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Abstract We study the thermodynamic limit of the anisotropic XYZ spin chain with non-diagonal integrable open boundary conditions. Although the U(1)-symmetry is broken, by using the new parametrization scheme, we exactly obtain the surface energy and the excitation energy of the system, which has solved the difficulty in the inhomogeneous T − Q relation. With the boundary parameters in the regions making the Hamiltonian Hermitian, we have obtained the distribution patterns of the zeros of the eigenvalue of the transfer matrix for the ground state and the excited ones. We find that the surface and excitation energies depend on the parities of sites number N, due to the long-range Neel order in the bulk. The easy-axis and thermodynamic limit for all the regions of boundary parameters are studied. We also obtain the physical quantities in the thermodynamic limit of boundary XXZ model by taking the trigonometric limit.

AIM: To investigate the clinical features of the ocular surface in patients with different degrees of myopia. METHODS: A cross-sectional study was conducted involving 122 participants with myopia in Beijing Tongren Hospital from February to June, 2023. After completing the Ocular Surface Disease Index (OSDI) score scale, measurements were taken for refraction, biometric parameters and ocular surface parameters. The prevalence, severity and related parameters of the dry eye among different groups based on axial length (AL) were compared. Correlation analysis was performed between ocular surface parameters and refraction/biometric measurement parameters. RESULTS: Statistically significant differences were observed in refractive error, corneal thickness, anterior chamber depth, and subfoveal choroidal thickness among the groups (all P

An exactly solvable strongly correlated electron model with two independent parameters is constructed in the frame of the quantum inverse scattering method, which can be seen as a generalization of the Bariev model. Through the Bethe ansatz method, a set of Bethe ansatz equations is derived. In the thermodynamic limit, to study the ground state of the model, we obtain the integral equations for the density of Bethe roots. Numerical validation is done to confirm the accuracy of our analytic results.

Abstract Background This study aims to investigate the morphologic features of the crystalline lens in Primary Angle Closure Disease (PACD) patients with zonular instability during cataract surgery using the swept-source CASIA 2 Anterior Segment-Optical Coherence Tomography (AS-OCT) system. Methods A total of 398 eyes (125 PACD eyes with zonular instability, 133 PACD eyes with zonular stability, and 140 cataract patient controls) of 398 patients who underwent cataract surgery combined or not glaucoma surgery between January 2021 and January 2023 were enrolled. The crystalline lens parameters were measured by CASIA2 AS-OCT. Then, logistic regression was performed to evaluate the risk factors associated with zonular instability. Results The results revealed that PACD eyes had a more anterior lens equator position, a steeper anterior curvature of lens, shorter Axial Length (AL), shallower Anterior Chamber Distance (ACD), higher Lens Vault (LV) and thicker Lens Thickness (LT), when compared to eyes in the cataract control group. Furthermore, PACD eyes in the zonular instability group had steeper front R, front Rs and Front Rf, flatter back Rf, thicker lens anterior part thickness, higher lens anterior-to-posterior part thickness ratios, shallower ACD, and greater LV, when compared to PACD eyes with zonular stability. The logistic regression analysis, which was adjusted for age and gender, revealed that zonular instability was positively correlated with anterior part thickness, lens anterior-to-posterior part thickness ratio, and LV, but was negatively correlated with lens anterior radius and ACD. Conclusion Steeper anterior curvature, increased lens anterior part thickness, higher anterior-to-posterior part thickness ratio, shallower ACD, and greater LV are the anatomic features of PACD eyes associated with zonular instability.

In this paper, we studied the exact solution of the C2(1) invariant quantum spin chain with off-diagonal open boundary condition. We obtain a solution of the reflection equation where the all matrix element of reflection matrix are nonzeros. By using the technique of fusion, we construct the fused transfer matrix and find the closed recursive relations among the transfer matrices. Based on the algebraic analysis, we obtain the eigenvalue of the system and express it as the inhomogeneous T−Q relation.

