Amplitude Of Order Parameter Research Articles

Transition to collective oscillations in finite Kuramoto ensembles.

We present an alternative approach to finite-size effects around the synchronization transition in the standard Kuramoto model. Our main focus lies on the conditions under which a collective oscillatory mode is well defined. For this purpose, the minimal value of the amplitude of the complex Kuramoto order parameter appears as a proper indicator. The dependence of this minimum on coupling strength varies due to sampling variations and correlates with the sample kurtosis of the natural frequency distribution. The skewness of the frequency sample determines the frequency of the resulting collective mode. The effects of kurtosis and skewness hold in the thermodynamic limit of infinite ensembles. We prove this by integrating a self-consistency equation for the complex Kuramoto order parameter for two families of distributions with controlled kurtosis and skewness, respectively.

Pairing fluctuations in a strongly interacting two-dimensional Fermi gas

We theoretically investigate strong-coupling properties of a two-dimensional Fermi gas in the normal phase. Including pairing fluctuations within the framework of a self-consistent T-matrix approximation, we calculate the single-particle density of states (DOS) near the observed Berezinskii-Kosterlitz-Thouless phase transition temperature TBKT. We show that the pseudogap phenomenon associated with strong pairing fluctuations appears as an almost fully gapped single-particle density of states in the strong-coupling regime, indicating that amplitude fluctuations of the order parameter are almost absent and the system is dominated by phase fluctuations of the order parameter, as expected in a two-dimensional system near TBKT. On the other hand, although pairing fluctuations are enhanced by the low-dimensionality of the system, they are found to be still not strong enough to open such a full gap in the intermediate coupling (crossover) regime. In this regime, a dip structure is only seen in DOS around ω = 0, indicating that amplitude of the superfluid order parameter is also fluctuating, in addition to phase fluctuations. In theoretically determining TBKT in a two-dimensional Fermi gas, it is frequently assumed that, while phase fluctuations exist, the amplitude of the superfluid order parameter is fixed. However, our results indicate the necessity of including both amplitude and phase fluctuations of the order parameter in evaluating TBKT in the crossover region. Since the BKT phase transition has recently been reported in a two-dimensional 6Li Fermi gas, our results would be useful for the theoretical study of this observed novel Fermi superfluid and effects of strong low-dimensional pairing fluctuations.

Topological Fulde-Ferrell and Larkin-Ovchinnikov states in spin-orbit-coupled lattice system

The spin-orbit coupled lattice system under Zeeman fields provides an ideal platform to realize exotic pairing states. Notable examples range from the topological superfluid/superconducting (tSC) state, which is gapped in the bulk but metallic at the edge, to the Fulde-Ferrell (FF) state (having a phase-modulated order parameter with a uniform amplitude) and the Larkin-Ovchinnikov (LO) state (having a spatially varying order parameter amplitude). Here, we show that the topological FF state with Chern number ($\mathcal{C}=-1$) (tFF$_{1}$) and topological LO state with $\mathcal{C}=2$ (tLO$_{2}$) can be stabilized in Rashba spin-orbit coupled lattice systems in the presence of both in-plane and out-of-plane Zeeman fields. Besides the inhomogeneous tSC states, in the presence of a weak in-plane Zeeman field, two topological BCS phases may emerge with $\mathcal{C}=-1$ (tBCS$_{1}$) far from half filling and $\mathcal{C}=2$(tBCS$_{2}$) near half filling. We show intriguing effects such as different spatial profiles of order parameters for FF and LO states, the topological evolution among inhomogeneous tSC states, and different non-trivial Chern numbers for the tFF$_{1}$ and tLO$_{1,2}$ states, which are peculiar to the lattice system. Global phase diagrams for various topological phases are presented for both half-filling and doped cases. The edge states as well as local density of states spectra are calculated for tSC states in a 2D strip.

