[Show abstract][Hide abstract] ABSTRACT: We study the R\'enyi mutual information $\tilde{I}_n$ of the ground state of
different critical quantum chains. The R\'enyi mutual information definition
that we use is based on the well established concept of the R\'enyi divergence.
We calculate this quantity numerically for several distinct quantum chains
having either discrete $Z(Q)$ symmetries (Q-state Potts model with $Q=2,3,4$
and $Z(Q)$ parafermionic models with $Q=5,6,7,8$ and also Ashkin-Teller model
with different anisotropies) or the $U(1)$ continuous symmetries(Klein-Gordon
field theory, XXZ and spin-1 Fateev-Zamolodchikov quantum chains with different
anisotropies). For the spin chains these calculations were done by expressing
the ground-state wavefunctions in two special basis. Our results indicate some
general behavior for particular ranges of values of the parameter $n$ that
defines $\tilde{I}_n$. For a system, with total size $L$ and subsystem sizes
$\ell$ and $L-\ell$, the$\tilde{I}_n$ has a logarithmic leading behavior given
by $\frac{\tilde{c}_n}{4}\log(\frac{L}{\pi}\sin(\frac{\pi \ell}{L}))$ where the
coefficient $\tilde{c}_n$ is linearly dependent on the central charge $c$ of
the underlying conformal field theory (CFT) describing the system's critical
properties.
Physical Review B 01/2015; 91(15). DOI:10.1103/PhysRevB.91.155122 · 3.74 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: A lattice model of critical dense polymers $O(0)$ is considered for the
finite cylinder geometry. Due to the presence of non-contractible loops with a
fixed fugacity $\xi$, the model 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 $\rho$ of
non-contractible loops is found for $N \rightarrow \infty$ and large $L$. The
results are compared with those obtained for the anisotropic quantum chain with
twisted boundary conditions. Using the latter method we obtained $\rho$ for any
$O(n)$ model and an arbitrary fugacity.
Physical Review E 11/2014; DOI:10.1103/PhysRevE.90.052138 · 2.29 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We study the Shannon and R\'enyi 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 $$\ell${}$ in a periodic chain with $L$ sites behaves as $\frac{c}{4}\frac{n}{n$-${}1}ln[\frac{L}{$\pi${}}sin(\frac{$\pi${}$\ell${}}{L})]$, when the ground-state wave function 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 ${b}_{n}ln[\frac{L}{$\pi${}}sin(\frac{$\pi${}$\ell${}}{L})]$ is a good fit to our numerical data, in general, there is no direct relation between ${b}_{n}$ 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 ${Z}_{Q}$ quantum chains with $Q=5,6,7$, and 8.
Physical Review B 08/2014; 90(7). DOI:10.1103/PhysRevB.90.075132 · 3.74 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We study the Shannon and R\'enyi mutual information (MI) in 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
$\ell$ in a periodic chain with $L$ sites behaves as
$\frac{c}{4}\frac{n}{n-1}\ln\Big{(}\frac{L}{\pi}\sin(\frac{\pi
\ell}{L})\Big{)}$, when the ground-state wave function 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
$b_n\ln\Big{(}\frac{L}{\pi}\sin(\frac{\pi \ell}{L})\Big{)}$ is a good fit to
our numerical data, in general there is no direct relation between $b_n$ 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 $Z(Q)$ quantum chains with $Q=5,6,7$ and $8$.
[Show abstract][Hide abstract] ABSTRACT: We study the Shannon and R\'enyi mutual information (MI) in 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 $\ell$ in a periodic chain with $L$ sites behaves as $\frac{c}{4}\frac{n}{n-1}\ln\Big{(}\frac{L}{\pi}\sin(\frac{\pi \ell}{L})\Big{)}$, when the ground-state wave function 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 $b_n\ln\Big{(}\frac{L}{\pi}\sin(\frac{\pi \ell}{L})\Big{)}$ is a good fit to our numerical data, in general there is no direct relation between $b_n$ 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 $Z(Q)$ quantum chains with $Q=5,6,7$ and $8$.
