Lattice Translation Research Articles

Catching the π-Stacks: Prediction of Aggregate Structures of Porphyrin.

π-π interactions decisively shape the supramolecular structure and functionality of π-conjugated molecular semiconductor materials. Despite the customizable molecular building blocks, predicting their supramolecular structure remains a challenge. Traditionally, force field methods have been used due to the complexity of these structures, but advances in computational power have enabled ab initio approaches such as density functional theory (DFT). DFT is particularly suitable for finding energetically favorable structures of dye aggregates, which are determined by a large number of different interactions, but a systematic aggregate search can still be very challenging due to the large number of possible geometries. In this work, we show ways to overcome this challenge. We investigate how finely translational and rotational lattices must be structured to identify all energetic minima of π-stack structures, focusing on porphyrins as a prototype challenge. Our approach involves single-point DFT calculations of systematically varied dimer geometries, identification of local energy minima, hierarchical grouping of geometrically similar structures, and optimization of the energetically favorable representatives of each geometric family. This ab initio method provides a general framework for the systematic prediction of aggregate structures and reveals geometrically diverse and energetically favorable dimers.

How periodic surfaces bend.

A periodic surface is one that is invariant by a two-dimensional lattice of translations. Deformation modes that stretch the lattice without stretching the surface are effective membrane modes. Deformation modes that bend the lattice without stretching the surface are effective bending modes. For periodic piecewise smooth simply connected surfaces, it is shown that the effective membrane modes are, in a sense, orthogonal to effective bending modes. This means that if a surface gains a membrane mode, it loses a bending mode, and conversely, in such a way that the total number of modes, membrane and bending combined, can never exceed 3. Various examples, inspired from curved-crease origami tessellations, illustrate the results.This article is part of the theme issue 'Origami/Kirigami-inspired structures: from fundamentals to applications'.

Lieb-Schultz-Mattis theorems for symmetry-protected topological phases

The Lieb-Schultz-Mattis (LSM) theorem and its generalizations are a class of powerful no-go theorems that rule out any short-range-entangled (SRE) symmetric ground state irrespective of the specific Hamiltonian, based only on certain microscopic inputs, such as symmetries and particle filling numbers. In this work, we introduce and provide physical arguments for a new class of LSM-type theorems, where any symmetry-allowed SRE ground state must be a symmetry-protected topological (SPT) phase with robust gapless edge states, such as topological insulators and superconductors. The key ingredient is to replace the lattice translation symmetry in usual LSM theorems by the magnetic translation symmetry. These theorems provide new insights into realistic models and experimental realizations of SPT phases in interacting bosons and fermions.

Lattice Hamiltonian for adjoint QCD2

We introduce a Hamiltonian lattice model for the (1 + 1)-dimensional SU(Nc) gauge theory coupled to one adjoint Majorana fermion of mass m. The discretization of the continuum theory uses staggered Majorana fermions. We analyze the symmetries of the lattice model and find lattice analogs of the anomalies of the corresponding continuum theory. An important role is played by the lattice translation by one lattice site, which in the continuum limit involves a discrete axial transformation. On a lattice with periodic boundary conditions, the Hilbert space breaks up into sectors labeled by the Nc-ality p = 0, … Nc − 1. Our symmetry analysis implies various exact degeneracies in the spectrum of the lattice model. In particular, it shows that, for m = 0 and even Nc, the sectors p and p′ are degenerate if |p − p′| = Nc/2. In the Nc = 2 case, we explicitly construct the action of the Hamiltonian on a basis of gauge-invariant states, and we perform both a strong coupling expansion and exact diagonalization for lattices of up to 12 lattice sites. Upon extrapolation of these results, we find good agreement with the spectrum computed previously using discretized light-cone quantization. One of our new results is the first numerical calculation of the fermion bilinear condensate.

