Spin-flip Symmetry Research Articles

Combining Complex Conjugation, Time-Reversal, and Spin-Flip Symmetry Projection of Coupled Cluster Wave Functions.

Complex conjugation symmetry breaking and restoration generates two nonorthogonal configurations at the Hartree-Fock level that can capture static correlation naturally. In conjunction with broken spin-symmetry coupled cluster theory, the symmetry-projected wave function shows good agreement with full configuration interaction in beryllium hydride insertion, lithium fluoride dissociation, and symmetric stretching of tetrahedral H4. By adding spin flip projection, we can also recover time reversal symmetry in the same coupled cluster framework. We also show results including point group symmetry projection.

Entanglement entropy along a massless renormalisation flow: the tricritical to critical Ising crossover

We study the Rényi entanglement entropies along the massless renormalisation group flow that connects the tricritical and critical Ising field theories. Similarly to the massive integrable field theories, we derive a set of bootstrap equations, from which we can analytically calculate the twist field form factors in a recursive way. Additionally, we also obtain them as a non-trivial ‘roaming limit’ of the sinh-Gordon theory. Then the Rényi entanglement entropies are obtained as expansions in terms of the form factors of these branch point twist fields. We find that the form factor expansion of the entanglement entropy along the flow organises in two different kind of terms. Those that couple particles with the same chirality, and reproduce the entropy of the infrared Ising theory, and those that couple particles with different chirality, which provide the ultraviolet contributions. The massless flow under study possesses a global ℤ2 spin-flip symmetry. We further consider the composite twist fields associated to this group, which enter in the study of the symmetry resolution of the entanglement. We derive analytical expressions for their form factors both from the bootstrap equations and from the roaming limit of the sinh-Gordon theory.

Symmetry Breaking for Voting Mechanisms.

Recently, Rowe and Aishwaryaprajna (2019) introduced a simple majority vote technique that efficiently solves Jump with large gaps, OneMax with large noise, and any monotone function with a polynomial-size image. In this paper, we identify a pathological condition for this algorithm: the presence of spin-flip symmetry in the problem instance. Spin-flip symmetry is the invariance of a pseudo-Boolean function to complementation. Many important combinatorial optimization problems admit objective functions that exhibit this pathology, such as graph problems, Ising models, and variants of propositional satisfiability. We prove that no population size exists that allows the majority vote technique to solve spin-flip symmetric functions of unitation with reasonable probability. To remedy this, we introduce a symmetry-breaking technique that allows the majority vote algorithm to overcome this issue for many landscapes. This technique requires only a minor modification to the original majority vote algorithm to force it to sample strings in {0,1}n from a dimension n-1 hyperplane. We prove a sufficient condition for a spin-flip symmetric function to possess in order for the symmetry-breaking voting algorithm to succeed, and prove its efficiency on generalized TwoMax, a spin-flip symmetric variant of Jump, and families of constructed 3-NAE-SAT and 2-XOR-SAT formulas. We also prove that the algorithm fails on the one-dimensional Ising model, and suggest different techniques for overcoming this. Finally, we present empirical results that explore the tightness of the runtime bounds and the performance of the technique on randomized satisfiability variants.

Symmetry-prohibited thermalization after a quantum quench

The observable long-time behavior of an isolated many-body system after a quantum quench is considered, i.e. an eigenstate (or an equilibrium ensemble) of some pre-quench Hamiltonian H serves as initial condition which then evolves in time according to some post-quench Hamiltonian H p . Absence of thermalization is analytically demonstrated for a large class of quite common pre- and post-quench spin Hamiltonians. The main requirement is that the pre-quench Hamiltonian must exhibit a Z 2 (spin-flip) symmetry, which would be spontaneously broken in the thermodynamic limit, though we actually focus on finite (but large) systems. On the other hand, the post-quench Hamiltonian must violate the Z 2 symmetry, but for the rest may be non-integrable and may obey the eigenstate thermalization hypothesis for (sums of) few-body observables.

