Unified theory of strongly correlated electron systems

In framework of eigen-functional bosonization method, we introduce an imaginary phase field to uniquely represent electron correlation, and demonstrate that the Landau Fermi liquid theory and the Tomonaga-Luttinger liquid theory can be unified. It is very clear in this framework that the TomonagaLuttinger liquid behavior of one-dimensional interacting electron gases originates from their Fermi structure, and the non-Landau-Fermi liquid behavior of 2D interacting electron gases is induced by the long-range electron interaction, while 3D interacting electron gases generally show the Landau Fermi liquid behavior. Typeset using REVTEX 1 Since the discovery of high Tc cuprate superconductors [1], the strongly correlated electron systems has been extensively studied theoretically [2–10]. Now a common consensus is reached that the low energy physics properties of cuprate superconductors, such as anomalous normal state behavior and high superconducting transition temperature, are determined by the strong electron correlation in their copper-oxide plane(s). However, up to now there is not a microscopic or phenomenal theory to successfully explain the normal and superconducting physical properties of the cuprate superconductors, because one cannot exactly and effectively treat the strong electron correlation of the systems. It is well-known that usual metals can be described by the Landau Fermi liquid theory [11,12], in which there is weak correlation among electrons, and near the Fermi surface there exist well-defined quasi-particles (holes), thus the fundamental assumption of the Landau Fermi liquid theory is satisfied, i.e., the states of an interacting electron gas can be put into a one-to-one correspondence via adiabatic continuation with those of the free electron gas. However, for strong electron correlation, this fundamental assumption of the Landau Fermi liquid theory may fail, and one does not have well-defined quasi-particles (holes) near the Fermi surface. A one-dimensional interacting electron gas is a good example, in which there is strong electron correlation even though for small electron interaction, thus one does not have well-defined quasi-particles (holes) near the Fermi levels ±kF (its Fermi surface is composed of two points, ±kF , defined by the Fermi momentum kF ). It is described by the Tomonaga-Luttinger liquid theory [13–17], where in low energy regime electron Green’s function and other correlation functions present power-law behavior, and the correlation exponents are not universal, and depend upon the electron interaction strength. Therefore, the one-dimensional interacting electron gas is a strongly correlated electron system even for weak electron interaction. Generally, the low energy behavior of the one-dimensional (1D) interacting electron gas is qualitatively different from that of three-dimensional (3D) interacting electron gas, the former is represented by the Tomonaga-Luttinger liquid theory, while the latter is represented by the Landau Fermi liquid theory (see below). This difference derives from their different 2 Fermi surface strctures. For 1D electron gas, its Fermi surface is composed of two points ±kF , defined by the Fermi momentum kF , and its Hilbert space is drastically suppressed. There are only two kinds of elementary excitation modes, one is the excitation modes near these two Fermi levels ±kF , respectively, where the energy spectrum is approximately linear; and another one is the excitation modes with large momentum (2nkF , n = 1, 2, ...) transfer between these Fermi levels ±kF , which usually make the system become an insulator ( for a 1D electron gas this kind of excitation is absent). This drastically suppressed Hilbert space will induce the strong electron correlation even for weak electron interaction, thus it is true that for a 1D interacting electron gas the strong electron correlation originates from its special Fermi surface structure, and the Tomonaga-Luttinger liquid theory is an universal theory of 1D electron gases. In contrast with the 1D electron gas, the Fermi surface of the 3D electron gas is a sphere with a radius kF , and its Hilbert space is enlarged comparing with that of the 1D electron gas. The low energy elementary excitation modes are quasiparticles and quasi-holes. It is well-known that even for long-range Coulomb interaction, the 3D electron gas still can be described by the Landau Fermi liquid theory, where strong electron interaction does not mean that there is the strong electron correlation, and it only produces weak and short-range electron correlation. Thus the Landau Fermi liquid theory is an universal theory of 3D electron gases. For 2D electron gas, the situation is different. Its Fermi surface is a circle with a radius kF , and for long-range Coulomb interaction, it shows non-Landau-Fermi liquid behavior in the low energy regime [18,19]. We shall demonstrate that this anomalous behavior of the 2D interacting electron gas derives from the two aspects, one is its Fermi structure, and another one is the long-range electron interaction. These both effects induce the strong electron correlation, thus the non-Landau-Fermi liquid behavior of the 2D strongly correlated electron gas cannot be completely represented by the Tomonaga-Luttinger liquid theory. It shows not only some characters of the Landau Fermi liquid, but also some characters of the Tomonaga-Luttinger liquid. It is very desirable to find an unified theory to represent not only 1D interacting electron 3 gases, but also 2D and 3D interacting electron gases. In the framework of the eigen-functional bosonization method [20,21], we try to give such the unified theory of strongly correlated electron gases, which not only reduces to the Tomonaga-Luttinger liquid theory for the 1D interacting electron gas and to the Landau Fermi liquid thoery for the 3D interacting electron gas, but also can represent the low energy behavior of the 2D interacting electron gas. We clearly demonstrate that an imaginary phase field which naturally appears in the eigen-functional bosonization, is a key parameter field hiden in the strongly correlated electron systems and represents the electron correlation, and the eigen-functionals can be used to define the states of the interacting electrons. By calculation the overlap of these eigen-functionals with the eigen-functions of the free electrons, we can judge whether or not there exist well-defined quasi-particles (holes) near the Fermi surface. The imaginary phase field is a very important quantity in our representation, and it determines the low energy behavior of the system. Moreover, as d ≥ 2 it does not contribute to the action of the system. In general, we consider the Hamiltonian (omitting spin label σ of the electron operators ck)