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arXiv:1108.2832 · International Journal of Modern Physics B

A SPIN CHAIN WITH SPIRAL ORDERS: PERSPECTIVES OF QUANTUM INFORMATION AND MECHANICAL RESPONSE

Shi-Jian Gu, Wing Chi Yu, Hai‐Qing Lin

Published Jul 9, 2013• 4 citations
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Abstract

In this paper, we study the ground state of a one-dimensional exactly solvable model with a spiral order. While the model's energy spectrum is the same as the one-dimensional transverse field Ising model, its ground state manifests a spiral order with various periods. The quantum phase transition from a spiral-order phase to a paramagnetic phase is investigated in the perspectives of quantum information science and mechanics. We show that the modes of the ground-state fidelity and its susceptibility can identify the change of periodicity around the critical point. We also study the spin torsion modulus which defines the coefficient of the potential energy stored under a small rotation. We find that at the critical point, it is a constant; while away from the critical point, the spin torsion modulus tends to zero.

DOI
10.1142/s0217979213501063

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In this paper, we study the ground state of a one-dimensional exactly solvable model with a spiral order. While the model's energy spectrum is the same as the one-dimensional transverse field Ising model, its ground state manifests a spiral order with various periods. The quantum phase transition from a spiral-order phase to a paramagnetic phase is investigated in the perspectives of quantum information science and mechanics. We show that the modes of the ground-state fidelity and its susceptibility can identify the change of periodicity around the critical point. We also study the spin torsion modulus which defines the coefficient of the potential energy stored under a small rotation. We find that at the critical point, it is a constant; while away from the critical point, the spin torsion modulus tends to zero.
The critical properties of the one-dimensional spin-1/2 transverse Ising model in the presence of a longitudinal magnetic field were studied by the quantum fidelity method. We used exact diagonalization to obtain the ground-state energies and corresponding eigenvectors for lattice sizes up to 24 spins. The maximum of the fidelity susceptibility was used to locate the various phase boundaries present in the system. The type of dominant spin ordering for each phase was identified by examining the corresponding ground-state eigenvector. For a given antiferromagnetic nearest-neighbor interaction J_{2}, we calculated the fidelity susceptibility as a function of the transverse field B_{x} and the longitudinal field B_{z}. The phase diagram in the (B_{x},B_{z})-plane shows three phases. These findings are in contrast with the published literature that claims that the system has only two phases. For B_{x}<1, we observed an antiferromagnetic phase for small values of B_{z} and a paramagnetic phase for large values of B_{z}. For B_{x}>1 and low B_{z}, we found a disordered phase that undergoes a second-order phase transition to a paramagnetic phase for large values of B_{z}.
The ground-state behaviour of the frustrated quantum spin-1/2 two-leg ladder with the Heisenberg intra-rung and Ising inter-rung interactions is examined in detail. The investigated model is transformed into the quantum Ising chain with composite spins in an effective transverse and longitudinal field by employing either the bond-state representation or the unitary transformation. It is shown that the ground state of the Heisenberg–Ising ladder can be descended from three exactly solvable models: the quantum Ising chain in a transverse field, the ‘classical’ Ising chain in a longitudinal field or the spin-chain model in a staggered longitudinal–transverse field. The last model serves in evidence of the staggered bond phase with alternating singlet and triplet bonds on the rungs of a two-leg ladder, which appears at moderate values of the external magnetic field and consequently leads to a fractional plateau at a half of the saturation magnetization. The ground-state phase diagram totally consists of five ordered and one quantum paramagnetic phase, which are separated from each other either by the lines of discontinuous or continuous quantum phase transitions. The order parameters are exactly calculated for all five ordered phases and the quantum paramagnetic phase is characterized through different short-range spin–spin correlations.
