Interacting boson model from energy density functionals

We consider a model of one-dimensional spin-1 bosons with repulsive density–density interactions and antiferromagnetic exchange. We show that the low energy effective field theory is given by a spin–charge separated theory of a Tomonaga–Luttinger Hamiltonian and the O(3) non-linear sigma model describing collective charge and spin excitations respectively. At a particular ratio of the density–density to spin–spin interaction the model is integrable, and we use the exact solutions to provide an independent derivation of the low energy effective theory. The system is in a superfluid phase made of singlet pairs of bosons, and we calculate the long-distance asymptotics of certain correlation functions.We consider a model of one-dimensional spin-1 bosons with repulsive density–density interactions and antiferromagnetic exchange. We show that the low energy effective field theory is given by a spin–charge separated theory of a Tomonaga–Luttinger Hamiltonian and the O(3) non-linear sigma model describing collective charge and spin excitations respectively. At a particular ratio of the density–density to spin–spin interaction the model is integrable, and we use the exact solutions to provide an independent derivation of the low energy effective theory. The system is in a superfluid phase made of singlet pairs of bosons, and we calculate the long-distance asymptotics of certain correlation functions.

arXiv:0811.0844Journal of Statistical Mechanics Theory and ExperimentFeb 10, 200921 citations

Microscopic approach to the interacting boson model

Martin R. Zirnbauer, D.M. Brink

Nuclear Physics AAug 1, 198266 citations

Towards a practical pair density-functional theory for many-electron systems

Filipp Furche

In pair density-functional theory, the only unknown piece of the energy is the kinetic energy $T$ as a functional of the pair density $P({x}_{1},{x}_{2})$ . Although $T\phantom{\rule{0.1em}{0ex}}[P]$ has a simpler structure than the Hohenberg-Kohn functional of conventional density-functional theory, computational requirements are still moderate. In the present work, a particularly convenient model system to represent many-electron pair densities is introduced. This ``boson pair model'' (BPM) approximately treats electron pairs as noninteracting bosons. The resulting explicit model for the kinetic energy ${T}_{2}[P]$ is shown to be exact for two-electron systems and a lower bound to $T\phantom{\rule{0.1em}{0ex}}[P]$ for more than two electrons. The one- and two-particle density matrices obtained from the BPM yield upper bounds for the corresponding many-electron quantities. This suggests a partitioning $T\phantom{\rule{0.1em}{0ex}}[P]={T}_{2}[P]+{T}_{\mathrm{eff}}\phantom{\rule{0.1em}{0ex}}[P]$ , where only the remainder ${T}_{\mathrm{eff}}\phantom{\rule{0.1em}{0ex}}[P]\ensuremath{\geqslant}0$ needs to be approximated. If the BPM is constrained to yield the exact ground-state pair density, a two-electron Schr\"odinger equation with an effective local two-particle potential results; the latter is identified as a sum of the bare Coulomb interaction and the functional derivative of ${T}_{\mathrm{eff}}\phantom{\rule{0.1em}{0ex}}[P]$ . This self-consistent scheme to minimize the energy with respect to $P$ is more efficient than previous procedures. Further information on the functional derivative of ${T}_{\mathrm{eff}}\phantom{\rule{0.1em}{0ex}}[P]$ is derived from a contracted Schr\"odinger equation. Since ${T}_{\mathrm{eff}}\phantom{\rule{0.1em}{0ex}}[P]$ is explicitly known in the two-electron and noninteracting (Hartree-Fock) limits, the present method provides an alternative to density-matrix functional theories, which can be exact in the same limits and are similar in computational cost.In pair density-functional theory, the only unknown piece of the energy is the kinetic energy $T$ as a functional of the pair density $P({x}_{1},{x}_{2})$. Although $T\phantom{\rule{0.1em}{0ex}}[P]$ has a simpler structure than the Hohenberg-Kohn functional of conventional density-functional theory, computational requirements are still moderate. In the present work, a particularly convenient model system to represent many-electron pair densities is introduced. This ``boson pair model'' (BPM) approximately treats electron pairs as noninteracting bosons. The resulting explicit model for the kinetic energy ${T}_{2}[P]$ is shown to be exact for two-electron systems and a lower bound to $T\phantom{\rule{0.1em}{0ex}}[P]$ for more than two electrons. The one- and two-particle density matrices obtained from the BPM yield upper bounds for the corresponding many-electron quantities. This suggests a partitioning $T\phantom{\rule{0.1em}{0ex}}[P]={T}_{2}[P]+{T}_{\mathrm{eff}}\phantom{\rule{0.1em}{0ex}}[P]$, where only the remainder ${T}_{\mathrm{eff}}\phantom{\rule{0.1em}{0ex}}[P]\ensuremath{\geqslant}0$ needs to be approximated. If the BPM is constrained to yield the exact ground-state pair density, a two-electron Schr\"odinger equation with an effective local two-particle potential results; the latter is identified as a sum of the bare Coulomb interaction and the functional derivative of ${T}_{\mathrm{eff}}\phantom{\rule{0.1em}{0ex}}[P]$. This self-consistent scheme to minimize the energy with respect to $P$ is more efficient than previous procedures. Further information on the functional derivative of ${T}_{\mathrm{eff}}\phantom{\rule{0.1em}{0ex}}[P]$ is derived from a contracted Schr\"odinger equation. Since ${T}_{\mathrm{eff}}\phantom{\rule{0.1em}{0ex}}[P]$ is explicitly known in the two-electron and noninteracting (Hartree-Fock) limits, the present method provides an alternative to density-matrix functional theories, which can be exact in the same limits and are similar in computational cost.

