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Acta Physica Sinica

Quantum pseudocritical point in the unbounded quasiperiodic transverse field Ising chain

Zhang Zhen-Jun, Li Wen-Juan, Zhu Xuan, Ye Xiong, Tong Pei-Qing

Published Jan 1, 2015• 2 citations
Publisher

Abstract

We study the quantum pseudocritical points in the unbounded quasiperiodic transverse field Ising chain of finite-size systematically. Firstly, we study the derivatives of averaged magnetic moment and the averaged concurrence with transverse fields. Both of them show two visible peaks, with are nearly not raised when the length of chain is increased. Moreover, the places where the peaks occur in the transverse field are obviously different from that of the quantum phase transition point in the thermodynamic limit. These results are very different from those of the bounded quasiperiodic transverse field Ising chain and the disordered transverse field Ising chain. Then, we analyze the origin of the two visible peaks. For that we study the derivative of magnetic moment for each spin with transverse field. For all spins, the single magnetic moment only show one peak. However, the places where the peaks occur are not random. The peaks always occur in two regions. Thus, the derivatives of averaged magnetic moment reveal two peaks. Furthermore, we study the probability distribution of the pseudocritical points through finding out the peaks of the single magnetic moment in 1000 samples. The distribution is not Guassian. This result is obviously different from that of the disordered case. Besides, the pseudocritical points nearly do not occur at the quantum phase transition point. Finally, we analyze the origin of the pseudocritical points. For that we study the relationship between the spin places and the corresponding places of pseudocritical points. It is found that the pseudocritical points are caused by the two groups that exist in the nearest neighboring interactions of the unbounded quasiperiodic structure. When a spin is in one group, this group will decide the probable place of the pseudocritical point. Through this study, we find that although the quantum phase transition behaviors of the unbounded quasiperiodic transverse field Ising chain and the disordered transverse field Ising chain belong to the same universal class in the thermodynamic limit, the thermodynamic behaviors of the two Ising chains are very different as in finite sizes. The differences are caused by the special structure in the unbounded quasiperiodic system.

DOI
10.7498/aps.64.190501

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We study the quantum pseudocritical points in the unbounded quasiperiodic transverse field Ising chain of finite-size systematically. Firstly, we study the derivatives of averaged magnetic moment and the averaged concurrence with transverse fields. Both of them show two visible peaks, with are nearly not raised when the length of chain is increased. Moreover, the places where the peaks occur in the transverse field are obviously different from that of the quantum phase transition point in the thermodynamic limit. These results are very different from those of the bounded quasiperiodic transverse field Ising chain and the disordered transverse field Ising chain. Then, we analyze the origin of the two visible peaks. For that we study the derivative of magnetic moment for each spin with transverse field. For all spins, the single magnetic moment only show one peak. However, the places where the peaks occur are not random. The peaks always occur in two regions. Thus, the derivatives of averaged magnetic moment reveal two peaks. Furthermore, we study the probability distribution of the pseudocritical points through finding out the peaks of the single magnetic moment in 1000 samples. The distribution is not Guassian. This result is obviously different from that of the disordered case. Besides, the pseudocritical points nearly do not occur at the quantum phase transition point. Finally, we analyze the origin of the pseudocritical points. For that we study the relationship between the spin places and the corresponding places of pseudocritical points. It is found that the pseudocritical points are caused by the two groups that exist in the nearest neighboring interactions of the unbounded quasiperiodic structure. When a spin is in one group, this group will decide the probable place of the pseudocritical point. Through this study, we find that although the quantum phase transition behaviors of the unbounded quasiperiodic transverse field Ising chain and the disordered transverse field Ising chain belong to the same universal class in the thermodynamic limit, the thermodynamic behaviors of the two Ising chains are very different as in finite sizes. The differences are caused by the special structure in the unbounded quasiperiodic system.
We study the distribution of finite-size pseudocritical points in a one-dimensional random quantum magnet with a quantum phase transition described by an infinite randomness fixed point. Pseudocritical points are defined in three different ways: the position of the maximum of the average entanglement entropy, the scaling behavior of the surface magnetization, and the energy of a soft mode. All three lead to a log-normal distribution of the pseudocritical transverse fields, where the width scales as ${L}^{\ensuremath{-}1∕\ensuremath{\nu}}$ with $\ensuremath{\nu}=2$ and the shift of the average value scales as ${L}^{\ensuremath{-}1∕{\ensuremath{\nu}}_{typ}}$ with ${\ensuremath{\nu}}_{typ}=1$, which we related to the scaling of average and typical quantities in the critical region.
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.
The infinite-disorder fixed point of the random transverse-field Ising model is expected to control the critical behavior of a large class of random quantum and stochastic systems having an order parameter with discrete symmetry. Here we study the model on the square lattice with a very efficient numerical implementation of the strong disorder renormalization group method, which makes us possible to treat finite samples of linear size up to L=2048L=2048L=2048. We have calculated sample dependent pseudocritical points and studied their distribution, which is found to be characterized by the same shift and width exponent: $\ensuremath{\nu}=1.24(2)$. For different types of disorder the infinite-disorder fixed point is shown to be characterized by the same set of critical exponents, for which we have obtained improved estimates: x=0.982(15)x=0.982(15)x=0.982(15) and $\ensuremath{\psi}=0.48(2)$. We have also studied the scaling behavior of the magnetization in the vicinity of the critical point as well as dynamical scaling in the ordered and disordered Griffiths phases.
The interplay between disorder, quantum fluctuations, and dissipation is studied in the random transverse Ising chain coupled to a dissipative Ohmic bath with a real space renormalization group. A typically very large length scale L* is identified above which the physics of frozen clusters dominates. Below L* a strong-disorder fixed point determines scaling at a pseudocritical point. In a Griffiths-McCoy region frozen clusters produce already a finite magnetization resulting in a classical low temperature behavior of the susceptibility and specific heat. These override the confluent singularities that are characterized by a continuously varying exponent z and are visible above a temperature T*approximately L*-z.
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.
We have studied the antiferromagnetic Ising chain in a transverse magnetic field hx{h}_{x}hx​ and uniform longitudinal field hz.{h}_{z}.hz​. Using the density-matrix renormalization group calculation combined with a finite-size scaling the ground-state phase diagram in (hx,hz){(h}_{x}{,h}_{z})(hx​,hz​) plane is determined. It is shown that there is an order-disordered transition line in this plane and the critical properties belong to the universality class of the two-dimensional Ising model. Based on the perturbation theory in hz{h}_{z}hz​ the scaling behavior of the mass gap in the vicinity of the critical point (hx=1/2,hz=0){(h}_{x}{=1/2,h}_{z}=0)(hx​=1/2,hz​=0) is established. It is found that the form of the transition line near the classical multicritical point (hx=0,hz=1){(h}_{x}{=0,h}_{z}=1)(hx​=0,hz​=1) is linear. The connection of the considered quantum model with the quasi-one-dimensional classical Ising model in the magnetic field is discussed.