Nonanalytic Points Research Articles

Topological Characterization of Extended Quantum Ising Models.

We show that a class of exactly solvable quantum Ising models, including the transverse-field Ising model and anisotropic XY model, can be characterized as the loops in a two-dimensional auxiliary space. The transverse-field Ising model corresponds to a circle and the XY model corresponds to an ellipse, while other models yield cardioid, limacon, hypocycloid, and Lissajous curves etc. It is shown that the variation of the ground state energy density, which is a function of the loop, experiences a nonanalytical point when the winding number of the corresponding loop changes. The winding number can serve as a topological quantum number of the quantum phases in the extended quantum Ising model, which sheds some light upon the relation between quantum phase transition and the geometrical order parameter characterizing the phase diagram.

Quantum entanglement investigation on impurity effects in the transverse Ising chain

The impurity effects of the transverse Ising spin model with a magnetic impurity are investigated via the correlation function and entanglement on the basis of an exact solution. The non-analytical transition point from an ordered phase to a disorder phase is significantly shifted by the impurity coupling with local host spins. The maximal entanglement point which usually appears around the critical point shows strong dependence on the impurity coupling strength with local host spin and the independent transverse field upon the impurity itself. Quantum entanglement in disordered spin phase is much stronger than that in ordered spin phase. The exact concurrence of impurity spin with local host spin suddenly increases to a large value from zero at a threshold value. The maximal entanglement point between impurity and local host spin increases as the impurity coupling strength increases. So the impurity coupling strength and local transverse field on the impurity can control position of critical point and maximal entanglement point.

Singular behavior of fluctuations in a relaxation process

Carrying out explicitly the computation in a paradigmatic model of non-interacting systems, the Gaussian model, we show the existence of a singular point in the probability distribution P(M) of an extensive variable M. Interpreting P(M) as a thermodynamic potential of a dual system obtained from the original one by applying a constraint, we discuss how the non-analytical point of P(M) is the counterpart of a phase-transition in the companion system. We show the generality of such mechanism by considering both the system in equilibrium or in non-equilibrium state following a temperature quench.

Direct Observation of a Pulse Peak Using a Peak-Removed Gaussian Optical Pulse in a Superluminal Medium

A series of experiments is performed to examine the arrival of a pulse peak, using a Gaussian-shaped temporal wave packet as the input pulse and truncating it at various positions on or before the peak of the packet. When the truncating point is within the negative group delay limit of the fast light medium, a smooth Gaussian peak is observed at the exit port, despite the absence of an input pulse peak. The experimental results explicitly demonstrate that the superluminal propagation of a smooth Gaussian-shaped pulse peak is an analytic continuation over time of the earlier portion of the input pulse envelope. To investigate the physical meaning of the pulse peak further, we also examine the propagation of triangular-shaped pulses, for which the pulse peak can be recognized as a nonanalytical point.

Causal propagation of nonanalytical points in fast- and slow-light media

We examined the propagation of bending nonanalytical points encoded on temporally smooth Gaussian-shaped optical pulses in fast- and slow-light media, using an optical ring resonator. For bending nonanalytical points, the pulse function is continuous, but the first-order derivative is discontinuous, in contrast to traditional nonanalytical points in which the pulse function itself is discontinuous. The nonanalytical points were neither advanced nor delayed, but appeared at the same instance as they entered the ring resonator, in good accordance with the fact that the information velocity was equal to the light velocity in vacuum or the background medium. The differences between the traditional discontinuous and the present bending nonanalytical points were investigated.