Commuting-projector Hamiltonians for chiral topological phases built from parafermions

We introduce a family of commuting-projector Hamiltonians whose degrees of freedom involve $\mathbb{Z}_{3}$ parafermion zero modes residing in a parent fractional-quantum-Hall fluid. The two simplest models in this family emerge from dressing Ising-paramagnet and toric-code spin models with parafermions; we study their edge properties, anyonic excitations, and ground-state degeneracy. We show that the first model realizes a symmetry-enriched topological phase (SET) for which $\mathbb{Z}_2$ spin-flip symmetry from the Ising paramagnet permutes the anyons. Interestingly, the interface between this SET and the parent quantum-Hall phase realizes symmetry-enforced $\mathbb{Z}_3$ parafermion criticality with no fine-tuning required. The second model exhibits a non-Abelian phase that is consistent with $\text{SU}(2)_{4}$ topological order, and can be accessed by gauging the $\mathbb{Z}_{2}$ symmetry in the SET. Employing Levin-Wen string-net models with $\mathbb{Z}_{2}$-graded structure, we generalize this picture to construct a large class of commuting-projector models for $\mathbb{Z}_{2}$ SETs and non-Abelian topological orders exhibiting the same relation. Our construction provides the first commuting-projector-Hamiltonian realization of chiral bosonic non-Abelian topological order.

Symmetrized density matrix renormalization group algorithm for low-lying excited states of conjugated carbon systems: Application to 1,12-benzoperylene and polychrysene

The symmetry adapted density matrix renormalization group (SDMRG) technique has been an efficient method for studying low-lying eigenstates in one- and quasi-one-dimensional electronic systems. However, the SDMRG method had bottlenecks involving the construction of linearly independent symmetry adapted basis states as the symmetry matrices in the DMRG basis were not sparse. We have developed a modified algorithm to overcome this bottleneck. The new method incorporates end-to-end interchange symmetry $({C}_{2})$, electron-hole symmetry $(J)$, and parity or spin-flip symmetry $(P)$ in these calculations. The one-to-one correspondence between direct-product basis states in the DMRG Hilbert space for these symmetry operations renders the symmetry matrices in the new basis with maximum sparseness, just one nonzero matrix element per row. Using methods similar to those employed in the exact diagonalization technique for Pariser-Parr-Pople (PPP) models, developed in the 1980s, it is possible to construct orthogonal SDMRG basis states while bypassing the slow step of the Gram-Schmidt orthonormalization procedure. The method together with the PPP model which incorporates long-range electronic correlations is employed to study the correlated excited-state spectra of 1,12-benzoperylene and a narrow mixed graphene nanoribbon with a chrysene molecule as the building unit, comprising both zigzag and cove-edge structures.

Three-dimensional Ising model of polarity formation in molecular crystals

Polarity formation in a three-dimensional array of molecules is described as a symmetry breaking effect of a generalized Ising Hamiltonian. Geometrical constraints in conjunction with asymmetric multipole interactions are able to break the spin flip symmetry generating a non-vanishing average local polarization.

A bilayer Double Semion model with symmetry-enriched topological order

We construct a new model of two-dimensional quantum spin systems that combines intrinsic topological orders and a global symmetry called flavour symmetry. It is referred as the bilayer Doubled Semion model (bDS) and is an instance of symmetry-enriched topological order. A honeycomb bilayer lattice is introduced to combine a Double Semion Topological Order with a global spin–flavour symmetry to get the fractionalization of its quasiparticles. The bDS model exhibits non-trivial braiding self-statistics of excitations and its dual model constitutes a Symmetry-Protected Topological Order with novel edge states. This dual model gives rise to a bilayer Non-Trivial Paramagnet that is invariant under the flavour symmetry and the well-known spin flip symmetry.

