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RENORMALIZATION OF THE ONE-DIMENSIONAL QUANTUM ISING MODEL

Amalio F. Pacheco

Published Jan 1, 1978• 0 citations
OpenAlex

Abstract

Using the framework of the SLAC lattice approach it is shown that a symmetric block spinning between link and site terms in the Hamiltonian of the one-dimensional quantum Ising Model provides exact known results for both critical coupling constant and thermal exponent.The ground state energy density calculated in this way exhibits in the critical point a divergence in its second derivative.

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RENORMALIZATION APPROACH TO THE TWO-DIMENSIONAL QUANTUM ISING MODEL

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Physical review. D/Physical review. D.•Jan 1, 1978•0 citations

Comment on the SLAC renormalization-group approach to the Ising chain in a transverse magnetic field

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It is shown that the critical parameters obtained for the Ising chain in a transverse field by using the SLAC naive renormalization-group approach can be easily and considerably improved. This is obtained after doing a convenient trick so that site and link terms of the Hamiltonian of the theory are treated in a symmetric way. In particular, the critical coupling constant and the critical thermal exponent obtained with our method are the exact known results of this model.

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Using the framework of the SLAC lattice approach it is shown that a symmetric block spinning between link and site terms in the Hamiltonian of the one-dimensional quantum Ising Model provides exact known results for both critical coupling constant and thermal exponent.The ground state energy density calculated in this way exhibits in the critical point a divergence in its second derivative.
The results of a new hybrid method for calculating ground-state energies for lattice Hamiltonians are presented. This method combines the ttt expansion with the real-space renormalization-group approach, with the hope of extracting infinite-volume physics at $t\ensuremath{\rightarrow}\ensuremath{\infty}$ from calculations of only a few powers of ttt. We calculate the ground-state energy for the (1+1)-dimensional anisotropic Heisenberg model and the (1+1)-dimensional Ising model. Using a blocking that treats the sites and links symmetrically, we were able to determine the exact critical point of the Ising system. In both of these systems we see that the operator renormalization-group method substantially improves upon the results of either the ttt expansion or real-space renormalization-group methods used separately.
We use the quantum renormalization-group (QRG) method to study the entanglement and quantum phase transition (QPT) in the one-dimensional spin-1/2 Heisenberg—Ising model [Lieb E, Schultz T and Mattis D 1961 Ann. Phys. (N.Y.) 16 407]. We find the quantum phase boundary of this model by investigating the evolution of concurrence in terms of QRG iterations. We also investigate the scaling behavior of the system close to the quantum critical point, which shows that the minimum value of the first derivative of concurrence and the position of the minimum scale with an exponent of the system size. Also, the first derivative of concurrence between two blocks diverges at the quantum critical point, which is directly associated with the divergence of the correlation length.
We analyze the fidelity per lattice site for two different ground states of the one-dimensional quantum Ising model in a transverse field near the critical point. It is found that, in the thermodynamic limit, the fidelity per lattice site is singular, and the derivative of its logarithmic function with respect to the transverse field strength is logarithmically divergent at the critical point. The scaling behavior is confirmed numerically by performing a finite-size scaling analysis for systems of different sizes, consistent with the conformal invariance at the critical point. This allows us to extract the correlation length critical exponent, which turns out to be universal in the sense that the correlation length critical exponent does not depend on either the anisotropic parameter or the transverse field strength.
The scaling of the transition temperature into an ordered phase close to a quantum critical point as well as the order parameter fluctuations inside the quantum critical region provide valuable information about universal properties of the underlying quantum critical point. Here, we employ quantum Monte Carlo simulations to examine these relations in detail for two-dimensional quantum systems that exhibit a finite-temperature Ising-transition line in the vicinity of a quantum critical point that belongs to the universality class of either (i) the three-dimensional Ising model for the case of the quantum Ising model in a transverse magnetic field on the square lattice or (ii) the chiral Ising transition for the case of a half-filled system of spinless fermions on the honeycomb lattice with nearest-neighbor repulsion. While the first case allows large-scale simulations to assess the scaling predictions to a high precision in terms of the known values for the critical exponents at the quantum critical point, for the later case, we extract values of the critical exponents $\ensuremath{\nu}$ and $\ensuremath{\eta}$, related to the order parameter fluctuations, which we discuss in relation to other recent estimates from ground-state quantum Monte Carlo calculations as well as analytical approaches.