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:0805.1753 · Physical Review B

Quantum transitions of the isotropic XY model with long-range interactions on the inhomogeneous periodic chain

J.P. de Lima, L.L. Gonçalves

Published Jun 18, 2008• 10 citations
arXivHTMLPDF

Abstract

The isotropic $XY$ model $(s=1/2)$ in a transverse field, with uniform long-range interactions among the transverse components of the spins, on the inhomogeneous periodic chain, is studied. The model, composed of $N$ segments with $n$ different exchange interactions and magnetic moments, is exactly solved by introducing the integral Gaussian transformation and the generalized Jordan--Wigner transformation, which reduce the problem to the diagonalization of a finite matrix of $n\text{th}$ order. The quantum transitions induced by the transverse field are determined by analyzing the induced magnetization of the cell and the equation of state. The phase diagrams for the quantum transitions, in the space generated by the transverse field and the interaction parameters, are presented. As expected, the model presents multiple, first- and second-order quantum transitions induced by the transverse field, and it corresponds to an extension of the models recently considered by the authors. Detailed results are also presented, at $T=0$, for the induced magnetization and isothermal susceptibility ${\ensuremath{\chi}}_{T}^{zz}$ as a function of the transverse field.

DOI
10.1103/physrevb.77.214424

Related papers

Anisotropic XY model on the inhomogeneous periodic chain

J.P. de Lima, L.L. Gonçalves, T. F. A. Alves

The static and dynamic properties of the anisotropic $XY$ model $(s=1∕2)$ on the inhomogeneous periodic chain, composed of $N$ cells with $n$ different exchange interactions and magnetic moments, in a transverse field $h$, are determined exactly at arbitrary temperatures. The properties are obtained by introducing the Jordan-Wigner fermionization and by reducing the problem to a diagonalization of a finite matrix of $n$th order. The quantum transitions are determined exactly by analyzing, as a function of the field, the induced magnetization $1∕n{\ensuremath{\sum}}_{m=1}^{n}{\ensuremath{\mu}}_{m}⟨{S}_{j,m}^{z}⟩$ ($j$ denotes the cell, $m$ the site within the cell, ${\ensuremath{\mu}}_{m}$ the magnetic moment at site $m$ within the cell) and the spontaneous magnetization $1∕n{\ensuremath{\sum}}_{m=1}^{n}⟨{S}_{j,{m}^{\ensuremath{'}}}^{x}⟩$ which is obtained from the correlations $⟨{S}_{j,m}^{x}{S}_{j+r,m}^{x}⟩$ for large spin separations. These results, which are obtained for infinite chains, correspond to an extension of the ones obtained by Tong and Zhong [Physica B 304, 91 (2001)]. The dynamic correlations, $⟨{S}_{j,m}^{z}(t){S}_{{j}^{\ensuremath{'}},{m}^{\ensuremath{'}}}^{z}(0)⟩$, and the dynamic susceptibility, ${\ensuremath{\chi}}_{q}^{zz}(\ensuremath{\omega})$, are also obtained at arbitrary temperatures. Explicit results are presented in the limit $T=0$, where the critical behavior occurs, for the static susceptibility ${\ensuremath{\chi}}_{q}^{zz}(0)$ as a function of the transverse field $h$, and for the frequency dependency of dynamic susceptibility ${\ensuremath{\chi}}_{q}^{zz}(\ensuremath{\omega})$.

arXiv:cond-mat/0701630•Physical Review B•Jun 5, 2007•14 citations

Study of the XY chain with long-range interactions

Jialiang Li, Shuguo Lei, (1)常熟理工学院江苏省新型功能材料重点建设实验室,常熟 215500；常熟理工学院物理与电子科学系，常熟 215500; (2)南京工业大学应用物理系，南京 210009

Using the mean-field Jordan-Wigner transformation analysis, we have studied the one-dimensional spin-1/2 XY chain with Z-directional uniform long-range interactions in the external magnetic field. We have obtained such results as the Helmholtz free energy, internal energy, specific heat, magnetization and isothermal susceptibility. We also discuss the first and second-order phase transitions of the system at a finite temperature. Our results agree with results published in literature.

