron compounds., Comment: 14 pages, 12 figures

We study the ground-state phase diagram of the half-filled extended Haldane-Hubbard model on the honeycomb lattice with sublattice-dependent on-site repulsion ($U_{\text{A/B}}$) using the exact diagonalization (ED) and mean-field (MF) methods. The resulting phase diagram shows that there is a topologically nontrivial phase with the Chern number $C=1$, emerging via the development of the imbalance between $U_{\text{A}}$ and $U_{\text{B}}$. In this phase, the antiferromagnetic correlations are observed in the ED calculation, in line with the finite antiferromagnetic order obtained by the MF method. The spontaneous symmetry breaking of SU(2) spin rotation in the phase is also identified in the MF level. Distinct from previous studies in which the exotic $C=1$ phase relies on the interplay between sublattice-dependent potentials and electronic interactions, our paper presents an alternative way by solely tuning the on-site interactions., Comment: 8 pages, 5 figures

Exploring the influence of frustration on the phases and related phase transitions in condensed matter physics is of fundamental importance in uncovering the role played by frustration. In the two-dimensional square lattice, a minimal frustration has been formulated in 1981 as the Shastry-Sutherland (SS) model described by competitions between the nearest-neighbor bond ($J_1$) and the next-nearest-neighbor one ($J_2$). In the two limits of $\alpha=J_2/J_1$, i.e. $\alpha \ll 1$ and $\alpha \gg 1$, the corresponding phases are the N{\'e}el antiferromagnet (AFM) and the dimer-singlet(DS). Unfortunately, the intermediate regime remains controversial, and the nature of transition from the N{\'e}el AFM to the intermediate state is also unclear. Here we provide a pattern language to explore the SS model and take the lattice size $L=4 \times4$ with periodic boundary condition. We firstly diagonalize the Hamiltonian in an operator space to obtain all fundamental spin-patterns and then analyze their energy and occupancy evolutions with the frustration parameter $\kappa=\alpha / (1+\alpha)$. Our results indicate that the intermediate regime is characterized by diagonal two-domain spin-pattern while the N{\'e}el AFM state has a diagonal single-domain and the DS has mixings of diagonal single- and four-domain. While the transition from the DS to the intermediate phase occurred around $\alpha_c = 1.5$ is the first-order in nature, consistent with that in literature, the one from the intermediate phase to the AFM is clearly seen around $\alpha_c = 1.277$, where it has a reversal of the contributions from the single- and two-domain patterns to the ground state. The result indicates that the pattern language is powerful in identifying the possible phases in frustrated models., Comment: 6 pages, 4 figures

Using the exact-diagonalization (ED) and mean-field (MF) approaches, we investigate the ground-state phase diagram of the interacting Haldane model on the honeycomb lattice, incorporating spin-dependent sublattice potentials $\Delta_{\sigma,\alpha}$. Here $\alpha=\text{A}$,$\text{B}$ and $\sigma=\uparrow$,$\downarrow$ denote the sublattice and spin components, respectively. Setting $\Delta_{\sigma,\text{A}}=+\Delta$ ($-\Delta$) and $\Delta_{\sigma,\text{B}}$$=-\Delta$ ($+\Delta$) for $\sigma=\uparrow$ ($\downarrow$) results in the system favoring a spin ordered state. Conversely, introducing the nearest-neighbor Coulomb interaction can induce charge ordering in the system. Due to the competition between these factors, we observe that in both ED and MF approaches, an exotic state with Chern number $C=1$ survives amidst two locally ordered phases and a topologically ordered phase with $C=2$. In the ED method, various properties, such as the fidelity metric, the excitation gap and the structure factors, are employed to identify critical points. In the MF method, using a sufficiently large lattice size, we define the local order parameters and band gaps to characterize the phase transitions. The interacting Haldane model and the spin-dependent lattice potential may be experimentally realized in an ultracold atom gas, providing a potential means to detect this intriguing state., Comment: 8 pages, 8 figures

