Thermodynamical properties of quantum double-well systems

or 30). Calculated specific heats are shown to be rather different from counterparts of a harmonic oscillator. In particular, specific heats of symmetric double-well systems at very low temperatures have the Schottky-type anomaly, which is rooted to a small energy gap in low-lying two-level eigenstates induced by a tunneling through the potential barrier. The Schottky-type anomaly is removed when an asymmetry is introduced into the double-well potential. It has been pointed out that the specific-heat calculation of a double-well system reported by Feranchuk, Ulyanenkov and Kuz'min [Chem. Phys. 157, 61 (1991)] is misleading because the zeroth-order operator method they adopted neglects crucially important off-diagonal contributions.

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Specific heats of quantum double-well systems

Hideo Hasegawa

Specific heats of quantum systems with symmetric and asymmetric double-well potentials have been calculated. In numerical calculations of their specific heats, we have adopted the combined method, which takes into account not only eigenvalues of ε(n) for 0 ≤ n ≤ N(m) obtained by the energy-matrix diagonalization but also their extrapolated ones for N(m) + 1 ≤ n <∞ (N(m) = 20 or 30). Calculated specific heats are shown to be rather different from counterparts of a harmonic oscillator. In particular, specific heats of symmetric double-well systems at very low temperatures have the Schottky-type anomaly, which is rooted to a small energy gap in low-lying two-level eigenstates induced by a tunneling through the potential barrier. The Schottky-type anomaly is removed when an asymmetry is introduced into the double-well potential. It has been pointed out that the specific-heat calculation of a double-well system reported by Feranchuk, Ulyanenkov, and Kuz'min [Chem. Phys. 157, 61 (1991)] is misleading because the zeroth-order operator method they adopted neglects crucially important off-diagonal contributions.Specific heats of quantum systems with symmetric and asymmetric double-well potentials have been calculated. In numerical calculations of their specific heats, we have adopted the combined method, which takes into account not only eigenvalues of ε(n) for 0 ≤ n ≤ N(m) obtained by the energy-matrix diagonalization but also their extrapolated ones for N(m) + 1 ≤ n <∞ (N(m) = 20 or 30). Calculated specific heats are shown to be rather different from counterparts of a harmonic oscillator. In particular, specific heats of symmetric double-well systems at very low temperatures have the Schottky-type anomaly, which is rooted to a small energy gap in low-lying two-level eigenstates induced by a tunneling through the potential barrier. The Schottky-type anomaly is removed when an asymmetry is introduced into the double-well potential. It has been pointed out that the specific-heat calculation of a double-well system reported by Feranchuk, Ulyanenkov, and Kuz'min [Chem. Phys. 157, 61 (1991)] is misleading because the zeroth-order operator method they adopted neglects crucially important off-diagonal contributions.

arXiv:1205.2058Physical Review EDec 5, 201214 citations

Thermodynamics of the Two-Level Richardson Model

Anna Ciechan, Karol Izydor Wysokiński

Dedicated to Professor Józef Spa lek on the occasion of his 60th birthdayThe two-level version of the Richardson model presents a unique possibility to calculate numerically exactly thermodynamic properties of the system it describes.The point is that all energies and the degeneracies of the many-body system can be easily calculated.The energies are given by the eigenvalues of the small (of order of N × N , where 2N is a number of electrons in the system) tridiagonal matrices.Here we numerically obtain a complete spectrum of the interacting two-level model and calculate the specific heat and the pairing energy of the small system at finite temperatures.Dedicated to Professor Józef Spa lek on the occasion of his 60th birthdayThe two-level version of the Richardson model presents a unique possibility to calculate numerically exactly thermodynamic properties of the system it describes.The point is that all energies and the degeneracies of the many-body system can be easily calculated.The energies are given by the eigenvalues of the small (of order of N × N , where 2N is a number of electrons in the system) tridiagonal matrices.Here we numerically obtain a complete spectrum of the interacting two-level model and calculate the specific heat and the pairing energy of the small system at finite temperatures.

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Approximation of the Quantum Anharmonic Oscillator: Investigating the Accuracy of Effective Models for the Double Well Potential

Linda Nordmark, Jacob Pirinen

The report investigates how the quantum mechanical model of a particle in one dimension underthe influence of the double well potential V (x) = λ(x^2 − x_0^2)^2 can be approximated using aneffective model, where the system is approximated with a system of truncated dimension. Theinfinite dimensional space of energy eigenfunctions to the double well potential was estimatedby two finite dimensional spaces, ones which were spanned by the ground- and first excitedeigenstates of the harmonic oscillators corresponding to the two minima of the potential. Theenergy eigenvalues given by these models were then studied. Additionally, the effects on thepartition function, the internal energy and the heat capacity of the system were checked tofurther study the accuracy of the models. To determine the accuracy of the approximation ahigh order, highly resolved finite element method was used for comparison as stand in for anexact, analytical solution. All investigated quantities were found to be better approximated forlarger λ and lower temperatures, with the effective model of larger dimension achieving a betterapproximation. The result with respect to λ likely results from that, for low energy, the doublewell potential resembles two harmonic oscillator potentials more the higher the energy barrier,i.e. the higher the λ. The effective models being better approximations for lower temperaturescan be explained by the fact that at low temperature, the lowest energies have the strongestimpact on the behaviour of the system.The report investigates how the quantum mechanical model of a particle in one dimension underthe influence of the double well potential V (x) = λ(x^2 − x_0^2)^2 can be approximated using aneffective model, where the system is approximated with a system of truncated dimension. Theinfinite dimensional space of energy eigenfunctions to the double well potential was estimatedby two finite dimensional spaces, ones which were spanned by the ground- and first excitedeigenstates of the harmonic oscillators corresponding to the two minima of the potential. Theenergy eigenvalues given by these models were then studied. Additionally, the effects on thepartition function, the internal energy and the heat capacity of the system were checked tofurther study the accuracy of the models. To determine the accuracy of the approximation ahigh order, highly resolved finite element method was used for comparison as stand in for anexact, analytical solution. All investigated quantities were found to be better approximated forlarger λ and lower temperatures, with the effective model of larger dimension achieving a betterapproximation. The result with respect to λ likely results from that, for low energy, the doublewell potential resembles two harmonic oscillator potentials more the higher the energy barrier,i.e. the higher the λ. The effective models being better approximations for lower temperaturescan be explained by the fact that at low temperature, the lowest energies have the strongestimpact on the behaviour of the system.