Acta Physica Sinica•Jan 1, 2008•1 citation

Quantum phase transitions of one-dimensional period-two anisotropic XY models in a transverse field

Song Jia-Li, Ming Zhong, Tong Pei-Qing

The quantum phase transitions of one-dimensional period-two anisotropic XY models in a transverse field with the Hamiltonian where the anisotropy parameters i take and alternately, are studied. The Hamiltonian can be reduced to the diagonal form by Jordan-Wigner and Bogoliubov transformations. The long-range correlations Cx and Cy are calculated numerically. The phase with Cx Cy0 (or Cy Cx0) is referred to as the ferromagnetic (FM) phase along the x (or y) direction, while the phase with Cx=Cy=0 is the paramagnetic (PM) phase. It is found that the phase diagrams with the ratio -1 and =-1 are different obviously. For the case with -1, the line h=hc1=1-[(1-)/2]2 separates an FM phase from a PM phase, while the line =0 is the boundary between a ferromagnetic phase along the x direction (FMx) and a ferromagnetic phase along the y direction (FMy). These are similar to those of the uniform XY chains in a transverse field (i.e., =1). When =-1, the FMx and FMy phases disappear and there appears a new FM phase. The line h=hc2=1-2 separates this new FM phase from the PM phase. The new phase is gapless with two zeros in single particle energy spectrum. This is due to the new symmetry in the system with =-1, i.e., the Hamiltonian is invariant under the transformation 2ix 2i+1y,2iy 2i+1x. The correlation function between the 2i-1 and 2i lattice points along the x (y) direction is equal to that between the 2i and 2i+1 lattice points along the y (x) direction. As a result, the long-range correlation functions along two directions are equivalent. In order to facilitate the description, we call this gapless phase the isotropic ferromagnetic (FMxx) phase. Finally, the relationship between quantum entanglement and quantum phase transitions of the system is studied. The scaling behaviour of the von Neumann entropy at each point in the FMxx phase is SL~1/3log2L+ Const, which is similar to that at the anisotropic phase transition point of the uniform XY model in a transverse field, and different from those in the FMx and FMy phases.

Acta Physica Sinica•Jan 1, 2017•1 citation

Quantum phase transitions in long-range interacting hyperuniform spin chains in a transverse field

Amartya Bose, Salvatore Torquato A

Hyperuniform states of matter are characterized by anomalous suppression of long-wavelength density fluctuations. While most of the interesting cases of disordered hyperuniformity are provided by complex many-body systems such as liquids or amorphous solids, classical spin chains with certain long-range interactions have been shown to demonstrate the same phenomenon. Such spin-chain systems are ideal models for exploring the effects of quantum mechanics on hyperuniformity. It is well-known that the transverse field Ising model shows a quantum phase transition (QPT) at zero temperature. Under the quantum effects of a transverse magnetic field, classical hyperuniform spin chains are expected to lose their hyperuniformity. High-precision simulations of these cases are complicated because of the presence of highly nontrivial long-range interactions. We perform an extensive analysis of these systems using density matrix renormalization group simulations to study the possibilities of phase transitions and the mechanism by which they lose hyperuniformity. Even for a spin chain of length 30, we see discontinuous changes in properties like the ``$\ensuremath{\tau}$ order metric'' of the ground state, the measure of hyperuniformity, and the second cumulant of the total magnetization along the $x$-direction, all suggestive of first-order QPTs. An interesting feature of the phase transitions in these disordered hyperuniform spin chains is that, depending on the parameter values, the presence of a transverse magnetic field may lead remarkably to an increase in the order of the ground state as measured by the ``$\ensuremath{\tau}$ order metric,'' even if hyperuniformity is lost. Therefore, it would be possible to design materials to target specific novel quantum behaviors in the presence of a transverse magnetic field. Our numerical investigations suggest that these spin chains can show no more than two QPTs. We further analyze the long-range interacting spin chains via the Jordan-Wigner mapping onto a system of spinless fermions, showing that under the pairwise-interaction approximation and a mean-field treatment, there can be at most two quantum phase transitions. Based on these numerical and theoretical explorations, we conjecture that for spin chains with long-range pair interactions that have convergent cosine transforms, there can be a maximum of two quantum phase transitions at zero temperature.

