Invariant Spin Models Research Articles

Information spreading and scrambling in disorder-free multiple-spin-interaction models

Tripartite mutual information (TMI) is an efficient observable to quantify the ability of a scrambler for the unitary time-evolution operator with a quenched many-body Hamiltonian. In this paper we give numerical demonstrations of the TMI in disorder-free (translationally invariant) spin models with three-body and four-body multiple-spin interactions. The dynamical behavior of the TMI of these models does not exhibit a linear light cone for sufficiently strong interactions. In early-time evolution, the TMI displays a distinct negative increase fitted by a logarithmiclike function. This is in contrast to the conventional linear light-cone behavior present in the $XXZ$ model and its near-integrable vicinity. The late-time evolution of the TMI in finite-size systems is also numerically investigated. The multiple-spin interactions make the system nearly integrable and weakly suppress the spread of information and scrambling. The observation of the late-time value of the TMI indicates that the scrambling nature of the system changes by interactions and this change can be characterized by a phase-transition-like behavior of the TMI, reflecting the system's integrability and violating the eigenstate thermalization hypothesis.

Ultraslow dynamics in a translationally invariant spin model for multiplication and factorization

The authors introduce a lattice model of classical Ising spins inspired by a reversible classical computational circuit designed to factor semi-primes. The model is disorder- and frustration-free and lacks a thermodynamic phase transition, yet it exhibits unprecedentedly slow relaxation times that scale as a double exponential of the inverse temperature.

Emergence of chiral spin liquids via quantum melting of noncoplanar magnetic orders

Quantum spin liquids (QSLs) are long-range entangled states of quantum magnets which lie beyond the Landau paradigm of classifying phases of matter via broken symmetries. A physical route to arriving at QSLs is via frustration-induced quantum melting of ordered states such as valence bond crystals or magnetic orders. Here, we show, using extensive exact diagonalization (ED) and density-matrix renormalization group (DMRG) studies of concrete $SU(2)$ invariant spin models on honeycomb, triangular and square lattices, that chiral spin liquids (CSLs) emerge as descendants of triple-$Q$ spin crystals with tetrahedral magnetic order and a large scalar spin chirality. Such ordered-to-CSL melting transitions may yield lattice realizations of effective Chern-Simons-Higgs field theories. Our work provides a distinct unifying perspective on the emergence of CSLs, and suggests that materials with magnetic skyrmion crystal order might provide a good starting point to search for CSLs.

Anatomy of fermionic entanglement and criticality in Kitaev spin liquids

We analyse in detail the effect of non-trivial band topology on the area law behaviour of the entanglement entropy in Kitaev's honeycomb model. By mapping the translationally invariant 2D spin model into 1D fermionic subsystems, we identify those subsystems responsible for universal entanglement contributions in the gapped phases and those responsible for critical entanglement scaling in the gapless phases. For the gapped phases we analytically show how the topological edge states contribute to the entanglement entropy and provide a universal lower bound for it. For the gapless semi-metallic phases and topological phase transitions the identification of the critical subsystems shows that they fall always into the Ising or the XY universality classes. As our study concerns the fermionic degrees of freedom in the honeycomb model, qualitatively similar results are expected to apply also to generic topological insulators and superconductors.

Localization from Superselection Rules in Translationally Invariant Systems.

The cubic code model is studied in the presence of arbitrary extensive perturbations. Below a critical perturbation strength, we show that most states with finite energy are localized; the overwhelming majority of such states have energy concentrated around a finite number of defects, and remain so for a time that is near exponential in the distance between the defects. This phenomenon is due to an emergent superselection rule and does not require any disorder. Local integrals of motion for these finite energy sectors are identified as well. Our analysis extends more generally to systems with immobile topological excitations.

