Physical review. B./Physical review. B

Quantum wetting transition in the one-dimensional transverse-field Ising model with random bonds

Kun Hu, Xintian Wu

Published Oct 27, 2021• 3 citations

Abstract

The quantum wetting transition in the one-dimensional transverse-field Ising model with random bonds is studied. Opposite boundary fields are applied at the two ends of the Ising chain. The interface between two domains with oppositely oriented magnetization is localized near the boundary or in the middle of the chain. The wetting transition refers to the phase transition of localization and delocalization of the interface. The tuning parameter ${h}_{L}$ is the boundary field at one end, e.g., the left end. At the transition point ${h}_{L}={h}_{w}$, the wetting transition occurs. First, we study the wetting transition with one defect bond. It is a first-order transition, and the interface jumps from the left end of the chain to the defect bond at the transition point. Second, we study the wetting transition with two defect bonds and find that the weaker defect bond dominates the phase transition. The wetting transition is still first order, and the interface is localized at the left end of the chain or at the weaker defect bond. Lastly, the random bond case is studied. The random bonds have a rectangular distribution. The wetting transition is still first order. The finite-size effects are studied. The statistics of the transition points and the energy gaps are obtained. We study lattices with sizes $N=100$, 150, 200, 250, 300, and 350. The deviation of the phase transition points ${h}_{w}$ does not decrease as the lattice size increases. It is argued that the transition point is sample dependent, i.e., the variation in the transition point ${h}_{w}$ does not approach zero, even at the thermodynamic limit.

DOI: 10.1103/physrevb.104.134430

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Physical review. B./Physical review. B

Quantum wetting transition in the one-dimensional transverse-field Ising model with random bonds

Kun Hu, Xintian Wu

Published Oct 27, 2021• 3 citations

Publisher

Abstract

{h}_{L}

is the boundary field at one end, e.g., the left end. At the transition point

{h}_{L}={h}_{w}

, the wetting transition occurs. First, we study the wetting transition with one defect bond. It is a first-order transition, and the interface jumps from the left end of the chain to the defect bond at the transition point. Second, we study the wetting transition with two defect bonds and find that the weaker defect bond dominates the phase transition. The wetting transition is still first order, and the interface is localized at the left end of the chain or at the weaker defect bond. Lastly, the random bond case is studied. The random bonds have a rectangular distribution. The wetting transition is still first order. The finite-size effects are studied. The statistics of the transition points and the energy gaps are obtained. We study lattices with sizes

N=100

, 150, 200, 250, 300, and 350. The deviation of the phase transition points

{h}_{w}

does not decrease as the lattice size increases. It is argued that the transition point is sample dependent, i.e., the variation in the transition point

{h}_{w}

does not approach zero, even at the thermodynamic limit.

DOI: 10.1103/physrevb.104.134430

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First- and second-order quantum phase transitions in the one-dimensional transverse-field Ising model with boundary fields

Kun Hu, Xintian Wu

The one-dimensional transverse field Ising model with boundary fields is studied analytically and numerically. The phase diagram in the ordered state is obtained. We find that there exist two types of phase transitions. The longitudinal boundary fields applied to the left and right ends of the Ising chain are ${h}_{L}$ and ${h}_{R}$ , respectively. For $|{h}_{L}|,|{h}_{R}|<\sqrt{1\ensuremath{-}g}$, where $g$ is the transverse field and ${h}_{L}{h}_{R}<0$ , a first-order phase transition occurs with the changing ${h}_{L}$ or ${h}_{R}$ . The energy gap and boundary magnetization are solved exactly. The analytical expressions of the finite-size scaling for the first-order phase transition are obtained. For $|{h}_{R}|>\sqrt{1\ensuremath{-}g}$ and ${h}_{L}{h}_{R}<0$ , there exists a continuous phase transition with changing ${h}_{L}$ and vice versa. This transition is identified as a quantum wetting transition. The singularity of the boundary magnetization in this phase transition is explicitly shown. A simple computational procedure with high accuracy and efficiency is proposed to calculate the magnetization. The magnetization profile, correlation functions, and wetting layer thickness are studied numerically.The one-dimensional transverse field Ising model with boundary fields is studied analytically and numerically. The phase diagram in the ordered state is obtained. We find that there exist two types of phase transitions. The longitudinal boundary fields applied to the left and right ends of the Ising chain are ${h}_{L}$ and ${h}_{R}$, respectively. For $|{h}_{L}|,|{h}_{R}|<\sqrt{1\ensuremath{-}g}$, where $g$ is the transverse field and ${h}_{L}{h}_{R}<0$, a first-order phase transition occurs with the changing ${h}_{L}$ or ${h}_{R}$. The energy gap and boundary magnetization are solved exactly. The analytical expressions of the finite-size scaling for the first-order phase transition are obtained. For $|{h}_{R}|>\sqrt{1\ensuremath{-}g}$ and ${h}_{L}{h}_{R}<0$, there exists a continuous phase transition with changing ${h}_{L}$ and vice versa. This transition is identified as a quantum wetting transition. The singularity of the boundary magnetization in this phase transition is explicitly shown. A simple computational procedure with high accuracy and efficiency is proposed to calculate the magnetization. The magnetization profile, correlation functions, and wetting layer thickness are studied numerically.

