Resolving the two-dimensional ANNNI model using transfer matrices Yi Hu1 and Patrick Charbonneau1, 2, ∗ 1Department of Chemistry, Duke University, Durham, North Carolina 27708, USA 2Department of Physics, Duke University, Durham, North Carolina 27708, USA (Dated: December 18, 2020) The phase diagram of the two-dimensional ANNNI model has long been theoretically debated. Extremely long structural correlations and relaxation times further result in numerical simulations making contradictory predictions. Here, we introduce a numerical transfer matrix treatment that bypasses these problems, and thus overcome various ambiguities in the phase diagram. In particular, we confirm the transition temperatures and clarify the order of the transition to the floating incom- mensurate phase. Our approach motivates considering transfer matrices for resolving long-standing problems in related statistical physics models. I. INTRODUCTION matrix (TM) approach should be able to resolve all of these issues.29,30 This approach indeed provides ex- act solutions of semi-infinite systems, which can then Patterned and modulated phases robustly form when be extrapolated to the thermodynamic limit by finite- the components of a system interact via competing short- size scaling. Because both computational and memory range attractive and long-range repulsive (SALR) inter- complexity grow exponentially with system size, how- actions.1{4 Such phases have indeed been observed in ever, the accessible size range has long been too nar- materials ranging from magnetic alloys,5,6 to lipidic sur- row for physical insight to emerge from such an analy- factants,7,8 and biological tissues.9,10 Frustration, how- sis. Thanks to dramatic improvements in methodology, ever, is also associated with slowly decaying finite-size computer hardware and eigensolvers31 the TM approach corrections and to complex relaxation processes, which has recently been applied to more complex (quasi) one- both severely impede the study of equilibrium phases by dimensional continuum-space systems, including SALR numerical simulations. Specialized sampling techniques models with up to third-nearest-neighbor interaction,32 are thus needed to study even minimal microphase- and hard spheres in cylindrical confinement up to next- formers,11{14 let along more realistic ones. nearest-neighbor interaction.33{35 For two-dimensional Lattice models with SALR interactions were first for- lattice models with frustration, sufficiently large sys- mulated forty years ago, the simplest being the ax- tems have also recently become accessible to the TM 15 ial next-nearest-neighbor Ising (ANNNI) model. Yet approach to determine transition temperatures on the re- even in two dimensions, this small perturbation to the 36 lated J1−J2 model. In this article, we push the effective Ising model makes an analytical solution out of reach. use of the TM formalism to resolve various physical am- Different approximation methods have thus been at- biguities of the somewhat more complex ANNNI model. 16 tempted, including Hamiltonian limit, free fermion ap- In particular, we determine the phase boundaries for the 17 18 proximation, high temperature series expansion, clus- floating IC phase, and critically assess proposals for the 19 20,21 ter variational method and others. Although the Lifshitz point and the IC phase reentrance. physics of both the small frustration regime and the en- ergetic ground states have long been resolved, the finite temperature-strong frustration regime has not. Between II. TRANSFER MATRIX APPROACH the standard high-temperature paramagnetic phase and low-temperature modulated antiphase, a floating incom- The ANNNI model Hamiltonian for spin variables s = mensurate (IC) phase intercalates. From field-theory, i ±1 reads this critical phase is expected to be of the Kosterlitz- Thouless (KT) type, and thus to belong to the XY uni- X X X HANNNI = −J1 sisj + J2 sisj − J0 si; (1) versality class.22,23 Numerical validation, however, has hi;ji [i;j] i remained elusive, as has whether the IC phase persists axial 21 at large frustrations or disappears at a Lifshitz point. where the coupling constant J = J1 > 0, the frus- arXiv:2012.09747v1 [cond-mat.stat-mech] 17 Dec 2020 Both high and low transition temperatures (Tc1 and Tc2, tration along the axial next-nearest-neighbor direction respectively) are indeed challenging to determine in sim- κ = J2=J1 > 0 and the external field h = J0=J are 12,24{28 ulations (see Ref. 12). The proposed reentrance scaled. For κ = 0, the model reduces to the standard 19,20 of the IC phase around the multiphase point also Ising model; for the T = 0 ground state, ferromagnetic remains to be confirmed. Because these features are cen- order dominates until κ < 1=2, and the periodic an- tral to