Gravitational waves at aLIGO and vacuum stability with a scalar singlet extension of the standard model

A new gauge singlet scalar field can undergo a strongly first-order phase transition (PT) leading to gravitational waves (GW) potentially observable at aLIGO and simultaneously stabilize the electroweak vacuum. aLIGO (O5) is potentially sensitive to cosmological PTs at $10^7$-$10^8$ GeV, which coincides with the requirement that the singlet scale is less than the Standard Model (SM) vacuum instability scale, which is between $10^8$ GeV and $10^{14}$ GeV. After sampling its parameter space, we identify three benchmark points with a PT at about $T\approx 10^7$ GeV in a gauge singlet extension of the SM. We calculate the nucleation temperature, order parameter, characteristic timescale, and peak amplitude and frequency of GW from bubble collisions during the PT for the benchmarks and find that, in an optimistic scenario, GW from such a PT may be in reach of aLIGO (O5). We confirm that the singlet stabilizes the electroweak vacuum whilst remaining consistent with zero-temperature phenomenology as well. Thus, this scenario presents an intriguing possibility that aLIGO may detect traces of fundamental physics motivated by vacuum stability at an energy scale that is well above the reach of any other experiment.

Self-consistent Bogoliubov–de Gennes theory of the vortex lattice state in a two-dimensional strongly type-II superconductor at high magnetic fields

A self-consistent Bogoliubov deGennes theory of the vortex lattice state in a 2D strong type-II superconductor at high magnetic fields reveals a novel quantum mixed state around the semiclassical Hc2, characterized by a well-defined Landau--Bloch band structure in the quasi-particle spectrum and suppressed order-parameter amplitude, which sharply crossover into the well-known semiclassical (Helfand-Werthamer) results upon decreasing magnetic field. Application to the 2D superconducting state observed recently on the surface of the topological insulator Sb2Te3, accounts well for the experimental data, revealing a strong type-II superconductor, with unusually low carrier density and very small cyclotron mass, which can be realized only in the strong coupling superconductor limit.

Controlling many-body states by the electric-field effect in a two-dimensional material

To understand the complex physics of a system with strong electron-electron interactions, the ideal is to control and monitor its properties while tuning an external electric field applied to the system (the electric-field effect). Indeed, complete electric-field control of many-body states in strongly correlated electron systems is fundamental to the next generation of condensed matter research and devices. However, the material must be thin enough to avoid shielding of the electric field in the bulk material. Two-dimensional materials do not experience electrical screening, and their charge-carrier density can be controlled by gating. Octahedral titanium diselenide (1T-TiSe2) is a prototypical two-dimensional material that reveals a charge-density wave (CDW) and superconductivity in its phase diagram, presenting several similarities with other layered systems such as copper oxides, iron pnictides, and crystals of rare-earth elements and actinide atoms. By studying 1T-TiSe2 single crystals with thicknesses of 10 nanometres or less, encapsulated in two-dimensional layers of hexagonal boron nitride, we achieve unprecedented control over the CDW transition temperature (tuned from 170 kelvin to 40 kelvin), and over the superconductivity transition temperature (tuned from a quantum critical point at 0 kelvin up to 3 kelvin). Electrically driving TiSe2 over different ordered electronic phases allows us to study the details of the phase transitions between many-body states. Observations of periodic oscillations of magnetoresistance induced by the Little-Parks effect show that the appearance of superconductivity is directly correlated with the spatial texturing of the amplitude and phase of the superconductivity order parameter, corresponding to a two-dimensional matrix of superconductivity. We infer that this superconductivity matrix is supported by a matrix of incommensurate CDW states embedded in the commensurate CDW states. Our results show that spatially modulated electronic states are fundamental to the appearance of two-dimensional superconductivity.

Theory of disordered unconventional superconductors

In contrast to conventional s-wave superconductivity, unconventional (e.g., p- or d-wave) superconductivity is strongly suppressed even by relatively weak disorder. Upon approaching the superconductormetal transition, the order parameter amplitude becomes increasingly inhomogeneous, leading to effective granularity and a phase ordering transition described by the Mattis model of spin glasses. One consequence of this is that at sufficiently low temperatures, between the clean unconventional superconducting and the diffusive metallic phases, there is necessarily an intermediate superconducting phase that exhibits s-wave symmetry on macroscopic scales.

