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In this paper, we investigate the dynamical quantum phase transitions appearing in the Loschmidt echo and the time-dependent order parameter of a quantum system of harmonically coupled degenerate bosons as a function of the power-law decay $\sigma$ of long-range interactions. Following a sudden quench, the nonequilibrium dynamics of this system are governed by a set of nonlinear coupled Ermakov equations. To solve them, we develop an analytical approximation valid at late times. Based on this approximation, we show that the emergence of a dynamical quantum phase transition hinges on the generation of a finite mass gap following the quench, starting from a massless initial state. In general, we can define two distinct dynamical phases characterized by the finiteness of the post-quench mass gap. The Loschmidt echo exhibits periodical nonanalytic cusps whenever the initial state has a vanishing mass gap and the final state has a finite mass gap. These cusps are shown to coincide with the maxima of the time-dependent long-range correlations., Comment: 10 pages, 5 figures

Nontrivial band topology combined with magnetic order can lead to rich emergent phenomena, including quantized anomalous Hall effect and axion insulator state. Here we use scanning tunneling microscopy to image the surface Dirac fermions of the newly discovered magnetic topological insulators $\mathrm{Mn}{\mathrm{Bi}}_{2}{\mathrm{Te}}_{4}$ and $\mathrm{Mn}{\mathrm{Bi}}_{4}{\mathrm{Te}}_{7}$. We have determined the energy dispersion and helical spin texture of the surface states through quasiparticle interference patterns far above Dirac energy. Approaching the Dirac point, the native defects in the $\mathrm{Mn}{\mathrm{Bi}}_{2}{\mathrm{Te}}_{4}$ septuple layer give rise to resonance states which extend spatially and potentially hinder the detection of a mass gap in the spectra. Our results impose tight constraints on the magnitude of the possible mass gap at the nanoscale and provide key ingredients for a comprehensive understanding of the electronic structure in this class of fascinating materials.

Describing the experimental signatures of quantum spin ice has been the focus of many theoretical efforts, as definitive experimental verification of this candidate quantum spin liquid is yet to be achieved. Gapped excitations known as visons have largely eluded those efforts. We provide a theoretical framework, which captures their dynamics and predicts new signatures in the magnetic response. We achieve this by studying the ring-exchange Hamiltonian of quantum spin ice in the large-$s$ approximation, taking into account the compact nature of the emergent U(1) gauge theory. We find the stationary solutions of the action---the instantons---which correspond to visons tunneling between lattice sites. By integrating out the instantons, we calculate the effective vison Hamiltonian, including their mass. We show that in the ground state virtual vison pairs simply renormalize the speed of light and give rise to an inelastic continuum of excitations. At low temperatures, however, thermally activated visons form a Debye plasma and introduce a mass gap in the photon spectrum, equal to the plasma frequency, which we calculate as a function of temperature. We demonstrate that this dynamical mass gap should be visible in energy-resolved neutron scattering spectra but not in the energy-integrated ones. We also show that it leads to the vanishing of the susceptibility of an isolated system, through a mechanism analogous to the Meissner effect, but that it does not lead to confinement of static spinons.

Recently, Haldane's conjecture about SU(2) chains has been generalized to SU($n$) chains in the symmetric representations. For a rank-$p$ representation, a gapless phase is predicted when $p$ and $n$ are coprime; otherwise, a finite energy gap is present above the ground state. In this work, we offer an intuitive explanation of this behavior based on fractional topological excitations, which are able to generate a mass gap except when $p$ and $n$ have no common divisor. This is a generalization of an older work in SU(2), which explains the generation of the Haldane gap in terms of merons in the O(3) nonlinear sigma model., 11 pages + 2 pages of appendices

Motivated by the recent experimental observation of the half-quantized thermal Hall conductivity for the candidate material of the Kitaev spin liquid, $\alpha$-RuCl$_3$[ Y. Kasahara et al., Nature 559, 227 (2018)], we investigate effects of non-Kitaev exchange interactions, which exist in the real material, on the gapped chiral spin liquid state realized in the case with an applied magnetic field. It is found that off-diagonal exchange interactions enhance significantly the mass gap of Majorana fermions. This result provides a possible explanation for robust quantization of the thermal Hall conductivity observed in the above-mentioned experiment. Furthermore, we demonstrate that if the Kitaev spin liquid state and the zigzag antiferromagnetic order coexist around the border of these two phases, off-diagonal exchange interactions can induce Fermi surfaces of Majorana fermions, leading to a state similar to the U(1) spin liquid., Comment: 9 pages, 7 figures, some clarifications and references are added

