The momentum distribution function and spin-correlation functions are studied for the one-dimensional large-U Hubbard model with an emphasis on the effects of the magnetic field. A direct calculation of the correlation functions is carried out with the exact wave function, whose explicit form for the large-U limit was derived in a previous paper. Singularities are detected at ${\mathit{k}}_{\mathit{F}\mathrm{\ensuremath{\sigma}}}$ and ${\mathit{k}}_{\mathit{F}\mathrm{\ensuremath{\sigma}}}$+2${\mathit{k}}_{\mathit{F}\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\sigma}}}$ in the momentum distribution function ${\mathit{n}}_{\mathrm{\ensuremath{\sigma}}}$(k), where ${\mathit{k}}_{\mathit{F}\mathrm{\ensuremath{\sigma}}}$ is the Fermi wave number of the noninteracting electrons with a given magnetization. Similarly, the longitudinal spin-correlation function ${\mathit{S}}^{\mathit{z}\mathit{z}}$ (k) has singularities at 2${\mathit{k}}_{\mathit{F}\mathrm{\ensuremath{\uparrow}}}$ and 2${\mathit{k}}_{\mathit{F}\mathrm{\ensuremath{\downarrow}}}$, while the transverse one ${\mathit{S}}^{\mathit{x}\mathit{x}}$ (k) is singular at ${\mathit{k}}_{\mathit{F}\mathrm{\ensuremath{\uparrow}}}$+${\mathit{k}}_{\mathit{F}\mathrm{\ensuremath{\downarrow}}}$ and ${\mathit{k}}_{\mathit{F}\mathrm{\ensuremath{\uparrow}}}$-${\mathit{k}}_{\mathit{F}\mathrm{\ensuremath{\downarrow}}}$. The nature of the singularities (i.e., critical exponents) is determined from an analysis of the size dependence. As a complementary approach, those critical exponents for the U/t\ensuremath{\rightarrow}\ensuremath{\infty} limit are explicitly calculated with conformal field theory. In this limit the exponents depend only on the magnetization and are independent of the electron density. The resulting exponents are consistent with the first results.The momentum distribution function and spin-correlation functions are studied for the one-dimensional large-U Hubbard model with an emphasis on the effects of the magnetic field. A direct calculation of the correlation functions is carried out with the exact wave function, whose explicit form for the large-U limit was derived in a previous paper. Singularities are detected at ${\mathit{k}}_{\mathit{F}\mathrm{\ensuremath{\sigma}}}$ and ${\mathit{k}}_{\mathit{F}\mathrm{\ensuremath{\sigma}}}$+2${\mathit{k}}_{\mathit{F}\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\sigma}}}$ in the momentum distribution function ${\mathit{n}}_{\mathrm{\ensuremath{\sigma}}}$(k), where ${\mathit{k}}_{\mathit{F}\mathrm{\ensuremath{\sigma}}}$ is the Fermi wave number of the noninteracting electrons with a given magnetization. Similarly, the longitudinal spin-correlation function ${\mathit{S}}^{\mathit{z}\mathit{z}}$(k) has singularities at 2${\mathit{k}}_{\mathit{F}\mathrm{\ensuremath{\uparrow}}}$ and 2${\mathit{k}}_{\mathit{F}\mathrm{\ensuremath{\downarrow}}}$, while the transverse one ${\mathit{S}}^{\mathit{x}\mathit{x}}$(k) is singular at ${\mathit{k}}_{\mathit{F}\mathrm{\ensuremath{\uparrow}}}$+${\mathit{k}}_{\mathit{F}\mathrm{\ensuremath{\downarrow}}}$ and ${\mathit{k}}_{\mathit{F}\mathrm{\ensuremath{\uparrow}}}$-${\mathit{k}}_{\mathit{F}\mathrm{\ensuremath{\downarrow}}}$. The nature of the singularities (i.e., critical exponents) is determined from an analysis of the size dependence. As a complementary approach, those critical exponents for the U/t\ensuremath{\rightarrow}\ensuremath{\infty} limit are explicitly calculated with conformal field theory. In this limit the exponents depend only on the magnetization and are independent of the electron density. The resulting exponents are consistent with the first results.

Physical review. B, Condensed matterApr 1, 199155 citations

The renormalization group and two dimensional multicritical effective scalar field theory

Tim R. Morris

arXiv:hep-th/9410141Physics Letters BFeb 1, 199586 citations

Harris criterion for direct and orthogonal quenched randomness

A. Nihat Berker

Critical-behavior stability is studied, with respect to the introduction of quenched randomness that is direct (e.g., field randomness at field approach to criticality) or orthogonal (e.g., field randomness at temperature approach to criticality). Rather than resorting to renormalization-group theory, an argument is conducted by considering the effective uniform field per spin within a correlated region. By comparing the correlation lengths limited by this effective uniform field and by the actual uniform deviation from criticality, it is recovered that nonrandom critical behavior is not maintained under random fields when its susceptibility critical exponent \ensuremath{\gamma} is positive, and conversely.Critical-behavior stability is studied, with respect to the introduction of quenched randomness that is direct (e.g., field randomness at field approach to criticality) or orthogonal (e.g., field randomness at temperature approach to criticality). Rather than resorting to renormalization-group theory, an argument is conducted by considering the effective uniform field per spin within a correlated region. By comparing the correlation lengths limited by this effective uniform field and by the actual uniform deviation from criticality, it is recovered that nonrandom critical behavior is not maintained under random fields when its susceptibility critical exponent \ensuremath{\gamma} is positive, and conversely.

