Entanglement has become central for the characterization of quantum matter both in and out of equilibrium. In a dynamical context entanglement exhibits universal linear temporal growth in generic systems, which stems from the underlying linear light cones as they occur in planar geometries. Inhomogeneous spacetimes can lead, however, to strongly bent trajectories. While such bent trajectories crucially impact correlation spreading and therefore the light-cone structure, it has remained elusive how this influences the entanglement dynamics. In this work we investigate the real-time evolution of the entanglement entropy in one-dimensional quantum systems after quenches which change the underlying spacetime background of the Hamiltonian. Concretely, we focus on the Rindler space describing the spacetime in close vicinity to a black hole. As a main result we find that entanglement grows sublinearly in a generic fashion both for interacting and noninteracting quantum matter. We further observe that the asymptotic relaxation becomes exponential, as opposed to algebraic for planar Minkowski spacetimes, and that in the vicinity of the black hole the relaxation time for large subsystems becomes independent of the subsystem size. We study entanglement dynamics both for the case of noninteracting fermions, allowing for exact numerical solutions, and for random unitary circuits representing a paradigmatic class of ergodic systems.
Inhomogeneous quantum critical systems in one spatial dimension have been studied by using conformal field theory in static curved backgrounds. Two interesting examples are the free fermion gas in the harmonic trap and the inhomogeneous XX spin chain called rainbow chain. For conformal field theories defined on static curved spacetimes characterised by a metric which is Weyl equivalent to the flat metric, with the Weyl factor depending only on the spatial coordinate, we study the entanglement hamiltonian and the entanglement spectrum of an interval adjacent to the boundary of a segment where t…
The thermal entanglement of a two-qubit anisotropic Heisenberg $XYZ$ chain under an inhomogeneous magnetic field b is studied. It is shown that when inhomogeneity is increased to certain value, the entanglement can exhibit a larger revival than that of less values of b. The property is both true for zero temperature and a finite temperature. The results also show that the entanglement and critical temperature can be increased by increasing inhomogeneous exteral magnetic field.
Conformal field theories in curved backgrounds have been used to describe inhomogeneous one-dimensional systems, such as quantum gases in trapping potentials and non-equilibrium spin chains. This approach provided, in a elegant and simple fashion, non-trivial analytic predictions for quantities, such as the entanglement entropy, that are not accessible through other methods. Here, we generalise this approach to low-lying excited states, focusing on the entanglement and relative entropies in an inhomogeneous free-fermionic system. Our most important finding is that the universal scaling functio…
We investigate entanglement of strongly interacting fermions in spatially inhomogeneous environments. To quantify entanglement in the presence of spatial inhomogeneity, we propose a local-density approximation (LDA) to the entanglement entropy, and a nested LDA scheme to evaluate the entanglement entropy on inhomogeneous density profiles. These ideas are applied to models of electrons in superlattice structures with different modulation patterns, electrons in a metallic wire in the presence of impurities, and phase-separated states in harmonically confined many-fermion systems, such as electro…
Entanglement dynamics after quenches with inhomogeneous Hamiltonians
Dipartimento di Fisica dell’Università di Pisa（比萨大学物理系）；Dipartimento di Fisica e Astronomia “Galileo Galilei”, Università di Padova（帕多瓦大学伽利略·伽利莱物理与天文学系）；INFN Sezione di Pisa（意大利国家核物理研究所比萨分部）；Istituto di Informatica e Telematica, CNR（意大利国家研究委员会信息学与电信研究所）
We give a pedagogical introduction to the Generalized Hydrodynamic approach to inhomogeneous quenches in integrable many-body quantum systems. We review recent applications of the theory, focusing in particular on two classes of problems: bipartitioning protocols and trap quenches, which represent two prototypical examples of broken translational symmetry in either the system initial state or post-quench Hamiltonian. We report on exact results that have been obtained for generic time-dependent correlation functions and entanglement evolution, and discuss in detail the range of applicability of…
We develop a general framework to compute the scaling of entanglement entropy in inhomogeneous one-dimensional quantum systems belonging to the Luttinger liquid universality class. While much insight has been gained in homogeneous systems by making use of conformal field theory techniques, our focus is on systems for which the Luttinger parameter $K$ depends on position, and conformal invariance is broken. An important point of our analysis is that contributions stemming from the UV cutoff have to be treated very carefully, since they now depend on position. We show that such terms can be remo…
Spacetime picture for entanglement generation in noisy fermion chains
Rudolf Peierls Centre for Theoretical Physics（鲁道夫·皮尔斯理论物理中心）；Laboratoire de Physique de l’École Normale Supérieure, CNRS, ENS & Université PSL, Sorbonne Université, Université Paris Cité（巴黎高等师范学院物理实验室，法国国家科学研究中心，巴黎高等师范学院及巴黎文理研究大学，索邦大学，巴黎西岱大学）
arXiv:1205.2953 defines an entropy for a gaussian scalar field $ϕ$ in an arbitrary region of either a causal set or a continuous spacetime, given only the correlator $\langleϕ(x)ϕ(y)\rangle$ within the region. As a first application, we compute numerically the entanglement entropy in two cases where the asymptotic form is known or suspected from conformal field theory, finding excellent agreement when the required ultraviolet cutoff is implemented as a truncation on spacetime mode-sums. We also show how the symmetry of entanglement entropy reflects the fact that $RS$ and $SR$ share the same ei…
Timelike Entanglement from Spacetime Density Matrices: A Lattice Realization
School of Physics, Huazhong University of Science and Technology（华中科技大学物理学院）；Center for Gravitational Physics and Quantum Information, Yukawa Institute for Theoretical Physics, Kyoto University（京都大学汤川理论物理研究所引力物理与量子信息中心）
We consider a molecular dynamics method, the so-called flea gas for computing the evolution of entanglement after inhomogeneous quantum quenches in an integrable quantum system. In such systems the evolution of local observables is described at large space-time scales by the Generalized Hydrodynamics approach, which is based on the presence of stable, ballistically propagating quasiparticles. Recently it was shown that the GHD approach can be joined with the quasiparticle picture of entanglement evolution, providing results for entanglement growth after inhomogeneous quenches. Here we apply th…
The discovery of novel entanglement patterns in quantum many-body systems is a prominent research direction in contemporary physics. Here we provide the example of a spin chain with random and inhomogeneous couplings that in the ground state exhibits a very unusual area law violation. In the clean limit, i.e., without disorder, the model is the rainbow chain and has volume law entanglement. We show that, in the presence of disorder, the entanglement entropy exhibits a power-law growth with the subsystem size, with an exponent 1/2. By employing the Strong Disorder Renormalization Group (SDRG) f…
A more complete understanding of entanglement entropy in a covariant manner could inform the search for quantum gravity. We build on work in this direction by extending previous results to disjoint regions in $1+1$D. We investigate the entanglement entropy of a scalar field in disjoint intervals within the causal set framework, using the spacetime commutator and correlator, $i\mathbfΔ$ and $\mathbf{W}$ (or the Pauli-Jordan and Wightman functions), respectively. A new truncation scheme for disjoint causal diamonds is presented, which follows from the single diamond truncation scheme. We investi…