Non-invertible symmetries and LSM-type constraints on a tensor product Hilbert space

We discuss the exact non-invertible Kramers-Wannier symmetry of 1+1d lattice models on a tensor product Hilbert space of qubits. This symmetry is associated with a topological defect and a conserved operator, and the latter can be presented as a matrix product operator. Importantly, unlike its continuum counterpart, the symmetry algebra involves lattice translations. Consequently, it is not described by a fusion category. In the presence of this defect, the symmetry algebra involving parity/time-reversal is realized projectively, which is reminiscent of an anomaly. Different Hamiltonians with the same lattice non-invertible symmetry can flow in their continuum limits to infinitely many different fusion categories (with different Frobenius-Schur indicators), including, as a special case, the Ising CFT. The non-invertible symmetry leads to a constraint similar to that of Lieb-Schultz-Mattis, implying that the system cannot have a unique gapped ground state. It is either in a gapless phase or in a gapped phase with three (or a multiple of three) ground states, associated with the spontaneous breaking of the lattice non-invertible symmetry.

Hyperbolic Non-Abelian Semimetal.

We extend the notion of topologically protected semi-metallic band crossings to hyperbolic lattices in a negatively curved plane. Because of their distinct translation group structure, such lattices are associated with a high-dimensional reciprocal space. In addition, they support non-Abelian Bloch states which, unlike conventional Bloch states, acquire a matrix-valued Bloch factor under lattice translations. Combining diverse numerical and analytical approaches, we uncover an unconventional scaling in the density of states at low energies, and illuminate a nodal manifold of codimension five in the reciprocal space. The nodal manifold is topologically protected by a nonzero second Chern number, reminiscent of the characterization of Weyl nodes by the first Chern number.

Lieb-Schultz-Mattis anomalies as obstructions to gauging (non-on-site) symmetries

We study ’t Hooft anomalies of global symmetries in 1+1d lattice Hamiltonian systems. We consider anomalies in internal and lattice translation symmetries. We derive a microscopic formula for the “anomaly cocycle” using topological defects implementing twisted boundary conditions. The anomaly takes value in the cohomology group H^3(G,U(1)) × H^2(G,U(1))H3(G,U(1))×H2(G,U(1)). The first factor captures the anomaly in the internal symmetry group G, and the second factor corresponds to a generalized Lieb-Schultz-Mattis anomaly involving G and lattice translation. We present a systematic procedure to gauge internal symmetries (that may not act on-site) on the lattice. We show that the anomaly cocycle is the obstruction to gauging the internal symmetry while preserving the lattice translation symmetry. As an application, we construct anomaly-free chiral lattice gauge theories. We demonstrate a one-to-one correspondence between (locality-preserving) symmetry operators and topological defects, which is essential for the results we prove. We also discuss the generalization to fermionic theories. Finally, we construct non-invertible lattice translation symmetries by gauging internal symmetries with a Lieb-Schultz-Mattis anomaly.

Majorana chain and Ising model - (non-invertible) translations, anomalies, and emanant symmetries

We study the symmetries of closed Majorana chains in 1+1d, including the translation, fermion parity, spatial parity, and time-reversal symmetries. The algebra of the symmetry operators is realized projectively on the Hilbert space, signaling anomalies on the lattice, and constraining the long-distance behavior. In the special case of the free Hamiltonian (and small deformations thereof), the continuum limit is the 1+1d free Majorana CFT. Its continuum chiral fermion parity (-1)^{F_L}(−1)FL emanates from the lattice translation symmetry. We find a lattice precursor of its mod 8 ’t Hooft anomaly. Using a Jordan-Wigner transformation, we sum over the spin structures of the lattice model (a procedure known as the GSO projection), while carefully tracking the global symmetries. In the resulting bosonic model of Ising spins, the Majorana translation operator leads to a non-invertible lattice translation symmetry at the critical point. The non-invertible Kramers-Wannier duality operator of the continuum Ising CFT emanates from this non-invertible lattice translation of the transverse-field Ising model.

Synthesis, crystal structure and thermal behavior of tetra-kis-(3-cyano-pyridine N-oxide-κO)bis-(thio-cyanato-κN)cobalt(II), which shows strong pseudo-symmetry.