Control of qubits by shaped pulses of finite duration

We consider the interaction of a two-state quantum system with a class of pulses of finite temporal duration. The pulse shape function $f(t)$ of such a pulse is a nonanalytic function of time, with discontinuous derivatives at the turn-on and turn-off times. The excitation line width---the excited-state population versus the detuning---is determined primarily by the magnitude of the jumps of the derivative ${f}^{(n)}(t)$ at the points of nonanalyticity, where $n$ is the order of the first discontinuous derivative; this nonanalyticity shows up in the $n$th superadiabatic basis. The excitation line width for such pulses exhibits weak power broadening---it scales up as ${\ensuremath{\Omega}}_{0}^{1/(n+1)}$, where ${\ensuremath{\Omega}}_{0}$ is the peak Rabi frequency of the transition: $\ensuremath{\Omega}(t)={\ensuremath{\Omega}}_{0}f(t)$. As a specific example, we consider the power-of-sine class $f(t)={\mathrm{sin}}^{n}(\ensuremath{\pi}t/T)$ ($0\ensuremath{\leqq}t\ensuremath{\leqq}T$) and a truncated Gaussian pulse, and we compare their excitation line widths with the well-known excitation profile of the rectangular pulse (the Rabi formula). We find that, because of the reduced power broadening, the ${\mathrm{sin}}^{n}$ and truncated Gaussian pulses may accelerate manipulation of qubits compared to rectangular pulses. The reason is that the lower power broadening allows one to use higher Rabi frequency, and hence shorter pulse duration, without affecting significantly other closely lying states.

Background field approach to electromagnetic properties of baryons

We investigate the self-energies of particles in an external magnetic field $B$. The dependence is generally of the type $\sqrt{P(B)}$ with $P$ a polynomial in $B$ and the participating masses. The non-analytic point depends on the mass and charge constellations, is unproblematic for stable particles but constrains the linear energy shift approximation for resonances. We calculate the $B$ dependent self-energies of the nucleon and $\Delta(1232)$-isobar in the SU(2) covariant chiral perturbation theory and outline a way to obtain finite volume corrections to the nucleon anomalous magnetic moment without using the three point function method. We show that finite volume corrections might explain present discrepancies of lattice QCD and chiral perturbation theory results in the small pion mass region.

Power-law Kohn anomaly in undoped graphene induced by Coulomb interactions

Phonon dispersions generically display non-analytic points, known as Kohn anomalies, due to electron-phonon interactions. We analyze this phenomenon for a zone boundary phonon in undoped graphene. When electron-electron interactions with coupling constant $\beta$ are taken into account, one observes behavior demonstrating that the electrons are in a critical phase: the phonon dispersion and lifetime develop power law behavior with $\beta$ dependent exponents. The observation of this signature would allow experimental access to the critical properties of the electron state, and would provide a measure of its proximity to an excitonic insulating phase.

Analytical behavior of the fourth and fifth virial coefficients of a hard-sphere mixture

This paper focus on the study of the composition-dependent fourth and fifth virial coefficient of a hard-sphere binary mixture, namely, B(4)(s) and B(5)(s). On one side, the analytical expression of B(4)(s) is revised. In particular, I analyzed the composition-independent term B(4)[1](s) included in B(4)(s) and found a simplified form of its exact expression, which has the advantage over that previously published of highlighting the geometric properties of this term. On the other side, using a geometrical approach, I found a set of nonanalytic points of B(4)(s) and B(5)(s) as functions of s by analyzing each of their composition-independent terms.

Causal information velocity in fast and slow pulse propagation in an optical ring resonator

We examined the propagation of nonanalytical points encoded on temporally Gaussian-shaped optical pulses in fast and slow light in an optical ring resonator at $\ensuremath{\lambda}$ = 1.5 \ensuremath{\mu}m. The temporal peak of the Gaussian pulse was either advanced or delayed, reflecting anomalous or normal dispersions in the ring resonator, relevant to under- or overcoupling conditions, respectively. The nonanalytical points were neither advanced nor delayed but appeared as they entered the ring resonator. The nonanalytical points could be interpreted as information; therefore, the experimental results suggested that information velocity is equal to the light velocity in vacuum or the background medium, independent of the group velocity. The transient behaviors at the leading and trailing edges of the nonanalytical points are discussed in terms of optical precursors.