Engineering SU(2) invariant spin models to mimic quantum dimer physics on the square lattice

We consider the spin-1/2 hamiltonians proposed by Cano and Fendley [J. Cano and P. Fendley, Phys. Rev. Lett. 105, 067205 (2010)] which were built to promote the well-known Rokshar-Kivelson (RK) point of quantum dimer models to spin-1/2 wavefunctions. We first show that these models, besides the exact degeneracy of RK point, support gapless spinless excitations as well as a spin gap in the thermodynamic limit, signatures of an unusual spin liquid. We then extend the original construction to create a continuous family of SU(2) invariant spin models that reproduces the phase diagram of the quantum dimer model, and in particular show explicit evidences for existence of columnar and staggered phases. The original models thus appear as multicritical points in an extended phase diagram. Our results are based on the use of a combination of numerical exact simulations and analytical mapping to effective generalized quantum dimer models.

Bell inequality in an exactly soluble spin-1/2 XX chain displaying spin-Peierls transition

Quantum nonlocality is an experimentally measurable quantum correlation indicated by the violation of Bell inequality. Previous studies show that bipartite Bell inequality is not violated in various translationally invariant spin models. In this paper, we study bipartite Bell inequality in an exactly soluble spin-1/2 XX chain displaying spin-Peierls transition, where the system is in a uniform phase at high temperatures and in a dimerized phase in low temperatures. The dimerization is induced by spin-phonon interactions in the system. We find that the Bell inequality is violated in the dimerized phase, even if the dimerization parameter is very small. It suggests that two-qubit quantum nonlocality may be widely present in real materials in which spin-phonon interactions cannot be ignored. The relation between Bell inequality and the thermal phase transition in the chain is also studied.

Bridging Lattice-Scale Physics and Continuum Field Theory with Quantum Monte Carlo Simulations

We discuss designer Hamiltonians—lattice models tailored to be free from sign problems (“de-signed”) when simulated with quantum Monte Carlo (QMC) methods but which still host complex many-body states and quantum phase transitions of interest in condensed matter physics. We focus on quantum spin systems in which competing interactions lead to nonmagnetic ground states. These states and the associated quantum phase transitions can be studied in great detail, enabling direct access to universal properties and connections with low-energy effective quantum field theories. As specific examples, we discuss the transition from a Néel antiferromagnet to either a uniform quantum paramagnet or a spontaneously symmetry-broken valence-bond solid (VBS) in SU(2) and SU(N) invariant spin models. We also discuss anisotropic (XXZ) systems harboring topological Z2 spin liquids and the XY* transition. We briefly review recent progress on QMC algorithms, including ground-state projection in the valence-bond basis and direct computation of the Renyi variants of the entanglement entropy.

Type-II matrices in weighted Bose–Mesner algebras of ranks 2 and 3

Type-II matrices are nonzero complex matrices that were introduced in connection with spin models for link invariants. Type-II matrices have been found in connection with symmetric designs, sets of equiangular lines, strongly regular graphs, and some distance regular graphs. We investigate weighted complete and strongly regular graphs, and show that type-II matrices arise in this setting as well.

Type II Matrices and Their Bose-Mesner Algebras

Type II matrices were introduced in connection with spin models for link invariants. It is known that a pair of Bose-Mesner algebras (called a dual pair) of commutative association schemes are naturally associated with each type II matrix. In this paper, we show that type II matrices whose Bose-Mesner algebras are imprimitive are expressed as so-called generalized tensor products of some type II matrices of smaller sizes. As an application, we give a classification of type II matrices of size at most 10 except 9 by using the classification of commutative association schemes.

Statistics of energy levels and zero temperature dynamics for deterministic spin models with glassy behaviour

We consider the zero-temperature dynamics for the infinite-range, non translation invariant one-dimensional spin model introduced by Marinari, Parisi and Ritort to generate glassy behaviour out of a deterministic interaction. It is argued that there can be a large number of metastable (i.e., one-flip stable) states with very small overlap with the ground state but very close in energy to it, and that their total number increases exponentially with the size of the system.

Mean-field expansion for spin models with medium-range interactions

We study the critical crossover between the Gaussian and the Wilson-Fisher fixed point for general O( N)-invariant spin models with medium-range interactions. We perform a systematic expansion around the mean-field solution, obtaining the universal crossover curves and their leading corrections. In particular we show that, in three dimensions, the leading correction scales as R −3, R being the range of the interactions. We compare our results with the existing numerical ones obtained by Monte Carlo simulations and present a critical discussion of other approaches.

