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We generalize the notions of the St\"ackel transform and the coupling constant metamorphosis to quasi-exactly solvable systems. We discover that for a variety of one-dimensional and separable multidimensional quasi-exactly solvable systems, their $sl(2)$ algebraizations can only be achieved via coupling constant metamorphosis after appropriate St\"ackel transformations. This discovery has interesting applications, allowing us to derive algebraizations and energies for a wide class of quasi-exactly solvable systems, such as Hooke's atoms in magnetic fields and Newtonian cosmology. The approach of coupling constant metamorphosis was successfully applied previously in the context of exactly solvable, integrable and superintegrable systems. To our knowledge, the present work is the first to apply the idea and approach in the context of quasi-exactly solvable systems.

We study the quantum integrable spin chain model associated with the twisted $D_2^{(2)}$ algebra (or simply the $D_2^{(2)}$ model) under generic open boundary conditions. The Hamiltonian of this model can be factorized into the sum of two staggered XXZ spin chains. Applying the $t$-$W$ method, we derive the homogeneous Bethe ansatz equations for the zeros of the transfer matrix eigenvalues and the patterns of the corresponding zeros of the staggered XXZ spin chain with generic integrable boundaries. Based on these results, we analytically compute the surface energies and excitation energies of the $D_2^{(2)}$ model in different regimes of boundary parameters.

The medium-temperature $T$ dependence of the jet transport coefficient $\hat q$ was studied via the nuclear modification factor $R_{AA}(p_{\rm T})$ and elliptical flow parameter $v_2(p_{\rm T})$ for large transverse momentum $p_{\rm T}$ hadrons in high-energy nucleus-nucleus collisions. Within a next-to-leading-order perturbative QCD parton model for hard scatterings with modified fragmentation functions due to jet quenching controlled by $\hat q$, we check the suppression and azimuthal anisotropy for large $p_{\rm T}$ hadrons, and extract $\hat q$ by global fits to $R_{AA}(p_{\rm T})$ and $v_2(p_{\rm T})$ data in A + A collisions at RHIC and LHC, respectively. The numerical results from the best fits show that $\hat q/T^3$ goes down with local medium temperature $T$ in the parton jet trajectory. Compared with the case of a constant $\hat{q}/T^3$, the going-down $T$ dependence of $\hat{q}/T^3$ makes a hard parton jet to lose more energy near $T_{\rm c}$ and therefore strengthens the azimuthal anisotropy for large $p_{\rm T}$ hadrons. As a result, $v_2(p_{\rm T})$ for large $p_{\rm T}$ hadrons was enhanced by approximately 10% to better fit the data at RHIC/LHC. Considering the first-order phase transition from QGP to the hadron phase and the additional energy loss in the hadron phase, $v_2(p_{\rm T})$ is again enhanced by 5%-10% at RHIC/LHC.

The Izergin-Korepin model is an integrable model with the simplest twisted quantum affine algebra $U_q(A_2^{(2)})$ symmetry. Applying the $t-W$ method, we derive the homogeneous zero roots Bethe ansatz equations and the corresponding zero root patterns of the Izergin-Korepin model with generic integrable boundaries. Based on these results, we analytically compute the surface energies and boundary excitations in different regimes of boundary parameters of the model. It is shown that in some regimes, correlation effect appears between two boundary fields.

We provide a classification and explicit formulas for the elements that span the centralizer of Cartan subalgebras of complex semisimple Lie algebras of non-exceptional type in their universal enveloping algebra, and show that these generate polynomial (Poisson) algebras. The precise structure for the case of rank three simple Lie algebras is provided, and the inclusion relations between the corresponding polynomial algebras illustrated. We develop the ideas of constructing algebraic superintegrable systems and their integrals from the generators of the centralizer subalgebras and their symmetry algebra. In particular, we present explicitly one example the superintegrable system corresponding to the Cartan centralizer of $\mathfrak{so}(6,\mathbb{C})$.

Twist between neighboring layers and variation of interlayer distance are two extra ways to control the physical properties of stacked two-dimensional van der Waals materials without alteration of chemical compositions or application of external fields, compared to their monolayer counterparts. In this work, we explored the dependence of the magnetic states of the untwisted and twisted bilayer 1T-VX$_2$ (X = S, Se) on the interlayer distance by density functional theory calculations. We find that, while a magnetic phase transition occurs from interlayer ferromagnetism to interlayer antiferromagnetism either as a function of decreasing interlayer distance for the untwisted bilayer 1T-VX$_2$ or after twist, richer magnetic phase transitions consecutively take place for the twisted bilayer 1T-VX$_2$ as interlayer distance is gradually reduced. Besides, the critical pressures for the phase transition are greatly reduced in twisted bilayer 1T-VX$_2$ compared with the untwisted case. We derived the Heisenberg model with intralayer and interlayer exchange couplings to comprehend the emergence of various magnetic states. Our results point out an easy access towards tunable two-dimensional magnets., Comment: 8 pages, 4 figures

