Axial Next-nearest Neighbour Ising Research Articles

A simple theory for quantum quenches in the ANNNI model

In a recent numerical study by Haldar et al. [Phys. Rev. X 11, 031062] it was shown that signatures of proximate quantum critical points can be observed at early and intermediate times after certain quantum quenches. Said work focused mainly on the case of the axial next-nearest neighbour Ising (ANNNI) model. Here we construct a simple time-dependent mean-field theory that allows us to obtain a quantitatively accurate description of these quenches at short times, which for reasons we explain remains a fair approximation at late times (with some caveats). Our approach provides a simple framework for understanding the reported numerical results as well as fundamental limitations on detecting quantum critical points through quench dynamics. We moreover explain the origin of the peculiar oscillatory behaviour seen in various observables as arising from the formation of a long-lived bound state.

Communication: Weakening the critical dynamical slowing down of models with SALR interactions.

In systems with frustration, the critical slowing down of the dynamics severely impedes the numerical study of phase transitions for even the simplest of lattice models. In order to help sidestep the gelation-like sluggishness, a clearer understanding of the underlying physics is needed. Here, we first obtain generic insight into that phenomenon by studying one-dimensional and Bethe lattice versions of a schematic frustrated model, the axial next-nearest neighbor Ising (ANNNI) model. Based on these findings, we formulate two cluster algorithms that speed up the simulations of the ANNNI model on a 2D square lattice. Although these schemes do not eliminate the critical slowing own, speed-ups of factors up to 40 are achieved in some regimes.

Ising-like models for stacking faults in a free electron metal.

We propose an extension of the axial next nearest neighbour Ising (ANNNI) model to a general number of interactions between spins. We apply this to the calculation of stacking fault energies in magnesium—particularly challenging due to the long-ranged screening of the pseudopotential by the free electron gas. We employ both density functional theory (DFT) using highest possible precision, and generalized pseudopotential theory (GPT) in the form of an analytic, long ranged, oscillating pair potential. At the level of first neighbours, the Ising model is reasonably accurate, but higher order terms are required. In fact, our ‘ ANNNI model’ is slow to converge—an inevitable feature of the free electron-like electronic structure. In consequence, the convergence and internal consistency of the ANNNI model is problematic within the most precise implementation of DFT. The GPT shows the convergence and internal consistency of the DFT bandstructure approach with electron temperature, but does not lead to loss of precision. The GPT is as accurate as a full implementation of DFT but carries the additional benefit that damping of the oscillations in the ANNNI model parameters are achieved without entailing error in stacking fault energies. We trace this to the logarithmic singularity of the Lindhard function.

First Principles Investigation on Thermodynamic Properties and Stacking Fault Energy of Paramagnetic Nickel at High Temperatures

Reliable data on the temperature dependence of thermodynamic properties of alloy phases are very useful for modeling the behavior of high-temperature materials such as nickel-based superalloys. Moreover, for predicting the mechanical properties of such alloys, additional information on the energy of lattice defects (e.g., stacking faults) at high temperatures is highly desirable, but difficult to obtain experimentally. In this study, we use first-principles calculations, in conjunction with a quasi-harmonic Debye model, to evaluate the Helmholtz free energy of paramagnetic nickel as a function of temperature and volume, taking into account the electronic, magnetic, and vibrational contributions. The thermodynamic properties of Ni, such as the equilibrium lattice parameter and elastic moduli, are derived from the free energy in the temperature range from 800 to 1600 K and compared with available experimental data. The derived temperature dependence of the lattice parameter is then used for calculating the energies of intrinsic and extrinsic stacking faults in paramagnetic Ni. The stacking fault energies have been evaluated according to three different methodologies, the axial-next-nearest-neighbor Ising (ANNNI) model, the tilted supercell approach, and the slab supercell approach. The results show that the elastic moduli and stacking fault energies of Ni decrease with increasing temperature. This “softening” effect of temperature on the mechanical properties of nickel is mainly due to thermal expansion, and partly due to magnetic free energy contribution.

Unveiling phase transitions with machine learning

The classification of phase transitions is a central and challenging task in condensed matter physics. Typically, it relies on the identification of order parameters and the analysis of singularities in the free energy and its derivatives. Here, we propose an alternative framework to identify quantum phase transitions, employing both unsupervised and supervised machine learning techniques. Using the axial next-nearest neighbor Ising (ANNNI) model as a benchmark, we show how unsupervised learning can detect three phases (ferromagnetic, paramagnetic, and a cluster of the antiphase with the floating phase) as well as two distinct regions within the paramagnetic phase. Employing supervised learning we show that transfer learning becomes possible: a machine trained only with nearest-neighbour interactions can learn to identify a new type of phase occurring when next-nearest-neighbour interactions are introduced. All our results rely on few and low dimensional input data (up to twelve lattice sites), thus providing a computational friendly and general framework for the study of phase transitions in many-body systems.

