Physical review. B, Condensed matter

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

B. Gerlach, Hartmut Löwen

Published Mar 15, 1987• 29 citations

Publisher

Abstract

DOI: 10.1103/physrevb.35.4297

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

B. Gerlach, Hartmut Löwen

Physical review. B, Condensed matterMar 15, 198763 citations

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

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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.

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Moving acoustical-mode polarons

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Progress of Theoretical PhysicsApr 1, 19615 citations

Physical review. B, Condensed matter

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

B. Gerlach, Hartmut Löwen

Published Mar 15, 1987• 29 citations

Publisher

Abstract

{\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.

DOI: 10.1103/physrevb.35.4297

Related papers

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

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 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.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.

Physical review. B, Condensed matterMar 15, 198763 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.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 matterJun 15, 198258 citations

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François Maria Peeters, J. T. Devreese

physica status solidi (b)Jul 1, 198268 citations

Moving acoustical-mode polarons

John Thomchick

A general class of acoustical-phonon mode induced-polaron systems is considered. The electron-phonon interaction for these systems is described by a Fr\"ohlich-interaction Hamiltonian involving Fourier components proportional to ${q}^{\ensuremath{\gamma}}$, where $\ensuremath{\gamma}$ is an arbitrary real number. The energy versus total momentum ($E\ensuremath{-}P$) relations for these polarons are investigated. At weak coupling, in the context of the intermediate coupling theory, it is found that when $\ensuremath{-}1<\ensuremath{\gamma}\ensuremath{\le}\ensuremath{-}\frac{1}{2}$ a linear structure is present in the $E\ensuremath{-}P$ relations. As $\ensuremath{\gamma}$ decreases from -1/2 to -1, this linear structure becomes apparent only for larger and larger values of $P$ . For $\ensuremath{\gamma}>\ensuremath{-}\frac{1}{2}$, this linear structure is not present, but rather the $E\ensuremath{-}P$ relations cut off at a critical value of $P$ . When $\ensuremath{-}1<\ensuremath{\gamma}\ensuremath{\le}0$, the polaron mass for these systems grows large as the polaron velocity $v$ approaches the speed of sound $s$ , but for $\ensuremath{\gamma}>0$ the polaron mass has a finite value as $v\ensuremath{\rightarrow}s$. As a function of $\ensuremath{\gamma}$, the self-energy reaches a maximum value for $\ensuremath{\gamma}\ensuremath{\approx}\ensuremath{-}0.73$ and the $v\ensuremath{\rightarrow}0$ polaron mass reaches a minimum value for $\ensuremath{\gamma}\ensuremath{\approx}\ensuremath{-}0.25$, independent of coupling strength. From a strong-coupling theory appropriate for moving polarons it is found that a linear structure in the $E\ensuremath{-}P$ relations is obtained for all $\ensuremath{\gamma}>\ensuremath{-}1$. At both weak and strong coupling it is unclear that the $E\ensuremath{-}P$ relations are meaningful when $\ensuremath{\gamma}\ensuremath{\le}\ensuremath{-}1$.A general class of acoustical-phonon mode induced-polaron systems is considered. The electron-phonon interaction for these systems is described by a Fr\"ohlich-interaction Hamiltonian involving Fourier components proportional to ${q}^{\ensuremath{\gamma}}$, where $\ensuremath{\gamma}$ is an arbitrary real number. The energy versus total momentum ($E\ensuremath{-}P$) relations for these polarons are investigated. At weak coupling, in the context of the intermediate coupling theory, it is found that when $\ensuremath{-}1<\ensuremath{\gamma}\ensuremath{\le}\ensuremath{-}\frac{1}{2}$ a linear structure is present in the $E\ensuremath{-}P$ relations. As $\ensuremath{\gamma}$ decreases from -1/2 to -1, this linear structure becomes apparent only for larger and larger values of $P$. For $\ensuremath{\gamma}>\ensuremath{-}\frac{1}{2}$, this linear structure is not present, but rather the $E\ensuremath{-}P$ relations cut off at a critical value of $P$. When $\ensuremath{-}1<\ensuremath{\gamma}\ensuremath{\le}0$, the polaron mass for these systems grows large as the polaron velocity $v$ approaches the speed of sound $s$, but for $\ensuremath{\gamma}>0$ the polaron mass has a finite value as $v\ensuremath{\rightarrow}s$. As a function of $\ensuremath{\gamma}$, the self-energy reaches a maximum value for $\ensuremath{\gamma}\ensuremath{\approx}\ensuremath{-}0.73$ and the $v\ensuremath{\rightarrow}0$ polaron mass reaches a minimum value for $\ensuremath{\gamma}\ensuremath{\approx}\ensuremath{-}0.25$, independent of coupling strength. From a strong-coupling theory appropriate for moving polarons it is found that a linear structure in the $E\ensuremath{-}P$ relations is obtained for all $\ensuremath{\gamma}>\ensuremath{-}1$. At both weak and strong coupling it is unclear that the $E\ensuremath{-}P$ relations are meaningful when $\ensuremath{\gamma}\ensuremath{\le}\ensuremath{-}1$.