The paramagnetic phase diagrams of the half-filled Hubbard model on a twisted bilayer square lattice are investigated using coherent potential approximation. Besides the conventional metallic, band insulating, and Mott insulating phases, we find two site-selective insulating phases where certain sites exhibit band insulating behaviors while the others display Mott insulating behaviors. These phases are identified by the band gap, the double occupancy, the density of states, as well as the imaginary part of self-energy. Furthermore, we examine the effect of on-site potential on the stability of the site-selective insulating phases. Our results indicate that fruitful site-selective phases can be engineered by twisting., Comment: 12 pages, 6 figures

We propose that a weakly spontaneous charge-ordered insulating state probably exists in Bernal-stacked bilayer graphene which can account for experimentally observed non-monotonic behavior of resistance as a function of the gated field, namely, the gap closes and reopens at a critical gated field. The underlying physics is demonstrated by a simple model on a corresponding lattice that contains the nearest intralayer and interlayer hoppings, electric field, and staggered potential between different sublattices. Combining density functional theory calculations with model analyses, we argue that the interlayer van der Waals interactions cooperating with ripples may be responsible for the staggered potential which induces a charge-ordered insulating state in the absence of the electric field., Comment: 10 pages, 7 figures

At large commensurate angles, twisted bilayer graphene which holds even parity under sublattice exchange exhibits a tiny gap. Here, we point out a way to tune this tiny gap into a large gap. We start from comprehensive understanding of the physical origin of gap opening by density functional theory calculations. We reveal that the effective inter-layer hopping, intra-layer CDW, or inter-layer charge imbalance favors a gap. Then, on the basis of tight-binding calculations, we suggest that a periodic transverse inhomogeneous pressure, which can tune inter-layer hoppings in specific regions of the moi$\rm\acute{r}$e supercell, may open a gap of over $100$~meV, which is further confirmed by first-principles calculations. Our results provide a theoretical guidance for experiments to open a large gap in twisted bilayer graphene., Comment: 9 pages, 9 figures

An improved particle swarm optimization algorithm is proposed and its superiority over standard particle swarm optimization algorithm is tested on two typical benchmark functions. By employing this algorithm to search for the magnetic ground states of the Hubbard model on the real-space square lattice with finite size based on the mean-field approximation, two new magnetic states, namely the double striped-type antiferromagnetic state and the triple antiferromagnetic state, are found. We further perform mean-field calculations in the thermodynamical limit to confirm that these two new magnetic states are not a result of a finite-size effect, where the properties of the double striped-type antiferromagnetic state are also presented., Comment: 8 pages, 7 figures

We study the thermodynamics of the antiperiodic XXZ chain with anisotropy parameter {\eta}=i{\pi}/3 by means of the t-W method. We parameterize the eigenvalues of both the transfer matrix and the corresponding fused transfer matrix by their zero points instead of Bethe roots. Based on the patterns of the zero points distribution and the reconstructed entropy, we obtain the nonlinear integral equations (NLIEs) describing the thermodynamics of the model and compute its free energy at a finite temperature.

We study Hawking radiation and wave scattering of charged scalar fields in a charged $C$-metric black hole background. The conformally invariant wave equation for charged scalar fields can be separated into radial and angular parts, each with five singularities. We first show that the radial and angular equations can be respectively transformed into the general Heun equation, and then we explore exact solutions of the radial Heun equation in terms of the local Heun functions and connection coefficients. Exact behaviors of the asymptotic wave functions are determined without approximations. We further apply the exact results to derive Hawking radiation, quasinormal modes and superradiance., Comment: V3: 28 pages; eliminated material regarding numerical analysis and implemented other minor corrections; version published in PRD

We investigate the thermodynamic limit and exact surface energy of the isotropic spin-1 Heisenberg chain with integrable generic open boundary conditions by a novel Bethe ansatz method. We obtain the homogeneous Bethe ansatz equations for the zero roots of the transfer matrix. Based on the patterns of the zero roots, we analytical calculate the densities of zero roots and the surface energies of the model in all regimes of the boundary parameters.