Pseudo-character expansions forU(N)-invariant spin models onCP N−1

We define a set of orthogonal functions on the complex projective spaceCPN−1, and compute their Clebsch-Gordan coefficients as well as a large class of 6-j symbols. We also provide all the needed formulae for the generation of high-temperature expansions forU(N)-invariant spin models defined onCPN−1.

Continuum limits and exact finite-size-scaling functions for one-dimensionalO(N)-invariant spin models

We solve exactly the general one-dimensionalO(N)-invariant spin model taking values in the sphereS N−1, with nearest-neighbor interactions, in finite volume with periodic boundary conditions, by an expansion in hyperspherical harmonics. The possible continuum limits are discussed for a general one-parameter family of interactions and an infinite number of universality classes is found. For these classes we compute the finite-size-scaling functions and the leading corrections to finite-size scaling. A special two-parameter family of interactions (which includes the mixed isovector/isotensor model) is also treated and no additional universality classes appear. In the appendices we give new formulae for the Clebsch-Gordan coefficients and 6−j symbols of theO(N) group, and some new generalizations of the Poisson summation formula; these may be of independent interest.

Application of the O( N)-hyperspherical harmonics to the study of the continuum limits of one-dimensional σ-models and to the generation of high-temperature expansions in higher dimensions

In this talk we present the exact solution of the most general one-dimensional $O(N)$-invariant spin model taking values in the sphere $S^{N-1}$, with nearest-neighbour interactions, and we discuss the possible continuum limits. All these results are obtained using a high-temperature expansion in terms of hyperspherical harmonics. Applications in higher dimensions of the same technique are then discussed.

On Spin Models, Triply Regular Association Schemes, and Duality

Motivated by the construction of invariants of links in 3-space, we study spin models on graphs for which all edge weights (considered as matrices) belong to the Bose-Mesner algebra of some association scheme. We show that for series-parallel graphs the computation of the partition function can be performed by using series-parallel reductions of the graph appropriately coupled with operations in the Bose-Mesner algebra. Then we extend this approach to all plane graphs by introducing star-triangle transformations and restricting our attention to a special class of Bose-Mesner algebras which we call exactly triply regular. We also introduce the following two properties for Bose-Mesner algebras. The planar duality property (defined in the self-dual case) expresses the partition function for any plane graph in terms of the partition function for its dual graph, and the planar reversibility property asserts that the partition function for any plane graph is equal to the partition function for the oppositely oriented graph. Both properties hold for any Bose-Mesner algebra if one considers only series-parallel graphs instead of arbitrary plane graphs. We relate these notions to spin models for link invariants, and among other results we show that the Abelian group Bose-Mesner algebras have the planar duality property and that for self-dual Bose-Mesner algebras, planar duality implies planar reversibility. We also prove that for exactly triply regular Bose-Mesner algebras, to check one of the above properties it is sufficient to check it on the complete graph on four vertices. A number of applications, examples and open problems are discussed.

Strongly regular graphs and spin models for the Kauffman polynomial

We study spin models for invariants of links as defined by Jones [22]. We consider the two algebras generated by the weight matrices of such models under ordinary or Hadamard product and establish an isomorphism between them. When these algebras coincide they form the Bose-Mesner algebra of a formally self-dual association scheme. We study the special case of strongly regular graphs, which is associated to a particularly interesting link invariant, the Kauffman polynomial [27]. This leads to a classification of spin models for the Kauffman polynomial in terms of formally self-dual strongly regular graphs with strongly regular subconstituents [7]. In particular we obtain a new model based on the Higman-Sims graph [17].

Exact solution of Z(N) invariant spin models on a triangular lattice

Based on Baxter's ideas the authors present a method to solve spin models on triangular lattices. If the model has only nearest neighbour interaction and it satisfies the star-triangle relation then the exact free energy can be obtained. They apply the method to reobtain the critical triangular Potts model and to extend the self-dual Z(N) solutions of Fateev and Zamolodchikov to the triangular lattice.