Commensurate-incommensurate Transition Research Articles

Probing quantum floating phases in Rydberg atom arrays

The floating phase, a critical incommensurate phase, has been theoretically predicted as a potential intermediate phase between crystalline ordered and disordered phases. In this study, we investigate the different quantum phases that arise in ladder arrays comprising up to 92 neutral-atom qubits and experimentally observe the emergence of the quantum floating phase. We analyze the site-resolved Rydberg state densities and the distribution of state occurrences. The site-resolved measurement reveals the formation of domain walls within the commensurate ordered phase, which subsequently proliferate and give rise to the floating phase with incommensurate quasi-long-range order. By analyzing the Fourier spectra of the Rydberg density-density correlations, we observe clear signatures of the incommensurate wave order of the floating phase. Furthermore, as the experimental system sizes increase, we show that the wave vectors approach a continuum of values incommensurate with the lattice. Our work motivates future studies to further explore the nature of commensurate-incommensurate phase transitions and their non-equilibrium physics.

Study of Localization-Delocalization Transition of Light in Photonic Moiré Lattices Fabricated with Saturable Nonlinear Materials

Recently, Moiré lattices have received much attention from physicists and materials scientists. These structures have opened the door to the exploration of numerous physical phenomena such as superconductivity, the commensurate-incommensurate transition, the appearance of quasicrystals at special rotation angles, or the two-dimensional localization-delocalization transition of light in linear systems. In this study, we propose photonic Moiré lattices induced by saturable nonlinear materials. After performing numerical simulations, it is observed that there exists a transition between delocalized and localized formation of laser beams under different geometrical conditions, commensurate and incommensurate lattices. The results suggest that Moiré lattices with their compactness and tunability would be utilized to control the light patterns in integrated optical devices.

Observation of the sliding phason mode of the incommensurate magnetic texture in Fe/Ir(111)

The nanoscopic magnetic texture forming in a monolayer of iron on the (111) surface of iridium, Fe/Ir(111), is spatially modulated and uniaxially incommensurate with respect to the crystallographic periodicities. As a consequence, a low-energy magnetic excitation is expected that corresponds to the sliding of the texture along the incommensurate direction, i.e., a phason mode, which we explicitly confirm with atomistic spin simulations. Using scanning tunneling microscopy (STM), we succeed to observe this phason mode experimentally. It can be excited by the STM tip, which leads to a random telegraph noise in the tunneling current that we attribute to the presence of two minima in the phason potential due to the presence of disorder in our sample. This provides the prospect of a floating phase in cleaner samples and, potentially, a commensurate-incommensurate transition as a function of external control parameters.

Probing Interlayer Interactions and Commensurate-Incommensurate Transition in Twisted Bilayer Graphene through Raman Spectroscopy.

Twisted 2D layered materials have garnered much attention recently as a class of 2D materials whose interlayer interactions and electronic properties are dictated by the relative rotation/twist angle between the adjacent layers. In this work, we explore a prototype of such a twisted 2D system, artificially stacked twisted bilayer graphene (TBLG), where we probe, using Raman spectroscopy, the changes in the interlayer interactions and electron-phonon scattering pathways as the twist angle is varied from 0° to 30°. The long-range Moiré potential of the superlattice gives rise to additional intravalley and intervalley scattering of the electrons in TBLG, which has been investigated through their Raman signatures. Density functional theory (DFT) calculations of the electronic band structure of the TBLG superlattices were found to be in agreement with the resonant Raman excitations across the van Hove singularities in the valence and conduction bands predicted for TBLG due to hybridization of bands from the two layers. We also observe that the relative rotation between the graphene layers has a marked influence on the second order overtone and combination Raman modes signaling a commensurate-incommensurate transition in TBLG as the twist angle increases. This serves as a convenient and rapid characterization tool to determine the degree of commensurability in TBLG systems.

Regulating interfacial thermal conductance with commensurate-incommensurate transitions at atomic-scale silicon/silicon interfaces.

Interfacial thermal conductance (ITC) between two contact surfaces is an important factor in accurately measuring energy transfer and heat dissipation at the interface; however, it is still not fully resolved how to more effectively modulate the ITC and unravel the related inner mechanisms. In this study, the contribution of commensurability and normal load to ITC at the atomic-scale silicon/silicon interface is disclosed. The results manifest that the ITC gradually reduces with the transition from commensurability to incommensurability. This is because the reduced force constant at the incommensurate interface decreases the transmittance of phonons, leading to the suppression of high-frequency phonon excitation and a red shift in the phonon spectrum, thereby weakening the ITC. We further discovered that increasing the normal loads can significantly enhance the ITC in both contact states, and the reason is that the interlayer distance decreases with increasing normal loads, which strengthens the interfacial potential and force constant, consequently resulting in greater heat transfer efficiency. This paper reveals that interfacial thermal transport can be regulated by applying normal loads and changing the interfacial contact states.

