Reply to "Comment on 'Quantum fidelity approach to the ground-state properties of the one-dimensional axial next-nearest-neighbor Ising model in a transverse field' ".

We disagree with the objections raised by Nemati et al. regarding the phase transitions reported in our paper, where we used the fidelity method. Contrary to their claims, our fidelity calculations do not depend on energy level crossing between excited states. We obtain the same results just by analyzing the second derivative of the ground-state energy with respect to the interaction energy coupling J_{2}.

Ground-state properties of the one-dimensional transverse Ising model in a longitudinal magnetic field.

The critical properties of the one-dimensional spin-1/2 transverse Ising model in the presence of a longitudinal magnetic field were studied by the quantum fidelity method. We used exact diagonalization to obtain the ground-state energies and corresponding eigenvectors for lattice sizes up to 24 spins. The maximum of the fidelity susceptibility was used to locate the various phase boundaries present in the system. The type of dominant spin ordering for each phase was identified by examining the corresponding ground-state eigenvector. For a given antiferromagnetic nearest-neighbor interaction J_{2}, we calculated the fidelity susceptibility as a function of the transverse field B_{x} and the longitudinal field B_{z}. The phase diagram in the (B_{x},B_{z})-plane shows three phases. These findings are in contrast with the published literature that claims that the system has only two phases. For B_{x}<1, we observed an antiferromagnetic phase for small values of B_{z} and a paramagnetic phase for large values of B_{z}. For B_{x}>1 and low B_{z}, we found a disordered phase that undergoes a second-order phase transition to a paramagnetic phase for large values of B_{z}.

Quantum fidelity approach to the ground-state properties of the one-dimensional axial next-nearest-neighbor Ising model in a transverse field.

In this work we analyze the ground-state properties of the s=1/2 one-dimensional axial next-nearest-neighbor Ising model in a transverse field using the quantum fidelity approach. We numerically determined the fidelity susceptibility as a function of the transverse field B_{x} and the strength of the next-nearest-neighbor interaction J_{2}, for systems of up to 24 spins. We also examine the ground-state vector with respect to the spatial ordering of the spins. The ground-state phase diagram shows ferromagnetic, floating, and 〈2,2〉 phases, and we predict an infinite number of modulated phases in the thermodynamic limit (L→∞). Paramagnetism only occurs for larger magnetic fields. The transition lines separating the modulated phases seem to be of second order, whereas the line between the floating and the 〈2,2〉 phases is possibly of first order.

Quantum phase transitions in a chain with two- and four-spin interactions in a transverse field.

We use entanglement entropy and finite-size scaling methods to investigate the ground-state properties of a spin-1/2 Ising chain with two-spin (J(2)) and four-spin (J(4)) interactions in a transverse magnetic field (B). We concentrate our study on the unexplored critical region B=1 and obtain the phase diagram of the model in the (J(4)-J(2)) plane. The phases found include ferromagnetic (F), antiferromagnetic (AF), as well as more complex phases involving spin configurations with multiple periodicity. The system presents both first- and second-order transitions separated by tricritical points. We find an unusual phase boundary on the semi-infinite segment (J(4)<-1,J(2)=0) separating the F and AF phases.

Critical behavior of a quantum chain with four-spin interactions in the presence of longitudinal and transverse magnetic fields.

We study the ground-state properties of a spin-1/2 model on a chain containing four-spin Ising-like interactions in the presence of both transverse and longitudinal magnetic fields. We use entanglement entropy and finite-size scaling methods to obtain the phase diagrams of the model. Our numerical calculations reveal a rich variety of phases and the existence of multicritical points in the system. We identify phases with both ferromagnetic and antiferromagnetic orderings. We also find periodically modulated orderings formed by a cluster of like spins followed by another cluster of opposite like spins. The quantum phases in the model are found to be separated by either first- or second-order transition lines.

Phase transitions in the two-dimensional superantiferromagnetic Ising model with next-nearest-neighbor interactions.

We use Monte Carlo and transfer matrix methods in combination with extrapolation schemes to determine the phase diagram of the two-dimensional superantiferromagnetic (SAF) Ising model with next-nearest-neighbor (NNN) interactions in a magnetic field. The interactions between nearest-neighbor (NN) spins are ferromagnetic along x, and antiferromagnetic along Y. We find that for sufficiently low temperatures and fields, there exists a region limited by a critical line of second-order transitions separating a SAF phase from a magnetically induced paramagnetic phase. We did not find any region with either first-order transition or with reentrant behavior. The NNN couplings produce either an expansion or a contraction of the SAF phase. Expansion occurs when the interactions are antiferromagnetic, and contraction when they are ferromagnetic. There is a critical ratio R(c)=1/2 between NNN and NN couplings, beyond which the SAF phase no longer exists.