We carry out a detailed study of medium modifications on momentum and angular correlations between a large transverse momentum hadron and a $Z/\gamma$ trigger in relativistic heavy-ion collisions within a perturbative QCD parton model improved by the Sudakov resummation technique. The total energy loss of a hard parton propagating inside the medium is employed to modify the fragmentation function, while the medium-induced transverse momentum broadening is included in the resummation approach, and both of them are related to the jet transport parameter and obtained by the high-twist formalism. We obtain good agreements with the existing data on transverse momentum and azimuthal angular correlations for the $Z/\gamma$-hadron pairs in $pp$ and $AA$ collisions, and predict the correlations for the $\gamma$-hadron in central $PbPb$ collisions at 5.02 TeV. The numerical analyses for the $Z/\gamma$-hadron in central $PbPb$ collisions show that the normalized angular distribution is decorrelated due to the medium-induced transverse momentum broadening, however, the angular correlation is enhanced due to the parton energy loss, namely anti-broadening. The observed modification of the angular correlation is a result of the competition between the broadening and the anti-broadening. This work provides a reliable theoretical tool for a comprehensive and precise study of jet quenching in relativistic heavy-ion collisions., Comment: 19 pages, 20 figures

The construction of analytic solutions for quasi-exactly solvable systems is an interesting problem. We revisit a class of models for which the odd solutions were largely missed previously in the literature: the anharmonic oscillator, the singular anharmonic oscillator, the generalized quantum isotonic oscillator, non-polynomially deformed oscillator, and the Schr\"odinger system from the kink stability analysis of $\phi^6$-type field theory. We present a systematic and unified treatment for the odd and even sectors of these models. We find generic closed-form expressions for constraints to the allowed model parameters for quasi-exact solvability, the corresponding energies and wavefunctions. We also make progress in the analysis of solutions to the Bethe ansatz equations in the spaces of model parameters and provide insight into the curves/surfaces of the allowed parameters in the parameter spaces. Most previous analyses in this aspect were on a case-by-case basis and restricted to the first excited states. We present analysis of the solutions (i.e. roots) of the Bethe ansatz equations for higher excited states (up to levels $n$=30 or 50). The shapes of the root distributions change drastically across different regions of model parameters, illustrating phenomena analogous to phase transition in context of integrable models. Furthermore, we also obtain the $sl(2)$ algebraization for the class of models in their respective even and odd sectors in a unified way.

The construction of superintegrable systems based on Lie algebras and their universal enveloping algebras has been widely studied over the past decades. However, most constructions rely on explicit differential operator realisations and Marsden-Weinstein reductions. In this paper, we develop an algebraic approach based on the subalgebras of the 2D conformal algebra $\mathfrak{c}(2)$. This allows us to classify the centralisers of the enveloping algebra of the conformal algebra and construct the corresponding Hamiltonians with integrals in algebraic form. It is found that the symmetry algebras underlying these algebraic Hamiltonians are six-dimensional quadratic algebras. The Berezin brackets and commutation relations of the quadratic algebraic structures are closed without relying on explicit realisations or representations. We also give the Casimir invariants of the symmetry algebras. Our approach provides algebraic perspectives for the recent work by Fordy and Huang on the construction of superintegrable systems in the Darboux spaces.

We review three different approaches to polynomial symmetry algebras underlying superintegrable systems in Darboux spaces. The first method consists of using deformed oscillator algebra to obtain finite-dimensional representations of quadratic algebras. This allow one to gain information on the spectrum of the superintegrable systems. The second method has similarities with the induced module construction approach in the context of Lie algebras and can be used to construct infinite dimensional representations of the symmetry algebras. Explicit construction of these representations is a non-trivial task due to the non-linearity of the polynomial algebras. This method allows the construction of states of the superintegrable systems beyond the reach of separation of variables. As a result, we are able to construct a large number of states in terms of Airy, Bessel and Whittaker functions which would be difficult to obtain in other ways. We also discuss the third approach which is based on the notion of commutants of subalgebras in the enveloping algebra of a Poisson algebra or a Lie algebra. This allows us to discover new superintegrable models in the Darboux spaces and to construct their integrals and symmetry algebras via polynomials in the enveloping algebras.

We review some aspects of the Racah algebra $R(n)$, including the closure relations, pointing out their role in superintegrability, as well as in the description of the symmetry algebra for several models with coalgebra symmetry. The connection includes the generic model on the $(n-1)$ sphere. We discuss an algebraic scheme of constructing Hamiltonians, integrals of the motion and symmetry algebras. This scheme reduces to the Racah algebra $R(n)$ and the model on the $(n-1)$ sphere only for the case of specific differential operator realizations. We review the method, which allows us to obtain the commutant defined in the enveloping algebra of $\mathfrak{sl}(n)$ in the classical setting. The related $\mathcal{A}_3$ polynomial algebra is presented for the case $\mathfrak{sl}(3)$. An explicit construction of the quantization of the scheme for $\mathcal{A}_3$ by symmetrization of the polynomial and the replacement of the Berezin bracket by commutator and symmetrization of the polynomial relations is presented. We obtain the additional quantum terms. These explicit relations are of interest not only for superintegrability, but also for other applications in mathematical physics.