Gap Exponent Research Articles

Quantum wetting transition in the cluster Ising model

The wetting transition in the one-dimensional cluster Ising model with opposite boundary fields is studied. Tuning one boundary field while fixing another leads to the occurrence of phase transitions, where the transition points depend on the cluster coupling. Furthermore, the phase diagram is divided into three regions with different phase transition. For weak and strong cluster coupling, the phase transition is continuous and belongs to the same universality of transverse Ising model with boundary fields. For intermediate cluster coupling, the phase transition is first order. In the strong cluster coupling region, the critical region becomes exponentially small and the asymptotic behavior is absent even for lattice size up to 104. A numerical method to solve the energy gap and the correlation length is proposed on an infinite long spin chain. With this method, one can get the critical behavior as close to the critical point as possible provided that the numerical accuracy is high enough. In the light of this method, we clearly show that there is a preasymptotic regime in which the apparent critical exponents depend on the cluster coupling. Moreover, we obtain the accurate energy gap exponent zν and the correlation length exponent ν in the asymptotic critical region.

2D in situ determination of soot optical band gaps in flames using hyperspectral absorption tomography

During the formation of soot, the particles undergo a fine structural transformation, akin to graphitization, from “young” to “mature” soot. This ageing process is accompanied by a change in optical properties, which manifests in the soot optical band gap and dispersion exponent, Eg and ζ. Although both quantities are regularly determined via broadband extinction measurements, they have a nonlinear relationship to the local extinction spectrum, so line-of-sight integrated measurements introduce errors into estimates of Eg and ζ. We report the development of an absorption tomography sensor that enables 2D in situ mapping of Eg and ζ. Our sensor employs broadband back-illumination, fiber-coupled collection optics, and an imaging spectrograph to simultaneously acquire 24 spectrally-resolved absorption signals, suitable for tomographic reconstruction. We perform proof-of-concept experiments to characterize soot formation in non-premixed steady flames from a Gülder burner, using ethylene and propane as fuels. Reconstructions based on axisymmetric and non-axisymmetric absorption tomography are compared, and we explore how our sensor can be adapted to perform temporally-resolved measurements in a turbulent flame.

First-order and continuous quantum phase transitions in the anisotropic quantum Rabi-Stark model

Various quantum phase transitions in the anisotropic Rabi-Stark model with both the nonlinear Stark coupling and the linear dipole coupling between a two-level system and a single-mode cavity are studied in this work. The first-order quantum phase transitions are detected by the level crossing of the ground-state and the first-excited state with the help of the pole structure of the transcendental functions derived by the Bogoliubov operators approach. As the nonlinear Stark coupling is the same as the cavity frequency, this model can be solved by mapping to an effective quantum oscillator. All energy levels close at the critical coupling in this case, indicating continuous quantum phase transitions. The critical gap exponent is independent of the anisotropy as long as the counter-rotating wave coupling is present, but essentially changed if the counter-rotating wave coupling disappears completely. It is suggested that the gapless Goldstone mode excitations could appear above a critical coupling in the present model in the rotating-wave approximation.

Anomalous bulk behavior in the free parafermion Z(N) spin chain.

We demonstrate using direct numerical diagonalization and extrapolation methods that boundary conditions have a profound effect on the bulk properties of a simple Z(N) model for N≥3 for which the model Hamiltonian is non-Hermitian. For N=2 the model reduces to the well-known quantum Ising model in a transverse field. For open boundary conditions, the Z(N) model is known to be solved exactly in terms of free parafermions. Once the ends of the open chain are connected by considering the model on a ring, the bulk properties, including the ground-state energy per site, are seen to differ dramatically with increasing N. Other properties, such as the leading finite-size corrections to the ground-state energy, the mass gap exponent, and the specific-heat exponent, are also seen to be dependent on the boundary conditions. We speculate that this anomalous bulk behavior is a topological effect.

Numerical evidence of hyperscaling violation in wetting transitions of the random-bond Ising model in d=2 dimensions.