Physical Review AAug 30, 200464 citations

The Interacting Boson Model

Abraham Klein

ScienceApr 2, 19821 citation

Microscopic study of interacting boson model

高浩水崎

Jan 1, 19931 citation

Auxiliary boson approach for electronic states in the two-dimensional Hubbard model

Taiichirou Saikawa, Álvaro Antônio Bandeira Ferraz, P. E. de Brito +1 more

Electronic states in the two-dimensional Hubbard model are studied in the doped paramagnetic states by use of the auxiliary boson approach. Four auxiliary bosons are introduced by means of the Hubbard-Stratonovich transformation within the functional integral treatment. These bosons correspond to one charge and three spin fluctuations. An effective model is formulated. Going beyond the boson fluctuations around the saddle point, the single-electron Green's function including one-loop self-energy effects of fermion-boson interaction is derived. The behavior of the fermion self-energy, spectral functions, and density of states are investigated for several values of the Coulomb interaction $U$ within the limit imposed by the Stoner criterion and also for small and moderate doping concentrations $\ensuremath{\delta}$. For small doping $(\ensuremath{\delta}=0.2)$, as $U$ approaches a value determined by the Stoner criterion, the spin-boson spectrum has a striking low-energy enhancement around $\mathbf{q}\ensuremath{\sim}2{\mathbf{k}}_{F}.$ This enhancement of the spin fluctuation causes a non-Fermi-liquid-like low-energy $\ensuremath{\omega}$ linear dependence in the imaginary part of the self-energy. Due to the self-energy effects, band splittings and a band narrowing are produced in the spectral functions. As a consequence of this, the density of states has pseudogap structures around the quasiparticle band with a narrow bandwidth on the Fermi energy. These features seem to be precursors of the metal-insulator (Mott-Hubbard) transition and they might be related to the spin-gap phenomena observed in high- ${T}_{c}$ materials. Our results for the density of states, the spectral function, and the band dispersions show qualitative agreement with the data of finite-size cluster simulations.Electronic states in the two-dimensional Hubbard model are studied in the doped paramagnetic states by use of the auxiliary boson approach. Four auxiliary bosons are introduced by means of the Hubbard-Stratonovich transformation within the functional integral treatment. These bosons correspond to one charge and three spin fluctuations. An effective model is formulated. Going beyond the boson fluctuations around the saddle point, the single-electron Green's function including one-loop self-energy effects of fermion-boson interaction is derived. The behavior of the fermion self-energy, spectral functions, and density of states are investigated for several values of the Coulomb interaction $U$ within the limit imposed by the Stoner criterion and also for small and moderate doping concentrations $\ensuremath{\delta}$. For small doping $(\ensuremath{\delta}=0.2)$, as $U$ approaches a value determined by the Stoner criterion, the spin-boson spectrum has a striking low-energy enhancement around $\mathbf{q}\ensuremath{\sim}2{\mathbf{k}}_{F}.$ This enhancement of the spin fluctuation causes a non-Fermi-liquid-like low-energy $\ensuremath{\omega}$ linear dependence in the imaginary part of the self-energy. Due to the self-energy effects, band splittings and a band narrowing are produced in the spectral functions. As a consequence of this, the density of states has pseudogap structures around the quasiparticle band with a narrow bandwidth on the Fermi energy. These features seem to be precursors of the metal-insulator (Mott-Hubbard) transition and they might be related to the spin-gap phenomena observed in high-${T}_{c}$ materials. Our results for the density of states, the spectral function, and the band dispersions show qualitative agreement with the data of finite-size cluster simulations.