The title compound, [Co(SCN)2(C6H4N2O)4], was prepared by the reaction of cobalt(II)thio-cyanate with 3-cyano-pyridine N-oxide in ethanol. In the crystal, the cobalt(II) cations are octa-hedrally coordinated by two terminal N-bonded thio-cyanate anions and four O-bonded 3-cyano-pyridine N-oxide coligands, forming discrete complexes that are located on centers of inversion, hence forming trans-CoN2O4 octa-hedra. The structure refinement was performed in the monoclinic space group P21/n, for which a potential lattice translation and new symmetry elements with a fit of 100% is suggested. The structure can easily be refined in the space group I2/m, where the complexes have 2/m symmetry. However, nearly all of the reflections that violate the centering are observed with significant intensity and the refinement in P21/n leads to significantly lower R(F) values (0.027 versus 0.033). Moreover, in I2/m much larger components of the anisotropic displacement parameters are observed and therefore, the crystal structure is presented in the primitive unit cell. IR investigations confirm that the anionic ligands are only terminally bonded and that the cyano group is not involved in the metal coordination. PXRD investigations show that a pure crystalline phase has been obtained and measurements using simultaneously thermogravimetry and differential thermoanalysis reveal that the compound decomposes in an exothermic reaction upon heating, without the formation of a coligand-deficient inter-mediate phase.

Lieb-Schultz-Mattis, Luttinger, and 't Hooft - anomaly matching in lattice systems

We analyze lattice Hamiltonian systems whose global symmetries have ’t Hooft anomalies. As is common in the study of anomalies, they are probed by coupling the system to classical background gauge fields. For flat fields (vanishing field strength), the nonzero spatial components of the gauge fields can be thought of as twisted boundary conditions, or equivalently, as topological defects. The symmetries of the twisted Hilbert space and their representations capture the anomalies. We demonstrate this approach with a number of examples. In some of them, the anomalous symmetries are internal symmetries of the lattice system, but they do not act on-site. (We clarify the notion of “on-site action.”) In other cases, the anomalous symmetries involve lattice translations. Using this approach we frame many known and new results in a unified fashion. In this work, we limit ourselves to 1+1d systems with a spatial lattice. In particular, we present a lattice system that flows to the c=1c=1 compact boson system with any radius (no BKT transition) with the full internal symmetry of the continuum theory, with its anomalies and its T-duality. As another application, we analyze various spin chain models and phrase their Lieb-Shultz-Mattis theorem as an ’t Hooft anomaly matching condition. We also show in what sense filling constraints like Luttinger theorem can and cannot be viewed as reflecting an anomaly. As a by-product, our understanding allows us to use information from the continuum theory to derive some exact results in lattice model of interest, such as the lattice momenta of the low-energy states.

Phases of the hard-plate lattice gas on a three-dimensional cubic lattice.

We study the phase diagram of a lattice gas of 2×2×1 hard plates on the three-dimensional cubic lattice. Each plate covers an elementary plaquette of the cubic lattice, with the constraint that a site can belong to utmost one plate. We focus on the isotropic system, with equal fugacities for the three orientations of plates. We show, using grand canonical Monte Carlo simulations, that the system undergoes two phase transitions when the density of plates is increased: the first from a disordered fluid phase to a layered phase, and the second from the layered phase to a sublattice-ordered phase. In the layered phase, the system breaks up into disjoint slabs of thickness two along one spontaneously chosen Cartesian direction, corresponding to a twofold (Z_{2}) symmetry breaking of translation symmetry along the layering direction. Plates with normals perpendicular to this layering direction are preferentially contained entirely within these slabs, while plates straddling two adjacent slabs have a lower density, thus breaking the symmetry between the three types of plates. We show that the slabs exhibit two-dimensional power-law columnar order even in the presence of a nonzero density of vacancies. In contrast, interslab correlations of the two-dimensional columnar order parameter decay exponentially with the separation between the slabs. In the sublattice-ordered phase, there is twofold symmetry breaking of lattice translation symmetry along all three Cartesian directions. We present numerical evidence that the disordered to layered transition is continuous and consistent with universality class of the three-dimensional O(3) model with cubic anisotropy, while the layered to sublattice transition is first-order in nature.

