The purpose of this review is to provide a comprehensive pedagogical introduction into Keldysh technique for interacting out-of-equilibrium fermionic and bosonic systems. The emphasis is placed on a functional integral representation of the underlying microscopic models. A large part of the review is devoted to derivation and applications of the non-linear σ-model for disordered metals and superconductors. We discuss topics such as transport properties, mesoscopic effects, counting statistics, interaction corrections, kinetic equations, etc. The section devoted to disordered superconductors includes the Usadel equation, fluctuation corrections, time-dependent Ginzburg–Landau theory, proximity and Josephson effects, etc.

We present an analytic theory of quantum criticality in quasi-one-dimensional topological Anderson insulators. We describe these systems in terms of two parameters $(g,\chi )$ representing localization and topological properties, respectively. Certain critical values of $\chi$ (half-integer for $\mathbb{Z}$ classes, or zero for ${\mathbb{Z}}_2$ classes) define phase boundaries between distinct topological sectors. Upon increasing system size, the two parameters exhibit flow similar to the celebrated two-parameter flow of the integer quantum Hall insulator. However, unlike the quantum Hall system, an exact analytical description of the entire phase diagram can be given in terms of the transfer-matrix solution of corresponding supersymmetric nonlinear sigma models. In ${\mathbb{Z}}_2$ classes we uncover a hidden supersymmetry, present at the quantum critical point.

We study the short-time behavior of the probability distribution $\mathcal{P}(H,t)$ of the surface height $h(x=0,t)=H$ in the Kardar-Parisi-Zhang (KPZ) equation in $1+1$ dimension. The process starts from a stationary interface: $h(x,t=0)$ is given by a realization of two-sided Brownian motion constrained by $h(0,0)=0$. We find a singularity of the large deviation function of $H$ at a critical value $H=H_c$. The singularity has the character of a second-order phase transition. It reflects spontaneous breaking of the reflection symmetry $x\leftrightarrow -x$ of optimal paths $h(x,t)$ predicted by the weak-noise theory of the KPZ equation. At $\left|H\right|\gg |H_c|$ the corresponding tail of $\mathcal{P}\left(H\right)$ scales as $-ln\mathcal{P}\sim {\left|H\right|}^{3/2}/t^{1/2}$ and agrees, at any $t>0$, with the proper tail of the Baik-Rains distribution, previously observed only at long times. The other tail of $\mathcal{P}$ scales as $-ln\mathcal{P}\sim {\left|H\right|}^{5/2}/t^{1/2}$ and coincides with the corresponding tail for the sharp-wedge initial condition.

We study the probability distribution $\mathcal{P}(H,t,L)$ of the surface height $h(x=0,t)=H$ in the Kardar-Parisi-Zhang (KPZ) equation in $1+1$ dimension when starting from a parabolic interface, $h(x,t=0)=x^2/L$. The limits of $L\to \infty$ and $L\to 0$ have been recently solved exactly for any $t>0$. Here we address the early-time behavior of $\mathcal{P}(H,t,L)$ for general $L$. We employ the weak-noise theory—a variant of WKB approximation—which yields the optimal history of the interface, conditioned on reaching the given height $H$ at the origin at time $t$. We find that at small $H\mathcal{P}(H,t,L)$ is Gaussian, but its tails are non-Gaussian and highly asymmetric. In the leading order and in a proper moving frame, the tails behave as $-ln\mathcal{P}=f_{+}{\left|H\right|}^{5/2}/t^{1/2}$ and $f_{-}{\left|H\right|}^{3/2}/t^{1/2}$. The factor $f_{+}(L,t)$ monotonically increases as a function of $L$, interpolating between time-independent values at $L=0$ and $L=\infty$ that were previously known. The factor $f_{-}$ is independent of $L$ and $t$, signaling universality of this tail for a whole class of deterministic initial conditions.

We investigate existence of replica off-diagonal solutions in the field-theoretical description of Sachdev-Ye-Kitaev model. To this end we evaluate a set of local and non-local dynamic correlation functions in the long time limit. We argue that the structure of the soft-mode Schwarzian action is qualitatively different in replica-diagonal vs. replica-off-diagonal scenarios, leading to distinct long-time predictions for the correlation functions. We then evaluate the corresponding correlation functions numerically and compare the simulations with analytical predictions of replica-diagonal and replica-off-diagonal calculations. We conclude that all our numerical results are in a quantitative agreement with the theory based on the replica-diagonal saddle point plus Schwarzian and massive Gaussian fluctuations (the latter do contain replica off-diagonal components). This seems to exclude any contributions from replica-off-diagonal saddle points, at least on the time scales shorter than the inverse many-body level spacing.

We study the short-time distribution $\mathcal{P}\left(H,L,t\right)$ of the two-point two-time height difference $H=h(L,t)-h(0,0)$ of a stationary Kardar-Parisi-Zhang interface in 1+1 dimension. Employing the optimal-fluctuation method, we develop an effective Landau theory for the second-order dynamical phase transition found previously for $L=0$ at a critical value $H=H_c$. We show that $\left|H\right|$ and $L$ play the roles of inverse temperature and external magnetic field, respectively. In particular, we find a first-order dynamical phase transition when $L$ changes sign, at supercritical $H$. We also determine analytically $\mathcal{P}\left(H,L,t\right)$ in several limits away from the second-order transition. Typical fluctuations of $H$ are Gaussian, but the distribution tails are highly asymmetric. The tails $-ln\mathcal{P}\sim {\left|H\right|}^{3/2}/\sqrt{t}$ and $-ln\mathcal{P}\sim {\left|H\right|}^{5/2}/\sqrt{t}$, previously found for $L=0$, are enhanced for $L\ne 0$. At very large $\left|L\right|$ the whole height-difference distribution $\mathcal{P}\left(H,L,t\right)$ is time-independent and Gaussian in $H, -ln\mathcal{P}\sim {\left|H\right|}^2/\left|L\right|$, describing the probability of creating a ramplike height profile at $t=0$.