[Show abstract][Hide abstract] ABSTRACT: We consider the Shannon mutual information of subsystems of critical quantum
chains in their ground states. Our results indicate a universal leading
behavior for large subsystem sizes. Moreover, as happens with the entanglement
entropy, its finite-size behavior yields the conformal anomaly $c$ of the
underlying conformal field theory governing the long distance physics of the
quantum chain. We studied analytically a chain of coupled harmonic oscillators
and numerically the Q-state Potts models ($Q = 2$; 3 and 4), the XXZ quantum
chain and the spin-1 Fateev-Zamolodchikov model. The Shannon mutual information
is a quantity easily computed, and our results indicate that for relatively
small lattice sizes its finite-size behavior already detects the universality
class of quantum critical behavior.
[Show abstract][Hide abstract] ABSTRACT: We study the entanglement entropies in one-dimensional open critical systems,
whose effective description is given by a conformal field theory with
boundaries. We show that for pure-state systems formed by the ground state or
by the excited states associated to primary fields, the entanglement entropies
have a finite-size behavior that depends on the correlation of the underlying
field theory. The analytical results are checked numerically, finding excellent
agreement for the quantum chains ruled by the theories with central charge
$c=1/2$ and $c=1$.
[Show abstract][Hide abstract] ABSTRACT: 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 (4L states) and derive the effective Hamiltonian in the strong repulsion (large U) regime. This Hamiltonian acts on 3L states. We show that the spectrum of the latter Hamiltonian (not the degeneracies) coincides with the spectrum of the anisotropic Heisenberg chain (XX Z model) in the presence of a Z field (2L states). The wave functions of the 3L-state system are obtained explicitly from those of the 2L-state system, and the degeneracies can be understood in terms of irreducible representations of .
International Journal of Modern Physics A 01/2012; 09(19). DOI:10.1142/S0217751X94001370 · 1.70 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: Using the density matrix renormalization group, we calculated the finite-size
corrections of the entanglement $\alpha$-Renyi entropy of a single interval for
several critical quantum chains. We considered models with U(1) symmetry like
the spin-1/2 XXZ and spin-1 Fateev-Zamolodchikov models, as well models with
discrete symmetries such as the Ising, the Blume-Capel and the three-state
Potts models. These corrections contain physically relevant information. Their
amplitudes, that depend on the value of $\alpha$, are related to the dimensions
of operators in the conformal field theory governing the long-distance
correlations of the critical quantum chains. The obtained results together with
earlier exact and numerical ones allow us to formulate some general conjectures
about the operator responsible for the leading finite-size correction of the
$\alpha$-Renyi entropies. We conjecture that the exponent of the leading
finite-size correction of the $\alpha$-Renyi entropies is
$p_{\alpha}=2X_{\epsilon}/\alpha$ for $\alpha>1$ and $p_{1}=\nu$, where
$X_{\epsilon}$ is the dimensions of the energy operator of the model and
$\nu=2$ for all the models.
[Show abstract][Hide abstract] ABSTRACT: Finite-size scaling analysis turns out to be a powerful tool to calculate the
phase diagram as well as the critical properties of two dimensional classical
statistical mechanics models and quantum Hamiltonians in one dimension. The
most used method to locate quantum critical points is the so called crossing
method, where the estimates are obtained by comparing the mass gaps of two
distinct lattice sizes. The success of this method is due to its simplicity and
the ability to provide accurate results even considering relatively small
lattice sizes. In this paper, we introduce an estimator that locates quantum
critical points by exploring the known distinct behavior of the entanglement
entropy in critical and non critical systems. As a benchmark test, we use this
new estimator to locate the critical point of the quantum Ising chain and the
critical line of the spin-1 Blume-Capel quantum chain. The tricritical point of
this last model is also obtained. Comparison with the standard crossing method
is also presented. The method we propose is simple to implement in practice,
particularly in density matrix renormalization group calculations, and provides
us, like the crossing method, amazingly accurate results for quite small
lattice sizes. Our applications show that the proposed method has several
advantages, as compared with the standard crossing method, and we believe it
will become popular in future numerical studies.