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The characterization of quantum phase transitions is a fundamental task for the understanding of quantum phases of matter, with a number of potential applications in quantum technologies. In this work, we use digital quantum simulation as a resource to theoretically and experimentally study quantum phase transitions. More specifically, we implement the variational quantum eigensolver (VQE) algorithm for the one-dimensional spin- $1/2$ transverse-field Ising chain in the presence of boundary magnetic fields. Such fields can induce a rich phase diagram, including a first-order line and a continuous wetting transition, which is a quantum version of the classical wetting surface phenomenon. We present results for noiseless simulations of the associated quantum circuits as well as hardware results taken from a superconducting quantum processor. For different regions of the phase diagram, the quantum algorithm allows us to predict the critical value of the magnetic fields responsible for either the first- or second-order transitions occurring in the system.The characterization of quantum phase transitions is a fundamental task for the understanding of quantum phases of matter, with a number of potential applications in quantum technologies. In this work, we use digital quantum simulation as a resource to theoretically and experimentally study quantum phase transitions. More specifically, we implement the variational quantum eigensolver (VQE) algorithm for the one-dimensional spin-$1/2$ transverse-field Ising chain in the presence of boundary magnetic fields. Such fields can induce a rich phase diagram, including a first-order line and a continuous wetting transition, which is a quantum version of the classical wetting surface phenomenon. We present results for noiseless simulations of the associated quantum circuits as well as hardware results taken from a superconducting quantum processor. For different regions of the phase diagram, the quantum algorithm allows us to predict the critical value of the magnetic fields responsible for either the first- or second-order transitions occurring in the system.

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Abstract A finite length ferromagnetic chain with opposite spin polarization imposed at its two ends is one of the simplest frustrated spin models. In the clean classical limit the domain wall inserted on account of the boundary conditions resides with equal probability on any one of the bonds, and the degeneracy is precisely equal to the number of bonds. If quantum mechanics is introduced via a transverse field, the domain wall will behave as a particle in a box, and prefer to be nearer the middle of the chain rather than the ends. A simple characteristic of a real quantum annealer is therefore which of these limits obtains in practice. Here we have used the ferromagnetic chain with antiparallel boundary spins to test a real flux qubit quantum annealer and discover that contrary to both expectations, the domain walls found are non-uniformly distributed on account of effective random longitudinal fields present notwithstanding tuning carried out to zero out such fields when the couplings between qubits are nominally zero. We present a simple derivation of the form of the distribution function for the domain walls, and show also how the effect we have discovered can be used to determine the strength of the effective random fields (noise) characterizing the annealer. The noise measured in this fashion is smaller than what is seen during the single-qubit tuning process, but nonetheless qualitatively affects the outcome of the simulation performed by the annealer.Abstract A finite length ferromagnetic chain with opposite spin polarization imposed at its two ends is one of the simplest frustrated spin models. In the clean classical limit the domain wall inserted on account of the boundary conditions resides with equal probability on any one of the bonds, and the degeneracy is precisely equal to the number of bonds. If quantum mechanics is introduced via a transverse field, the domain wall will behave as a particle in a box, and prefer to be nearer the middle of the chain rather than the ends. A simple characteristic of a real quantum annealer is therefore which of these limits obtains in practice. Here we have used the ferromagnetic chain with antiparallel boundary spins to test a real flux qubit quantum annealer and discover that contrary to both expectations, the domain walls found are non-uniformly distributed on account of effective random longitudinal fields present notwithstanding tuning carried out to zero out such fields when the couplings between qubits are nominally zero. We present a simple derivation of the form of the distribution function for the domain walls, and show also how the effect we have discovered can be used to determine the strength of the effective random fields (noise) characterizing the annealer. The noise measured in this fashion is smaller than what is seen during the single-qubit tuning process, but nonetheless qualitatively affects the outcome of the simulation performed by the annealer.