Topology of nonsymmorphic crystalline insulators and superconductors

Topological classification in our previous paper [K. Shiozaki and M. Sato, Phys. Rev. B ${\bf 90}$, 165114 (2014)] is extended to nonsymmorphic crystalline insulators and superconductors. Using the twisted equivariant $K$-theory, we complete the classification of topological crystalline insulators and superconductors in the presence of additional order-two nonsymmorphic space group symmetries. The order-two nonsymmorphic space groups include half lattice translation with $Z_2$ flip, glide, two-fold screw, and their magnetic space groups. We find that the topological periodic table shows modulo-2 periodicity in the number of flipped coordinates under the order-two nonsymmorphic space group. It is pointed out that the nonsymmorphic space groups allow $\mathbb{Z}_2$ topological phases even in the absence of time-reversal and/or particle-hole symmetries. Furthermore, the coexistence of the nonsymmorphic space group with the time-reversal and/or particle-hole symmetries provides novel $\mathbb{Z}_4$ topological phases, which have not been realized in ordinary topological insulators and superconductors. We present model Hamiltonians of these new topological phases and the analytic expression of the $\mathbb{Z}_2$ and $\mathbb{Z}_4$ topological invariants. The half lattice translation with $Z_2$ spin flip and glide symmetry are compatible with the existence of the boundary, leading to topological surface gapless modes protected by such order-two nonsymmorphic symmetries. We also discuss unique features of these gapless surface modes.

Quantum state transfer in XXZ spin chains: A measurement induced transport method

We study the information transferring ability of a spin-1∕2 XXZ Hamiltonian for two different proposals of state transfer, namely, the well-studied attaching scenario and the recently proposed measurement induced transport. The latter one has been inspired by recent achievements in optical lattice experiments for local addressability of individual atoms and their time evolution when only local rotations and measurements are available and local control of the Hamiltonian is very limited. We show that while the both scenarios performs with almost similar quality in the case of non-interacting free fermionic XX phase, the difference become more pronounced around the isotropic Heisenberg point. Our study shows that the presence of spin-flip symmetry plays a key point in the quality of state transfer and each scenario which benefits from this symmetry transfers the quantum states with higher fidelity. In fact, for even chains this symmetry exists only for the measurement induced dynamics which then gives higher transport quality and for odd chains the spin-flip symmetry is only valid for the attaching scenarios which become more superior. We also study the effect of thermal fluctuations and environmental interactions on both scenarios.

Moments of Wishart-Laguerre and Jacobi ensembles of random matrices: application to the quantum transport problem in chaotic cavities

We collect explicit and user-friendly expressions for one-point densities of the real eigenvalues $\{\lambda_i\}$ of $N\times N$ Wishart-Laguerre and Jacobi random matrices with orthogonal, unitary and symplectic symmetry. Using these formulae, we compute integer moments $\tau_n=<\sum_{i=1}^N\lambda_i^n>$ for all symmetry classes without any large $N$ approximation. In particular, our results provide exact expressions for moments of transmission eigenvalues in chaotic cavities with time-reversal or spin-flip symmetry and supporting a finite and arbitrary number of electronic channels in the two incoming leads.

Quantum discord in matrix product systems

We consider a class of quantum systems with spin-flip symmetry and derive the quantum correlation measured by the quantum discord (QD). As an illustration, we investigate the QD in a three-body interaction model and an XYZ interaction model, whose ground states can be expressed as matrix product states, and the QD is exactly soluble. We show that the QD behaves differently than the quantum entanglement (QE) in many ways; for example, they may show opposite monotonicity and completely different finite-size effects. Furthermore, we compare the capability of the QD and the QE to detect quantum phase transitions (QPTs) and find that the QD is more reliable than the QE for signaling QPTs in these models: In the three-body interaction model, the QE is singular at the quantum critical point, however, it exhibits an additional singularity in the noncritical region, while the analyticity of the QD can be used to identify the quantum critical point perfectly; and in the XYZ interaction model, the QE vanishes in the thermodynamic limit, thus losing its ability to detect QPTs, while the QD still functions very well.

