Wide Critical Region Research Articles

Critical slowing-down and field-dependent paramagnetic fluctuations in the skyrmion host EuPtSi: μSR and NMR studies

We report the results of muon spin relaxation ($\ensuremath{\mu}\mathrm{SR}$) and $^{29}\mathrm{Si}$ NMR measurements carried out on the cubic chiral magnet EuPtSi. This compound exhibits a helimagnetic transition at ${T}_{\mathrm{N}}=4$ K, and hosts a unique skyrmion phase characterized by a short modulation period and strong anisotropy under magnetic fields. Our zero-field $\ensuremath{\mu}\mathrm{SR}$ experiments revealed the development of critical slowing down of Eu spin fluctuations over a relatively wide critical region $(T\ensuremath{-}{T}_{\mathrm{N}})/{T}_{\mathrm{N}}<5$ above ${T}_{\mathrm{N}}$. We also found from $^{29}\mathrm{Si}$ NMR that the spin fluctuations are strongly suppressed by magnetic field in the paramagnetic state above 20 K. These characteristic spin dynamics observed over a wide region of temperature and magnetic field suggest the presence of magnetic frustration in the spin system. Such frustration would underlie the mechanism stabilizing the short-period skyrmion lattice observed in this compound.

Topological superconductivity of spin- 3/2 carriers in a three-dimensional doped Luttinger semimetal

We investigate topological Cooper pairing, including gapless Weyl and fully gapped class DIII superconductivity, in a three-dimensional doped Luttinger semimetal. The latter describes effective spin-3/2 carriers near a quadratic band touching and captures the normal-state properties of the 227 pyrochlore iridates and half-Heusler alloys. Electron-electron interactions may favor non-$s$-wave pairing in such systems, including even-parity $d$-wave pairing. We argue that the lowest energy $d$-wave pairings are always of complex (e.g., $d + i d$) type, with nodal Weyl quasiparticles. This implies $\varrho(E) \sim |E|^2$ scaling of the density of states (DoS) at low energies in the clean limit, or $\varrho(E) \sim |E|$ over a wide critical region in the presence of disorder. The latter is consistent with the $T$-dependence of the penetration depth in the half-Heusler compound YPtBi. We enumerate routes for experimental verification, including specific heat, thermal conductivity, NMR relaxation time, and topological Fermi arcs. Nucleation of any $d$-wave pairing also causes a small lattice distortion and induces an $s$-wave component; this gives a route to strain-engineer exotic $s+d$ pairings. We also consider odd-parity, fully gapped $p$-wave superconductivity. For hole doping, a gapless Majorana fluid with cubic dispersion appears at the surface. We invent a generalized surface model with $\nu$-fold dispersion to simulate a bulk with winding number $\nu$. Using exact diagonalization, we show that disorder drives the surface into a critically delocalized phase, with universal DoS and multifractal scaling consistent with the conformal field theory (CFT) SO($n$)${}_\nu$, where $n \rightarrow 0$ counts replicas. This is contrary to the naive expectation of a surface thermal metal, and implies that the topology tunes the surface renormalization group to the CFT in the presence of disorder.

Thouless energy and multifractality across the many-body localization transition

Thermal and many-body localized phases are separated by a dynamical phase transition of a new kind. We analyze the distribution of off-diagonal matrix elements of local operators across the many-body localization transition (MBLT) in a disordered spin chain, and use it to characterize the breakdown of the eigenstate thermalization hypothesis and to extract the many-body Thouless energy. We find a wide critical region around the MBLT, where Thouless energy becomes smaller than the level spacing, while matrix elements show critical dependence on the energy difference. In the same region, matrix elements, viewed as amplitudes of a fictitious wave function, exhibit strong multifractality. Our findings show that the correlation length becomes larger than the accessible system sizes in a broad range of disorder strength values, and shed light on the critical behaviour of MBL systems.