[Show abstract][Hide abstract] ABSTRACT: Using the density matrix renormalization group, we investigate the Rényi entropy of the anisotropic spin-s Heisenberg chains in a z-magnetic field. We considered the half-odd-integer spin-s chains, with s=1/2, 3/2, and 5/2, and periodic and open boundary conditions. In the case of the spin-1/2 chain we were able to obtain accurate estimates of the new parity exponents pα(p) and pα(o) that gives the power-law decay of the oscillations of the α-Rényi entropy for periodic and open boundary conditions, respectively. We confirm the relations of these exponents with the Luttinger parameter K, as proposed by Calabrese et al. [ Phys. Rev. Lett. 104 095701 (2010)]. Moreover, the predicted periodicity of the oscillating term was also observed for some nonzero values of the magnetization m. We show that for s>1/2 the amplitudes of the oscillations are quite small and get accurate estimates of pα(p) and pα(o) become a challenge. Although our estimates of the new universal exponents pα(p) and pα(o) for the spin-3/2 chain are not so accurate, they are consistent with the theoretical predictions.
[Show abstract][Hide abstract] ABSTRACT: We present four estimators of the shared information (or interdepency) in ground states given that the coefficients appearing in the wave function are all real non-negative numbers and therefore can be interpreted as probabilities of configurations. Such ground states of Hermitian and non-Hermitian Hamiltonians can be given, for example, by superpositions of valence bond states which can describe equilibrium but also stationary states of stochastic models. We consider in detail the last case, the system being a classical not a quantum one. Using analytical and numerical methods we compare the values of the estimators in the directed polymer and the raise and peel models which have massive, conformal invariant and nonconformal invariant massless phases. We show that like in the case of the quantum problem, the estimators verify the area law with logarithmic corrections when phase transitions take place.
[Show abstract][Hide abstract] ABSTRACT: We present four estimators of the entanglement (or interdepency) of ground-states in which the coefficients are all real nonnegative and therefore can be interpreted as probabilities of configurations. Such ground-states of hermitian and non-hermitian Hamiltonians can be given, for example, by superpositions of valence bond states which can describe equilibrium but also stationary states of stochastic models. We consider in detail the last case. Using analytical and numerical methods we compare the values of the estimators in the directed polymer and the raise and peel models which have massive, conformal invariant and non-conformal invariant massless phases. We show that like in the case of the quantum problem, the estimators verify the area law and can therefore be used to signal phase transitions in stationary states. Comment: 4 pages 3figures
[Show abstract][Hide abstract] ABSTRACT: In one-component Abelian sandpile models, the toppling probabilities are independent quantities. This is not the case in multicomponent models. The condition of associativity of the underlying Abelian algebras imposes nonlinear relations among the toppling probabilities. These relations are derived for the case of two-component quadratic Abelian algebras. We show that Abelian sandpile models with two conservation laws have only trivial avalanches.
[Show abstract][Hide abstract] ABSTRACT: We analyze the finite size corrections to entanglement in quantum critical systems. By using conformal symmetry and density functional theory, we discuss the structure of the finite size contributions to a general measure of ground state entanglement, which are ruled by the central charge of the underlying conformal field theory. More generally, we show that all conformal towers formed by an infinite number of excited states (as the size of the system $L \to \infty$) exhibit a unique pattern of entanglement, which differ only at leading order $(1/L)^2$. In this case, entanglement is also shown to obey a universal structure, given by the anomalous dimensions of the primary operators of the theory. As an illustration, we discuss the behavior of pairwise entanglement for the eigenspectrum of the spin-1/2 XXZ chain with an arbitrary length $L$ for both periodic and twisted boundary conditions.
