We present a study of the magnetic-field effect on spin correlations in the charge ordered triangular Ising system LuFe2O4+δ through single crystal neutron diffraction. In the absence of a magnetic field, the strong diffuse neutron scattering observed below the Neel temperature (TN = 240 K) indicates that LuFe2O4+δ shows short-range, two-dimensional (2D) correlations in the FeO5 triangular layers, characterized by the development of a magnetic scattering rod along the 1/3 1/3 L direction, persisting down to 5 K. We also found that on top of the 2D correlations, a long range ferromagnetic component associated with the propagation vector k1 = 0 sets in at around 240 K. On the other hand, an external magnetic field applied along the c-axis effectively favours a three-dimensional (3D) spin correlation between the FeO5 bilayers evidenced by the increase of the intensity of satellite reflections with propagation vector k2 = (1/3, 1/3, 3/2). This magnetic modulation is identical to the charge ordered superstructure, highlighting the field-promoted coupling between the spin and charge degrees of freedom. Formation of the 3D spin correlations suppresses both the rod-type diffuse scattering and the k1 component. Simple symmetry-based arguments provide a natural explanation of the observed phenomenon and put forward a possible charge redistribution in the applied magnetic field.
We report thermodynamic and neutron scattering measurements of the triangular-lattice quantum Ising magnet TmMgGaO 4 in longitudinal magnetic fields. Our experiments reveal a quasi-plateau state induced by quantum fluctuations. This state exhibits an unconventional non-monotonic field and temperature dependence of the magnetic order and excitation gap. In the high field regime where the quantum fluctuations are largely suppressed, we observed a disordered state with coherent magnon-like excitations despite the suppression of the spin excitation intensity. Through detailed semi-classical calcul…
Frustrated Ising magnets host exotic excitations, such as magnetic monopoles in spin ice. The ground state (GS) in this case is characterized by an extensive degeneracy and associated residual entropy going back to the pioneering work by G. Wannier who established large residual entropy of nearly 50%Rln2 per mole spins in a triangular Ising antiferromagnet (TIAF) already in 1950. Here, we endeavor to verify this result experimentally using TmMgGaO4, a novel rare-earth-based frustrated antiferromagnet with Ising spins arranged on a perfect triangular lattice. Contrary to theoretical expectation…
We report the single-crystal growth and the fundamental magnetic and thermodynamic properties of a rare-earth triangular lattice antiferromagnet CeCd$_3$As$_3$. In this rare-earth antiferromagnet, the Ce local moments form a perfect triangular lattice. Due to the spin-orbital-entangled nature of the Ce local moments, the compound exhibits extremely anisotropic antiferromagnetic couplings along the c direction and in the ab plane respectively. We show that CeCd$_3$As$_3$ represents a {\it rare} experimental realization of an antiferromagnetic Ising model on a two dimensional triangular lattice …
A mixed Ising-Heisenberg spin system consisting of triangular XXZ-Heisenberg spin clusters assembled into a chain by alternating with Ising spins interacting to all three spins in the triangle is considered. The exact solution of the model is given in terms of the generalized decoration--iteration map and within the transfer-matrix technique. Exact expressions for thermodynamic functions are derived. Ground state phase diagrams, thermodynamic and magnetic properties of the system are examined.