Quantum Spin Liquid in Spin 1/2J1–J2Heisenberg Model on Square Lattice: Many-Variable Variational Monte Carlo Study Combined with Quantum-Number Projections

The nature of quantum spin liquids is studied for the spin-$1/2$ antiferromagnetic Heisenberg model on a square lattice containing exchange interactions between nearest-neighbor sites, $J_1$, and those between next-nearest-neighbor sites, $J_2$. We perform variational Monte Carlo simulations together with the quantum-number-projection technique and clarify the phase diagram in the ground state together with its excitation spectra. We obtain the nonmagnetic phase in the region $0.4< J_2/J_1\le 0.6$ sandwiched by the staggered and stripe antiferromagnetic (AF) phases. Our direct calculations of the spin gap support the notion that the triplet excitation from the singlet ground state is gapless in the range of $0.4 < J_2/J_1 \le 0.5$, while the gapped valence-bond-crystal (VBC) phase is stabilized for $0.5 < J_2/J_1 \le 0.6$. The VBC order is likely to have the columnar symmetry with a spontaneous symmetry breaking of the $C_{4v}$ symmetry. The power-law behaviors of the spin-spin and dimer-dimer correlation functions in the gapless region are consistent with the emergence of the algebraic quantum-spin-liquid phase (critical phase). The exponent of the spin correlation $\langle S(0)S(r)\rangle \propto 1/r^{z+\eta}$ at a long distance $r$ appears to increase from $z+\eta\sim 1$ at $J_2/J_1\sim0.4$ toward the continuous transition to the VBC phase at $J_1/J_1\sim0.5$. Our results, however, do not fully exclude the possibility of a direct quantum transition between the staggered AF and VBC phases with a wide critical region and deconfined criticality.

Thermodynamic properties near the onset of loop-current order in high-Tcsuperconducting cuprates

We have performed large-scale Monte Carlo simulations on a two-dimensional generalized Ashkin-Teller model to calculate the thermodynamic properties in the critical region near its transitions. The Ashkin-Teller model has a pair of Ising spins at each site which interact with neighboring spins through pair-wise and four-spin interactions. The model represents the interactions between orbital current loops in ${\text{CuO}}_{2}$ plaquettes of high-${T}_{c}$ cuprates, which order with a staggered magnetization ${M}_{s}$ inside each unit cell in the underdoped region of the phase diagram below a temperature ${T}^{\ensuremath{\ast}}(x)$ which depends on doping. The pair of Ising spins per unit cell represents the directions of the currents in the links of the current loops. The generalizations are the inclusion of anisotropy in the pair-wise nearest-neighbor current-current couplings consistent with the symmetries of a square lattice and the next-nearest-neighbor pair-wise couplings. We use the Binder cumulant to estimate the correlation length exponent $\ensuremath{\nu}$ and the order-parameter exponent $\ensuremath{\beta}$. Our principal results are that in a range of parameters; the Ashkin-Teller model as well as its generalization has an order-parameter susceptibility which diverges as $T\ensuremath{\rightarrow}{T}^{\ensuremath{\ast}}$ and an order parameter below ${T}^{\ensuremath{\ast}}$. Importantly, however, there is no divergence in the specific heat. This puts the properties of the model in accord with the experimental results in the underdoped cuprates. We also calculate the magnitude of the ``bump'' in the specific heat in the critical region to put limits on its observability. Finally, we show that the staggered magnetization couples to the uniform magnetization ${M}_{0}$ such that the latter has a weak singularity at ${T}^{\ensuremath{\ast}}$ and also displays a wide critical region, also in accord with recent experiments.

Critical Slowing Down of Triangular Lattice Spin-3/2 Heisenberg Antiferromagnet Li7RuO6 via 7Li NMR

We report on 7 Li NMR studies of single crystals of the triangular-lattice Heisenberg antiferromagnet Li 7 RuO 6 . A slow critical divergence with a wide critical region of | T / T N -1|≤7 was observed in 7 Li nuclear spin–lattice relaxation rate. The slowing down of staggered spin fluctuations was analyzed in a renormalized classical region of a two-dimensional triangular-lattice nonlinear sigma model. Spin stiffness constant was found to decrease to about 20% of its value in spin-wave approximation. The effect of spin frustration, e.g., Z 2 vortex excitations on critical phenomena, is suggested.

