Quantum Entanglement Under Lorentz Transformation

We study the properties of quantum entanglement in moving frames, with a non-maximally entangled bipartite state: $|\phi\rangle=\sqrt{1-\varepsilon}|{\uparrow\uparrow}\rangle +\sqrt{\varepsilon}|{\downarrow\downarrow}\rangle$ , (0<ε<1). We calculate the concurrence of this state under Lorentz transformation and show that if the momenta part of the spin-1/2 pair is appropriately correlated, the concurrence is invariant ( $\mathcal {C}(\rho)=2\sqrt{\varepsilon-\varepsilon^{2}}$ ); despite the entanglement of this state is not maximal, there is no transfer of entanglement between spin and momentum.     

On systems having Poincaré and Galileo symmetry

Using the wave equation in d≥1$d\ge 1$ space dimensions it is illustrated how dynamical equations may be simultaneously Poincaré and Galileo covariant with respect to different sets of independent variables. This provides a method to obtain dynamics-dependent representations of the kinematical symmetries. When the field is a displacement function both symmetries have a physical interpretation. For d=1$d=1$ the Lorentz structure is utilized to reveal hitherto unnoticed features of the non-relativistic Chaplygin gas including a relativistic structure with a limiting case that exhibits the Carroll group, and field-dependent symmetries and associated Noether charges. The Lorentz transformations of the potentials naturally associated with the Chaplygin system are given. These results prompt the search for further symmetries and it is shown that the Chaplygin equations support a nonlinear superposition principle. A known spacetime mixing symmetry is shown to decompose into label-time and superposition symmetries. It is shown that a quantum mechanical system in a stationary state behaves as a Chaplygin gas. The extension to d>1$d>1$ is used to illustrate how the physical significance of the dual symmetries is contingent on the context by showing that Maxwell’s equations exhibit an exact Galileo covariant formulation where Lorentz and gauge transformations are represented by field-dependent symmetries. A natural conceptual and formal framework is provided by the Lagrangian and Eulerian pictures of continuum mechanics.     

Relativistic particles and the geometry of 4-D null curves

We study actions in (d+1)$(d+1)$-dimensions associated with null curves, mainly when d=3$d=3$, whose Lagrangian is a linear function on the curvature of the particle path, showing that null helices are always possible trajectories of the particles. We find Killing vector fields along critical curves of the action which correspond to the linear and the angular momenta of the particle. They provide two constants of the motion which can be interpreted in terms of the mass and the spin of the system. Moreover, we are able to integrate both the Euler–Lagrange equations and the Cartan equations in cylindrical coordinates around a certain plane.     

Quantum criticality and Yang–Mills gauge theory

We present a family of nonrelativistic Yang–Mills gauge theories in D+1$D+1$ dimensions whose free-field limit exhibits quantum critical behavior with gapless excitations and dynamical critical exponent z=2$z=2$. The ground state wavefunction is intimately related to the partition function of relativistic Yang–Mills in D   dimensions. The gauge couplings exhibit logarithmic scaling and asymptotic freedom in the upper critical spacetime dimension, equal to 4+1$4+1$. The theories can be deformed in the infrared by a relevant operator that restores Poincaré invariance as an accidental symmetry. In the large-N limit, our nonrelativistic gauge theories can be expected to have weakly curved gravity duals.     

Ground State,Magnetization Process and Bipartite Quantum Entanglement of a Spin-1/2 Ising–Heisenberg Model on Planar Lattices of Interconnected Trigonal Bipyramids

The ground state, magnetization scenario and the local bipartite quantum entanglement of a mixed spin-1/2 Ising–Heisenberg model in a magnetic field on planar lattices formed by identical corner-sharing bipyramidal plaquettes is examined by combining the exact analytical concept of generalized decoration-iteration mapping transformations with Monte Carlo simulations utilizing the Metropolis algorithm. The ground-state phase diagram of the model involves six different phases, namely, the standard ferrimagnetic phase, fully saturated phase, two unique quantum ferrimagnetic phases, and two macroscopically degenerate quantum ferrimagnetic phases with two chiral degrees of freedom of the Heisenberg triangular clusters. The diversity of ground-state spin arrangement is manifested themselves in seven different magnetization scenarios with one, two or three fractional plateaus whose values are determined by the number of corner-sharing plaquettes. The low-temperature values of the concurrence demonstrate that the bipartite quantum entanglement of the Heisenberg spins in quantum ferrimagnetic phases is field independent, but twice as strong if the Heisenberg spin arrangement is unique as it is two-fold degenerate.     

