Nonlocal Order Parameters Research Articles

Quantum phases and transitions of spin-1/2 quantum compass chain

Quantum phases (QPs) and quantum phase transitions (QPTs) are very important parts of the strongly correlated quantum many-body systems in condensed matter. To study the QPs and QPTs, the systems should include rich quantum phase diagram. In this sense, the corresponding quantum spin models should have strong quantum fluctuation, strong geometric frustration, complicated spin-spin exchange or orbital degrees of freedom, which induces a variety of spontaneous symmetry breaking (SSB) or hidden spontaneous symmetry breaking. The QPs induced by the SSB can be characterized by local order parameters, a concept that originates from Landau-Ginzburg-Wilson paradigm (LGW). However, there is also a novel class of topological QPs beyond LGW, which has aroused one’s great interest since the Haldane phase was found. Such QPs can be characterized only by topological long-range nonlocal string correlation order parameters instead of local order parameters. In this paper, we investigate a spin-1/2 quantum compass chain model (QCC) with orbital degrees of freedom in <i>x</i>, <i>y</i> and <i>z</i> components. The prototype of QCC is the quantum compass model including novel topological QPs beyond LGW, and consequently one can also anticipate the existence of novel topological QPs in QCC. However, very little attention has been paid to the QPs and QPTs for QCC, which deserves to be further investigated. By using the infinite time evolving block decimation in the presentation of matrix product states, we study the QPs and QPTs of QCC. To characterize QPs and QPTs of QCC, the ground state energy, local order parameter, topological long-range nonlocal string correlation order parameters, critical exponent, correlation length and central charge are calculated. The results show the phase diagram of QCC including local antiferromagnetic phase, local stripe antiferromagnetic phase, oscillatory odd Haldane phase and monotonic odd Haldane phase. The QPTs from oscillatory odd Haldane phase to local stripe antiferromagnetic phase and from local antiferromagnetic phase to monotonic odd Haldane phase are continuous; on the contrary, QPTs from local stripe antiferromagnetic phase to local antiferromagnetic phase and from oscillatory odd Haldane phase to monotonic odd Haldane phase are discontinuous. The crossing point where the line of continuous QPTs meets with the line of discontinuous QPTs is the multiple critical point. The critical exponents <i>β</i> of local antiferromagnetic order parameter, local stripe antiferromagnetic order parameter, topological long-range nonlocal oscillatory odd string correlation order parameter, and topological long-range nonlocal monotonic odd string correlation order parameter are all equal to 1/8. Moreover, <inline-formula><tex-math id="M3">\begin{document}$\beta =1/8$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20211433_M3.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20211433_M3.png"/></alternatives></inline-formula> and the central charges <inline-formula><tex-math id="M4">\begin{document}$c = 1/2$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20211433_M4.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20211433_M4.png"/></alternatives></inline-formula> at the critical points show that the QPTs from local phases to nonlocal phases belong to the Ising-type universality class.

Matrix product states and projected entangled pair states: Concepts, symmetries, theorems

The theory of entanglement provides a fundamentally new language for describing interactions and correlations in many body systems. Its vocabulary consists of qubits and entangled pairs, and the syntax is provided by tensor networks. We review how matrix product states and projected entangled pair states describe many-body wavefunctions in terms of local tensors. These tensors express how the entanglement is routed, act as a novel type of non-local order parameter, and we describe how their symmetries are reflections of the global entanglement patterns in the full system. We will discuss how tensor networks enable the construction of real-space renormalization group flows and fixed points, and examine the entanglement structure of states exhibiting topological quantum order. Finally, we provide a summary of the mathematical results of matrix product states and projected entangled pair states, highlighting the fundamental theorem of matrix product vectors and its applications.

Converting electrons into emergent fermions at a superconductor–Kitaev spin liquid interface

We study an interface between a Kitaev spin liquid (KSL) in the chiral phase and a non-chiral superconductor. When the coupling across the interface is sufficiently strong, the interface undergoes a transition into a phase characterized by a condensation of a bound state of a Bogoliubov quasiparticle in the superconductor and an emergent fermionic excitation in the spin liquid. In the condensed phase, electrons in the superconductor can coherently convert into emergent fermions in the spin liquid and vice versa. As a result, the chiral Majorana edge mode of the spin liquid becomes visible in the electronic local density of states at the interface, which can be measured in scanning tunneling spectroscopy experiments. We demonstrate the existence of this phase transition, and the non-local order parameter that characterizes it, using density matrix renormalization group simulations of a KSL strip coupled at its edge to a superconductor. An analogous phase transition can occur in a simpler system composed of a one-dimensional spin chain with a spin-flip $\mathbb{Z}_2$ symmetry coupled to a superconductor.