Ginzburg-Landau theory for three-dimensional Josephson junction arrays

We have derived a mean-field theory and a Ginzburg-Landau (GL) theory for classical three-dimensional Josephson junction arrays. We show that the full GL theory for arrays is formally equivalent to the GL theory for a bulk superconductor, demonstrating that for low fields and close to the critical temperature, the two systems are electrodynamically equivalent. We derive expressions for the penetration depth and coherence length, the effective superfluid density, critical fields, and the current-phase relationship for arrays. Using the Ginzburg criterion, we show that the critical region is very wide and that three-dimensional critical fluctuations should be experimentally accessible. Although the wide critical region casts doubt on the strict validity of the GL theory, suitable modifications of the coefficients should allow it to be used at lower temperatures, as is done for bulk superconductors.

Air Motions Accompanying the Development of a Planetary Wave Critical Layer

Horizontal air motions accompanying the development of a planetary wave critical layer are investigated on the sphere in the equivalent barotropic framework. For small wave amplitude or strong shear in the neighborhood of the zero wind line, the critical region is confined laterally to a narrow zone adjacent to the critical latitude. Increasing the wave amplitude or reducing the zonal shear near the zero wind line expands the critical region poleward. Stirring within the critical region is controlled by a competition between wave advection, which promotes instability by rearranging potential vorticity, and dissipation, which limits the steepening of gradients and damps instability. Eddy advection strains bodies of air inside the critical region down to small dimensions, until eventually dissipation becomes efficient and anomalies in potential vorticity and other tracers “dissolve.” For a narrow critical region, eddy stirring tends to homogenize potential vorticity near the zero wind line. The reduced gradient of potential vorticity resulting under weakly damped conditions lead to reflection of wave activity that suppresses wave generation and allows the flow to approach equilibrium. For a wide critical region, the flow can adjust nonlinearly directly at the forcing, making eddy stirring at low latitudes less important. The picture in that case is one of “global-scale entrainment,” where material lines fold across much of the hemisphere initially in westerlies. Air injected into the polar cap spins up anticyclonically, sweeping easterlies across mid-latitudes that shield the critical line from wave activity. Once this occurs, the eddy field at low latitudes collapses and easterlies advance over the forcing to suppress wave generation and restore equilibrium. The preliminary development of the critical region resembles the familiar “wave breaking” signature in stratospheric potential vorticity maps and is reproduced quite well in linear and quasi-linear restrictions of the complete system. Although the nonlinear planetary wave critical layer captures a number of observed features, this paradigm of stratospheric air motions breaks down quickly with realistic wave amplitudes. Without a restoring mechanism to counteract the disruptive influence of eddy advection, the vortex is seriously compromised after only a month.

Statistical mechanics of a fractal lattice: Renormalization-group analysis of the Sierpinski gasket.

Exact renormalization-group recursion relations are constructed for several thermodynamic properties of the Sierpinski gasket (SG). A model with one, two, and three spin interactions is studied. The interactions are restricted to be explicitly part of the fractal geometry. The specific heat, paramagnetic susceptibility, and near-neighbor correlation function are calculated exactly both near and away from criticality which occurs at zero temperature (infinite near-neighbor coupling). With one exception, our results agree with the asymptotic critical properties of the SG as discussed by Gefen et al. As a finitely ramified structure, the critical exponents of the SG are essentially one dimensional. The nonuniversal properties of the SG, however, are markedly different from those in one dimension. The scale-invariant, fractal structure of the SG leads to strong critical fluctuations and, therefore, to a rather wide critical region. We find a logarithmic correction to the leading singularity in the correlation length which is necessary to account for the approach to the critical region. The specific heat of the SG is qualitatively different from that in one dimension. A continuous line of fixed points at zero temperature is also found. This is due to a marginal operator associated with a competition between the two (odd-spin) symmetry-breaking fields. In the absence of a two-spin interaction, the SG becomes frustrated when the two odd-spin interactions compete directly. The phase diagram in this limit is also calculated.

Critical Fluctuation near Threshold of Gunn Instability

The low frequency noise and the oscillatory microwave power spectrum have been simultaneously measured on the Gunn diode in order to investigate the critical phenomenon near the threshold of the Gunn instability. In a rather wide bias range an anomalously large low frequency noise and a noisy power spectrum are observed. The appearance of a wide critical region is discussed with the use of an equation for the negative-resistance oscillator.