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The one-dimensional spin-$s$ ${\rm SU}(2)$ ferromagnetic Heisenberg model, as a paradigmatic example for spontaneous symmetry breaking (SSB) with type-B Goldstone modes (GMs), is expected to exhibit an abstract fractal underlying the ground state subspace. This intrinsic abstract fractal is here revealed from a systematic investigation into the entanglement entropy for a linear combination of factorized (unentangled) ground states on a fractal decomposable into a set of the Cantor sets. The entanglement entropy scales logarithmically with the block size, with the prefactor being half the fractal dimension of a fractal, as long as the norm for the linear combination scales as the square root of the number of the self-similar building blocks kept at each step $k$ for a fractal, under an assumption that the maximum absolute value of the coefficients in the linear combination is chosen to be around one, and the coefficients in the linear combination are almost constants within the building blocks. Actually, the set of the fractal dimensions for all the Cantor sets forms a {\it dense} subset in the interval $[0,1]$. As a consequence, the ground state subspace is separated into a disjoint union of countably infinitely many regions, each of which is labeled by a decomposable fractal. Hence, the interpretation of the prefactor as half the fractal dimension is valid for any support beyond a fractal, which in turn leads to the identification of the fractal dimension with the number of type-B GMs for the orthonormal basis states. Our argument may be extended to any quantum many-body systems undergoing SSB with type-B GMs., Comment: 23 pages, 8 figures

G352.7$-$0.1 is a mixed-morphology (MM) supernova remnant (SNR) with multiple radio arcs and has a disputed supernova origin. We conducted a spatially resolved spectroscopic study of the remnant with XMM-Newton X-ray data to investigate its explosion mechanism and explain its morphology. The global X-ray spectra of the SNR can be adequately reproduced using a metal-rich thermal plasma model with a temperature of $\sim 2$ keV and ionization timescale of $\sim 3\times 10^{10}~{\rm cm^{-3}~s}$. Through a comparison with various supernova nucleosynthesis models, we found that observed metal properties from Mg to Fe can be better described using core-collapse supernova models, while thermonuclear models fail to explain the observed high Mg/Si ratio. The best-fit supernova model suggests a $\sim 13$ $M_\odot$ progenitor star, consistent with previous estimates using the wind bubble size. We also discussed the possible mechanisms that may lead to SNR G352.7$-$0.1 being an MMSNR. By dividing the SNR into several regions, we found that the temperature and abundance do not significantly vary with regions, except for a decreased temperature and abundance in a region interacting with molecular clouds. The brightest X-ray emission of the SNR spatially matches with the inner radio structure, suggesting that the centrally filled X-ray morphology results from a projection effect., Comment: 12 pages, 19 figures; Accepted for publication in MNRAS

Exact matrix product state representations for a type of scale-invariant states are presented, which describe highly degenerate ground states arising from spontaneous symmetry breaking with type-B Goldstone modes in one-dimensional quantum many-body systems. As a possible application, such a representation offers a convenient but powerful means for evaluating the norms of highly degenerate ground states. This in turn allows us to perform a universal finite system-size scaling analysis of the entanglement entropy. Moreover, this approach vividly explains why the entanglement entropy does not depend on what types of the boundary conditions are adopted, either periodic boundary conditions or open boundary conditions. Illustrative examples include the ${\rm SU}(2)$ spin-$s$ Heisenberg ferromagnetic model, the ${\rm SU}(2s+1)$ ferromagnetic model, and the staggered ${\rm SU}(3)$ spin-1 ferromagnetic biquadratic model., Comment: 12 pages, 5 figures, 3 tables

Spontaneous symmetry breaking (SSB) with type-B Goldstone modes is investigated in the macroscopically degenerate phase for a quantum spin-1 many-body system with competing dimer and trimer interactions. The SSB involves three distinct patterns. The first occurs at the dimer point, with the pattern from staggered ${\rm SU}(3)$ to ${\rm U}(1)\times{\rm U}(1)$. The second occurs at the trimer point, with the pattern from uniform ${\rm SU}(3)$ to ${\rm U}(1)\times{\rm U}(1)$. The third occurs in the dimer-trimer regime, with the pattern from uniform ${\rm SU}(2)$ to ${\rm U}(1)$. The number of type-B Goldstone modes is thus two, two and one for the three patterns, respectively. The ground state degeneracies arising from the three patterns are exponential with the system size, which may be recognized as sequences of integers relevant to self-similar logarithmic spirals. This in turn is attributed to the presence of an emergent symmetry operation tailored to a specific degenerate ground state. As a consequence, the residual entropy is non-zero, which measures the disorder present in a unit cell of highly degenerate ground state generated from a generalized highest weight state. An exact Schmidt decomposition exists for the highly degenerate ground states, thus exposing the self-similarities underlying an abstract fractal, described by the fractal dimension. The latter is extracted from performing a universal finite system-size scaling analysis of the entanglement entropy, which is identical to the number of type-B Goldstone modes. The model under investigation thus accommodates an exotic scale invariant quantum state of matter., Comment: 14 pages, 3 figures, 2 tables