Quantized edge magnetizations and their symmetry protection in one-dimensional quantum spin systems

The bulk electric polarization works as a nonlocal order parameter that characterizes topological quantum matters. Motivated by a recent paper [H. Watanabe \textit{et al.}, Phys. Rev. B {\bf 103}, 134430 (2021)], we discuss magnetic analogs of the bulk polarization in one-dimensional quantum spin systems, that is, quantized magnetizations on the edges of one-dimensional quantum spin systems.The edge magnetization shares the topological origin with the fractional edge state of the topological odd-spin Haldane phases. Despite this topological origin, the edge magnetization can also appear in topologically trivial quantum phases. We develop straightforward field theoretical arguments that explain the characteristic properties of the edge magnetization. The field theory shows that a U(1) spin-rotation symmetry and a site-centered or bond-centered inversion symmetry protect the quantization of the edge magnetization. We proceed to discussions that quantum phases on nonzero magnetization plateaus can also have the quantized edge magnetization that deviates from the magnetization density in bulk. We demonstrate that the quantized edge magnetization distinguishes two quantum phases on a magnetization plateau separated by a quantum critical point. The edge magnetization exhibits an abrupt stepwise change from zero to $1/2$ at the quantum critical point because the quantum phase transition occurs in the presence of the symmetries protecting the quantization of the edge magnetization. We also show that the quantized edge magnetization can result from the spontaneous ferrimagnetic order.

From Ising Model to Kitaev Chain

In this general article, we map the one-dimensional transverse field quantum Ising model of ferromagnetism to Kitaev's one-dimensional p-wave superconductor, which has its application in fault-tolerant topological quantum computing. Mapping Pauli's spin operators of transverse Ising chain to spinless fermionic creation and annihilation operators by Inverse Jordan-Wigner transformation leads to a Hamiltonian form closely related to Kitaev Chain, which exhibits topological phase transition where phases are characterized by different topological invariant that changes discontinuously at the transition point. Kitaev Chain supports two Majorana zero modes (MZMs) in the non-trivial topological phase, while none is in the trivial phase. The doubly degenerate ground state of the transverse Ising in ferromagnetic phase corresponds to non-local free fermion degree made from MZMs. The quasi-particle excitations of Ising chain, viz., domain wall formation in the ferromagnetic phase and spin-flip in paramagnetic phase maps to Bogoliubon excitations. The mapping suggests that a non-local order parameter can be defined for Kitaev Chain to work with the usual paradigm of Landau's theory.

Reflection of plane wave at free boundary of micro-polar nonlocal semiconductor medium

This work presents the reflection phenomena of thermo-elastic waves in nonlocal micro-polar semi-conducting solid. The thermo-elasticity theory of factional-order time derivative three phase lag thermo-elastic model is applied to study the thermal effect on the waves. Governing equations of the problem are decomposed by using the Helmholtz decomposition. Two frequency dispersion relations corresponding to couple longitudinal and transverse waves are obtained. It is observed that three coupled longitudinal and two coupled transverse waves are propagating through the elastic solid. Relations for amplitude ratios, Energy flux balance are obtained analytically. All the results obtained mathematically are represented graphically for the selected material. The silicon crystal has been selected for computational purpose to check the variation of amplitude ratios against incident angle. The dependence of reflection coefficients upon nonlocal and fractional order parameters is been represented graphically.

Nonlocal discrete time crystals in periodically driven surface codes

Discrete time crystals (DTCs) are nonequilibrium phases of matter characterized by robust subharmonic order parameter dynamics. We report a type of DTC in a periodically driven surface code, the subharmonic signature of which is only observable with respect to some nonlocal order parameter. By modifying the commonly used metrics for characterizing ordinary DTCs, i.e., the spectral functions, spin-glass order parameters, and two-point correlators, we further numerically demonstrate the rich phases of the proposed system. Specifically, depending on the system parameters and boundary conditions, we find that the system may fall into either a ``surface code'' phase, trivial paramagnet phase, nonlocal period-doubling DTC phase, or nonlocal period-quadrupling DTC phase. Our work thus demonstrates the prospect of exploring topological codes for discovering novel phases of matter.

