Toeplitz Determinants Research Articles

N-body correlation of Tonks–Girardeau gas

For the well-known exponential complexity it is a giant challenge to calculate the correlation function for general many-body wave function. We investigate the ground state $n$th-order correlation functions of the Tonks-Girardeau (TG) gases. Basing on the wavefunction of free fermions and Bose-Fermi mapping method we obtain the exact ground state wavefunction of TG gases. Utilizing the properties of Vandermonde determinant and Toeplitz matrix, the $n$th-order correlation function is formulated as $(N-n)$-order Toeplitz determinant, whose element is the integral dependent on 2$(N-n)$ sign functions and can be computed analytically. By reducing the integral on domain $[0,2\pi]$ into the summation of the integral on several independent domains, we obtain the explicit form of the Toeplitz matrix element ultimately. As the applications we deduce the concise formula of the reduced two-body density matrix and discuss its properties. The corresponding natural orbitals and their occupation distribution are plotted. Furthermore, we give a concise formula of the reduced three-body density matrix and discuss its properties. It is shown that in the successive second measurements, atoms appear in the regions where atoms populate with the maximum probability in the first measurement.

Coefficient inequalities for a subfamily of Sakaguchi functions

In this paper, we explore a subfamily of starlike functions with respect to symmetric points allied with the hyperbolic cosine function. We study Hermitian–Toeplitz determinants of third and fourth orders for the functions belonging to this subfamily. Further, we calculate estimates on the initial successive inverse coefficients as well as logarithmic coefficients and related functionals for such functions. In addition, we also determine bounds on Hankel determinants of third order and symmetric Toeplitz determinants of second and third orders.

Toeplitz Determinants for the Class of Functions with Bounded Turning

In this paper, we obtain the upper bounds of the Toeplitz determinants for the class of functions with bounded turning. We also present some consequences of our main results. Some estimates obtained on Toeplitz determinants are sharp.

Brane order and quantum magnetism in modulated anisotropic ladders

Two-leg spin-$1/2$ ladders with anisotropy and two different dimerization patterns are analyzed at zero temperature. This model is equivalent to a modulated interacting (Kitaev) ladder. The Hartree-Fock mean-field approximation reduces the model to a sum of two quadratic effective Majorana Hamiltonians, which are dual to two quantum transverse XY chains. The mapping between the effective Hamiltonian of the ladder and a pair of chains considerably simplifies calculations of the order parameters and analysis of the hidden symmetry breaking. The ground-state phase diagram of the staggered ladder contains nine phases, four of them are conventional antiferromagnets, while the other five possess nonlocal brane orders. Using the dualities and the newly found exact results for the local and string order parameters of the transverse XY chains, we were able to find analytically all the magnetizations and the brane order parameters for the staggered case, as functions of the renormalized couplings of the effective Hamiltonian. The columnar ladder has three ground-state phases and does not possess magnetic long-ranged order. The brane order parameters for these three phases are calculated numerically from the Toeplitz determinants. All brane-ordered phases are spin liquids with identified distinct order parameters, winding numbers, and sets of the Majorana edge modes. Disorder lines and the special points of disentanglement are found in both dimerization patterns. We expect this study to motivate the search for the real spin-Peierls anisotropic ladder compounds which manifest predicted properties.

HANKEL DETERMINANT OF CERTAIN ORDERS FOR SOME SUBCLASSES OF HOLOMORPHIC FUNCTIONS

In this paper, we are introducing certain subfamilies of holomorphic functions and making an attempt to obtain an upper bound (UB) to the second and third order Hankel determinants by applying certain lemmas, Toeplitz determinants, for the normalized analytic functions belong to these classes, defined on the open unit disc in the complex plane. For one of the inequality, we have obtained sharp bound.