We performed extensive simulations of the random-bond Ising model confined between walls where competitive surface fields act. By properly taking the thermodynamic limit we unambiguously determined wetting transition points of the system, as extrapolation of localization-delocalization transitions of the interface between domains of different orientation driven by the respective fields. The finite-size scaling theory for wetting with short-range fields [E. V. Albano and K. Binder, Phys. Rev. Lett. 109, 036101 (2012)PRLTAO0031-900710.1103/PhysRevLett.109.036101] establishes that the average magnetization of the sample, with critical exponent β, is the proper order parameter for the study of wetting. While the hyperscaling relationship given by γ+2β=ν_{∥}+ν_{⊥} requires β=1/2 (γ=4,ν_{∥}=3, and ν_{⊥}=2), the thermodynamic scaling establishes that Δ_{s}=γ+β, which in contrast requires β=0 (Δ_{s}=4), where γ,ν_{∥},ν_{⊥}, and Δ_{s} are the critical exponents of the susceptibility, the correlation lengths parallel and perpendicular to the interface, and the gap exponent, respectively. So, we formulate a finite-size scaling theory for wetting without hyperscaling and perform numerical simulations that provide strong evidence of hyperscaling violation (i.e., β=0) and a direct measurement of the susceptibility critical exponent γ/ν_{⊥}=2.0±0.2, in agreement with theoretical results for the strong fluctuation regime of wetting transitions with quenched noise.

Random matrix theory and critical phenomena in quantum spin chains.

We compute critical properties of a general class of quantum spin chains which are quadratic in the Fermi operators and can be solved exactly under certain symmetry constraints related to the classical compact groups U(N),O(N), and Sp(2N). In particular we calculate critical exponents s,ν, and z, corresponding to the energy gap, correlation length, and dynamic exponent, respectively. We also compute the ground state correlators 〈σ_{i}^{x}σ_{i+n}^{x}〉_{g},〈σ_{i}^{y}σ_{i+n}^{y}〉_{g}, and 〈∏_{i=1}^{n}σ_{i}^{z}〉_{g}, all of which display quasi-long-range order with a critical exponent dependent upon system parameters. Our approach establishes universality of the exponents for the class of systems in question.

Critical charge fluctuations in a pseudogap Anderson model

The Anderson impurity model with a density of states $\rho(\varepsilon) \propto |\varepsilon|^r$ containing a power-law pseudogap centered on the Fermi energy ($\varepsilon = 0$) features for $0<r<1$ a Kondo-destruction quantum critical point (QCP) separating Kondo-screened and local-moment phases. The observation of mixed valency in quantum critical $\beta$-YbAlB$_4$ has prompted study of this model away from particle-hole symmetry. The critical spin response associated with all Kondo destruction QCPs has been shown to be accompanied, for $r=0.6$ and noninteger occupation of the impurity site, by a divergence of the local charge susceptibility on both sides of the QCP. In this work, we use the numerical renormalization-group method to characterize the Kondo-destruction charge response using five critical exponents, which are found to assume nontrivial values only for $0.55\lesssim r < 1$. For $0 < r \lesssim 0.55$, by contrast, the local charge susceptibility shows no divergence at the QCP, but rather exhibits nonanalytic corrections to a regular leading behavior. Both the charge critical exponents and the previously obtained spin critical exponents satisfy a set of scaling relations derived from an ansatz for the free energy near the QCP. These critical exponents can all be expressed in terms of just two underlying exponents: the correlation-length exponent $\nu(r)$ and the gap exponent $\Delta(r)$. The ansatz predicts a divergent local charge susceptibility for $\nu<2$, which coincides closely with the observed range $0.55\lesssim r<1$. Many of these results are argued to generalize to interacting QCPs that have been found in other quantum impurity models.

Induced effects of the Dzyaloshinskii-Moriya interaction on the thermal entanglement in spin-1/2 Heisenberg chains

The nearest-neighbor spins in the one-dimensional spin-1/2 $XX$ model with the added Dzyaloshinskii-Moriya (DM) interaction are entangled at zero temperature. In the presence of a transverse magnetic field (TF) they remain entangled up to a quantum critical field, ${h}_{c}$. Using the fermionization technique, we have studied the mutual effect of the DM interaction and a TF on the thermal entanglement (TE) in this model. The critical temperature where the entanglement disappears is specified. It is found that the TE in the finite-temperature neighborhood of the quantum critical field shows a scaling behavior with the critical exponent equal to the critical gap exponent. We also argue that thermodynamical properties like the specific heat and the magnetocaloric effect (instead of the usual internal energy and the magnetization) can detect the mentioned quantum entanglement in solid systems. In addition, we suggest a tactic to find all critical temperatures, which is based on the derivative of the entanglement witness with respect to the temperature.

