Conformal field theory (CFT) has been extremely successful in describing large-scale universal effects in one-dimensional (1D) systems at quantum critical points. Unfortunately, its applicability in condensed matter physics has been limited to situations in which the bulk is uniform because CFT describes low-energy excitations around some energy scale, taken to be constant throughout the system. However, in many experimental contexts, such as quantum gases in trapping potentials and in several out-of-equilibrium situations, systems are strongly inhomogeneous. We show here that the powerful CFT methods can be extended to deal with such 1D situations, providing a few concrete examples for non-interacting Fermi gases. The system's inhomogeneity enters the field theory action through parameters that vary with position; in particular, the metric itself varies, resulting in a CFT in curved space. This approach allows us to derive exact formulas for entanglement entropies which were not known by other means.

Abstract. Motivated by quantum quenches in spin chains, a one-dimensional toy-model of fermionic particles evolving in imaginary-time from a domain-wall initial state is solved. The main interest of this toymodel is that it exhibits the arctic circle phenomenon, namely a spatial phase separation between a critically fluctuating region and a frozen region. Large-scale correlations inside the critical region are expressed in terms of correlators in a (euclidean) two-dimensional massless Dirac field theory. It is observed that this theory is inhomogenous: the metric is position-dependent, so it is in fact a Dirac theory in curved space. The technique used to solve the toy-model is then extended to deal with the transfer matrices of other models: dimers on the honeycomb and square lattice, and the six-vertex model at the free fermion point (∆ = 0). In all cases, explicit expressions are given for the long-range correlations in the critical region, as well as for the underlying Dirac action. Although the setup developed here is heavily based on fermionic observables, the results can be translated into the language of height configurations and of the gaussian free field, via bosonization. Correlations close to the phase boundary and the generic appearance of Airy processes in all these models are also briefly revisited in the appendix.

We consider the out-of-equilibrium dynamics generated by joining two domains with arbitrary opposite magnetisations. We study the stationary state which emerges by the unitary evolution via the spin 1/2 XXZ Hamiltonian, in the gapless regime, where the system develops a stationary spin current. Using the generalized hydrodynamic approach, we present a simple formula for the space-time profile of the spin current and the magnetisation exact in the limit of large times. As a remarkable effect, we show that the stationary state has a strongly discontinuous dependence on the strength of interaction. This feature allows us to give a qualitative estimation for the transient behavior of the current which is compared with numerical simulations. Moreover, we analyse the behavior around the edge of the magnetisation profile and we argue that, unlike the XX free-fermionic point, interactions always prevent the emergence of a Tracy-Widom scaling.Introduction.-Recent experimental developments with cold atoms [1] have given a new perspective to the study of non-equilibrium transport under coherent evolution. As an example, the measurement of conductances well beyond the regime of linear response has provided clear examples of the thermoelectric effect [2,3]. The simplest protocol to induce an out-of-equilibrium behavior is the one of quantum quenches, in which the system is prepared in an equilibrium state of the initial Hamiltonian H 0 , which is suddenly switched to H, thus inducing a non-trivial time-evolution [4][5][6][7]. Then, in describing the long-time dynamics, a fundamental role is played by the conserved quantities of H, i.e. the set of local (or quasi-local [8]) operators {Q k } satisfying [Q k , H] = 0. As the system is isolated, the expectation value of these conserved quantities remains constant during the evolution. For homogeneous systems, these conditions are sufficient to predict the exact behavior of any local observable at long times: this is based on assuming equilibration to the generalized Gibbs ensemble (GGE), which results from the maximization of entropy given the constraints imposed by conserved quantities [9]. This principle suggests a dichotomy between generic models, for which a finite number of conserved quantities exist (i.e. the Hamiltonian and few others) and integrable ones, which instead present an infinite number of them [10]. Nowadays, the GGE scheme has been validated by several experiments [11][12][13][14][15][16][17][18] and theoretical works, employing free theories [19][20][21], integrability [22][23][24][25][26][27] and numerical methods [9,[28][29][30].

We consider the non-equilibrium physics induced by joining together two tight binding fermionic chains to form a single chain. Before being joined, each chain is in a many-fermion ground state. The fillings (densities) in the two chains might be the same or different. We present a number of exact results for the correlation functions in the non-interacting case. We present a short-time expansion, which can sometimes be fully resummed, and which reproduces the so-called 'light cone' effect or wavefront behavior of the correlators. For large times, we show how all interesting physical regimes may be obtained by stationary phase approximation techniques. In particular, we derive semiclassical formulas in the case when both time and positions are large, and show that these are exact in the thermodynamic limit. We present subleading corrections to the large-time behavior, including the corrections near the edges of the wavefront. We also provide results for the return probability or Loschmidt echo. In the maximally inhomogeneous limit, we prove that it is exactly gaussian at all times. The effects of interactions on the Loschmidt echo are also discussed.PACS numbers: 75.10. Fd, 75.45.Gm, 05.60.Gg Introduction -Local quantum quenches are a particularly neat setup to address theoretical questions about non-equilibrium current-carrying stationary states as well as transport properties of many-body isolated quantum systems. The characterization of the long-time behavior of local correlation functions unveils universal features of the quantum dynamics [1][2][3][4][5] and paves the way to the construction of effective field theories capable of capturing them. Analytic results are relevant as benchmarks for cold atom experiments that very recently [6][7][8] started to investigate particle and energy transport under a unitary dynamics.In this work, we study a quench which, although injecting an extensive amount of energy into the system, has mostly local effects. We take two uniformly filled long tight binding chains in their respective ground-states, but with a different number of fermions. At time t = 0 the two edges are connected so that it turns into a single chainThe hopping and interactions between sites j = 0 and j = 1 are initially absent; the initial state |ψ 0 = |ψ l |ψ r is the tensor product of the ground states (with specified occupancies) of the two decoupled chains. For unequal fillings, one expects some particle current at time t > 0. We denote by k l F (k r F ) the Fermi momenta on the left (right), so that the particle number is2 ) on the left (right). For simplicity we focus on the symmetric case k l F + k r F = π, but extension to other values is straightforward. It is useful to keep two limits in mind. When the fillings are equal k l F = k r F = π/2 there is no particle current. This particular quench was studied in [1][2][3]9], using low-energy field theory, and belongs to the class of Fermi-edge problems [10,11]. The other simple limit is k The motivation for the present study is two-fold. First, ou...