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:2210.09380 · Condensed Matter Physics

Random Ising chain in transverse and longitudinal fields: Strong disorder RG study

Tamás Pető, Ferenc Iglói, István A. Kovács

Published Jan 1, 2023• 3 citations
arXivHTMLPDF

Abstract

Motivated by the compound LiHoxY1-xF4, we consider the Ising chain with random couplings and in the presence of simultaneous random transverse and longitudinal fields, and study its low-energy properties at zero temperature by the strong disorder renormalization group approach. In the absence of longitudinal fields, the system exhibits a quantum-ordered and a quantum-disordered phase separated by a critical point of infinite disorder. When the longitudinal random field is switched on, the ordered phase vanishes and the trajectories of the renormalization group are attracted to two disordered fixed points: one is characteristic of the classical random field Ising chain, the other describes the quantum disordered phase. The two disordered phases are separated by a separatrix that starts at the infinite disorder fixed point and near which there are strong quantum fluctuations.

DOI
10.5488/cmp.26.13102

Related papers

Random transverse and longitudinal field Ising chains

Tamás Pető, Ferenc Iglói, István A. Kovács

Abstract Motivated by experimental results on compounds like LiHo x Y 1 − x F 4 , we consider an Ising chain with random bonds in the simultaneous presence of random transverse and longitudinal fields. We study the low-energy properties of the model at zero temperature by the strong disorder renormalisation group method. In the absence of random longitudinal fields, the model showcases a trivial quantum-ordered and quantum-disordered fixed-point and a non-trivial infinite disorder critical point. In the absence of random transverse fields, the behaviour is dictated by the classical random-field Ising fixed-point. In the simultaneous presence of both a longitudinal and transverse random field, the RG trajectories are attracted to a set of disordered fixed-points, in which the disorder is either due to random quantum fluctuations, or due to classical random-field effects. Between the two regimes there is a smooth cross-over, which becomes sharp at the infinite disorder fixed-point. This local separatrix defines the relevant scaling direction, where the correlation-length is shown to diverge with an exponent ν h ≈ 1 .

arXiv:2501.02643•Journal of Statistical Mechanics Theory and Experiment•Sep 1, 2025•0 citations

Reentrant random quantum Ising antiferromagnet

Péter Lajkó, Jean-Christian Anglès d’Auriac, Heiko Rieger +1 more

We consider the quantum Ising chain with uniformly distributed random antiferromagnetic couplings $(1\ensuremath{\le}{J}_{i}\ensuremath{\le}2)$ and uniformly distributed random transverse fields $({\mathrm{\ensuremath{\Gamma}}}_{0}\ensuremath{\le}{\mathrm{\ensuremath{\Gamma}}}_{i}\ensuremath{\le}2{\mathrm{\ensuremath{\Gamma}}}_{0})$ in the presence of a homogeneous longitudinal field, $h$. Using different numerical techniques (density-matrix renormalization group, combinatorial optimization, and strong disorder renormalization group methods) we explore the phase diagram, which consists of an ordered and a disordered phase. At one end of the transition line $(h=0,\phantom{\rule{0.16em}{0ex}}{\mathrm{\ensuremath{\Gamma}}}_{0}=1)$ there is an infinite disorder quantum fixed point, while at the other end $(h=2,\phantom{\rule{0.16em}{0ex}}{\mathrm{\ensuremath{\Gamma}}}_{0}=0)$ there is a classical random first-order transition point. Close to this fixed point, for $h>2$ and ${\mathrm{\ensuremath{\Gamma}}}_{0}>0$ there is a reentrant ordered phase, which is the result of quantum fluctuations by means of an order through disorder phenomenon.

arXiv:1912.06035•Physical review. B./Physical review. B•Jan 30, 2020•9 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

The random transverse field Ising model ind= 2: analysis via boundary strong disorder renormalization

Cécile Monthus, Thomas Garel

To avoid the complicated topology of surviving clusters induced by standard strong disorder RG in dimension d > 1, we introduce a modified procedure called 'boundary strong disorder RG' where the order of decimations is chosen a priori . We apply this modified procedure numerically to the random transverse field Ising model in dimension d = 2. We find that the location of the critical point, the activated exponent ψ ≃ 0.5 of the infinite-disorder scaling, and the finite-size correlation exponent ν FS ≃ 1.3 are compatible with the values obtained previously using standard strong disorder RG. Our conclusion is thus that strong disorder RG is very robust with respect to changes in the order of decimations. In addition, we analyze the RG flows within the two phases in more detail, to show explicitly the presence of various correlation length exponents: we measure the typical correlation exponent ν typ ≃ 0.64 for the disordered phase (this value is very close to the correlation exponent of the pure two-dimensional quantum Ising model), and the typical exponent ν h ≃ 1 for the ordered phase. These values satisfy the relations between critical exponents imposed by the expected finite-size scaling properties at infinite-disorder critical points. We also measure, within the disordered phase, the fluctuation exponent ω ≃ 0.35 which is compatible with the directed polymer exponent in (1 + 1) dimensions.