The ground-state phase diagram is mapped out for an alternative anisotropic extension of quantum spin-1 ferromagnetic biquadratic model, which accommodates twelve distinct phases: three degenerate fractal phases, six Luttinger liquid phases and three symmetry-protected trivial phases. It is found that distinct types of quantum phase transitions are involved between them. In particular, one type arises from an instability of a Luttinger liquid towards a degenerate fractal phase, and the other type describes spontaneous symmetry breaking with type-B Goldstone modes from one degenerate fractal phase to another degenerate fractal phase, with the fractal dimension $d_f$ being identical to the number of the type-B Goldstone modes, both of which turn out to be one. In addition, quantum phase transitions from the Luttinger liquid phases to the symmetry-protected trivial phases are identified to be in the Kosterlitz-Thouless universality class, with central charge being one., Comment: 11 pages, 8 figures, 2 tables

An emergent gapless Goldstone mode originates from continuous spontaneous symmetry breaking, which has become a doctrine since the pioneering work by Goldstone [J. Goldstone, Nuovo Cimento \textbf{19}, 154 (1961)]. However, we argue that it is possible for a continuous symmetry group $\rm{U}(1)$ to make an exceptional case, simply due to the well-known mathematical result that a continuous symmetry group $\rm{U}(1)$ may be regarded as a limit of a discrete symmetry group $Z_q$ when $q$ tends to infinity. As a consequence, spontaneous symmetry breaking for such a continuous symmetry group $\rm{U}(1)$ does not necessarily lead to any gapless Goldstone mode. This is explicitly explained for an anisotropic extension of the ferromagnetic spin-1 biquadratic model. In a sense, this model provides an illustrative example regarding the dichotomy between continuity and discreteness., Comment: 13 pages, 9 figures, 9 tables

Let $G$ be a graph with an adjacent matrix $A(G)$. The multiplicity of an arbitrary eigenvalue $\lambda$ of $A(G)$ is denoted by $m_\lambda(G)$. In \cite{Wong}, the author apply the Pater-Wiener Theorem to prove that if the diameter of $T$ at least $4$, then $m_\lambda(T)\leq \beta'(T)-1$ for any $\lambda\neq0$. Moreover, they characterized all trees with $m_\lambda(T)=\beta'(T)-1$, where $\beta'(G)$ is the induced matching number of $G$. In this paper, we intend to extend this result from trees to any connected graph. Contrary to the technique used in \cite{Wong}, we prove the following result mainly by employing algebraic methods: For any non-zero eigenvalue $\lambda$ of the connected graph $G$, $m_\lambda(G)\leq \beta'(G)+c(G)$, where $c(G)$ is the cyclomatic number of $G$, and the equality holds if and only if $G\cong C_3(a,a,a)$ or $G\cong C_5$, or a tree with the diameter is at most $3$. Furthermore, if $\beta'(G)\geq3$, we characterize all connected graphs with $m_\lambda(G)=\beta'(G)+c(G)-1$.

We present numerical results for three-dimensional (3D) solitons with symmetries of the semi-vortex (SV) and mixed-mode (MM) types, which can be created in spinor Bose-Einstein condensates of Rydberg atoms under the action of the spin-orbit coupling (SOC). By means of systematic numerical computations, we demonstrate that the interplay of SOC and long-range spherically symmetric Rydberg interactions stabilize the 3D solitons, improving their resistance to collapse. We find how the stability range depends on the strengths of the SOC and Rydberg interactions and the soft-core atomic radius., Comment: to be published in Scientific Reports