Tuning the stability of a model quasicrystal and its approximants with a periodic substrate.

Quasicrystals and their periodic approximants are complex crystalline phases. They have now been observed in many metallic alloys, soft matter systems, and particle simulations. In recent experiments of thin-film perovskites on solid substrates, the type of complex phase was found to change depending on thermodynamic conditions and the type of substrate used. Here, we investigate the effect of a substrate on the relative thermodynamic stability of a two-dimensional model quasicrystal and its approximants. Our simulation model is particles interacting via the Lennard-Jones-Gauss potential. Our numerical methods are molecular dynamics simulations and free energy calculations that take into account phason flips explicitly. For substrates interacting weakly with the particles, we observe an incommensurate-commensurate transition, in which a continuous series of quasicrystal approximants locks into a small number of approximants. Interestingly, we observe that the 3/2 approximant exhibits phason mode fluctuations in thermodynamic equilibrium. Such fluctuations are reminiscent of random tiling and a phenomenon usually associated only with quasiperiodic order. For stronger substrates, we find an enhancement of the stability of the dodecagonal quasicrystal and variants of square lattices. We explain all observed phenomena by the interplay of the model system with the substrate. Our results demonstrate that designing novel complex periodic and quasiperiodic structures by choice of suitable substrates is a promising strategy.

A peculiar topological Hall effect in noncentrosymmetric ternary carbide GdCoC2

A peculiar topological Hall effect (THE) is reported in a noncentrosymmetric ternary carbide GdCoC2. The GdCoC2 reveals a magnetic ordering transition from paramagnetic to ferromagnetic state at the Curie temperature TC = 15.6 K, followed by a commensurate-incommensurate phase transition at Tt = 14.1 K. Below Tt, the competition between an external magnetic field and magnetic interactions, including the Dzyaloshinskii–Moriya interaction, would give rise to characteristic spin textures with non-zero scalar spin chirality χijk, or even topologically protected spin configurations like skyrmions, for which the exotic spin-charge coupled phenomena are induced. Besides a large negative magnetoresistance (MR) up to ∼−52% at 14 K, a remarkably sharp topological Hall signal is observed with almost no anomalous Hall resistivity below 10 K. The topological Hall resistivity (ρxyT) of GdCoC2 reaches a maximum value of ∼0.23 μΩ cm at 3 K under μ0H = 0.6 T. The mechanism underlying the exceptional THE with relatively large ρxyT value in GdCoC2 is discussed in detail.

Injection and nucleation of topological defects in the quench dynamics of the Frenkel-Kontorova model

Topological defects have strong impact on both elastic and inelastic properties of materials. In this article, we investigate the possibility to controllably inject topological defects in quantum simulators of solid state lattice structures. We investigate the quench dynamics of a Frenkel-Kontorova chain, which is used to model discommensurations of particles in cold atoms and trapped ionic crystals. The interplay between an external periodic potential and the inter-particle interaction makes lattice discommensurations, the topological defects of the model, energetically favorable and can tune a commensurate-incommensurate structural transition. Our key finding is that a quench from the commensurate to incommensurate phase causes a controllable injection of topological defects at periodic time intervals. We employ this mechanism to generate quantum states which are a superposition of lattice structures with and without topological defects. We conclude by presenting concrete perspectives for the observation and control of topological defects in trapped ion experiments.

Magnetic field induced deformation of the spin density wave microphases in Ca3Co2O6

The frustrated triangular Ising magnet Ca$_3$Co$_2$O$_6$ has long been known for an intriguing combination of extremely slow spin dynamics and peculiar magnetic orders, such as the evenly-spaced non-equilibrium metamagnetic magnetization steps and the long-wavelength spin density wave (SDW) order, the latter of which is essentially an emergent crystal of solitons. Recently, an elaborate field-cooling protocol to bypass the low-field SDW phase was proposed to overcome the extraordinarily long timescale of spin relaxation that impeded previous experimental studies in equilibrium, which may point to a deep connection between the low-temperature slow relaxation and the cooling process passing through the low-field SDW phase. As the first step to elucidate the conjectured connection, we investigate the magnetic field-induced deformation of the SDW state and incommensurate-commensurate transitions, thereby mapping out the equilibrium in-field phase diagram for a realistic three-dimensional lattice spin model by using Monte Carlo simulations. We also discuss Ginzburg-Landau theory that includes several Umklapp terms as well as an effective sine-Gordon model, which can qualitatively explain the observed magnetic field-induced deformation of the SDW microphases.

