On the tunneling perturbation of the discrete spectrum

We calculate perturbatively the tunneling decay rate $\ensuremath{\Gamma}$ of the metastable phase in the quantum $d=1$ Ising model in a skew magnetic field near the coexistence line $0<{h}_{x}<1, {h}_{z}\ensuremath{\rightarrow}\ensuremath{-}0$ at $T=0.$ It is shown that $\ensuremath{\Gamma}$ oscillates in the magnetic field ${h}_{z}$ due to discreteness of the excitation energy spectrum. After mapping of the obtained results onto the extreme anisotropic $d=2$ Ising model at $T<{T}_{c},$ we verify in the latter model the droplet theory predictions for the free energy analytically continued to the metastable phase. We also find evidence for the discrete-lattice corrections in this metastable phase free energy.We calculate perturbatively the tunneling decay rate $\ensuremath{\Gamma}$ of the metastable phase in the quantum $d=1$ Ising model in a skew magnetic field near the coexistence line $0<{h}_{x}<1, {h}_{z}\ensuremath{\rightarrow}\ensuremath{-}0$ at $T=0.$ It is shown that $\ensuremath{\Gamma}$ oscillates in the magnetic field ${h}_{z}$ due to discreteness of the excitation energy spectrum. After mapping of the obtained results onto the extreme anisotropic $d=2$ Ising model at $T<{T}_{c},$ we verify in the latter model the droplet theory predictions for the free energy analytically continued to the metastable phase. We also find evidence for the discrete-lattice corrections in this metastable phase free energy.

arXiv:cond-mat/9904059Physical review. B, Condensed matterDec 1, 199949 citations

Instanton and noninstanton tunneling in periodically perturbed barriers: Semiclassical and quantum interpretations

Kin’ya Takahashi, Kensuke S. Ikeda

In multidimensional barrier tunneling, there exist two different types of tunneling mechanisms, instanton-type tunneling and noninstanton tunneling. In this paper we investigate transitions between the two tunneling mechanisms from the semiclassical and quantum viewpoints taking two simple models: a periodically perturbed Eckart barrier for the semiclassical analysis and a periodically perturbed rectangular barrier for the quantum analysis. As a result, similar transitions are observed with change of the perturbation frequency ω for both systems, and we obtain a comprehensive scenario from both semiclassical and quantum viewpoints for them. In the middle range of ω, in which the plateau spectrum is observed, noninstanton tunneling dominates the tunneling process, and the tunneling amplitude takes the maximum value. Noninstanton tunneling explained by stable-unstable manifold guided tunneling (SUMGT) from the semiclassical viewpoint is interpreted as multiphoton-assisted tunneling from the quantum viewpoint. However, in the limit ω→0, instanton-type tunneling takes the place of noninstanton tunneling, and the tunneling amplitude converges on a constant value depending on the perturbation strength. The spectrum localized around the input energy is observed, and there is a scaling law with respect to the width of the spectrum envelope, i.e., the width ∝ℏω. In the limit ω→∞, the tunneling amplitude converges on that of the unperturbed system, i.e., the instanton of the unperturbed system.In multidimensional barrier tunneling, there exist two different types of tunneling mechanisms, instanton-type tunneling and noninstanton tunneling. In this paper we investigate transitions between the two tunneling mechanisms from the semiclassical and quantum viewpoints taking two simple models: a periodically perturbed Eckart barrier for the semiclassical analysis and a periodically perturbed rectangular barrier for the quantum analysis. As a result, similar transitions are observed with change of the perturbation frequency ω for both systems, and we obtain a comprehensive scenario from both semiclassical and quantum viewpoints for them. In the middle range of ω, in which the plateau spectrum is observed, noninstanton tunneling dominates the tunneling process, and the tunneling amplitude takes the maximum value. Noninstanton tunneling explained by stable-unstable manifold guided tunneling (SUMGT) from the semiclassical viewpoint is interpreted as multiphoton-assisted tunneling from the quantum viewpoint. However, in the limit ω→0, instanton-type tunneling takes the place of noninstanton tunneling, and the tunneling amplitude converges on a constant value depending on the perturbation strength. The spectrum localized around the input energy is observed, and there is a scaling law with respect to the width of the spectrum envelope, i.e., the width ∝ℏω. In the limit ω→∞, the tunneling amplitude converges on that of the unperturbed system, i.e., the instanton of the unperturbed system.

Physical Review ENov 13, 201211 citations

Problems with Discrete Energy Spectrum

WORLD SCIENTIFIC eBooksSep 27, 20160 citations

We calculate perturbatively the tunneling decay rate $\ensuremath{\Gamma}$ of the metastable phase in the quantum

d=1

Ising model in a skew magnetic field near the coexistence line $0<{h}_{x}<1, {h}_{z}\ensuremath{\rightarrow}\ensuremath{-}0$ at

T=0.

It is shown that $\ensuremath{\Gamma}$ oscillates in the magnetic field

{h}_{z}

due to discreteness of the excitation energy spectrum. After mapping of the obtained results onto the extreme anisotropic

d=2

Ising model at

T<{T}_{c},

we verify in the latter model the droplet theory predictions for the free energy analytically continued to the metastable phase. We also find evidence for the discrete-lattice corrections in this metastable phase free energy.