We calculate the Gibbs potential $\ensuremath{\Gamma}(M)$ of a one-dimensional metal at constant magnetization $M$ to second order in the screened electron-electron interaction $U$. At zero temperature we find that $\ensuremath{\Gamma}(M)$ contains nonanalytic corrections proportional to ${M}^{2}\phantom{\rule{0.2em}{0ex}}\mathrm{ln}\ensuremath{\mid}M\ensuremath{\mid}$ and ${\ensuremath{\mid}M\ensuremath{\mid}}^{3}$, implying that a possible paramagnetic-ferromagnetic quantum phase transition in one-dimensional metals must be first order.
We calculate the Gibbs potential \ensuremathΓ(M) of a one-dimensional metal at constant magnetization M to second order in the screened electron-electron interaction U. At zero temperature we find that \ensuremathΓ(M) contains nonanalytic corrections proportional to M2ln\ensuremath∣M\ensuremath∣ and \ensuremath∣M\ensuremath∣3, implying that a possible paramagnetic-ferromagnetic quantum phase transition in one-dimensional metals must be first order.
We study the Hubbard model in infinite dimension d= infinity within an approximation reproducing simultaneously the atomic limit and the weak-coupling limit up to second order in the interaction strength U. Depending on U we find a paramagnetic metal, a paramagnetic insulator, or an antiferromagnetic insulator. The stability of the antiferromagnetic phase is investigated by calculating the grand canonical potential. We present the phase diagram in the T-U plane and results for the magnetization m as a function of temperature T. For small values of U the antiferromagnetic phase transition is continuous, but for larger values of U the approximation yields a first-order magnetic phase transition. Results for the density of states N( epsilon ) are presented; it either exhibits one single band or an upper and a lower Hubbard band separated by the Mott gap. The upper and lower Hubbard band may exhibit additional magnetic structure, depending on the value of T and U.
Alexander Shashkin Institute of Solid State Physics, Chernogolovka, Moscow District 142432, Russia Sergey Kravchenko Physics Department, Northeastern University, Boston, Massachusetts 02115, U.S.A. Two-dimensional (2D) electron systems are realized when the electrons are free to move in a plane but their motion perpendicular to the plane is quantized in a confining potential well. Quantum phase transitions realized experimentally in such systems so far include metal-insulator transitions in perpendicular magnetic fields, metal-insulator transition in zero magnetic field, and a possible transition to a Wigner crystal. The first transition is governed by the externally controlled electron density or magnetic field, while the other two are governed by the electron density. At low electron densities in 2D systems, the strongly-interacting limit is reached because the kinetic energy is overwhelmed by the energy of electron-electron interactions. The interaction strength is characterized by the ratio between the Coulomb energy and the Fermi energy, r∗s = Eee/EF . Assuming that the effective electron mass is equal to the band mass, the interaction parameter r∗s in the single-valley case reduces to the Wigner-Seitz radius, rs = 1/(πns)1/2aB, and therefore increases as the electron density, ns, decreases (here aB is the Bohr radius in the semiconductor). Possible candidates for the ground state of the system include a Wigner crystal characterized by spatial and spin ordering [1], a ferromagnetic Fermi liquid with spontaneous spin ordering [2], a paramagnetic Fermi liquid [3], etc. In the strongly-interacting limit (rs 1), no analytical theory has been developed to date. According to numerical simulations [4], Wigner crystallization is expected in a very dilute regime, when rs reaches approximately 35. Refined numerical simulations [5] have predicted that prior to the crystallization, in the range of the interaction parameter 25 ≤ rs ≤ 35, the ground state of the system is a strongly correlated ferromagnetic Fermi liquid. At higher electron densities, rs ∼ 1, the electron liquid is expected to be paramagnetic, with the effective mass, m, and Lande´ g factor renormalized by interactions. Apart from the ferromagnetic Fermi liquid, other intermediate phases between the Wigner crystal and the paramagnetic Fermi liquid may also exist.
