Classical Ground State Research Articles

Emergent criticality and Friedan scaling in a two-dimensional frustrated Heisenberg antiferromagnet

We study a two-dimensional frustrated Heisenberg antiferromagnet on the windmill lattice consisting of triangular and dual honeycomb lattice sites. In the classical ground state the spins on different sublattices are decoupled, but quantum and thermal fluctuations drive the system into a coplanar state via an "order from disorder" mechanism. We obtain the finite temperature phase diagram using renormalization group approaches. In the coplanar regime, the relative U$(1)$ phase between the spins on the two sublattices decouples from the remaining degrees of freedom, and is described by a six-state clock model with an emergent critical phase. At lower temperatures the system enters a $\mathbb{Z}_6$ broken phase with long-range phase correlations. We derive these results by two distinct renormalization group approaches to two-dimensional magnetism: by Wilson-Polyakov scaling and by Friedan's geometric approach to nonlinear sigma models where the scaling of the spin-stiffnesses is governed by the Ricci flow of a 4D metric tensor.

Quantum theory of spin waves in finite chiral spin chains

We calculate the effect of spin waves on the properties of finite size spin chains with a chiral spin ground state observed on bi-atomic Fe chains deposited on Iridium(001). The system is described with a Heisenberg model supplemented with a Dzyaloshinskii-Moriya (DM) coupling and a uniaxial single ion anisotropy that presents a chiral spin ground state. Spin waves are studied using the Holstein-Primakoff (HP) boson representation of spin operators. Both the renormalized ground state and the elementary excitations are found by means of Bogoliubov transformation, as a function of the two variables that can be controlled experimentally, the applied magnetic field and the chain length. Three main results are found. First, because of the non-collinear nature of the classical ground state, there is a significant zero point reduction of the ground state magnetization of the spin spiral. Second, the two lowest energy spin waves are edge modes in the spin spiral state that, above a critical field the results into a collinear ferromagnetic ground state, become confined bulk modes. Third, in the spin spiral state, the spin wave spectrum exhibits oscillatory behavior as function of the chain length with the same period of the spin helix.

Designer spin systems via inverse statistical mechanics. II. Ground-state enumeration and classification

In the first paper of this series (DiStasio, Jr., Marcotte, Car, Stillinger, and Torquato, Phys. Rev. B 88, 134104 (2013)), we applied inverse statistical-mechanical techniques to study the extent to which targeted spin configurations on the square lattice can be ground states of finite-ranged radial spin-spin interactions. In this sequel, we enumerate all of the spin configurations within a unit cell on the one-dimensional integer lattice and the two-dimensional square lattice up to some modest size under periodic boundary conditions. We then classify these spin configurations into those that can or cannot be unique classical ground states of the aforementioned radial pair spin interactions and found the relative occurrences of these ground-state solution classes for different system sizes. As a result, we also determined the minimal radial extent of the spin-spin interaction potentials required to stabilize those configurations whose ground states are either unique or degenerate (i.e., those sharing the same radial spin-spin correlation function). This enumeration study has established that unique ground states are not limited to simple target configurations. However, we also found that many simple target spin configurations cannot be unique ground states.

