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Physical review. B, Condensed matter

Proof of the nonexistence of (formal) phase transitions in polaron systems. I

B. Gerlach, Hartmut Löwen

Published Mar 15, 1987• 63 citations
Publisher

Abstract

In recent literature it was frequently claimed that free polarons should undergo a (formal) phase transition from a mobile to a localized state, if the electron-phonon coupling constant \ensuremath{\alpha} exceeds some critical value ${\ensuremath{\alpha}}_{c}$. For \ensuremath{\alpha}=${\ensuremath{\alpha}}_{c}$, the formal free energy should be nonanalytic in \ensuremath{\alpha}. In this paper we prove that for optical polarons no such transition exists for finite temperatures. To do so, we use functional-integration techniques and discuss perturbation theory in \ensuremath{\alpha} to infinite order.

DOI
10.1103/physrevb.35.4291

Related papers

Proof of the nonexistence of (formal) phase transitions in polaron systems. II

B. Gerlach, Hartmut Löwen

In recent literature it was frequently claimed that free polarons should undergo a (formal) phase transition from a mobile to a localized state, if the electron-phonon coupling parameter \ensuremath{\alpha} exceeds some critical value ${\ensuremath{\alpha}}_{c}$. For \ensuremath{\alpha}=${\ensuremath{\alpha}}_{c}$, the formal free energy should be nonanalytic in \ensuremath{\alpha}. In this paper we prove that for nonoptical polarons no such transition exists for finite temperatures. Our results can be generalized to the case of lower spatial dimensions as well as to polarons in homogeneous electromagnetic fields. We include some comments on related problems; in particular, the partition function of an electron moving in a Gaussian random potential is considered.

Physical review. B, Condensed matter•Mar 15, 1987•29 citations

The optical polaron bound in a Coulomb potential and its phase diagram

Naoaki Tokuda, H Shoji, K Yoneya

The optical polaron bounded in a Coulomb potential is investigated within the modified variational scheme of the Lee-Low-Pines theory (1953) which is valid for the whole range of the electron-phonon coupling. The ground state energy, the mean number of phonons in the cloud around the electron and the spatial localisation of the electron for the optical polaron change abruptly at certain values of the electron-phonon coupling constant alpha and of the Coulomb potential beta . The polaron changes its state according to the values of the parameters alpha and beta . Such features are summarised in the phase diagram.

Journal of Physics C Solid State Physics•Oct 20, 1981•19 citations

Localization of acoustic polarons at low temperatures: A path-integral Monte Carlo approach

Riccardo Fantoni

We calculate the low-temperature properties of an acoustic polaron in three dimensions in thermal equilibrium at a given temperature using a specialized path-integral Monte Carlo method. In particular we find numerical evidence that the chosen Hamiltonian for the acoustic polaron describes a phase transition from a localized state to an unlocalized state for the electron as the phonon-electron coupling constant decreases. The phase transition manifests itself with a jump discontinuity in the potential energy as a function of the coupling constant. In the weak coupling regime the electron is in an extended state whereas in the strong coupling regime it is found in a self-trapped state.

arXiv:1209.5975•Physical Review B•Oct 8, 2012•18 citations

On the Existence of a Phase Transition for the Fröhlich Polaron

François Maria Peeters, J. T. Devreese

Abstract The ground state energy and its first and second derivative with respect to the electron‐phonon coupling are calculated for the polaron theories of Feynman, Lépine and Matz, Mańka, Pekar, and the Gaussian approximation. It is suggested that the phase transition, from the mobile to the localized polaron state, found by some authors is a feature of their approximation rather than an intrinsic property of the Fröhlich Hamiltonian.

physica status solidi (b)•Jul 1, 1982•68 citations

Holstein polaron

J. Bonča, S. A. Trugman, I. Batistić

We describe a variational method to solve the Holstein model for an electron coupled to dynamical, quantum phonons on an infinite lattice. The variational space can be systematically expanded to achieve high accuracy with modest computational resources (12-digit accuracy for the one-dimensional polaron energy at intermediate coupling). We compute ground-state and low-lying excited-state properties of the model at continuous values of the wave vector k in essentially all parameter regimes. Our results for the polaron energy band, effective mass, and correlation functions compare favorably with those of other numerical techniques, including the density-matrix renormalization-group technique, the global-local method, and the exact diagonalization technique. We find a phase transition for the first excited state between a bound and unbound system of a polaron and an additional phonon excitation. The phase transition is also treated in strong-coupling perturbation theory.

arXiv:cond-mat/9812252•Physical review. B, Condensed matter•Jul 15, 1999•281 citations

Statistical properties of polarons in a magnetic field. II. Numerical results and discussion on the phase transition

François Maria Peeters, J. T. Devreese

A detailed numerical analysis is made of the analytic results presented in paper I. Numerical results are presented for the mass of the Feynman polaron, parallel (${M}_{\ensuremath{\parallel}}$) and perpendicular (${M}_{\ensuremath{\perp}}$) to the magnetic field, and for the following thermodynamic quantities: the magnetization, the susceptibility, the internal energy, the entropy, and the specific heat. Those quantities are studied for different values of the electron-phonon coupling ($\ensuremath{\alpha}$), temperature ($T$), and magnetic field strength ($\mathcal{H}$). We found that an ideal gas of polarons undergoes a phase transition. In the physical parameter space ($\frac{1}{\ensuremath{\alpha}},\frac{1}{\mathcal{H}},T$) the points of first-order phase transition define a two-dimensional surface which is circumscribed by a line of second-order phase transitions. At the transition point the polaron transforms in the direction perpendicular to the magnetic field, and with increasing magnetic field strength, from a polaron state (${M}_{\ensuremath{\parallel}}\ensuremath{\approx}{M}_{\ensuremath{\perp}}$) to an almost free Landau state (${M}_{\ensuremath{\parallel}}>>{M}_{\ensuremath{\perp}}\ensuremath{\sim}1$). This transition can be viewed as a magnetic-field-induced two-dimensional stripping of the polaron. The experimental consequences of this phase transition on the thermodynamic quantities and the magneto-optical absorption spectrum are discussed.

