Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:0704.1194 · Physical Review B

Finite-size scaling of pseudocritical point distributions in the random transverse-field Ising chain

Ferenc Iglói, Yu‐Cheng Lin, Heiko Rieger, Cécile Monthus

Published Aug 16, 2007• 26 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.76.064421

Related papers

Quantum pseudocritical point in the unbounded quasiperiodic transverse field Ising chain

Zhang Zhen-Jun, Li Wen-Juan, Zhu Xuan +2 more

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.

Acta Physica Sinica•Jan 1, 2015•2 citations

Renormalization group study of the two-dimensional random transverse-field Ising model

István A. Kovács, Ferenc Iglói

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 $L=2048$. 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: $x=0.982(15)$ 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.

arXiv:1005.4740•Physical Review B•Aug 31, 2010•105 citations

Finite-size scaling properties of random transverse-field Ising chains: Comparison between canonical and microcanonical ensembles for the disorder

Cécile Monthus

The random transverse-field Ising chain is the simplest disordered model presenting a quantum phase transition at $T=0.$ We compare analytically its finite-size scaling properties in two different ensembles for the disorder: (i) the canonical ensemble, where the disorder variables are independent and (ii) the microcanonical ensemble, where there exists a global constraint on the disorder variables. The observables under study are the surface magnetization, the correlation of the two surface magnetizations, the gap and the end-to-end spin-spin correlation $C(L)$ for a chain of length L. At criticality, each observable decays typically as ${e}^{\ensuremath{-}w\sqrt{L}}$ in both ensembles, but the probability distributions of the rescaled variable w are different in the two ensembles, in particular, in their asymptotic behaviors. As a consequence, the dependence on L of averaged observables differs in the two ensembles. For instance, the correlation $C(L)$ decays algebraically as $1/L$ in the canonical ensemble, but subexponentially as ${e}^{\ensuremath{-}{\mathrm{cL}}^{1/3}}$ in the microcanonical ensemble. Off criticality, probability distributions of rescaled variables are governed by the critical exponent $\ensuremath{\nu}=2$ in both ensembles, but the following observables are governed by the exponent $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\nu}}=1$ in the microcanonical ensemble, instead of the exponent $\ensuremath{\nu}=2$ in the canonical ensemble: (a) In the disordered phase, the averaged surface magnetization, the averaged correlation of the two surface magnetizations and the averaged end-to-end spin-spin correlation; (b) in the ordered phase, the averaged gap. In conclusion, the measure of the rare events that dominate various averaged observables can be very sensitive to the microcanonical constraint.

arXiv:cond-mat/0309029•Physical Review B•Feb 27, 2004•26 citations

Long-range random transverse-field Ising model in three dimensions

István A. Kovács, Róbert Juhász, Ferenc Iglói

We consider the random transverse-field Ising model in $d=3$ dimensions with long-range ferromagnetic interactions which decay as a power $\ensuremath{\alpha}>d$ with the distance. Using a variant of the strong-disorder renormalization group method we study numerically the phase-transition point from the paramagnetic side. We find that the fixed point controlling the transition is of the strong-disorder type, and based on experience with other similar systems, we expect the results to be qualitatively correct, but probably not asymptotically exact. The distribution of the (sample dependent) pseudocritical points is found to scale with $1/lnL, L$ being the linear size of the sample. Similarly, the critical magnetization scales with ${(lnL)}^{\ensuremath{\chi}}/{L}^{d}$ and the excitation energy behaves as ${L}^{\ensuremath{-}\ensuremath{\alpha}}$. Using extreme-value statistics we argue that extrapolating from the ferromagnetic side the magnetization approaches a finite limiting value and thus the transition is of mixed order.

arXiv:1601.04206•Physical review. B./Physical review. B•May 31, 2016•29 citations

Finite-size scaling of the entanglement entropy of the quantum Ising chain with homogeneous, periodically modulated and random couplings

Ferenc Iglói, Yu-Cheng Lin

Using free-fermionic techniques we study the entanglement entropy of a block of contiguous spins in a large finite quantum Ising chain in a transverse field, with couplings of different types: homogeneous, periodically modulated and random. We carry out a systematic study of finite-size effects at the quantum critical point, and evaluate subleading corrections both for open and for periodic boundary conditions. For a block corresponding to a half of a finite chain, the position of the maximum of the entropy as a function of the control parameter (e.g. the transverse field) can define the effective critical point in the finite sample. On the basis of homogeneous chains, we demonstrate that the scaling behavior of the entropy near the quantum phase transition is in agreement with the universality hypothesis, and calculate the shift of the effective critical point, which has different scaling behaviors for open and for periodic boundary conditions.

arXiv:0803.3610•Journal of Statistical Mechanics Theory and Experiment•Jun 3, 2008•61 citations

Critical behavior and entanglement of the random transverse-field Ising model between one and two dimensions

István A. Kovács, Ferenc Iglói

We consider disordered ladders of the transverse-field Ising model and study their critical properties and entanglement entropy for varying width, $w\ensuremath{\le}20$, by numerical application of the strong disorder renormalization group method. We demonstrate that the critical properties of the ladders for any finite $w$ are controlled by the infinite disorder fixed point of the random chain and the correction to scaling exponents contain information about the two-dimensional model. We calculate sample dependent pseudocritical points and study the shift of the mean values as well as scaling of the width of the distributions and show that both are characterized by the same exponent, $\ensuremath{\nu}(2d)$. We also study scaling of the critical magnetization, investigate critical dynamical scaling as well as the behavior of the critical entanglement entropy. Analyzing the $w$ dependence of the results we have obtained accurate estimates for the critical exponents of the two-dimensional model: $\ensuremath{\nu}(2d)=1.25(8)$, $x(2d)=0.996(15)$, and $\ensuremath{\psi}(2d)=0.51(3)$.