Phase Transitions in the Distribution of the Andreev Conductance of Superconductor-Metal Junctions with Multiple Transverse Modes

We compute analytically the full distribution of Andreev conductance G(NS) of a metal-superconductor interface with a large number N(c) of transverse modes, using a random matrix approach. The probability distribution P(G(NS),N(c) in the limit of large N(c) displays a Gaussian behavior near the average value <G(NS)>=(2-√2)N(c) and asymmetric power-law tails in the two limits of very small and very large G(NS). In addition, we find a novel third regime sandwiched between the central Gaussian peak and the power-law tail for large G(NS). Weakly nonanalytic points separate these four regimes-these are shown to be consequences of three phase transitions in an associated Coulomb gas problem.

A simple topological model with continuous phase transition

In the area of topological and geometric treatment of phase transitions and symmetrybreaking in Hamiltonian systems, some general sufficient conditions for these phenomena in-symmetric systems (i.e. invariant under reflection of coordinates) were found in a recentpaper. In this paper we present a simple topological model satisfying the above conditions,hoping to shed light on the mechanism which causes this phenomenon in more generalphysical models. The symmetry breaking is proved by a continuous magnetizationwith a nonanalytic point in correspondence with a critical temperature whichdivides the broken symmetry phase from the unbroken one. A particularity withrespect to the common pictures of a phase transition is that the nonanalyticityof the magnetization is not accompanied by a nonanalytic behavior of the freeenergy.

Probability distributions of linear statistics in chaotic cavities and associated phase transitions

We establish large deviation formulas for linear statistics on the $N$ transmission eigenvalues ${{T}_{i}}$ of a chaotic cavity, in the framework of random matrix theory. Given any linear statistics of interest $A={\ensuremath{\sum}}_{i=1}^{N}a({T}_{i})$, the probability distribution ${\mathcal{P}}_{A}(A,N)$ of $A$ generically satisfies the large deviation formula ${\text{lim}}_{N\ensuremath{\rightarrow}\ensuremath{\infty}}[\ensuremath{-}2\text{ }\text{log}\text{ }{\mathcal{P}}_{A}(Nx,N)/\ensuremath{\beta}{N}^{2}]={\ensuremath{\Psi}}_{A}(x)$, where ${\ensuremath{\Psi}}_{A}(x)$ is a rate function that we compute explicitly in many cases (conductance, shot noise, and moments) and $\ensuremath{\beta}$ corresponds to different symmetry classes. Using these large deviation expressions, it is possible to recover easily known results and to produce new formulas, such as a closed form expression for $v(n)={\text{lim}}_{N\ensuremath{\rightarrow}\ensuremath{\infty}}\text{ }\text{var}({\mathcal{T}}_{n})$ (where ${\mathcal{T}}_{n}={\ensuremath{\sum}}_{i}{T}_{i}^{n}$) for arbitrary integer $n$. The universal limit ${v}^{\ensuremath{\star}}={\text{lim}}_{n\ensuremath{\rightarrow}\ensuremath{\infty}}\text{ }v(n)=1/2\ensuremath{\pi}\ensuremath{\beta}$ is also computed exactly. The distributions display a central Gaussian region flanked on both sides by non-Gaussian tails. At the junction of the two regimes, weakly nonanalytical points appear, a direct consequence of phase transitions in an associated Coulomb gas problem. Numerical checks are also provided, which are in full agreement with our asymptotic results in both real and Laplace space even for moderately small $N$. Part of the results have been announced by Vivo et al. [Phys. Rev. Lett. 101, 216809 (2008)].