Shape-preserving machining produces gradient nanolaminate medium entropy alloys with high strain hardening capability

A high density of grain boundaries can potentially increase structural materials' strength, but at the expense of losing the materials' strain hardening ability at high flow stress levels. However, endowing materials with grain size gradients and a high density of internal interfaces can simultaneously increase the strength and strain hardening ability. This applies particularly for through-thickness gradients of nanoscale interface structures. Here we apply a machining method that produces metals with nanoscale interface gradients. Conventional bulk plastic deformation such as rolling, a process applied annually to about 2 billion tons of material, aims to reduce the metal thickness. We have modified this process by introducing severe strain path changes, realized by leading the sheet through a U-turn while preserving its shape, an approach known as ‘hard turning’. We applied this process at both room temperature and 77 K to a NiCrCo medium entropy alloy. Micropillar compression was conducted to evaluate the mechanical response. After hard turning at room temperature, the surface microstructure obtained a ∼50% increase in yield stress (0.9 GPa) over the original state with homogeneous grain size (0.4 GPa), but the initial strain hardening rate did not show significant improvement. However, after hard turning at 77 k, the gradient nanolaminate structure tripled in yield stress and more than doubled its initial strain hardening rate. The improvements were achieved by introducing a specific microstructure that consists of gradient nanolaminates in the form of nanospaced twins and martensite in the face center cubic (fcc) phase. This microstructure was formed only at cryogenic temperature. It was found after turning at room temperature that only nanospaced twins were present in the fcc phase inside nanolaminates that had formed at the surface. The origin of the enhanced strain hardening mechanism was studied. Joint density functional theory (DFT) and axial next nearest neighbor Ising (ANNNI) models were used to explain the temperature-dependent phase formation of the NiCrCo nanolaminate at the surface of the hard-turned material.

Coexistence of magnetic fluctuations and long-range order in the one-dimensional J1−J2 zigzag chain materials BaDy2O4 and BaHo2O4

The compounds BaDy$_2$O$_4$ and BaHo$_2$O$_4$ are part of a family of frustrated systems exhibiting interesting properties, including spin liquid-type ground states, magnetic field-induced phases, and the coexistence of short- and long-range magnetic orders, with dominant one-dimensional correlations, which can be described as Ising $J_1-J_2$ zigzag chains along the $c$-axis. We have investigated polycrystalline samples of BaDy$_2$O$_4$ and BaHo$_2$O$_4$ with both neutron diffraction and neutron spectroscopy, coupled to detailed crystalline electric field calculations. The latter points to site-dependent anisotropic magnetism in both materials, which is corroborated by the magnetic structures we determined. The two systems show the coexistence of two different long-range orders --- two double N\'eel $\uparrow\uparrow\downarrow\downarrow$ orders in the $ab$-plane with propagation vectors $\mathbf{k}_1$ = ($\frac{1}{2}$,0,$\frac{1}{2}$) and $\mathbf{k}_2$ = ($\frac{1}{2}$,$\frac{1}{2}$,$\frac{1}{2}$) for BaDy$_2$O$_4$, and two distinct arrangements of simple N\'eel $\uparrow\downarrow\uparrow\downarrow$ orders along the $c$-axis, both with the propagation vector $\mathbf{k}_0$ = (0,0,0) for BaHo$_2$O$_4$. The order for both wave vectors in BaDy$_2$O$_4$ occurs at $T_\mathrm{N}$ = 0.48 K, while in BaHo$_2$O$_4$, the first order sets in at $T_\mathrm{N}\sim$ 1.3 K and the second one has a lower ordering temperature of 0.84 K. Both compounds show extensive diffuse scattering which we successfully modeled with a one-dimensional axial next-nearest neighbor Ising (ANNNI) model. In both materials, strong diffusive scattering persists to temperatures well below where the magnetic order is fully saturated. The ANNNI model fits indicate the presence of sites which do not order with moments in the $ab$-plane.

