Matrix Elements Of Spin Operators Research Articles

Relaxation of antiferromagnetic order and growth of Rényi entropy in a generalized Heisenberg star

Interacting central spin systems, in which a central spin is coupled to a strongly correlated spin bath with intrabath interaction, consist of an important class of spin systems beyond the usual Gaudin magnet. These systems are relevant to several realistic setups and serve as an interesting platform to study interaction controlled decoherence and frustration induced instability of magnetic order. Using an equations-of-motion method based on analytical representations of spin-operator matrix elements in the XX chain, we obtain exact long-time dynamics of a generalized Heisenberg star consisting of a spin-$S$ central spin and an inhomogeneously coupled XXZ chain of $N\ensuremath{\le}16$ bath spins. In contrast to previous studies where the central spin dynamics is mainly concerned, we focus on the influence of the central spin on the dynamics of magnetic orders within the spin bath. By preparing the XXZ bath in a N\'eel state, we find that in the gapless phase of the bath even weak system-bath coupling could lead to nearly perfect relaxation of the antiferromagnetic order. In the gapped phase, the staggered magnetization decays rapidly and approaches a steady value that increases with increasing anisotropy parameter. These findings suggest the possibility of controlling internal dynamics of strongly correlated many-spin systems by certain coupled auxiliary systems of even few degrees of freedom. We also study the dynamics of the R\'enyi entanglement entropy of the central spin when the bath is prepared in the ground state. Both the overall profile and initial growth rate of the R\'enyi entropy are found to exhibit minima at the critical point of the XXZ bath.

Hilbert Space Structure of the Low Energy Sector of U(N) Quantum Hall Ferromagnets and Their Classical Limit

Using the Lieb–Mattis ordering theorem of electronic energy levels, we identify the Hilbert space of the low energy sector of U(N) quantum Hall/Heisenberg ferromagnets at filling factor M for L Landau/lattice sites with the carrier space of irreducible representations of U(N) described by rectangular Young tableaux of M rows and L columns, and associated with Grassmannian phase spaces U(N)/U(M)×U(N−M). We embed this N-component fermion mixture in Fock space through a Schwinger–Jordan (boson and fermion) representation of U(N)-spin operators. We provide different realizations of basis vectors using Young diagrams, Gelfand–Tsetlin patterns and Fock states (for an electron/flux occupation number in the fermionic/bosonic representation). U(N)-spin operator matrix elements in the Gelfand–Tsetlin basis are explicitly given. Coherent state excitations above the ground state are computed and labeled by complex (N−M)×M matrix points Z on the Grassmannian phase space. They adopt the form of a U(N) displaced/rotated highest-weight vector, or a multinomial Bose–Einstein condensate in the flux occupation number representation. Replacing U(N)-spin operators by their expectation values in a Grassmannian coherent state allows for a semi-classical treatment of the low energy (long wavelength) U(N)-spin-wave coherent excitations (skyrmions) of U(N) quantum Hall ferromagnets in terms of Grasmannian nonlinear sigma models.

Determinant representations of spin-operator matrix elements in the XX spin chain and their applications

For the one-dimensional spin-1/2 XX model with either periodic or open boundary conditions, it is shown by using a fermionic approach that the matrix element of the spin operator $S^-_j$ ($S^-_{j}S^+_{j'}$) between two eigenstates with numbers of excitations $n$ and $n+1$ ($n$ and $n$) can be expressed as the determinant of an appropriate $(n+1)\times (n+1)$ matrix whose entries involve the coefficients of the canonical transformations diagonalizing the model. In the special case of a homogeneous periodic XX chain, the matrix element of $S^-_j$ reduces to a variant of the Cauchy determinant that can be evaluated analytically to yield a factorized expression. The obtained compact representations of these matrix elements are then applied to two physical scenarios: (i) Nonlinear optical response of molecular aggregates, for which the determinant representation of the transition dipole matrix elements between eigenstates provides a convenient way to calculate the third-order nonlinear responses for aggregates from small to large sizes compared with the optical wavelength, and (ii) real-time dynamics of an interacting Dicke model consisting of a single bosonic mode coupled to a one-dimensional XX spin bath. In this setup, full quantum calculation up to $N\leq 16$ spins for vanishing intrabath coupling shows that the decay of the reduced bosonic occupation number approaches a finite plateau value (in the long-time limit) that depends on the ratio between the number of excitations and the total number of spins. Our results can find useful applications in various "system-bath" systems, with the system part inhomogeneously coupled to an interacting XX chain.

Spin operator matrix elements in the quantum Ising chain: fermion approach

Using some modification of the standard fermion technique we derive factorized formulasfor spin operator matrix elements (form factors) between general eigenstates of theHamiltonian of the quantum Ising chain in a transverse field of finite length. Thederivation is based on the approach recently used to derive factorized formulas forZN-spin operator matrix elements between ground eigenstates of the Hamiltonian of theZN-symmetric superintegrable chiral Potts quantum chain. The obtained factorized formulasfor the matrix elements of the Ising chain coincide with the corresponding expressionsobtained by the separation of variables method.

Spin Operator Matrix Elements in the Superintegrable Chiral Potts Quantum Chain

We derive spin operator matrix elements between general eigenstates of the superintegrable ℤ N -symmetric chiral Potts quantum chain of finite length. Our starting point is the extended Onsager algebra recently proposed by Baxter. For each pair of spaces (Onsager sectors) of the irreducible representations of the Onsager algebra, we calculate the spin matrix elements between the eigenstates of the Hamiltonian of the quantum chain in factorized form, up to an overall scalar factor. This factor is known for the ground state Onsager sectors. For the matrix elements between the ground states of these sectors we perform the thermodynamic limit and obtain the formula for the order parameters. For the Ising quantum chain in a transverse field (N=2 case) the factorized form for the matrix elements coincides with the corresponding expressions obtained recently by the Separation of Variables method.

Yang-Mills Correlation Functions from Integrable Spin Chains

The relation between the dilatation operator of N=4 Yang-Mills theory and integrable spin chains makes it possible to compute the one-loop anomalous dimensions of all operators in the theory. In this paper we show how to apply the technology of integrable spin chains to the calculation of Yang-Mills correlation functions by expressing them in terms of matrix elements of spin operators on the corresponding spin chain. We illustrate this method with several examples in the SU(2) sector described by the XXX_1/2 chain.

Matrix elements of spin operators in exchange coupled tetrameric metal clusters

An analytical evaluation of matrix elements of single spin operators in tetranuclear mixed-valence metal clusters is presented. The case of a ground state of the cluster with one strongly anti-ferromagnetic coupled spin pair and the second pair with undefined spin state is considered in detail. Such spin states have been discussed as a model for the water-oxidizing complex in photosystem II. Consequences of the structure of the ground state apparent in the hyperfine structure observed by EPR are discussed. The spin coupling algebra is presented and evaluated using the Mathematica software system.

Graphical unitary group approach to spin–spin interaction

AbstractA detailed exposition of spin–spin operator matrix elements is presented in the context of the graphical unitary group approach (GUGA) to atomic and molecular physics and quantum chemistry. A compendium of subgraph types and formulae is given. Aspects of computer implementation within the structure of the Columbus CI programs is discussed.