Frustrated spin-12Heisenberg antiferromagnet on a chevron-square lattice

The coupled cluster method (CCM) is used to study the zero-temperature properties of a frustrated spin-half ($s={1}{2}$) $J_{1}$--$J_{2}$ Heisenberg antiferromagnet (HAF) on a 2D chevron-square lattice. Each site on an underlying square lattice has 4 nearest-neighbor exchange bonds of strength $J_{1}>0$ and 2 next-nearest-neighbor (diagonal) bonds of strength $J_{2} \equiv x J_{1}>0$, with each square plaquette having only one diagonal bond. The diagonal bonds form a chevron pattern, and the model thus interpolates smoothly between 2D HAFs on the square ($x=0$) and triangular ($x=1$) lattices, and also extrapolates to disconnected 1D HAF chains ($x \to \infty$). The classical ($s \to \infty$) version of the model has N\'{e}el order for $0 < x < x_{{\rm cl}}$ and a form of spiral order for $x_{{\rm cl}} < x < \infty$, where $x_{{\rm cl}} = {1}{2}$. For the $s={1}{2}$ model we use both these classical states, as well as other collinear states not realized as classical ground-state (GS) phases, as CCM reference states, on top of which the multispin-flip configurations resulting from quantum fluctuations are incorporated in a systematic truncation scheme, which we carry out to high orders and extrapolate to the physical limit. We calculate the GS energy, GS magnetic order parameter, and the susceptibilities of the states to various forms of valence-bond crystalline (VBC) order, including plaquette and two different dimer forms. We find that the $s={1}{2}$ model has two quantum critical points, at $x_{c_{1}} \approx 0.72(1)$ and $x_{c_{2}} \approx 1.5(1)$, with N\'{e}el order for $0 < x < x_{c_{1}}$, a form of spiral order for $x_{c_{1}} < x < x_{c_{2}}$ that includes the correct three-sublattice $120^{\circ}$ spin ordering for the triangular-lattice HAF at $x=1$, and parallel-dimer VBC order for $x_{c_{2}} < x < \infty$.

Probing the limitations of isotropic pair potentials to produce ground-state structural extremes via inverse statistical mechanics

Inverse statistical-mechanical methods have recently been employed to design optimized short-range radial (isotropic) pair potentials that robustly produce novel targeted classical ground-state many-particle configurations. The target structures considered in those studies were low-coordinated crystals with a high degree of symmetry. In this paper, we further test the fundamental limitations of radial pair potentials by targeting crystal structures with appreciably less symmetry, including those in which the particles have different local structural environments. These challenging target configurations demanded that we modify previous inverse optimization techniques. In particular, we first find local minima of a candidate enthalpy surface and determine the enthalpy difference ΔH between such inherent structures and the target structure. Then we determine the lowest positive eigenvalue λ(0) of the Hessian matrix of the enthalpy surface at the target configuration. Finally, we maximize λ(0)ΔH so that the target structure is both locally stable and globally stable with respect to the inherent structures. Using this modified optimization technique, we have designed short-range radial pair potentials that stabilize the two-dimensional kagome crystal, the rectangular kagome crystal, and rectangular lattices, as well as the three-dimensional structure of the CaF(2) crystal inhabited by a single-particle species. We verify our results by cooling liquid configurations to absolute zero temperature via simulated annealing and ensuring that such states have stable phonon spectra. Except for the rectangular kagome structure, all of the target structures can be stabilized with monotonic repulsive potentials. Our work demonstrates that single-component systems with short-range radial pair potentials can counterintuitively self-assemble into crystal ground states with low symmetry and different local structural environments. Finally, we present general principles that offer guidance in determining whether certain target structures can be achieved as ground states by radial pair potentials.

Spin line groups

Spin line groups describe the symmetries of spin arrangements in quasi-one-dimensional systems. These groups are derived for the first family of line groups. Among them, magnetic groups are singled out as a special case. Spin arrangements generated by the derived groups are first discussed for single-orbit systems and then the conclusions are extended to multi-orbit cases. The results are illustrated by the examples of a CuO2 zigzag chain, a (13)C nanotube and the hexaferrite Ba2Mg2Fe12O22. Applications to neutron diffraction and classical ground-state determination are indicated.

Cluster Luttinger Liquids of Rydberg-Dressed Atoms in Optical Lattices

We investigate the zero-temperature phases of bosonic and fermionic gases confined to one dimension and interacting via a class of finite-range soft-shoulder potentials (i.e., soft-core potentials with an additional hard-core onsite interaction). Using a combination of analytical and numerical methods, we demonstrate the stabilization of critical quantum liquids with qualitatively new features with respect to the Tomonaga-Luttinger liquid paradigm. These features result from frustration and cluster formation in the corresponding classical ground state. Characteristic signatures of these liquids are accessible in state-of-the-art experimental setups with Rydberg-dressed ground-state atoms trapped in optical lattices.