Torus amplitudes in minimal Liouville gravity and matrix models

We evaluate one-point correlation numbers on the torus in the Liouville theory coupled to the conformal matter M(2,2p+1)$M(2,2p+1)$. We find agreement with the recent results obtained in the matrix model approach.     

Quantum spherical model with competing interactions

We analyse the phase diagram of a quantum mean spherical model in terms of the temperature T$T$, a quantum parameter g$g$, and the ratio p=−_J2/_J1$p=-J_2/J_1$, where _J1>0$J_1>0$ refers to ferromagnetic interactions between first-neighbour sites along the d$d$ directions of a hypercubic lattice, and _J2<0$J_2<0$ is associated with competing antiferromagnetic interactions between second neighbours along m≤d$m\le d$ directions. We regain a number of known results for the classical version of this model, including the topology of the critical line in the g=0$g=0$ space, with a Lifshitz point at p=1/4$p=1/4$, for d>2$d>2$, and closed-form expressions for the decay of the pair correlations in one dimension. In the T=0$T=0$ phase diagram, there is a critical border, _gc=_gc(p)$g_c=g_c\left(p\right)$ for d≥2$d\ge 2$, with a singularity at the Lifshitz point if d<(m+4)/2$d<(m+4)/2$. We also establish upper and lower critical dimensions, and analyse the quantum critical behavior in the neighborhood of p=1/4$p=1/4$.     

The entropic boundary law in BF theory

We compute the entropy of a closed bounded region of space for pure 3d Riemannian gravity formulated as a topological BF theory for the gauge group SU(2)$\mathrm{SU}(2)$ and show its holographic behavior. More precisely, we consider a fixed graph embedded in space and study the flat connection spin network state without and with particle-like topological defects. We regularize and compute exactly the entanglement for a bipartite splitting of the graph and show it scales at leading order with the number of vertices on the boundary (or equivalently with the number of loops crossing the boundary). More generally these results apply to BF theory with any compact gauge group in any space–time dimension.     

Spin vortices in the Abelian–Higgs model with cholesteric vacuum structure

We continue the study of U(1)$U\left(1\right)$ vortices with cholesteric vacuum structure. A new class of solutions is found which represent global vortices of the internal spin field. These spin vortices are characterized by a non-vanishing angular dependence at spatial infinity, or winding. We show that despite the topological Z₂${\mathbb{Z}}_2$ behavior of SO(3)$SO\left(3\right)$ windings, the topological charge of the spin vortices is of the Z$\mathbb{Z}$ type in the cholesteric. We find these solutions numerically and discuss the properties derived from their low energy effective field theory in 1+1$1+1$ dimensions.     

Optimal symmetric economical phase-covariant quantum cloning

We derive the explicit transformation of the optimal symmetric economical 1→M$1\to M$ phase-covariant quantum cloning machine (EPCCM), which works without ancilla, as well as the corresponding fidelities in 2-dimension. The fidelity (1→M=2k+1 EPCCM$1\to M=2k+1\text{EPCCM}$) is covered with previous contributions [G.M. D'Ariano, et al., Phys. Rev. A 71 (2005) 042327]. Meanwhile, the explicit transformation of the optimal symmetric 1→M$1\to M$ phase-covariant quantum cloning machine (PCCM) (working with ancilla) as well as the corresponding fidelities in 2-dimension is also derived. The results of the PCCM agree with previous contributions [H. Fan, et al., Phys. Rev. A 65 (2001) 012304].     

Hydrogen atom in space with a compactified extra dimension and potential defined by Gauss’ law

We investigate the consequences of one extra spatial dimension for the stability and energy spectrum of the non-relativistic hydrogen atom with a potential defined by Gauss’ law, i.e.    proportional to 1/|x|²$1/{\left|x\right|}^2$. The additional spatial dimension is considered to be either infinite or curled-up in a circle of radius R$R$. In both cases, the energy spectrum is bounded from below for charges smaller than the same critical value and unbounded from below otherwise. As a consequence of compactification, negative energy eigenstates appear: if R$R$ is smaller than a quarter of the Bohr radius, the corresponding Hamiltonian possesses an infinite number of bound states with minimal energy extending at least to the ground state of the hydrogen atom.     