Quantum phase transitions take place between distinct phases of matter at zero temperature. Near the transition point, exotic quantum symmetries can emerge that govern the excitation spectrum of the system. A symmetry described by the E8 Lie group with a spectrum of eight particles was long predicted to appear near the critical point of an Ising chain. We realize this system experimentally by using strong transverse magnetic fields to tune the quasi-one-dimensional Ising ferromagnet CoNb2O6 (cobalt niobate) through its critical point. Spin excitations are observed to change character from pairs of kinks in the ordered phase to spin-flips in the paramagnetic phase. Just below the critical field, the spin dynamics shows a fine structure with two sharp modes at low energies, in a ratio that approaches the golden mean predicted for the first two meson particles of the E8 spectrum. Our results demonstrate the power of symmetry to describe complex quantum behaviors.
We have found the exact ground state for two electronic models on a linear chain. The first model describes a half-filling electron system at the ferromagnet-antiferromagnet transition point. In the singlet ground state the spin correlators show giant spiral magnetic ordering, with the period of a spiral equal to the system size. The second electronic model describes the point where the ground state has giant spiral off-diagonal long-range order and is, therefore, superconducting. We suggest the formation of a ground state with giant spiral order (ferromagnetic or off-diagonal) as a probable scenario of the subsequent destruction of the ferromagnetism and the superconductivity.
We consider the insulating spin-orbit coupled Bose atoms confined within one-dimensional optical lattices and explore their ground-state magnetic phase transitions. Under strong interactions, the charge degrees of atoms are frozen and the system can be described by an anisotropic XXZ Heisenberg chain with Dzyaloshinskii-Moriya interaction and transverse field. We apply the matrix product state method to obtain the ground states, and analyze spiral correlation functions, spin-spin correlation functions, and first energy gap. Under weakly transverse fields, the ground state is a ferromagnetic phase if the interspin sss-wave interaction strength is stronger than that of the intraspin ones, otherwise, it is a paramagnetic phase, an antiferromagnetic phase, or a gapless spiral phase with algebraic decaying spin correlations. When the transverse field is strengthened, the gapless spiral phase is broken and a new spiral phase with long-range spin-spin correlations emerges.
In this study we investigate the ground-state phase diagram of the quantum spin-1/2 anisotropic XZ chain using the infinite time-evolving block decimation algorithm. The model exhibits four distinct phases: FM_{x}, FM_{z}, AFM_{z}, and PM. In terms of the transverse magnetic field h and anisotropy parameter J, our analysis focuses on the lines h=0 and 0.5 with J≥0, where different types of quantum phase transitions (QPTs) occur. We compute the bifurcation behavior of the ground-state fidelity per lattice site. On the line h=0, continuous behavior at the bifurcation point identifies a continuous QPT between the ordered FM_{x} and FM_{z} phases at the critical point J_{c1}, with bifurcations on both sides reflecting Z_{2} symmetry breaking in both phases. On the line h=0.5, a continuous bifurcation indicates a QPT between the ordered FM_{x} and disordered PM phases at J_{c2}, with bifurcation occurring only on one side, corresponding to Z_{2} symmetry breaking in a single phase. We further compute the ground-state fidelity susceptibility per lattice site. When the ground state is obtained from random initial states, the fidelity susceptibility exhibits oscillatory behavior in the symmetry-broken phase, providing a clear signature of ground-state degeneracy. These fidelity-based measures serve as effective tools for identifying critical points and characterizing the nature of symmetry breaking. To complement these results, we evaluate the two-spin Wigner-Yanase skew information and local quantum uncertainty (LQU) as indicators of quantum correlations. While both detect QPTs, we find that not all nonanalytic features in the LQU correspond to genuine phase transitions. A critical behavior analysis is carried out at J_{c1} and J_{c2} to extract the critical exponents η^{I}, η^{C}, η^{x}, and η^{z}, associated with mutual information I(r), classical correlation C(r), and spin-spin correlations 〈σ_{i}^{x}σ_{j}^{x}〉, 〈σ_{i}^{z}σ_{j}^{z}〉. Our results demonstrate that η^{I}=η^{C} and η^{x/z}=η^{α}/2 for α=I,C, consistent with known universal scaling relations. We also extract the order parameter exponent β, correlation length exponent ν, dynamic exponent z, and the central charge c. A transition at J_{c1} between the FM_{x} and FM_{z} phases is identified as a Gaussian transition with central charge c=1, involving the breaking of two Z_{2} symmetries and lying beyond the conventional Landau-Ginzburg-Wilson symmetry-breaking paradigm. The transition at J_{c2} between the FM_{x} and PM phases belongs to the quantum Ising universality class, characterized by c=1/2 and breaking of a single Z_{2} symmetry.