Disorder lines, modulation, and partition function zeros in free fermion models

The modulation is analyzed from the analytical properties of zeros of free fermionic partition function on the complex plane of wave numbers. It is shown how these properties are related to the oscillations of correlation functions. This approach can be used for analysis of phase transitions with local or nonlocal order parameters, as well as for the disorder lines. We find an infinite cascade of disorder lines at finite temperature in the quantum $XY$ chain (equivalent to free fermions). The well-known ground state factorization on the disorder line, and consequently, disentanglement, is shown to follow directly from analytical properties of this model on the complex plane. From the quantum-classical correspondence the results for the chain are used to detect the disorder lines in several frustrated 2D Ising models. The present formalism can be applied to other fermionic models in two and three spatial dimensions. In particular, we find the temperature-dependent Fermi wave vector of oscillations in the degenerate gas of 3D fermions, which naturally leads in the limit $T \to 0$ to the definition of the Fermi energy as the surface of quantum criticality. The modulation is a very common phenomenon, and it occurs in a large variety of models. The important point is that all these modulation transitions can be related to the complex zeros of partition functions, as done in the present study.

Higher-form symmetry breaking at Ising transitions

In recent years, new phases of matter that are beyond the Landau paradigm of symmetry breaking have been accumulating, and to catch up with this fast development, new notions of global symmetry are introduced. Among them, the higher-form symmetry, whose symmetry charges are spatially extended, can be used to describe topologically ordered phases as the spontaneous breaking of the symmetry, and consequently unify the unconventional and conventional phases under the same conceptual framework. However, such conceptual tools have not been put into quantitative tests except for certain solvable models, therefore limiting their usage in the more generic quantum many-body systems. In this work, we study ${\mathbb{Z}}_{2}$ higher-form symmetry in a quantum Ising model, which is dual to the global (zero-form) Ising symmetry. We compute the expectation value of the Ising disorder operator, which is a nonlocal order parameter for the higher-form symmetry, analytically in free scalar theories and through unbiased quantum Monte Carlo simulations for the interacting fixed point in $(2+1)d$. From the scaling form of this extended object, we confirm that the higher-form symmetry is indeed spontaneously broken inside the paramagnetic, or quantum disordered phase (in the Landau sense), but remains symmetric in the ferromagnetic or ordered phase. At the Ising critical point, we find that the disorder operator also obeys a ``perimeter'' law scaling with possibly multiplicative power-law corrections. We discuss examples where both the global zero-form symmetry and the dual higher-form symmetry are preserved, in systems with a codimension-1 manifold of gapless points in momentum space. These results provide nontrivial working examples of higher-form symmetry operators, including the direct computation of one-form order parameter in an interacting conformal field theory, and open the avenue for their generic implementation in quantum many-body systems.

Transmission phenomenon at the interface between isotropic and semiconductor nanostructure based on nonlocal theory

In this article, we analyzed the response of elastic waves after striking the interface between two elastic mediums. The upper-medium is isotropic while the lower medium is semiconductor nanostructure in nature. Both the mediums are set on rotation with some fix angular frequency . A fractional ordered heat conduction model of three-phase lag along with the nonlocal theory of elasticity is adopted to formulate the problem. The incident wave after striking the interface results in five coupled quasireflected waves and three elastic transmitted waves. The ratios of the amplitude of reflection and transmitted waves with incident wave are calculated. Variations in obtained amplitude ratios with respect to incident angle have been shown graphically for silicon (Si) semiconductor. The effects of nonlocal parameter, rotational frequency and fractional order parameter on the reflection and transmitted coefficients in the presence of voids parameters are studied. The results obtained through the current analysis may be applicable in the field of seismology and semiconductors nanostructure appliances.

Topology in quasicrystals [Invited

Topological phases of matter have sparked an immense amount of activity in recent decades. Topological materials are classified by topological invariants that act as a non-local order parameter for any symmetry and condition. As a result, they exhibit quantized bulk and boundary observable phenomena, motivating various applications that are robust to local disorder and imperfections. In this review, we explore such a topological classification for quasiperiodic systems, and detail recent experimental activity using photonic metamaterials.

Topological Dynamic Matter.