Maxwell equal area law is an important and traditional analytical tool in thermodynamic phase transition, especially in the calculation of gas-liquid phase transition, which reconciles the theoretical calculation with the experimental results. Undoubtedly, its importance is also self-evident for the black hole thermodynamic system. In this study, we construct a generalized Maxwell equal area law, which allows different states of thermodynamic systems to be within the generalized free energy. The black hole thermodynamic characteristics are spontaneously emerged in the free energy landscape. Furthermore, by analytic continuation, we utilize the properties of analytical functions to investigate some universal characteristics of thermodynamic phase transitions in black holes, and preliminarily establish the counterpart of thermodynamic phase transitions in the complex domain.

AIM: To analyze the differences, agreements, and correlation among total corneal power parameters generated by different instruments after myopic keratorefractive surgery. METHODS: The prospective cross-sectional study included patients who underwent myopic keratorefractive surgery and received measurements of corneal power 3mo after surgery. Automated keratometer was used for the measurement of simulated keratometry (SimK), swept-source optical coherence tomography (SS-OCT) based biometer for total keratometry (TK), anterior segment-OCT for real keratometry (RK), and Scheimpflug keratometer for the true net power (TNP), the total corneal refractive power (TCRP) and equivalent K-readings (EKR). The differences among these parameters were analyzed, and the agreements and correlation between SimK and other total corneal power parameters were investigated. RESULTS: A total of 70 eyes of 70 patients after myopic keratorefractive surgery were included. The evaluated corneal power parameters were as follows: SimK 38.32±1.93 D, TK 37.54±2.12 D, RK 36.64±2.09 D, TNP 36.56±1.97 D, TCRP 36.70±2.01 D, and EKR 37.55±2.00 D. Pairwise comparison showed that there were significant differences (P

In this study, the Ruppeiner geometry and fluctuations of a four-dimensional charged Anti-de Sitter (AdS) black hole within the context of extensive thermodynamics have been investigated. By fixing the AdS radius in Vissers' construction, an extensive thermodynamics description of the RN-AdS black hole is established, with the central charge playing the role of the particle number, introducing the concept of a subsystem into black hole thermodynamics. By employing the density of thermodynamic variables, the phase transition of the black hole is reexamined, and this result holds for black holes with different AdS backgrounds. Furthermore, the authors delve into a detailed analysis of the Ruppeiner geometry using fluctuation theory. Interestingly, an increase in the particle number of the smaller subsystem is found to induce a reduction in the thermodynamic curvature scalar.

Triple-negative breast cancer (TNBC) is the most aggressive subtype of breast cancer, with a high degree of malignancy and poor prognosis. Tumor-associated macrophages (TAMs) have been identified as significant contributors to the growth and metastasis of TNBC through the secretion of various growth factors and chemokines. Salvianolic acid A (SAA) has been shown to have anti-cancer activities. However, the potential activity of SAA on re-polarized TAMs remains unclear. As there is a correlation between the TAMs and TNBC, this study investigates the effect of SAA on TAMs in the TNBC microenvironment. For that purpose, M2 TAM polarization was induced by two kinds of TNBC-conditioned medium (TNBC-TCM) in the absence or presence of SAA. The gene and protein expression of TAM markers were analyzed by qPCR, FCM, IF, ELISA, and Western blot. The protein expression levels of ERK and p-ERK in M2-like TAMs were analyzed by Western blot. The migration and invasion properties of M2-like TAMs were analyzed by Transwell assays. Here, we demonstrated that SAA increased the expression levels of CD86, IL-1β, and iNOS in M2-like TAMs and, conversely, decreased the expression levels of Arg-1 and CD206. Moreover, SAA inhibited the migration and invasion properties of M2-like TAMs effectively and decreased the protein expression of TGF-β1 and p-ERK in a concentration-dependent manner, as well as TGF-β1 gene expression and secretion. Our current findings for the first time demonstrated that SAA inhibits macrophage polarization to M2-like TAMs by inhibiting the ERK pathway and promotes M2-like TAM re-polarization to the M1 TAMs, which may exert its anti-tumor effect by regulating M1/M2 TAM polarization. These findings highlight SAA as a potential regulator of M2 TAMs and the possibility of utilizing SAA to reprogram M2 TAMs offers promising insights for the clinical management of TNBC.