We study the effect of symmetry-breaking perturbations in the one-dimensional $\mathrm{SU}(4)$ spin-orbital model. We allow the exchange in spin ${(J}_{1})$ and orbital ${(J}_{2})$ channel to be different and thus reduce the symmetry to $\mathrm{SU}(2)\ensuremath{\bigotimes}\mathrm{SU}(2).$ A magnetic field $h$ along the ${S}^{z}$ direction is also applied. Using the formalism developed by Azaria et al. [Phys. Rev. Lett. $83,$ 624 (1999)] we extend their analysis of the isotropic ${J}_{1}{=J}_{2},$ $h=0$ case and obtain the low-energy effective theory near the $\mathrm{SU}(4)$ point in the generic case ${J}_{1}\ensuremath{\ne}{J}_{2},$ $h\ensuremath{\ne}0.$ In zero magnetic field, we retrieve the same qualitative low-energy physics as in the isotropic case. In particular, the isotropic massless behavior found on the line ${J}_{1}{=J}_{2}lK/4$ extends in a large anisotropic region. We discover, however, that the anisotropy plays its trick in allowing nontrivial scaling behaviors of the physical quantities. For example, the mass gap $M$ has two different scaling behaviors depending on the anisotropy. In addition, we show that in some regions, the anisotropy is responsible for anomalous finite-size effects and may change qualitatively the shape of the computed critical line in a finite system. When a magnetic field is present the effect of the anisotropy is striking. In addition to the usual commensurate-incommensurate phase transition that occurs in the spin sector of the theory, we find that the field may induce a second transition of the KT type in the remaining degrees of freedom to which it does not couple directly. In this sector, we find that the effective theory is that of an SO(4) Gross-Neveu model with an $h\ensuremath{-}\mathrm{dependent}$ coupling that may change its sign as $h$ varies.

It has been proposed that the effective Hamiltonian describing high ${T}_{c}$ superconductivity in cuprate materials has an approximate SO(5) symmetry relating the superconducting (SC) and antiferromagnetic (AF) phases of these systems. We show that robust consequences of this proposal are potentially large optical conductivities and Raman scattering rates in the AF phase, due to the electromagnetic response of the doubly charged pseudo Goldstone bosons which must exist there. This provides potentially strong constraints on the properties of the bosons, such as their mass gap and velocity, at any given value of the doping.

The effect of a magnetic field on the charged vacuum is investigated. The field dependence of the energy levels causes jumps in the total vacuum charge that occur whenever an energy level crosses the Fermi level, and this leads to reentrant cycles of vacuum charging and discharging. In atomic systems these effects require astrophysical magnetic fields of around ${10}^{8}$ T, but in graphene with a mass gap they occur in laboratory fields of about 1 T or lower. It is suggested that an electrostatic graphene quantum dot defined by a gate electrode provides a solid state model of the as yet unobserved charged vacuum as well as a model of an atomic system in an extreme astrophysical environment. Phase diagrams are computed to show how the total vacuum charge depends on the confining potential strength and applied magnetic field. The vacuum charge density is also investigated and experimental consequences are discussed.

We theoretically study the electromagnetic interaction in Dirac systems with $N$ nodes by using the renormalization group, which is relevant to the quantum critical phenomena of topological phase transition ($N=1$) and Weyl semimetals ($N=4$ or $N=12$). Compared with the previous work for $N=1$ [H. Isobe and N. Nagaosa, Phys. Rev. B 86, 165127 (2012)], we obtained the analytic solution for the large $N$ limit, which differs qualitatively for the scaling of the speed of light $c$ and that of electron $v$; i.e., $v$ does not change while $c$ is reduced to $v$. We also found a reasonably accurate approximate analytic solution for generic $N$, which well interpolates between $N=1$ and large $N$ limit, and it concludes that ${c}^{2}{v}^{N}$ is almost unrenormalized. The temperature dependence of the physical properties, the dielectric constant, magnetic susceptibility, spectral function, DC conductivity, and mass gap are discussed based on these results.

