Spin Chains in an External Magnetic Field. Closure of the Haldane Gap and Effective Field Theories

We investigate both numerically and analytically the behaviour of a spin-1 antiferromagnetic (AFM) isotropic Heisenberg chain in an external magnetic field. Extensive DMRG studies of chains up to N=80 sites extend previous analyses and exhibit the well known phenomenon of the closure of the Haldane gap at a lower critical field H_c1. We obtain an estimate of the gap below H_c1. Above the lower critical field, when the correlation functions exhibit algebraic decay, we obtain the critical exponent as a function of the net magnetization as well as the magnetization curve up to the saturation (upper critical) field H_c2. We argue that, despite the fact that the SO(3) symmetry of the model is explicitly broken by the field, the Haldane phase of the model is still well described by an SO(3) nonlinear sigma-model. A mean-field theory is developed for the latter and its predictions are compared with those of the numerical analysis and with the existing literature.Comment: 11 pages, 4 eps figure

Rapidly-converging methods for the location of quantum critical points from finite-size data

We analyze in detail, beyond the usual scaling hypothesis, the finite-size convergence of static quantities toward the thermodynamic limit. In this way we are able to obtain sequences of pseudo-critical points which display a faster convergence rate as compared to currently used methods. The approaches are valid in any spatial dimension and for any value of the dynamic exponent. We demonstrate the effectiveness of our methods both analytically on the basis of the one dimensional XY model, and numerically considering c = 1 transitions occurring in non integrable spin models. In particular, we show that these general methods are able to locate precisely the onset of the Berezinskii-Kosterlitz-Thouless transition making only use of ground-state properties on relatively small systems.Comment: 9 pages, 2 EPS figures, RevTeX style. Updated to published versio

Quantum criticality as a resource for quantum estimation

We address quantum critical systems as a resource in quantum estimation and derive the ultimate quantum limits to the precision of any estimator of the coupling parameters. In particular, if L denotes the size of a system and \lambda is the relevant coupling parameters driving a quantum phase transition, we show that a precision improvement of order 1/L may be achieved in the estimation of \lambda at the critical point compared to the non-critical case. We show that analogue results hold for temperature estimation in classical phase transitions. Results are illustrated by means of a specific example involving a fermion tight-binding model with pair creation (BCS model).Comment: 7 pages. Revised and extended version. Gained one author and a specific exampl

Stable particles in anisotropic spin-1 chains

Motivated by field-theoretic predictions we investigate the stable excitations that exist in two characteristic gapped phases of a spin-1 model with Ising-like and single-ion anisotropies. The sine-Gordon theory indicates a region close to the phase boundary where a stable breather exists besides the stable particles, that form the Haldane triplet at the Heisenberg isotropic point. The numerical data, obtained by means of the Density Matrix Renormalization Group, confirm this picture in the so-called large-D phase for which we give also a quantitative analysis of the bound states using standard perturbation theory. However, the situation turns out to be considerably more intricate in the Haldane phase where, to the best of our data, we do not observe stable breathers contrarily to what could be expected from the sine-Gordon model, but rather only the three modes predicted by a novel anisotropic extension of the Non-Linear Sigma Model studied here by means of a saddle-point approximation.Comment: 8 pages, 7 eps figures, svjour clas

Optimal quantum estimation in spin systems at criticality

It is a general fact that the coupling constant of an interacting many-body Hamiltonian do not correspond to any observable and one has to infer its value by an indirect measurement. For this purpose, quantum systems at criticality can be considered as a resource to improve the ultimate quantum limits to precision of the estimation procedure. In this paper, we consider the one-dimensional quantum Ising model as a paradigmatic example of many-body system exhibiting criticality, and derive the optimal quantum estimator of the coupling constant varying size and temperature. We find the optimal external field, which maximizes the quantum Fisher information of the coupling constant, both for few spins and in the thermodynamic limit, and show that at the critical point a precision improvement of order $L$ is achieved. We also show that the measurement of the total magnetization provides optimal estimation for couplings larger than a threshold value, which itself decreases with temperature.Comment: 8 pages, 4 figure

Renormalization of the vacuum angle in quantum mechanics, Berry phase and continuous measurements

The vacuum angle $\theta$ renormalization is studied for a toy model of a quantum particle moving around a ring, threaded by a magnetic flux $\theta$. Different renormalization group (RG) procedures lead to the same generic RG flow diagram, similar to that of the quantum Hall effect. We argue that the renormalized value of the vacuum angle may be observed if the particle's position is measured with finite accuracy or coupled to additional slow variable, which can be viewed as a coordinate of a second (heavy) particle on the ring. In this case the renormalized $\theta$ appears as a magnetic flux this heavy particle sees, or the Berry phase, associated with its slow rotation.Comment: 4 pages, 2 figure

Luttinger liquid behavior in spin chains with a magnetic field

Antiferromagnetic Heisenberg spin chains in a sufficiently strong magnetic field are Luttinger liquids, whose parameters depend on the actual magnetization of the chain. Here we present precise numerical estimates of the Luttinger liquid dressed charge $Z$, which determines the critical exponents, by calculating the magnetization and quadrupole operator profiles for $S=1/2$ and S=1 chains using the density matrix renormalization group method. Critical amplitudes and the scattering length at the chain ends are also determined. Although both systems are Luttinger liquids the characteristic parameters differ considerably.Comment: Final version, 6 pages, 6 EPS figure

Numerical Calculation of the Fidelity for the Kondo and the Friedel-Anderson Impurities

The fidelities of the Kondo and the Friedel-Anderson (FA) impurities are calculated numerically. The ground states of both systems are calculated with the FAIR (Friedel artificially inserted resonance) theory. The ground state in the interacting systems is compared with a nullstate in which the interaction is zero. The different multi-electron states are expressed in terms of Wilson states. The use of N Wilson states simulates the use of a large effective number N_{eff} of states. A plot of ln(F) versus N\proptoln(N_{eff}) reveals whether one has an Anderson orthogonality catastrophe at zero energy. The results are at first glance surprising. The ln(F)-ln(N_{eff}) plot for the Kondo impurity diverges for large N_{eff}. On the other hand, the corresponding plot for the symmetric FA impurity saturates for large N_{eff} when the level spacing at the Fermi level is of the order of the singlet-triplet excitation energy. The behavior of the fidelity allows one to determine the phase shift of the electron states in this regime. PACS: 75.20.Hr, 71.23.An, 71.27.+a, 05.30.-

Particle Content of the Nonlinear Sigma Model with Theta-Term: a Lattice Model Investigation

Using new as well as known results on dimerized quantum spin chains with frustration, we are able to infer some properties on the low-energy spectrum of the O(3) Nonlinear Sigma Model with a topological theta-term. In particular, for sufficiently strong coupling, we find a range of values of theta where a singlet bound state is stable under the triplet continuum. On the basis of these results, we propose a new renormalization group flow diagram for the Nonlinear Sigma Model with theta-term.Comment: 10 pages, 5 figures .eps, iopart format, submitted to JSTA