The noncollinear spin textures provide promising avenues to stabilize exotic magnetic phases and excitations. They have attracted vast attention owning to their nontrivial band topology in the past decades. Distinct from the conventional route of involving the Dzyaloshinskii-Moriya interaction in a honeycomb magnet, the interplay of bond-dependent Kitaev and $\Gamma$ interactions, originating from the spin-orbit coupling and octahedra crystal field in real materials, has demonstrated to be another source to generate noncollinear spin textures with multiple spins in a magnetic unit cell. Notably, earlier works have revealed a triple-meron crystal (TmX) consisting of eighteen spins in the frustrated Kitaev-$\Gamma$ model. Aligning with previous efforts, here we attempt to identify that the TmX hosts several peculiar features with the help of the linear spin-wave theory. To begin with, the symmetric anisotropic exchanges are beneficial for the existence of nonreciprocal magnons, which are stabilized by an external magnetic field. Further, within the regime of TmX, successive topological phase transitions occur, accompanied by the changes of Chern number in value and thermal Hall conductivity in sign. In addition, topological nature of magnons is also verified by the onset of chiral edge modes in a nanoribbon geometry. Our findings pave the way to study topological phenomena of noncollinear spin textures in potential Kitaev materials., Comment: 11 pages, 6+3 figures

The quantum to classical transition (QCT) is one of the central mysteries in quantum physics. This process is generally interpreted as state collapse from measurement or decoherence from interacting with the environment. Here we define the quantumness of a Hamiltonian by the free energy difference between its quantum and classical descriptions, which vanishes during QCT. We apply this criterion to the many-body Rabi model and study its scaling law across the phase transition, finding that not only the temperature and Planck constant, but also all the model parameters are important for this transition. We show that the Jaynes-Cummings and anti Jaynes-Cummings models exhibit greater quantumness than the Rabi model. Moreover, we show that the rotating wave and anti-rotating wave terms in this model have opposite quantumness in QCT. We demonstrate that the quantumness may be enhanced or suppressed at the critical point. Finally, we estimate the quantumness of the Rabi model in current trapped ion experiments. The quantumness provides an important tool to characterize the QCT in a vast number of many-body models., Comment: 6 pages, 5 figures

The Haldane-Hubbard model is a prime example of the combined effects of band topology and electronic interaction. We revisit its spinful phase diagram at half-filling as a consensus on the presence of SU($2$) symmetry is currently lacking. To start, we utilize the Hartree-Fock mean-field method, which offers a direct understanding of symmetry breaking through the effective mass term that can acquire spin dependence. Our results, in agreement with previous studies, provide an instructive insight into the regime where the Chern number $C=1$, with only one spin species remaining topological. Besides that, we numerically study the phase diagram of the Haldane-Hubbard model via a large-scale infinite-density matrix renormalization group (iDMRG) method. The phase boundaries are determined by the Chern number and the correlation lengths obtained from the transfer-matrix spectrum. Unlike previous studies, the iDMRG method investigates the Haldane-Hubbard model on a thin and infinitely long cylinder and examines scenarios consistent with the two-dimensional thermodynamic limit. Here, the phase diagram we obtained qualitatively goes beyond the Hartree-Fock scope, particularly in the $C=1$ region, and serves as a quantitative benchmark for further theoretical and experimental investigations., Comment: 12 pages, 9 figures

The frustrated spin-$1/2$ $J_1-J_2$ Heisenberg model on the square lattice has been extensively studied since 1988 because of its close relationship to the high-temperature superconductivity in cuprates and more importantly involved novel phase of matter in its own right, namely, quantum spin liquid (QSL), one of hot topics in condensed matter physics in recent years. However, the phase diagram of the model, particularly in the maximally frustrated regime $J_2/J_1 \sim 0.5$, is quite controversial, and more seriously the nature of the QSL is not clear at all. Here we provide a pattern picture, on one hand, to show explicitly how the system evolves from the N\'eel antiferromagnetic (AFM) state at small $J_2$ to the striped AFM one at large $J_2$; on the other hand, to uncover the nature of the QSL if it exists in the intermediate $J_2$ coupling regime. For simplicity, we show our results by taking the square lattice $L=L_x \times L_y$ with size $L_x=L_y=4$ here and periodic boundary condition is considered, and furthermore, exact diagonalization is employed to confirm the correctness of our picture. Our results indicate that the highly frustration regime is characterized by diagonal two-domain, while the N\'eel AFM state has a diagonal single-domain and the striped AFM state shows itself as a diagonal four-domain, namely, completely diagonal antiferromagnetic order, in the present case. Increasing the system size, the number of the diagonal domains increases correspondingly, but the diagonal single-domain for the N\'eel AFM state and the diagonal $L_{x(y)}$-domain for the striped AFM state remain unchanged. Our results shed light on the understanding of the QSL., Comment: 6 pages, 4 figures

