Papers by Francisco Alcaraz
Random instances of feedforward Boolean circuits are studied both analytically and numerically. E... more Random instances of feedforward Boolean circuits are studied both analytically and numerically. Evaluating these circuits is known to be a Pcomplete problem and thus, in the worst case, believed to be impossible to perform, even given a massively parallel computer, in a time much less than the depth of the circuit. Nonetheless, it is found that, for some ensembles of random circuits, saturation to a fixed truth value occurs rapidly so that evaluation of the circuit can be accomplished in much less parallel time than the depth of the circuit. For other ensembles saturation does not occur and circuit evaluation is apparently hard. In particular, for some random circuits composed of connectives with five or more inputs, the number of true outputs at each level is a chaotic sequence. Finally, while the average case complexity depends on the choice of ensemble, it is shown that for all ensembles it is possible to simultaneously construct a typical circuit together with its solution in polylogarithmic parallel time.
Journal of Statistical Mechanics: Theory and Experiment, 2015
We consider the problem of correlation functions in the stationary states of onedimensional stoch... more We consider the problem of correlation functions in the stationary states of onedimensional stochastic models having conformal invariance. If one considers the space dependence of the correlators, the novel aspect is that although one considers systems with periodic boundary conditions, the observables are described by boundary operators. From our experience with equilibrium problems one would have expected bulk operators. Boundary operators have correlators having critical exponents being half of those of bulk operators. If one studies the space-time dependence of the two-point function, one has to consider one boundary and one bulk operators. The Raise and Peel model has conformal invariance as can be shown in the spin 1/2 basis of the Hamiltonian which gives the time evolution of the system. This is an XXZ quantum chain with twisted boundary condition and local interactions. This Hamiltonian is integrable and the spectrum is known in the finite-size scaling limit. In the stochastic base in which the process is defined, the Hamiltonian is not local anymore. The mapping into an SOS model, helps to define new local operators. As a byproduct some new properties of the SOS model are conjectured. The predictions of conformal invariance are discussed in the new framework and compared with Monte Carlo simulations.
Journal of Statistical Mechanics: Theory and Experiment, 2018
The raise and peel model (RPM) is a nonlocal stochastic model describing the space and time fluct... more The raise and peel model (RPM) is a nonlocal stochastic model describing the space and time fluctuations of an evolving one dimensional interface. Its relevant parameter $u$ is the ratio between the rates of local adsorption and nonlocal desorption processes (avalanches) processes. The model at $u=1$ give us the first example of a conformally invariant stochastic model. For small values $u u_0$ it is critical. By calculating the structure function of the height profiles in the reciprocal space we confirm with good precision that indeed $u_0=1$. We establish that at the conformal invariant point $u=1$ the RPM has a roughness transition with dynamical and roughness critical exponents $z=1$ and $\alpha=0$, respectively. For $u>1$ the model is critical with an $u$-dependent dynamical critical exponent $z(u)$ that tends towards zero as $u\to \infty$. However at $1/u=0$ the RPM is exactly mapped into the totally asymmetric exclusion problem (TASEP). This last model is known to be noncr...
Physical Review E, 2014
A lattice model of critical dense polymers O(n) is considered for the finite cylinder geometry. D... more A lattice model of critical dense polymers O(n) is considered for the finite cylinder geometry. Due to the presence of non-contractible loops with a fixed fugacity ξ, the model at n = 0 is a generalization of the critical dense polymers solved by Pearce, Rasmussen and Villani. We found the free energy for any height N and circumference L of the cylinder. The density ρ of non-contractible loops is obtained for N → ∞ and large L. The results are compared with those found for the anisotropic quantum chain with twisted boundary conditions. Using the latter method we derived ρ for any O(n) model and an arbitrary fugacity.
Journal of Physics A: Mathematical and General, 1997
The critical behaviour of anisotropic Heisenberg models with two kinds of antiferromagnetically e... more The critical behaviour of anisotropic Heisenberg models with two kinds of antiferromagnetically exchange-coupled centers are studied numerically by using finite-size calculations and conformal invariance. These models exhibit the interesting property of ferrimagnetism instead of antiferromagnetism. Most of our results are centered in the mixed Heisenberg chain where we have at even (odd) sites a spinS (S ′) SU(2) operator interacting with a XXZ like interaction (anisotropy ∆). Our results indicate universal properties for all these chains. The whole phase, 1 > ∆ > −1, where the models change from ferromagnetic (∆ = 1) to ferrimagnetic (∆ = −1) behaviour is critical. Along this phase the critical fluctuations are ruled by a c = 1 conformal field theory of Gaussian type. The conformal dimensions and critical exponents, along this phase, are calculated by studying these models with several boundary conditions.
