String Order Parameter Research Articles

Detection of symmetry-protected topological phases in one dimension

A topological phase is a phase of matter which cannot be characterized by a local order parameter. It has been shown that gapped symmetric phases in one-dimensional (1D) systems can be completely characterized using tools related to projective representations of the symmetry groups. We explain two ways to detect these symmetry protected topological phases in 1D. First, we give a numerical approach for directly extracting the projective representations from a matrix-product state representation. Second, we derive nonlocal order parameters for time-reversal and inversion symmetry, and discuss a generalized string order for local symmetries for which the regular string-order parameter cannot be applied. We furthermore point out that the nonlocal order parameter for these topological phases is actually related to topological surfaces.

Matrix product state and quantum phase transitions in the one-dimensional extended quantum compass model

The matrix product state (MPS) is utilized to study the ground state properties and quantum phase transitions (QPTs) of the one-dimensional quantum compass model (QCM). The MPS wavefunctions are argued to be very efficient descriptions of QCM ground states, and are numerically determined by imaginary time projections. The ground state energy, correlations, quantum entanglement and its spectrum, local and nonlocal order parameters, etc., are calculated and studied in details. It is revealed that the bipartite and block entanglement entropies, as well as the nearest neighbor correlation functions can be used to detect the second-order QPTs, but not the first-order ones, while fidelity detections can recognize both. The entanglement spectrum is extracted from the MPS wavefunction, and found to be doubly degenerate in disordered phases of QCM, where non-local string order parameters exist. Moreover, with linearized tensor renormalization group method, the specific heat curves are evaluated and their low temperature behaviors are investigated.

Accurate determination of the Gaussian transition in spin-1 chains with single-ion anisotropy

The Gaussian transition in the spin-one Heisenberg chain with single-ion anisotropy is extremely difficult to treat, both analytically and numerically. We introduce an improved DMRG procedure with strict error control, which we use to access very large systems. By considering the bulk entropy, we determine the Gaussian transition point to 4-digit accuracy, $D_{c}/J = 0.96845(8)$, resolving a long-standing debate in quantum magnetism. With this value, we obtain high-precision data for the critical behavior of quantities including the ground-state energy, gap, and transverse string-order parameter, and for the critical exponent, $\nu = 1.472(2)$. Applying our improved technique at $J_{z} = 0.5$ highlights essential differences in critical behavior along the Gaussian transition line.

Intermediate Haldane phase in spin-2 quantum chains with uniaxial anisotropy

We provide evidence of an intermediate Haldane phase in a spin-2 quantum chain. By combining effective field theory and numerical approaches, we show that the phase diagram of the proposed model includes SO(5) Haldane, intermediate Haldane, and large-D phases. We determine the characteristic properties of these phases, including edge states, string order parameters, and degeneracies in the entanglement spectrum. The symmetries responsible for the degeneracy patterns observed in the entanglement spectrum are also discussed.

Hidden order and dynamics in supersymmetric valence-bond solid states: Super-matrix product state formalism

Supersymmetric valence-bond solid models are extensions of the VBS model, a paradigmatic model of ``solvable'' gapped quantum antiferromagnets, to the case with doped fermionic holes. In this paper, we present a detailed analysis of physical properties of the models. For systematic studies, a supersymmetric version of the matrix-product formalism is developed. On 1D chains, we exactly evaluate the hole-doping behavior of various physical quantities, such as the spin or the charge excitation spectrum, the superconducting order parameter. A generalized hidden order is proposed, and the corresponding string nonlocal order parameter is also calculated. The behavior of the string-order parameter is discussed in the light of the entanglement spectrum.

Numerical study of critical properties and hidden orders in dimerized spin ladders

Dimerized antiferromagnetic spin-1/2 ladders are known to exhibit a quantum critical phase transition in the ground state, the existence or absence of which is dependent on the dimerization pattern of the ladder. The gapped phases cannot be distinguished by the conventional Landau long-range order parameter. However, they possess a non-local (hidden) string order parameter, which is non-zero in one phase and vanishes in the other. We use an exact diagonalization technique to calculate ground state energies, energy gaps and string order parameters of dimerized two- and three-leg Heisenberg ladders, as well as a critical scaling analysis to yield estimates of the critical exponents nu and beta.