We study the triangular-lattice Ising model with dipolar interactions, inspired by its realisation in artificial arrays of nanomagnets. We show that a classical spin-liquid forms at intermediate temperatures, and that its behaviour can be tuned by temperature and/or a small lattice distortion between a string Luttinger liquid and a domain-wall-network state. At low temperature there is a transition into a magnetically ordered phase, which can be first-order or continous with a crossover in the critical behaviour between Pokrovsky-Talapov and 2D-Ising universality. When the Pokrovsky-Talapov cr…
Phase diagram of an Ising-spin Kondo lattice model on a triangular lattice near 1/3-filling is investigated by Monte Carlo simulation. We identify a partially disordered phase with coexistence of magnetic order and paramagnetic moments, which was unstable in two-dimensional Ising models with localized spins only. The partial disorder emerges in the competing regime between a twosublattice stripe phase and three-sublattice ferrimagnetic phase, at finite temperatures above an electronic phase separation. The peculiar magnetic structure accompanies a charge order and develops a gap in the electro…
We investigate the effects of diamagnetic doping in the solid-solution series $Tb_{1-x}Y_{x}(HCO_{2})_{3}$, in which the parent $Tb(HCO_{2})_{3}$ phase has previously been shown to host a combination of frustrated and quasi-1D physics, giving rise to a triangular Ising antiferromagnetic ground state that lacks long range 3D order. Heat capacity measurements show three key features: (i) a low temperature Schottky anomaly is observed, which is constant as a function of x; (ii) the transition temperature and associated entropy change are both surprisingly robust to diamagnetic doping; and (iii) a…
We present a detailed magnetic study of the triangular antiferromagnet ErMgGaO$_4$. A point charge calculation under the single ion approximation reveals a crystal field ground state doublet with a strong Ising-like behavior of the Er$^{3+}$ moment along the c axis. Magnetic susceptibility and specific heat measurements indicate no presence of magnetic transitions above 0.5~K and no evidence of residual entropy as temperature approaching zero. Zero field (ZF) $μ$SR measurements shows no sign of static uniform or random field and longitudinal field (LF) $μ$SR measurements exhibit persistent spi…
We study multilayer triangular lattice Ising antiferromagnets with interlayer interactions that are weak and frustrated in an abc stacking. By analysing a coupled height model description of these systems, we show that they exhibit a classical spin liquid regime at low temperature, in which both intralayer and interlayer correlations are strong but there is no long range order. Diffuse scattering in this regime is concentrated on a helix in reciprocal space, as observed for charge-ordering in the materials LuFe$_2$O$_4$ and YbFe$_2$O$_4$.
We study the Ising antiferromagnet on the triangular lattice which are interacting along three different directions via two $J$-bonds and one anisotropic $J'$-bond with $J' \ge J$. Although its finite temperature state has long been considered as simply disordered, we find a systematic generation of a ferromagnetic $J'$-bonds named "good defects" each carrying the energy $2δ_J\equiv 2(J'-J)$. They exhibit an eminent correlation which is regarded as a network or a soft lattice. The specific heat shows a universal peak at $T^* \sim δ_J$ and the corresponding energy follows the activation type of…
Low-temperature magnetization curves and thermodynamics of a frustrated spin-1/2 Heisenberg triangular bilayer with the antiferromagnetic intradimer interaction and either ferromagnetic or antiferromagnetic interdimer interaction are investigated in a highly frustrated parameter region, where localized many-magnon eigenstates provide the most dominant contribution to magnetothermodynamics. Low-energy states of the highly frustrated spin-1/2 Heisenberg triangular bilayer can be accordingly found from a mapping correspondence with an effective triangular-lattice spin-1/2 Ising model in a field. …
We study the $S = 1$ triangular lattice Ising antiferromagnet by Monte Carlo simulations. Frustrations between a major antiferromagnetic third-neighbor interaction $J_3$ and a minor ferromagnetic nearest-neighbor interaction $J_1$ cause incommensurate short-range orders at intermediate temperatures. At low temperatures (below $T/J_3 \lesssim 0.2$ for $J_1/J_3 = -1/3$), the system exhibits fourfold periodic ordered state. In the short-range order phase, the system shows a glassy two-step relaxation. We demonstrate that the features of the short-range order are attributed to the cooperation betw…