arXiv:2012.06545•Physical review. B./Physical review. B•Jan 29, 2021•2 citations

Mean-field approach to midspectrum eigenstates of long-range interacting quantum systems

Bojan Žunkovič, Pedro Ribeiro

We study the equilibrium properties of the spin-$1/2$ XY chain with an infinite-range transverse interaction. At zero temperature, competition between the XY- and $z$-ordered phases induced by the infinite-range interactions gives rise to a first-order transition upon increasing the transverse coupling. We show that the two gapless points of the XY model behave in fundamentally different ways: isotropic spin chains experience a first-order transition at finite coupling; maximal anisotropic chains overcome a nonalgebraic phase transition at zero coupling strength. The phase diagram depicts a first-order reentrant transition that turns second order along a tricritical line separating a paramagnetic phase from an ordered one at finite temperature. The mean-field approach captures the local properties of the eigenstates and reveals the appearance of a magnetization gap in the spectrum. Global properties, e.g., entanglement entropy, are well approximated only at spectral boundaries. The mean entanglement entropy and the level-spacing ratio deviate from the Gaussian results, revealing the interacting nature of the problem.

arXiv:2310.09803•Physical review. B./Physical review. B•Sep 26, 2024•0 citations

Quantum phases of spin-1/2 extended XY model in transverse magnetic field

Rakesh Kumar Malakar, Asim Kumar Ghosh

Abstract In this study, a spin-1/2 extended anisotropic XY chain has been introduced in which both time reversal and SU(2) symmetries are broken but Z 2 symmetry is preserved. Magnetic and topological phase diagrams in the parameter space have been drawn in the presence of transverse magnetic field. Entanglement measures like mutual information and quantum discord are also evaluated and it indicates that these transitions are second order in nature. Quantum phase transition is noted at zero magnetic field, as well as magnetic long range order is found to withstand magnetic field of any strength. Exact analytic results for spin-spin correlation functions have been obtained in terms of Jordan-Wigner fermionization. Existence of long range magnetic order has been investigated numerically by finding correlation functions as well as the Binder cumulant in the ground state. Dispersion relation, ground state energy, and energy gap are obtained analytically. In order to find the topologically nontrivial phase, sign of Pfaffian invariant and value of winding number have been evaluated. Both magnetic and topological phases are robust against the magnetic field and found to move coercively in the parameter space with the variation of its strength. Long range orders along two orthogonal directions and two different topological phases are found and their one-to-one correspondence has been found. Finally casting the spinless fermions onto Majorana fermions, properties of zero energy edge states are studied. Three different kinds of Majorana pairings are noted. In the trivial phase, next-nearest-neighbor Majorana pairing is found, whereas two different types of nearest-neighbor Majorana pairings are identified in the topological superconducting phase.