Strong-Coupling Approach to Antiferromagnetic Ordering Driven by Paramagnetic Pair-Breaking in d-Wave Superconducting Phase

The field-induced antiferromagnetic (AFM) ordering in the $d_{x^2-y^2}$-paired superconducting phase, which has been recently found in the weak-coupling approach as a basic mechanism due to the Pauli paramagnetic pair-breaking (PPB) in relation to the high-field behaviors in CeCoIn_5, is studied in the strong-coupling approach taking account of the electron correlation. Applying the fluctuation-exchange (FLEX) approximation to the two-dimensional Hubbard model including the Zeeman term, it is shown that the PPB-induced AFM ordering in the superconducting (SC) phase and the first order SC transition on H_{c2}(T) are realized in the strong-coupling approach as well as those in the weak-coupling model, and that the AFM ordering is affected by the quasiparticle renormalization and the amplitude of the SC order parameter. This AFM ordering may appear in a wide range of materials close to an AFM quantum critical point (QCP).

Competing superfluid orders in spin-orbit-coupled fermionic cold-atom optical lattices

The Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase, a superconducting state with non-zero total momentum Cooper pairs in a large magnetic field, was first predicted about 50 years ago, and since then became an important concept in many branches of physics. Despite intensive search in various materials, unambiguous experimental evidence for the FFLO phase is still lacking in experiments. In this paper, we show that both FF (uniform order parameter with plane-wave phase) and LO phase (spatially varying order parameter amplitude) can be observed using fermionic cold atoms in spin-orbit coupled optical lattices. The increasing spin-orbit coupling enhances the FF phase over the LO phase. The coexistence of superfluid and magnetic orders is also found in the normal BCS phase. The pairing mechanism for different phases is understood by visualizing superfluid pairing densities in different spin-orbit bands. The possibility of observing similar physics using spin-orbit coupled superconducting ultra-thin films is also discussed.

Theory of Antiferromagnetic Order in High-Tc Oxides: An Approach Based on Ginzburg–Landau Expansion

The mean-field phase diagram of antiferromagnetic order in t-J model has been examined, using the free energy obtained by Ginzburg-Landau (GL) expansion. We extended the usual GL theory in two ways: First, we have included higher order terms with respect to the spatial derivative (or wave number) to incorporate the incommensurate antiferromagnetic order. Second, we have also included higher order terms with respect to the order parameter amplitude, in order to treat the first order phase transition between paramagnetic and antiferromagnetic phase, which appears at some doping rates. We found the possibility of tricritical point and critical endpoint in the magnetic phase diagram of the high-Tc oxides associated with the commensurate and incommensurate antiferromagnetic order. The possible effects of thermal fluctuations and randomness (spin glass) are also discussed qualitatively based on the GL free energy.

Superconductivity and the Pseudogap in the Two-Dimensional Hubbard Model

Recently developed numerical methods have enabled the explicit construction of the superconducting state of the Hubbard model of strongly correlated electrons in parameter regimes where the model also exhibits a pseudogap and a Mott insulating phase. d(x(2)-y(2)) symmetry superconductivity is found to occur in proximity to the Mott insulator, but separated from it by a pseudogapped nonsuperconducting phase. The superconducting transition temperature and order parameter amplitude are found to be maximal at the onset of the normal-state pseudogap. The emergence of superconductivity from the normal state pseudogap leads to a decrease in the excitation gap. All of these features are consistent with the observed behavior of the copper-oxide superconductors.

Unlocking of frozen dynamics in the complex Ginzburg-Landau equation

We present results for pattern formation and related dynamics in the two-dimensional complex Ginzburg-Landau equation. Both single and multispiral morphologies have been considered. For the former, Hagan's solution has been tested. In case of the multispiral morphology, at a late time, depending upon certain parameter values, the dynamics is found to be frozen. However, upon introduction of disorder in these parameters the frozen dynamics is observed to be unlocked. This latter result is counterintuitive considering our current knowledge of dynamics in disorder systems. We also present results for the role of shocks (the regions where Hagan's solution is violated) in the multispiral dynamics. It is observed that the suppression of the order-parameter amplitude, in this region, to the value allowed by Hagan's single-spiral solution, also unlocks the dynamical freezing. In this case, both the pattern and dynamics are observed to be very similar to the the dynamical XY model.