It is found that the staggered $\rm SU(4)$ ferromagnetic spin-orbital model accommodates highly degenerate ground states arising from spontaneous symmetry breaking with type-B Goldstone modes. The spontaneous symmetry breaking pattern is ${\rm SU(4)} \rightarrow {\rm U(1)} \times {\rm U(1)} \times {\rm U(1)}$, with three type-B Goldstone modes. An abstract fractal underlies the ground state subspace with the fractal dimension identified with the number of type-B Goldstone modes. This connection is established by evaluating the entanglement entropy, which exhibits a universal finite system-size scaling behaviour. The latter in turn implies that the entanglement entropy scales logarithmically with the block size in the thermodynamic limit. In addition, the ground state degeneracies, depending on the boundary conditions adopted, constitute the two Fibonacci-Lucas sequences. In the limit of large system size their asymptotic forms become a self-similar logarithmic spiral. As a result, the model has a non-zero residual entropy $S_{\!r} = -2 \ln R $, where $R=(\! \sqrt{6}-\!\sqrt{2})/2$., Comment: 16 pages, 5 figures, 1 table. arXiv admin note: substantial text overlap with arXiv:2302.13126

A generic scheme is proposed to perform a finite-entanglement scaling analysis for scale-invariant states, which appear to be highly degenerate ground states arising from spontaneous symmetry breaking with type-B Goldstone modes. This allows us to extract the number of type-B Goldstone modes and the dynamical critical exponent, in combination with a finite block-size scaling analysis, from numerical simulations of quantum many-body systems in the context of tensor network representations. The number of type-B Goldstone modes is identical to the fractal dimension, thus reflecting an abstract fractal underlying the ground state subspace. As illustrative examples, we investigate the spin-$s$ Heisenberg ferromagnetic model, the $\rm{SU}(3)$ ferromagnetic model and the $\rm{SO}(4)$ spin-orbital model., Comment: 14 pages, 24 figures, 11 tables

Ordered electronic states have been extensively explored in cuprates and iron-based unconventional superconductors, but seldom observed in the epitaxial FeSe/SrTiO3(001) monolayer (FeSe/STO) with an enhanced superconducting transition temperature (Tc). Here, by using scanning tunneling microscopy/ spectroscopy (STM/STS), we reveal a checkerboard charge order in the epitaxial FeSe/STO monolayer, with a period of four times the inter-Fe-atom distance along two perpendicular directions of the Fe lattice. This ordered state is uniquely present in the superconducting FeSe/STO monolayer, even at liquid nitrogen temperature, but absent in the non-superconducting FeSe monolayer or bilayer. Quasiparticle interference (QPI) measurements further confirm it as a static order without an energy-dependent dispersion and gapped out within the superconductivity gap. The intensity of the charge order shows an enhancement near the superconducting transition temperature, thus implying a correlation with the high-Tc superconductivity in the FeSe/STO monolayer. This study provides a new basis for exploring the ordered electronic states and their interplay with high-Tc superconductivity in the FeSe monolayer., Comment: 22 pages, 4 figures

The long-standing Gaussian product inequality (GPI) conjecture states that, for any centered $\mathbb{R}^n$-valued Gaussian random vector $(X_1, \dots, X_n)$ and any positive reals $\alpha_1, \dots, \alpha_n$, ${\bf E}[\prod_{j=1}^{n}|X_j|^{\alpha_j}]\ge \prod_{j=1}^{n}{\bf E}[|X_j|^{\alpha_j}]$. In this paper, we present some related inequalities for centered $\mathbb{R}^n$-valued Gaussian random vector $(X_1, \dots, X_n)$ when $\{\alpha_1, \dots, \alpha_n\}$ contains both positive and negative numbers., Comment: 15 pages

Let $B(n,p)$ denote a binomial random variable with parameters $n$ and $p$. Chv\'{a}tal's theorem says that for any fixed $n\geq 2$, as $m$ ranges over $\{0,\ldots,n\}$, the probability $q_m:=P(B(n,m/n)\leq m)$ is the smallest when $m$ is closest to $\frac{2n}{3}$. Motivated by this theorem, in this paper we consider the infimum value of the probability $P(X\leq \kappa E[X])$, where $\kappa$ is a positive real number, and $X$ is a random variable whose distribution belongs to some infinitely divisible distributions including the inverse Gaussian, log-normal, Gumbel and logistic distributions., Comment: 10 pages