arXiv:2310.11243•Physica Scripta•Jan 31, 2024•5 citations
The isotropic XYXYXY model (s=1/2)(s=1/2)(s=1/2) in a transverse field, with uniform long-range interactions among the transverse components of the spins, on the inhomogeneous periodic chain, is studied. The model, composed of NNN segments with nnn different exchange interactions and magnetic moments, is exactly solved by introducing the integral Gaussian transformation and the generalized Jordan--Wigner transformation, which reduce the problem to the diagonalization of a finite matrix of nthn\text{th}nth order. The quantum transitions induced by the transverse field are determined by analyzing the induced magnetization of the cell and the equation of state. The phase diagrams for the quantum transitions, in the space generated by the transverse field and the interaction parameters, are presented. As expected, the model presents multiple, first- and second-order quantum transitions induced by the transverse field, and it corresponds to an extension of the models recently considered by the authors. Detailed results are also presented, at T=0T=0T=0, for the induced magnetization and isothermal susceptibility \ensuremathχTzz{\ensuremath{\chi}}_{T}^{zz}\ensuremathχTzz​ as a function of the transverse field.
The static and dynamic properties of the anisotropic XYXYXY model (s=1∕2)(s=1∕2)(s=1∕2) on the inhomogeneous periodic chain, composed of NNN cells with nnn different exchange interactions and magnetic moments, in a transverse field hhh, are determined exactly at arbitrary temperatures. The properties are obtained by introducing the Jordan-Wigner fermionization and by reducing the problem to a diagonalization of a finite matrix of nnnth order. The quantum transitions are determined exactly by analyzing, as a function of the field, the induced magnetization $1∕n{\ensuremath{\sum}}_{m=1}^{n}{\ensuremath{\mu}}_{m}⟨{S}_{j,m}^{z}⟩$ (jjj denotes the cell, mmm the site within the cell, ${\ensuremath{\mu}}_{m}$ the magnetic moment at site mmm within the cell) and the spontaneous magnetization $1∕n{\ensuremath{\sum}}_{m=1}^{n}⟨{S}_{j,{m}^{\ensuremath{'}}}^{x}⟩$ which is obtained from the correlations ⟨Sj,mxSj+r,mx⟩⟨{S}_{j,m}^{x}{S}_{j+r,m}^{x}⟩⟨Sj,mx​Sj+r,mx​⟩ for large spin separations. These results, which are obtained for infinite chains, correspond to an extension of the ones obtained by Tong and Zhong [Physica B 304, 91 (2001)]. The dynamic correlations, $⟨{S}_{j,m}^{z}(t){S}_{{j}^{\ensuremath{'}},{m}^{\ensuremath{'}}}^{z}(0)⟩$, and the dynamic susceptibility, ${\ensuremath{\chi}}_{q}^{zz}(\ensuremath{\omega})$, are also obtained at arbitrary temperatures. Explicit results are presented in the limit T=0T=0T=0, where the critical behavior occurs, for the static susceptibility ${\ensuremath{\chi}}_{q}^{zz}(0)$ as a function of the transverse field hhh, and for the frequency dependency of dynamic susceptibility ${\ensuremath{\chi}}_{q}^{zz}(\ensuremath{\omega})$.
The quantum phase transitions of one-dimensional period-two anisotropic XY models in a transverse field with the Hamiltonian where the anisotropy parameters i take and alternately, are studied. The Hamiltonian can be reduced to the diagonal form by Jordan-Wigner and Bogoliubov transformations. The long-range correlations Cx and Cy are calculated numerically. The phase with Cx Cy0 (or Cy Cx0) is referred to as the ferromagnetic (FM) phase along the x (or y) direction, while the phase with Cx=Cy=0 is the paramagnetic (PM) phase. It is found that the phase diagrams with the ratio -1 and =-1 are different obviously. For the case with -1, the line h=hc1=1-[(1-)/2]2 separates an FM phase from a PM phase, while the line =0 is the boundary between a ferromagnetic phase along the x direction (FMx) and a ferromagnetic phase along the y direction (FMy). These are similar to those of the uniform XY chains in a transverse field (i.e., =1). When =-1, the FMx and FMy phases disappear and there appears a new FM phase. The line h=hc2=1-2 separates this new FM phase from the PM phase. The new phase is gapless with two zeros in single particle energy spectrum. This is due to the new symmetry in the system with =-1, i.e., the Hamiltonian is invariant under the transformation 2ix 2i+1y,2iy 2i+1x. The correlation function between the 2i-1 and 2i lattice points along the x (y) direction is equal to that between the 2i and 2i+1 lattice points along the y (x) direction. As a result, the long-range correlation functions along two directions are equivalent. In order to facilitate the description, we call this gapless phase the isotropic ferromagnetic (FMxx) phase. Finally, the relationship between quantum entanglement and quantum phase transitions of the system is studied. The scaling behaviour of the von Neumann entropy at each point in the FMxx phase is SL~1/3log2L+ Const, which is similar to that at the anisotropic phase transition point of the uniform XY model in a transverse field, and different from those in the FMx and FMy phases.
