Temperature-dependent Correlation Functions Research Articles

Time and temperature-dependent correlation function of an impurity in one-dimensional Fermi and Tonks–Girardeau gases as a Fredholm determinant

We investigate a free one-dimensional spinless Fermi gas, and the Tonks–Girardeau gas interacting with a single impurity particle of equal mass. We obtain a Fredholm determinant representation for the time-dependent correlation function of the impurity particle. This representation is valid for an arbitrary temperature and an arbitrary repulsive or attractive impurity-gas δ-function interaction potential. It includes, as particular cases, the representations obtained for zero temperature and arbitrary repulsion in (Gamayun et al 2015 Nucl. Phys. B 892 83–104), and for arbitrary temperature and infinite repulsion in (Izergin and Pronko 1998 Nucl. Phys. B 520 594–632).

Two approaches for describing the Casimir interaction in graphene: Density-density correlation function versus polarization tensor

The comparison studies of theoretical approaches to the description of the Casimir interaction in layered systems including graphene is performed. It is shown that at zero temperature the approach using the polarization tensor leads to the same results as the approach using the longitudinal density-density correlation function of graphene. An explicit expression for the zero-temperature transverse density-density correlation function of graphene is provided. We further show that the computational results for the Casimir free energy of graphene-graphene and graphene-Au plate interactions at room temperature, obtained using the temperature-dependent polarization tensor, deviate significantly from those using the longitudinal density-density correlation function defined at zero temperature. We derive both the longitudinal and transverse density-density correlation functions of graphene at nonzero temperature. The Casimir free energy in layered structures including graphene, computed using the temperature-dependent correlation functions, is exactly equal to that found using the polarization tensor.

Thermal form factors of the XXZ chain and the large-distance asymptotics of its temperature dependent correlation functions

We derive expressions for the form factors of the quantum transfer matrix of the spin- XXZ chain which are suitable for taking the infinite Trotter number limit. These form factors determine the finitely many amplitudes in the leading asymptotics of the finite-temperature correlation functions of the model. We consider form factor expansions of the longitudinal and transversal two-point functions. Remarkably, the formulae for the amplitudes are in both cases of the same form. We also explain how to adapt our formulae to the description of ground-state correlation functions of the finite chain. The usefulness of our novel formulae is demonstrated by working out explicit results in the high- and low-temperature limits. We obtain, in particular, the large-distance asymptotics of the longitudinal two-point functions for small temperatures by summing up the asymptotically most relevant terms in the form factor expansion of a generating function of the longitudinal correlation functions. As expected, the leading term in the expansion of the corresponding two-point functions is in accordance with conformal field theory predictions. Here it is obtained for the first time by a direct calculation.

Nonlinear phenomena, optical and quantum solitons

Research in the theory of solitons and its applications has undergone impressive development in recent years and has affected a broad range of fields, including theory of special functions, quantum integrable systems, numerical analysis, cellular automata, representations of quantum groups and

Quantum nonlocality of Heisenberg XX model with site-dependent coupling strength

We show that the generalized Bell inequality is violated in the extended Heisenberg model when the temperature is below a threshold value. The threshold temperature values are obtained by constructing exact solutions of the model using the temperature-dependent correlation functions. The effect due to the presence of external magnetic field is also illustrated.

Singlet superconductivity phase in carbon nanotubes

We discuss phase diagrams obtained by calculating temperature dependent correlation functions for superconductivity and density waves for a single (5,0) carbon nanotube (CN). We use one-loop renormalization group method within logarithmic accuracy. In this system we must specify scattering channels in terms of momentum along the circumferential direction as well as the axis direction, since (5,0) CN has two degenerate bands crossing the Fermi energy with circumferential momenta. We find singlet-superconducting phase with $5\hbar$ circumferential angular momentum. Critical exponents for each correlation functions are also calculated.

Temperature-dependent correlation function of magnetoconductance quantum fluctuations in a stadium quantum dot

The temperature dependence of weak localization and conductance fluctuations in a ballistic quantum dot stadium was studied, where the experimental data of measurements could be best fitted by including both dephasing effects of electron scatterings and thermal smearing of the states around the Fermi energy. We observed a crossover change in correlation function of fluctuations from a Lorentzian-squared form to a simple Lorentzian form as temperature was changed from low-temperature (50 mK) to high-temperature (300 mK) regimes.