An Interpretation for Defect-Induced Structural Transformation in SiC

Silicon carbide (SiC) has been considered as one of the alternative power semiconductor materials due to its excellent properties such as wide band gap, high break down field, and high thermal conductivity. Although SiC with 3C (zinc blende) is suitable for high-frequency power devices from its high electron mobility and high electron-saturation-velocity, SiC is well known as having dozens of polytypes including 4H and 6H in addition to 2H (wurtzite) and 3C that consist of different ordered stacking sequences of SiC atomic double layers. The 3C-SiC is believed to be more stable than the hexagonal structure at low temperatures. In contrast, 2H-SiC, which has the simplest stacking sequence, is rarely observed at high temperatures. Moreover, point defects also affect the stable polytype in SiC. Heavily Nitrogen (N)-doped single crystals of 6H-SiC were completely transformed into 3C-SiC by annealing in vacuum at presence of Si vapor at 2080 K and 2180 K (hereafter Case I). The 3C-SiC-like bands was also observed in 4H-SiC epilayers with heavily N-doping after thermal annealing (hereafter Case II). Presuming that vacancy formation is related to temperature, point defects such as vacancy and N in SiC are crucial factor for the structural transformation between hexagonal and cubic structures. In this study, the structural transformation in SiC is simply interpreted by considering point defect formation with the aid of ab initio calculations. Ab initio total energy calculations in a √3×√3×12 unit cell imply that stable stacking for SiC is 6H with Si-vacancy (VSi), 4H with C-vacancy (VC) and N substituting for Si (NSi), and 3C with N substituting for C (NC), whereas 2H-stacking does not appear as a stable stacking in these systems. Furthermore, the formation energy calculations reveal that vacancy formation depends on C chemical potential m C, where VSi and VC formations are favorable at large m C (corresponding to C-rich) and small m C (Si-rich), respectively. On the other hand, NC formation is more favorable than NSi formation over the entire range of m C and independent of conditions because of very small difference in covalent radius between N and C. Comparing these calculated results with the experimental finding in the Case I for transformation from 6H to 3C, thermal treatment with presence of Si vapor at high temperatures suppresses VSi formation favoring 6H-stacking while it enhances VC formation to be occupied by N (NC) in addition to N substitution for C (NC) that transform 6H to 3C. The Case II can be also interpreted by considering NC. Thermal annealing for N-doped 4H-SiC favored by VC easily produces NC with N occupying VC and substituting for C to realize 3C-SiC bands at high N-doped region in 4H-SiC epilayers. These results suggest that NC is crucial for the transformation from hexagonal stacking such as 6H and 4H to cubic stacking 3C. In order to give physical interpretation for the transformation to 3C-SiC with NC, the interaction energies Jn between nth neighbor SiC double layers are estimated in the range of n ≤ 3 using the axial next nearest neighbor Ising (ANNNI) model and calculated total energies for SiC polytypes obtained in this study. The calculated results elucidate that the absolute value of positive J 1 (51.70 meV) with J 3 (9.92 meV) favoring 3C-stacking is much larger than that of negative J 2 (-16.66 meV) inducing hexagonal stacking. It should be noted that the Jn values are one order of magnitude greater than those of perfect SiC such as J 1 (2.33 meV), J 2 (-1.41 meV), and J 3 (0.51 meV). This implies that the layer interaction in SiC becomes strong and long-range due to N-doping. Considering coordinate of (|J 2|/J 1, J 3/J 1) in the ANNNI phase diagram, it is also found that (0.31, 0.19) for SiC with NC is located in the stable region of 3C and is more distant from the multi-phase degeneracy point (0.5, 0) inducing polytype and phase boundary between 3C and 4H than (0.61, 0.21) for perfect SiC. This clarifies that the large J 1 in SiC with NC stabilizes 3C-stacking to dominate the structural transformation to 3C due to N-doping. Furthermore, atomistic interaction between the nearest neighbor layers corresponding to J 1 is also qualitatively discussed in terms of electrostatic interaction between bond charges located at the center of interatomic bonds and that between ionic charges located at the lattice sites on the basis of a simple energy description and charge distribution obtained by ab initio calculations.

Stability of multiple Shockley type basal plane stacking faults in heavily nitrogen-doped 4H-SiC crystals

The stability of various types of multiple Shockley type basal plane stacking fault in heavily nitrogen-doped 4H-SiC crystals was theoretically investigated on the basis of the quantum well action (QWA) mechanism. The energy levels of confined electrons within multiple Shockley stacking faults (SSFs) were calculated using the simple quantum well model and compared to the experimental values estimated from photoluminescence measurements. The formation energies of the stacking faults were also calculated on the basis of the axial next nearest neighbor Ising (ANNNI) model for SiC polytypes. Based on these physical and electronic parameters, the stability of multiple SSFs in heavily nitrogen-doped 4H-SiC crystals was discussed, and it was revealed that the stacking sequence of multiple SSFs largely affects their stability; SSFs having a cubic stacking sequence of medium thickness (five to six Si-C bilayer thickness) are stable in heavily nitrogen-doped 4H-SiC crystals at the typical growth and device processing temperatures of SiC, i.e., 1500–2600 K.

