Solvable Spin Research Articles

Correlation Function and Simplified TBA Equations for XXZ Chain

The calculation of the correlation functions of Bethe ansatz solvable models is very difficult problem. Among these solvable models spin 1/2 XXX chain has been investigated for a long time. Even for this model only the nearest neighbor and the second neighbor correlations were known. In 1990's multiple integral formula for the general correlations is derived. But the integration of this formula is also very difficult problem. Recently these integrals are decomposed to products of one dimensional integrals and at zero temperature, zero magnetic field and isotropic case, correlation functions are expressed by $\log 2$ and Riemann's zeta functions with odd integer argument $\zeta(3),\zeta(5),\zeta(7),...$. We can calculate density sub-matrix of successive seven sites. Entanglement entropy of seven sites is calculated. These methods can be extended to XXZ chain up to $n=4$. Correlation functions are expressed by the generalized zeta functions. Several years ago I derived new thermodynamic Bethe ansatz equation for XXZ chain. This is quite different with Yang-Yang type TBA equations and contains only one unknown function. This equation is very useful to get the high temperature expansion. In this paper we get the analytic solution of this equation at $\Delta=0$.

Topological order in an exactly solvable 3D spin model

We study a 3D generalization of the toric code model introduced recently by Chamon. This is an exactly solvable spin model with six-qubit nearest-neighbor interactions on an FCC lattice whose ground space exhibits topological quantum order. The elementary excitations of this model which we call monopoles can be geometrically described as the corners of rectangular-shaped membranes. We prove that the creation of an isolated monopole separated from other monopoles by a distance R requires an operator acting on Ω( R 2) qubits. Composite particles that consist of two monopoles (dipoles) and four monopoles (quadrupoles) can be described as end-points of strings. The peculiar feature of the model is that dipole-type strings are rigid, that is, such strings must be aligned with face-diagonals of the lattice. For periodic boundary conditions the ground space can encode 4 g qubits where g is the greatest common divisor of the lattice dimensions. We describe a complete set of logical operators acting on the encoded qubits in terms of closed strings and closed membranes.

Manipulating unpaired Majorana fermions in a quantum spin chain

We analyze an exactly solvable spin-1/2 chain which is a generalized version of Kitaev's honeycomb model. We show that every state of the system has a ${2}^{N/4}$-fold degeneracy, where $N$ is the number of sites. We present analytic solutions for the zero energy modes of the Majorana fermions. Localized, unpaired Majorana modes occur even in the bulk of the chain and they are bound to kink (antikink) ${Z}_{2}$ flux configurations. The unpaired Majorana modes can therefore be created and manipulated if the ${Z}_{2}$ flux configurations can be controlled. We delineate the regions in parameter space for homogenous chains where the zero modes occur. We further show that there is a large parameter space for inhomogenous chains where the unpaired modes occur and that their wave functions can be tuned if the couplings of the model can be tuned.

Exact renormalization in quantum spin chains

We introduce a real-space exact renormalization group method to find exactly solvable quantum spin chains and their ground states. This method allows us to provide a complete list for exact solutions within SU(2) symmetric quantum spin chains with $S\leq 4$ and nearest-neighbor interactions, as well as examples with S=5. We obtain two classes of solutions: One of them converges to the fixed points of renormalization group and the ground states are matrix product states. Another one does not have renormalization fixed points and the ground states are partially ferromagnetic states.

Quantum Spin Metal State on a Decorated Honeycomb Lattice

We present a modification of the exactly solvable spin-(1/)2 Kitaev model on the decorated honeycomb lattice, with a ground state of "spin metal" type. The model is diagonalized in terms of Majorana fermions; the latter form a 2D gapless state with a Fermi circle whose size depends on the ratio of exchange couplings. Low-temperature heat capacity C(T) and dynamic spin susceptibility χ(ω,T) are calculated in the case of small Fermi circle. Whereas, C(T)∼T at low temperatures as it is expected for a Fermi liquid, spin excitations are gapped and χ(ω,T) demonstrates unusual behavior with a power-law peak near the resonance frequency. The corresponding exponent as well as the peak shape are calculated.

Channel capacities of an exactly solvable spin-star system

We calculate the entanglement-assisted and unassisted channel capacities of an exactly solvable spin star system, which models the quantum dephasing channel. The capacities for this non-Markovian model exhibit a strong dependence on the coupling strengths of the bath spins with the system, the bath temperature, and the number of bath spins. For equal couplings and bath frequencies, the channel becomes periodically noiseless.