Symmetry-Resolved Entanglement Entropy in Critical Free-Fermion Chains

The symmetry-resolved Rényi entanglement entropy is the Rényi entanglement entropy of each symmetry sector of a density matrix rho . This experimentally relevant quantity is known to have rich theoretical connections to conformal field theory (CFT). For a family of critical free-fermion chains, we present a rigorous lattice-based derivation of its scaling properties using the theory of Toeplitz determinants. We consider a class of critical quantum chains with a microscopic U(1) symmetry; each chain has a low energy description given by N massless Dirac fermions. For the density matrix, rho _A, of subsystems of L neighbouring sites we calculate the leading terms in the large L asymptotic expansion of the symmetry-resolved Rényi entanglement entropies. This follows from a large L expansion of the charged moments of rho _A; we derive mathrm {tr}(mathrm {e}^{mathrm {i}alpha Q_A} rho _A^n)~=~a mathrm {e}^{mathrm {i}alpha langle Q_Arangle } (sigma L)^{-x}(1+O(L^{-mu })), where a, x and mu are universal and sigma depends only on the N Fermi momenta. We show that the exponent x corresponds to the expectation from CFT analysis. The error term O(L^{-mu }) is consistent with but weaker than the field theory prediction O(L^{-2mu }). However, using further results and conjectures for the relevant Toeplitz determinant, we find excellent agreement with the expansion over CFT operators.

Symmetry-resolved entanglement in a long-range free-fermion chain

We investigate the symmetry resolution of entanglement in the presence of long-range couplings. To this end, we study the symmetry-resolved entanglement entropy in the ground state of a fermionic chain that has dimerised long-range hoppings with power-like decaying amplitude—a long-range generalisation of the Su–Schrieffer–Heeger model. This is a system that preserves the number of particles. The entropy of each symmetry sector is calculated via the charged moments of the reduced density matrix. We exploit some recent results on block Toeplitz determinants generated by a discontinuous symbol to obtain analytically the asymptotic behaviour of the charged moments and of the symmetry-resolved entropies for a large subsystem. At leading order we find entanglement equipartition, but comparing with the short-range counterpart its breaking occurs at a different order and it does depend on the hopping amplitudes.

Analytic Property of a Generalized Poisson Distribution

Basic properties of probability with Poisson distribution is used in obtaining the coefficient bound by subordination principle which is the fundamental purpose of this work. A class of analytic function $f:\xi \rightarrow \mathbb{C}$ with unit disc $\xi :\{z\in\mathbb{C}:|z|<1\}$ is established. Likewise known results of Fekete-Szegö inequalities type and the second bound of Toeplitz determinant are obtained.

Asymptotics of Bordered Toeplitz Determinants and Next-to-Diagonal Ising Correlations

We prove the analogue of the strong Szegő limit theorem for a large class of bordered Toeplitz determinants. In particular, by applying our results to the formula of Au-Yang and Perk (Physica A 144:44–104, 1987) for the next-to-diagonal correlations \(\langle \sigma _{0,0}\sigma _{N-1,N} \rangle \) in the square lattice Ising model, we rigorously justify that the next-to-diagonal long-range order is the same as the diagonal and horizontal ones in the low temperature regime. We also confirm the leading and subleading terms in an asymptotic formula of Cheng and Wu (Phys Rev 164:719–735, 1967) for \(\langle \sigma _{0,0}\sigma _{M,N} \rangle \) when \(M=N\) and \(M=N-1\), thereby establishing the anisotropy-dependence of the subleading term in the asymptotics of the next-to-diagonal correlations. We use Riemann-Hilbert and operator theory techniques, independently and in parallel, to prove these results.

Sharp Bounds of the Hermitian Toeplitz Determinants for Certain Close-to-Star Functions

The sharp lower and upper bounds of the Hermitian Toeplitz determinants of the second and third order for certain close-to-star functions are computed.

"Upper bounds of Toeplitz determinants for a subclass of alpha-close-to-convex functions"

"Let A be the class of P analytic functions in the unit disc U which are of the form $f(z)=z+\sum_{n=2}^{\infty}a_nz^n$. For 0 ≤ α < 1, let C_α, be the class of all functions f ∈ A satisfying the condition ${Re}{f'(z)+αzf''(z)}>0$. We consider the Toeplitz matrices whose elements are the coefficients an of the function f in the class C_α. In this paper we obtain upper bounds for the Toeplitz determinants. "

Some properties of certain close-to-convex harmonic mappings

In this paper, we determine the sharp estimates for Toeplitz determinants of a subclass of close-to-convex harmonic mappings. Moreover, we obtain an improved version of Bohr’s inequalities for a subclass of close-to-convex harmonic mappings, whose analytic parts are Ma-Minda convex functions.