We predict a tunable and nonadiabatic Berry phase effect of Dirac fermions, which is an electronic analog of the Pancharatnam phase of polarized light. The Berry phase occurs as a scattering phase shift in a single scattering event of transmission or reflection of Dirac fermions at a junction with a spatially nonuniform mass gap, unveiling the topological aspects of scattering of chiral Dirac fermions. This geometric phase plays different roles in solids as compared with the Pancharatnam phase in optics. It provides a unique approach of detecting the Chern number of the insulator side in a metal-insulator junction of Dirac fermions, implying a different type of bulk-edge correspondence at the boundary between a metal and an insulator. This phase also modifies the quantization rule of Dirac fermions, suggesting geometric-phase devices with nontrivial charge and spin transport such as a topological waveguide and a topological transistor.

For a topological antiferromagnet on a square lattice, with the standard Hartree-Fock, spin-density-wave decoupling of the on-site Hubbard interaction, there is an exact mapping of the low-energy one-electron excitation spectrum to a relativistic Dirac continuum field theory. In this field theory, the Dirac mass gap is precisely the Mott-Hubbard charge gap and the continuum field variable is an eight-component Dirac spinor describing the components of physical electron-spin amplitude on each of the four sites of the elementary plaquette in the original Hubbard model. Within this continuum model we derive explicitly the existence of hedgehog Skyrmion textures as local minima of the classical magnetic energy. These magnetic solitons carry a topological winding number [mu] associated with the vortex rotation of the background magnetic moment field by a phase angle 2[pi][mu] along a path encircling the soliton. Such solitons also carry a spin flux of [mu][pi] through the plaquette on which they are centered. The [mu]=1 hedgehog Skyrmion describes a local transition from the topological (antiperiodic) sector of the one-electron Hilbert space to the nontopological sector. We derive from first principles the existence of deep level localized electronic states within the Mott-Hubbard charge gap for the [mu]=1 and 2 solitons. The spectrummore » of localized states is symmetric about [ital E]=0 and each subgap electronic level can be occupied by a pair of electrons in which one electron resides primarily on one sublattice and the second electron on the other sublattice. It is suggested that flux-carrying solitons and the subgap electronic structure which they induce are important in understanding the physical behavior of doped Mott insulators.« less

We investigate the thermal Casimir interaction of a suspended graphene described by the Dirac model with a plate made of dielectric or metallic materials. The reflection coefficients on graphene expressed in terms of a temperature-dependent polarization tensor are used. We demonstrate that for a graphene with nonzero mass gap parameter the Casimir free energy remains nearly constant (and the thermal correction negligibly small) over some temperature interval. For the interaction of graphene with metallic plate, the free energy is nearly the same, irrespective of whether the metal is nonmagnetic or magnetic and whether it is described using the Drude- or plasma-model approaches. The free energy computed using the Dirac model was compared with that computed using the hydrodynamic model of graphene and big differences accessible for experimental observation have been found. For dielectric and nonmagnetic metallic plates described by the Drude model these differences vanish with increasing temperature (separation). However, for nonmagnetic metals described by the plasma model and for magnetic metals, a severe dependence on the chosen theoretical description of graphene remains even at high temperature. In all cases the analytic asymptotic expressions for the free energy at high temperature are obtained and found in a very good agreement with the results of numerical computations., Comment: 26 pages, 12 figures

The amplitude mode is a ubiquitous collective excitation in condensed matter systems with broken continuous symmetry. It is expected in antiferromagnets, short coherence length superconductors, charge density waves, and lattice Bose condensates. Its detection is a valuable test of the corresponding field theory, and its mass gap measures the proximity to a quantum critical point. However, since the amplitude mode can decay into low-energy Goldstone modes, its experimental visibility has been questioned. Here we show that the visibility depends on the symmetry of the measured susceptibility. The longitudinal susceptibility diverges at low frequency as \chi_{\sigma\sigma} ~ i/\omega (d=2) or log(1/|\omega|) (d=3), which can completely obscure the amplitude peak. In contrast, the scalar susceptibility is suppressed by four extra powers of frequency, exposing the amplitude peak throughout the ordered phase. We discuss experimental setups for measuring the scalar susceptibility. The conductivity of the O(2) theory (relativistic superfluid) is a scalar response and therefore exhibits suppressed absorption below the Higgs mass threshold, \sigma ~ \omega^{2d+1}. In layered, short coherence length superconductors, (relevant e.g. to cuprates) this threshold is raised by the interlayer plasma frequency., Comment: 17 pages, 9 figures