Field Dependence of Magnetocaloric Properties in La 0.6Pr 0.4Fe 10.7Co 0.8Si 1.5

In this paper, the field dependence of magnetocaloric properties of La 0.6Pr 0.4Fe 10.7Co 0.8Si 1.5 with second-order phase transition material is studied using a phenomenological model. The model parameters were determined from the magnetization data adjustment and were used to give better fits to magnetic transition and to calculate the magnetocaloric thermodynamic quantities. The entropy curves have been observed to behave as an asymmetrical broadening of ΔS M peak with increasing magnetic field. For larger fields, the peak shifts to higher temperatures, while the overall shape of the curve broadens over a wide temperature range. The values of maximum magnetic entropy change, full width at half maximum, and relative cooling power, at several magnetic field variations, were calculated. The maximum magnetic entropy changes of 3.957(5) and 14.197(4) J kg −1 K −1 and the relative cooling power (RCP) values of 95.420(3) and 392.729(2) J kg −1 are obtained for 1 and 5 T, respectively. The theoretical calculations are compared with the available experimental data. The critical exponents associated with ferromagnetic transition have been determined from magnetocaloric effect (MCE) methods. By using the field dependence of ΔS M max≈a(μ 0 H) n and the distance (T peak−T c)≈b(μ 0 H)1/Δ, we have investigated the critical behavior of La 0.6Pr 0.4Fe 10.7Co 0.8Si 1.5. From the analysis of the relationship between the local exponent n and the gap exponent Δ, we have calculated other exponents: β, γ, and δ. The large MCE, relatively high RCP, high magnetization, and low cost jointly make the present compound a promising candidate for magnetic refrigerant near room temperature.

Effect of anisotropy on the field-induced quantum critical properties of the three-dimensionals=12Heisenberg model

The field induced quantum critical properties of the three dimensional spin-1/2 anisotropic antiferromagnetic Heisenberg model has been studied. We have investigated the quantum phase transition between the spiral order and field induced ferromagnetic order by means of Bose-Einstein condensation of magnons in terms of a bosonic representation. The effect of in-plane anisotropy on the critical properties has been studied via the bosonic model by Green's function approach. We have found an analytic expression for the gap exponent in addition to numerical results for the critical magnetic field in terms of anisotropy parameter. The in-plane anisotropy breaks the U(1) symmetry explicitly which changes the universal behavior by a drastic change on the gap exponent. Moreover, the critical magnetic field depends strongly on the in-plane anisotropies. The divergence of the transverse structure factor at the antiferromagnetic wave vector confirms the onset of the magnetic order which scales with the negative value of gap exponent as the magnetic field approaches the critical one. The transverse staggered magnetization as an order parameter vanishes with exponent $\beta=0.5$ when the magnetic field reaches its critical value in low field region.

Effect of long-range connections on an infinite randomness fixed point associated with the quantum phase transitions in a transverse Ising model

We study the effect of long-range connections on the infinite-randomness fixed point associated with the quantum phase transitions in a transverse Ising model (TIM). The TIM resides on a long-range connected lattice where any two sites at a distance r are connected with a non-random ferromagnetic bond with a probability that falls algebraically with the distance between the sites as 1/r^{d+\sigma}. The interplay of the fluctuations due to dilutions together with the quantum fluctuations due to the transverse field leads to an interesting critical behaviour. The exponents at the critical fixed point (which is an infinite randomness fixed point (IRFP)) are related to the classical "long-range" percolation exponents. The most interesting observation is that the gap exponent \psi is exactly obtained for all values of \sigma and d. Exponents depend on the range parameter \sigma and show a crossover to short-range values when \sigma >= 2 -\eta_{SR} where \eta_{SR} is the anomalous dimension for the conventional percolation problem. Long-range connections are also found to tune the strength of the Griffiths phase.

Gap exponent of theXXZmodel in a transverse field

We have calculated numerically the gap exponent of the anisotropic Heisenberg model in the presence of the transverse magnetic field. We have implemented the modified Lanczos method to obtain the excited states of our model with the same accuracy of the ground state. The coefficient of the leading term in the perturbation expansion diverges in the thermodynamic limit (N --> infinity). We have obtained the relation between this divergence and the scaling behaviour of the energy gap. We have found that the opening of gap in the presence of transverse field scales with a critical exponent which depends on the anisotropy parameter (Delta). Our numerical results are in well agreement with the field theoretical approach in the whole range of the anisotropy parameter, -1 < Delta < 1.

