LUTTINGER LIQUIDS WITH MULTIPLE FERMI EDGES: GENERALIZED FISHER-HARTWIG CONJECTURE AND NUMERICAL ANALYSIS OF TOEPLITZ DETERMINANTS
I.V. Protopopov^a,b, D.B. Gutman^c, and A.D. Mirlin^a,d,e
aInstitut für Nanotechnologie, Karlsruhe Institute of Technology, 76021 Karlsruhe, Germany
E-mail: alexander.mirlin@kit.edu
^bL.D. Landau Institute for Theoretical Physics RAS, 119334 Moscow, Russia
^cDepartment of Physics, Bar Ilan University, Ramat Gan 52900, Israel
^dInstitut für Theorie der kondensierten Materie and DFG Center for Functional Nanostructures, Karlsruhe Institute of Technology, 76128 Karlsruhe, Germany
^ePetersburg Nuclear Physics Institute, 188300 St. Petersburg, Russia

Received 28 March 2012; accepted 7 June 2012

It has been shown that solutions of a number of many-body problems out of equilibrium can be expressed in terms of Toeplitz determinants with Fisher-Hartwig (FH) singularities. In the present paper, such Toeplitz determinants are studied numerically. Results of our numerical calculations fully agree with the FH conjecture in an extended form that includes a summation over all FH representations (corresponding to different branches of the logarithms). As specific applications, we consider problems of Fermi edge singularity and tunneling spectroscopy of Luttinger liquid with multiple-step energy distribution functions, including the case of population inversion. In the energy representation, a sum over FH branches produces power-law singularities at multiple edges.