TeV-range cosmic ray electrons and positrons (CREs) have been directly measured in the search for new physics or unknown astrophysical sources. CREs can inverse-Compton scatter solar photons and boost their energies into gamma ray bands. Any potential CRE excess would enhance the resultant inverse Compton emission spectrum in the relevant energy range, offering a new window to verify the measured CRE spectrum. In this paper, we show that an excess in the TeV range of the CRE spectrum, such as the one indicated by the DAMPE experiment, can induce a characteristic solar gamma ray signal. Accounting for contamination from extragalactic gamma ray backgrounds (EGB), we forecast the DAMPE feature is testable ($\gtrsim 4 \sigma$) with a $\sim 10^{5}\,\mathrm{m}^2\,{\rm yr}$ exposure in the off-disk direction. This can be achieved by long-exposure observations of water Cherenkov telescopes, such as LHAASO (7.2 years) and HAWC (25.9 years)., Comment: 6 pages, 4 figures

We present a comprehensive investigation of quantum oscillations (QOs) in the strongly-correlated Falicov-Kimball model (FKM). The FKM is a particularly suitable platform for probing the non-Fermi liquid state devoid of quasiparticles, affording exact Monte Carlo simulation across all parameter spaces. In the high-correlation regime, we report the presence of prominent QOs in magnetoresistance and electron density at low temperatures within the phase separation state. The frequency behavior of these oscillations uncovers a transition in the Fermi surface as electron density diminishes, switching from hole-like to electron-like. Both types of Fermi surfaces are found to conform to the Onsager relation, establishing a connection between QOs frequency and Fermi surface area. Upon exploring the temperature dependence of QOs amplitude, we discern a strong alignment with the Lifshitz-Kosevich (LK) theory, provided the effective mass is suitably renormalized. Notwithstanding, the substantial enhancement of the overall effective mass results in a notable suppression of the QOs amplitude within the examined temperature scope, a finding inconsistent with Fermi liquid predictions. For the most part, the effective mass diminishes as the temperature increases, but an unusual increase is observed at the proximity of the second-order phase transition instigated by thermal effects. As the transition ensues, the regular QOs disappear, replaced by irregular ones in the non-Fermi liquid state under a high magnetic field. We also uncover significant QOs in the insulating charge density wave state under weak interactions ($0 < U < 1$), a phenomenon we elucidate through analytical calculations. Our findings shed light on the critical role of quasiparticles in the manifestation of QOs, enabling further understanding of their function in this context., Comment: 10 pages, 9 figures

In a conventional boson system, the ground state can either be an insulator or a superfluid (SF) due to the duality between particle number and phase. This paper reveals that the long-sought Bose metal (BM) state can be realized in an exactly solvable interacting bosonic model, i.e. the Bose-Hatsugai-Kohmoto (BHK) model, which acts as the nontrivial extension of Bose-Hubbard (BH) model. By tuning the parameters such as bandwidth $W$, chemical potential $\mu$, and interaction strength $U$, a BM state without any symmetry-breaking can be accessed for a generic $W/U$ ratio, while a Mott insulator (MI) with integer boson density is observed at small $W/U$. The quantum phase transition between the MI and BM states belongs to the universality class of the Lifshitz transition, which is further confirmed by analyzing the momentum-distribution function, the Drude weight, and the superfluid density. Additionally, our investigation at finite temperature reveals similarities between the BM state and the Fermi liquid, such as a linear-$T$ dependent heat capacity ($Cv\sim \gamma T$) and a saturated charge susceptibility ($\chi_{c} \sim $ constant) as $T$ approaches zero. Comparing the BM state with the SF state in the standard BH model, we find that the key feature of the BM state is a compressible total wavefunction accompanied by an incompressible zero-momentum component. Given that the BM state prevails over the SF state at any finite $U$ in the BHK model, our work suggests the possibility of realizing the BM state with on-site repulsion interactions in momentum space., Comment: 10 pages, 8 figures