Galactic supernova remnants (SNRs) play an important role in our understanding of supernovae and their feedback on the interstellar environment. SNR G352.7$-$0.1 is special for its thermal composite morphology and double-ring structure. We have performed spectroscopic mapping in $^{12}$CO and $^{13}$CO $J=2$-1 lines toward G352.7$-$0.1 with the Atacama Pathfinder Experiment telescope. Broad $^{12}$CO lines are found in the northeastern ring at a local-standard-of-rest velocity range of $\sim-50$-$-30$ km s$^{-1}$, suggesting that the remnant is interacting with molecular clouds (MCs) at $\sim-51$ km s$^{-1}$. Thus, we adopt a distance of $\sim10.5$ kpc for this SNR. The momentum and kinetic energy of the shocked gas along the line of sight are estimated to be $\sim10^2{\rm M_{sun}}$ km s$^{-1}$ and $\sim10^{46}$ erg, respectively. We also find an expanding structure around the remnant, which is possibly related to the wind-blown bubble of the progenitor star. From the Fermi-LAT data in an energy range of 0.1-500 GeV, we find no gamma-ray counterparts of G352.7$-$0.1., Comment: 12 pages, 10 figures

With the rapid development of high-power petawatt class lasers worldwide, exploring physics in the strong field QED regime will become one of the frontiers for laser-plasma interactions research. Particle-in-cell codes, including quantum emission processes, are powerful tools for predicting and analyzing future experiments where the physics of relativistic plasma is strongly affected by strong-field QED processes. The spin/polarization dependence of these quantum processes has been of recent interest. In this article, we perform a parametric study of the interaction of two laser pulses with an ultrarelativistic electron beam. The first pulse is optimized to generate high-energy photons by nonlinear Compton scattering and efficiently decelerate the electron beam through quantum radiation reaction. The second pulse is optimized to generate electron-positron pairs by nonlinear Breit-Wheeler decay of the photons with the maximum polarization dependence. This may be experimentally realized as a verification of the strong field QED framework, including the spin/polarization rates., Comment: 15 pages, 13 figures

The weak charge of the proton $Q_{\textrm{w}}$ is one of the most fundamental quantities in physics. It can be determined by measuring the parity asymmetry $A_{\textrm{PV}}$ in elastic $ep$ scattering, where the $\gamma Z$-exchange contributions are crucial. For the past fifteen years, dispersion relations (DRs) in the forward limit have been widely used as a model-independent method to estimate these contributions. In this work, we study corrections to these forward-limit DRs. We first estimate these corrections using pointlike interactions as an illustrative example. We then estimate the $\gamma Z$-exchange contributions for the upcoming P2 experiment through both direct calculation and the forward-limit DRs, within the framework of low-energy effective interactions. The results indicate that the correction to the forward-limit DR for $\Box_{\gamma Z}^{V}$ is around 47\% for the upcoming P2 experiment, which will significantly modify the extracted value of $Q_{\textrm{w}}$., Comment: 4 figures

For a graph $G$, let $\mathcal{S}(G)$ be the set consisting of Hermitian matrices whose graph is $G$. Denoted by $m_B(G,\lambda)$ the multiplicity of an eigenvalue $\lambda$ of $B(G)\in \mathcal{S}(G)$, we show that $m_B(G,\lambda)\le 2\theta(G)+p(G)$ where $\theta(G)$ and $p(G)$ are the cyclomatic number and the number of pendent vertices of $G$ respectively, and characterize the graphs attaining the equality. This is a generalization of a result on adjacency matrix by Wang et al.\cite{Wang1}. Moreover, they arose an open problem in \cite{Wang1}: \textit{characterize all graphs with $m_A(G,\lambda)=2\theta(G)+p(G)-1$ for any eigenvalue $\lambda$ of its adjacency matrix.} In this paper, we completely characterize the graphs with $m_B(G,\lambda)=2\theta(G)+p(G)-1$ for any eigenvalue $\lambda$ of an arbitrary Hermitian matrix $B(G)\in \mathcal{S}(G)$. This result provides a stronger answer to the above problem, and encompasses some previous known works considering $\lambda=-1$ or $0$ on the problem.

Let $B(n,p)$ denote a binomial random variable with parameters $n$ and $p$. Chv\'{a}tal's theorem says that for any fixed $n\geq 2$, as $m$ ranges over $\{0,\ldots,n\}$, the probability $q_m:=P(B(n,m/n)\leq m)$ is the smallest when $m$ is closest to $\frac{2n}{3}$. Motivated by this theorem, we consider the minimum value problem on the probability that a random variable is at most its expectation, when its distribution is the Weibull distribution or the Pareto distribution in this note., Comment: 8 pages