Dynamical Correlation Functions for an Impenetrable Bose Gas with Neumann or Dirichlet Boundary Conditions

We study the time and temperature dependent correlation functions for an impenetrable Bose gas with Neumann or Dirichlet boundary conditions ψ(x 1, 0)ψ †(x 2 , t) ±,T . We derive the Fredholm determinant formulae for the correlation functions, by means of the Bethe Ansatz. For the special case x 1 = 0, we express correlation functions with Neumann boundary conditions ψ(0, 0)ψ †(x 2 , t) +,T , in terms of solutions of nonlinear partial differential equations which were introduced in [1] as a generalization of the nonlinear Schrödinger equations. We generalize the Fredholm minor determinant formulae of ground state correlation functions ψ(x 1)ψ †(x 2) ±,0 in [2], to the Fredholm determinant formulae for the time and temperature dependent correlation functions ψ(x 1, 0)ψ †(x 2 , t) ±,T , t ∈ R, T ≥ 0.

Temperature correlators in the two-component one-dimensional gas

The quantum non-relativistic two-component Bose and Fermi gases with infinitely strong point-like coupling between particles in one space dimension are considered. Time- and temperature-dependent correlation functions are represented in the thermodynamic limit as Fredholm determinants of integrable linear integral operators.

Time and temperature dependent correlation functions of 1D models of quantum statistical mechanics

We consider exactly solvable, gapless models of statistical mechanics. We found the universal formula which describes the asymptotics of correlation functions at any temperature and an arbitrary coupling constant.

The Maxwell - Bloch equation and correlation functions for the penetrable Bose gas

We consider the quantum nonlinear Schr\"odinger equation in one space and one time dimension. We are interested in the non-free-fermionic case. We consider static temperature-dependent correlation functions. The determinant representation for correlation functions simplifies in the small mass limit of the Bose particle. In this limit we describe the correlation functions by the vacuum expectation value of a boson-valued solution for Maxwell-Bloch differential equation. We evaluate long-distance asymptotics of correlation functions in the small mass limit.

Completely Integrable Equation for the Quantum Correlation Function of Nonlinear Schrödinger Equation

Correlation functions of exactly solvable models can be described by differential equation [Barough, McCoy, Wu]. In this paper we show that for non free fermionic case differential equations should be replaced by integro-differential equations. We derive an integro-differential equation, which describes time and temperature dependent correlation function $<\psi(0,0)\psi^\dagger(x,t)>_T$ of penetrable Bose gas. The integro-differential equation turns out be the continuum generalization of classical nonlinear Schr\"odinger equation.

Temperature Correlations of Quantum Spins

This is a historical note. In 1993 we calculated space, time and temperature dependent correlation function in isotropic version of one dimensional XY spin chain. The correlation function decays exponentially with time and space separation. The rate of exponential decay was evaluated explicitly. Since that time similar results were obtained in other models: Bose gas with delta interaction, Ising model and strongly correlated electrons.

Correlators in the Heisenberg XXO chain as Fredholm determinants

The two-point time and temperature dependent correlation functions for the XXO one-dimensional model in constant magnetic field are represented (in the thermodynamical limit) as Fredholm determinants of linear integral operators.

On the nature of supercooled and glassy states of matter

The temperature dependence of relaxation behavior in glass-forming liquids is formulated in terms of the temperature variations of a critical lower limit z ∗ to the sizes of cooperative regions that can rearrange into another configuration independent of its environment. The temperature dependence of the size of this cooperatively rearranging region is determined by dividing the total volume of the system into blocks, each of volume L 3, and then relating the number of particles or molecular segments in each block to the entropy per block, to the Gibbs free energy per block, to the molar entropy of the macroscopic sample, and to the temperature derivative of block length L. The result of the molecular-kinetic theory is a relation for the relaxation time which almost coincides with the empirical WLF and VTF equations. The theory is applied to viscosimetric experiments. This permits evaluation of the ratio of the kinetic glass temperature T g to the equilibrium temperature T 2, where the size of the rearranging region diverges. Predictions are made concerning the temperature dependence of relaxation times and the activation energy for viscous flow as T→ T g, and the ratio of the enthalpy of the glass to the enthalpy of supercooled liquid. The values of T g/ T 2 obtained for fourteen widely different materials were found to be nearly the same, i.e., 1.28 ± 5.78%. The non-linearity of the glassy state is found to be governed by properties of the equilibrium melt above T g. Finally, for an arbitrary one-component many-body system we obtain a lower bound to the entropy density at low temperatures. The entropy per unit volume is related to integrals over certain temperature-dependent correlation functions of the system. An operator product expansion of the correlation functions leads to a necessary condition for the existence of the Kauzmann temperature.