Phase diagram of the spin-1 Heisenberg model with three-site interactions on the square lattice

We study the spin S=1 antiferromagnetic Heisenberg model on the square lattice with, in addition to the nearest-neighbor interaction, a three-site interaction of the form (S_i S_j)*(S_j S_k) + h.c.. This interaction appears naturally in a strong coupling exansion of the two-orbital, half-filled Hubbard model. For spin 1/2, this model reduces to a Heisenberg model with bilinear interactions up to third neighbors, with a second-neighbor interaction twice as large the third-neighbor one, a very frustrated model with an infinite family of helical classical ground states in a large parameter range. Using a variety of analytical and numerical methods, we show that the spin-1 case is also very frustrated, and that its phase diagram is even richer, with possibly the succession of seven different phases as a function of the ratio of the three-site interaction to the bilinear one. The phases are either purely magnetic phases with collinear order, or of mixed magnetic and quadrupolar character with helical order.

Physical Invariant Constitutive Equation for Soft Tissues

Principal axis formulations are regularly used in isotropic elasticity but they are not often used in dealing with anisotropic problems. In this paper, based on a principal axis technique, we develop a physical invariant constitutive equation for incompressible transversely isotropic solids, where it contains only a one variable (general) function. The corresponding strain energy function depends on four invariants that have immediate physical interpretation. These invariants are useful in facilitating an experiment to obtain a specific constitutive equation for a particular type of materials. The explicit appearance of the classical ground state constants in the constitutive equation simplifies the calculation for their admissible values. A specific constitutive model is proposed for soft tissues and the model fits reasonably well with existing experimental data; it is also able to accurately predict experiment data.

Nonplanar ground states of frustrated antiferromagnets on an octahedral lattice

We consider methods to identify the classical ground state for an exchange-coupled Heisenberg antiferromagnet on a non-Bravais lattice with interactions $J_{ij}$ to several neighbor distances. Here we apply this to the unusual "octahedral" lattice in which spins sit on the edge midpoints of a simple cubic lattice. Our approach is informed by the eigenvectors of $J_{ij}$ with largest eigenvalues. We discovered two families of non-coplanar states: (i) two kinds of commensurate state with cubic symmetry, each having twelve sublattices with spins pointing in (1,1,0) directions in spin space (modulo a global rotation); (ii) varieties of incommensurate conic spiral. The latter family is addressed by projecting the three-dimensional lattice to a one-dimensional chain, with a basis of two (or more) sites per unit cell.

Optimal and Near Optimal Configurations on Lattices and Manifolds

Optimal configurations of points arise in many contexts, for example classical ground states for interacting particle systems, Euclidean packings of convex bodies, as well as minimal discrete and continuous energy problems for general kernels. Relevant questions in this area include the understanding of asymptotic optimal configurations, of lattice and periodic configurations, the development of algorithmic constructions of near optimal configurations, and the application of methods in convex optimization such as linear and semidefinite programming.

Observing Geometric Frustration with Thousands of Coupled Lasers

Geometric frustration, the inability of an ordered system to find a unique ground state plays a key role in a wide range of systems. We present a new experimental approach to observe large-scale geometric frustration with 1500 negatively coupled lasers arranged in a kagome lattice. We show how dissipation drives the lasers into a phase-locked state that directly maps to the classical XY spin Hamiltonian ground state. In our system, frustration is manifested by the lack of long range phase ordering. Finally, we show how next-nearest-neighbor coupling removes frustration and restores order.

