Describing systems of interacting fermions by boson models: exact mapping in arbitrary dimension and applications arXiv:1001.1552v2 [cond-mat.str-el] 26 Sep 2010 K. B. Efetov $^{1}$, C. P´ epin $^{2}$, and H. Meier1 $^{1}$Theoretische Physik III, Ruhr-Universit¨ at Bochum, 44780 Bochum, Germany $^{2}$IPhT, CEA-Saclay, L’Orme des Merisiers, 91191 Gif-sur-Yvette, France (Dated: November 15, 2018) We develop a new method that allows us to map models of interacting fermions onto bosonic models describing collective excitations in an arbitrary dimension. This mapping becomes exact in the thermodynamic limit in the presence of a bath. The boson models can be written either in the form of a model of non-interacting bosons in a fluctuating auxiliary field or in the form of a superfield theory of interacting bosons. We show how one can study the latter version using perturbation theory. Using the developed diagrammatic technique we compared the first two orders of perturbation theory with the corresponding results for the original fermion model and found a perfect agreement. As concerns the former representation, we suggest a scheme that may be suitable for Monte Carlo simulations and demonstrate that it is free of the fermionic sign problem. We discuss in details the properties of the bosonic representation and argue that there should not be any obstacles preventing from an efficient computation. PACS numbers: 71.10.Ay, 71.10.Pm, 75.40.Cx Many body electron systems with interaction tradi- tionally attract a strong attention. Numerous interest- ing phenomena of the condensed matter physics originate from the electron-electron, electron-phonon and other types of the interaction and do not exist in the ideal Fermi gas of the electrons. Generally, models with interactions cannot be solved exactly in the space of dimensionality d > 1 and one has to use approximate schemes of calculation. The most de- veloped method is the perturbation theory with respect to the interaction. Within this theory one starts with the ideal Fermi gas and calculates corrections to physical quantities assuming that the interaction is weak. Some- times it is sufficient to compute only the first several or- ders of this perturbation theory, but very often in order to capture important physics, one has to sum certain se- ries. The formalism of the perturbation theory for the in- teracting electron gas is well developed (see, e.g., Ref. 1). Each term of the perturbation theory is represented di- agrammatically, which enables one to carry out quite complicated calculations. Many interesting phenomena like, e.g., superconductivity are not seen in the lowest orders of the diagrammatic expansion and show up after summation of certain ladder diagrams. Consideration of chain diagrams is necessary in the microscopic theory of the Landau Fermi liquid and there are many examples of this type. At the same time, such sequences of diagrams corre- spond to physically well-defined collective bosonic exci- tations. In the Fermi liquid theory, the chain diagrams describe particle density and spin excitations that can be spoken of as bosonic quasiparticles. From this and many other examples, where a well-defined physical quasiparti- cle is represented by an infinite sequence of diagrams, one Another method based on quasiclassical equations has been developed recently for an arbitrary interaction $^{15}$. Such an approach called bosonization is well known for one dimensional (1 D ) systems. A huge number of pub- lications are devoted to it (see, for a review, Refs. 4,5). Attempts to bosonize higher dimensional fermionic sys- tems have been undertaken in the past starting from the work by Luther 6 where the Fermi surface of a special form (square or cubic) was considered. This idea was fur- ther developed by Haldane 7 who suggested to patch the Fermi surface by a large number of segments and contin- ued in a number of publications $^{8–14}$. All these bosoniza- tion schemes are based on the assumption that only small momenta are transferred by the electron-electron inter- action such that, after any scattering event, the electron remains in the same segment of the Fermi surface. This case is, of course, relevant to a long range interaction but a generalization to an arbitrary interaction and arbi- trary momentum transfer is hardly possible within this scheme. At the end one can reproduce results of the random phase approximation (RPA) that can be inter- preted in terms of quasiparticles but the interaction be- tween these quasiparticles cannot be taken into account properly. The inconveniences of the conventional perturbative methods are especially evident in situations when the in- teraction between the quasiparticles is important. This happens in one-dimensional systems, d = 1 , but mod- els in higher dimensions, d > 1, used for describing high T$_{c}$ superconducting cuprates $^{2}$, heavy fermions$^{3}$, etc., can hardly be successfully studied by conventional diagram- matic approaches either. These difficulties call for con- structing different approaches dealing with collective ex- citations rather than with single particles. can conclude that the conventional diagrammatic tech- nique is not always the most convenient technique for the description of interacting many body systems. I. INTRODUCTION