Master equation approach to computing RVB bond amplitudes

We describe a ``master equation'' analysis for the bond amplitudes $h(\mathbf{r})$ of an RVB wave function. Starting from any initial guess, $h(\mathbf{r})$ evolves---in a manner dictated by the spin Hamiltonian under consideration---toward a steady-state distribution representing an approximation to the true ground state. Unknown transition coefficients in the master equation are treated as variational parameters. We illustrate the method by applying it to the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ antiferromagnetic Heisenberg model. Without frustration $({J}_{2}=0)$, the amplitudes are radially symmetric and fall off as $1/{r}^{3}$ in the bond length. As the frustration increases, there are precursor signs of columnar or plaquette valence bond solid order: the bonds preferentially align along the axes of the square lattice and weight accrues in the nearest-neighbor bond amplitudes. The Marshall sign rule holds over a large range of couplings, ${J}_{2}/{J}_{1}\ensuremath{\lesssim}0.418$. It fails when the $\mathbf{r}=(2,1)$ bond amplitude first goes negative, a point also marked by a cusp in the ground-state energy. A nonrigorous extrapolation of the staggered magnetic moment (through this point of nonanalyticity) shows it vanishing continuously at a critical value ${J}_{2}/{J}_{1}\ensuremath{\approx}0.447$. This may be preempted by a first-order transition to a state of broken translational symmetry.

Electron correlation and electron momenta in polyacetylene

In a uniform noninteracting electron gas, (here are two nonanalytic points in the structure factor S(k), which is the Fourier transform of the electron pair correlation function g(r). One of these is the origin of k or momentum space, while the other occurs at k = 2kf, where pf= hkf is the Fermi momentum. In a Peierls insulator, attention is most often focused on the nonanalytic region around k = 2kf, which in turn is modified by the Peierls distortion. By studying here the interpretation of the linear π-plasmon dispersion observed by Ritsko et al. in polyacetylene (CH)x and also, more recently, in a specimen containing a polydiacetylene backbone, attention focuses naturally on the zero momentum non-analyticity. Using a simple model of the π-electron Fermi liquid in (CH) x paralleling that used by Feynman for the Bose liquid 4He at T = 0, the observed plasmon dispersion is related to the structure factor S(k) of the interacting π-electron liquid. Analysis of the resulting form of S(k) makes it plain that both Pauli Principle correlations between parallel spin π-electrons, and Coulomb repulsions, are essential ingredients in interpreting the linear dispersion of the π-plasmon.

Effective pair potentials for spherical nanoparticles

An effective description for rigid spherical nanoparticles in a fluid of point particles ispresented. The points inside the nanoparticles and the point particles are assumed tointeract via spherically symmetric additive pair potentials, while the distribution of pointsinside the nanoparticles is taken to be spherically symmetric and smooth. The resultingeffective pair interactions between a nanoparticle and a point particle, as well as betweentwo nanoparticles, are then given by spherically symmetric potentials. If overlap betweenparticles is allowed, as can occur for some forms of the pair potentials, the effectivepotential generally has non-analytic points. It is shown that for each effective potential theexpressions for different overlapping cases can be written in terms of one analytic auxiliarypotential. Even when only non-overlapping situations are possible, the auxiliary potentialsfacilitate the formulation of the effective potentials. Effective potentials for hollownanoparticles (appropriate e.g. for buckyballs) are also considered and shown to berelated to those for solid nanoparticles. For hollow nanoparticles overlap is morephysical, since this covers the case of a smaller particle embedded in a larger, hollownanoparticle. Finally, explicit expressions are given for the effective potentials derivedfrom basic pair potentials of power law and exponential form, as well as from thecommonly used London–van der Waals, Morse, Buckingham, and Lennard-Jonespotentials. The applicability of the latter is demonstrated by comparison with anatomic description of nanoparticles with an internal face centered cubic structure.

Pinning and wetting transition for (1+1)-dimensional fields with Laplacian interaction

