Quantum sine-Gordon Equation Research Articles

Quantum simulation of expanding space–time with tunnel-coupled condensates

We consider two weakly interacting quasi-1D condensates of cold bosonic atoms. It turns out that a time-dependent variation of the tunnel-coupling between those condensates is equivalent to the spatial expansion of a one-dimensional toy-Universe, with regard to the dynamics of the relative phase field. The dynamics of this field is governed by the quantum sine-Gordon equation. Thus, this analogy could be used to ‘quantum simulate’ the dynamics of a scalar, interacting quantum field on an expanding background. We discuss how to observe the ‘freezing’ of quantum fluctuations during an accelerating expansion in a possible experiment. We also analyze an experimental protocol to study the formation of sine-Gordon breathers in the relative phase field, seeded by quantum fluctuations.

Localized Phase Structures Growing Out of Quantum Fluctuations in a Quench of Tunnel-coupled Atomic Condensates

We investigate the relative phase between two weakly interacting 1D condensates of bosonic atoms after suddenly switching on the tunnel coupling. The following phase dynamics is governed by the quantum sine-Gordon equation. In the semiclassical limit of weak interactions, we observe the parametric amplification of quantum fluctuations leading to the formation of breathers with a finite lifetime. The typical lifetime and density of these "quasibreathers" are derived employing exact solutions of the classical sine-Gordon equation. Both depend on the initial relative phase between the condensates, which is considered as a tunable parameter.

Collective rotational tunneling of methyl groups and quantum solitons in 4-methylpyridine: Neutron scattering studies of single crystals

The structure of the 4-methylpyridine crystal has been determined at 10 and 260 K with the single-crystal neutron-diffraction technique. The space-group symmetries are ${I4}_{1}/a$ and ${I4}_{1}/amd,$ respectively. In both cases, there are eight molecular entities in the unit cell. The rotational axes of the methyl groups are aligned along the c axis. The shortest intermolecular distances occur between face-to-face methyl groups. The next shortest distances correspond to infinite chains of rotors parallel to the orthogonal axes a and b. The angular probability densities of the methyl groups are clear evidence of orientational disorder. The incoherent scattering function ${S(Q}_{a}{,Q}_{b},\ensuremath{\omega})$ has been measured with the inelastic neutron scattering technique at 1.7 K. The energy-transfer ranges, $\ensuremath{\Elzxh}\ensuremath{\omega}=\ifmmode\pm\else\textpm\fi{}(500\ifmmode\pm\else\textpm\fi{}60)\ensuremath{\mu}\mathrm{eV},$ encompass transitions due to rotational tunneling. The anisotropy in $\mathbf{Q}$ is distinctive of dynamics in one dimension. These are represented with the quantum sine-Gordon equation that is an approximation to the Hamiltonian for an infinite chain of coupled rotors. Spots of intensity observed for neutron-energy loss at $|{Q}_{a}|$ or $|{Q}_{b}|\ensuremath{\approx}1.55{\mathrm{\AA{}}}^{\ensuremath{-}1}$ are distinctive of stationary states for breathers. Weaker peaks at $|{Q}_{a}|$ or $|{Q}_{b}|\ensuremath{\approx}1.0{\mathrm{\AA{}}}^{\ensuremath{-}1}$ reveal collective tunneling. This is a thermally activated process arising from rather heavy pseudoparticles composed of large numbers (from 22 to 25) of kinks or antikinks. Quantization of the kinetic momentum arises from the chain discreteness and from conservation of the angular momentum. Traveling states are stationary when the kinetic energy is within the tunneling energy band. In the ground state, the chain dynamics are represented with a single collective angular coordinate. The incoherent scattering function for neutron-energy-gain reveals bound excited states with lifetimes of several days.

Sine-Gordon breather form factors and quantum field equations

Using the results of previous investigations on sine-Gordon form factors exact expressions of all breather matrix elements are obtained for several operators: all powers of the fundamental bose field, general exponentials of it, the energy momentum tensor and all higher currents. Formulae for the asymptotic behavior of bosonic form factors are presented which are motivated by Weinberg's power counting theorem in perturbation theory. It is found that the quantum sine-Gordon field equation holds and an exact relation between the ``bare'' mass and the renormalized mass is obtained. Also a quantum version of a classical relation for the trace of the energy momentum is proven. The eigenvalue problem for all higher conserved charges is solved. All results are compared with perturbative Feynman graph expansions and full agreement is found.

The exact quantum sine-Gordon field equation and other non-perturbative results

Using the methods of the “form factor program” exact expressions of all matrix elements are obtained for several operators of the quantum sine Gordon model: all powers of the fundamental bose field, general exponentials of it, the energy momentum tensor and all higher currents. It is found that the quantum sine-Gordon field equation holds with an exact relation between the “bare” mass and the renormalized mass. Also a relation for the trace of the energy momentum is obtained. The eigenvalue problem for all higher conserved charges is solved. All results are compared with Feynman graph expansions and full agreement is found.

Deformation of a Virasoro algebra and the integrals of motion of the quantum sine-Gordon equation

A special deformation of a Virasoro algebra such that the screening operator is not deformed (the space where it operates is deformed) is studied. This deformation leads to a 3-index algebra. The residue of the generating function of the generators of this algebra is a generating function of the integrals of motion for the quantum sine-Gordon model. The algebra of generating functions is calculated. Explicit formulas are presented for the first few integrals of motion.