In this paper, we study the exact solution of the q-deformed D3(1) quantum lattice model with non-diagonal open boundary condition. We demonstrate the crossing symmetry of the transfer matrix and obtain the quantum determinant. We construct the independent transfer matrix fusion identities and show that the fusion processes can be closed. Based on the fusion hierarchies and polynomial analysis, we obtain the inhomogeneous T−Q relations, exact energy spectrum and Bethe ansatz equations of the system.

Abstract Exact solution of the quantum integrable D 2 2 $$ {D}_2^{(2)} $$ spin chain with generic integrable boundary fields is constructed. It is found that the transfer matrix of this model can be factorized as the product of those of two open staggered anisotropic XXZ spin chains. Based on this identity, the eigenvalues and Bethe ansatz equations of the D 2 2 $$ {D}_2^{(2)} $$ model are derived via off-diagonal Bethe ansatz.

Abstract We generalize the nested off-diagonal Bethe ansatz method to study the quantum chain associated with the twisted D 3 2 $$ {D}_3^{(2)} $$ algebra (or the D 3 2 $$ {D}_3^{(2)} $$ model) with either periodic or integrable open boundary conditions. We obtain the intrinsic operator product identities among the fused transfer matrices and find a way to close the recursive fusion relations, which makes it possible to determinate eigenvalues of transfer matrices with an arbitrary anisotropic parameter η. Based on them, and the asymptotic behaviors and values at certain points, we construct eigenvalues of transfer matrices in terms of homogeneous T − Q relations for the periodic case and inhomogeneous ones for the open case with some off-diagonal boundary reflections. The associated Bethe ansatz equations are also given. The method and results in this paper can be generalized to the D n + 1 2 $$ {D}_{n+1}^{(2)} $$ model and other high rank integrable models.

Abstract Finding out root patterns of quantum integrable models is an important step to study their physical properties in the thermodynamic limit. Especially for models without U(1) symmetry, their spectra are usually given by inhomogeneous T − Q relations and the Bethe root patterns are still unclear. In this paper with the antiperiodic XXZ spin chain as an example, an analytic method to derive both the Bethe root patterns and the transfer-matrix root patterns in the thermodynamic limit is proposed. Based on them the ground state energy and elementary excitations in the gapped regime are derived. The present method provides an universal procedure to compute physical properties of quantum integrable models in the thermodynamic limit.

The exact solution of the D2(1) quantum spin chain with generic non-diagonal boundary reflections is obtained. It is found that the generating functional of conserved quantities of the system can be factorized as the product of transfer matrices of two anisotropic XXZ spin chains with open boundary conditions. By using the factorization identities and the fusion technique, the eigenvalues and the Bethe ansatz equations of the model are obtained. The eigenvalues are also parameterized by the zero roots of the transfer matrix, and the patterns of root distributions are obtained. Based on them, ground states energy and the surface energies induced by the twisted boundary magnetic fields in the thermodynamic limit are obtained. These results are checked by the numerical calculations. The corresponding isotropic limit is also discussed. The results given in this paper are the foundation to study the exact physical properties of high rank Dn(1) model by using the nested processes.

Non-degenerate rogue waves (RWs) and multiple transitions between RWs and solitons arising from vector three-wave resonant interaction are studied analytically and numerically. In contrast to the conventional degenerate RWs, such non-degenerate RWs consist of two fundamental RWs each with individual Lax spectral parameter. We show distinctive continuous transitions from bright (dark) RWs to four-petal RWs to dark (bright) solitons as the relative frequency between the wave components increases. The underlying mechanism of such processes is the non-monotonic variation of the energy exchanges between different components of the waves. We further reveal the transition dynamics of non-degenerate RWs. Finally, we confirm numerically the robustness of the transition dynamics in the presence of spontaneous modulation instability induced by white noise. Our results provide insights into the RW formation and the multiple transitions in systems with three-wave resonant interaction, and may offer the possibility of experimental observations in multi-component resonant processes.