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A generalized semiclassical expression for the eigenvalues of multiple well potentials

Fabiano M. Andrade, Bin Kang Cheng, Marcus W. Beims +1 more

From the poles of a generalized semiclassical Green's function we derive expressions for the eigenvalues of 1D multiple well potentials. In the case of asymmetric and symmetric double wells, we also obtain analytical formulae for, respectively, the shift and splitting of energies. Our results are better than some approximations in the literature because they take more properly into account the tunnelling through the barriers forming the multiple well and depend on energy-dependent Maslov indices. We illustrate the good numerical precision of the method by discussing some case tests on double wells.From the poles of a generalized semiclassical Green's function we derive expressions for the eigenvalues of 1D multiple well potentials. In the case of asymmetric and symmetric double wells, we also obtain analytical formulae for, respectively, the shift and splitting of energies. Our results are better than some approximations in the literature because they take more properly into account the tunnelling through the barriers forming the multiple well and depend on energy-dependent Maslov indices. We illustrate the good numerical precision of the method by discussing some case tests on double wells.

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Thermodynamic Properties of Bosons in Symmetric Double-well Potentials

J. Choy, K.L. Liu, Chi‐Fai Lo +1 more

We study the thermodynamic properties and the Bose-Einstein condensation (BEC) for a finite num-ber N of identical non-interacting bosons in the field of a deep symmetric double-well potential (SDWP). The temperature dependence of the heat capacity C(T) at low temperatures is analyzed, and we derive several generic results which are valid when the energy difference between the first two excited states is sufficiently large. We also investigate numerically the properties of non-interacting bosons in three-dimensional superpositions of deep quartic SDWP's. At low temperatures, we find that C(T) displays microstructures which are sensitive to the value of N and the thermal variation of the condensate frac-tion shows a characteristic plateau. The origin of these features is discussed, and some general conclu-sions are drawn.We study the thermodynamic properties and the Bose-Einstein condensation (BEC) for a finite num-ber N of identical non-interacting bosons in the field of a deep symmetric double-well potential (SDWP). The temperature dependence of the heat capacity C(T) at low temperatures is analyzed, and we derive several generic results which are valid when the energy difference between the first two excited states is sufficiently large. We also investigate numerically the properties of non-interacting bosons in three-dimensional superpositions of deep quartic SDWP's. At low temperatures, we find that C(T) displays microstructures which are sensitive to the value of N and the thermal variation of the condensate frac-tion shows a characteristic plateau. The origin of these features is discussed, and some general conclu-sions are drawn.

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Double-peak specific heat anomaly and correlations in the Bose-Hubbard model

Eduardo O. Rizzatti, Marco Aurélio A. Barbosa, Márcia C. Barbosa

Considering the thermodynamics of bosons in a lattice described by the Bose-Hubbard Hamiltonian, we report the occurrence of anomalous double peaks in their specific heat dependence on temperature. This feature, usually associated with a high geometrical frustration, can also be a consequence of a purely energetic competition. By employing self-energy functional calculations combined with finite-temperature perturbation theory, we propose a mechanism based on ground-state degeneracies expressed as residual entropies. A general decomposition of the specific heat regarding all possible transitions between the system's eingenvalues provides an insight into the nature of each maximum. Furthermore, we address how the model parameters modify the structure of these peaks based on its spectral properties and atom-atom correlation function.Considering the thermodynamics of bosons in a lattice described by the Bose-Hubbard Hamiltonian, we report the occurrence of anomalous double peaks in their specific heat dependence on temperature. This feature, usually associated with a high geometrical frustration, can also be a consequence of a purely energetic competition. By employing self-energy functional calculations combined with finite-temperature perturbation theory, we propose a mechanism based on ground-state degeneracies expressed as residual entropies. A general decomposition of the specific heat regarding all possible transitions between the system's eingenvalues provides an insight into the nature of each maximum. Furthermore, we address how the model parameters modify the structure of these peaks based on its spectral properties and atom-atom correlation function.

arXiv:2010.06560Oct 13, 20205 citations

\epsilon_n

for

0 \leq n \leq N_m

obtained by the energy-matrix diagonalization but also their extrapolated ones for

N_m+1 \leq n < \infty

(

N_m=20