our understanding of the floating IC phase in mi- tiphase (with periodicity h2i) takes overs for κ > 1=2. crophase formers, resolving these questions is particularly Note that in lattice-gas representation, this model corre- important. sponds to SALR interacting particles, and h then plays As was recognized already in the mid-1980s, a transfer- the role of an effective chemical potential. The ANNNI 2 model is thus clearly a minimal model for layered mi- III. PHASE DIAGRAM FOR h = 0 crophases. The finite-temperature, finite-frustration phase behav- We first consider results from the ?TM route (Fig. 1). ior of semi-infinite strips is obtained by a TM approach For κ < 1=2, the energy curves for different L robustly with each layer s having L spins s1; s2; :::; sL. (Setting cross at well-defined critical point Tc(κ). For the Ising, sL+1 ≡ s1 imposes periodic boundary conditions.) Be- 39,40 κ = 0, limit u(Tc) is perfectly invariant with L, while cause the interaction in the ANNNI model is anisotropic, for 0 < κ 1=2, small systems, L 10, exhibit a correc- 29 . the TM can be propagated either perpendicular (?TM) tion of at most 0:1%. From this identification of T (κ) 30 c or parallel (kTM) to the axial next-nearest-neighbor in- we confirm that the peak of c grows logarithmically with teraction direction. Both matrices can be decomposed L, as expected for the Ising universality class. into intra-layer, T , and inter-layer, T , contributions, x y For κ & 1=2 a markedly different behavior is observed. 27 1 1 A pronounced step in u(T ) gives rise to a sharp c peak. 2 2 T = Tx TyTx : (2) At first glance, these features might suggest a simple first- order transition, in contrast to the Pokrovsky-Talapov In TM, row and column indices correspond to neigh- ? scenario,24 but the single step height scales as 1=L (not boring layer configurations s and s0, shown) and is thus projected to vanish in the thermody- 8 h i namic, 1=L ! 0, limit. Furthermore, a second peak ap- T (s) = exp J PL (s s − κs s + hs ) ; <? x i=1 i i+1 i i+2 i pears for L ≥ 24, a third one for L ≥ 32, and it is reason- 0 PL 0 :?Ty(s; s ) = exp J i=1 sisi ; able to expect that more such peaks eventually do. This (3) behavior is related to the stepwise change to the modula- L L tion block Nmod from the h2i antiphase (Fig. 1(c)). This which makes ?T a 2 × 2 symmetric dense matrix. In TM, row and column indices correspond to two subse- change in modulation has long been considered a finite- k size echo of the thermodynamic floating IC phase.41 quent layers fs; s0g and fs0; s00g, respectively, and then Although the transition temperatures identified by the step-wise steps and heat capacity peaks shift with L, 8 h i 0 PL 0 finite-size results clearly suggest an exponential scaling, <kTx(s; s ) = exp J (sisi+1 + sisi + hsi) ; i=1 c ∼ exp(aL)=L (up to c ≥ 103) (Fig. 1(d)), and thus a L (4) 00 P 00 42 :kTy(s; s ) = exp −κJ i=1 sisi ; first-order transition. At finite L, the modulation takes up available fractions of the system size, and as 1=L ! 0, L L which makes kT a 4 × 4 non-symmetric sparse matrix infinite commensurate phases are separated by infinites- with 8L nonzero entries. imal temperature intervals. As a result the system re- In both cases the leading eigenvalue, λ0, provides the mains critical everywhere, which is a hallmark of the free energy per spin, f = − log λ0=(βL), and the product floating IC phase. The transition at Tc2 being discontin- of left and right leading eigenvectors, P (s) = '−1(s)'(s), uous (rather than critical), as was proposed in Ref. 27, provides the equilibrium probability of a layer config- is thus here confirmed. Extrapolating the temperature uration. Equilibrium configurations can thus be effi- of the first peak using a quadratic form further gives 37 ciently planted. Thermal properties can be obtained Tc2(κ = 0:6) = 0:90(1), which is fully consistent with the by taking partial derivatives of f, e.g., the energy u = most recent simulations.28 The second lowest transition is 2 −kBT @(βf)=@T and specific heat c = @u=@T per spin. projected to merge with the first as 1=L ! 0. The mod- The leading correlation length can also be obtained from ulation wavenumber q = Nmod=L thus seemingly shifts 0 the spectrum gap, ξ1 = 1= log(λ0=jλ1j), albeit only along from q0 = 1=4 to some q < q0, resulting in an abrupt the direction of layer propagation. Hence, although the change in q at Tc2. Admittedly, the alternative scenario compactness of ?TM brings larger L within computa- that q could non-smoothly yet continuously change at Tc2 tional reach, the kTM geometry is more informative cannot be excluded, but the first-order transition (discon- about the modulation direction, which is of greater phys- tinuity in u) proposal appears marginally more consistent ical interest.