To gain a better understanding of the interplay between frustrated long-range interactions and zero-temperature quantum fluctuations, we investigate the ground-state phase diagram of the transverse-field Ising model with algebraically-decaying long-range Ising interactions on quasi one-dimensional infinite-cylinder triangular lattices. Technically, we apply various approaches including low- and high-field series expansions. For the classical long-range Ising model, we investigate cylinders with an arbitrary even circumference. We show the occurrence of gapped stripe-ordered phases emerging out…
Using a specially designed Monte Carlo algorithm with directed loops, we investigate the triangular lattice Ising antiferromagnet with coupling beyond nearest neighbour. We show that the first-order transition from the stripe state to the paramagnet can be split, giving rise to an intermediate nematic phase in which algebraic correlations coexist with a broken symmetry. Furthermore, we demonstrate the emergence of several properties of a more topological nature such as fractional edge excitations in the stripe state, the proliferation of double domain walls in the nematic phase, and the Kastel…
We investigate the nature of a $Z_2$-invariant XY ring-exchange interaction with a frustrated Ising coupling on the triangular lattice. In the limit of pure XY ring-exchange interaction, we show that the classical ground state is degenerate resulting from the $Z_2$-invariance of the Hamiltonian. Quantum fluctuations lift these classical degenerate ground states and produce an unusual state whose excitation spectrum exhibits a gapped maximum quadratic dispersion near ${\bf k}=0$ and vanishes at the midpoints of each side of the Brillouin zone. This result is in contrast to a gapless quadratic d…
In this work, we study the magnetization behaviors of the classical Ising model on the triangular lattice using Monte Carlo simulations, and pay particular attention to the effect of further-neighbor interactions. Several fascinating spin states are identified to be stabilized in certain magnetic field regions, respectively, resulting in the magnetization plateaus at 2/3, 5/7, 7/9 and 5/6 of the saturation magnetization MS, in addition to the well known plateaus at 0, 1/3 and 1/2 of MS. The stabilization of these interesting orders can be understood as the consequence of the competition betwee…
Inelastic neutron scattering study on the spin-chain compound Sr$_3$NiIrO$_6$ reveals gapped quasi-1D magnetic excitations. The observed one-magnon band between 29.5 and 39 meV consists of magnon modes of the Ni$^{2+}$ ions. The fitting of the spin wave spectrum reveals strongly coupled Ising-like chains along the $c$-axis that are weakly coupled into a frustrated triangular lattice in the $ab$-plane. The magnetic excitations survive up to 200 K well above the magnetic ordering temperature of $T_N \sim 75$ K, also indicating a quasi-1D nature of the magnetic interactions in Sr$_3$NiIrO$_6$. Ou…
The rare-earth magnet TmMgGaO$_{4}$ is proposed to be an intrinsic quantum Ising magnet described by the antiferromagnetic transverse field Ising model (TFIM) on a triangular lattice, where the relevant degrees of freedom are the non-degenerate dipole-multipole doublets of the Tm$^{3+}$ ions and the transverse field has an intrinsic origin from the weak splitting of the doublet. We compare this special doublet of Tm$^{3+}$ with the dipole-octupole Kramers doublet. We study the proposed effective model for the Tm-based triangular lattice and consider the effects of external magnetic fields and …
Motivated by the experimental realisation of the two-dimensional triangular spin lattice at the National Instate of Standards and Technology in 2012, we developed a new algorithm which allows a time evolution of an MPS with a Hamiltonian with long-range interactions. The algorithm is based on generalised Suzuki-Trotter decompositions with Krylov subspace expansion steps. In the benchmark tests, it predicted the correct local magnetisation behaviour within 5 decimal places up to t\sim 8, it provided the ground state energy of Haldane-Shastry odel correctly up to 4 decimal places for evolution w…
Motivated by recent experiments with a Penning ion trap quantum simulator, we perform numerically exact Stochastic Series Expansion quantum Monte Carlo simulations of long-range transverse-field Ising models on a triangular lattice for different decay powers $α$ of the interactions. The phase boundary for the ferromagnet is obtained as a function of $α$. For antiferromagnetic interactions, there is strong indication that the transverse field stabilizes a clock ordered phase with sublattice magnetization $(M,-\frac{M}{2}, -\frac{M}{2})$ with unsaturated $M < 1$ in a process known as "order by d…
Disordered magnetic states known as spin liquids are of paramount importance in both fundamental and applied science. A classical state of this kind was predicted for the Ising antiferromagnetic triangular model more than half a century ago, while additional non-commuting exchange terms were proposed to induce its quantum version -- a quantum spin liquid. These predictions have not been yet confirmed experimentally. Here we report a discovery of such a state in the structurally perfect triangular-lattice antiferromagnet NdTa$_7$O$_{19}$. Its magnetic ground state is characterized by spin-1/2 d…