Laser preparation of polarized nuclei: Absorption of one polarized photon is enough and no high spectral resolution is needed to achieve 100% polarization

A new method to prepare photoions with the polarized nuclear spin is proposed. Selected total electron momentum state |J,mJ〉$|J,m_J〉$ is excited by short (pico- or (sub)nanosecond) spectrally broad laser pulse which does not resolve a hyperfine structure thus preparing a superposition of all sublevels of the total angular momentum F   of an atom. Initially unpolarized nuclear spin state becomes highly polarized in a course of subsequent free quantum evolution, and at appropriate time an atom is ionized by another short laser pulse. For the case of nuclear spin I=1/2$I=1/2$, absorption of only one polarized photon is needed to achieve 100% nuclear polarization.     

Optical Abelian lattice gauge theories

We discuss a general framework for the realization of a family of Abelian lattice gauge theories, i.e., link models or gauge magnets, in optical lattices. We analyze the properties of these models that make them suitable for quantum simulations. Within this class, we study in detail the phases of a U(1)$U\left(1\right)$-invariant lattice gauge theory in 2+1$2+1$ dimensions, originally proposed by P. Orland. By using exact diagonalization, we extract the low-energy states for small lattices, up to 4×4$4\times 4$. We confirm that the model has two phases, with the confined entangled one characterized by strings wrapping around the whole lattice. We explain how to study larger lattices by using either tensor network techniques or digital quantum simulations with Rydberg atoms loaded in optical lattices, where we discuss in detail a protocol for the preparation of the ground-state. We propose two key experimental tests that can be used as smoking gun of the proper implementation of a gauge theory in optical lattices. These tests consist in verifying the absence of spontaneous (gauge) symmetry breaking of the ground-state and the presence of charge confinement. We also comment on the relation between standard compact U(1)$U\left(1\right)$ lattice gauge theory and the model considered in this paper.     

Duality,magnetic space group and their applications to quantum phases and phase transitions on bipartite lattices in several experimental systems

By using a dual vortex method, we study phases such as superfluid, solids, supersolids and quantum phase transitions in a unified scheme in extended boson Hubbard models at and slightly away from half filling on bipartite optical lattices such as honeycomb and square lattice. We also map out its global phase diagram at T=0$T=0$ of chemical potential versus the ratio of kinetic energy over the interaction. We stress the importance of the self-consistence condition on the saddle point structure of the dual gauge fields in the translational symmetry breaking insulating sides, especially in the charge density wave side. We find that in the translational symmetry breaking side, different kinds of supersolids are generic possible states slightly away from half filling. We propose a new kind of supersolid: valence bond supersolid (VB-SS). In this VB-SS, the density fluctuation at any site is very large indicating its superfluid nature, but the boson kinetic energies on bonds between two sites are given and break the lattice translational symmetries indicating its valence bound nature. We show that the quantum phase transitions from solids to supersolids driven by a chemical potential are in the same universality class as that from a Mott insulator to a superfluid, therefore have exact exponents z=2$z=2$, ν=1/2$\nu =1/2$, η=0$\eta =0$ with a logarithmic correction. Comparisons with previous quantum Monte Carlo (QMC) simulations on a square lattice are made. Implications on possible future QMC simulations in both bipartite lattices are given. All these phases and phase transitions can be potentially realized in ultra-cold atoms loaded on optical bipartite lattices. Then we apply our results to investigate the reentrant “superfluid” in a narrow region of coverages in the second layer of ⁴He adsorbed on graphite and the low temperature phase diagram of hydrogen physisorbed on krypton-preplated graphite (H₂/Kr/graphite) near half filling. We suggest that ⁴He and H₂ lattice supersolids maybe responsible for the experimental signals in the two systems. Finally, we suggest Cooper supersolid is repressible for the phase diagram of La_2−x Ba_xCuO₄ near x=1/8$x=1/8$.     

Entanglement entropy and variational methods: Interacting scalar fields

We develop a variational approximation to the entanglement entropy for scalar ?⁴${?}^4$ theory in 1+1$1+1$, 2+1$2+1$, and 3+1$3+1$ dimensions, and then examine the entanglement entropy as a function of the coupling. We find that in 1+1$1+1$ and 2+1$2+1$ dimensions, the entanglement entropy of ?⁴${?}^4$ theory as a function of coupling is monotonically decreasing and convex. While ?⁴${?}^4$ theory with positive bare coupling in 3+1$3+1$ dimensions is thought to lead to a trivial free theory, we analyze a version of ?⁴${?}^4$ with infinitesimal negative bare coupling, an asymptotically free theory known as precarious  ?⁴${?}^4$ theory, and explore the monotonicity and convexity of its entanglement entropy as a function of coupling. Within the variational approximation, the stability of precarious ?⁴${?}^4$ theory is related to the sign of the first and second derivatives of the entanglement entropy with respect to the coupling.