Brazilian Journal of Physics, 2000
Physical Review B, 1999
The exact solution is obtained for the eigenvalues and eigenvectors for two models of strongly co... more The exact solution is obtained for the eigenvalues and eigenvectors for two models of strongly correlated particles with single-particle correlated and uncorrelated pair hoppings. The asymptotic behavior of correlation functions are analysed in different regions, where the models exhibit different physical behavior.//
Journal of Physics A: Mathematical and Theoretical, 2017
Results are given for the ground state energy and excitation spectrum of a simple N-state Z N spi... more Results are given for the ground state energy and excitation spectrum of a simple N-state Z N spin chain described by free parafermions. The model is non-Hermitian for N ≥ 3 with a real ground state energy and a complex excitation spectrum. Although having a simpler Hamiltonian than the superintegrable chiral Potts model, the model is seen to share some properties with it, e.g., the specific heat exponent α = 1 − 2/N and the anisotropic correlation length exponents ν = 1 and ν ⊥ = 2/N .
Journal of Physics A: Mathematical and General, 1999
We consider the exact solution of a model of correlated particles, which is presented as a system... more We consider the exact solution of a model of correlated particles, which is presented as a system of interacting XX and Fateev-Zamolodchikov chains. This model can also be considered as a generalization of the multiband anisotropic t − J model in the case we restrict the site occupations to at most two electrons. The exact solution is obtained for the eigenvalues and eigenvectors using the Bethe-ansatz method.
Physical Review B, 2014
We study the Shannon and Rényi mutual information (MI) in the ground state (GS) of different crit... more We study the Shannon and Rényi mutual information (MI) in the ground state (GS) of different critical quantum spin chains. Despite the apparent basis dependence of these quantities we show the existence of some particular basis (we will call them conformal basis) whose finite-size scaling function is related to the central charge c of the underlying conformal field theory of the model. In particular, we verified that for large index n, the MI of a subsystem of size ℓ in a periodic chain with L sites behaves as c 4 n n−1 ln L π sin(πℓ L) , when the ground-state wavefunction is expressed in these special conformal basis. This is in agreement with recent predictions. For generic local basis we will show that, although in some cases bn ln L π sin(πℓ L) is a good fit to our numerical data, in general there is no direct relation between bn and the central charge of the system. We will support our findings with detailed numerical calculations for the transverse field Ising model, Q = 3, 4 quantum Potts chain, quantum Ashkin-Teller chain and the XXZ quantum chain. We will also present some additional results of the Shannon mutual information (n = 1), for the parafermionic ZQ quantum chains with Q = 5, 6, 7 and 8.
We demonstrate using direct numerical diagonalization and extrapolation methods that boundary con... more We demonstrate using direct numerical diagonalization and extrapolation methods that boundary conditions have a profound effect on the bulk properties of a simple Z(N) model for N > 3 for which the model hamiltonian is non-hermitian. For N=2 the model reduces to the well known quantum Ising model in a transverse field. For open boundary conditions the Z(N) model is known to be solved exactly in terms of free parafermions. Once the ends of the open chain are connected by considering the model on a ring, the bulk properties, including the ground-state energy per site, are seen to differ dramatically with increasing N. Other properties, such as the leading finite-size corrections to the ground-state energy, the mass gap exponent and the specific heat exponent, are also seen to be dependent on the boundary conditions. We speculate that this anomalous bulk behaviour is a topological effect.
Physical Review B, 1985
... VOLUME 32, NUMBER 11 Hamiltonian studies of the Blume-Emery-Griffiths model FC Alcaraz Depart... more ... VOLUME 32, NUMBER 11 Hamiltonian studies of the Blume-Emery-Griffiths model FC Alcaraz Departamento de Fisica, Universidade Federal de Sdo Carlos, Caixa Postal 616, CEP-13560Sdo Carlos, Sdo Paulo, Brazil JR Drugowich de Felicio and R. K6berle Departamento ...
Physics Letters A, 1986
A stochastic reorganization model is simulated on a two-dimensional square lattice, where we inve... more A stochastic reorganization model is simulated on a two-dimensional square lattice, where we investigate finite size effects, and on dusters generated by diffusion limited aggregation. We propose a simple mean-field model, which accounts well for the data.
We present a numerical method for the evaluation of the mass gap, and the low-lying energy gaps, ... more We present a numerical method for the evaluation of the mass gap, and the low-lying energy gaps, of a large family of free-fermionic and free-parafermionic quantum chains. The method is suitable for some generalizations of the quantum Ising and XY models with multispin interactions. We illustrate the method by considering the Ising quantum chains with uniform and random coupling constants. The mass gaps of these quantum chains are obtained from the largest root of a characteristic polynomial. We also show that the Laguerre bound, for the largest root of a polynomial, used as an initial guess for the largest root in the method, gives us estimates for the mass gaps sharing the same leading finite-size behavior as the exact results. This opens an interesting possibility of obtaining precise critical properties very efficiently which we explore by studying the critical point and the paramagnetic Griffiths phase of the quantum Ising chain with random couplings. In this last phase, we obt...