High-pressure structural transition and magnetic phase diagram of CuB2O4

Noncentrosymmetric $\mathrm{Cu}{\mathrm{B}}_{2}{\mathrm{O}}_{4}$ is a prototypical system with interacting spin sublattices of different dimensionality, exhibiting diverse magnetic structures and magnetic phase transitions. Here we report pressure-induced structural evolution of $\mathrm{Cu}{\mathrm{B}}_{2}{\mathrm{O}}_{4}$ and its effect on the magnetic phase diagram. Powder x-ray diffraction (XRD), Raman scattering, and infrared phonon measurements have been performed under high pressure at room temperature. Although XRD shows lattice stability up to $\ensuremath{\sim}25\phantom{\rule{0.16em}{0ex}}\mathrm{GPa}$, Raman and IR phonon measurements reveal structural anomaly near 3 GPa, followed by a structural phase transition at $\ensuremath{\sim}12\phantom{\rule{0.16em}{0ex}}\mathrm{GPa}$. Pressure evolution of the zero-phonon lines in the low-temperature optical absorption studies helps identifying the symmetry change of the Cu sites modifying the crystal-field energy levels at the structural transition. High-pressure magnetization measurements have been performed with magnetic field applied perpendicular to the tetragonal $c$ axis. With increasing pressure the saturation magnetization of the commensurate weak ferromagnetic phase increases with systematic reduction of its turnup critical field. Pressure-induced systematic suppression of the dc susceptibility in this phase is attributed to the reduced critical field, indicating pressure-driven enhanced stability of the weak ferromagnetic phase at high pressure. Whereas, below 10 K the critical field of the incommensurate-commensurate transition increases with pressure, indicating enhanced stability of the incommensurate helical order at high pressure.

From Kosterlitz-Thouless to Pokrovsky-Talapov transitions in spinless fermions and spin chains with next-nearest-neighbor interactions

We investigate the nature of the quantum phase transition out of charge-density-wave phase in the spinless fermion model with nearest- and next-nearest-neighbor interaction at one-third filling. Using an extensive Density Matrix Renormalization Group (DMRG) simulations we show that the transition changes it nature. We show that for weak next-nearest-neighbor coupling the transition is of Kosterlitz-Thouless type in agreement with bosonisation predictions. We also provide strong numerical evidences that for large next-nearest-neighbor repulsion the transition belongs to the Pokrovsky-Talapov univerality class describing a non-conformal commensurate-incommensurate transition. Finally, we argue that the change of the nature of the transition is a result of incommensurability induced by frustration and realized even at zero doping. The implications in the context of XXZ chain with next-nearest-neighbor Ising interaction is briefly discussed.

Quantum phase transitions in the matrix product states of the one-dimensional boson Hubbard model

We study quantum phase transitions in the matrix product states of the one-dimensional boson Hubbard model, whose amount of entanglement is limited by the size of the matrices used in the representation of the states. By measuring entanglement entropy and other physical properties, we observe that the Mott-insulator-to-superfluid transitions occur sharp and continuous, accompanied by shifting transition points that are not blurred by finite entanglement effects. This strongly suggests that the transition always occurs between the Mott insulator and a mean-field-like compressible state, followed by the more entangled superfluid state. Both $O(2)$ and the commensurate-incommensurate transitions are studied, whose properties can be characterized by entanglement spectra and critical exponents.