Physical review. B, Condensed matterAug 15, 19974 citations

In pair density-functional theory, the only unknown piece of the energy is the kinetic energy

T

as a functional of the pair density

P({x}_{1},{x}_{2})

. Although

T\phantom{\rule{0.1em}{0ex}}[P]

has a simpler structure than the Hohenberg-Kohn functional of conventional density-functional theory, computational requirements are still moderate. In the present work, a particularly convenient model system to represent many-electron pair densities is introduced. This ``boson pair model'' (BPM) approximately treats electron pairs as noninteracting bosons. The resulting explicit model for the kinetic energy

{T}_{2}[P]

is shown to be exact for two-electron systems and a lower bound to

T\phantom{\rule{0.1em}{0ex}}[P]

for more than two electrons. The one- and two-particle density matrices obtained from the BPM yield upper bounds for the corresponding many-electron quantities. This suggests a partitioning

T\phantom{\rule{0.1em}{0ex}}[P]={T}_{2}[P]+{T}_{\mathrm{eff}}\phantom{\rule{0.1em}{0ex}}[P]

, where only the remainder ${T}_{\mathrm{eff}}\phantom{\rule{0.1em}{0ex}}[P]\ensuremath{\geqslant}0$ needs to be approximated. If the BPM is constrained to yield the exact ground-state pair density, a two-electron Schr\"odinger equation with an effective local two-particle potential results; the latter is identified as a sum of the bare Coulomb interaction and the functional derivative of

{T}_{\mathrm{eff}}\phantom{\rule{0.1em}{0ex}}[P]

. This self-consistent scheme to minimize the energy with respect to

P

is more efficient than previous procedures. Further information on the functional derivative of

{T}_{\mathrm{eff}}\phantom{\rule{0.1em}{0ex}}[P]

is derived from a contracted Schr\"odinger equation. Since

{T}_{\mathrm{eff}}\phantom{\rule{0.1em}{0ex}}[P]

is explicitly known in the two-electron and noninteracting (Hartree-Fock) limits, the present method provides an alternative to density-matrix functional theories, which can be exact in the same limits and are similar in computational cost.

U

within the limit imposed by the Stoner criterion and also for small and moderate doping concentrations $\ensuremath{\delta}$. For small doping $(\ensuremath{\delta}=0.2)$, as

U

approaches a value determined by the Stoner criterion, the spin-boson spectrum has a striking low-energy enhancement around $\mathbf{q}\ensuremath{\sim}2{\mathbf{k}}_{F}.$ This enhancement of the spin fluctuation causes a non-Fermi-liquid-like low-energy $\ensuremath{\omega}$ linear dependence in the imaginary part of the self-energy. Due to the self-energy effects, band splittings and a band narrowing are produced in the spectral functions. As a consequence of this, the density of states has pseudogap structures around the quasiparticle band with a narrow bandwidth on the Fermi energy. These features seem to be precursors of the metal-insulator (Mott-Hubbard) transition and they might be related to the spin-gap phenomena observed in high-

{T}_{c}

materials. Our results for the density of states, the spectral function, and the band dispersions show qualitative agreement with the data of finite-size cluster simulations.