The Hubbard and related models serve as a fundamental starting point in understanding the novel experimental phenomena in correlated electron materials, such as superconductivity, Mott insulator, magnetism and stripe phases. Recent numerical simulations indicate that the emergence of superconductivity is connected with the next nearest-neighbor hopping $t^\prime$ in the Hubbard model. However, the impacts of complex inter-site electron interaction in the $t^\prime$-Hubbard model are less explored. Utilizing the state-of-art density-matrix renormalization group method, we investigate the $t^\prime$-Hubbard model on a two-leg ladder with inter-site interactions extended to the fourth neighbor sites. The accurate numerical results show that the quasi-long-range superconducting correlation remains stable under the repulsive nearest-neighbor and the next nearest-neighbor interactions though these interactions are against the superconductivity. The ground state properties are also undisturbed by the longer-range repulsive interactions. In addition, inspired by recent experiments on one-dimensional cuprates chain $\mathrm{Ba}_{2-x}\mathrm{Sr}_x\mathrm{CuO}_{3+\delta}$, which implies an effective attraction between the nearest neighbors may exist in the cuprates superconductors, we also show that the attractive interaction between the nearest neighbors significantly enhances the superconducting correlation when it is comparable to the strength of the nearest-neighbor hopping $t^\prime$. Stronger attraction drives the system into a Luther-Emery liquid phase. Nevertheless, with the attraction further increasing, the system enters an electron-hole phase separation and the superconducting correlation is destroyed. Finally, we investigate the effects of on-site Coulomb interaction on superconductivity.

The nature of phase transitions involving the questions why and how phase transitions take place has not been sufficiently touched in the literature. In contrast, the current attention to certain extent still focus on the description of critical phenomena and the classification of the associated phase transition along with the Ginzburg-Landau-Wilson paradigm, where the key issue is to identify phenomenologically order parameters and related symmetries. This brings the question to topological phase transitions (TPTs), where no obvious order parameter and the broken symmetry are identified. Here we present a unified pattern description of the second-order quantum phase transition (QPT) and TPT, both involved in the one-dimensional anisotropic quantum XY model in a transverse field, which contains the transverse Ising model (TIM) as a limit case. Away from the TIM, the XY model enters the ferromagnetic phase (marked by a second-order QPT or a direct TPT) as increasing ferromagentic exchange coupling, a series of TPTs occur, which are absent in the TIM. The TPTs behave like the first-order QPTs. In the isotropic and large exchange coupling cases, the ground state of the XY model is dominated by two topologically different vortices along positive and negative direction of the transverse field. We confirm the above conclusion by analyzing the energy contributions of the patterns to the ground state and calculating the ground state pattern occupations of the XY model. The results have been obtained in a unified and self-evident way and answer the questions why and how the QPT and TPTs take place in the XY model., Comment: 5 pages, 5 figures

When non-magnetic impurity immerses in Fermi sea, a regular modulation of charge density around impurity will appear and such phenomena is called Friedel oscillation (FO). Although both Luttinger liquid and Landau Fermi liquid show such characteristic oscillation, FO in generic non-Fermi liquid (NFL) phase is still largely unknown. Here, we show that FO indeed exists in NFL state of an exactly solvable model, i.e. the Hatsugai-Kohmoto model which has been intensively explored in recent years. Combining T-matrix approximation and linear-response-theory, an interesting picture emerges, if two interaction-induced quasi-particles bands in NFL are partially occupied, FO in this situation is determined by a novel structure in momentum space, i.e. the 'average Fermi surface' (average over two quasi-particle Fermi surface), which highlights the inter-band particle-hole excitation. We hope our study here provides a counterintuitive example in which FO with Fermi surface coexists with NFL quasi-particle, and it may be useful to detect hidden 'average Fermi surface' structure in other correlated electron systems., Comment: 11 pages, 13 figures. Typos are fixed and more discussions are added