The effect of pure state structure on nonequilibrium dynamics

Motivated by short-range Ising spin glasses, we review some rigorous results and theirconsequences for the relation between the number/nature of equilibrium pure states andnonequilibrium dynamics. Two of the consequences for spin glass dynamics following aninstantaneous deep quench to a temperature with broken spin flip symmetry are: (1) almostall initial configurations lie on the boundary between the basins of attraction of multiplepure states; (2) unless there are uncountably many pure states with almost all pairs havingzero overlap, there can be no equilibration to a pure state as time . We discuss the relevance of these results to the difficulty of equilibration of spin glasses.We also review some results concerning the ‘nature versus nurture’ problem of whether thelarge-t behavior of both ferromagnets and spin glasses following a deep quench is determined moreby the initial configuration (nature) or by the dynamics realization (nurture).

Eightfold way from dynamical first principles in strongly coupled lattice quantum chromodynamics

We obtain from first principles, i.e., from the quark-gluon dynamics, the Gell’Mann-Ne’eman baryonic eightfold way energy momentum spectrum exactly in an imaginary-time functional integral formulation of strongly coupled lattice quantum chromodynamics in 3+1 dimensions, with local SU(3)c gauge and global SU(3)f flavor symmetries. We take the hopping parameter κ and the pure gauge coupling β satisfying the strong coupling regime condition 0⩽β⪡κ⪡1. The form of the 56 baryon fields emerges naturally from the dynamics and is unveiled using the hyperplane decoupling method. There is no a priori guesswork. In the associated physical quantum mechanical Hilbert space H, spectral representations are derived for the two-baryon functions, which are used to rigorously detect the particles in the energy-momentum spectrum. Using the SU(3)f symmetry, the 56 baryon states admit a decomposition into 8×2 states associated with a spin 1∕2 octet and 10×4 states associated with a spin 3∕2 decuplet. The states are labeled by the quantum numbers of total hypercharge Y, total isospin I, its third component I3, and the value of the quadratic Casimir of SU(3)f; they also carry a label of total spin J and its z component Jz. The total spin operators are defined using π∕2 rotations about the spatial coordinate axes and for improper zero momentum baryon states agree with the infinitesimal generators of the continuum. We show there is a partial restoration of continuous rotational symmetry which implies that all the octet (decuplet) masses are the same. For β=0, the masses of the 56 baryon states have the form M=−3lnκ−3κ3∕4+κ6r(κ), with r(κ) analytic. There is no mass splitting within the octet (decuplet). However, we find an octet-decuplet mass splitting given by 3κ6∕4+O(κ7). For β=0,[M(κ,β)−(−3lnκ)], the non-singular part of the masses, is analytic in κ and β and the mass splitting persists for β≠0. For spatial momentum p≠0, p=(p1,p2,p3)∊(−π,π]3, the 56 baryon dispersion curves have the form w(κ,p)=−3lnκ−3κ3∕4+κ3∑j=1,2,3(1−cospj)∕4+r(κ,p), where r(κ,p) is of O(κ6). For the octet, r(κ,p) is jointly analytic in κ and in each pj for small ∣Impj∣. For each baryon, there is an antibaryon related to it by charge conjugation and with identical spectral properties. It is shown that the spectrum associated with baryons and antibaryons is the only spectrum in the subspace of H with an odd number of quarks, up to near the meson-baryon energy threshold of ≈−5lnκ. A new time reflection is found which is used to define a local spin flip symmetry. The spin flip symmetry, together with the usual parity, time reversal, and spatial π∕2 rotation symmetries and analytic implicit function arguments, are used to obtain these results. Our method extends to the SU(N)f case to uncover (2N+2)!∕[3!(2N−1)!] baryon states and also to treat mesons. Coupling our baryon results with our similar results for the eightfold mesons (of asymptotic mass −2lnκ) shows that the model exhibits confinement up to near the two-meson threshold.