We present results from lattice simulations of a monolayer graphene model at non-zero temperature. At low temperatures for sufficiently strong coupling the model develops an excitonic condensate of particle-hole pairs corresponding to an insulating phase. The Berezinskii-Kosterlitz-Thouless phase transition temperature is associated with the value of the coupling where the critical exponent delta governing the response of the order parameter at criticality to an external source has a value close to 15. The critical coupling on a lattice with temporal extent N_t=32 (T=1/(N_t a_t) where a_t is the temporal lattice spacing) and spatial extent N_s=64 is very close to infinite coupling. The value of the transition temperature normalized with the zero temperature fermion mass gap Delta_0 is given by T_BKT/Delta_0=0.055(2). This value provides an upper bound on the transition temperature, because simulations closer to the continuum limit where the full U(4) symmetry is restored may result in an even lower value. In addition, we measured the helicity modulus Upsilon and the fermion thermal mass Delta_T(T), the later providing evidence for a pseudogap phase with Delta_T>0 extending to arbitrarily high T., Comment: 20 pages, 12 figures, version accepted by Phys. Rev. B

The $O(3)$ nonlinear $\sigma$ model is studied in the disordered phase, using the techniques of the effective action and finite temperature field theory. The nonlinear constraint is implemented through a Lagrange multiplier. The finite temperature effective potential for this multiplier is calculated at one loop. The existence of a nontrivial minimum for this potential is the signal of a disordered phase in which the lowest excited state is a massive triplet. The mass gap is easily calculated as a function of temperature in dimensions 1, 2 and 3. In dimension 1, this gap is known as the Haldane gap, and its temperature dependence is compared with experimental results., Comment: 10 pages (RevTeX 3.0), 3 PostScript figures (errors fixed) appended, preprint CRPS-92-01

We consider the 2D Hubbard model on the honeycomb lattice, as a model for single layer graphene with screened Coulomb interactions; at half filling and weak coupling, we construct its ground state correlations by a convergent multiscale expansion, rigorously excluding the presence of magnetic or superconducting instabilities or the formation of a mass gap. The Fermi velocity, which can be written in terms of a convergent series expansion, remains close to its non-interacting value and turns out to be isotropic. On the contrary, the interaction produces an asymmetry between the two components of the charge velocity, in contrast with the predictions based on relativistic or continuum approximations., 4 pages, 1 figure; version published on Phys. Rev. B; erratum added

The spin-1/2 and spin-1 Heisenberg antiferromagnets on a square lattice are studied via series expansions around the Ising limit. Series are calculated for the ground-state energy, staggered magnetization, transverse susceptibility, staggered parallel susceptibility, and mass gap. Extrapolating these series to the isotropic limit, we find extremely good agreement with the predictions of spin-wave theory.

Motivated by recent experiments and a theoretical analysis of the gap equation for the propagator of Dirac quasiparticles, we assume that the physics underlying the recently observed removal of sublattice and spin degeneracies in graphene in a strong magnetic field is connected with the generation of both Dirac masses and spin gaps. The consequences of such a scenario for the existence of the gapless edge states with zigzag and armchair boundary conditions are discussed. In the case of graphene on a half-plane with a zigzag edge, there are gapless edge states in the spectrum only when the spin gap dominates over the mass gap. In the case of an armchair edge, however, the existence of the gapless edge states depends on the specific type of mass gaps., 14 pages, 7 figures

We have performed Monte Carlo simulations for the three-dimensional Ising model. Using histogram techniques, we calculate the density of states on ${L}^{3}$ block lattices up to size L=14. Statistical jackknife methods are employed to perform a thorough error analysis. We obtain high-precision estimates for the leading zeros of the partition function, which, using finite-size scaling, translate into \ensuremath{\nu}=0.6285\ifmmode\pm\else\textpm\fi{}0.0019. Along a different line of approach following recent work in lattice-gauge theories, we accurately determine the mass gap m=1/\ensuremath{\xi} (\ensuremath{\xi} correlation length) for cylindrical ${L}^{2}$${L}_{z}$ lattices (with ${L}_{z}$=256 and L up to 12). The finite-size-scaling analysis of the mass-gap data leads to \ensuremath{\nu}=0.6321\ifmmode\pm\else\textpm\fi{}0.0019.