We study the exact physical quantities of a competing spin chain which contains many interesting and meaningful couplings including the nearest neighbor, next nearest neighbor, chiral three spins, Dzyloshinsky-Moriya interactions and unparallel boundary magnetic fields in the thermodynamic limit. We obtain the density of zero roots, surface energies and elementary excitations in different regimes of model parameters. Due to the competition of various interactions, the surface energy and excited spectrum show many different pictures from those of the Heisenberg spin chain.

Abstract The phase transition and thermodynamic geometry of a 4-dimensional AdS topological charged black hole in de Rham, Gabadadze and Tolley (dRGT) massive gravity have been studied. After introducing a normalized thermodynamic scalar curvature, it is speculated that its value is related to the interaction between the underlying black hole molecules if the black hole molecules exist. We show that there does exist a crucial parameter given in terms of the topology, charge, and massive parameters of the black hole, which characterizes the thermodynamic properties of the black hole. It is found that when the parameter is positive, the singlet large black hole phase does not exist for sufficient low temperature and there is a weak repulsive interaction dominating for the small black hole which is similar to the Reissner–Nordström AdS black hole; when the parameter is negative, an additional phase region describing large black holes also implies a dominant repulsive interaction. These constitute the distinguishable features of dRGT massive topological black hole from those of the Reissner–Nordström AdS black hole as well as the Van der Waals fluid system.

Abstract A new nonlinear integral equation (NLIE) describing the thermodynamics of the Heisenberg spin chain is derived based on the t − W relation of the quantum transfer matrices. The free energy of the system in a magnetic field is thus obtained by solving the NLIE. This method can be generalized to other lattice quantum integrable models. Taking the SU(3)-invariant quantum spin chain as an example, we construct the corre- sponding NLIEs and compute the free energy. The present results coincide exactly with those obtained via other methods previously.

In this paper, we study the surface energy of the one-dimensional supersymmetric t−Jmodel with nonparallel boundary magnetic fields, which is a typical U(1)-symmetry broken quantum integrable strongly correlated electron system. It is shown that at the ground state, the contribution of inhomogeneous term in the Bethe ansatz solution of eigenvalues of transfer matrix satisfies the finite size scaling law Lβwhere β

In this paper, we investigate the effect of self-awareness (interlayer interaction) for information-epidemic dynamics with simplicial complexes both near and away from the epidemic threshold. It is shown that, contrary to previous views, self-awareness plays a key role near the epidemic threshold. In small homogeneous networks, multiple susceptibility peaks can emerge in the susceptibility of the epidemic layer under the combined effect of simplicial complexes and self-awareness, even two types of multiple susceptibility peaks with completely opposite mechanisms. This means that one needs to be very careful when obtaining epidemic thresholds based on susceptibility. Moreover, the self-awareness can regulate the presence or absence of bistable phenomena both in epidemic prevalence and epidemic threshold. We also found that the time series of disease may be nonmonotonic, with a peak, and that self-awareness is one of the factors controlling the relative height between maximum and steady state. In addition, we modify heterogeneous mean-field theory and partial effective degree theory to accommodate the introduction of simplicial complexes in dynamics. We believe that our study has implications for other dynamics concerning higher-order and interlayer interactions.

Abstract Based on its off-diagonal Bethe ansatz solution, we study the thermodynamic limit of the spin- 1 2 $$ \frac{1}{2} $$ XYZ spin chain with the antiperiodic boundary condition. The key point of our method is that there exist some degenerate points of the crossing parameter η m,l , at which the associated inhomogeneous T − Q relation becomes a homogeneous one. This makes extrapolating the formulae deriving from the homogeneous one to an arbitrary η with O(N −2) corrections for a large N possible. The ground state energy and elementary excitations of the system are obtained. By taking the trigonometric limit, we also give the results of antiperiodic XXZ spin chain within the gapless region in the thermodynamic limit, which does not have any degenerate points.

