Boundary Quantum Entanglement of the XXZ Spin Chain with Boundary Impurities

The boundary quantum entanglement for the s = 1/2 X X Z spin chain with boundary impurities is studied via the density matrix renormalization group （DMRG） method. It is shown that the entanglement entropy of the boundary bond （the impurity and the chain spin next to it） behaves differently in different phases. The relationship between the singular points of the boundary entropy and boundary quantum critical points is discussed.     

Boundary effects in the critical scaling of entanglement entropy in 1D systems

We present exact diagonalization and density matrix renormalization group results for the entanglement entropy of critical spin-1/2 XXZ chains. We find that open boundary conditions induce an alternating term in both the energy density and the entanglement entropy which are approximately proportional, decaying away from the boundary with a power law. The power varies with anisotropy along the critical line and is corrected by a logarithmic factor, which we calculate analytically, at the isotropic point. A heuristic resonating valence bond explanation is suggested.     

Boundary entropy can increase under bulk RG flow

The boundary entropy log(g)$\log (g)$ of a critical one-dimensional quantum system (or two-dimensional conformal field theory) is known to decrease under renormalization group (RG) flow of the boundary theory. We study instead the behavior of the boundary entropy as the bulk theory flows between two nearby critical points. We use conformal perturbation theory to calculate the change in g   due to a slightly relevant bulk perturbation and find that it has no preferred sign. The boundary entropy log(g)$\log (g)$ can therefore increase during appropriate bulk flows. This is demonstrated explicitly in flows between minimal models. We discuss the applications of this result to D-branes in string theory and to impurity problems in condensed matter.     

Singular effects of impurities near the ferromagnetic quantum-critical point

Systematic theoretical results for the effects of a dilute concentration of magnetic impurities on the thermodynamic and transport properties in the region around the quantum critical point of a ferromagnetic transition are obtained. In the quasiclassical regime, the dynamical spin fluctuations enhance the Kondo temperature. This energy scale decreases rapidly in the quantum fluctuation regime, where the properties are those of a line of critical points of the multichannel Kondo problem with the number of channels increasing as the critical point is approached, except at unattainably low temperatures where a single channel wins out.     

The thermodynamic one-particle Green function in the renormalized spin wave approximation for isotropic cubic ferromagnets

The thermodynamic one-particle Green function in the renormalized spin wave approximation for isotropic cubic ferromagnetic insulators with Dyson's spin wave theory as a base is derived. In quantitative respect, dynamic and kinematic effects of spin waves are approximated by the graphs deficient in the energy denominators, wherefore at low temperature kinematic interaction turns out to be too strong. As against the one-particle Green function for independent spin waves, dynamic interaction of ferromagnons is shown to effect the renormalization of the spin wave energy, whereas kinematic interaction directly modifies the average ferromagnon population numbers. In the matter of magnetization, its formula based on the Green function assumes a similar form as in the spin wave theory without interactions on the understanding that it remains valid within the entire range of temperatures from absolute zero up to the critical point.     

The running coupling at finite temperature

We propose the ensemble averaged running coupling as the meaningful measure of the coupling strength of an equilibrium gas and consider its scaling behaviour. We show that the high temperature limit of the average coupling is obtained by keeping the temperature to the renormalization point ratio,T/μ, fixed. In hot QCD gas this implies asymptotic freedom and a beta function which is independent ofT up to two loops. In the MOM scheme a minimal sensitivity to temperature is obtained with the choice μ=2.6T.     

Dynamics and Universality of Unimodal Mappings with Infinite Criticality

We consider infinitely renormalizable unimodal mappings with topological type which are periodic under renormalization. We study the limiting behavior of fixed points of the renormalization operator as the order of the critical point increases to infinity. It is shown that a limiting dynamics exists, with a critical point that is flat, but still having a well-behaved analytic continuation to a neighborhood of the real interval pinched at the critical point. We study the dynamics of limiting maps and prove their rigidity. In particular, the sequence of fixed points of renormalization for finite criticalities converges, uniformly on the real domain, to a mapping of the limiting type.Both authors were supported by Grant No. 2002062 from the United States-Israel Binational Science Foundation (BSF), Jerusalem, Israel.Partially supported by NSF grant DMS-0245358.     