Avalanche behavior in an absorbing state Oslo model

Self-organized criticality can be translated into the language of absorbing state phase transitions. Most models for which this analogy is established have been investigated for their absorbing state characteristics. In this paper, we transform the self-organized critical Oslo model into an absorbing state Oslo model and analyze the avalanche behavior. We find that the resulting gap exponent D is consistent with its value in the self-organized critical model. For the avalanche size exponent tau an analysis of the effect of the external drive and the boundary conditions is required.

Low-energy fixed points of random Heisenberg models

The effect of quenched disorder on the low-energy and low-temperature properties of various two- and three-dimensional Heisenberg models is studied by a numerical strong disorder renormalization group method. For strong enough disorder we have identified two relevant fixed points, in which the gap exponent, omega, describing the low-energy tail of the gap distribution, P(Delta) ~ Delta^omega is independent of disorder, the strength of couplings and the value of the spin. The dynamical behavior of non-frustrated random antiferromagnetic models is controlled by a singlet-like fixed point, whereas for frustrated models the fixed point corresponds to a large spin formation and the gap exponent is given by omega ~ 0. Another type of universality classes is observed at quantum critical points and in dimerized phases but no infinite randomness behavior is found, in contrast to one-dimensional models.

Field dependent susceptibility of the 2D ferromagnet (CH 3NH 3) 2CuCl 4 in the paramagnetic phase

The temperature dependence of the ac susceptibility ( χ) at constant applied magnetic field was investigated in the paramagnetic region of the quasi-2D ferromagnet (CH 3NH 3) 2CuCl 4. Above the Curie temperature ( T C=8.85 K) a maximum in the χ( T, H) curves was observed at T m( H). The temperature at the maximum increases with increasing applied field. This anomaly is related to short range fluctuations close the order transition. The behavior of T m( H) is governed by the gap exponent of the scaling function ( Δ= γ+ β). We found Δ=2.2±0.1 in very good agreement with the previously known values of γ and β.

Critical fluctuations in MnP near the Curie temperature

The temperature dependence of the AC susceptibility ( χ) at constant applied magnetic field was investigated in the paramagnetic region close to the ferromagnetic transition in MnP. Above the curie temperature ( T c=291.9 K), a maximum in the χ( T, H) curves was observed at T m( H). The temperature of the maximum increases with increasing the applied field. The behavior of T m( H) is governed by the gap exponent of the scaling function ( Δ= γ+ β) . From the fitting of our data we found that Δ=1.51±0.03 which is in excellent agreement with the theoretical predictions that are in the range 1.49< Δ<1.56.

A Monte Carlo test of the Fisher–Nakanishi–Scaling theory for the capillary condensation critical point

Extending the Swendsen–Wang cluster algorithm to include both bulk (H) and surface fields (H1) in L×L×D Ising films of thickness D and two free L×L surfaces, a Monte Carlo study of the capillary condensation critical point of the model is presented. Applying a finite-size scaling analysis where the lateral linear dimension L is varied over a wide range, the critical temperature Tc(D) and the associated critical field Hc(D) are estimated for 4⩽D⩽32 lattice spacings, for a choice of the surface field H1 small enough that the dependence of Hc(D) on H1 is still linear. It is shown that the results are consistent with the power laws predicted by Fisher and Nakanishi [M. E. Fisher and H. Nakanishi, J. Chem. Phys. 75, 5857 (1981)], namely Tc(∞)−Tc(D)∝D−1/ν, Hc(D)∝D−(Δ−Δ1)/ν, where ν is the bulk correlation length exponent of the three-dimensional Ising model, and Δ, Δ1 are the corresponding “gap exponents” associated with bulk and surface fields, respectively. As expected, the order parameter of the thin film near its critical point exhibits critical behavior compatible with the universality class of the two-dimensional Ising model.

Series expansion study of quantum percolation on the square lattice

We study the site and bond quantum percolation model on the two-dimensional square lattice using series expansion in the low concentration limit. We calculate series for the averages of , where T ij (E) is the transmission coefficient between sites i and j, for k=0, 1, , 5 and for several values of the energy E near the center of the band. In the bond case the series are of order p14 in the concentration p(some of those have been formerly available to order p10) and in the site case of order p16. The analysis, using the Dlog-Pade approximation and the techniques known as M1 and M2, shows clear evidence for a delocalization transition (from exponentially localized to extended or power-law-decaying states) at an energy-dependent threshold p q(E) in the range , confirming previous results (e.g. and for bond and site percolation) but in contrast with the Anderson model. The divergence of the series for different kis characterized by a constant gap exponent, which is identified as the localization length exponent from a general scaling assumption. We obtain estimates of . These values violate the bound of Chayes et al.