We propose a novel characterization of the exact solution of the one-dimensional Hubbard model with unparallel boundary magnetic fields. The non-diagonal boundary terms break the U(1) symmetry of the system and the number of electrons with fixed spin is not conserved. Thus it is difficult to study the physical quantities in the thermodynamic limit based on the previously obtained inhomogeneous T−Q relation and Bethe ansatz equations. In order to solve this problem, we propose the t−W scheme. The basic idea is that the eigenvalues of the transfer matrix can be expressed by its zero-roots instead of the usual Bethe roots. This method allows us to study thermodynamic properties of the system such as the ground state, surface energy, charge and spin excitations. The t−W scheme is universal and can be applied to the systems either with or without U(1) symmetry.

The connection between the shadow radius and the Ruppeiner geometry of a charged static spherically symmetric black hole is investigated. The normalized curvature scalar is adopted, and its close relation to the Van der Waals-like and Hawking-Page phase transition of the Reissner-Nordström AdS black hole is studied. The results show that the shadow radius is a useful tool to reveal the correct information of the phase structure and the underlying microstructure of the black hole, which opens a new window to investigate the strong gravity system from the observational point of view.

Abstract Due to the asymptotic structure of the black hole solution, there are two different thermodynamic schemes for the charged Banados–Teitelboim–Zanelli (BTZ) black hole. In one scheme, the charged BTZ black hole is super-entropic, while in the other, it is not (the reverse isoperimetric inequality is saturated). In this paper, we investigate the thermodynamic curvature of the charged BTZ black hole in different coordinate spaces. We find that in both schemes, the thermodynamic curvature is always positive, which may be related to the information of repulsive interaction between black hole molecules for the charged BTZ black hole if we accept an empirical relationship between the thermodynamic curvature and interaction of a system. More importantly, we provide a diagnosis for the discrimination of the two schemes from the point of view of the thermodynamics geometry. For the charged BTZ black hole, when the reverse isoperimetric inequality is saturated, the thermodynamic curvature of an extreme black hole tends to be infinity, while when the reverse isoperimetric inequality is violated, the thermodynamic curvature of the extreme black hole goes to a finite value.

We study the thermodynamic limit of the antiperiodic XXZ spin chain with the anisotropic parameter η=πi3. We parameterize eigenvalues of the transfer matrix by their zero points instead of Bethe roots. We obtain patterns of the distribution of zero points. Based on them, we calculate the ground state energy and the elementary excitations in the thermodynamic limit. We also obtain the two-body scattering matrix of elementary excitations. Two types of elementary excitations and three types of scattering processes are discussed in detailed.

Abstract The exact solutions of the D 3 1 $$ {D}_3^{(1)} $$ model (or the so(6) quantum spin chain) with either periodic or general integrable open boundary conditions are obtained by using the off-diagonal Bethe Ansatz. From the fusion, the complete operator product identities are obtained, which are sufficient to enable us to determine spectrum of the system. Eigenvalues of the fused transfer matrices are constructed by the T - Q relations for the periodic case and by the inhomogeneous T- Q one for the non-diagonal boundary reflection case. The present method can be generalized to deal with the D n 1 $$ {D}_n^{(1)} $$ model directly.

Commonly, the classical formulas for designing TE21-mode coupler are suitable for the operating frequency under Ka-band by virtue of ignoring wall thickness. However, in real application of TE21-mode coupler design beyond Ka-band, the new formulas need to consider the effect of wall thickness. Herein, we demonstrate a design measure of a TE21-mode coupler beyond Ka-band via computer simulation, giving attention to wall thickness. The experimental results show that a measure of linear second weighted sum for coupling hole size could improve coupler performance. Amazingly, such coupler shows 10% bandwidth with less than 0.5 ​dB coupling loss and more than 40 ​dB isolation is possible.