Entropy majorization, thermal adiabatic theorem, and quantum phase transitions

Let a general quantum many-body system at a low temperature adiabatically cross through the vicinity of the system’s quantum critical point. We show that the system’s temperature is significantly suppressed due to both the entropy majorization theorem in quantum information science and the entropy conservation law in reversible adiabatic processes. We take the one-dimensional transverse-field Ising model and the spinless fermion system as concrete examples to show that the inverse temperature might become divergent around the systems’ critical points. Since the temperature is a measurable quantity in experiments, it can be used, via reversible adiabatic processes at low temperatures, to detect quantum phase transitions in the perspectives of quantum information science and quantum statistical mechanics.     

C function representation of the Local Potential Approximation

Within the Local Potential Approximation to Wilson's, or Polchinski's, exact renormalization group, and for general spacetime dimension, we construct a function, c, of the coupling constants; it has the property that (for unitary theories) it decreases monotonically along flows, and is stationary only at fixed points — where it ‘counts degrees of freedom’, i.e. is extensive, counting one for each Gaussian scalar. Furthermore, by choosing restrictions to some sub-manifold of coupling constant space, we arrive at a very promising variational approximation method.     

Fractal dimension of Siegel disc boundaries

Using renormalization techniques, we provide rigorous computer-assisted bounds on the Hausdorff dimension of the boundary of Siegel discs. Specifically, for Siegel discs with golden mean rotation number and quadratic critical points we show that the Hausdorff dimension is less than 1.08523. This is done by exploiting a previously found renormalization fixed point and expressing the Siegel disc boundary as the attractor of an associated Iterated Function System. Received: 26 January 1998 / Received in final form: 5 June 1998 / Accepted: 11 June 1998     

Critical points of the Bose–Hubbard model with three-body local interaction

Using the density matrix renormalization group method, we study a one-dimensional system of bosons that interact with a local three-body term. We calculate the phase diagram for higher densities, where the Mott insulator lobes are surrounded by the superfluid phase. We also show that the Mott insulator lobes always grow as a function of the density. The critical points of the Kosterlitz–Thouless transitions were determined through the von Neumann block entropy, and its dependence on the density is given by a power law with a negative exponent.     

The Casimir effect for fermions in one dimension

We study the Casimir problem for a fermion coupled to a static background field in one space dimension. We examine the relationship between interactions and boundary conditions for the Dirac field. In the limit that the background becomes concentrated at a point (a “Dirac spike”) and couples strongly, it implements a confining boundary condition. We compute the Casimir energy for a masslike background and show that it is finite for a stepwise continuous background field. However the total Casimir energy diverges for the Dirac spike. The divergence cannot be removed by standard renormalization methods. We compute the Casimir energy density of configurations where the background field consists of one or two sharp spikes and show that the energy density is finite except at the spikes. Finally we define and compute an interaction energy density and the force between two Dirac spikes as a function of the strength and separation of the spikes.     

Field-induced quantum criticality of systems with ferromagnetically coupled structural spin units

The field-induced quantum criticality of compounds with ferromagnetically coupled structural spin units (as dimers and ladders) is explored by applying Wilson's renormalization group framework to an appropriate effective action. We determine the low-temperature phase boundary and the behavior of relevant quantities decreasing the temperature with the applied magnetic field fixed at its quantum critical point value. In this context, a plausible interpretation of some recent experimental results is also suggested.     

Logarithmic terms in entanglement entropies of 2D quantum critical points and Shannon entropies of spin chains

Universal logarithmic terms in the entanglement entropy appear at quantum critical points (QCPs) in one dimension (1D) and have been predicted in 2D at QCPs described by 2D conformal field theories. The entanglement entropy in a strip geometry at such QCPs can be obtained via the "Shannon entropy" of a 1D spin chain with open boundary conditions. The Shannon entropy of the XXZ chain is found to have a logarithmic term that implies, for the QCP of the square-lattice quantum dimer model, a logarithm with universal coefficient ±0.25. However, the logarithm in the Shannon entropy of the transverse-field Ising model, which corresponds to entanglement in the 2D Ising conformal QCP, is found to have a singular dependence on the replica or Rényi index resulting from flows to different boundary conditions at the entanglement cut.     

Hyperscaling above the upper critical dimension

Above the upper critical dimension, the breakdown of hyperscaling is associated with dangerous irrelevant variables in the renormalization group formalism at least for systems with periodic boundary conditions. While these have been extensively studied, there have been only a few analyses of finite-size scaling with free boundary conditions. The conventional expectation there is that, in contrast to periodic geometries, finite-size scaling is Gaussian, governed by a correlation length commensurate with the lattice extent. Here, detailed numerical studies of the five-dimensional Ising model indicate that this expectation is unsupported, both at the infinite-volume critical point and at the pseudocritical point where the finite-size susceptibility peaks. Instead the evidence indicates that finite-size scaling at the pseudocritical point is similar to that in the periodic case. An analytic explanation is offered which allows hyperscaling to be extended beyond the upper critical dimension.     