Neutron scattering study of ferroelectric Sn2P2S6under pressure

Ferroelectric phase transition in the semiconductor Sn${}_{2}$P${}_{2}$S${}_{6}$ single crystal has been studied by means of neutron scattering in the pressure-temperature range adjacent to the anticipated tricritical Lifshitz point ($p\ensuremath{\approx}0.18$ GPa, $T\ensuremath{\approx}296$ K). The observations reveal a direct ferroelectric-paraelectric phase transition in the whole investigated pressure range ($0.18--0.6$ GPa). These results are in a clear disagreement with phase diagrams assumed in numerous earlier works, according to which a hypothetical intermediate incommensurate phase extends over several or even tens of degrees in the 0.5 GPa pressure range. Temperature dependence of the anisotropic quasielastic diffuse scattering suggests that polarization fluctuations present above ${T}_{\mathrm{C}}$ are strongly reduced in the ordered phase. Still, the temperature dependence of the ($\overline{2}$00) Bragg reflection intensity at $p=0.18$ GPa can be remarkably well modeled assuming the order-parameter amplitude growth according to the power law with logarithmic corrections predicted for a uniaxial ferroelectric transition at the tricritical Lifshitz point.

Coexistence of superfluid gap and pseudogap in the BCS-BEC crossover regime of a trapped Fermi gas belowTc

We investigate strong pairing fluctuations and effects of a harmonic trap in the superfluid phase of an ultracold Fermi gas. Including amplitude and phase fluctuations of the inhomogeneous superfluid order parameter $\Delta(r)$ in a trap within a combined $T$-matrix theory with the local density approximation, we examine local properties of single-particle excitations and a thermodynamic quantity in the BCS-BEC crossover region. Below the superfluid phase transition temperature $T_{\rm c}$, we show that inhomogeneous pairing fluctuations lead to a shell structure of the gas cloud in which the spatial region where the ordinary BCS-type superfluid density of states appears is surrounded by the region where the pseudogap associated with strong pairing fluctuations dominates single-particle excitations. The former spatial region enlarges to eventually cover the whole gas cloud far below $T_{\rm c}$. We also examine how this shell structure affects the photoemission spectrum, as well as the local pressure. Since a cold Fermi gas is always trapped in a harmonic potential, our results would be useful for the study of strong-coupling superfluid physics, including this realistic situation.

Disorder-driven loss of phase coherence in a quasi-2D cold atom system

We study the order parameter of a quasi-two-dimensional (quasi-2D) gas of ultracold atoms trapped in an optical potential in the presence of controllable disorder. Our results show that disorder drives phase fluctuations without significantly affecting the amplitude of the quasi-condensate order parameter. This is evidence that disorder can drive phase fluctuations in 2D systems, relevant to the phase-fluctuation mechanism for the superconductor-to-insulator phase transition (SIT) in disordered 2D superconductors.

Reconstruction of the Charge Density Wave State Under the Applied Electric Field

Charge density wave (CDW) under an applied electric field in constraint geometry experience stresses, which can easily exceed a plastic threshold. The stress is resolved by the ground state reconstruction which proceed via creation of topological defects like solitons and dislocations—the CDW vortices. These states can be observed experimentally either in average at macroscopic scales of X-ray and multijunction space resolved studies, at mesoscopic scales of coherent X-ray micro-diffraction and nano-junctions or individually as by the STM. Here, we report numerical modeling taking into account multiple fields in their mutual nonlinear interactions: the phase and the amplitude of the CDW order parameter, distributions of the electric field, of the density and the current of normal carriers. Following events of creation and the subsequent evolution of dislocations, we find that vortices are formed in the junction when the voltage across, or the current through, exceed a threshold. The number of vortices remnant in the reconstructed ground state increases stepwise—in agreement with experiments. The vortex core concentrates the voltage drop across the junction giving rise to observed peaks of the interlayer tunneling. The studied reconstruction in junctions of CDWs may be relevant to modern efforts of the field-effect transformations in other correlated electronic systems.