Field dependence of the magnetic correlations of the frustrated magnetSrDy2O4

The frustrated magnet SrDy$_2$O$_4$ exhibits a field-induced phase with a magnetization plateau at $1/3$ of the saturation value for magnetic fields applied along the $b$-axis. We report here a neutron scattering study of the nature and symmetry of the magnetic order in this field-induced phase. Below $T\approx 0.5$ K, there are strong hysteretic effects, and the order is short or long ranged for zero-field and field cooling, respectively. We find that the long-range ordered magnetic structure within the zig-zag chains is identical to that expected for the one-dimensional axial next-nearest neighbour Ising (ANNNI) model in longitudinal field. The long-range ordered structure in field contrasts with the short-range order found at zero field, and is probably reached through enhanced quantum fluctuations with increasing fields.

Absence of long-range order in the frustrated magnet SrDy2O4 due to trapped defects from a dimensionality crossover

Magnetic frustration and low dimensionality can prevent long range magnetic order and lead to exotic correlated ground states. SrDy$_2$O$_4$ consists of magnetic Dy$^{3+}$ ions forming magnetically frustrated zig-zag chains along the c-axis and shows no long range order to temperatures as low as $T=60$ mK. We carried out neutron scattering and AC magnetic susceptibility measurements using powder and single crystals of SrDy$_2$O$_4$. Diffuse neutron scattering indicates strong one-dimensional (1D) magnetic correlations along the chain direction that can be qualitatively accounted for by the axial next-nearest neighbour Ising (ANNNI) model with nearest-neighbor and next-nearest-neighbor exchange $J_1=0.3$ meV and $J_2=0.2$ meV, respectively. Three-dimensional (3D) correlations become important below $T^*\approx0.7$ K. At $T=60$ mK, the short range correlations are characterized by a putative propagation vector $\textbf{k}_{1/2}=(0,\frac{1}{2},\frac{1}{2})$. We argue that the absence of long range order arises from the presence of slowly decaying 1D domain walls that are trapped due to 3D correlations. This stabilizes a low-temperature phase without long range magnetic order, but with well-ordered chain segments separated by slowly-moving domain walls.

Modulated phases and devil’s staircases in a layered mean-field version of the ANNNI model

We investigate the phase diagram of a spin-1/2 Ising model on a cubic lattice, with competing interactions between nearest and next-nearest neighbors along an axial direction, and fully connected spins on the sites of each perpendicular layer. The problem is formulated in terms of a set of noninteracting Ising chains in a position-dependent field. At low temperatures, as in the standard mean-field version of the Axial-Next-Nearest-Neighbor Ising (ANNNI) model, there are many distinct spatially commensurate phases that spring from a multiphase point of infinitely degenerate ground states. As temperature increases, we confirm the existence of a branching mechanism associated with the onset of higher-order commensurate phases. We check that the ferromagnetic phase undergoes a first-order transition to the modulated phases. Depending on a parameter of competition, the wave number of the striped patterns locks in rational values, giving rise to a devil’s staircase. We numerically calculate the Hausdorff dimension D0 associated with these fractal structures, and show that D0 increases with temperature but seems to reach a limiting value smaller than D0=1.

The partition function zeros of the anisotropic Ising model with multisite interactions on a zigzag ladder

Abstract It is shown that the spin- 1 2 anisotropic Ising model with multisite interactions on a zigzag ladder may be mapped into the one dimensional spin- 1 2 Axial-Next-Nearest-Neighbor Ising (ANNNI) model with multisite interactions. The partition function zeros of the ANNNI model with multisite interactions are investigated. A comprehensive analysis of the partition function zeros of the ANNNI model with and without three-site interactions on a zigzag ladder is done using the transfer matrix method. Analytical equations for the distribution of the partition function zeros in the complex magnetic field (Yang–Lee zeros) and temperature (Fisher zeros) planes are derived. The Yang–Lee and Fisher zeros distributions are studied numerically for a variety of values of the model parameters. The densities of the Yang–Lee and Fisher zeros are studied and the corresponding edge singularity exponents are calculated. It is shown that the introduction of three-site interaction terms in the ANNNI model leads to a simpler distribution of the partition function zeros. For example, the Yang–Lee zeros tend to a circular distribution when increasing by modulus the three-site interactions term coefficient. It is found that the Yang–Lee and Fisher edge singularity exponents are universal and equal to each other, σ = − 1 2 .