The description and understanding of the consequences of competing interactions in various systems, both classical and quantum, are notoriously difficult due to insufficient information involved in conventional concepts, for example, order parameters and/or correlation functions. Here we go beyond these conventional language and present a pattern picture to describe and understand the frustration physics by taking the one-dimensional (1D) axial next-nearest-neighbor Ising (ANNNI) model in a transverse field as an example. The system is dissected by the patterns, obtained by diagnonalizing the model Hamiltonian in an operator space with a finite lattice size $4n$ ($n$: natural number) and periodic boundary condition. With increasing the frustration parameter, the system experiences successively various phases/metastates, identified respectively as those with zero, two, four, $\cdots$, $2n$ domains/kinks, where the first is the ferromagnetic phase and the last the antiphase. Except for the ferromagnetic phase and antiphase, the others should be metastates, whose transitions are crossing over in nature. The results clarify the controversial issues about the phases in the 1D ANNNI model and provide a starting point to study more complicated situations, for example, the frustration systems in high dimensions., Comment: 5 pages, 5 figures

Despite of simplicity of the transverse antiferromagnetic Ising model with a uniform longitudinal field, its phases and involved quntum phase transitions (QPTs) are nontrivial in comparison to its ferromagnetic counterpart. For example, what is the nature of the mixed-order in such a model and does there exist a disorder phase? Here we use a pattern picture to explore the competitions between the antiferromagnetic Ising interaction, the transverse and longitudinal fields and uncover what kind of pattern takes responsibility of these three competing energy scales, thus determine the possible phases and their QPTs or crossovers. Our results not only unveil rich physics of this paradigmatic model, but also further stimulate quantum simulation by using current available experimental platforms., Comment: 6 pages, 5 figures. arXiv admin note: text overlap with arXiv:2301.02066

The investigation of the first-order quantum phase transition (QPT) is far from clarity in comparison to that of the second-order or continuous QPT, in which the order parameter and associated broken symmetry can be clearly identified and at the same time the concepts of universality class and critical scaling can be characterized by critical exponents. Here we present a compared study of these two kinds of QPT in the transverse Ising model. In the absence of a longitudinal field, the ground state of the model exhibits a second-order QPT from paramagnetic phase to ferromagnetic one, which is smeared out once the longitudinal field is applied. Surprisingly, the first excited state involves a first-order QPT as the longitudinal field increases, which has not been reported in literature. Within the framework of a pattern picture we clearly identify the difference between these two kinds of QPT: for the continuous QPT only the pattern flavoring ferromagnetic phase is always dominant over the others, and on the contrary, there exist at least two competitive patterns in the first-order QPT, which is further indicated by patterns' occupancies calculated by pattern projections on the ground and first excited states wavefunctions. Our result has not only a fundamental significance in the understandings of the nature of QPTs, but also a practical interest in quantum simulations used to test the present finding., Comment: 5 pages, 5 figures

Irrespective of the fact that a complete theoretical description of critical phenomena in connection with phase transition has been well-established through the renormalization group formalism, the understanding of the phase transition itself remains incomplete. For example, the questions like why and how the phase transition happens are still unclear. Here we provide a pattern picture to dissect the quantum phase transition occurring in the transverse Ising model for a finite lattice. After the validity of the pattern formulation obtained is confirmed, the energy contributions of different patterns to the ground state energy provide a sufficient detail to show why and how the phase transition takes place. Furthermore, a histogram of patterns' occupancy calculated by the projections of ground state wavefunction on the patterns also shows the detailed process of the phase transition. Our results are not only fundamental in understanding the mechanism of phase transition, but also of practical interest in quantum simulation platforms., Comment: 6 pages, 4 figures

The phase transition is both thermodynamically and quantum-mechanically ubiquitous in nature or laboratory and its understanding is still one of most active issues in modern physics and related disciplines. The Landau's theory provides a general framework to describe \textit{phenomenologically} the phase transition by the introduction of order parameters and the associated symmetry breakings; and is also taken as starting point to explore the critical phenomena in connection with phase transitions in renormalization group, which provides a complete theoretical description of the behavior close to the critical points. In this sense the microscopic mechanism of the phase transition remains still to be uncovered. Here we make a first attempt to explore the microscopic mechanism of the superradiant phase transition in the quantum Rabi model (QRM). We firstly perform a diagonalization in an operator space to obtain three fundamental patterns involved in the QRM and then analyze explicitly their energy evolutions with increasing coupling strengths. The characteristic behaviors found uncover the microscipic mechanism of the superradiant phase transition: one is active to drive the happening of phase transition, the second responses rapidly to the change of the active pattern and wakes up the third pattern to stablize the new phase. This kind of dissecting mechanism explains for the first time why and how happens the superradiant phase transition in the QRM and paves a way to explore the microscopic mechanism of the phase transitions happening popularly in nature., Comment: 6 pages, 5 figures