Hyperuniform states of matter are characterized by anomalous suppression of long-wavelength density fluctuations. While most of the interesting cases of disordered hyperuniformity are provided by complex many-body systems such as liquids or amorphous solids, classical spin chains with certain long-range interactions have been shown to demonstrate the same phenomenon. Such spin-chain systems are ideal models for exploring the effects of quantum mechanics on hyperuniformity. It is well-known that the transverse field Ising model shows a quantum phase transition (QPT) at zero temperature. Under the quantum effects of a transverse magnetic field, classical hyperuniform spin chains are expected to lose their hyperuniformity. High-precision simulations of these cases are complicated because of the presence of highly nontrivial long-range interactions. We perform an extensive analysis of these systems using density matrix renormalization group simulations to study the possibilities of phase transitions and the mechanism by which they lose hyperuniformity. Even for a spin chain of length 30, we see discontinuous changes in properties like the ``$\ensuremath{\tau}$ order metric'' of the ground state, the measure of hyperuniformity, and the second cumulant of the total magnetization along the xxx-direction, all suggestive of first-order QPTs. An interesting feature of the phase transitions in these disordered hyperuniform spin chains is that, depending on the parameter values, the presence of a transverse magnetic field may lead remarkably to an increase in the order of the ground state as measured by the ``$\ensuremath{\tau}$ order metric,'' even if hyperuniformity is lost. Therefore, it would be possible to design materials to target specific novel quantum behaviors in the presence of a transverse magnetic field. Our numerical investigations suggest that these spin chains can show no more than two QPTs. We further analyze the long-range interacting spin chains via the Jordan-Wigner mapping onto a system of spinless fermions, showing that under the pairwise-interaction approximation and a mean-field treatment, there can be at most two quantum phase transitions. Based on these numerical and theoretical explorations, we conjecture that for spin chains with long-range pair interactions that have convergent cosine transforms, there can be a maximum of two quantum phase transitions at zero temperature.
We study the equilibrium properties of the spin-1/21/21/2 XY chain with an infinite-range transverse interaction. At zero temperature, competition between the XY- and zzz-ordered phases induced by the infinite-range interactions gives rise to a first-order transition upon increasing the transverse coupling. We show that the two gapless points of the XY model behave in fundamentally different ways: isotropic spin chains experience a first-order transition at finite coupling; maximal anisotropic chains overcome a nonalgebraic phase transition at zero coupling strength. The phase diagram depicts a first-order reentrant transition that turns second order along a tricritical line separating a paramagnetic phase from an ordered one at finite temperature. The mean-field approach captures the local properties of the eigenstates and reveals the appearance of a magnetization gap in the spectrum. Global properties, e.g., entanglement entropy, are well approximated only at spectral boundaries. The mean entanglement entropy and the level-spacing ratio deviate from the Gaussian results, revealing the interacting nature of the problem.
Abstract In this study, a spin-1/2 extended anisotropic XY chain has been introduced in which both time reversal and SU(2) symmetries are broken but Z 2 symmetry is preserved. Magnetic and topological phase diagrams in the parameter space have been drawn in the presence of transverse magnetic field. Entanglement measures like mutual information and quantum discord are also evaluated and it indicates that these transitions are second order in nature. Quantum phase transition is noted at zero magnetic field, as well as magnetic long range order is found to withstand magnetic field of any strength. Exact analytic results for spin-spin correlation functions have been obtained in terms of Jordan-Wigner fermionization. Existence of long range magnetic order has been investigated numerically by finding correlation functions as well as the Binder cumulant in the ground state. Dispersion relation, ground state energy, and energy gap are obtained analytically. In order to find the topologically nontrivial phase, sign of Pfaffian invariant and value of winding number have been evaluated. Both magnetic and topological phases are robust against the magnetic field and found to move coercively in the parameter space with the variation of its strength. Long range orders along two orthogonal directions and two different topological phases are found and their one-to-one correspondence has been found. Finally casting the spinless fermions onto Majorana fermions, properties of zero energy edge states are studied. Three different kinds of Majorana pairings are noted. In the trivial phase, next-nearest-neighbor Majorana pairing is found, whereas two different types of nearest-neighbor Majorana pairings are identified in the topological superconducting phase.