Fourth‐Order Hankel Determinants and Toeplitz Determinants for Convex Functions Connected with Sine Functions

This article deals with the upper bound of fourth‐order Hankel and Toeplitz determinants for the convex functions which are defined by using the sine function. The main tools in this study are the coefficient inequalities for the class P of functions with positive real parts. Also, the investigation of the upper bound of the fourth‐order Hankel determinant for 3‐fold symmetric convex functions associated with the sine function is included.

Certain subclass of analytic functions based on $ q $-derivative operator associated with the generalized Pascal snail and its applications

<abstract><p>By the principle of differential subordination and the $ q $-derivative operator, we introduce the $ q $-analog $ \mathcal{SP}^{q}_{snail}(\lambda; \alpha, \beta, \gamma) $ of certain class of analytic functions associated with the generalized Pascal snail. Firstly, we obtain the coefficient estimates and Fekete-Szegö functional inequalities for this class. Meanwhile, we also estimate the corresponding symmetric Toeplitz determinant. Secondly, for all the above results we provide the corresponding results for the reduced classes $ \mathcal{SP}^{q}_{snail}(\alpha, \beta, \gamma) $ and $ \mathcal{RP}^{q}_{snail}(\alpha, \beta, \gamma) $. Thirdly, we characterize the Bohr radius problems for the function class $ \mathcal{SP}^{q}_{snail}(\alpha, \beta, \gamma) $. Lastly, we establish certain results for some new subclasses of functions defined by the neutrosophic Poisson distribution series.</p></abstract>

Coefficient Inequalities for certain Starlike and Convex functions

In this paper, we consider two Ma--Minda-type subclasses of starlike and convex functions associated with the normalized analytic function $\varphi_{Ne}(z)=1+z-z^3/3$ that maps an open unit disk onto the Nephroid shaped bounded domain in the right--half of the complex plane. We investigate convolution and quasi-Hadamard product properties for the functions belonging to such classes. In addition, we compute best possible estimates on third order Hermitian--Toeplitz determinant and non-sharp estimates on certain third order Hankel determinants for the starlike functions associated with the interior region of Nephroid.

Toeplitz Determinant for a Subclass of Tilted Starlike Functions with Respect to Conjugate Points

Toeplitz Determinant for a Subclass of Tilted Starlike Functions with Respect to Conjugate Points

A Sharp Inequality on the Exponentiation of Functions on the Sphere

In this paper we show a new inequality that generalizes to the unit sphere the Lebedev‐Milin inequality of the exponentiation of functions on the unit circle. It may also be regarded as the counterpart on the sphere of the second inequality in the Szegö limit theorem on the Toeplitz determinants on the circle. On the other hand, this inequality is also a variant of several classical inequalities of Moser‐Trudinger type on the sphere. The inequality incorporates the deviation of the center of mass from the origin into the optimal inequality of Aubin for functions with mass centered at the origin, and improves Onofri's inequality with the contribution of the shifting of the mass center explicitly expressed. © 2021 Wiley Periodicals LLC.

QFT, EFT and GFT

We explore the correspondence between geometric function theory (GFT) and quantum field theory (QFT). The crossing symmetric dispersion relation provides the necessary tool to examine the connection between GFT, QFT, and effective field theories (EFTs), enabling us to connect with the crossing-symmetric EFT-hedron. Several existing mathematical bounds on the Taylor coefficients of Typically Real functions are summarized and shown to be of enormous use in bounding Wilson coefficients in the context of 2-2 scattering. We prove that two-sided bounds on Wilson coefficients are guaranteed to exist quite generally for the fully crossing symmetric situation. Numerical implementation of the GFT constraints (Bieberbach-Rogosinski inequalities) is straightforward and allows a systematic exploration. A comparison of our findings obtained using GFT techniques and other results in the literature is made. We study both the three-channel as well as the two-channel crossing-symmetric cases, the latter having some crucial differences. We also consider bound state poles as well as massless poles in EFTs. Finally, we consider nonlinear constraints arising from the positivity of certain Toeplitz determinants, which occur in the trigonometric moment problem.

Hermitian–Toeplitz determinants for functions with bounded turning

Hermitian–Toeplitz determinants for functions with bounded turning

Coefficient Estimate of Toeplitz Determinant for a Certain Class of Close to Convex Functions

Let S denote the class of analytic and univalent functions in D, where D is defined as unit disk and having the Taylor representation form of S. We will determine the estimation for the Toeplitz determinants where the elements are the Taylor coefficients of the class close-to-convex functions in S.