We study the problem of a quantized elastic string in the presence of an impenetrable wall. This is a two-dimensional field theory of an N-component real scalar field $\phi$ which becomes interacting through the restriction that the magnitude of $\phi$ is less than $\phi_{\rm max}$, for a spherical wall of radius $\phi_{\rm max}$. The N=1 case is a string vibrating in a plane between two straight walls. We review a simple nonperturbative argument that there is a gap in the spectrum, with asymptotically-free behavior in the coupling (which is the reciprocal of $\phi_{\rm max}$) for N greater than or equal to one. This scaling behavior of the mass gap has been disputed in some of the recent literature. We find, however, that perturbation theory and the 1/N expansion each confirms that these theories are asymptotically free. The large N limit coincides with that of the O(N) nonlinear sigma model. A theta parameter exists for the N=2 model, which describes a string confined to the interior of a cylinder of radius $\phi_{\rm max}$., Comment: Text slightly improved, bibilography corrected, more typos corrected, still Latex 7 pages

We derive an effective polaronic interaction Hamiltonian, {\it exact to second order in perturbation}, for the spinless one-dimensional Holstein model. The small parameter is given by the ratio of the hopping term ($t$) to the polaronic energy ($g^2 \omega_0$) in all the region of validity for our perturbation; however, the exception being the regime of extreme anti-adiabaticity ($t/\omega_0 \le 0.1$) and small electron-phonon coupling ($g < 1$) where the small parameter is $t/\omega_0$. We map our polaronic Hamiltonian onto a next-to-nearest-neighbor interaction anisotropic Heisenberg spin model. By studying the mass gap and the power-law exponent of the spin-spin correlation function for our Heisenberg spin model, we analyze the Luttinger liquid to charge-density-wave transition at half-filling in the effective polaronic Hamiltonian. We calculate the structure factor at all fillings and find that the spin-spin correlation length decreases as one deviates from half-filling. We also extend our derivation of polaronic Hamiltonian to $d$-dimensions., Comment: Content changed. Accepted in Phys. Rev. B

We have studied the antiferromagnetic Ising chain in a transverse magnetic field $h_{x}$ and uniform longitudinal field $h_{z}$. Using the density matrix renormalization group calculation combined with a finite-size scaling the ground state phase diagram in ($h_{x},h_{z}$) plane is determined. It is shown that there is an order-disordered transition line in this plane and the critical properties belong to the universality class of the two-dimensional Ising model. Based on the perturbation theory in $h_{z}$ the scaling behavior of the mass gap in the vicinity of the critical point ($h_{x}=1/2,h_{z}=0$) is established. It is found that the form of the transition line near the classical multicritical point ($h_{x}=0,h_{z}=1$) is linear. The connection of the considered quantum model with the quasi-one-dimensional classical Ising model in the magnetic field is discussed., Comment: 15 pages, 6 figures

In this paper we propose an efficient routine to carry out finite size scaling studies of one-dimensional massive lattice models. Unlike massless systems, for which the lattice and continuum models yield the same results for low-lying excitations due to the infinite correlation length, the massive continuum models can at best approximate their lattice counterparts because of the intrinsic length scale xi similar to Delta -1, where Delta is the mass gap. On examples of antiferromagnetic gapped spin-1/2 XXZ chains we show explicitly that several relations between physical quantities like the mass gap, spin-wave velocity upsilon, and the correlation length xi, derived from the continuum models deviate significantly from the numerical results obtained for finite chains, when xi is comparable with the lattice spacing ao. On the other hand, we find if the dispersion for elementary excitations is modified from the "Lorentz invariant" form root Delta (2)+upsilon (2)p(2) to root Delta (2)+upsilon (2)sub(2)p, the spin-wave velocity upsilon becomes almost size independent, and the appropriately defined correlation length using that dispersion agrees very well with numerical results. In fact, this follows from the Bethe ansatz solution for the spin-1/2 XXZ chains. Based on our previous experience and these considerations we propose scaling equations to obtain the a-state energy, mass gap, correlation length, spin-wave velocity and scattering length between the massive elementary excitations. Only three low energy levels are needed in this method. Although the method is illustrated on the gapped XXZ quantum spins, it is applicable to all one-dimensional lattice models with massive relativistic low energy dispersions. To substantiate our statement we show explicitly that the numerical data for the Ising model in a transverse magnetic field fully agree with the finite-size scaling based on the correlation length newly defined from the exact analytic solution.