Quantum correlations in a clusterlike system

We discuss a clusterlike one-dimensional system with triplet interaction. We study the topological properties of this system. We find that the degeneracy depends on the topology of the system and is well protected against external local perturbations. All these facts show that the system is topologically ordered. We also find a string order parameter to characterize the quantum phase transition. Besides, we investigate two-site correlations including entanglement, quantum discord, and mutual information. We study the different divergence behaviors of the correlations. The quantum correlation decays exponentially in both topological and magnetic phases, and diverges in reversed power law at the critical point. And we find that in topological order systems, the global difference of topology induced by dimension can be reflected in local quantum correlations.

Asymmetric spin-12two-leg ladders: Analytical studies supported by exact diagonalization, DMRG, and Monte Carlo simulations

We consider asymmetric spin-1/2 two-leg ladders with non-equal antiferromagnetic (AF) couplings J_|| and \kappa J_|| along legs (\kappa <= 1) and ferromagnetic rung coupling, J_\perp. This model is characterized by a gap \Delta in the spectrum of spin excitations. We show that in the large J_\perp limit this gap is equivalent to the Haldane gap for the AF spin-1 chain, irrespective of the asymmetry of the ladder. The behavior of the gap at small rung coupling falls in two different universality classes. The first class, which is best understood from the case of the conventional symmetric ladder at \kappa=1, admits a linear scaling for the spin gap \Delta ~ J_\perp. The second class appears for a strong asymmetry of the coupling along legs, \kappa J_|| << J_\perp << J_|| and is characterized by two energy scales: the exponentially small spin gap \Delta ~ J_\perp \exp(-J_|| / J_\perp), and the bandwidth of the low-lying excitations induced by a Suhl-Nakamura indirect exchange ~ J_\perp^2 /J_|| . We report numerical results obtained by exact diagonalization, density matrix renormalization group and quantum Monte Carlo simulations for the spin gap and various spin correlation functions. Our data indicate that the behavior of the string order parameter, characterizing the hidden AF order in Haldane phase, is different in the limiting cases of weak and strong asymmetry. On the basis of the numerical data, we propose a low-energy theory of effective spin-1 variables, pertaining to large blocks on a decimated lattice.

One-Dimensional Quantum Compass Model in an Out-plane Magnetic Field

The one-dimensional quantum compass model (1D-QCM) in an out-plane magnetic field is investigated by numerical and analytical methods. The presence of the 1D-QCM in an area of the ground-state magnetic phase diagram where the odd couplings are antiferromagnetic and larger than the even couplings is studied. Using pseudo spin ladder operators indicates that the 1D-QCM in an out-plane magnetic field can be mapped to the one-dimensional Ising model in a transverse magnetic field (ITF). Analytical investigation of the effective ITF Hamiltonian predicts the occurrence of two quantum phase transitions with increasing magnetic field. Also, the critical fields are obtained from this method, which are completely in agreement with the results of the numerical experiment based on the Lanczos method. This numerical method accurately determined the behavior of the energy gap, magnetization and string order parameter, and therefore it is a very useful method for indicating the ground-state phases of the system.

A Novel Plateau in the Magnetization Curve of the Alternating Heisenberg Chain

At zero temperature, the effect of a uniform magnetic field on an alternating antiferromagnetic–ferromagnetic (AF–F) spin-1/2 chain is studied using the numerical Lanczos method. The alternating model is defined as an alternating AF–F Heisenberg chain with the space-modulated dominant antiferromagnetic exchange. A detailed numerical analysis of the low energy excitation spectrum is presented. The energy gap, the magnetization and the string order parameter of the system are calculated for finite chains. It is found that the space modulation on the antiferromagnetic exchange induced two new gaps in the spectrum. In these new gapped regions, the magnetization shows two nonzero plateaus at the values 1/3 and 2/3 of the saturation magnetization.