Single magnetic impurity in a spin-imbalanced superfluid Fermi gas

A spin-imbalanced superfluid Fermi gas harmonically trapped in a two-dimensional optical lattice with a single classical magnetic impurity is investigated by Bogoliubov-de Gennes equations. In both the spin-balanced and weak spin-imbalanced cases, we show that a strong magnetic impurity can change the sign of the pairing order parameter. The amplitude of the sign-changed order parameter caused by impurity is affected by the strength of the impurity potential, temperature, and particle density. Compared to the spin-balanced case, we find that an additional in-gap bound state can be induced by a strong magnetic impurity in the weak spin-imbalanced case. In the strong spin-imbalanced case where the Fulde-Ferrell-Larkin-Ovchinnikov state is established as the ground state, the impurity induces wide spatial oscillations of pairing order parameters and can enhance the order parameters by suppressing the local spin imbalance.

Phase fluctuations and the absence of topological defects in a photo-excited charge-ordered nickelate

The dynamics of an order parameter's amplitude and phase determines the collective behaviour of novel states emerging in complex materials. Time- and momentum-resolved pump-probe spectroscopy, by virtue of measuring material properties at atomic and electronic time scales out of equilibrium, can decouple entangled degrees of freedom by visualizing their corresponding dynamics in the time domain. Here we combine time-resolved femotosecond optical and resonant X-ray diffraction measurements on charge ordered La(1.75)Sr(0.25)NiO(4) to reveal unforeseen photoinduced phase fluctuations of the charge order parameter. Such fluctuations preserve long-range order without creating topological defects, distinct from thermal phase fluctuations near the critical temperature in equilibrium. Importantly, relaxation of the phase fluctuations is found to be an order of magnitude slower than that of the order parameter's amplitude fluctuations, and thus limits charge order recovery. This new aspect of phase fluctuations provides a more holistic view of the phase's importance in ordering phenomena of quantum matter.

Multiferroicity and magnetoelectric coupling inα-CaCr2O4

Ferroelectricity in the incommensurate helical magnetic phase (below TN, 43K) of alpha CaCr2O4 has been confirmed by pyroelectric measurements. Magnetoelectric and magnetodielectric coupling exist below TN and are proportional to the square of magnetic field. From symmetry analysis, we suggest that the presence of an external electric field destabilizes the symmetrical 2221' phase and stabilizes 21' symmetry. This provides a unique system in which polarization varies as the fourth-degree of the order-parameter amplitude and exhibit a vanishingly small value below the first-order transition at TN, as observed experimentally.

D-wave superconductivity on the checkerboard Hubbard model at weak and strong coupling

It has been argued that inhomogeneity generally can enhance superconductivity (SC) in the cuprate high-${T}_{c}$ materials. To check the validity of this claim, we study $d$-wave SC on the checkerboard Hubbard model on a square lattice using the cellular dynamical mean-field theory method with an exact diagonalization solver at zero temperature. The $d$-wave order parameter is computed for various inhomogeneity levels over the entire doping range of interest in both strong- and weak-coupling regimes. At a given doping, the size of the $d$-wave order parameter manifests itself directly in the height of the coherence peaks and, hence, is an appropriate measure of the strength of SC. The weak-coupling results reveal a suppression of the order parameter in the presence of inhomogeneity for small-to-intermediate hole dopings, while it is enhanced for large dopings. In contrast, for strong coupling, there is a monotonic decrease in the maximum amplitude of the SC order parameter with inhomogeneity over the entire doping range of interest. Furthermore, at moderately high inhomogeneity, the system undergoes a first-order transition from the SC to the normal state in the underdoped regime. In the overdoped regime, the change in the value of the SC order parameter correlates with the height of the lowest-energy peak in the spectral weight of antiferromagnetic spin fluctuations, confirming the connection between antiferromagnetic fluctuations and $d$-wave SC found in earlier papers on the homogeneous case. Our results are benchmarked by comparisons with numerically exact results on the checkerboard Hubbard ladder.