Physical Review E
It is well known the relationship between the eigenspectrum of the Ising and XY quantum chains. A... more It is well known the relationship between the eigenspectrum of the Ising and XY quantum chains. Although the Ising model has a Z(2) symmetry and the XY model a U (1) symmetry, both models are described in terms of free-fermionic quasi-particles. The fermionic quasi-energies are obtained by means of a Jordan-Wigner transformation. On the other hand, there exist in the literature a huge family of Z(N) quantum chains whose eigenspectra, for N > 2, are given in terms of free parafermions and are not derived from the standard Jordan-Wigner transformation. The first members of this family are the Z(N) free-parafermionic Baxter quantum chains. In this paper we introduce a family of XY models that beyond two-body also have N-multispin interactions. Similarly to the standard XY model they have a U (1) symmetry and are also solved by the Jordan-Wigner transformation. We show that with appropriate choices of the N-multispin couplings the eigenspectra of these XY models contains the eigenspectra of the Z(N) free-parafermionic quantum chains. The correspondence is established via the identification of the characteristic polynomial which fix the eigenspectrum. In the Z(N) free-parafermionic models the quasi-energies obey an exclusion circle principle that is not present in the related N-multispin XY models.
Physical Review B
We present a numerical method for the evaluation of the mass gap, and the low-lying energy gaps, ... more We present a numerical method for the evaluation of the mass gap, and the low-lying energy gaps, of a large family of free-fermionic and free-parafermionic quantum chains. The method is suitable for some generalizations of the quantum Ising and XY models with multispin interactions. We illustrate the method by considering the Ising quantum chains with uniform and random coupling constants. The mass gaps of these quantum chains are obtained from the largest root of a characteristic polynomial. We also show that the Laguerre bound, for the largest root of a polynomial, used as an initial guess for the largest root in the method, gives us estimates for the mass gaps sharing the same leading finite-size behavior as the exact results. This opens an interesting possibility of obtaining precise critical properties very efficiently which we explore by studying the critical point and the paramagnetic Griffiths phase of the quantum Ising chain with random couplings. In this last phase, we obtain the effective dynamical critical exponent as a function of the distance-tocriticality. Finally, we compare the mass gap estimates derived from the Laguerre bound and the strong-disorder renormalization-group method. Both estimates require comparable computational efforts, with the former having the advantage of being more accurate and also being applicable away from infinite-randomness fixed points. We believe this method is a new and relevant tool for tackling critical quantum chains with and without quenched disorder.
Physical Review B
We introduce a new a family of Z(N) multispins quantum chains with a free-fermionic (N = 2) or fr... more We introduce a new a family of Z(N) multispins quantum chains with a free-fermionic (N = 2) or free-parafermionic (N > 2) eigenspectrum. The models have (p + 1) interacting spins (p = 1, 2,. . .), being Hermitian in the Z(2) (Ising) case and non-Hermitian for N > 2. We construct a set of mutually commuting charges that allows us to derive the eigenenergies in terms of the roots of polynomials generated by a recurrence relation of order (p + 1). In the critical limit we identify these polynomials with certain hypergeometric polynomials p+1Fp. Also in the critical regime, we calculate the ground state energy in the bulk limit and verify that they are given in terms of the Lauricella hypergeometric series. The models with special couplings are self-dual and at the self-dual point show a critical behavior with dynamical critical exponent zc = p+1 N .
International Journal of Modern Physics A, 1993
We consider an extension of the (t-U) Hubbard model taking into account new interactions between ... more We consider an extension of the (t-U) Hubbard model taking into account new interactions between the numbers of up and down electrons. We confine ourselves to a one-dimensional open chain with L sites (4^L states) and derive the effective Hamiltonian in the strong repulsion (large U) regime. This Hamiltonian acts on 3^L states. We show that the spectrum of the
Physical Review B
We present a general study of a large family of exactly integrable quantum chains with multispin ... more We present a general study of a large family of exactly integrable quantum chains with multispin interactions. The exact integrability follows from the algebraic properties of the energy density operators defining the quantum chains. The Hamiltonians are characterized by a parameter p = 1, 2,. .. related to the number of interacting spins in the multispin interaction. In the general case, the quantum spins are of infinite dimension. In special cases, characterized by the parameter N = 2, 3,. . ., the quantum chains describe the dynamics of Z (N) quantum spin chains with open boundary conditions. The simplest case p = 1 corresponds to the free fermionic quantum Ising chain (N = 2) or the Z (N) free parafermionic quantum chain. The eigenenergies of the quantum chains are given in terms of the roots of special polynomials, and for general values of p the quantum chains are characterized by a free fermionic (N = 2) or free parafermionic (N > 2) eigenspectrum. The models have a special critical point when all coupling constants are equal. At this point, the ground-state energy is exactly calculated in the bulk limit, and our analytical and numerical analyses indicate that the models belong to universality classes of critical behavior with dynamical critical exponent z = (p + 1)/N and specific-heat exponent α = max{0, 1 − (p + 1)/N}.
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Papers by Francisco Alcaraz