Quantum phase transitions in three-leg spin tubes

We investigate the properties of a three-leg quantum spin tube using several techniques such as the density-matrix renormalization-group (DMRG) method, strong-coupling approaches and the nonlinear sigma model. For integer $S$, the model proves to exhibit a particularly rich phase diagram consisting of an ensemble of $2S$ phase transitions. They can be accurately identified by the behavior of a nonlocal string order parameter associated to the breaking of a hidden symmetry in the Hamiltonian. The nature of these transitions is further elucidated within the different approaches. We carry a detailed DMRG analysis in the specific cases $S=1$. The numerical data confirm the existence of two Haldane phases with broken hidden symmetry separated by a trivial singlet state. The study of the gap and of the von Neumann entropy suggest a first-order phase transition but at the close proximity of a tricritical point separating a gapless and a first-order transition line in the phase diagram of the quantum spin tube.

Link between the hierarchy of fractional quantum Hall states and Haldane’s conjecture for quantum spin chains

We study a strong coupling expansion of the $\ensuremath{\nu}=1/3$ fractional quantum Hall state away from the Tao-Thouless limit and show that the leading quantum fluctuations lead to an effective spin-1 Hamiltonian that lacks parity symmetry. By analyzing the energetics, discrete symmetries of low-lying excitations, and string order parameters, we demonstrate that the $\ensuremath{\nu}=1/3$ fractional quantum Hall state is adiabatically connected to both Haldane and large-$D$ phases and is characterized by a string order parameter which is dual to the ordinary one. This result indicates a close relation between (a generalized form of) the Haldane conjecture for spin chains and the fractional quantum Hall effect.

Spontaneous parity violation in a quantum spin chain

We report on a spontaneous breakdown of parity in the ground state of a spin Hamiltonian involving nearest-neighbor interactions. This occurs for a one-dimensional model where spins transform under the gauge field representation of QCD, the eight-dimensional adjoint representation of SU(3). The ground state spontaneously violates parity and is two-fold degenerate. In addition, the model possesses a non-vanishing topological string order parameter which we explicate analytically.

Tensor-entanglement-filtering renormalization approach and symmetry-protected topological order

We study the renormalization group flow of the Lagrangian for statistical and quantum systems by representing their path integral in terms of a tensor network. Using a tensor-entanglement-filtering renormalization (TEFR) approach that removes local entanglement and coarse grain the lattice, we show that the resulting renormalization flow of the tensors in the tensor network has a nice fixed-point structure. The isolated fixed-point tensors $T_{inv}$ plus the symmetry group $G_{sym}$ of the tensors (i.e. the symmetry group of the Lagrangian) characterize various phases of the system. Such a characterization can describe both the symmetry breaking phases and topological phases, as illustrated by 2D statistical Ising model, 2D statistical loop gas model, and 1+1D quantum spin-1/2 and spin-1 models. In particular, using such a $(G_{sym}, T_{inv}) $ characterization, we show that the Haldane phase for a spin-1 chain is a phase protected by the time-reversal, parity, and translation symmetries. Thus the Haldane phase is a symmetry protected topological phase. The $(G_{sym}, T_{inv})$ characterization is more general than the characterizations based on the boundary spins and string order parameters. The tensor renormalization approach also allows us to study continuous phase transitions between symmetry breaking phases and/or topological phases. The scaling dimensions and the central charges for the critical points that describe those continuous phase transitions can be calculated from the fixed-point tensors at those critical points.

Spatial reflection and associated string order in quantum spin chains

We investigate spatial reflection and associated nonlocal order in spin chain quantum systems. The proposed string order parameters, e.g., reflected via operations of the spatial reflection or combinations of it with spin reflection, are able to characterize a variety of physical systems and allow us to gain renewed insights to the statistical mechanism underlying phenomena such as the Haldane gap and quantum phase transitions. Besides revealing further the potential application of the generalized parity symmetry in numerical algorithm, we build an explicit scheme to determine the symmetry and the related string order for matrix product states so that one can construct ansatz models with presumed properties.