Order-by-disorder and long-range interactions in the antiferromagnetic transverse-field Ising model on the triangular lattice -- A perturbative point of view
We study the triangular lattice Ising model with a finite number of vertically stacked layers and demonstrate a low temperature reentrance of two Berezinskii-Kosterlitz-Thouless transitions, which results in an extended disordered regime down to $T=0$. Numerical results are complemented with the derivation of an effective low-temperature dimer theory. Contrary to order by disorder, we present a new scenario for fluctuation-induced ordering in frustrated spin systems. While short-range spin-spin correlations are enhanced by fluctuations, quasi-long-range ordering is precluded at low enough temp…
Field-tuned incommensurate fan phase in an Ising-like triangular antiferromagnet
University of Oxford（牛津大学）；Institute for Solid State Physics, University of Tokyo（东京大学固体物理研究所）；Rutherford Appleton Laboratory（鲁瑟福阿普尔顿实验室）；University of Tennessee（田纳西大学）
We present a Quantum Monte Carlo study of the Ising model in a transverse field on a square lattice with nearest-neighbor antiferromagnetic exchange interaction J and one diagonal second-neighbor interaction $J'$, interpolating between square-lattice ($J'=0$) and triangular-lattice ($J'=J$) limits. At a transverse-field of $B_x=J$, the disorder-line first introduced by Stephenson, where the correlations go from Neel to incommensurate, meets the zero temperature axis at $J'\approx 0.7 J$. Strong evidence is provided that the incommensurate phase at larger $J'$, at finite temperatures, is a floa…
Ca3Co2O6 is a frustrated magnet consisting of a triangular arrangement of chains of Ising spins. It shows regular magnetization steps vs magnetic field every 1.2 T that are metastable with very slow dynamics. This has puzzled the community for many years and given rise to numerous potential theories. Here we approach the problem by seeking the elusive magnetic equilibrium state at T = 2 K. To this end, we explore two approaches: (1) bypassing the slow dynamics produced by changing fields by instead field-cooling directly to the target temperature, and (2) quantum annealing in transverse magnet…
Magnetic and electronic properties of a Kondo lattice model with Ising localized spins are studied on an isotropic triangular lattice. By using Monte Carlo simulation, we present that the model shows a rich phase diagram with four dominant states: two-sublattice stripe, three-sublattice ferrimganetic, partially disordered, and Kosterlitz-Thouless like quasi-long-range ordered states. Among them, the partially disordered state and Kosterlitz-Thouless like state are intermediate phases induced by thermal fluctuations in the phase competing regime; they are present only at finite temperatures and…
Quantum Ising model on a triangular lattice hosts a finite temperature Berezinskii-Kosterlitz-Thouless (BKT) phase with emergent U(1) symmetry, and it will transit into an up-up-down (UUD) phase with $C_3$ symmetry breaking upon an infinitesimal external field along the longitudinal direction, but the overall phase diagram spanned by the axes of external field and temperature remains opaque due to the lack of systematic invesitgations with controlled methodologies. By means of quantum Monte Carlo at finite temperature and ground state density matrix renormalization group simulations, we map ou…
It is known that there is no phase transition down to zero temperature in the antiferromagnetic Ising model on spatially anisotropic triangular lattices, in which the exchange coupling of one direction is stronger than those of other two directions. In the model, the low-temperature physics is governed by domain-wall excitations (defects) residing on bonds of the strong-coupling direction. In this letter, we show that an additional small attractive interaction between defects (a ferromagnetic next-nearest-neighbor interaction in the weak-coupling direction) leads to a Berezinskii-Kosterlitz-Th…
We study Ising antiferromagnets that have nearest-neighbour interactions on multilayer triangular lattices with frustrated ($abc$ and $abab$) stacking, and make comparisons with the unfrustrated ($aaa$) stacking. If interlayer couplings are much weaker than in-plane ones, the paramagnetic phase of models with frustrated stackings has a classical spin-liquid regime at low temperature, in which correlations are strong both within and between planes, but there is no long-range order. We investigate this regime using Monte Carlo simulations and by mapping the spin models to coupled height models, …
The transverse-field Ising model on the triangular lattice is expected to host an intermediate finite-temperature Kosterlitz-Thouless (KT) phase through a mapping of the spins on each triangular unit to a complex order parameter. TmMgGaO$_4$ is a candidate material to realize such physics due to the non-Kramers nature of Tm$^{3+}$ ion and the resulting two-singlet single-ion ground state. Using inelastic neutron scattering, we confirm this picture by determining the leading parameters of the low-energy effective Hamiltonian of TmMgGaO$_4$. Subsequently, we track the predicted KT phase and rela…