Spontaneous layering and power-law order in the three-dimensional fully packed hard-plate lattice gas.

We obtain the phase diagram of fully packed hard plates on the cubic lattice. Each plate covers an elementary plaquette of the cubic lattice and occupies its four vertices, with each vertex of the cubic lattice occupied by exactly one such plate. We consider the general case with fugacities s_{μ} for "μ plates," whose normal is the μ direction (μ=x,y,z). At and close to the isotropic point, we find, consistent with previous work, a phase with long-range sublattice order. When two of the fugacities s_{μ_{1}} and s_{μ_{2}} are comparable, and the third fugacity s_{μ_{3}} is much smaller, we find a spontaneously layered phase. In this phase, the system breaks up into disjoint slabs of width two stacked along the μ_{3} axis. μ_{1} and μ_{2} plates are preferentially contained entirely within these slabs, while plates straddling two successive slabs have a lower density. This corresponds to a twofold breaking of translation symmetry along the μ_{3} axis. In the opposite limit, with μ_{3}≫μ_{1}∼μ_{2}, we find a phase with long-range columnar order, corresponding to simultaneous twofold symmetry breaking of lattice translation symmetry in directions μ_{1} and μ_{2}. The spontaneously layered phases display critical behavior, with power-law decay of correlations in the μ_{1} and μ_{2} directions when the slabs are stacked in the μ_{3} direction, and represent examples of "floating phases" discussed earlier in the context of coupled Luttinger liquids and quasi-two-dimensional classical systems. We ascribe this remarkable behavior to the constrained motion of defects in this phase, and we sketch a coarse-grained effective field theoretical understanding of the stability of power-law order in this unusual three-dimensional floating phase.

General Theory of Momentum-Space Nonsymmorphic Symmetry

As a fundamental concept of all crystals, space groups are partitioned into symmorphic groups and nonsymmorphic groups. Each nonsymmorphic group contains glide reflections or screw rotations with fractional lattice translations, which are absent in symmorphic groups. Although nonsymmorphic groups ubiquitously exist on real-space lattices, on the reciprocal lattices in momentum space, the ordinary theory only allows symmorphic groups. In this work, we develop a novel theory for momentum-space nonsymmorphic space groups ($k$-NSGs), utilizing the projective representations of space groups. The theory is quite general: Given any $k$-NSGs in any dimensions, it can identify the real-space symmorphic space groups ($r$-SSGs) and construct the corresponding projective representation of the $r$-SSG that leads to the $k$-NSG. To demonstrate the broad applicability of our theory, we show these projective representations and therefore all $k$-NSGs can be realized by gauge fluxes over real-space lattices. Our work fundamentally extends the framework of crystal symmetry, and therefore can accordingly extend any theory based on crystal symmetry, for instance, the classification crystalline topological phases.

Bootstrapping Lieb-Schultz-Mattis anomalies

We incorporate the microscopic assumptions that lead to a certain generalization of the Lieb-Schultz-Mattis (LSM) theorem for one-dimensional spin chains into the conformal bootstrap. Our approach accounts for the ``LSM anomaly'' possessed by these spin chains through a combination of modular bootstrap and correlator bootstrap of symmetry defect operators. Specifically, we obtain universal bounds on the local operator content of $(1+1)d$ conformal field theories (CFTs) that could describe translationally invariant lattice Hamiltonians with a ${\mathbb{Z}}_{N}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{N}$ symmetry realized projectively at each site. We assume, for such models, that in the CFT the lattice translation symmetry is realized as an emanant internal ${\mathbb{Z}}_{N}$ symmetry. We present bounds on local operators both with and without refinement by their global symmetry representations. Interestingly, we can obtain nontrivial bounds on charged operators when $N$ is odd, which turns out to be impossible with modular bootstrap alone. Our bounds exhibit distinctive kinks, some of which are approximately saturated by known theories and others that are unexplained. We discuss additional scenarios with the properties necessary for our bounds to apply, including certain multicritical points between $(1+1)d$ symmetry protected topological phases, where we argue that the anomaly studied in our bootstrap calculations should emerge.