The principles of topology in condensed matter physics have expanded to areas such as photonics, acoustics, electronics, and mechanics. Their extension to dynamic (soft) matter could enable the control and design of topological thermodynamic (micro)states and nonreciprocal dynamics, potentially leading to paradigmatic applications in molecular and thermal waveguiding, logics, and energy management. This Perspective explores distinct topological concepts for dynamic matter and prospective function. Topological tools are exemplified and discussed for the study of nonlocal order parameters or invariants in dynamic molecular matter, toward the engineering of assemblies, reactions, and system chemistry with unconventional global properties-a scope which has the potential to push the frontiers of physical chemistry and transform chemical topology from form to function.

Quantum critical phase transition between two topologically ordered phases in the Ising toric code bilayer

We demonstrate that two toric code layers on the square lattice coupled by an Ising interaction display two distinct phases with intrinsic topological order. The second-order quantum phase transition between the weakly-coupled $\mathbb{Z}_2\times\mathbb{Z}_2$ and the strongly-coupled $\mathbb{Z}_2$ topological order can be described by the condensation of bosonic quasiparticles from both sides and belongs to the 3d Ising$^*$ universality class. This can be shown by an exact duality transformation to the transverse-field Ising model on the square lattice, which builds on the existence of an extensive number of local $\mathbb{Z}_2$ conserved parities. These conserved quantities correspond to the product of two adjacent star operators on different layers. Notably, we show that the low-energy effective model derived about the limit of large Ising coupling is given by an effective single-layer toric code in terms of the conserved quantities of the Ising toric code bilayer. The two topological phases are further characterized by the topological entanglement entropy which serves as a non-local order parameter.

Reflection of thermo-elastic wave in semiconductor nanostructures nonlocal porous medium

The current work is an extension of the nonlocal elasticity theory to fractional order thermo-elasticity in semiconducting nanostructure medium with voids. The analysis is made on the reflection phenomena in context of three-phase-lag thermo-elastic model. It is observed that, four-coupled longitudinal waves and an independent shear vertical wave exist in the medium which is dispersive in nature. It is seen that longitudinal waves are damped, and shear wave is un-damped when angular frequency is less than the cut-off frequency. The voids, thermal and non-local parameter affect the dilatational waves whereas shear wave is only depending upon non-local parameter. It is found that reflection coefficients are affected by nonlocal and fractional order parameters. Reflection coefficients are calculated analytically and computed numerically for a material, silicon and discussed graphically in details. The results for local (classical) theory are obtained as a special case. The study may be useful in semiconductor nanostructure, geology and seismology in addition to semiconductor nanostructure devices.

Floating topological phases

While quasi-two-dimensional (layered) materials can be highly anisotropic, their asymptotic long-distance behavior generally reflects the properties of a fully three dimensional phase of matter. However, certain topologically ordered quantum phases with an emergent 2+1 dimensional gauge symmetry can be asymptotically impervious to interplane couplings. We discuss the stability of such "floating topological phases", as well as their diagnosis by means of a non-local order parameter. Such a phase can produce a divergent ratio $\rho_{\perp}/\rho_{\parallel}$ of the inter-layer to intra-layer resistivity as $T\to 0$, even in an insulator where both $\rho_{\perp}$ and $\rho_\parallel$ individually diverge. Experimental observation of such a divergence would constitute proof of the existence of a topological (e.g. spin liquid) phase.

Symmetry-protected trivial phases and quantum phase transitions in an anisotropic antiferromagnetic spin-1 biquadratic model

The ground state phase diagram is obtained for an antiferromagnetic spin-1 anisotropic biquadratic model. With the help of symmetry and duality transformations, three symmetry-protected trivial phases and one dimerized symmetry breaking phase are found. Local and nonlocal order parameters are identified to characterize these phases. Quantum phase transitions between the symmetry-protected trivial phases belong to the Gaussian universality class with central charge c = 1, and quantum phase transitions from the symmetry-protected trivial phases to the dimerized phase belong to the Ising universality class with central charge c = 1/2. In addition, the model admits three characteristic lines of factorized ground states, which are located in the symmetry-protected trivial phases instead of a symmetry breaking phase, in sharp contrast to other known cases.