The discrete quantum pendulum

We study a doubly discrete quantum sine-Gordon equation, which is derived from the quantum Volterra model solved by Faddeev and Volkov [Teor. Mat. Fiz. 92 (1992) 207]. The discrete quantum pendulum is obtained as the simplest special case. The eigenfunctions of its Hamiltonian, displayed as densities on the phase space, are very localized around the classical energy levels.

Critical properties of the quantum sine-Gordon equation.

We have developed a quantum-mechanical approach to study the critical properties of the quantum sine-Gordon equation describing a massless scalar field in 1+1 space-time dimensions with interaction density proportional to cos(${\mathrm{\ensuremath{\beta}}}_{0}$\ensuremath{\varphi}). The main feature of the present theory is to find an appropriate starting point for perturbation expansion, which can directly treat the infrared divergences encountered in the conventional perturbation theory. Exact critical conditions for the phase transition of the model are derived; and the quantum sine-Gordon Hamiltonian is found to correspond exactly to a free-field model. Near the critical regime, a gap will open near zero momentum in the excitation spectrum, and the corresponding ground-state wave function is a pairing quasiparticle state, analogous to the BCS superconducting state. It is proved that this ground state will possess quasi-long-range and quasi-off-diagonal long-range correlations. Comparisons with the Kosterlitz-Thouless classical theory of the two-dimensional Coulomb gas are also made.

Collective rotation of methyl groups in isotopic mixtures of 4-methylpyridine at low temperature. Inelastic neutron scattering spectra

Inelastic neutron scattering spectra in the 500 μeV region of a series of mixtures of totally hydrogenated and totally deuterated 4-methylpyridine (4MP- h 7 and 4MP- d 7) with relative 4MP- h 7 concentrations of 100%, 85%, 65%, 50%, 26%, 20% and 5% are presented at 2.5 K. In pure 4MP- h 7 the spectrum shows three partially resolved bands at 468, 510 and 535 μeV. The weaker peak at 468 and 535 μeV shift to 100 μeV in pure 4MP- d 7. They are assigned to the tunnelling of the methyl groups. The central peak shows a continuous frequency shift with increasing concentration of 4MP- d 7, indicating collective motions of the methyl groups. The methyl group dynamics is quantitatively represented by the quantum sine-Gordon equation describing a one-dimensional infinite chain of coupled methyl groups. The 510 μeV peak is due to excitation of the first quantized travelling state of the breather mode.

Soliton and breather states of the quantum sine-Gordon model in light cone coordinates through the exact QIST method

The quantum sine-Gordon equation in light cone coordinates is solved exactly through the quantum inverse scattering technique by finding a suitable gauge in which the model becomes ultralocal. The existence of breather bound states is shown and a consistent soliton representation is produced. Physical consequences are discussed along with a possible gauge invariance of the system at the quantum level.

One-Dimensional Fluctuations and the Chain-Ordering Transformation inHg3−δAsF6

Neutron-scattering behavior and other properties of the mercury chains in ${\mathrm{Hg}}_{3\ensuremath{-}\ensuremath{\delta}}\mathrm{As}{\mathrm{F}}_{6}$ are calculated from a simple microscopic model. The independent chains satisfy a quantum sine-Gordon equation. The theory accounts for existing observations and makes a number of testable predictions about properties of the disordered phase, the nature of the phase transition, evolution of long-range order, and the excitation spectrum at low temperatures.

Massless quantum sine–Gordon equation in two space–time dimensions: Correlation inequalities and infinite volume limit

We prove new correlation inequalities for the massive and massless quantum sine–Gordon equations. These results are then used to construct infinite volume limit theory for the massless (S–G)2 model that satisfies the Osterwalder–Schrader axioms. As consequences, infinite volume limit theories for the classical, neutral, statistical mechanical systems with two-body Coulomb potentials and for the massive Thirring model exist.

Quantum solitons in one-dimensional conductors

The excitation spectrum in several models of the one-dimensional electron gas is calculated. The models studied are the backward scattering model, charge transfer model, and strong-coupling spinless fermion model, with no restrictions on the interaction strengths. All can exhibit new types of elementary excitations which are identified as quantum solitons and bound solitons. These excitations can participate in transport, and are exponentially activated at low temperatures. The gap energy is calculated for arbitrary coupling strengths in the various models, using a procedure which exactly solves the quantum sine-Gordon equation in one dimension.

Generalized quantum sine-Gordon equation and its relation to the Thirring model in quantum field theory

We discuss a generalization of the conventional sine-Gordon quantum field theory by using methods recently developed by Coleman. As a result we can argue that the equivalence between the sine-Gordon theory and the massive Thirring model is unaffected if we perturb the sine-Gordon Hamiltonian by a bounded perturbation consisting of a continuous sum of sine-Gordon type of interactions.

Quantum sine-Gordon equation as the massive Thirring model

The sine-Gordon equation is the theory of a massless scalar field in one space and one time dimension with interaction density proportional to $cos\ensuremath{\beta}\ensuremath{\phi}$, where $\ensuremath{\beta}$ is a real parameter. I show that if ${\ensuremath{\beta}}^{2}$ exceeds $8\ensuremath{\pi}$, the energy density of the theory is unbounded below; if ${\ensuremath{\beta}}^{2}$ equals $4\ensuremath{\pi}$, the theory is equivalent to the zero-charge sector of the theory of a free massive Fermi field; for other values of $\ensuremath{\beta}$, the theory is equivalent to the zero-charge sector of the massive Thirring model. The sine-Gordon soliton is identified with the fundamental fermion of the Thirring model.