Physical review. B, Condensed matterNov 1, 199019 citations

Dynamic Critical Exponent from One- and Two-Particle Irreducible 1/N Expansions of Effective and Microscopic Theories

Osamu Morimatsu, Hirotsugu Fujii, Kazunori Itakura A +1 more

We study the dynamic critical exponent from effective and microscopic theories. We employ a simple TDGL model, or model A in the classification of Hohenberg and Halperin, as an effective theory and the imaginary time formalism of the finite-temperature filed theory as a microscopic theory. Taking an O(N) scalar model as an example and carrying out the 1/N expansion up to the NLO in the 1PI and 2PI effective actions, we compare the low-energy and low-momentum behavior of the response function in the effective theory and of the retarded Green's function in the microscopic theory. At the NLO of the 1PI 1/N expansion the low-energy and low-momentum behavior of the two-point function is very much different in the microscopic and effective theories: in the field theory it is dominated by the propagating mode while in model A it is dominated by the diffusive mode. Also, in the microscopic theory the dynamic critical exponent, z, depends on whether the kinematics is relativistic or nonrelativistic. In contrast, at the NLO of the 2PI 1/N expansion the microscopic and effective theories are equivalent. They satisfy exactly the same Kadanoff-Baym equation. Also, whether the kinematics is relativistic or nonrelativistic in the microscopic theory becomes irrelevant. This implies that the diffusive mode with z = 2 + O(1/N) is dominant at low energies and momenta even in the microscopic theory at the NLO of the 2PI 1/N expansion, though we do not explicitly solve the Kadanoff-Baym equation. We also try to improve the calculation of the dynamic critical exponent of model A by incorporating the static 2PI NLO correlations. The obtained critical exponent is slightly smaller than the previous result and its N dependence is also milder than the previous one.We study the dynamic critical exponent from effective and microscopic theories. We employ a simple TDGL model, or model A in the classification of Hohenberg and Halperin, as an effective theory and the imaginary time formalism of the finite-temperature filed theory as a microscopic theory. Taking an O(N) scalar model as an example and carrying out the 1/N expansion up to the NLO in the 1PI and 2PI effective actions, we compare the low-energy and low-momentum behavior of the response function in the effective theory and of the retarded Green's function in the microscopic theory. At the NLO of the 1PI 1/N expansion the low-energy and low-momentum behavior of the two-point function is very much different in the microscopic and effective theories: in the field theory it is dominated by the propagating mode while in model A it is dominated by the diffusive mode. Also, in the microscopic theory the dynamic critical exponent, z, depends on whether the kinematics is relativistic or nonrelativistic. In contrast, at the NLO of the 2PI 1/N expansion the microscopic and effective theories are equivalent. They satisfy exactly the same Kadanoff-Baym equation. Also, whether the kinematics is relativistic or nonrelativistic in the microscopic theory becomes irrelevant. This implies that the diffusive mode with z = 2 + O(1/N) is dominant at low energies and momenta even in the microscopic theory at the NLO of the 2PI 1/N expansion, though we do not explicitly solve the Kadanoff-Baym equation. We also try to improve the calculation of the dynamic critical exponent of model A by incorporating the static 2PI NLO correlations. The obtained critical exponent is slightly smaller than the previous result and its N dependence is also milder than the previous one.

arXiv:1411.1867Nov 7, 20141 citation

{\mathit{S}}^{\mathit{z}\mathit{z}}

(k) has singularities at 2${\mathit{k}}_{\mathit{F}\mathrm{\ensuremath{\uparrow}}}$ and 2${\mathit{k}}_{\mathit{F}\mathrm{\ensuremath{\downarrow}}}$, while the transverse one

{\mathit{S}}^{\mathit{x}\mathit{x}}

(k) is singular at ${\mathit{k}}_{\mathit{F}\mathrm{\ensuremath{\uparrow}}}$+${\mathit{k}}_{\mathit{F}\mathrm{\ensuremath{\downarrow}}}$ and ${\mathit{k}}_{\mathit{F}\mathrm{\ensuremath{\uparrow}}}$-${\mathit{k}}_{\mathit{F}\mathrm{\ensuremath{\downarrow}}}$. The nature of the singularities (i.e., critical exponents) is determined from an analysis of the size dependence. As a complementary approach, those critical exponents for the U/t\ensuremath{\rightarrow}\ensuremath{\infty} limit are explicitly calculated with conformal field theory. In this limit the exponents depend only on the magnetization and are independent of the electron density. The resulting exponents are consistent with the first results.