We report the neutron scattering results on magnetic orderings and excitations in $\mathrm{Ba}_2\mathrm{CoTeO}_6$ composed of two almost isolated subsystems A and B, which are described as an $S\,{=}\,1/2$ triangular Heisenberg-like antiferromagnet and a frustrated honeycomb Ising-like antiferromagnet, respectively. Stripy ordering of subsystem B was confirmed below $T_{\rm N1}\,{=}\,12.0$ K, whereas sharp streaks were observed along $(1/3, 1/3, L)$ and $(2/3, 2/3, L)$ at 0.3 K (${\ll}\,T_{\rm N2}\,{=}\,3.0$ K). This indicates the two-dimensional nature of ordering in subsystem A. It was found…
Obtaining quantitative ground-state behavior for geometrically-frustrated quantum magnets with long-range interactions is challenging for numerical methods. Here, we demonstrate that the ground states of these systems on two-dimensional lattices can be efficiently obtained using state-of-the-art translation-invariant variants of matrix product states and density-matrix renormalization-group algorithms. We use these methods to calculate the fully-quantitative ground-state phase diagram of the long-range interacting triangular Ising model with a transverse field on 6-leg infinite-length cylinder…
We consider the problem of the doped Ising antiferromagnet on the frustrated triangular lattice in the limit where the hole kinetic energy is much smaller than the Ising exchange. For a {\it single} hole we prove a ``frustrated Nagaoka theorem'' showing that the ground state is magnetized and breaks translational symmetry, in contrast to the parent insulating state that is unmagnetized and spatially homogenous. The extension of this physics to finite dopings depends on the strength of a density-density coupling that is inevitably present - we find either phase separation of the holes, or a sup…
We study a spin-1/2 model with triangular XXZ-clusters on the orthogonal-dimer chain in the presence of an external magnetic field. First, we discuss the case where the triangular clusters are coupled via intermediate "classical" Ising spins. Diagonalization of the triangular XXZ-clusters yields the exact ground states; finite-temperature properties are computed exactly by an additional transfer-matrix step. A detailed analysis reveals a large variety of ground states at magnetization M equal to fractions 0, 1/4, and 1/2 of the saturation magnetization M=1. Some of these ground states break tr…
Geometrical frustration and the enhancement of strong quantum fluctuations in two-dimensional triangular antiferromagnets can lead to various intriguing phenomena. Here, we studied the spin-1/2 triangular lattice antiferromagnet NdAuAl$_4$Ge$_2$. Thermodynamic and transport properties, such as magnetization and specific heat together with the resistivity measurements were performed. In zero field, two successive phase transitions were observed at $T_{\rm N1}=1.75\pm 0.02$ K and $T_{\rm N2}=0.49\pm 0.02$ K, respectively. Under magnetic field, $\rm XXZ$-type anisotropy was revealed, with the mom…
Residual entropy is a key feature associated with emergence in many-body systems. From a variety of frustrated magnets to the onset of spin-charge separation in Hubbard models and fermion-$Z_2$-flux variables in Kitaev models, the freezing of one set of degrees of freedom and establishment of local constraints are marked by a plateau in entropy as a function of temperature. Yet, with the exception of rare-earth pyrochlore family of spin-ice materials, evidence for such plateaus is rarely seen in real materials, raising questions about their robustness. Following recent experimental findings of…
Frustrated magnets host the promises of material realizations of new paradigm of quantum matter, while direct comparison of unbiased model calculations with experimental measurements is still very challenging. Here, we design and implement a protocol of employing many-body computation methodologies for accurate model calculation -- both equilibrium and dynamical properties -- of a frustrated rare-earth magnet TmMgGaO$_4$ (TMGO), which perfectly explains the corresponding experimental findings. Our results confirm TMGO is an ideal realization of triangular-lattice Ising model with an intrinsic …
The recently fabricated two-dimensional magnetic materials Cu9X2(cpa)6.xH2O (cpa=2-carboxypentonic acid; X=F,Cl,Br) have copper sites which form a triangular kagome lattice (TKL), formed by introducing small triangles (``a-trimers'') inside of each kagome triangle (``b-trimer''). We show that in the limit where spins residing on b-trimers have Ising character, quantum fluctuations of XXZ spins residing on the a-trimers can be exactly accounted for in the absence of applied field. This is accomplished through a mapping to the kagome Ising model, for which exact analytic solutions exist. We deri…
