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Physical review. B, Condensed matter

Ground state of a two-level system with phonon coupling

Vassilios Fessatidis, Jay D. Mancini, William J. Massano, Samuel P. Bowen

Published Feb 1, 2000• 7 citations
Publisher

Abstract

The ground state of a two-level system coupled to a dispersionless phonon bath is studied using both a connected moments expansion and a truncated Lanczos tridiagonal scheme. We consider the spin-boson Hamiltonian, $\ifmmode \hat{H}\else \^{H}\fi{}=\ensuremath{-}{\ensuremath{\delta}}_{o}{\ensuremath{\sigma}}_{x}+{\ensuremath{\sum}}_{\mathbf{k}}\ensuremath{\Elzxh}{\ensuremath{\omega}}_{\mathbf{k}}{a}_{\mathbf{k}}^{\ifmmode\dagger\else\textdagger\fi{}}{a}_{\mathbf{k}}+{\ensuremath{\sum}}_{\mathbf{k}}{g}_{\mathbf{k}}{(a}_{\mathbf{k}}^{\ifmmode\dagger\else\textdagger\fi{}}{+a}_{\mathbf{k}}){\ensuremath{\sigma}}_{z},$ where ${\ensuremath{\delta}}_{0}$ is the bare tunneling matrix element and ${g}_{\mathbf{k}}$ represents the coupling to the $\mathbf{k}$ phonon modes. Such systems have found relevance in applications to molecular polaron formation, exciton motion, and attenuation of sound in glasses. Our results are then compared to those of variational methods as well as an exact numerical diagonalization.

