A new framework for the reflection positivity, based on order-preserving operator inequalities, is proposed. This framework is utilized to investigate one-dimensional fermion–phonon systems, with a particular focus on the detailed examination of ground state properties. Our analysis reveals that the reflection positivity provides a consistent description of the charge-density-wave (CDW) order in such systems. Additionally, we establish the existence of CDW long-range order in the ground state when the Coulomb interaction is long range.
A new framework for the reflection positivity, based on order-preserving operator inequalities, is proposed. This framework is utilized to investigate one-dimensional fermion–phonon systems, with a particular focus on the detailed examination of ground state properties. Our analysis reveals that the reflection positivity provides a consistent description of the charge-density-wave (CDW) order in such systems. Additionally, we establish the existence of CDW long-range order in the ground state when the Coulomb interaction is long range.
For nearly a decade, quasi-one-dimensional conductors have been the subject of intensive study. Theoretically, much attention has been devoted to the development of one-dimensional Fermi gas models, some which may be solved exactly, and to the calculation of their response functions. After a review of this theory, a different approach is adopted in the investigation of two models. The dielectric response theory of the three-dimensional Coulomb gas has been applied to an anisotropic system in which the particles interact with an effective one-dimensional long-range potential. Within the framework of the approximation of Singwi, Tosi, Land, and Sjolander, the dielectric properties of the model are examined in order to determine the conditions under which it is unstable with respect to formation of a charge density wave state. It is found that the positive neutralizing background must be polarizable in order for such an instability to occur. The same approximation method, when applied to a one-dimensional fermion gas with a ʃ-function interaction may be compared with the exact solution of Yang. This solution, which exists in the form of coupled integral equations, has been calculated numerically, and, as predicted by the Lieb-Mattis theorem, the ground state is found to be non-magnetic. The approximation of Singwi et al. proves to give better correlation energies than other inexact methods, particularly at higher densities.
Theory of spin-reflection positivity developed in recent years is reviewed. This theory makes use of symmetries in an electron system and theory of matrix to investigate the ground state properties. Existences of anti- and ferromagnetic long-range orders in itinerant electron systems, and of off-diagonal long-range order are two successful applications of the theory. In this article, the author attempt to summarize exact results proved by utilizing this theory and related topics. First a general theory and basic theorems are introduced. Second, based on the band structures of conduction electrons, existences of a singlet state with strongly antiferromagnetic correlation, a state with both anti- and ferromagnetic long-range orders, and a fully saturated ferromagnetic state are proved. The theory is applied to several of the main theoretical models for strongly correlated electron systems, such as the Heisenberg model, the Hubbard model, the Anderson model, the single- and multichannel Kondo model, and the generalized Hubbard model, and a series of rigorous results are found in these models. Third, it is proved that off-diagonal long-range order and charge-density wave exist in the ground states of the attractive Hubbard model and the generalized Hubbard model. A relation between pseudospin symmetry and the uniform density theorem is introduced. Fourth, the theory is applied successfully to explain experimental observations of oscillatory interlayer magnetic coupling in ultrathin magnetic films. Finally several unsolved problems are discussed. All results introduced in this article are mathematically exact.
Standard Peierls theory predicts that the wave-vector of the charge density wave (CDW) induced by the electron-phonon coupling in the ground-state of a one-dimensional conductor is twice the Fermi wave-vector 2k F . At moderately large electron-phonon coupling, beyond the TBA (transition by breaking of analyticity), it is known that this CDW becomes an array of interacting bipolarons. In the simplest models, such as the 1D adiabatic Holstein model, the bipolaron interactions are repulsive and the ground state remains a 2k F CDW of equidistant bipolarons. We show that (apparently) minor changes in the phonon spectrum of the model induce extra elastic forces between the bipolarons, which could break the bipolaronic 2k F CDW ordering. This effect is studied in detail in the modified 1D Holstein model, where the optical phonon has a non-zero dispersion which is chosen positive in order that an extra force that appears between the bipolarons be attractive. The Coulomb forces between the bipolarons are initially neglected. An accurate numerical study modelled on the standard analysis of multiphase points confirms that, in a large part of the phase diagram, the CDW ground state separates in two phases, which correspond to two CDW with different 2k F wave-vectors (which are generally commensurate) and thus different electronic densities. These separations, into phases called either 'parent' or 'non-parent', are studied on the basis of the Farey construction of rational numbers. The long-range Coulomb interaction between the charged bipolarons, forbids any macroscopic phase separation in two phases with different electronic densities. The bipolaron structure then has to be a periodic sequence of alternate domains of the two CDW phases on the microscopic scale. The resulting structure is a new CDW with a modulation wave-vector that is not 2k F . We suggest that the aberrant wave-vector observed in the real CDW system (TaSe 4 ) 2 I could be interpreted along the ideas developed in this paper. Although this compound shares many of the usual properties of quasi-one-dimensional conductors with CDW, the component of the modulation wave-vector 0.085c* in the chain direction is, unusually, quite different from the value 2k F =c* expected from band filling. The CDW in (TaSe 4 ) 2 I could correspond to a nonstandard bipolaron ordering due to a strong attractive elastic interaction between the bipolarons along the chain competing with the long-range Coulomb forces.
The Holstein model describes the interaction between fermions and a collection of local (dispersionless) phonon modes. In the dilute limit, the phonon degrees of freedom dress the fermions, giving rise to polaron and bipolaron formation. At higher densities, the phonons mediate collective superconducting (SC) and charge-density wave (CDW) phases. Quantum Monte Carlo (QMC) simulations have considered both these limits but have not yet focused on the physics of more general phonon spectra. Here we report QMC studies of the role of phonon dispersion on SC and CDW order in such models. We quantify the effect of finite phonon bandwidth and curvature on the critical temperature T_{cdw} for CDW order and also uncover several novel features of diagonal long-range order in the phase diagram, including a competition between charge patterns at momenta q=(π,π) and q=(0,π) which lends insight into the relationship between Fermi surface nesting and the wave vector at which charge order occurs. We also demonstrate SC order at half filling in situations where a nonzero bandwidth sufficiently suppresses T_{cdw}.
We present exact wave functions for the ground state of a lattice gas of spinless one-dimensional fermions, at filling factors of 1/M. The one-body hopping operator decreases exponentially. The two-body interaction is Coulomb-like: A neutralizing background is required for the system to be extensive. There is a parameter in the problem, which can be taken to be the strength of the charge. All correlation functions are evaluated, and the system is shown to have long-ranged crystalline order. For filling factor 1/2, the momentum distribution and pair correlation function are explicitly evaluated. There is no Fermi surface. The crystalline order parameter is evaluated.
The ground-state phase diagram of mixtures of spin polarized fermions and bosons in a 1D periodic lattice is discussed in the limit of large fermion hopping and half filling of the fermions. Numerical simulations performed with the density matrix renormalization group (DMRG) show in addition to bosonic Mott insulating (MI), superfluid (SF), and charge density-wave phases (CDW) a yet unreported phase with spatial separation of MI and CDW regions. We derive an effective bosonic theory which allows for a complete understanding and quantitative prediction of the bosonic phase diagram. In particular the origin of CDW phase and the MI-CDW phase separation is revealed as an effective fermion-mediated long-range interaction between bosons with alternating sign.