1. Exact bosonization for interacting fermions in arbitrary dimensions. (New route to numerical and analytical calculations) K.B. Efetov, C. Pepin, H. Meier Bosonization: mapping of electron models onto a model describing collective excitations (charge, spin excitations, diffusion modes, etc). RUB Bochum, Germany, CEA Saclay, France 112 12

2. 17.11.20152 Origin of the word: 1D electron systems. The main idea: writing the electronic operators as A simple Hamiltonian H -Density fluctuations operator K- compressibility, N-average density The most general form conjectured by K.E. & A. Larkin (1975) Bosonization: for Tomonaga- Luttinger model (long range interaction) (Luttinger, Tomonaga (196?)) Microscopic theory Haldane (1982) ………………… Importance of long wave length excitations!

3. 17.11.20153 Formal replacement of electron Green functions by propagators for collective excitations! Equivalent representation Spin excitations Transformation from electrons to collective excitations: Bosonization Very often direct expansions with electronic Green functions are not efficient (infrared divergences, high energy cutoffs respecting symmetries).

4. 17.11.20154 Difficulties in Monte Carlo simulations for fermionic systems: negative sign problem exponential growth of the computation time with the size of the system. (From a lecture by M. Troyer) NP-nondeterministic polynomial

5. 17.11.20155 Can one bosonize in higher dimensions? Earlier attempts: A. Luther 1979: Special form of Fermi surface (square, cube, etc.). Almost 1D. F.D.M. Haldane 1992: Patching of the Fermi surface, no around corner scattering Further development of the patching idea: A. Houghton & Marston 1993; A.H. Castro Neto & E. Fradkin 1994; P. Kopietz & Schonhammer 1996; Khveshchenko, R. Hlubina, T.M. Rice 1994 et al; C. Castellani, Di Castro, W. Metzner 1994…… Main assumption of all these works: long range interaction.

6. 17.11.20156 I.L. Aleiner & K. B. Efetov 2006, Method of quasiclassical Green functions supplemented by integration over supervectors Logarithmic contributions to specific heat and susceptibility are found. Good agreement with known results in 1D. No restriction on the range of interaction but not a full account of effects of the Fermi surface curvature in d>1. In all the approaches only low energy excitations are considered: no chance for using in numerics. Present work: Exact mapping fermion models onto bosonic ones. New possibilities for both analytical and numerical computations.

7. 17.11.20157 Starting Hamiltonian H The interaction, tunneling and dimensionality are arbitrary! Small simplification of the formulas

8. 17.11.20158 Hubbard-Stratonovich Transformation with a real field

9. 17.11.20159 Another representation for Basis of auxiliary field Monte Carlo simulations (Blankenbecler, Scalapino, Sugar (1981)) Procedure: subdividing the interval into time slices and calculation for each slice recursively. However, can in such a procedure be both positive and negative: sign problem! Not convenient for analytical calculations either.

10. 17.11.201510 Good news: can exactly be obtained from a bosonic model! Derivation of the model (main steps): -fermionic Green function in the external field. Boundary conditions

11. 17.11.201511 Looking for the solution: (in spirits of a Schwinger Ansatz) is the Green function of the ideal Fermi gas Equations for and Bosonic periodic boundary conditions: Solving these equations is not convenient. Solution is not unambiguous.

12. 17.11.201512 New function A: -Fourier transform of the Fermi distribution or Then, the “partition function” is What is good in that? How to find A without solving equation for T?

13. 17.11.201513 “Free lunch”: a) b) an operator 1. Act with on a) and multiply by from the right. 2. Multiply b) by from the left. 3. Subtract the obtained equations from each other.

14. 17.11.201514 Final equation for A Partition function Boundary conditions Purely bosonic problem! Linear (almost separable) real equation for A New possibilities for both numerical and analytical studies!

15. 17.11.201515 How can one calculate numerically ? (Non-expert’s view) Introduce the Green function g for the linear equation and represent it in the form: where the operator P is Possibility to do recursion in time (as usual)! Compensation of the Bose-denominator no singularities in g. In the absence of singularities, the exponent remains real for any subdivision of the interval into slices is positive: no sign problem!

16. 17.11.201516 Perturbation theory: expanding in and subsequent integration over this field. Main order: RPA-like formula (first order in the expansion in collective excitations).

17. 17.11.201517 Analytical calculations: BRST (Becchi, Rouet, Stora, Tyutin) -possibility of integration over the auxiliary field before doing approximations How to calculate if is the solution of the equation ? A well known trick: Next step: -Grassmann variables

18. 17.11.201518 Description with a supersymmetric action and superfields Introducing new Grassmann variables and superfields is anticommuting (!) The “partition function” as the functional integral over Gaussian averaging over can immediately be performed!

19. 17.11.201519 - partition function of the ideal Fermi gas Final superfield theory (still exact). is the bare action (fully supersymmetric)

20. 17.11.201520 Logarithmic contributions exist in any dimensions and they can be studied by RG. Reduction of the exact model to a low energy one is convenient. Variety of phenomena for, e.g., cuprates and other strongly correlated systems can be attacked in this way. The terms and are invariant under the transformation of the fields : -anticommuting variables (Almost) supersymmetry transformation. What to calculate?

21. 17.11.201521 Conclusions. The model of interacting fermions can be bosonized in any dimension for any reasonable interaction: free lunch but….. What about a free dinner? Can the bosonization be the key to the ultimate solution of the sign problem? Can one solve non-trivial models (e.g., models for high temperature cuprates or for quantum phase transitions) using the supersymetric model for collective excitations? To be answered