arXiv:1206.6997•Journal of Statistical Mechanics Theory and Experiment•Sep 28, 2012•9 citations

Transverse spin correlations of the random transverse-field Ising model

Ferenc Iglói, István A. Kovács

The critical behavior of the random transverse-field Ising model in finite-dimensional lattices is governed by infinite disorder fixed points, several properties of which have already been calculated by the use of the strong disorder renormalization-group (SDRG) method. Here we extend these studies and calculate the connected transverse-spin correlation function by a numerical implementation of the SDRG method in $d=1,2$, and 3 dimensions. At the critical point an algebraic decay of the form $\ensuremath{\sim}{r}^{\ensuremath{-}{\ensuremath{\eta}}_{t}}$ is found, with a decay exponent being approximately ${\ensuremath{\eta}}_{t}\ensuremath{\approx}2+2d$. In $d=1$ the results are related to dimer-dimer correlations in the random antiferromagnetic $XX$ chain and have been tested by numerical calculations using free-fermionic techniques.

arXiv:1712.07467•Physical review. B./Physical review. B•Mar 21, 2018•4 citations

Quantum criticality in Ising chains with random hyperuniform couplings

Philip J. D. Crowley, Chris R. Laumann, Sarang Gopalakrishnan

We study quantum phase transitions in transverse-field Ising spin chains in which the couplings are random but hyperuniform, in the sense that their large-scale fluctuations are suppressed. We construct a one-parameter family of disorder models in which long-wavelength fluctuations are increasingly suppressed as a parameter $\ensuremath{\alpha}$ is tuned. For $\ensuremath{\alpha}=0$, one recovers the familiar infinite-randomness critical point. For $0<\ensuremath{\alpha}<1$, we find a line of infinite-randomness critical points with continuously varying critical exponents; however, the Griffiths phases that flank the critical point at $\ensuremath{\alpha}=0$ are absent at any $\ensuremath{\alpha}>0$. When $\ensuremath{\alpha}>1$, randomness is a dangerously irrelevant perturbation at the clean Ising critical point, leading to a state we call the critical Ising insulator. In this state, thermodynamics and equilibrium correlation functions behave as in the clean system. However, all finite-energy excitations are localized, thermal transport vanishes, and autocorrelation functions remain finite in the long-time limit. We characterize this line of hyperuniform critical points using a combination of perturbation theory, renormalization-group methods, and exact diagonalization.