Probing Quantum Frustrated Systems via Factorization of the Ground State

The existence of definite orders in frustrated quantum systems is related rigorously to the occurrence of fully factorized ground states below a threshold value of the frustration. Ground-state separability thus provides a natural measure of frustration: strongly frustrated systems are those that cannot accommodate for classical-like solutions. The exact form of the factorized ground states and the critical frustration are determined for various classes of nonexactly solvable spin models with different spatial ranges of the interactions. For weak frustration, the existence of disentangling transitions determines the range of applicability of mean-field descriptions in biological and physical problems such as stochastic gene expression and the stability of long-period modulated structures.

Entanglement and local extremes at an infinite-order quantum phase transition

The characterization of an infinite-order quantum phase transition (QPT) by entanglement measures is analyzed. To this aim, we consider two closely related solvable spin-1/2 chains, namely, the Ashkin-Teller and the staggered XXZ models. These systems display a distinct pattern of eigenstates but exhibit the same thermodynamics, i.e. the same energy spectrum. By performing exact diagonalization, we investigate the behavior of pairwise and block entanglement in the ground state of both models. In contrast with the XXZ chain, we show that pairwise entanglement fails in the characterization of the infinite-order QPT in the Ashkin-Teller model, although it can be achieved by analyzing the distance of the pair state from the separability boundary. Concerning block entanglement, we show that both XXZ and Ashkin-Teller models exhibit identical von Neumann entropies as long as a suitable choice of blocks is performed. Entanglement entropy is then shown to be able to identify the quantum phase diagram, even though its local extremes (either maximum or minimum) may also appear in the absence of any infinite-order QPT.

Reentrant phenomenon in the exactly solvable mixed spin-1/2 and spin-1 Ising-Heisenberg model on diamond-like decorated planar lattices

Ground-state and finite-temperature behaviour of the mixed spin-1/2 and spin-1 Ising-Heisenberg model on decorated planar lattices consisting of inter-connected diamonds is investigated by means of the generalised decoration-iteration mapping transformation. The obtained exact results clearly point out that this model has a rather complex ground state composed of two unusual quantum phases, which is valid regardless of the lattice topology as well as the spatial dimensionality of the investigated system. It is shown that the diamond-like decorated planar lattices with a sufficiently high coordination number may exhibit a striking critical behaviour including reentrant phase transitions with two or three consecutive critical points.

An Exactly Solvable Spin-1 Model under External Magnetic Field

A two-fold Cayley tree with fully q-coordinated sites is constructed and the spin-1 Blume–Capel model in the presence of an external magnetic field is solved exactly. The relevant properties such as magnetization m and square magnetic moment q are investigated as functions of temperature and external magnetic field both for ferromagnetic and antiferromagnetic couplings. In the study, we propose a possible mechanism of the plateau based on the exchange couplings.

Exact spinon excitations in an integrable ferrimagnetic spin chain

An exactly solvable one-dimensional ferrimagnetic quantum spin chain model with alternating 1/2-spins and 1-spins is proposed and solved via algebraic Bethe ansatz. It is found that the ground state of this model is a ferrimagnetic state with a residual magnetization. It is proven that the low-lying elementary excitations of the system are the varieties of gapless spinons rather than spin waves.

Algebraic Spin Liquid in an Exactly Solvable Spin Model

We have proposed an exactly solvable quantum spin-3/2 model on a square lattice. Its ground state is a quantum spin liquid with a half-integer spin per unit cell. The fermionic excitations, dubbed as "spinons", are gapless with a linear dispersion, while the topological "vison" excitations are gapped. Moreover, these massless fermionic spinon excitations are topologically stable. Thus, this model is, to the best of our knowledge, the first exactly solvable model of half-integer spins whose ground state is an "algebraic spin liquid."

Nonlinear integral equations and determinant formulae of the open XXZ spin chain

We derive a nonlinear integral equation for the Bethe ansatz solvable open XXZ spin chain of arbitrary length describing the lowest lying state with zero magnetization. For this case we show how to combine the integral representation with the known determinant formula of norms and scalar products.

Entanglement in spin chains with gradients

We study solvable spin chains where either fields or couplings vary linearly in space and createa sandwich-like structure of the ground state. We find that the entropy of entanglementbetween two halves of a chain varies logarithmically with the interface width. After quenchingto a homogeneous critical system, the entropy grows logarithmically in time in theXX model, but quadratically in the transverse Ising chain. We explain this behaviour andindicate generalizations to other power laws.