Magnetoelastic coupling behaviour of nanocrystalline ε-Fe2O3

Preparation of a ceramic sample which preserved the nanoscale grain sizes of ∼10–30 nm has allowed elastic and anelastic properties of ε-Fe2O3 to be investigated as functions of temperature and magnetic field strength by Resonant Ultrasound Spectroscopy in the frequency range ∼0.1–1 MHz. Formal analysis of spontaneous strains, e, from previously obtained high resolution lattice parameter data confirmed that magnetic ordering below ∼500 K is accompanied by strains of up to ∼±0.002 due to coupling with the order parameter, m, according to λem2. However, this coupling does not result in elastic softening that would be expected if the order parameter relaxed in response to an induced strain on the timescale of the acoustic resonances. Stiffening or softening during heating through ∼450–480 K of the initial as prepared material, which acquired a distinct magnetisation parallel to the stress applied during Spark Plasma Sintering, is attributed to changes in magnetic domain configurations involving preferred local alignments of individual moments of monodomain crystals. The crystallographic space group, Pna21, and magnetic space group at room temperature, Pna'21', allow ε-Fe2O3 to be both piezoelectric and piezomagnetic. It is postulated that changes in the effective coefficients of these properties for bulk samples are responsible for the observed variations in acoustic resonance frequencies according to the magnetic domain structure present. No elastic or anelastic anomalies were observed at the commensurate-incommensurate transition near 110 K. Instead, acoustic resonance frequencies changed in response to an applied magnetic field, consistent with viscous motion of magnetic domain walls and the effect of poling on piezo coefficients. Ceramic samples of ε-Fe2O3 could have potentially functional piezoelectric and/or piezomagnetic properties that are tunable by choice of magnetic and thermal history.

Fidelity susceptibility as a diagnostic of the commensurate-incommensurate transition: A revisit of the programmable Rydberg chain

In recent years, programmable Rydberg-atom arrays have been widely used to simulate new quantum phases and phase transitions, generating great interest among theorists and experimentalists. Based on the large-scale density matrix renormalization group method, the ground-state phase diagram of one-dimensional Rydberg chains is investigated with fidelity susceptibility as an efficient diagnostic method. We find that the competition between Rydberg blockade and external detuning produces unconventional phases and phase transitions. As the Rydberg blockade radius increases, the phase transition between disordered and density-wave ordered phases changes from the standard Potts universality class to an unconventional chiral one. As the radius increases further (above Potts point but still close to the tip of the lobe), a very narrow intermediate floating-phase region begins to appear. A concise physical picture is also provided to illustrate the numerical results. Compared with previous studies, this work brings more evidence for commensurate-incommensurate quantum phase transitions in programmable quantum simulators from the perspective of quantum information, showing that fidelity susceptibility can be used to study such phase transitions.

Adsorption of molecular iodine on the Ag(111) surface: Phase transitions, silver reconstruction, and iodide growth.

Room temperature adsorption of molecular iodine on Ag(111) has been studied by scanning tunneling microscopy (STM), low energy electron diffraction, Auger electron spectroscopy with factor analysis, and density functional theory (DFT). At the chemisorption stage, iodine first forms a (3×3)R30° structure. Further iodine dosing leads to continuous commensurate-incommensurate phase transition, taking place via the formation of striped superheavy domain walls. As a result, the uniaxially compressed (13 ×3-R30°) phase is formed at an iodine coverage (θ) of 0.38 ML. At θ > 0.38 ML, first-order phase transition begins, leading to the formation of hexagonal moiré-like phases, which exhibit an anomalously large corrugation in STM (0.8-2.3 Å). In the range of 0.40-0.43 ML, the compression of hexagonal phases occurs, which ends at the formation of the (7 × 7)R21.8° structure at saturation. The DFT calculations allow us to explain the anomalous atomic corrugation of the hexagonal phases by the strong violation of the atomic structure of the substrate including up to ten layers of silver. Iodine dosing above 0.43 ML leads to the growth of 2D islands of silver iodide. The STM images of the silver iodide surface demonstrate a clear visible hexagonal superstructure with a periodicity of 25 Å superimposed with a quasi-hexagonal atomic modulation. DFT calculations of the atomic structure of AgI islands point to the formation of a sandwich-like double layer honeycomb structure similar to the case of I/Ag(100).

Possible quadrupole-order-driven commensurate-incommensurate phase transition in B20 CoGe

The B20-type cobalt germanide CoGe was investigated by measuring the specific heat, resistivity, and $^{59}$Co nuclear magnetic resonance (NMR). We observed a phase transition at $T_Q=13.7$ K, evidenced by a very narrow peak of the specific heat and sharp changes of the nuclear spin-spin ($T_2^{-1}$) and spin-lattice ($T_1^{-1}$) relaxation rates. The fact that the entropy release is extremely small and the Knight shift is almost independent of temperature down to low temperatures as anticipated in a paramagnetic metal indicates that the $T_Q$ transition is of non-magnetic origin. In addition, we detected a crossover scale $T_0\sim30$ K below which the resistivity and the NMR linewidth increase, and $T_1^{-1}$ is progressively distributed in space, that is, a static and dynamical spatial inhomogeneity develops. While the order parameter for the $T_Q$ transition remains an open question, a group-theoretical analysis suggests that the finite electric quadrupole density arising from the low local site symmetry at cobalt sites could drive the crystal symmetry lowering from the P2$_1$3 symmetry that is commensurate to the R3 symmetry with an incommensurate wavevector, which fairly well accounts for the $T_Q$ transition. The quadrupole-order-driven commensurate-incommensurate phase transition may be another remarkable phenomenon arising from the structural chirality inherent in the noncentrosymmetric B20 family.