Ground states of spin-12triangular antiferromagnets in a magnetic field

We use a combination of numerical density matrix renormalization group calculations and several analytical approaches to comprehensively study a simplified model for a spatially anisotropic spin-$\frac{1}{2}$ triangular lattice Heisenberg antiferromagnet: the three-leg triangular spin tube (TST). The model is described by three Heisenberg chains, with exchange constant $J$, coupled antiferromagnetically with exchange constant ${J}^{\ensuremath{'}}$ along the diagonals of the ladder system, with periodic boundary conditions in the shorter direction. Here, we determine the full phase diagram of this model as a function of both spatial anisotropy (between the isotropic and decoupled chain limits) and magnetic field. We find a rich phase diagram, which is remarkably dominated by quantum states: the phase corresponding to the classical ground state appears only in an exceedingly small region. Among the dominant phases generated by quantum effects are commensurate and incommensurate coplanar quasiordered states, which appear in the vicinity of the isotropic region for most fields, and in the high-field region for most anisotropies. The coplanar states, while not classical ground states, can at least be understood semiclassically. Even more strikingly, the largest region of phase space is occupied by a spin density wave phase, which has incommensurate collinear correlations along the field. This phase has no semiclassical analog, and may be ascribed to enhanced one-dimensional fluctuations due to frustration. Cutting across the phase diagram is a magnetization plateau, with a gap to all excitations and ``up-up-down'' spin order, with a quantized magnetization equal to $\frac{1}{3}$ of the saturation value. In the TST, this plateau extends almost but not quite to the decoupled chains limit. Most of the above features are expected to carry over to the two-dimensional system, which we also discuss. At low field, a dimerized phase appears, which is particular to the one-dimensional nature of the TST, and which can be understood from quantum Berry phase arguments.

Parity effect in the ground state localization of antiferromagnetic chains coupled to a ferromagnet.

We investigate the ground states of antiferromagnetic Mn nanochains on Ni(110) by spin-polarized scanning tunneling microscopy in combination with theory. While the ferrimagnetic linear trimer experimentally shows the predicted collinear classical ground state, no magnetic contrast was observed for dimers and tetramers where noncollinear structures were expected based on abinitio theory. This striking observation can be explained by zero-point energy motion for even-numbered chains derived within a classical equation of motion leading to nonclassical ground states. Thus, depending on the parity of the chain length, the system shows a classical or a quantum behavior.

Supergoop dynamics

We initiate a systematic study of the dynamics of multi-particle systems with supersymmetric Van der Waals and electron-monopole type interactions. The static interaction allows a complex continuum of ground state configurations, while the Lorentz interaction tends to counteract this configurational fluidity by magnetic trapping, thus producing an exotic low temperature phase of matter aptly named supergoop. Such systems arise naturally in $\mathcal{N}=2$ gauge theories as monopole-dyon mixtures, and in string theory as collections of particles or black holes obtained by wrapping D-branes on internal space cycles. After discussing the general system and its relation to quiver quantum mechanics, we focus on the case of three particles. We give an exhaustive enumeration of the classical and quantum ground states of a probe in an arbitrary background with two fixed centers. We uncover a hidden conserved charge and show that the dynamics of the probe is classically integrable. In contrast, the dynamics of one heavy and two light particles moving on a line shows a nontrivial transition to chaos, which we exhibit by studying the Poincar\'e sections. Finally we explore the complex dynamics of a probe particle in a background with a large number of centers, observing hints of ergodicity breaking. We conclude by discussing possible implications in a holographic context.

Geometry fluctuations and Casimir effect in a quantum antiferromagnet

We show the presence of a Casimir type force between domain walls in a two dimensional Heisenberg antiferromagnet subject to geometrical fluctuations. The type of fluctuations that we consider, called phason flips, are well known in quasicrystals, but less so in periodic structures. As the classical ground state energy of the antiferromagnet is unaffected by this type of fluctuation, energy changes are purely of quantum origin. We calculate the effective interaction between two parallel domain walls, defining a slab of thickness d, in such an antiferromagnet within linear spin wave theory. The interaction is anisotropic, and for a particular orientation of the slab we find that it decays as 1/d, thus, more slowly than the electromagnetic Casimir effect in the same geometry.