We consider a random field $\varphi:\{1,...,N\}\to\mathbb{R}$ as a model for a linear chain attracted to the defect line $\varphi=0$, that is, the x-axis. The free law of the field is specified by the density $\exp(-\sum_iV(\Delta\varphi_i))$ with respect to the Lebesgue measure on $\mathbb{R}^N$, where $\Delta$ is the discrete Laplacian and we allow for a very large class of potentials $V(\cdot)$. The interaction with the defect line is introduced by giving the field a reward $\varepsilon\ge0$ each time it touches the x-axis. We call this model the pinning model. We consider a second model, the wetting model, in which, in addition to the pinning reward, the field is also constrained to stay nonnegative. We show that both models undergo a phase transition as the intensity $\varepsilon$ of the pinning reward varies: both in the pinning ($a=\mathrm{p}$) and in the wetting ($a=\mathrm{w}$) case, there exists a critical value $\varepsilon_c^a$ such that when $\varepsilon>\varepsilon_c^a$ the field touches the defect line a positive fraction of times (localization), while this does not happen for $\varepsilon<\varepsilon_c^a$ (delocalization). The two critical values are nontrivial and distinct: $0<\varepsilon_c^{\mat hrm{p}}<\varepsilon_c^{\mathrm{w}}<\infty$, and they are the only nonanalyticity points of the respective free energies. For the pinning model the transition is of second order, hence the field at $\varepsilon=\varepsilon_c^{\mathrm{p}}$ is delocalized. On the other hand, the transition in the wetting model is of first order and for $\varepsilon=\varepsilon_c^{\mathrm{w}}$ the field is localized. The core of our approach is a Markov renewal theory description of the field.

Phase transitions and configuration space topology

Equilibrium phase transitions may be defined as nonanalytic points of thermodynamic functions, e.g., of the canonical free energy. Given a certain physical system, it is of interest to understand which properties of the system account for the presence, or the absence, of a phase transition, and an investigation of these properties may lead to a deeper understanding of the physical phenomenon. One possible way to approach this problem, reviewed and discussed in the present paper, is the study of topology changes in configuration space which are found to be related to equilibrium phase transitions in classical statistical mechanical systems. For the study of configuration space topology, one considers the subsets ${\mathcal{M}}_{v}$, consisting of all points from configuration space with a potential energy per particle equal to or less than a given $v$. For finite systems, topology changes of ${\mathcal{M}}_{v}$ are intimately related to nonanalytic points of the microcanonical entropy. In the thermodynamic limit, a more complex relation between nonanalytic points of thermodynamic functions (i.e., phase transitions) and topology changes is observed. For some class of short-range systems, a topology change of the ${\mathcal{M}}_{v}$ at $v={v}_{t}$ was proven to be necessary, but not sufficient, for a phase transition to take place at a potential energy ${v}_{t}$. In contrast, phase transitions in systems with long-range interactions or in systems with nonconfining potentials need not be accompanied by such a topology change. Instead, for such systems the nonanalytic point in a thermodynamic function is found to have some maximization procedure at its origin. These results may foster insight into the mechanisms which lead to the occurrence of a phase transition, and thus may help to explore the origin of this physical phenomenon.

Partial equivalence of statistical ensembles and kinetic energy

The phenomenon of partial equivalence of statistical ensembles is illustrated by discussing two examples, the mean-field XY and the mean-field spherical model. The configurational parts of these systems exhibit partial equivalence of the microcanonical and the canonical ensemble. Furthermore, the configurational microcanonical entropy is a smooth function, whereas a nonanalytic point of the configurational free energy indicates the presence of a phase transition in the canonical ensemble. In the presence of a standard kinetic energy contribution, partial equivalence is removed and a nonanalyticity arises also microcanonically. Hence in contrast to the common belief, kinetic energy, even though a quadratic form in the momenta, has a nontrivial effect on the thermodynamic behaviour. As a by-product we present the microcanonical solution of the mean-field spherical model with kinetic energy for finite and infinite system sizes.

Is computation reversible?

Recent studies have suggested that computation is essentially reversible, provided no information is lost. This is a consequence of Landauer’s principle which only requires energy expenditure and entropy increase for information deletion. In this paper we propose to treat information as being intrinsic to points of non-analyticity, so that the movement of information is always associated with the dissipation of heat. This allows us to construct a theory consistent with causality, and the second law of thermodynamics. Since computation requires the movement of information bits through finite volume gates, energy is dissipated even when information is not destroyed, thus indicating that computation is fundamentally non-reversible.