Physical review. B, Condensed matterFeb 15, 19783 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.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:1209.5975Physical Review BOct 8, 201218 citations

Theory of Polaron Mobility

Yukio Ôsaka

The polaron mobility is calculated by making use of the general theory of electrical conductivity. We take the states determined by Feynman's trial action as the unperturbed states and treat the difference between the true action and the trial action as a perturbation. Numerical values of the polaron mobility at very .low temperatures are given and are dis· cussed in comparison with the results obtained by Shultz and .by Morita. § I. Introduction We have investigated the static properties of polaron at finite temperatures in the previous paper1> (hereafter referred to as I). We shall investigB;te the polaron mobility in the present paper. In the same way as in I, we calculate the polaron mobility by using Feynman's path-integral method. In the polaron problem, the interaction between an electron and lattice vibrations is so large that the usual perturbation theoretic treatment fails. Therefore the calculation of the mobility in the present paper will be based on the general theory of electrical conductivity recently developed by many authors,2 > which is applicable to a system which does not allow us to set up the Boltzmann equation. We take the states determined by Feynman's trial action as unperturbed states and treat the difference between the trial action and the true action as a perturba tion which yields a decay of electronic current correlation. The following assumptions will be made in the present paper. i) The electrical conductivity is determined by the asymptotic form of a correlation function of electronic current. ii) This asymptotic form has the property of exponential decay in time. These assumptions are valid in the case of weak interaction. It is not sure whether or not they are applicable to the present problem which includes the case of strong interaction. In the present treat ment, however, ~he states determined by Feynman's trial action instead of the states of a free electron are chosen as the unperturbed states. It is expected that the perturbation in the former case may be made weaker than in the latter case and that the above mentioned assumptions may be valid. In fact the self energy of polaron at 0°K calculated by using these assumptions agrees with that obtained by Feynman3> as will be shown _in § 3. Therefore, the approxiThe polaron mobility is calculated by making use of the general theory of electrical conductivity. We take the states determined by Feynman's trial action as the unperturbed states and treat the difference between the true action and the trial action as a perturbation. Numerical values of the polaron mobility at very .low temperatures are given and are dis· cussed in comparison with the results obtained by Shultz and .by Morita. § I. Introduction We have investigated the static properties of polaron at finite temperatures in the previous paper1> (hereafter referred to as I). We shall investigB;te the polaron mobility in the present paper. In the same way as in I, we calculate the polaron mobility by using Feynman's path-integral method. In the polaron problem, the interaction between an electron and lattice vibrations is so large that the usual perturbation theoretic treatment fails. Therefore the calculation of the mobility in the present paper will be based on the general theory of electrical conductivity recently developed by many authors,2 > which is applicable to a system which does not allow us to set up the Boltzmann equation. We take the states determined by Feynman's trial action as unperturbed states and treat the difference between the trial action and the true action as a perturba tion which yields a decay of electronic current correlation. The following assumptions will be made in the present paper. i) The electrical conductivity is determined by the asymptotic form of a correlation function of electronic current. ii) This asymptotic form has the property of exponential decay in time. These assumptions are valid in the case of weak interaction. It is not sure whether or not they are applicable to the present problem which includes the case of strong interaction. In the present treat ment, however, ~he states determined by Feynman's trial action instead of the states of a free electron are chosen as the unperturbed states. It is expected that the perturbation in the former case may be made weaker than in the latter case and that the above mentioned assumptions may be valid. In fact the self energy of polaron at 0°K calculated by using these assumptions agrees with that obtained by Feynman3> as will be shown _in § 3. Therefore, the approxi

Progress of Theoretical PhysicsApr 1, 19615 citations

{\ensuremath{\alpha}}_{c}

. For \ensuremath{\alpha}=

{\ensuremath{\alpha}}_{c}

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.

P

. For $\ensuremath{\gamma}>\ensuremath{-}\frac{1}{2}$, this linear structure is not present, but rather the $E\ensuremath{-}P$ relations cut off at a critical value of

P

. When $\ensuremath{-}1<\ensuremath{\gamma}\ensuremath{\le}0$, the polaron mass for these systems grows large as the polaron velocity

v

approaches the speed of sound

s

, but for $\ensuremath{\gamma}>0$ the polaron mass has a finite value as $v\ensuremath{\rightarrow}s$. As a function of $\ensuremath{\gamma}$, the self-energy reaches a maximum value for $\ensuremath{\gamma}\ensuremath{\approx}\ensuremath{-}0.73$ and the $v\ensuremath{\rightarrow}0$ polaron mass reaches a minimum value for $\ensuremath{\gamma}\ensuremath{\approx}\ensuremath{-}0.25$, independent of coupling strength. From a strong-coupling theory appropriate for moving polarons it is found that a linear structure in the $E\ensuremath{-}P$ relations is obtained for all $\ensuremath{\gamma}>\ensuremath{-}1$. At both weak and strong coupling it is unclear that the $E\ensuremath{-}P$ relations are meaningful when $\ensuremath{\gamma}\ensuremath{\le}\ensuremath{-}1$.