The upper bound on the ground state energy of the two-component charged Bose gas derived in \[Comm. Math. Phys. 266 (2006), 797–818] is extended to the more general case, where the mass of the positive bosons can be different from the mass of the negative bosons. The analysis in principle is quite similar, however the formerly scalar problem becomes two-dimensional (one dimension for the $+$ and one for the $-$ case). This leads to a non-linear $2\times 2$ -matrix minimization problem. Its minimizer is calculated and results in a non-linear mass-dependent coefficient in the upper bound. These new results agree in the equal-mass case with the formerly known results. It is also discussed how the result here interpolates between the equal mass case and the case where one mass is infinite.The upper bound on the ground state energy of the two-component charged Bose gas derived in \[Comm. Math. Phys. 266 (2006), 797–818] is extended to the more general case, where the mass of the positive bosons can be different from the mass of the negative bosons. The analysis in principle is quite similar, however the formerly scalar problem becomes two-dimensional (one dimension for the $+$ and one for the $-$ case). This leads to a non-linear $2\times 2$-matrix minimization problem. Its minimizer is calculated and results in a non-linear mass-dependent coefficient in the upper bound. These new results agree in the equal-mass case with the formerly known results. It is also discussed how the result here interpolates between the equal mass case and the case where one mass is infinite.

EMS Press eBooksJun 29, 20220 citations

Ground state energy for a model gas in one dimension

S. L. Wolff

The European Physical Journal CJun 1, 19812 citations

Approximate determination of the number and energies of the bound states of a physical system

Luciano Fonda, G. C. Ghirardi

Nuovo cimento della Società italiana di fisica. A, Nuclei, particles and fieldsNov 1, 19665 citations

Energy spectrum of the optical polaron at finite total momentum

B. Gerlach, F. Kalina

In the following discussion we are concerned with the standard Fr\"ohlich model for an optical polaron. We clarify the qualitative properties of the energy spectrum for arbitrary total momentum Q. Concerning the ground-state energy, we establish an effective lower bound. Until now, we have to assume that the electron-phonon coupling parameter $\ensuremath{\alpha}$ does not exceed a specified positive value. Using this bound, we demonstrate that the ground-state energy coincides with the continuum edge for $|Q|>~|{Q}_{C}|,$ ${Q}_{C}$ being finite. Consequently, it is only for $|Q|<|{Q}_{C}|$ that an isolated ground state exists at all. This behavior is strikingly different from that of the corresponding system in lower dimensions, which has been analyzed previously by other authors, the discussion of the three-dimensional case remaining incomplete. Concerning the overall behavior of the ground-state energy as a function of Q and $\ensuremath{\alpha},$ we find an increase (strict decrease) with increasing $|Q|(\ensuremath{\alpha}).$ In addition, we present an approach to the excited states. Interestingly enough, this can be based entirely on the knowledge of the ground-state energy and ground-state wave function.In the following discussion we are concerned with the standard Fr\"ohlich model for an optical polaron. We clarify the qualitative properties of the energy spectrum for arbitrary total momentum Q. Concerning the ground-state energy, we establish an effective lower bound. Until now, we have to assume that the electron-phonon coupling parameter $\ensuremath{\alpha}$ does not exceed a specified positive value. Using this bound, we demonstrate that the ground-state energy coincides with the continuum edge for $|Q|>~|{Q}_{C}|,$ ${Q}_{C}$ being finite. Consequently, it is only for $|Q|<|{Q}_{C}|$ that an isolated ground state exists at all. This behavior is strikingly different from that of the corresponding system in lower dimensions, which has been analyzed previously by other authors, the discussion of the three-dimensional case remaining incomplete. Concerning the overall behavior of the ground-state energy as a function of Q and $\ensuremath{\alpha},$ we find an increase (strict decrease) with increasing $|Q|(\ensuremath{\alpha}).$ In addition, we present an approach to the excited states. Interestingly enough, this can be based entirely on the knowledge of the ground-state energy and ground-state wave function.

Physical review. B, Condensed matterOct 15, 19999 citations

Certifying Ground-State Properties of Many-Body Systems

Jie Wang, Jacopo Surace, Irénée Frérot +4 more

A ubiquitous problem in quantum physics is to understand the ground-state properties of many-body systems. Confronted with the fact that exact diagonalization quickly becomes impossible when increasing the system size, variational approaches are typically employed as a scalable alternative: Energy is minimized over a subset of all possible states and then different physical quantities are computed over the solution state. Despite remarkable success, rigorously speaking, all that variational methods offer are upper bounds on the ground-state energy. On the other hand, so-called relaxations of the ground-state problem based on semidefinite programming represent a complementary approach, providing lower bounds to the ground-state energy. However, in their current implementation, neither variational nor relaxation methods offer provable bound on other observables in the ground state beyond the energy. In this work, we show that the combination of the two classes of approaches can be used to derive certifiable bounds on the value of any observable in the ground state, such as correlation functions of arbitrary order, structure factors, or order parameters. We illustrate the power of this approach in paradigmatic examples of 1D and 2D spin- 1 / 2 Heisenberg models. To improve the scalability of the method, we exploit the symmetries and sparsity of the considered systems to reach sizes of hundreds of particles at much higher precision than previous works. Our analysis therefore shows how to obtain certifiable bounds on many-body ground-state properties beyond energy in a scalable way. Published by the American Physical Society 2024A ubiquitous problem in quantum physics is to understand the ground-state properties of many-body systems. Confronted with the fact that exact diagonalization quickly becomes impossible when increasing the system size, variational approaches are typically employed as a scalable alternative: Energy is minimized over a subset of all possible states and then different physical quantities are computed over the solution state. Despite remarkable success, rigorously speaking, all that variational methods offer are upper bounds on the ground-state energy. On the other hand, so-called relaxations of the ground-state problem based on semidefinite programming represent a complementary approach, providing lower bounds to the ground-state energy. However, in their current implementation, neither variational nor relaxation methods offer provable bound on other observables in the ground state beyond the energy. In this work, we show that the combination of the two classes of approaches can be used to derive certifiable bounds on the value of any observable in the ground state, such as correlation functions of arbitrary order, structure factors, or order parameters. We illustrate the power of this approach in paradigmatic examples of 1D and 2D spin- 1 / 2 Heisenberg models. To improve the scalability of the method, we exploit the symmetries and sparsity of the considered systems to reach sizes of hundreds of particles at much higher precision than previous works. Our analysis therefore shows how to obtain certifiable bounds on many-body ground-state properties beyond energy in a scalable way. Published by the American Physical Society 2024

arXiv:2310.05844Physical Review XJul 11, 202415 citations

+

and one for the

-

case). This leads to a non-linear

2\times 2

-matrix minimization problem. Its minimizer is calculated and results in a non-linear mass-dependent coefficient in the upper bound. These new results agree in the equal-mass case with the formerly known results. It is also discussed how the result here interpolates between the equal mass case and the case where one mass is infinite.

|Q|>~|{Q}_{C}|,

{Q}_{C}

being finite. Consequently, it is only for

|Q|<|{Q}_{C}|

that an isolated ground state exists at all. This behavior is strikingly different from that of the corresponding system in lower dimensions, which has been analyzed previously by other authors, the discussion of the three-dimensional case remaining incomplete. Concerning the overall behavior of the ground-state energy as a function of Q and $\ensuremath{\alpha},$ we find an increase (strict decrease) with increasing $|Q|(\ensuremath{\alpha}).$ In addition, we present an approach to the excited states. Interestingly enough, this can be based entirely on the knowledge of the ground-state energy and ground-state wave function.