Surface effects on phase transitions of modulated phases and at Lifshitz points: A mean field theory of the ANNNI model

The semi-infinite axial next nearest neighbor Ising (ANNNI) model in the disordered phase is treated within the molecular field approximation, as a prototype case for surface effects in systems undergoing transitions to both ferromagnetic and modulated phases. As a first step, a discrete set of layerwise mean field equations for the local order parameter mn in the nth layer parallel to the free surface is derived and solved, allowing for a surface field H1 and for interactions JS in the surface plane which differ from the interactions J0 in the bulk, while only in the z-direction perpendicular to the surface competing nearest neighbor ferromagnetic exchange (J1) and next nearest neighbor antiferromagnetic exchange (J 2 ) occurs. We show that for $$\kappa \equiv - {J_2}/{J_1} < {\kappa _L} = 1/4$$ and temperatures in between the critical point of the bulk $$({T_{cb}}(\kappa ))$$ and the disorder line $$({T_{d}}(\kappa ))$$ the decay of the profile is exponential with two competing lengths $$\xi + ,\xi \_$$ with $$\xi + \propto {[T/{T_{cb}}(\kappa ) - 1]^{ - 1/2}}$$ while $$\xi \_$$ stays finite at Tcb. The amplitudes of these exponentials $$\exp ( - na/\xi \pm )$$ (a is the lattice spacing) are obtained from boundary conditions that follow from the molecular field equations. For $$\kappa < {\kappa _L}$$ but $$T > {T_d}(\kappa )$$ , as well as at the Lifshitz point $$(\kappa = {\kappa _L} = 1/4)$$ and in the modulated region $$(\kappa > {\kappa _L})$$ , we obtain a modulated profile $${m_{n + 1}} = A\cos (naq + \psi )$$ , where again the amplitude A and the phase $$\Psi $$ can be found from the boundary conditions. As a further step, replacing differences by differentials we derive a continuum description, where the familiar differential equation in the bulk (which contains both terms of order $${\partial ^2}m/\partial {z^2}$$ and $${\partial ^2}m/\partial {z^4}$$ here) is supplemented by two boundary conditions, which both contain terms up to order $${\partial ^2}m/\partial {z^4}$$ . It is shown that the solution of the continuum theory reproduces the lattice model only when both the leading correlation length ( $${\xi ^ + }$$ or $$\xi $$ , respectively) and the second characteristic length ( $${\xi _ - }$$ or the wavelength of the modulation $$\lambda = 2\pi /q$$ , respectively) are very large. We obtain for $${J_s} > {J_{cs}}(\kappa )$$ a surface transition, with a two-dimensional ferromagnetic order occurring at a transition $${T_{cs}}(\kappa )$$ exceeding the transition of the bulk, and calculate the associated critical exponents within mean field theory. In particular, we show that at the Lifshitz point $${T_{cs}}({\kappa _L}) \propto {({J_s} - {J_{sc}})^{1/\phi L}}$$ with $$\kappa \ne {\kappa _L}$$ while for $$\kappa \ne {\kappa _L}$$ the crossover exponent is $$\phi = 1/2$$ . We also consider the “ordinary transition” $$({J_s} < {J_{sc}}(\kappa ))$$ and obtain the critical exponents and associated critical amplitudes (the latter are often singular when $$\kappa \to {\kappa _L}$$ ). At the Lifshitz point, the exponents of the surface layer and surface susceptibilities take the values $$\gamma _{11}^L = - 1/4,\gamma _1^L = 1/2,\gamma _s^L = 5/4$$ , while from scaling relations the surface “gap exponent” is found to be $$\Delta _1^L = 3/4$$ and the surface order parameter exponents are $$\beta _1^L = 1,\beta _s^l = 1/4$$ . Open questions and possible applications are discussed briefly.

Ground State Properties of One DimensionalS=1/2Heisenberg Model with Dimerization and Quadrumerization

The one dimensional S=1/2 Heisenberg model with dimerization and quadrumerization is studied by means of the numerical exact diagonalization of finite size systems. Using the phenomenological renormalization group and finite size scaling law, the ground state phase diagram is obtained in the isotropic case. It exhibits a variety of the ground states which contains the S=1 Haldane state, S=1 dimer state and S=1/2 dimer state as limiting cases. The gap exponent $\nu$ is also calculated which coincides with the value for the dimerization transition of the isotropic Heisenberg chain. In the XY limit, the phase diagram is obtained analytically and the comparison is made with the isotropic case.