Electrically controllable spin polarization in collinear antiferromagnetic junctions

Antiferromagnetic spintronics is a rapidly growing subfield of spintronics in condensed-matter physics and information technology. Electrical current in collinear antiferromagnetic materials is typically spin unpolarized, limiting the realization of antiferromagnetic spintronics effects. Here we study the transport in the collinear antiferromagnetic junctions by applying a transverse electric field E y to the antiferromagnets (AFs). The band structures of the collinear AFs may become spin-polarized when the combined time reversal and lattice translation symmetry is broken by E y . The separation between spin-up and spin-down bands is controlled by E y . Full spin polarization originating from spin-polarized states near the band gap’s edges is observed at high exchange energy. In particular, as E y increases, the region capable of generating high spin polarization broadens due to the increased separation between spin-up and spin-down bands. The amplitude and sign of spin polarization can be controlled by E y . These characteristics indicate that collinear AF materials are ideal for future spintronics applications.

Fermi surface symmetric mass generation

Symmetric mass generation is a novel mechanism to give gapless fermions a mass gap by non-perturbative interactions without generating any fermion bilinear condensation. The previous studies of symmetric mass generation have been limited to Dirac/Weyl/Majorana fermions with zero Fermi volume in the free fermion limit. In this work, we generalize the concept of symmetric mass generation to Fermi liquid with a finite Fermi volume and discuss how to gap out the Fermi surfaces by interactions without breaking the U(1) loop group symmetry or developing topological orders. We provide examples of Fermi surface symmetric mass generation in both (1+1)D and (2+1)D Fermi liquid systems when several Fermi surfaces together cancel the Fermi surface anomaly. However, the U(1) loop group symmetry in these cases is still restrictive enough to rule out all possible fermion bilinear gapping terms, such that a non-perturbative interaction mechanism is the only way to gap out the Fermi surfaces. This symmetric Fermi surface reconstruction is in contrast to the conventional symmetry-breaking mechanism to gap the Fermi surfaces. As a side product, our model provides a pristine 1D lattice regularization for the (1+1)D U(1) symmetric chiral fermion model (e.g., the 3-4-5-0 model) by utilizing a lattice translation symmetry as an emergent U(1) symmetry at low energy. This opens up the opportunity for efficient numerical simulations of chiral fermions in their own dimensions without introducing mirror fermions under the domain wall fermion construction.

A global prediction of the Kataura plot for chiral carbon nanotubes: Topological family effect revealed in the natural helical crystal lattice scheme

The optical transition energies of chiral carbon nanotubes (CNTs) are intimately related to their geometrical structures, and are frequently applied in the recognition and isolation of single nanotubes. Conventional translational crystal lattice (TCL) scheme failed to offer an accurate description of the optical properties of the chiral CNTs due to the formidable unit cell size constraint as well as the complexity of the band structures. Most fitting equations for the Kataura plot, usually based on the empirical correlations with the chiral indices of the CNTs, lack a clear physical interpretation. In this study, we apply the natural helical crystal lattice (NHCL) scheme to systematically analyze the band gap variation of arbitrary chiral CNTs. We derive a global analytical formula that takes into account the comprehensive geometrical connection topologies for the band gap of all (n, m)-CNTs with a diameter larger than 1.2 nm. With the inclusion of the many-body effect, we are able to provide a global Kataura plot prediction with a minimal set of only two fitting parameters, and the results are consistent with the experimental Kataura plot within 0.67% deviation for over 1300 data points. Our formula precisely captures the observed experimental branching features and presents a clear physical rationalization for the topological family effect on the optical excitation energies of chiral CNTs.