Mott Insulators in the Strong Spin-Orbit Coupling Limit: From Heisenberg to a Quantum Compass and Kitaev Models

We study the magnetic interactions in Mott-Hubbard systems with partially filled t_{2g} levels and with strong spin-orbit coupling. The latter entangles the spin and orbital spaces, and leads to a rich variety of the low energy Hamiltonians that extrapolate from the Heisenberg to a quantum compass model depending on the lattice geometry. This gives way to "engineer" in such Mott insulators an exactly solvable spin model by Kitaev relevant for quantum computation. We, finally, explain "weak" ferromagnetism, with an anomalously large ferromagnetic moment, in Sr2IrO4.

Mechanism of a prototypical synthetic membrane-active antimicrobial: Efficient hole-punching via interaction with negative intrinsic curvature lipids

Phenylene ethynylenes comprise a prototypical class of synthetic antimicrobial compounds that mimic antimicrobial peptides produced by eukaryotes and have broad-spectrum antimicrobial activity. We show unambiguously that bacterial membrane permeation by these antimicrobials depends on the presence of negative intrinsic curvature lipids, such as phosphatidylethanolamine (PE) lipids, found in high concentrations within bacterial membranes. Plate-killing assays indicate that a PE-knockout mutant strain of Escherichia coli drastically out-survives the wild type against the membrane-active phenylene ethynylene antimicrobials, whereas the opposite is true when challenged with traditional metabolic antibiotics. That the PE deletion is a lethal mutation in normative environments suggests that resistant bacterial strains do not evolve because a lethal mutation is required to gain immunity. PE lipids allow efficient generation of negative curvature required for the circumferential barrel of an induced membrane pore; an inverted hexagonal H(II) phase, which consists of arrays of water channels, is induced by a small number of antimicrobial molecules. The estimated antimicrobial occupation in these water channels is nonlinear and jumps from approximately 1 to 3 per 4 nm of induced water channel length as the global antimicrobial concentration is increased. By comparing to exactly solvable 1D spin models for magnetic systems, we quantify the cooperativity of these antimicrobials.

Socio-economical dynamics as a solvable spin system on co-evolving networks

We consider social systems in which agents are not only characterized by their states but also have the freedom to choose their interaction partners to maximize their utility. We map such systems onto an Ising model in which spins are dynamically coupled by links in a dynamical network. In this model there are two dynamical quantitieswhich arrange towards a minimum energy state in the canonical framework:the spins, si, and the adjacency matrix elements, cij.The model is exactly solvable because microcanonical partition functions reduce to productsof binomial factors as a direct consequence of the cij minimizing energy. We solve the system for finite sizes and for the two possible thermodynamic limits and discussthe phase diagrams.

An exactly solvable supersymmetric spin chain of [formula omitted] type

We construct a new exactly solvable supersymmetric spin chain related to the BC N extended root system, which includes as a particular case the BC N version of the Polychronakos–Frahm spin chain. We also introduce a supersymmetric spin dynamical model of Calogero type which yields the new chain in the large coupling limit. This connection is exploited to derive two different closed-form expressions for the chain's partition function by means of Polychronakos's freezing trick. We establish a boson–fermion duality relation for the new chain's spectrum, which is in fact valid for a large class of (not necessarily integrable) spin chains of BC N type. The exact expressions for the partition function are also used to study the chain's spectrum as a whole, showing that the level density is normally distributed even for a moderately large number of particles. We also determine a simple analytic approximation to the distribution of normalized spacings between consecutive levels which fits the numerical data with remarkable accuracy. Our results provide further evidence that spin chains of Haldane–Shastry type are exceptional integrable models, in the sense that their spacings distribution is not Poissonian, as posited by the Berry–Tabor conjecture for “generic” quantum integrable systems.

Genuine tripartite entanglement monotone of(2⊗2⊗n)-dimensional systems

A genuine tripartite entanglement monotone is presented for $(2\ensuremath{\bigotimes}2\ensuremath{\bigotimes}n)$-dimensional tripartite pure states, obtained by introducing an entanglement measure for bipartite pure states. As an application, we consider the genuine tripartite entanglement of the ground state of the exactly solvable isotropic spin-$1/2$ chain with three-spin interaction. It is shown that the singular behavior of the genuine tripartite entanglement exactly signals a quantum phase transition.

Partially Solvable Spin Chains and QES Spin Models

In this paper we prove an extension of the usual freezing trick argument which can be applied to a number of quasi-exactly solvable spin models of Calogero–Sutherland type. In order to illustrate the application of this method we analyze a partially solvable spin chain presenting near-neighbors interactions which was introduced and studied in J. Phys. A: Math. Theor. 40 (2007) 1857–1883; Nucl. Phys. 789 (2008) 452–482. Our discussion focuses on the existence of integer eigenvalues.