arXiv:0909.4442•Physical Review B•Dec 17, 2009•46 citations
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∕\ensuremathν{L}^{\ensuremath{-}1∕\ensuremath{\nu}}L\ensuremath−1∕\ensuremathν with \ensuremathν=2\ensuremath{\nu}=2\ensuremathν=2 and the shift of the average value scales as L\ensuremath−1∕\ensuremathνtyp{L}^{\ensuremath{-}1∕{\ensuremath{\nu}}_{typ}}L\ensuremath−1∕\ensuremathνtyp​ with \ensuremathνtyp=1{\ensuremath{\nu}}_{typ}=1\ensuremathνtyp​=1, which we related to the scaling of average and typical quantities in the critical region.
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.
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 random transverse-field Ising chain is the simplest disordered model presenting a quantum phase transition at T=0.T=0.T=0. We compare analytically its finite-size scaling properties in two different ensembles for the disorder: (i) the canonical ensemble, where the disorder variables are independent and (ii) the microcanonical ensemble, where there exists a global constraint on the disorder variables. The observables under study are the surface magnetization, the correlation of the two surface magnetizations, the gap and the end-to-end spin-spin correlation C(L)C(L)C(L) for a chain of length L. At criticality, each observable decays typically as ${e}^{\ensuremath{-}w\sqrt{L}}$ in both ensembles, but the probability distributions of the rescaled variable w are different in the two ensembles, in particular, in their asymptotic behaviors. As a consequence, the dependence on L of averaged observables differs in the two ensembles. For instance, the correlation C(L)C(L)C(L) decays algebraically as 1/L1/L1/L in the canonical ensemble, but subexponentially as ${e}^{\ensuremath{-}{\mathrm{cL}}^{1/3}}$ in the microcanonical ensemble. Off criticality, probability distributions of rescaled variables are governed by the critical exponent $\ensuremath{\nu}=2$ in both ensembles, but the following observables are governed by the exponent $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\nu}}=1$ in the microcanonical ensemble, instead of the exponent $\ensuremath{\nu}=2$ in the canonical ensemble: (a) In the disordered phase, the averaged surface magnetization, the averaged correlation of the two surface magnetizations and the averaged end-to-end spin-spin correlation; (b) in the ordered phase, the averaged gap. In conclusion, the measure of the rare events that dominate various averaged observables can be very sensitive to the microcanonical constraint.
We consider the random transverse-field Ising model in d=3d=3d=3 dimensions with long-range ferromagnetic interactions which decay as a power $\ensuremath{\alpha}>d$ with the distance. Using a variant of the strong-disorder renormalization group method we study numerically the phase-transition point from the paramagnetic side. We find that the fixed point controlling the transition is of the strong-disorder type, and based on experience with other similar systems, we expect the results to be qualitatively correct, but probably not asymptotically exact. The distribution of the (sample dependent) pseudocritical points is found to scale with 1/lnL,L1/lnL, L1/lnL,L being the linear size of the sample. Similarly, the critical magnetization scales with ${(lnL)}^{\ensuremath{\chi}}/{L}^{d}$ and the excitation energy behaves as ${L}^{\ensuremath{-}\ensuremath{\alpha}}$. Using extreme-value statistics we argue that extrapolating from the ferromagnetic side the magnetization approaches a finite limiting value and thus the transition is of mixed order.
Using free-fermionic techniques we study the entanglement entropy of a block of contiguous spins in a large finite quantum Ising chain in a transverse field, with couplings of different types: homogeneous, periodically modulated and random. We carry out a systematic study of finite-size effects at the quantum critical point, and evaluate subleading corrections both for open and for periodic boundary conditions. For a block corresponding to a half of a finite chain, the position of the maximum of the entropy as a function of the control parameter (e.g. the transverse field) can define the effective critical point in the finite sample. On the basis of homogeneous chains, we demonstrate that the scaling behavior of the entropy near the quantum phase transition is in agreement with the universality hypothesis, and calculate the shift of the effective critical point, which has different scaling behaviors for open and for periodic boundary conditions.
We consider disordered ladders of the transverse-field Ising model and study their critical properties and entanglement entropy for varying width, $w\ensuremath{\le}20$, by numerical application of the strong disorder renormalization group method. We demonstrate that the critical properties of the ladders for any finite www are controlled by the infinite disorder fixed point of the random chain and the correction to scaling exponents contain information about the two-dimensional model. We calculate sample dependent pseudocritical points and study the shift of the mean values as well as scaling of the width of the distributions and show that both are characterized by the same exponent, $\ensuremath{\nu}(2d)$. We also study scaling of the critical magnetization, investigate critical dynamical scaling as well as the behavior of the critical entanglement entropy. Analyzing the www dependence of the results we have obtained accurate estimates for the critical exponents of the two-dimensional model: $\ensuremath{\nu}(2d)=1.25(8)$, x(2d)=0.996(15)x(2d)=0.996(15)x(2d)=0.996(15), and $\ensuremath{\psi}(2d)=0.51(3)$.