Roughening behavior of a one-dimensional crystal surface with inverse square potential

The discrete Gaussian chain with inverse square potential—which models a one-dimensional crystal surface—is studied by Monte-Carlo methods. The system is found to undergo a roughening transition, and the temperature-dependent correlation function exponent η( T) is shown to be discontinous at the critical point. These results can be transposed to the metal-insulator transition of a one-dimensional plasma of logarithmically interacting charges.

The discrete Gaussian chain with 1/r n interactions: Exact results

We show that the discrete Gaussian chain with interactionV(r) = 1/(r2−1/4) is self-dual. At the dual temperaturek B T = 1 we calculate the height-height correlation function and find that the system is rough. A duality relation is established for the temperature-dependent correlation function exponentη. We also consider interactionsV(r)−1/rn and show that absence of a phase transition for 2 <n < 3 implies absence of a phase transition for 1 <n < 2. All these results have their counterparts in a linear system of charges interacting through a potential which is asymptotically logarithmic (forn = 2) or power-law-like (forn ≠ 2.

On the minimum entropy of a large system at low temperatures

This paper examines the general question: How far can one set lower limits on the entropy density S( T) of a large system at low temperatures by using data of a different type, e.g., scattering data or static susceptibilities? It is shown for an arbitrary one-component system that if we make a single unproved but highly plausible assumption about the fluctuations of large subvolumes, we can obtain an inequality relating S( T) to certain integrals over the temperature-dependent correlation functions of the system; and with one further very weak assumption the zero-temperature correlation functions alone determine a lower limit on S( T). It is then shown that the fluctuations of any locally conserved quantity give rise to a contribution to S of at least a constant times T 3; in particular, by considering the density fluctuations we obtain the inequality S(T)⩾ 1 2 S D(T) , where S D( T) is the “Debye” entropy which would arise from longitudinal phonons propagating at the hydrodynamic sound velocity determined by the macroscopic compressibility. (This result does not assume the existence of such phonons as good elementary excitations of the system.) Some other general results are derived as a byproduct: for instance, it is shown that any system obeying a diffusion equation of a certain type must have an entropy at least proportional to T.

Standard-Basis Operator Method in the Green's-Function Technique of Many-Body Systems with an Application to Ferromagnetism

A new formalism, using standard-basis matrix operators, is presented for the study of the collective excitations and the thermodynamic properties of an ensemble of identical interacting quantum-mechanical systems subsequently called ions, each ion having discrete energy levels. A model Hamiltonian is formulated in terms of these operators. The Hamiltonian contains terms which express the interaction of the ion with the crystal field and the external fields as well as terms which arise from the mutual interaction of ions. Using the doubletime temperature-dependent Green's-function technique, the equations of motion of the Green's functions of standard-basis operators are developed in the random-phase decoupling approximation. It is demonstrated that the temperature-dependent correlation functions of standard-basis operators, which are obtained from associated spectral Green's functions, lead to a set of equations that can be solved for the occupation probabilities of the single-ion energy levels. Hence, one can calculate the thermal-average expectation value of any quantum-mechanical operator representing a microscopic observable of a single ion, or a pair of ions (correlations). An important feature of the standard-basis matrix-operator formalism is that the single-ion terms, such as crystal field, molecular field, or external fields, are always treated exactly in any Green's-function decoupling scheme. In contrast, Green's-function methods which use angular momentum operators often necessarily treat single-ion terms in a decoupling approximation. As an example, the general standard-basis operator theory is applied to the Heisenberg ferromagnet in the presence of uniaxial single-ion crystal-field anisotropy, which has received extensive theoretical attention previously, with widely varying results.

RPA instability and the gas-liquid transition. Part I

The possibility of a many particle system undergoing a classical gas-liquid phase transition, is shown to be reflected in the structure of the two-particle Green's function in the random phase approximation. When the interaction is sufficiently attractive this Green's function possesses forbidden poles on the imaginary axis of the complex frequency plane. An investigation of the temperature dependent correlation function reveals that the poles appear even in the classical limit, and that as they appear ( ϖP ϖV ) T vanishes and long range density correlations of the form usually associated with critical fluctuations arise. It is therefore concluded that the poles are symptomatic of the instabilities associated with the condensation of the system from a gaseous to a liquid phase.