Origin of the spin-triplet Andreev reflection at ferromagnet/s-wave superconductor interface

To understand the origin of the spin-triplet Andreev reflection which may appear at a ferromagnet (F)/s-wave superconductor (S) interface, we study tunneling conductance spectra of F/2DEG/s-wave S tunnel junctions, with 2DEG the two-dimensional electron gas with Rashba spin-orbit coupling and subject to a magnetic field. It is found that both spin-flip and broken time-reversal symmetry in the 2DEG are conditions necessary to giving rise to the novel Andreev reflection and spin-triplet pairing correlations.

Twelve sublattice ordered phase in theJ1−J2model on the kagomé lattice

Motivated by recent experiments on an S=1/2 antiferromagnet on the kagome lattice, we investigate the Heisenberg J1-J2 model with ferromagnetic J1 and antiferromagnetic J2. Classically the ground state displays Neel long range order with 12 non-coplanar sublattices. The order parameter has the symmetry of a cuboctahedron so that it fully breaks SO(3) as well as the spin-flip symmetry, and we expect from the latter a Z2 symmetry breaking pattern. As might be expected from the Mermin-Wagner theorem in two dimensions, the SO(3) symmetry is restored by thermal fluctuations while the Z2 symmetry breaking persists up to a finite temperature. A complete study of S=1/2 exact spectra reveals that the classical order subsists for quantum spins in a finite range of parameters. First order spin wave calculations give the range of existence of this phase and the renormalisations at T=0 of the order parameters associated to both symmetry breakings. This phase is destroyed by quantum fluctuations for a small but finite J2/|J1|\simeq3 consistently with exact spectra studies which indicate a gapped phase.

Surface Tension and Wulff Shape for a Lattice Model without Spin Flip Symmetry

We propose a new definition of surface tension and check it in a spin model of the Pirogov-Sinai class without symmetry. We study the model at low temperatures on the phase transitions line and prove: (i) existence of the surface tension in the thermodynamic limit, for any orientation of the surface and in all dimensions $ d \geq 2 $; (ii) the Wulff shape constructed with such a surface tension coincides with the equilibrium shape of the cluster which appears when fixing the total spin magnetization (Wulff problem).

Entanglement, mixedness, and spin-flip symmetry in multiple-qubit systems

A relationship between a recently introduced multipartite entanglement measure, state mixedness, and spin-flip symmetry is established for any finite number of qubits. It is also shown that, for those classes of states invariant under the spin-flip transformation, there is a complementarity relation between multipartite entanglement and mixedness. A number of example classes of multiple-qubit systems are studied in light of this relationship.

A Quantitative Clustering Approach to Ultrametricity in Spin Glasses

We discuss the problem of ultrametricity in mean field spin glasses by means of a hierarchical clustering algorithm. We complement the clustering approach with quantitative testing: we discuss both in some detail. We show that the elimination of the (in this context accidental) spin flip symmetry plays a crucial role in the analysis, since the symmetry hides the real nature of the data. We are able to use in the analysis disorder averaged quantities. We are able to exhibit a number of features of the low $T$ phase of the mean field theory, and to claim that the full hierarchical structure can be observed without ambiguities only on very large lattice volumes, not currently accessible by numerical simulations.

Spin-flip symmetry and synchronization.

In the context of optimization by evolutionary algorithms (EAs), epistasis, deception, and scaling are well-known examples of problem difficulty characteristics. The presence of one such characteristic in the representation of a search problem indicates a certain type of difficulty the EA is to encounter during its search for globally optimal configurations. In this paper, we claim that the occurrence of symmetry in the representation is another problem difficulty characteristic and discuss one particular form, spin-flip symmetry, characterized by fitness invariant permutations on the alphabet. Its usual effect on unspecialized EAs, premature convergence due to synchronization problems, is discussed in detail. We discuss five different ways to specialize EAs to cope with the symmetry: adapting the genetic operators, changing the fitness function, using a niching technique, using a distributed EA, and attaching a highly redundant genotype-phenotype mapping.