A transverse magnetic field $Γ$ is known to induce antiferromagnetic three-sublattice order of the Ising spins $σ^z$ in the triangular lattice Ising antiferromagnet at low enough temperature. This low-temperature order is known to melt on heating in a two-step manner, with a power-law ordered intermediate temperature phase characterized by power-law correlations at the three-sublattice wavevector ${\bf Q}$: $\langle σ^z(\vec{R}) σ^z(0)\rangle \sim \cos({\mathbf Q}\cdot \vec{R}) /|\vec{R}|^{η(T)}$ with the temperature-dependent power-law exponent $η(T) \in (1/9,1/4)$. Here, we use a newly devel…
The spin-half Heisenberg antiferromagnet (HAF) on the square and triangular lattices is studied using the coupled cluster method (CCM) technique of quantum many-body theory. The phase relations between different expansion coefficients of the ground-state wave function in an Ising basis for the square lattice HAF is exactly known via the Marshall-Peierls sign rule, although no equivalent sign rule has yet been obtained for the triangular lattice HAF. Here the CCM is used to give accurate estimates for the Ising-expansion coefficients for these systems, and CCM results are noted to be fully cons…
The anisotropic Heisenberg model close to the Ising limit: triangular lattice vs. effective models
Max Planck Institute for the Physics of Complex Systems（马克斯·普朗克复杂系统物理研究所）；Faculty of Civil and Geodetic Engineering, University of Ljubljana（卢布尔雅那大学土木与测量工程学院）；Jožef Stefan Institute（约瑟夫·斯特凡研究所）；Department of Applied Physics, Tokyo University of Science（东京科学大学应用物理系）
We identify and discuss the ground state of a quantum magnet on a triangular lattice with bond-dependent Ising-type spin couplings, that is, a triangular analog of the Kitaev honeycomb model. The classical ground-state manifold of the model is spanned by decoupled Ising-type chains, and its accidental degeneracy is due to the frustrated nature of the anisotropic spin couplings. We show how this subextensive degeneracy is lifted by a quantum order-by-disorder mechanism and study the quantum selection of the ground state by treating short-wavelength fluctuations within the linked cluster expansi…
Geometrical frustration in spin systems often results in a large number of degenerate ground states. In this work we study the antiferromagnetic Ising model on the three dimensional swedenborgite lattice which is a specific stacking of Kagomé and triangular layers. The model contains two exchange couplings, one within the Kagomé layer, another one in between Kagomé and triangular layers. We determine the phase diagram with and without easy axis magnetic field and calculate the ground state degeneracy explicitly in terms of the residual entropy. At zero field we find two different ground state …
The $S=1$ bilinear-biquadratic Heisenberg exchange model on the triangular lattice with a single-ion anisotropy has previously been shown to host a number of exotic magnetic and nematic orders [Moreno-Cardoner $\textit{et al.}$, Phys. Rev. B $\textbf{90}$, 144409 (2014)], including an extensive region of "supersolid" order. In this work, we amend the model by an XXZ anisotropy in the exchange interactions. Tuning to the limit of an exactly solvable $S=1$ generalized Ising-/Blume-Capel-type model provides a controlled limit to access phases at finite transverse exchange. Notably, we find an add…
We study the transverse field Ising model on a kagome and a triangular lattice using high-order series expansions about the high-field limit. For the triangular lattice our results confirm a second-order quantum phase transition in the 3d XY universality class. Our findings for the kagome lattice indicate a notable instance of a disorder by disorder scenario in two dimensions. The latter follows from a combined analysis of the elementary gap in the high- and low-field limit which is shown to stay finite for all fields h. Furthermore, the lowest one-particle dispersion for the kagome lattice is…
We propose a new quantum model interpolating between the fully frustrated spin-1/2 Ising model in a transverse field and a dimer model. This model contains a resonating-valence-bond phase, including a line with an exactly solvable ground state of the Rokhsar-Kivelson type. We discuss the phase diagrams of this model on the square and triangular (in terms of the dimer representation) lattices.
Motivated by the current interest in the quantum dimer model on the triangular lattice, we investigate the phase diagram of the closely related fully-frustrated transverse field Ising model on the honeycomb lattice using classical and semi-classical approximations. We show that, in addition to the fully polarized phase at large field, the classical model possesses a multitude of phases that break the translational symmetry which in the dimer language, correspond to a plaquette phase and a columnar phase separated by an infinite cascade of mixed phases. The modification of the phase diagram by …