We propose a novel characterization of the exact solution of the one-dimensional supersymmetric t−Jmodel with generic non-diagonal boundary conditions. Instead of using Bethe roots, we parameterize the eigenvalues of transfer matrix in the spin sector with their zero-roots. By constructing the t−Wrelation, we obtain the explicit forms of the eigenvalues of the system and the homogeneous Bethe ansatz equations. This scheme is universal and can be applied to other quantum integrable systems with or without U(1)symmetry.

We investigate the thermodynamic curvature in the Hawking-Page phase transition and the second-order phase transition of the AdS black hole. It is shown that the thermodynamic curvature has the same behavior in these two different phase transitions. Specifically, the thermodynamic curvature in the Hawking-Page phase transition is the power function of the Hawking-Page phase transition temperature with exponent related to the spacetime dimension, or is invariably proportional to the reciprocal of the Hawking-Page phase transition entropy. For the thermodynamic curvature in the second-order phase transition, the conclusion is similar. The more enlightening result is that the ratio of thermodynamic curvatures along the two different phase transitions tends to the natural constant in the limit that the number of spacetime dimensions is infinite. The universal and novel result provides an essential difference between the Hawking-Page phase transition and the second-order phase transition of the AdS black hole in the large dimension paradigm.

The thermodynamic properties of a shell of bosons with the inner surface locating at Planck length away from the horizon of Schwarzschild black holes by using statistical mechanics are studied. The covariant partition function of bosons is obtained, from which the Bose-Einstein condensation of bosons is found at a non-zero temperature in the curved spacetimes. As a special case of bosons, we analyze the entropy of photon gas near the horizon of the Schwarzschild black hole, which shows an area dependence similar to the Bekenstein-Hawking entropy. The results may offer new perspectives on the study of black hole thermodynamics. All these are extended to the D+1 dimensional spherically symmetric static spacetimes.

Abstract The off-diagonal Bethe ansatz method is generalized to the integrable model associated with the sp(4) (or C 2) Lie algebra. By using the fusion technique, we obtain the complete operator product identities among the fused transfer matrices. These relations, together with some asymptotic behaviors and values of the transfer matrices at certain points, enable us to determine the eigenvalues of the transfer matrices completely. For the periodic boundary condition case, we recover the same T − Q relations obtained via conventional Bethe ansatz methods previously, while for the off-diagonal boundary condition case, the eigenvalues are given in terms of inhomogeneous T − Q relations, which could not be obtained by the conventional Bethe ansatz methods. The method developed in this paper can be directly generalized to generic sp(2n) (i.e., C n ) integrable model.

We study the exact solutions of quantum integrable model associated with the Cn Lie algebra, with either a periodic or an open one with off-diagonal boundary reflections, by generalizing the nested off-diagonal Bethe ansatz method. Taking the C3 as an example we demonstrate how the generalized method works. We give the fusion structures of the model and provide a way to close fusion processes. Based on the resulted operator product identities among fused transfer matrices and some necessary additional constraints such as asymptotic behaviors and relations at some special points, we obtain the eigenvalues of transfer matrices and parameterize them as homogeneous T−Q relations in the periodic case or inhomogeneous ones in the open case. We also give the exact solutions of the Cn model with an off-diagonal open boundary condition. The method and results in this paper can be generalized to other high rank integrable models associated with other Lie algebras.

We study the nonlinear dynamics of a three-dimensional Bose-Einstein condensate (BEC) excited by a vortex ring phase imprinting. We identify independent, integrated, and stationary modes of the center-of-mass oscillation of the condensate with respect to the vortex ring movement. We show that the oscillation amplitude of the center-of-mass of the condensate depends strongly on the initial radius of the vortex ring, the nonlinear inter-atomic interaction, and the aspect ratio of the trap, while the oscillation frequency is fixed and equals to the frequency of the harmonic trap in the direction of the ring movement. However, when applying Kelvin wave perturbations on the vortex ring, the center-of-mass oscillation of the BEC is changed non-trivially with respect to the perturbation modes, the long-scale perturbation strength as well as the wave number of the perturbations. The parity of the wave number of the Kelvin perturbations plays an important role on the mode of the center-of-mass oscillation of the condensate.