Phase diagram of the three-dimensional axial next-nearest-neighbor Ising model

The three-dimensional axial-next-nearest-neighbor Ising (ANNNI) model is studied by a modified tensor product variational approach (TPVA). A global phase diagram is constructed with numerous commensurate and incommensurate magnetic structures. The devil's stairs behavior for the model is confirmed. The wavelength of the spin modulated phases increases to infinity at the boundary with the ferromagnetic phase. Widths of the commensurate phases are considerably narrower than those calculated by mean-field approximations.

Stacking faults in3C−, 4H−,and6H−SiCpolytypes investigated by anab initiosupercell method

Recent attempts to make SiC diodes have revealed a problem with stacking fault expansion in the material, leading to unstable devices. In this paper, we present detailed results from a density-functional supercell calculation on the electronic structure of stacking faults which result from glide of Shockley partials in $3C\ensuremath{-}, 4H\ensuremath{-}$ and $6H\ensuremath{-}\mathrm{SiC}.$ It was found [Phys. Rev. B 65, 033203 (2002)] that both types of stacking faults in $4H\ensuremath{-}\mathrm{SiC}$ and two types of stacking faults in $6H\ensuremath{-}\mathrm{SiC}$ give rise to band states, which are strongly localized (confined within around 10 \AA{}) in the direction orthogonal to the stacking fault plane. Based on estimates of the band offsets between different polytypes and a simple quantum-well theory, we show that it is possible to interpret this one-dimensional localization as a quantum-well confinement effect. We also find that the third type of stacking fault in $6H\ensuremath{-}\mathrm{SiC}$ and the only stacking fault in $3C\ensuremath{-}\mathrm{SiC}$ do not give rise to states clearly separated from the band edges, but instead give rise to rather strongly localized band states with energies very close to the band edges. We argue that these localized near band edge states are created by stacking fault induced changes in the dipole moment associated with the hexagonal symmetry. In addition, we have also calculated the stacking fault energies, using both the supercell method and the simpler ANNNI (axial next nearest-neighbor Ising) model. Both theories agree well with the low stacking fault energies found experimentally.

Gapless line for the anisotropic Heisenbergspin−12chain in a magnetic field and the quantum axial next-nearest-neighbor Ising chain

We study the anisotropic Heisenberg (XYZ) spin-1/2 chain placed in a magnetic field pointing along the x-axis. We use bosonization and a renormalization group analysis to show that the model has a non-trivial fixed point at a certain value of the XY anisotropy a and the magnetic field h. Hence, there is a line of critical points in the (a,h) plane on which the system is gapless, even though the Hamiltonian has no continuous symmetry. The quantum critical line corresponds to a spin-flop transition; it separates two gapped phases in one of which the Z_2 symmetry of the Hamiltonian is broken. Our study has a bearing on one of the transitions of the axial next-nearest neighbor Ising (ANNNI) chain in a transverse magnetic field. We also discuss the properties of the model when the magnetic field is increased further, in particular, the disorder line on which the ground state is a direct product of single spin states.

Ising thin films with modulations and surface defects

Properties of magnetic films are studied in the framework of Ising models. In particular, we discuss critical phenomena of ferromagnetic Ising films with straight lines of magnetic adatoms and straight steps on the surface as well as phase diagrams of the axial next–nearest neighbour Ising (ANNNI) model for thin films exhibiting various spatially modulated phases.

"Devil's staircase"-type phase transition in NaV2O5 under high pressure.

The "devil's staircase"-type phase transition in the quarter-filled spin-ladder compound NaV2O5 has been discovered at low temperature and high pressure by synchrotron radiation x-ray diffraction. A large number of transitions are found to successively take place among higher-order commensurate phases with 2a x 2b x zc type superstructures. The observed temperature and pressure dependence of modulation wave number q(c), defined by 1/z, is well reproduced by the axial next nearest neighbor Ising model. The q(c) is suggested to reflect atomic displacements presumably coupled with charge ordering in this system.

Spatially modulated magnetic structures in thin films

The axial next-nearest neighbour Ising (ANNNI) model is studied for thin films of up to $L= 10$ layers, with a distinct phase diagram for each film thickness. The systematics of the ordered phases, as obtained from mean-field theory, Monte Carlo simulations, and low temperature expansions, is discussed. Results are compared to those for the ANNNI model in the limit $L \longrightarrow \infty$.

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.