The ground state of quantum Rabi model (QRM) exhibits rich nonclassical states including squeezed state, cat state, and entangled state in different parameter regimes. In this paper, we firstly use the polaron picture to figure out the process of superradiant phase transition (SPT) in view of the nonclassical states. According to the polaron wave functions, the squeezed state extends to whole parameter regimes, and a small but non-zero weighted antipolaron is necessary to form novel semi-cat states. Moreover, the behavior of superradiance in the QRM can be viewed as the same displacement of the cat states from zero to a finite value, while the ground state becomes entangled state resulting from the emergence of spin-up state. On the other hand, the nonclassical states can be intuitively characterized by the Wigner functions with analytical expressions in the polaron picture, and the Wigner negativity is also evaluated to measure the nonclassicality. Based on the squeezing and superradiance, a classification of the coupling strength for the nonclassical states containing in the ground state is presented, and the process of the SPT is also revealed by the photon number distribution in Fock space, which is important to understand the fundamental quantum physics in the QRM. Our work provides a guidance for preparing the nonclassical states in experiments based on the QRM.

By using the tensor-network state algorithm, we study a spin-orbital model with SU(2)$\times$SU(2)$\times$U(1) symmetry on the triangular lattice. This model was proposed to describe some triangular $d^1$ materials and was argued to host a spin-orbital liquid ground state. In our work the trial wavefunction of its ground state is approximated by an infinite projected entangled simplex state and optimized by the imaginary-time evolution. Contrary to the previous conjecture, we find that the two SU(2) symmetries are broken, resulting in a stripe spin-orbital order with the same magnitude $m=0.085(10)$. This value is about half of that in the spin-1/2 triangular Heisenberg antiferromagnet. Our result demonstrates that although the long-sought spin-orbital liquid is absent in this model the spin-orbital order is significantly reduced due to the enhanced quantum fluctuation. This suggests that high-symmetry spin-orbital models are promising in searching for exotic states of matter in condensed-matter physics., Comment: 4 pages, 5 figures

Parity describing the symmetry of quantum mechanics wavefunction under space inversion transformation not only plays an essential role in solving quantum systems but also can be used to manipulate and measure the motional quantum states of such hybrid quantum systems as quantum Rabi model(QRM) and/or its variants through parity measurements. Here we address an exotic parity behavior of the QRM in its superradiant phase by numerical exact diagonalization, namely, the parities of eigenstates of the QRM behave irregular in the strong coupling regime but the sum of parities for each pair of eigenstates beginning from the ground state remains vanishing. It is found that this exotic behavior originates from the comparability of the photon distribution in the odd and even components of Fock basis when the eigenenergies of each pair of eigenstates approach enough to each other and physically is due to the emergent double-well potential induced by the strong coupling between the single-mode photon field and the two-level atom. The result not only uncovers the physics not known previously in the QRM but also makes an intrinsic limitation on the measurement precision of motional quantum states through parity measurements in modern quantum science and technologies., Comment: 7 pages, 4 figures

Recently, a superradiant phase transition first predicted theoretically in the quantum Rabi model (QRM) has been verified experimentally. This further stimulates the interest in the study of the process of phase transition and the nature of the superradiant phase since the fundamental role of the QRM in describing the interaction of light and matter, and more importantly, the QRM contains rich physics deserving further exploration despite its simplicity. Here we propose a scheme consisting of two successive diagonalization to accurately obtain the ground-state and excited states wavefunctions of the QRM in full parameter regime ranging from weak to deep-strong couplings. Thus one is able to see how the phase transition happens and how the photons populate in Fock space of the superradiant phase. We characterize the photon populations by borrowing the distribution concept in random matrix theory and find that the photon population follows a Poissonian-like distribution once the phase transition happens and further exhibits the statistics of Gaussian unitary ensemble as increasing coupling strength. More interestingly, the photons in the excited states behave even like the statistics of Gaussian orthogonal ensemble. Our results not only deepen understanding on the superradiant phase transition but also provide an insight on the nature of the superradiant phase of the QRM and related models., Comment: 7 pages, 4 figures plus SM