Phase diagram of an interacting staggered Su-Schrieffer-Heeger two-chain ladder close to a quantum critical point

We study the ground-state phase diagram of an interacting staggered Su-Schrieffer-Heeger (SSH) ladder in the vicinity of the Gaussian quantum critical point. The corresponding effective field theory is a double-frequency sine-Gordon (DSG) model which involves two perturbations at the Gaussian fixed point: the deviation from criticality and Umklapp scattering processes. A topological distinction between thermodynamically equivalent phases becomes only feasible when nonlocal fermionic fields, parity and string order parameter, are included into consideration. We prove that a noninteracting fermionic staggered SSH ladder is exactly equivalent to a O(2)-symmetric model of two decoupled Kitaev-Majorana chains, or two 1D p-wave superconductors. Close to the Gaussian fixed point the SSH ladder maps to an Ashkin-Teller like system when interactions are included. Thus, the topological order in the SSH ladder is related to broken-symmetry phases of the associated quantum spin-chain degrees of freedom. The obtained phase diagram includes a Tomonaga-Luttinger liquid state which, due to Umklapp processes, can become unstable against either spontaneous dimerization or the onset of a charge-density wave (CDW). In these gapped phases elementary bulk excitations are quantum kinks carrying the charge QF = 1/2. For sufficiently strong, long-range interactions, the phase diagram of the model exhibits a bifurcation of the Gaussian critical point into two outgoing Ising criticalities. The latter sandwich a mixed phase in which dimerization coexists with a site-diagonal CDW. In this phase elementary bulk excitations are represented by two types of topological solitons carrying different fermionic charges, which continuously interpolate between 0 and 1. This phase has also mixed topological properties with coexisting parity and string order parameters.

Haldane insulator in the 1D nearest-neighbor extended Bose-Hubbard model with cavity-mediated long-range interactions

In the one-dimensional Bose-Hubbard model with on-site and nearest-neighbor interactions, a gapped phase characterized by an exotic non-local order parameter emerges, the Haldane insulator. Bose-Hubbard models with cavity-mediated global range interactions display phase diagrams, which are very similar to those with nearest-neighbor repulsive interactions, but the Haldane phase remains elusive there. Here we study the one-dimensional Bose-Hubbard model with nearest-neighbor and cavity-mediated global-range interactions and scrutinize the existence of a Haldane Insulator phase. With the help of extensive quantum Monte-Carlo simulations we find that in the Bose-Hubbard model with only cavity-mediated global-range interactions no Haldane phase exists. For a combination of both interactions, the Haldane Insulator phase shrinks rapidly with increasing strength of the cavity-mediated global-range interactions. Thus, in spite of the otherwise very similar behavior the mean-field like cavity-mediated interactions strongly suppress the non-local order favored by nearest-neighbor repulsion in some regions of the phase diagram.Graphical abstract

Characterization of photoexcited states in the half-filled one-dimensional extended Hubbard model assisted by machine learning

Photoinduced nonequilibrium states can provide new insight into dynamical properties of strongly correlated electron systems. One of the typical and extensively studied systems is the half-filled one-dimensional extended Hubbard model (1DEHM). Here, we propose that the supervised machine learning (ML) can provide useful information for characterizing photoexcited states in 1DEHM. Using entanglement spectra as a training dataset, we construct neural network. Judging from the trained network, we find that bond-spin-density wave (BSDW) order can be enhanced in photoexcited states if the frequency of a driving pulse nearly resonates with gap. We separately calculate the time evolution of local and non-local order parameters and confirm that the correlation functions of BSDW are actually enhanced by photoexcitation as predicted by ML. The successful prediction of BSDW demonstrates the advantage of ML to assist characterizing photoexcited quantum states.

Nonlocal order parameters for states with topological electromagnetic response

Chern insulators are states of matter characterized by a quantized Hall conductance, gapless edge modes but also a singular response to monopole configurations of an external electromagnetic field. In this paper, we describe the nature of such a singular response and show how it can be used to define a class of operators acting as non-local order parameters. These operators characterize the Chern-insulator states in the following way: for a given state, there exists a corresponding operator which has an algebraically decaying two-point function in that particular state, while it decays exponentially in all other states. The behaviour of the order parameter is defined only in terms of the electromagnetic response, and not from any microscopic properties, and we therefore claim to have found a generic order parameter for the Chern insulating states. We support this claim by numerically evaluating the order parameters for different insulating states. We also show how our construction can be generalized to other states with topological electromagnetic response, and use the states with a quantized magnetoelectric effect in three dimensions as an example. Besides providing novel insights into topological states of matter, our construction can be exploited to efficiently diagnose such states numerically.