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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.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:1912.06035Physical review. B./Physical review. BJan 30, 20209 citations

{h}_{L}

is the boundary field at one end, e.g., the left end. At the transition point

{h}_{L}={h}_{w}

N=100

, 150, 200, 250, 300, and 350. The deviation of the phase transition points

{h}_{w}

does not decrease as the lattice size increases. It is argued that the transition point is sample dependent, i.e., the variation in the transition point

{h}_{w}

does not approach zero, even at the thermodynamic limit.

{h}_{L}

and

{h}_{R}

, respectively. For $|{h}_{L}|,|{h}_{R}|<\sqrt{1\ensuremath{-}g}$, where

g

is the transverse field and

{h}_{L}{h}_{R}<0

, a first-order phase transition occurs with the changing

{h}_{L}

{h}_{R}

. The energy gap and boundary magnetization are solved exactly. The analytical expressions of the finite-size scaling for the first-order phase transition are obtained. For $|{h}_{R}|>\sqrt{1\ensuremath{-}g}$ and

{h}_{L}{h}_{R}<0

, there exists a continuous phase transition with changing

{h}_{L}

and vice versa. This transition is identified as a quantum wetting transition. The singularity of the boundary magnetization in this phase transition is explicitly shown. A simple computational procedure with high accuracy and efficiency is proposed to calculate the magnetization. The magnetization profile, correlation functions, and wetting layer thickness are studied numerically.

The wetting transition in the transverse-field spin-

\frac{1}{2}

XY model with opposite boundary fields

{h}_{L}^{x}{h}_{R}^{x}<0

is studied analytically and numerically. We find that the phase diagram is complex and that the wetting transition is of three types: first, second, and fourth order. The energy gap is obtained analytically, and the magnetization profile, correlation functions, and wetting layer thickness are obtained numerically. For

|{h}_{L}^{x}|,|{h}_{R}^{x}|<{h}_{w}

, a first-order phase transition occurs at ${h}_{L}^{x}=\ensuremath{-}{h}_{R}^{x}$, where

{h}_{w}

is the continuous wetting transition point. For

|{h}_{R}^{x}|

larger than

{h}_{w}

, the continuous wetting transition occurs at

{h}_{L}^{x}={h}_{w}

, and vice versa. For $g\ensuremath{\ne}1\ensuremath{-}{\ensuremath{\gamma}}^{2}$, the wetting transition is second order, and commensurate and incommensurate phases occur for $g<1\ensuremath{-}{\ensuremath{\gamma}}^{2}$ and $g>1\ensuremath{-}{\ensuremath{\gamma}}^{2}$, respectively. For $g=1\ensuremath{-}{\ensuremath{\gamma}}^{2}$, the wetting transition is fourth order. For this fourth-order phase transition, the third derivative of the surface magnetization oscillates and diverges near the transition point. The correlation length exponent is $\ensuremath{\nu}=2$, and the dynamic exponent is

z=2

. Thus, this fourth-order transition belongs to a new universality class. The wetting behavior is induced by asymmetric boundary fields ${h}_{L}^{x}=\ensuremath{-}{h}_{R}^{x}$ and its finite-size scaling is discussed.

h

and the dilution

p

, the quantum phase diagram of both models is explored. Both quantum phase diagrams show long-range order for small

h

and small

p

. The ordered phase of each is separated from the disordered (quantum) Griffiths phase by second-order phase transitions on two critical lines touching at a multi-critical point. Using Binder ratios we locate quantum critical points with high accuracy. The order-parameter critical exponent

β

and the average correlation-length exponent

ν_{\mathrm{av}}

are determined along the critical lines and at the multi-critical points via finite-size scaling. We find three internally consistent sets of critical exponents and compare them with potentially connected universality classes. The quantum Griffiths phase in the vicinity of the phase transition lines is analyzed through the local susceptibility. Our results indicate that activated scaling occurs not only at the percolation transition, but also at the phase transition line for

p

smaller than the percolation threshold

p_c

1/2

transverse-field Ising chain in the presence of boundary magnetic fields. Such fields can induce a rich phase diagram, including a first-order line and a continuous wetting transition, which is a quantum version of the classical wetting surface phenomenon. We present results for noiseless simulations of the associated quantum circuits as well as hardware results taken from a superconducting quantum processor. For different regions of the phase diagram, the quantum algorithm allows us to predict the critical value of the magnetic fields responsible for either the first- or second-order transitions occurring in the system.

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.