arXiv:1912.06035 · Physical review. B./Physical review. B

Reentrant random quantum Ising antiferromagnet

Péter Lajkó, Jean-Christian Anglès d’Auriac, Heiko Rieger, Ferenc Iglói

Published Jan 30, 2020• 9 citations

Abstract

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.

DOI: 10.1103/physrevb.101.024203

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arXiv:1912.06035 · Physical review. B./Physical review. B

Reentrant random quantum Ising antiferromagnet

Péter Lajkó, Jean-Christian Anglès d’Auriac, Heiko Rieger, Ferenc Iglói

Published Jan 30, 2020• 9 citations

arXiv HTML PDF

Abstract

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.

DOI: 10.1103/physrevb.101.024203

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Jean-Christian Anglès d’Auriac, Ferenc Iglói

Using combinatorial optimization techniques we study the critical properties of the two- and three-dimensional Ising models with uniformly distributed random antiferromagnetic couplings $(1\ensuremath{\le}{J}_{i}\ensuremath{\le}2)$ in the presence of a homogeneous longitudinal field, $h$ , at zero temperature. In finite systems of linear size, $L$ , we measure the average correlation function, ${C}_{L}(\ensuremath{\ell},h)$, when the sites are either on the same sublattice, or they belong to different sublattices. The phase transition, which is of first order in the pure system, turns to mixed order in two dimensions with critical exponents $1/\ensuremath{\nu}\ensuremath{\approx}0.5$ and $\ensuremath{\eta}\ensuremath{\approx}0.7$. In three dimensions we obtain $1/\ensuremath{\nu}\ensuremath{\approx}0.7$, which is compatible with the value of the random-field Ising model, but we cannot discriminate between second-order and mixed-order transitions.Using combinatorial optimization techniques we study the critical properties of the two- and three-dimensional Ising models with uniformly distributed random antiferromagnetic couplings $(1\ensuremath{\le}{J}_{i}\ensuremath{\le}2)$ in the presence of a homogeneous longitudinal field, $h$, at zero temperature. In finite systems of linear size, $L$, we measure the average correlation function, ${C}_{L}(\ensuremath{\ell},h)$, when the sites are either on the same sublattice, or they belong to different sublattices. The phase transition, which is of first order in the pure system, turns to mixed order in two dimensions with critical exponents $1/\ensuremath{\nu}\ensuremath{\approx}0.5$ and $\ensuremath{\eta}\ensuremath{\approx}0.7$. In three dimensions we obtain $1/\ensuremath{\nu}\ensuremath{\approx}0.7$, which is compatible with the value of the random-field Ising model, but we cannot discriminate between second-order and mixed-order transitions.

arXiv:2206.03363Physical review. ESep 12, 20220 citations

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

(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.

The antiferromagnetic quantum Ising chain has a quantum critical point which belongs to the universality class of the transverse Ising model (TIM). When a longitudinal field (

h

) is switched on, the phase transition is preserved, which turns to first order for $h/\mathrm{\ensuremath{\Gamma}}\ensuremath{\rightarrow}\ensuremath{\infty}, \mathrm{\ensuremath{\Gamma}}$ being the strength of the transverse field. Here we will re-examine the critical properties along the phase transition line. During a quantum block renormalization group calculation, the TIM fixed point for $h/\mathrm{\ensuremath{\Gamma}}>0$ is found to be unstable. Using DMRG techniques, we calculated the entanglement entropy and the spin-spin correlation function, both of which signaled a divergent correlation length at the transition point with the TIM exponents. At the same time, the bulk correlation function has a jump and the end-to-end correlation function has a discontinuous derivative at the transition point. Consequently for finite $h/\mathrm{\ensuremath{\Gamma}}$ the transition is of mixed order.

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.

h

, at zero temperature. In finite systems of linear size,

L

, we measure the average correlation function, ${C}_{L}(\ensuremath{\ell},h)$, when the sites are either on the same sublattice, or they belong to different sublattices. The phase transition, which is of first order in the pure system, turns to mixed order in two dimensions with critical exponents $1/\ensuremath{\nu}\ensuremath{\approx}0.5$ and $\ensuremath{\eta}\ensuremath{\approx}0.7$. In three dimensions we obtain $1/\ensuremath{\nu}\ensuremath{\approx}0.7$, which is compatible with the value of the random-field Ising model, but we cannot discriminate between second-order and mixed-order transitions.