Maximized string order parameters in the valence bond solid states of quantum integer spin chains

We propose a set of maximized string order parameters to describe the hidden topological order in the valence bond solid states of quantum integer spin-S chains. These optimized string order parameters involve spin-twist angles corresponding to ZS+1 rotations around z or x-axes, suggesting a hidden ZS+1 × ZS+1 symmetry. Our results also suggest that a local triplet excitation in the valence bond solid states carries a ZS+1 topological charge measured by these maximized string order parameters.

Valence bond solid state induced by impurity frustration inCr8Ni

We provide a physically meaningful picture of the nature of the ground state of the ${\text{Cr}}_{8}\text{Ni}$ compound in the regime where it is a spin singlet. According to this picture, the anisotropy of the Ni atom in the Cr ring induces a dimerization in the molecule that makes the ground state to stabilize in a valence bond solid phase of virtual spins. We characterize rigorously this phase by means of a particular nonlocal order parameter that denoted the generalized string order parameter. In the completely antiferromagnetic regime, the system becomes frustrated. We have performed a numerical real-time evolution study of the correlations between the spin of the Ni impurity and the rest of the spins in order to show the reaction of the system under this frustration.

Finite-size scaling of string order parameters characterizing the Haldane phase

We have developed a numerical procedure to clarify the critical behavior near a quantum phase transition by analyzing a multipoint correlation function characterizing the ground state. This work presents a successful application of this procedure to the string order parameter of the $S=1$ $XXZ$ chain with uniaxial single-ion anisotropy. The finite-size string correlation function is estimated by the density-matrix renormalization-group method. We focus on the gradient of the inverse-system-size dependence of the correlation function on a logarithmic plot. This quantity shows that the finite-size scaling sensitively changes at the critical point. The behavior of the gradient with increasing system size is divergent, is stable at a finite value, or rapidly decreases to zero when the system is in the disordered phase, at the critical point, or in the ordered phase, respectively. The analysis of the finite-size string correlation functions allows precise determination of the boundary of the Haldane phase and estimation of the critical exponent of the correlation length. Our estimates of the transition point and the critical exponents, which are determined only by the ground-state quantities, are consistent with results obtained from the analysis of the energy-level structure. Our analysis requires only the correlation functions of several finite sizes under the same condition as a candidate for the long-range order. The quantity is treated in the same manner irrespective of the kind of elements which destroy the order concerned. This work will assist in the development of a method for directly observing quantum phase transitions.

String order and hidden topological symmetry in the SO(2n + 1) symmetric matrix product states

We have introduced a class of exactly soluble Hamiltonian with either SO(2n + 1) or SU(2) symmetry, whose ground states are the SO(2n + 1) symmetric matrix product states. The hidden topological order in these states can be fully identified and characterized by a set of nonlocal string order parameters. The Hamiltonian possesses a hidden (Z2 × Z2)n topological symmetry. The breaking of this hidden symmetry leads to 4n degenerate ground states with disentangled edge states in an open chain system. Such matrix product states can be regarded as cluster states, applicable to measurement-based quantum computation.

Class of exactly solvableSO(n)symmetric spin chains with matrix product ground states

We introduce a class of exactly solvable $SO(n)$ symmetric Hamiltonians with matrix product ground states. For an odd $n\ensuremath{\ge}3$ case, the ground state is a translational invariant Haldane gap spin liquid state; while for an even $n\ensuremath{\ge}4$ case, the ground state is a spontaneously dimerized state with twofold degeneracy. In the matrix product ground states for both cases, we identify a hidden antiferromagnetic order, which is characterized by nonlocal string order parameters. The ground-state phase diagram of a generalized $SO(n)$ symmetric bilinear-biquadratic model is discussed.