Contents. 1.1 Prerequisites and textbooks Physical phenomena and theoretical tools The path integrals... 9

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1 Preface v Chapter 1 Introduction Prerequisites and textbooks Physical phenomena and theoretical tools The path integrals Quantum statistical mechanics of a particle Few Gaussian integrals Another take on the path integral for the harmonic oscillator Chapter 2 Feynman diagrams in Quantum/Statistical mechanics Anharmonic oscillators alaqmtextbooks The first Feynman diagrams Disconnected diagrams Diagrams in! representation and their re-summation ix

2 x Chapter 3 Real scalar field and the renormalization group Path integrals for a scalar field The Matsubara sums lead to particle/hole interpretation of the diagrams Weakly interacting scalar field Symmetry breaking and the Landau theory of the second order phase transitions More on critical indices Renormalization group and Wilson s -expansion Chapter 4 Complex scalar fields Phase symmetry and its breaking The conserved charge and the chemical potential The non-relativistic approximation and normalization Weakly coupled Bose gas and Bose-Einstein Condensation Renormalized quasiparticles and condensates Summary of the perturbation theory for weakly coupled Bose gas Chapter 5 Liquid 4 He History Superfluidity in experiments Elementary excitations Quasiparticle decays, in weakly coupled Bose gas and in the He-II Landau s criterium of superfluidity

3 xi 5.6 Rotation and vortices Chapter 6 Bose-Einstein Condensation of trapped ultracold atoms The history of BEC Depletion of the condensate Gross-Pitaevsky equation and repulsive/attractive Bose gas Rotation Chapter 7 The electron gas Fermi systems to be discussed Path integrals and Matsubara sums for fermions Weakly interacting Fermi gas Cold electron gas The polarization diagram and the Lindhard function The strongly coupled electron fluid in graphene Chapter 8 The nuclear matter The nuclear forces The nuclear matter in the relativistic mean field model Nuclear forces and renormalization group Nuclear matter and the renormalization group Relativistic mean field and the optical potential Chapter 9 Cooper Pairing and the BCS theory of superconductivity177

4 xii 9.1 Superconducting metals Cooper pairing A continuous renormalization group (RG) Renormalization group and cold Fermi systems BCS Pairing and Gorkov s abnormal Green functions Chapter 10 Pairing in liquid 3 He and in the nuclear matter Liquid 3 He and Landau Fermi Liquids Zero sound and the Peierls instability from the re-summed Lindhard diagram Superfluidity in 3 He Pairing in nuclei and nuclear matter Chapter 11 Vortices and the topological matter The BKT phase transition of the vortex matter in two dimensions Vortices in superconductors Type I and II superconductors Phase transition in strongly rotating nuclei Glitches of the pulsars Chapter 12 Strongly coupled fermionic gases Weak and strong coupling regimes in trapped fermionic gases Global phase diagram

5 xiii 12.3 Thermodynamics and kinetics at strong coupling Chapter 13 Numerical evaluation of the path integrals Numerical evaluation of multi-dimensional integrals The first steps: PIMC for few-body systems with distinguishable particles Liquid 4 He PIMC simulations: Bose-clusters, condensation and the critical temperature Path integrals for fermions Chapter 14 Quantum field theories on the lattice and supercomputers History Gauge invariant Abelian theory on the lattice The non-abelian gauge theory on the lattice The confining potential Chapter 15 Theory of quark-gluonic plasma Overview and scales of QGP The perturbative formalism of QCD The polarization operator Cold quark matter and the perturbation theory Ring diagram re-summation

6 xiv 15.6 IR divergences and magnetic sector Where do the perturbative series converge? HTL re-summations and the quasiparticle gas Lattice gauge theory simulations at finite T Chapter 16 Quark-Gluon Plasma in experiment Experimental quest for quark-gluon plasma Mapping the phase diagram Collective flow and viscosity of QGP Relativistic Hydrodynamics The Little Bang versus the Big Bang : sound circles and fluctuations Phase transitions in the Big Bang: sound cascades and gravity waves. 348 Chapter 17 The QCD vacuum I. Monopoles and confinement Confinement and the flux tubes Monopoles in non-abelian theories Electric-magnetic duality and the RG flow Strongly coupled QGP as a dual plasma Chapter 18 The QCD vacuum II. The chiral symmetries and their breaking Light quarks and the chiral symmetries The (non-existent) U a (1) chiral symmetry

7 xv The chiral anomaly near the UV cuto Chiral anomalies, the IR approach The Nambu-Iona-Lasinio model Gauge topology and the chiral symmetry breaking Chapter 19 Chiral matter Electrodynamics in a CP violating matter Chiral magnetic e ect (CME) and the chiral anomaly Chiral vortical e ect The chiral waves Chapter 20 Cold quark matter and color superconductivity Quark Cooper pairs Topology-induced color superconductivity: the 2SC phase Three flavor QCD: the CFL phase Magnetic pairing in asymptotically dense matter Chapter 21 The QCD vacuum III: Instanton-dyons Non-zero holonomy and the finite-temperature Higgsing Instanton-dyons and their ensembles Modifying quark periodicity properties in the Euclidean time

8 xvi Chapter 22 Parting comments The Feynman diagrams and their re-summations Bose-Einstein Condensation and reality of the Cooper pairs Pairing can occur even without the Fermi surface! The topological matter Renormalization group flows and dualities Bibliography 499