DOI
10.1103/physrevb.61.3184

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Physical review. B, Condensed matter•Apr 1, 1983•236 citations
The ground state of a two-level system coupled to a dispersionless phonon bath is studied using both a connected moments expansion and a truncated Lanczos tridiagonal scheme. We consider the spin-boson Hamiltonian, \ifmmodeH^\elseHˆ\fi=\ensuremath−\ensuremathδo\ensuremathσx+\ensuremath∑k\ensuremath\Elzxh\ensuremathωkak\ifmmode†\else\textdagger\fiak+\ensuremath∑kgk(ak\ifmmode†\else\textdagger\fi+ak)\ensuremathσz,\ifmmode \hat{H}\else \^{H}\fi{}=\ensuremath{-}{\ensuremath{\delta}}_{o}{\ensuremath{\sigma}}_{x}+{\ensuremath{\sum}}_{\mathbf{k}}\ensuremath{\Elzxh}{\ensuremath{\omega}}_{\mathbf{k}}{a}_{\mathbf{k}}^{\ifmmode\dagger\else\textdagger\fi{}}{a}_{\mathbf{k}}+{\ensuremath{\sum}}_{\mathbf{k}}{g}_{\mathbf{k}}{(a}_{\mathbf{k}}^{\ifmmode\dagger\else\textdagger\fi{}}{+a}_{\mathbf{k}}){\ensuremath{\sigma}}_{z},\ifmmodeH^\elseHˆ\fi=\ensuremath−\ensuremathδo​\ensuremathσx​+\ensuremath∑k​\ensuremath\Elzxh\ensuremathωk​ak\ifmmode†\else\textdagger\fi​ak​+\ensuremath∑k​gk​(ak\ifmmode†\else\textdagger\fi​+ak​)\ensuremathσz​, where \ensuremathδ0{\ensuremath{\delta}}_{0}\ensuremathδ0​ is the bare tunneling matrix element and gk{g}_{\mathbf{k}}gk​ represents the coupling to the k\mathbf{k}k phonon modes. Such systems have found relevance in applications to molecular polaron formation, exciton motion, and attenuation of sound in glasses. Our results are then compared to those of variational methods as well as an exact numerical diagonalization.
A new variational approach is proposed to study the ground-state of a two-level system coupled to a dispersionless phonon bath. By the extended coherent state, where the more phonon correlations are easily incorporated, we can obtain very accurate ground state energy and the tunnelling reduction factor in all regime of tunnelling matrix element d0 and coupling parameter s. The relative difference between the present ones and those by exact numerical diagonalization is less then 0.001%. In addition, some simple analytical results are given in the limits of δ 0 /s ↠ 0 and ∞.
The ground state of a two-level system coupled to a dispersionless phonon bath is studied using the coherent approximation method. The ground-state energy and the tunneling reduction factor we obtain are in good agreement with the exact values in the whole parameter space. Since the real ground-state wave functions are well simulated, some other physical information about the system, such as the interaction energy and the interaction phonon numbers in each level and the whole system, are given. The interaction phonon numbers in the two-level system are found to increase with the coupling parameter s. In the intermediate coupling region, phenomenon of the population inversion and a maximum for the interaction phonon numbers versus ${\ensuremath{\delta}}_{0}/s$ have been found. Consisting with the exact results, our results show there is no evidence of the discontinuous localization-delocalization transition.
The thermodynamics of a two-level system coupled to a boson field is investigated by means of a functional-integral method. The model can be used to describe the motion of a small polaron between two sites on a molecule, or a spin-1/2 magnetic moment coupled to a phonon bath. The model is also relevant to the problem of dissipation in macroscopic quantum phenomena. We use a functional-integral method to investigate a two-level electronic system coupled linearly to a harmonic oscillator. The density matrix is cast as a functional integral over paths obeying one-sided boundary conditions at either 0 or \ensuremath{\beta}=1/kB{\mathit{k}}_{\mathit{B}}kB​T. The integration variables are c numbers for the oscillator and Grassmann numbers for the electrons. The oscillator variables are integrated out, leaving an effective density matrix for the electrons containing retarded interactions. The partition function is then calculated up to second order of an expansion about the exact fermionic classical paths. Then, the free energy, specific heat, and average hopping energy are obtained. We find that the most significant reduction in the hopping energy is manifested at intermediate temperatures.
While the concept of the entanglement spectrum has heretofore been utilized to address various many-body systems, the models describing an itinerant spinless-fermion excitation coupled to zero-dimensional bosons (e.g. dispersionless phonons) have as yet not received much attention in this regard. To fill this gap, the ground-state entanglement spectrum of a model that includes two of the most common types of short-ranged, nonlocal excitation-phonon interaction -- the Peierls- and breathing-mode couplings -- is numerically evaluated here. This model displays a sharp, level-crossing transition at a critical coupling strength, which signifies the change from a nondegenerate ground state at the quasimomentum Kgs=0K_{\textrm{gs}}=0Kgs​=0 to a twofold-degenerate one corresponding to a symmetric pair of nonzero quasimomenta. Another peculiarity of this model is that in the special case of equal Peierls- and breathing-mode coupling strengths the bare-excitation Bloch state with the quasimomentum 000 or πππ is its exact eigenstate. Moreover, below a critical coupling strength this state is the ground state of the model. Thus, the sharp transition between a bare excitation and a heavily phonon-dressed (polaronic) one can be thought of as a transition between vanishing and finite entanglement. It is demonstrated here that the smallest ground-state entanglement-spectrum eigenvalue to a large extent mimics the behavior of the entanglement entropy itself and vanishes in this special case of the model; by contrast, all the remaining eigenvalues diverge in this case. The implications of excitation-phonon entanglement for WWW-state engineering in superconducting and neutral-atom-based qubit arrays serving as analog simulators of this model are also discussed.
We study the nature of the ground state of a one-dimensional electron-phonon model for molecular crystals: The phonons are assumed dispersionless and couple to the local electronic density. We consider the half-filled-band sector and discuss the stability of the Peierls-dimerized ground state as a function of the phonon frequency ($\ensuremath{\omega}$), electron-phonon coupling constant ($\ensuremath{\lambda}$), and number of components of the electron spin (nnn). First, we discuss the properties of the model in the limiting cases of zero frequency and infinite frequency. We then perform a strong coupling expansion which maps the system onto a spinless fermion model with nearest-neighbor repulsion for both spinless and spin-\textonehalf{} electrons, but with different parameters in both cases. Finally, we perform a numerical study of the system using a Monte Carlo technique. We study the behavior of the order parameter and of correlation functions for various points in parameter space. We also perform a finite-size---scaling analysis of the numerical data. The conclusions of our study are the following: For the case of spinless electrons, there exists always a disordered phase for small coupling constant, and the system undergoes an infinite-order transition to a Peierls-dimerized state as the coupling constant increases beyond a critical value. The phase diagram is divided between a disordered and an ordered region by a line that connects the points ($\ensuremath{\omega}=0$, $\ensuremath{\lambda}=0$) and ($\ensuremath{\omega}=\ensuremath{\infty}$, $\ensuremath{\lambda}=\ensuremath{\infty}$). For the case of spin-\textonehalf{} electrons, the system is dimerized for arbitrary $\ensuremath{\lambda}$ and $\ensuremath{\omega}$ except in the limit $\ensuremath{\omega}=\ensuremath{\infty}$.