We show that, in the pure-quartic systems, modulation instability (MI) undergoes heteroclinic-structure transitions (HSTs) at two critical frequencies of ω_{c1} and ω_{c2} (ω_{c2}>ω_{c1}), which indicates that there are significant changes of the spatiotemporal behavior in the system. The complicated heteroclinic structure of instability obtained by the mode truncation method reveals all possible dynamic trajectories of nonlinear waves, which allows us to discover the various types of Fermi-Pasta-Ulam (FPU) recurrences and Akhmediev breathers (ABs). When the modulational frequency satisfies ω

The graded off-diagonal Bethe ansatz method is proposed to study supersymmetric quantum integrable models (i.e., quantum integrable models associated with superalgebras). As an example, the exact solutions of the SU(2|2) vertex model with both periodic and generic open boundary conditions are constructed. By generalizing the fusion techniques to the supersymmetric case, a closed set of operator product identities about the transfer matrices are derived, which allows us to give the eigenvalues in terms of homogeneous or inhomogeneous T−Q relations. The method and results provided in this paper can be generalized to other high rank supersymmetric quantum integrable models.

The exact solution of an integrable anisotropic Heisenberg spin chain with nearest-neighbor, next-nearest-neighbor and scalar chirality couplings is studied, where the boundary condition is the antiperiodic one. The detailed construction of Hamiltonian and the proof of integrability are given. The antiperiodic boundary condition breaks the U(1)-symmetry of the system and we use the off-diagonal Bethe Ansatz to solve it. The energy spectrum is characterized by the inhomogeneous T−Q relations and the contribution of the inhomogeneous term is studied. The ground state energy and the twisted boundary energy in different regions are obtained. We also find that the Bethe roots at the ground state form the string structure if the coupling constant J=−1 although the Bethe Ansatz equations are the inhomogeneous ones.

The thermodynamic limit and boundary energy of the isotropic spin-1 Heisenberg chain with non-diagonal boundary fields are studied. The finite size scaling properties of the inhomogeneous term in the $T-Q$ relation at the ground state are calculated by the density matrix renormalization group. Based on our findings, the boundary energy of the system in the thermodynamic limit can be obtained from Bethe ansatz equations of a related model with parallel boundary fields. These results can be generalized to the $SU(2)$ symmetric high spin Heisenberg model directly.

We investigate the thermodynamic limit of the inhomogeneous T−Q relation of the antiferromagnetic XXZ spin chain with antiperiodic boundary condition. It is shown that the contribution of the inhomogeneous term for the ground state can be neglected when the system-size N tends to infinity, which enables us to reduce the inhomogeneous Bethe ansatz equations (BAEs) to the homogeneous ones. Then the quantum numbers at the ground states are obtained, by which the system with arbitrary size can be studied. We also calculate the twisted boundary energy of the system.

Abstract We investigate the behavior of coherence in scattering quantum walk search on complete graph under the condition that the total number of vertices of the graph is significantly larger than the marked number of vertices we are searching, N ≫ v. We find that the consumption of coherence represents the increase of the success probability for the searching, also it is related to the efficiency of the algorithm in oracle queries. If no coherence is consumed or an incoherent state is utilized, the algorithm will behave as the classical blind search, implying that coherence is responsible for the speed-up in this quantum algorithm over its classical counterpart. The effect of noises, in particular of photon loss and random phase shifts, on the performance of algorithm is studied. Two types of noise are considered because they arise in the optical network used for experimental realization of scattering quantum walk. It is found that photon loss will reduce the coherence and random phase shifts will hinder the interference between the edge states, both leading to lower success probability compared with the noise-free case. We then conclude that coherence plays an essential role and is responsible for the speed-up in this quantum algorithm.

