Parity effects in the scaling of block entanglement in gapless spin chains

Universal Entanglement and Correlation Measure in Two-Dimensional Conformal Field Theories

We calculate the amount of entanglement shared by two intervals in the ground state of a (1+1)-dimensional conformal field theory (CFT), quantified by an entanglement measure E based on the computable cross norm (CCNR) criterion. Unlike negativity or mutual information, we show that E has a universal expression even for two disjoint intervals, which depends only on the geometry, the central charge c, and the thermal partition function of the CFT. We prove this universal expression in the replica approach, where the Riemann surface for calculating E at each order n is always a torus topologically. By analytic continuation, the result of n=1/2 gives the value of E. Furthermore, the results of other values of n also yield meaningful conclusions: The n=1 result gives a general formula for the two-interval purity, which enables us to calculate the Rényi-2 N-partite information for N≤4 intervals; while the n=∞ result bounds the correlation function of the two intervals. We verify our findings numerically in the spin-1/2 XXZ chain, whose ground state is described by the Luttinger liquid.

Negativity Hamiltonian: An Operator Characterization of Mixed-State Entanglement

In the context of ground states of quantum many-body systems, the locality of entanglement between connected regions of space is directly tied to the locality of the corresponding entanglement Hamiltonian: the latter is dominated by local, few-body terms. In this work, we introduce the negativity Hamiltonian as the (non-Hermitian) effective Hamiltonian operator describing the logarithm of the partial transpose of a many-body system. This allows us to address the connection between entanglement and operator locality beyond the paradigm of bipartite pure systems. As a first step in this direction, we study the structure of the negativity Hamiltonian for fermionic conformal field theories and a free-fermion chain: in both cases, we show that the negativity Hamiltonian assumes a quasilocal functional form, that is captured by simple functional relations.

Spin Models, Dynamics, and Criticality with Atoms in Tilted Optical Superlattices

We show that atoms in tilted optical superlattices provide a platform for exploring coupled spin chains of forms that are not present in other systems. In particular, using a period-2 superlattice in one dimension, we show that coupled Ising spin chains with XZ and ZZ spin coupling terms can be engineered. We use optimized tensor network techniques to explore the criticality and nonequilibrium dynamics in these models, finding a tricritical Ising point in regimes that are accessible in current experiments. These setups are ideal for studying low-entropy physics, as initial entropy is "frozen-out" in realizing the spin models, and provide an example of the complex critical behavior that can arise from interaction-projected models.

Symmetry-Resolved Entanglement in Many-Body Systems

Similarly to the system Hamiltonian, a subsystem's reduced density matrix is composed of blocks characterized by symmetry quantum numbers (charge sectors). We present a geometric approach for extracting the contribution of individual charge sectors to the subsystem's entanglement measures within the replica trick method, via threading appropriate conjugate Aharonov-Bohm fluxes through a multisheet Riemann surface. Specializing to the case of 1+1D conformal field theory, we obtain general exact results for the entanglement entropies and spectrum, and apply them to a variety of systems, ranging from free and interacting fermions to spin and parafermion chains, and verify them numerically. We find that the total entanglement entropy, which scales as lnL, is composed of sqrt[lnL] contributions of individual subsystem charge sectors for interacting fermion chains, or even O(L^{0}) contributions when total spin conservation is also accounted for. We also explain how measurements of the contribution to the entanglement from separate charge sectors can be performed experimentally with existing techniques.

Entanglement convertibility by sweeping through the quantum phases of the alternating bonds XXZ chain

We study the entanglement structure and the topological edge states of the ground state of the spin-1/2 XXZ model with bond alternation. We employ parity-density matrix renormalization group with periodic boundary conditions. The finite-size scaling of Rényi entropies S2 and S∞ are used to construct the phase diagram of the system. The phase diagram displays three possible phases: Haldane type (an example of symmetry protected topological ordered phases), Classical Dimer and Néel phases, the latter bounded by two continuous quantum phase transitions. The entanglement and non-locality in the ground state are studied and quantified by the entanglement convertibility. We found that, at small spatial scales, the ground state is not convertible within the topological Haldane dimer phase. The phenomenology we observe can be described in terms of correlations between edge states. We found that the entanglement spectrum also exhibits a distinctive response in the topological phase: the effective rank of the reduced density matrix displays a specifically large "susceptibility" in the topological phase. These findings support the idea that although the topological order in the ground state cannot be detected by local inspection, the ground state response at local scale can tell the topological phases apart from the non-topological phases.

Adiabatic tracking of a state: a new route to nonequilibrium physics

We present a novel numerical approach to track the response of a quantum system to an external perturbation that is progressively switched on. The method is applied, within the framework of the density matrix renormalization group technique, to track current-carrying states of interacting fermions in one dimension and in the presence of an Aharonov-Bohm magnetic flux. This protocol allows us to access highly excited states. We also discuss the connection with the entanglement entropy of these excited states.

Shell-filling effect in the entanglement entropies of spinful fermions

We consider the von Neumann and Rényi entropies of the one-dimensional quarter-filled Hubbard model. We observe that for periodic boundary conditions the entropies exhibit an unexpected dependence on system size: for L=4 mod 8 the results are in agreement with expectations based on conformal field theory, while for L=0 mod 8 additional contributions arise. We explain this observation in terms of a shell-filling effect and develop a conformal field theory approach to calculate the extra term in the entropies. Similar shell-filling effects in entanglement entropies are expected to be present in higher dimensions and for other multicomponent systems.

Entanglement negativity in quantum field theory

We develop a systematic method to extract the negativity in the ground state of a 1+1 dimensional relativistic quantum field theory, using a path integral formalism to construct the partial transpose ρ(A)(T(2) of the reduced density matrix of a subsystem [formula: see text], and introducing a replica approach to obtain its trace norm which gives the logarithmic negativity E=ln//ρ(A)(T(2))//. This is shown to reproduce standard results for a pure state. We then apply this method to conformal field theories, deriving the result E~(c/4)ln[ℓ(1)ℓ(2)/(ℓ(1)+ℓ(2))] for the case of two adjacent intervals of lengths ℓ(1), ℓ(2) in an infinite system, where c is the central charge. For two disjoint intervals it depends only on the harmonic ratio of the four end points and so is manifestly scale invariant. We check our findings against exact numerical results in the harmonic chain.

Entanglement entropy of one-dimensional gases

We introduce a systematic framework to calculate the bipartite entanglement entropy of a spatial subsystem in a one-dimensional quantum gas which can be mapped into a noninteracting fermion system. To show the wide range of applicability of the proposed formalism, we use it for the calculation of the entanglement in the eigenstates of periodic systems, in a gas confined by boundaries or external potentials, in junctions of quantum wires, and in a time-dependent parabolic potential.

Entanglement of low-energy excitations in conformal field theory

In a quantum critical chain, the scaling regime of the energy and momentum of the ground state and low-lying excitations are described by conformal field theory (CFT). The same holds true for the von Neumann and Rényi entropies of the ground state, which display a universal logarithmic behavior depending on the central charge. In this Letter we generalize this result to those excited states of the chain that correspond to primary fields in CFT. It is shown that the nth Rényi entropy is related to a 2n-point correlator of primary fields. We verify this statement for the critical XX and XXZ chains. This result uncovers a new link between quantum information theory and CFT.