Physical Review A 09/2008; 78:032319. DOI:10.1103/PhysRevA.78.032319 · 2.81 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: The raise and peel model is a stochastic model of a fluctuating interface separating a substrate covered with clusters of matter of different sizes and a rarefied gas of tiles. The stationary state is obtained when adsorption compensates the desorption of tiles. This model is generalized to an interface with defects (D) . The defects are either adjacent or separated by a cluster. If a tile hits the end of a cluster with a defect nearby, the defect hops at the other end of the cluster, changing its shape. If a tile hits two adjacent defects, the defects annihilate and are replaced by a small cluster. There are no defects in the stationary state. This model can be seen as describing the reaction D+D-->0 , in which the particles (defects) D hop at long distances, changing the medium, and annihilate. Between the hops the medium also changes (tiles hit clusters, changing their shapes). Several properties of this model are presented and some exact results are obtained using the connection of our model with a conformally invariant quantum chain.
[Show abstract][Hide abstract] ABSTRACT: We present a new conjecture for the SUq(N) Perk–Schultz models. This conjecture extends a conjecture presented in our article (Alcaraz F C and Stroganov Yu G J. Phys. A: Math. Gen.35 6767–87).
Journal of Physics A General Physics 12/2004; 37(48):11725-11727. DOI:10.1088/0305-4470/37/48/013
[Show abstract][Hide abstract] ABSTRACT: We discuss entanglement in the spin-1/2 anisotropic ferromagnetic Heisenberg chain in the presence of a boundary magnetic field generating domain walls. By increasing the magnetic field, the model undergoes a first-order quantum phase transition from a ferromagnetic to a kink-type phase, which is associated to a jump in the content of entanglement available in the system. Above the critical point, pairwise entanglement is shown to be non-vanishing and independent of the boundary magnetic field for large chains. Based on this result, we provide an analytical expression for the entanglement between arbitrary spins. Moreover the effects of the quantum domains on the gapless region and for antiferromagnetic anisotropy are numerically analysed. Finally multiparticle entanglement properties are considered, from which we establish a characterization of the critical anisotropy separating the gapless regime from the kink-type phase. Comment: v3: 7 pages, including 4 figures and 1 table. Published version. v2: One section (V) added and references updated
Physical Review A 09/2004; 70:032333. DOI:10.1103/PhysRevA.70.032333 · 2.81 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: Basing on the numerical observations of the eigenspectra of the SU(N)
Perk-Schultz model at the special value of the anisotropy
q=exp(iπ(N-1)/N), we formulate a set of conjectures concerning the
existence of the free-fermion like eigenenergies. We prove analytically
a part of these conjectures.
International Journal of Modern Physics A 05/2004; 19(supp02):1-15. DOI:10.1142/S0217751X04020270 · 1.70 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We conjecture that the free-fermion part of the eigenspectrum observed recently for the SUq(N) Perk–Schultz spin chain Hamiltonian in a finite lattice with q = exp(iπ(N − 1)/N) is a consequence of the existence of a special simple eigenvalue for the transfer matrix of the auxiliary inhomogeneous SUq(N − 1) vertex model which appears in the nested Bethe ansatz approach. We prove that this conjecture is valid for the case of the SUq(3) spin chain with periodic boundary condition. In this case we obtain a formula for the components of the eigenvector of the auxiliary inhomogeneous 6-vertex model (q = exp(2iπ/3)), which permits us to find one by one all components of this eigenvector and consequently to find the eigenvectors of the free-fermion part of the eigenspectrum of the SUq(3) spin chain. Similarly, as in the known case of the SUq(2) case at q = exp(i2π/3) our numerical and analytical studies induce some conjectures for special rates of correlation functions.
Journal of Physics A General Physics 02/2003; 36(10):2381. DOI:10.1088/0305-4470/36/10/301