Discovery of electric devil's staircase in perovskite antiferroelectric.

The devil’s staircase, describing step-like function for two competing frequencies, is well known over a wide range of dynamic systems including Huyghens’ clocks, Josephson junction, and chemical reaction. In condensed matter physics, the devil’s staircase has been observed in spatially modulated structures, such as magnetic ordering. It draws widespread attentions because it plays a crucial role in the fascinating phenomena including phase-locking behaviors, commensurate-incommensurate phase transition, and spin-valve effect. Here, we report the observation of polymorphic phase transitions consisting of several steps in PbZrO3-based system—namely, electric devil’s staircase—originated from competing ferroelectric and antiferroelectric interactions. We fully characterize a specific electric dipole configuration by decomposing this competitive interaction in terms of basic structure and modulation function. Of particular interest is that the occurrence of many degenerate electric dipole configurations in devil’s staircase enables superior energy storage performance. These observations are of great significance for exploring more substantive magnetic-electric correspondence and engineering practical high-power antiferroelectric capacitors.

Inhomogeneous superconducting states in two weakly linked superconducting ultrathin films

A sufficiently large parallel magnetic field will generate staggered supercurrent loops and superfluid density wave in two weakly linked superconducting (SC) ultrathin films, resulting in an inhomogeneous Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state. The SC order parameter of such an FFLO state is characterized by Bloch wave functions, called the "Bloch SC state". The staggered supercurrent loops form an array of Josephson vortex-antivortex pairs, instead of the usual Josephson vortex lattice. Enclosing a unit cell of the array, the London's fluxoid is quantized as $\Phi^{\prime}=\Phi_0=hc/2e$, while the net orbital magnetization caused by the staggered supercurrent is zero. Meanwhile, a small parallel magnetic field gives rise to an Fulde-Ferrell (FF) state that has uniform superfluid density. The phase transition between the Bloch SC state and the FF state belongs to the universality class of two-dimensional commensurate-incommensurate transitions. An analytical solution in terms of Jacobian elliptic functions is found to be an excellent approximation to the Bloch SC order parameter.

Commensurate-incommensurate transition in the chiral Ashkin-Teller model

We investigate the classical chiral Ashkin-Teller model on a square lattice with the corner transfer matrix renormalisation group (CTMRG) algorithm. We show that the melting of the period-4 phase in the presence of a chiral perturbation takes different forms depending on the coefficient of the four-spin term in the Ashkin-Teller model. Close to the clock limit of two decoupled Ising models, the system undergoes a two-step commensurate-incommensurate transition as soon as the chirality is introduced, with an intermediate critical floating phase bounded by a Kosterlitz-Thouless transition at high temperature and a Pokrovsky-Talapov transition at low temperature. By contrast, close to the four-states Potts model, we argue for the existence of a unique commensurate-incommensurate transition that belongs to the chiral universality class, and for the presence of a Lifshitz point where the ordered, disordered and floating phases meet. Finally, we map the whole phase diagram, which turns out to be in qualitative agreement with the 40 year old prediction by Huse and Fisher.

Heterobilayer moiré magnets: Moiré skyrmions and commensurate-incommensurate transitions

We study untwisted heterobilayers of ferromagnetic and antiferromagnetic van der Waals materials, with in particular a Dzyaloshinskii-Moriya interaction in the ferromagnetic layer. A continuum low energy field theory is utilized to study such systems. We develop a phase diagram as a function of the strength of inter-layer exchange and Dzyaloshinskii-Moriya interactions, combining perturbative and strong coupling analyses with numerical simulations using Landau-Lifshitz-Gilbert equations. Various moir\'e-periodic commensurate phases are found, and the commensurate-incommensurate transition is discussed. Among the commensurate phases, we observe an interesting skyrmion lattice phase wherein each moir\'e unit cell hosts one skyrmion.