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Physical review. B, Condensed matter

Phase transition and scaling properties of the energy spectrum for a hierarchical quantum Ising model

Zhifang Lin, Ruibao Tao

Published Jul 1, 1990• 0 citations
Publisher

Abstract

We study a one-dimensional quantum Ising model where the exchange couplings and the transverse fields are hierarchically distributed. Exact analytical results for the critical line, energy gap, and dispersion relation of the low-energy excitations are obtained. It is shown that only in the case of ${\mathit{R}}_{1}$=${\mathit{R}}_{2}$=R>${\mathit{R}}_{\mathit{c}}$, where ${\mathit{R}}_{1}$ and ${\mathit{R}}_{2}$ are the hierarchical parameters for the exchange couplings and the transverse fields, does the system preserve the logarithmic singularity in the specific heat. Using renormalization-group techniques and an iteration map for the transfer matrix, we have presented the analytical results for the scaling properties for the energy spectra of the models with ${\mathit{R}}_{1}$=${\mathit{R}}_{2}$ and ${\mathit{R}}_{2}$=1. From knowledge of the scaling properties of the energy spectra, we have exhibited a close relationship between the disappearance of the logarithmic singularity in the specific heat and the anomalous scaling behavior at the lower band edges.

DOI
10.1103/physrevb.42.648

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We study a one-dimensional quantum Ising model where the exchange couplings and the transverse fields are hierarchically distributed. Exact analytical results for the critical line, energy gap, and dispersion relation of the low-energy excitations are obtained. It is shown that only in the case of R1{\mathit{R}}_{1}R1​=R2{\mathit{R}}_{2}R2​=R>Rc{\mathit{R}}_{\mathit{c}}Rc​, where R1{\mathit{R}}_{1}R1​ and R2{\mathit{R}}_{2}R2​ are the hierarchical parameters for the exchange couplings and the transverse fields, does the system preserve the logarithmic singularity in the specific heat. Using renormalization-group techniques and an iteration map for the transfer matrix, we have presented the analytical results for the scaling properties for the energy spectra of the models with R1{\mathit{R}}_{1}R1​=R2{\mathit{R}}_{2}R2​ and R2{\mathit{R}}_{2}R2​=1. From knowledge of the scaling properties of the energy spectra, we have exhibited a close relationship between the disappearance of the logarithmic singularity in the specific heat and the anomalous scaling behavior at the lower band edges.
A quantum Ising chain with both the exchange couplings and the transverse fields arranged in a hierarchical way is considered. Exact analytical results for the critical line and dispersion relation of the low-energy excitations are obtained. It is shown that only when R1{\mathit{R}}_{1}R1​=R2{\mathit{R}}_{2}R2​>Rc{\mathit{R}}_{\mathit{c}}Rc​, where R1{\mathit{R}}_{1}R1​ and R2{\mathit{R}}_{2}R2​ are the hierarchical parameters for the exchange couplings and the transverse fields, respectively, does the system preserve a logarithmic singularity in the specific heat.
We study the one-dimensional quantum Ising model in a transverse magnetic field where the exchange couplings are ordered according to the Thue-Morse (TM) sequence. At zero temperature, this model is equivalent to a two-dimensional classical Ising model in a magnetic field with TM aperiodicity along one direction. We compute the order parameter (magnetization) of the chain and the scaling behavior of the energy spectrum when the system undergoes a phase transition. Analogous to the quasiperiodic (QP) quantum Ising chain, the onset of long-range order is signaled by a nonanaliticity in the exponent \ensuremath{\delta} which describes the scaling of the total bandwidth with the size of the chain. The critical spin-coupling can be computed analytically and it is found to be lower than the QP case. Furthermore, the energy bands are found to be narrower than the corresponding QP chain. The former and latter results are consistent with the fact that the present structure has a degree of ordering intermediate between QP and random.
A real-space renormalization-group treatment of random transverse-field Ising spin chains that was introduced previously is developed and extensively analyzed. It yields results that are asymptotically exact in the critical region near the zero-temperature para-to-ferromagnetic quantum phase transition. In particular, the exact scaling function is obtained for the magnetization as a function of a uniform applied magnetic field and the distance to the critical point, and up to the solution of a linear ordinary differential equation whose solution can be exhaustively analyzed, the scaling function of the average spin-spin correlation function is also obtained. Thus more exact information is obtainable about the critical behavior for this random model than is known for the pure version which is equivalent to the two-dimensional Ising model. The basic reason for this is the extremely broad distribution of energy scales that occurs at low energies near the critical point of the random system. For the random chain the distribution of the magnetization of the first spin in a semi-infinite system is also studied and the results found to agree in the scaling limit with results of McCoy obtained from the exact solution of the closely related McCoy-Wu Ising model; this provides strong justification for the validity of the present approach. The singular properties of the weakly ordered and weakly disordered ``Griffiths' phases'' that occur at zero temperature near the critical point are also studied, as well as the behavior at low but nonzero temperature. Possible extensions of the results and general lessons drawn from them for other random systems are briefly discussed.