Perspectives A critique of two metals

I argue that Anderson's identification of the conflict between the fermi-liquid and non-fermi-liquid metallic states as the central issue of cuprate superconductivity is fundamentally wrong. All experimental evidence points to adiabatic continuability of the strange metal into a conventional one, and thus to one metallic phase rather than two, and all attempts to account theoretically for the existence of a luttinger-liquid at zero temperature in spatial dimension greater than one have failed. I discuss the underlying reasons for this failure and then argue that the true higher-dimensional generalization of the luttinger-liquid behavior is a propensity of the system to order. This implies that the central issue is actually the conflict between different kinds of order, i.e. exactly the idea implicit in Zhang's paper. I then speculate about how the conflict between antiferromagnetism and superconductivity, the two principal kinds of order in this problem, might result in both the observed zero-temperature phase diagram of the cuprates and the luttinger-liquid phenomenology, i.e. the breakup of the electron into spinons and holons in certain regimes of doping and energy. The key idea is a quantum critical point regulating a first-order transition between these phases, and toward which one is first attracted under renormalization before bifurcating between the two phases. I speculate that this critical point lies on the insulating line, and that the difference between the Mott-insulator and fermi-liquid approaches to the high TC problem comes down to whether or not the superconducting states made by nand p-type doping can be continued into each other. A candidate for the second fixed point required for distinct superconducting phases is the P- and T-violating chiral spin liquid state invented by me.     

Entropy production in open volume-preserving systems

We describe a mechanism leading to positive entropy production in volume-preserving systems under nonequilibrium conditions. We consider volume-preserving systems sustaining a diffusion process like the multibaker map or the Lorentz gas. A continuous flux of particles is imposed across the system resulting in a steady gradient of concentration. In the limit where such flux boundary conditions are imposed at arbitrarily separated boundaries for a fixed gradient, the invariant measure becomes singular. For instance, in the multibaker map, the limit invariant measure has a cumulative function given in terms of the nondifferentiable Takagi function. Because of this singularity of the invariant measure, the entropy must be defined as a coarse-grained entropy instead of the fined-grained Gibbs entropy, which would require the existence of a regular measure with a density. The coarse-grained entropy production is then shown to be asymptotically positive and, moreover, given by the entropy production expected from irreversible thermodynamics.     

Entanglement in quantum spin chains, symmetry classes of random matrices, and conformal field theory

We compute the entropy of entanglement between the first N spins and the rest of the system in the ground states of a general class of quantum spin chains. We show that under certain conditions the entropy can be expressed in terms of averages over ensembles of random matrices. These averages can be evaluated, allowing us to prove that at critical points the entropy grows like kappalog(2N+kappa as N-->infinity, where kappa and kappa are determined explicitly. In an important class of systems, kappa is equal to one-third of the central charge of an associated Virasoro algebra. Our expression for kappa therefore provides an explicit formula for the central charge.     

Tricritical point and wing structure in the itinerant ferromagnet UGe₂

Precise resistivity measurements on the ferromagnetic superconductor UGe2 under pressure p and magnetic field H reveal a previously unobserved change of the anomaly at the Curie temperature. Therefore, the tricritical point (TCP) where the paramagnetic-to-ferromagnetic transition changes from a second order to a first order transition is located in the p-T phase diagram. Moreover, the evolution of the TCP can be followed under the magnetic field in the same way. It is the first report of the boundary of the first order plane which appears in the p-T-H phase diagram of weak itinerant ferromagnets. This line of critical points starts from the TCP and will terminate at a quantum critical point. These measurements provide the first estimation of the location of the quantum critical point in the p-H plane and will inspire similar studies of the other weak itinerant ferromagnets.     

Comprehensive diagnosis of growth rates of the ablative Rayleigh-Taylor instability

The growth rate of the ablative Rayleigh-Taylor instability is approximated by gamma = square root[kg/(1 + kL)] - beta km/rho(a), where k is the perturbation wave number, g the gravity, L the density scale length, m the mass ablation rate, and rho(a) the peak target density. The coefficient beta was evaluated for the first time by measuring all quantities of this formula except for L, which was taken from the simulation. Although the experimental value of beta = 1.2+/-0.7 at short perturbation wavelengths is in reasonably good agreement with the theoretical prediction of beta = 1.7, it is found to be larger than the prediction at long wavelengths.