arXiv:1809.04595•Physical review. B./Physical review. B•Oct 17, 2019•21 citations
Motivated by the compound LiHoxY1-xF4, we consider the Ising chain with random couplings and in the presence of simultaneous random transverse and longitudinal fields, and study its low-energy properties at zero temperature by the strong disorder renormalization group approach. In the absence of longitudinal fields, the system exhibits a quantum-ordered and a quantum-disordered phase separated by a critical point of infinite disorder. When the longitudinal random field is switched on, the ordered phase vanishes and the trajectories of the renormalization group are attracted to two disordered fixed points: one is characteristic of the classical random field Ising chain, the other describes the quantum disordered phase. The two disordered phases are separated by a separatrix that starts at the infinite disorder fixed point and near which there are strong quantum fluctuations.
Abstract Motivated by experimental results on compounds like LiHo x Y 1 − x F 4 , we consider an Ising chain with random bonds in the simultaneous presence of random transverse and longitudinal fields. We study the low-energy properties of the model at zero temperature by the strong disorder renormalisation group method. In the absence of random longitudinal fields, the model showcases a trivial quantum-ordered and quantum-disordered fixed-point and a non-trivial infinite disorder critical point. In the absence of random transverse fields, the behaviour is dictated by the classical random-field Ising fixed-point. In the simultaneous presence of both a longitudinal and transverse random field, the RG trajectories are attracted to a set of disordered fixed-points, in which the disorder is either due to random quantum fluctuations, or due to classical random-field effects. Between the two regimes there is a smooth cross-over, which becomes sharp at the infinite disorder fixed-point. This local separatrix defines the relevant scaling direction, where the correlation-length is shown to diverge with an exponent ν h ≈ 1 .
We consider the quantum Ising chain with uniformly distributed random antiferromagnetic couplings $(1\ensuremath{\le}{J}_{i}\ensuremath{\le}2)$ and uniformly distributed random transverse fields $({\mathrm{\ensuremath{\Gamma}}}_{0}\ensuremath{\le}{\mathrm{\ensuremath{\Gamma}}}_{i}\ensuremath{\le}2{\mathrm{\ensuremath{\Gamma}}}_{0})$ in the presence of a homogeneous longitudinal field, hhh. Using different numerical techniques (density-matrix renormalization group, combinatorial optimization, and strong disorder renormalization group methods) we explore the phase diagram, which consists of an ordered and a disordered phase. At one end of the transition line $(h=0,\phantom{\rule{0.16em}{0ex}}{\mathrm{\ensuremath{\Gamma}}}_{0}=1)$ there is an infinite disorder quantum fixed point, while at the other end $(h=2,\phantom{\rule{0.16em}{0ex}}{\mathrm{\ensuremath{\Gamma}}}_{0}=0)$ there is a classical random first-order transition point. Close to this fixed point, for h>2h>2h>2 and ${\mathrm{\ensuremath{\Gamma}}}_{0}>0$ there is a reentrant ordered phase, which is the result of quantum fluctuations by means of an order through disorder phenomenon.
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.
To avoid the complicated topology of surviving clusters induced by standard strong disorder RG in dimension d > 1, we introduce a modified procedure called 'boundary strong disorder RG' where the order of decimations is chosen a priori . We apply this modified procedure numerically to the random transverse field Ising model in dimension d = 2. We find that the location of the critical point, the activated exponent ψ ≃ 0.5 of the infinite-disorder scaling, and the finite-size correlation exponent ν FS ≃ 1.3 are compatible with the values obtained previously using standard strong disorder RG. Our conclusion is thus that strong disorder RG is very robust with respect to changes in the order of decimations. In addition, we analyze the RG flows within the two phases in more detail, to show explicitly the presence of various correlation length exponents: we measure the typical correlation exponent ν typ ≃ 0.64 for the disordered phase (this value is very close to the correlation exponent of the pure two-dimensional quantum Ising model), and the typical exponent ν h ≃ 1 for the ordered phase. These values satisfy the relations between critical exponents imposed by the expected finite-size scaling properties at infinite-disorder critical points. We also measure, within the disordered phase, the fluctuation exponent ω ≃ 0.35 which is compatible with the directed polymer exponent in (1 + 1) dimensions.
The critical behavior of the random transverse-field Ising model in finite-dimensional lattices is governed by infinite disorder fixed points, several properties of which have already been calculated by the use of the strong disorder renormalization-group (SDRG) method. Here we extend these studies and calculate the connected transverse-spin correlation function by a numerical implementation of the SDRG method in d=1,2d=1,2d=1,2, and 3 dimensions. At the critical point an algebraic decay of the form $\ensuremath{\sim}{r}^{\ensuremath{-}{\ensuremath{\eta}}_{t}}$ is found, with a decay exponent being approximately ${\ensuremath{\eta}}_{t}\ensuremath{\approx}2+2d$. In d=1d=1d=1 the results are related to dimer-dimer correlations in the random antiferromagnetic XXXXXX chain and have been tested by numerical calculations using free-fermionic techniques.
We study quantum phase transitions in transverse-field Ising spin chains in which the couplings are random but hyperuniform, in the sense that their large-scale fluctuations are suppressed. We construct a one-parameter family of disorder models in which long-wavelength fluctuations are increasingly suppressed as a parameter $\ensuremath{\alpha}$ is tuned. For $\ensuremath{\alpha}=0$, one recovers the familiar infinite-randomness critical point. For $0<\ensuremath{\alpha}<1$, we find a line of infinite-randomness critical points with continuously varying critical exponents; however, the Griffiths phases that flank the critical point at $\ensuremath{\alpha}=0$ are absent at any $\ensuremath{\alpha}>0$. When $\ensuremath{\alpha}>1$, randomness is a dangerously irrelevant perturbation at the clean Ising critical point, leading to a state we call the critical Ising insulator. In this state, thermodynamics and equilibrium correlation functions behave as in the clean system. However, all finite-energy excitations are localized, thermal transport vanishes, and autocorrelation functions remain finite in the long-time limit. We characterize this line of hyperuniform critical points using a combination of perturbation theory, renormalization-group methods, and exact diagonalization.