Abstract We investigate the thermodynamic limit of the exact solution, which is given by an inhomogeneous T − Q relation, of the one-dimensional supersymmetric t − J model with unparallel boundary magnetic fields. It is shown that the contribution of the inhomogeneous term at the ground state satisfies the L −1 scaling law, where L is the system-size. This fact enables us to calculate the surface (or boundary) energy of the system. The method used in this paper can be generalized to study the thermodynamic limit and surface energy of other models related to rational R-matrices.

By combining the algebraic Bethe ansatz and the off-diagonal Bethe ansatz, we investigate the trigonometric SU(3) model with generic open boundaries. The eigenvalues of the transfer matrix are given in terms of an inhomogeneous T−Q relation, and the corresponding eigenstates are expressed in terms of nested Bethe-type eigenstates which have well-defined homogeneous limit. This exact solution provides a basis for further analyzing the thermodynamic properties and correlation functions of the anisotropic models associated with higher rank algebras.

We propose a convenient orthogonal basis of the Hilbert space for the quantum spin chain associated with the A2(2) algebra (or the Izergin–Korepin model). It is shown that compared with the original basis the monodromy-matrix elements acting on this basis take relatively simple forms, which is quite similar as that for the quantum spin chain associated with An algebra in the so-called F-basis. As an application of our general results, we present the explicit recursive expressions of the Bethe states in this basis for the Izergin–Korepin model.

Abstract We compute the two-loop anomalous dimension matrix in the scalar sector of planar N = 3 $$ \mathcal{N}=3 $$ flavored ABJM theory. Using coordinate Bethe ansatz, we obtain the reflection matrices and confirm that the boundary Yang-Baxter equations are satisfied. This establishes the integrability of this theory in the scalar sector at the two-loop order.

The periodic Zn-Belavin model on a lattice with an arbitrary number of sites N is studied via the off-diagonal Bethe Ansatz method (ODBA). The eigenvalues of the corresponding transfer matrix are given in terms of an unified inhomogeneous T−Q relation. In the special case of N=nl with l being also a positive integer, the resulting T−Q relation recovers the homogeneous one previously obtained via algebraic Bethe Ansatz.

Abstract By combining the algebraic Bethe ansatz and the off-diagonal Bethe ansatz, we investigate the supersymmetric t − J model with generic open boundaries. The eigenvalues of the transfer matrix are given in terms of an inhomogeneous T − Q relation, and the corresponding eigenstates are expressed in terms of nested Bethe states which have welldefined homogeneous limit. This exact solution provides basis for further analyzing the thermodynamic properties and correlation functions of the model.

We investigate the thermodynamic limit of the su(n)-invariant spin chain models with unparallel boundary fields. It is found that the contribution of the inhomogeneous term in the associated T–Q relation to the ground state energy does vanish in the thermodynamic limit. This fact allows us to calculate the boundary energy of the system. Taking the su(2) (or the XXX) spin chain and the su(3) spin chain as concrete examples, we have studied the corresponding boundary energies of the models. The method used in this paper can be generalized to study the thermodynamic properties and boundary energy of other high rank models with non-diagonal boundary fields.

The so(5) (i.e., B2) quantum integrable spin chains with both periodic and non-diagonal boundaries are studied via the off-diagonal Bethe Ansatz method. By using the fusion technique, sufficient operator product identities (comparing to those in [1]) to determine the spectrum of the transfer matrices are derived. For the periodic case, we recover the results obtained in [1], while for the non-diagonal boundary case, a new inhomogeneous T−Q relation is constructed. The present method can be directly generalized to deal with the so(2n+1) (i.e., Bn) quantum integrable spin chains with general boundaries.

We propose an integrable spin-1 chain with a magnetic impurity whose spin is 1/2. The integrability of the model is based on an operator solution of the associated reflection equation. Due to the existence of the impurity, the SU(3) symmetry of the bulk is broken. By using the nested algebraic Bethe ansatz, we obtain the exact solution of the system. The eigenvalues, eigenstates, and Bethe ansatz equations are given explicitly. The method provided in this paper can be generalized to other high-rank quantum integrable systems with magnetic impurities where the spins of the bulk and of impurities are different.