Atomic-level quantification of twinning shears in magnesium alloy

Unlike dislocation slips with lattice translation vectors, the shear vector of twinning dislocation or disconnection (TD) is a non-lattice translation vector. Corresponding to the crystallography of twinning, geometrical analysis such as topological model has been widely adopted to define possible TDs that have both step and dislocation character. More importantly, multiple TDs can be defined to achieve the same crystal reorientation but generate different twinning shears. Thus, an accurate measurement of twinning shear allows us to define the character of elementary TD. Here, we quantitatively measured twinning shear of three twinning modes at an atomic level through tracing the twinning induced shear in nanoscale lamellar precipitates under laser shock peening and correspondingly determined the character of elementary TD. According to the measured angle for characteristic planes of nanoscale lamellar precipitates before and after twinning, we determined the twinning shear of 0.617 for{112¯1} twinning, which corresponds to a 1/2-layer TD with the Burgers vector of 0.024<1¯1¯26> as identified in the topological model. Using the same method, a twinning shear of 0.129 is determined for {101¯2} twinning, which corresponds to a two-layer TD with the Burgers vector of 0.080<1¯011>. More importantly, this result rules out the so-called pure-shuffle mechanism for {101¯2} twinning. A twinning shear of 0.136 is determined for {101¯1} twinning, which corresponds to a four-layer TD (or a reassembly of two-layer TDs) with the Burgers vector of 0.113<101¯2¯> as identified in topological model. These findings may obviate the dissent of twinning mechanisms, and provide an effective method to determine other complex shear mechanisms at an atomic level.

Metal to Wigner-Mott insulator transition in two-leg ladders

We study theoretically the quantum phase transition from a metal to a Wigner-Mott insulator at fractional commensurate filling on a two-leg ladder. We show that a continuous transition out of a symmetry-preserving Luttinger liquid metal is possible where the onset of insulating behavior is accompanied by the breaking of the lattice translation symmetry. At fillings $\nu = 1/m$ per spin per unit cell, we find that the spin degrees of freedom also acquire a gap at the Wigner-Mott transition for odd integer $m$. In contrast for even integer $m$, the spin sector remains gapless and the resulting insulator is a ladder analog of the two-dimensional spinon surface state. In both cases, a charge neutral spinless mode remains gapless across the Wigner-Mott transition. We discuss physical properties of these transitions, and comment on insights obtained for thinking about continuous Wigner-Mott transitions in two-dimensional systems which are being studied in moire materials.

Discrete chiral symmetry and mass shift in the lattice Hamiltonian approach to the Schwinger model

We revisit the lattice formulation of the Schwinger model using the Kogut-Susskind Hamiltonian approach with staggered fermions. This model, introduced by Banks et al. Phys. Rev. D 13, 1043 (1976), contains the mass term ${m}_{\mathrm{lat}}{\ensuremath{\sum}}_{n}{(\ensuremath{-}1)}^{n}{\ensuremath{\chi}}_{n}^{\ifmmode\dagger\else\textdagger\fi{}}{\ensuremath{\chi}}_{n}$, and setting it to zero is often assumed to provide the lattice regularization of the massless Schwinger model. We instead argue that the relation between the lattice and continuum mass parameters should be taken as ${m}_{\mathrm{lat}}=m\ensuremath{-}\frac{1}{8}{e}^{2}a$. The model with $m=0$ is shown to possess a discrete chiral symmetry that is generated by the unit lattice translation accompanied by the shift of the $\ensuremath{\theta}$ angle by $\ensuremath{\pi}$. While the mass shift vanishes as the lattice spacing $a$ approaches zero, we find that including this shift greatly improves the rate of convergence to the continuum limit. We demonstrate the faster convergence using both numerical diagonalizations of finite lattice systems, as well as extrapolations of the lattice strong coupling expansions.