Physical review. B, Condensed matter•Jun 15, 1982•58 citations
In recent literature it was frequently claimed that free polarons should undergo a (formal) phase transition from a mobile to a localized state, if the electron-phonon coupling constant \ensuremath{\alpha} exceeds some critical value \ensuremathαc{\ensuremath{\alpha}}_{c}\ensuremathαc​. For \ensuremath{\alpha}=\ensuremathαc{\ensuremath{\alpha}}_{c}\ensuremathαc​, the formal free energy should be nonanalytic in \ensuremath{\alpha}. In this paper we prove that for optical polarons no such transition exists for finite temperatures. To do so, we use functional-integration techniques and discuss perturbation theory in \ensuremath{\alpha} to infinite order.
In recent literature it was frequently claimed that free polarons should undergo a (formal) phase transition from a mobile to a localized state, if the electron-phonon coupling parameter \ensuremath{\alpha} exceeds some critical value ${\ensuremath{\alpha}}_{c}$. For \ensuremath{\alpha}=${\ensuremath{\alpha}}_{c}$, the formal free energy should be nonanalytic in \ensuremath{\alpha}. In this paper we prove that for nonoptical polarons no such transition exists for finite temperatures. Our results can be generalized to the case of lower spatial dimensions as well as to polarons in homogeneous electromagnetic fields. We include some comments on related problems; in particular, the partition function of an electron moving in a Gaussian random potential is considered.
The optical polaron bounded in a Coulomb potential is investigated within the modified variational scheme of the Lee-Low-Pines theory (1953) which is valid for the whole range of the electron-phonon coupling. The ground state energy, the mean number of phonons in the cloud around the electron and the spatial localisation of the electron for the optical polaron change abruptly at certain values of the electron-phonon coupling constant alpha and of the Coulomb potential beta . The polaron changes its state according to the values of the parameters alpha and beta . Such features are summarised in the phase diagram.
We calculate the low-temperature properties of an acoustic polaron in three dimensions in thermal equilibrium at a given temperature using a specialized path-integral Monte Carlo method. In particular we find numerical evidence that the chosen Hamiltonian for the acoustic polaron describes a phase transition from a localized state to an unlocalized state for the electron as the phonon-electron coupling constant decreases. The phase transition manifests itself with a jump discontinuity in the potential energy as a function of the coupling constant. In the weak coupling regime the electron is in an extended state whereas in the strong coupling regime it is found in a self-trapped state.
We describe a variational method to solve the Holstein model for an electron coupled to dynamical, quantum phonons on an infinite lattice. The variational space can be systematically expanded to achieve high accuracy with modest computational resources (12-digit accuracy for the one-dimensional polaron energy at intermediate coupling). We compute ground-state and low-lying excited-state properties of the model at continuous values of the wave vector k in essentially all parameter regimes. Our results for the polaron energy band, effective mass, and correlation functions compare favorably with those of other numerical techniques, including the density-matrix renormalization-group technique, the global-local method, and the exact diagonalization technique. We find a phase transition for the first excited state between a bound and unbound system of a polaron and an additional phonon excitation. The phase transition is also treated in strong-coupling perturbation theory.
A detailed numerical analysis is made of the analytic results presented in paper I. Numerical results are presented for the mass of the Feynman polaron, parallel (${M}_{\ensuremath{\parallel}}$) and perpendicular (${M}_{\ensuremath{\perp}}$) to the magnetic field, and for the following thermodynamic quantities: the magnetization, the susceptibility, the internal energy, the entropy, and the specific heat. Those quantities are studied for different values of the electron-phonon coupling ($\ensuremath{\alpha}$), temperature (TTT), and magnetic field strength (H\mathcal{H}H). We found that an ideal gas of polarons undergoes a phase transition. In the physical parameter space ($\frac{1}{\ensuremath{\alpha}},\frac{1}{\mathcal{H}},T$) the points of first-order phase transition define a two-dimensional surface which is circumscribed by a line of second-order phase transitions. At the transition point the polaron transforms in the direction perpendicular to the magnetic field, and with increasing magnetic field strength, from a polaron state (${M}_{\ensuremath{\parallel}}\ensuremath{\approx}{M}_{\ensuremath{\perp}}$) to an almost free Landau state (${M}_{\ensuremath{\parallel}}>>{M}_{\ensuremath{\perp}}\ensuremath{\sim}1$). This transition can be viewed as a magnetic-field-induced two-dimensional stripping of the polaron. The experimental consequences of this phase transition on the thermodynamic quantities and the magneto-optical absorption spectrum are discussed.