Zero-temperature or quantum phase transitions in itinerant electronic systems both with and without quenched disordered are discussed. Phase transitions considered include, the ferromagnetic transition, the antiferromagnetic transition, the superconductor-metal transition, and various metal-insulator transitions. Emphasis is placed on how to determine the universal properties that characterize these quantum phase transitions. For the first three of the phase transitions listed above, one of the main physical ideas established is that in zero-temperature systems there are soft or slow modes that exist in addition to the soft order parameter fluctuations, and that these modes can couple to the critical modes. These extra soft modes are shown to have a profound effect on the quantum critical properties. For quantum phase transitions involving zero wavenumber order parameters, i.e., the ferromagnetic and superconductor-metal transitions, these extra modes effectively lead to long-ranged effective interactions between order parameter fluctuations, which in turn lead to exactly soluble critical behaviors. For the antiferromagnetic case, we argue that while in low enough dimensions disorder fluctuation effects tend to destroy long-range order, quantum fluctuations counteract this effect and in some parameter regions manage to re-establish antiferromagnetic long-range order. For the metal-insulator transition, some recent new ideas are reviewed. In particular, it is pointed out that for interacting disordered electrons, one expects that in high dimensions the metal-insulator transition is related to the phase transition that occurs in random-field magnets in high dimensions. If the analogy also holds in three dimensions this suggests that the metal-insulator transition might have glassy characteristics.
Ferromagnetism in one dimension is a observation which has been reported in a recent work [Gambardella et al., Nature (London) 416, 301 (2002)], and it is thought that anisotropy barriers are responsible in that relevant effect. In the present work, transitions between two different magnetic ordering phases are obtained as a result of an alternative approach. The critical temperature has been estimated by the Binder method. Ferromagnetic long-range interactions have been included in a special Hamiltonian through a power law that decays at large interparticle distance r as ${r}^{\ensuremath{-}\ensuremath{\alpha}}$ for $\ensuremath{\alpha}\ensuremath{\geqslant}0$. If the range of interactions decreases $(\ensuremath{\alpha}\ensuremath{\rightarrow}\ensuremath{\infty})$, the trend of the critical temperature disappears, but if the range of interactions increases $(\ensuremath{\alpha}\ensuremath{\rightarrow}0)$, the trend of the critical temperature approaches to the mean field approximation. The crossover between these two limiting situations is discussed.
Antiferromagnetic and metal to insulator transitions in the ground state of two-dimensional half-filled and nearly half-filled Hubbard models with the nearest neighbor and the second nearest neighbor transfers (- t and t ′ , respectively) are studied by using the Green's function approach with the self-energy up to the second order in the coupling constant U . For a choice of the parameter value t ′ / t =0.2 the half-filled model undergoes the first order phase transition between paramagnetic metallic (PM) phase and antiferromagnetic insulator (AFI) phase at U / t =4.0 while the first order transition between PM and antiferromagnetic metallic (AFM) phases takes place for nearly half-filled band. The one-electron spectral density is calculated for the PM, AFM, and AFI phases near the phase boundaries. In the PM phase the quasi-particle dispersion indicates the band narrowing from the Hartree-Fock results and the flat band behavior as was indicated by previous stadies and was observed in high- T c cuprates. The quasi-particle dispersion in the AFM phase is seen to be close to the Hartree-Fock results.
We use quantum Monte Carlo methods to determine T=0 Green functions, $G(\stackrel{\ensuremath{\rightarrow}}{r},\ensuremath{\omega})$, on lattices up to $16\ifmmode\times\else\texttimes\fi{}16$ for the 2D Hubbard model at U/t=4. For chemical potentials $\ensuremath{\mu}$ within the Hubbard gap $|\ensuremath{\mu}|<{\ensuremath{\mu}}_{c}$ and at long distances $\stackrel{\ensuremath{\rightarrow}}{r}$, $G(\stackrel{\ensuremath{\rightarrow}}{r},\ensuremath{\omega}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{\mu})\ensuremath{\sim}{e}^{\ensuremath{-}|\stackrel{\ensuremath{\rightarrow}}{r}|/{\ensuremath{\xi}}_{l}}$ with critical behavior ${\ensuremath{\xi}}_{l}\ensuremath{\sim}|\ensuremath{\mu}\ensuremath{-}{\ensuremath{\mu}}_{c}{|}^{\ensuremath{-}\ensuremath{\nu}}$, $\ensuremath{\nu}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.26\ifmmode\pm\else\textpm\fi{}0.05$. This result stands in agreement with the assumption of hyperscaling with correlation exponent $\ensuremath{\nu}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1/4$ and dynamical exponent z=4. In contrast, the generic band insulator as well as the metal-insulator transition in the 1D Hubbard model are characterized by $\ensuremath{\nu}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1/2$ and z=2.