Neutron-scattering measurements on the one-dimensional gapped $S=1$ antiferromagnet, ${\mathrm{CsNiCl}}_{3},$ have shown that excitation corresponding to the Haldane mass gap $\ensuremath{\Delta}$ at low temperatures persists as a resonant feature to high temperatures. We find that the strong upward renormalization of the gap excitation, by a factor of three between 5 and 70 K, is more than enough to overcome its decreasing lifetime. We find that the gap lifetime is substantially shorter than that predicted by the scaling theory of Damle and Sachdev based on a classical dispersion for the Haldane excitations, but when we include their expected relativistic dispersion, the theory, in its low-temperature range of validity, gives a good account of experiment. The upward gap renormalization agrees with the nonlinear sigma model at low temperatures and even up to T of order $2\ensuremath{\Delta}$ provided an upper momentum cutoff is included.

Highly precise numerical calculations using a variation of the Lanczos method devised by Paige were performed for s=1/2 antiferromagnetic Heisenberg finite chains up to sizes N=28. Several interesting physical quantities, including the ground-state energy, the mass gap, and the spin-wave velocity, were computed and fitted for logarithmic finite-size corrections, as suggested by conformal invariance and Bethe ansatz calculations. A crossover size for a maximum of the spin-wave velocity is predicted for N\ensuremath{\sim}50, well above the current available sizes in a typical Lanczos simulation. A list of our numerical results is given along with extrapolated values using standard algorithms.

The influence of boundary conditions on the weak coupling behavior of the mass gap of the (1+1)-dimensional Ising model on a finite chain is analyzed by random walk and perturbation theoretic arguments. The behavior with lattice size found in recent numerical calculations is recovered.

Charge fluctuations of solitons with arbitrary fractional fermion number in 1 + 1 dimensions are calculated, generalizing the result of Kivelson and Schrieffer for solitons with fermion number \textonehalf{}. The soliton charge is measured by a sampling function $f(x)$ such that $f(x)\ensuremath{\simeq}1$ over a region of width $L$ around the soliton and then falls to zero in a distance $l$. It is shown that vacuum fluctuations vanish as ${l}^{\ensuremath{-}1}$ for large $l$ while the additional fluctuations due to the presence of a soliton vanish as either $\mathrm{exp}(\ensuremath{-}\frac{L}{\ensuremath{\xi}})$ or $\mathrm{exp}(\ensuremath{-}2{\ensuremath{\Delta}}_{0}L)$; $\ensuremath{\xi}$ is the soliton width and ${\ensuremath{\Delta}}_{0}$ is the mass gap of a ground state. This result establishes that the fractional fermion charge is a well-defined observable.

We consider the effects of repulsive electron-electron interactions between electrons of like spin in conducting polymers. Because the ratio W/2m of the physical bandwidth W to the band gap 2m is large in these systems, they may be represented by a field-theory model with a cutoff. When only interactions between like spins are included, the resulting model is equivalent to the massive Thirring model which has been exactly solved by Bethe's ansatz. From the exact solution we are able to conclude that perturbation theory in the interaction strength g is quite accurate so long as (g/..pi..)ln(W/2m) or approx. =1, results for the ground-state energy and mass gap are sensitive to the band structure far from the Fermi surface, and no simple field-theory model is adequate.

Under an external staggering field, \ensuremath{\epsilon}${\mathcal{J}}_{n}$${(\mathrm{\ensuremath{-}}1)}^{n}$${s}_{n}^{z}$, or a dimerizing interaction, \ensuremath{\beta}${\mathcal{J}}_{n}$${\mathrm{s}}_{n}$\ensuremath{\cdot}${\mathrm{s}}_{n+1}$, a mass gap opens up according to m\ensuremath{\sim}${\ensuremath{\epsilon}}^{\ensuremath{\eta}}$ or m\ensuremath{\sim}${\ensuremath{\beta}}^{\ensuremath{\nu}}$. Spronken, Fourcade, and L\'epine have shown that if logarithmic corrections are taken into account the finite-size-scaling result for \ensuremath{\nu} agrees with the theoretical prediction of (2/3). In this paper we show that the finite-size-scaling result for \ensuremath{\eta} is 0.720\ifmmode\pm\else\textpm\fi{}0.005 when logarithmic corrections are accounted for. This is in disagreement with the theoretical prediction of (2/3). Possible explanations for this discrepancy are discussed.