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Physical review. B, Condensed matter

Quantum hard rods: Critical behavior and conformal invariance

Ferenc Iglói

Published Aug 1, 1989• 12 citations
Publisher

Abstract

The critical properties of one-dimensional quantum Ising models with near-neighbor exclusion (hard rods) are studied by finite-size scaling and conformal invariance. For dimers (rods with length m=2) the system exhibits an Ising-type critical point, while for m\ensuremath{\ge}3 the system undergoes a first-order phase transition. The operator content of the critical Hamiltonian of hard dimers has been determined for different boundary conditions and identification with the sectors of the Ising model has been done. Using our results, we propose a phase diagram for the quantum Ising model with multispin interaction in the presence of transverse and longitudinal fields.

DOI
10.1103/physrevb.40.2362

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The critical properties of one-dimensional quantum Ising models with near-neighbor exclusion (hard rods) are studied by finite-size scaling and conformal invariance. For dimers (rods with length m=2) the system exhibits an Ising-type critical point, while for m\ensuremath{\ge}3 the system undergoes a first-order phase transition. The operator content of the critical Hamiltonian of hard dimers has been determined for different boundary conditions and identification with the sectors of the Ising model has been done. Using our results, we propose a phase diagram for the quantum Ising model with multispin interaction in the presence of transverse and longitudinal fields.
A one-dimensional quantum Ising model with near-neighbor exclusion (``hard rods'') is mapped onto an anisotropic two-dimensional classical model using the Trotter formula. The critical properties of this classical equivalent model are studied by Monte Carlo techniques and finite-size scaling. A first-order transition is detected if the length of the rods is larger than three, while for dimers a second-order Ising-like transition is found. The type of transition in the borderline case for trimers is not completely settled: It may be second order or very weakly first order, with our data slightly in favor of a first-order transition.
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.
We investigate two kinds of quantum phase transitions observed in the one-dimensional half-filled Peierls-Hubbard model with the next-nearest-neighbor hopping integral in the strong-coupling region $U\ensuremath{\gg}t$, ${t}^{\ensuremath{'}}$ [ttt ${(t}^{\ensuremath{'}})$, nearest- (next-nearest-) neighbor hopping; UUU, on-site Coulomb repulsion]. In the uniform case, with the help of the conformal field theory prediction, we numerically determine a phase boundary ${t}_{\mathrm{c}}^{\ensuremath{'}}(U/t)$ between the spin-fluid and the dimer states, where a bare coupling of the marginal operator vanishes and the low-energy and long-distance behaviors of the spin part are described by a free-boson model. To exhibit the conformal invariance of the systems on the phase boundary, a multiplet structure of the excitation spectrum of finite-size systems and a value of the central charge are also examined. The critical phenomenological aspect of the spin-Peierls transitions accompanied by the lattice dimerization is then argued for the systems on the phase boundary; the existence of logarithmic corrections to the power-law behaviors of the energy gain and the spin gap (i.e., the Cross-Fisher scaling law) are discussed.
With conformal-invariance methods, Burkhardt, Guim, and Xue studied the critical Ising model, defined on the upper half plane y>0 with different boundary conditions a and b on the negative and positive x axes. For ab=-+ and f+, they determined the one- and two-point averages of the spin σ and energy ε. Here +,-, and f stand for spin-up, spin-down, and free-spin boundaries, respectively. The case +-+-+⋯, where the boundary condition switches between + and - at arbitrary points, ζ_{1},ζ_{2},⋯ on the x axis was also analyzed. In the first half of this paper a similar study is carried out for the alternating boundary condition +f+f+⋯ and the case -f+ of three different boundary conditions. Exact results for the one- and two-point averages of σ,ε, and the stress tensor T are derived with conformal-invariance methods. From the results for 〈T〉, the critical Casimir interaction with the boundary of a wedge-shaped inclusion is derived for mixed boundary conditions. In the second half of the paper, arbitrary two-dimensional critical systems with mixed boundary conditions are analyzed with boundary-operator expansions. Two distinct types of expansions-away from switching points of the boundary condition and at switching points-are considered. Using the expansions, we express the asymptotic behavior of two-point averages near boundaries in terms of one-point averages. We also consider the strip geometry with mixed boundary conditions and derive the distant-wall corrections to one-point averages near one edge due to the other edge. Finally we confirm the consistency of the predictions obtained with conformal-invariance methods and with boundary-operator expansions, in the the first and second halves of the paper.
The fidelity per site between two ground states of a quantum lattice system corresponding to different values of the control parameter defines a surface embedded in a Euclidean space. The Gaussian curvature naturally quantifies quantum fluctuations that destroy orders at transition points. It turns out that quantum fluctuations wildly distort the fidelity surface near the transition points, at which the Gaussian curvature is singular in the thermodynamic limit. As a concrete example, the one-dimensional quantum Ising model in a transverse field is analyzed. We also perform a finite size scaling analysis for the transverse Ising model of finite sizes. The scaling behavior for the Gaussian curvature is numerically checked and the correlation length critical exponent is extracted, which is consistent with the conformal invariance at the critical point.