Intermediate magnetization plateaus in the spin-12Ising-Heisenberg and Heisenberg models on two-dimensional triangulated lattices

The ground state and zero-temperature magnetization process of the spin-1/2 Ising-Heisenberg model on two-dimensional triangles-in-triangles lattices is exactly calculated using eigenstates of the smallest commuting spin clusters. Our ground-state analysis of the investigated classical--quantum spin model reveals three unconventional dimerized or trimerized quantum ground states besides two classical ground states. It is demonstrated that the spin frustration is responsible for a variety of magnetization scenarios with up to three or four intermediate magnetization plateaus of either quantum or classical nature. The exact analytical results for the Ising-Heisenberg model are confronted with the corresponding results for the purely quantum Heisenberg model, which were obtained by numerical exact diagonalizations based on the Lanczos algorithm for finite-size spin clusters of 24 and 21 sites, respectively. It is shown that the zero-temperature magnetization process of both models is quite reminiscent and hence, one may obtain some insight into the ground states of the quantum Heisenberg model from the rigorous results for the Ising-Heisenberg model even though exact ground states for the Ising-Heisenberg model do not represent true ground states for the pure quantum Heisenberg model.

Interplay between spin frustration and thermal entanglement in the exactly solved Ising–Heisenberg tetrahedral chain

The spin-1/2 Ising-Heisenberg tetrahedral chain is exactly solved using its local gauge symmetry, which enables one to establish a rigorous mapping with the corresponding chain of composite Ising spins tractable within the transfer-matrix approach. Exact results derived for spin-spin correlation functions are employed to obtain the frustration temperature, at which a product of correlation functions along an elementary triangular plaquette becomes negative and the relevant spins experience a spin frustration. In addition, we have exactly calculated a concurrence quantifying thermal entanglement along with a threshold temperature, above which concurrence as a measure of thermal entanglement vanishes. It is shown that the frustration and threshold temperature coincide at sufficiently low temperatures, while they exhibit a very different behavior in the high-temperature region when tending towards completely different asymptotic limits. The threshold temperature additionally shows a notable reentrant behavior when it extends over a narrow temperature region above the classical ground state without any quantum correlations. It is demonstrated that the specific heat may display temperature dependence with or without an anomalous low-temperature peak for a relatively strong or weak Heisenberg interaction, respectively.

Communication: Designed diamond ground state via optimized isotropic monotonic pair potentials

We apply inverse statistical-mechanical methods to find a simple family of optimized isotropic, monotonic pair potentials (that may be experimentally realizable) whose classical ground state is the diamond crystal for the widest possible pressure range, subject to certain constraints (e.g., desirable phonon spectra). We also ascertain the ground-state phase diagram for a specific optimized potential to show that other crystal structures arise for pressures outside the diamond stability range. Cooling disordered configurations interacting with our optimized potential to absolute zero frequently leads to the desired diamond crystal ground state, revealing that the capture basin for the global energy minimum is large and broad relative to the local energy minima basins.

Exotic Ground States of Directional Pair Potentials via Collective-Density Variables

Collective-density variables have proved to be a useful tool in the prediction and manipulation of how spatial patterns form in the classical many-body problem. Previous work has employed properties of collective-density variables along with a robust numerical optimization technique to find the classical ground states of many-particle systems subject to radial pair potentials in one, two and three dimensions. That work led to the identification of ordered and disordered classical ground states. In this paper, we extend these collective-coordinate studies by investigating the ground states of directional pair potentials in two dimensions. Our study focuses on directional potentials whose Fourier representations are non-zero on compact sets that are symmetric with respect to the origin and zero everywhere else. We choose to focus on one representative set which has exotic ground-state properties: two circles whose centers are separated by some fixed distance. We obtain ground states for this "two-circle" potential that display large void regions in the disordered regime. As more degrees of freedom are constrained the ground states exhibit a collapse of dimensionality characterized by the emergence of filamentary structures and linear chains. This collapse of dimensionality has not been observed before in related studies.