Two-dimensional quantum spin hamiltonian: Spectral and dynamical domplexities

We report the comparative study of the energy-level properties of the Ising, the Heisenberg, and the dipolar quantum Hamiltonians of 1/2 spins on a two-dimensional (2D) square lattice. We evaluate the number of constants of motion by the dimension of the null space of the von Neumann operator. In this way, we show that the three Hamiltonians are ordered according to: Ising---Heisenberg---dipolar, with respect to their integrability properties. Extra dynamical symmetries persist beyond the geometric symmetries in the 2D Heisenberg Hamiltonian as evidenced by a Poisson statistics for the level spacings of the Hamiltonian desymmetrized by the geometric symmetries. On the other hand, the dipolar Hamiltonian has no extra symmetries beyond the geometric ones because the desymmetrized Hamiltonian shows Wigner spacing statistics characteristic of orthogonal ensembles of random matrices.We report the comparative study of the energy-level properties of the Ising, the Heisenberg, and the dipolar quantum Hamiltonians of 1/2 spins on a two-dimensional (2D) square lattice. We evaluate the number of constants of motion by the dimension of the null space of the von Neumann operator. In this way, we show that the three Hamiltonians are ordered according to: Ising---Heisenberg---dipolar, with respect to their integrability properties. Extra dynamical symmetries persist beyond the geometric symmetries in the 2D Heisenberg Hamiltonian as evidenced by a Poisson statistics for the level spacings of the Hamiltonian desymmetrized by the geometric symmetries. On the other hand, the dipolar Hamiltonian has no extra symmetries beyond the geometric ones because the desymmetrized Hamiltonian shows Wigner spacing statistics characteristic of orthogonal ensembles of random matrices.

Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topicsJan 1, 199430 citations

Dynamics of Two-Dimensional Quantum Spin Systems

M. S. Makivić, Mark Jarrell

Springer proceedings in physicsJan 1, 19930 citations

Dynamical properties in low-dimensional quantum spin systems

貴則杉本

Jan 1, 20120 citations

Jordan block structure of two-dimensional quantum neural cellular network

Bi Qiao, Harry E. Ruda

We present a Jordan type spectral decomposition for the Hamiltonian operator of a two-dimensional (2D) Ising model driven by a local external field. The results reveal that the external field induces a dynamic contribution to the Jordan block structure, while the interaction between lattice spins contributes to the energy spectrum, tending to attract the system to a minimum energy state. Both interactions contributing to the principle vectors of the Hamiltonian operator reflect their correlation. Since we adopt a first-principle approach the results are useful to general 2D Ising spin systems subject to time-independent external fields irrespective of whether the system is in equilibrium.We present a Jordan type spectral decomposition for the Hamiltonian operator of a two-dimensional (2D) Ising model driven by a local external field. The results reveal that the external field induces a dynamic contribution to the Jordan block structure, while the interaction between lattice spins contributes to the energy spectrum, tending to attract the system to a minimum energy state. Both interactions contributing to the principle vectors of the Hamiltonian operator reflect their correlation. Since we adopt a first-principle approach the results are useful to general 2D Ising spin systems subject to time-independent external fields irrespective of whether the system is in equilibrium.

Journal of Applied PhysicsJul 1, 19992 citations

Statics and dynamics of quantum XY and Heisenberg systems on graphs

Tobias J. Osborne

We consider the statics and dynamics of distinguishable spin- $\frac{1}{2}$ systems on an arbitrary graph $G$ with $N$ vertices. In particular, we consider systems of quantum spins evolving according to one of two Hamiltonians: (i) the $XY$ Hamiltonian ${H}_{XY}$ , which contains an $XY$ interaction for every pair of spins connected by an edge in $G$ ; and (ii) the Heisenberg Hamiltonian ${H}_{\mathrm{Heis}}$ , which contains a Heisenberg interaction term for every pair of spins connected by an edge in $G$ . We find that the action of the $XY$ (respectively, Heisenberg) Hamiltonian on state space is equivalent to the action of the adjacency matrix (respectively, combinatorial Laplacian) of a sequence ${G}_{k}$ , $k=0,\dots{},N$ of graphs derived from $G$ (with ${G}_{1}=G$ ). This equivalence of actions demonstrates that the dynamics of these two models is the same as the evolution of a free particle hopping on the graphs ${G}_{k}$ . Thus we show how to replace the complicated dynamics of the original spin model with simpler dynamics on a more complicated geometry. A simple corollary of our approach allows us to write an explicit spectral decomposition of the $XY$ model in a magnetic field on the path consisting of $N$ vertices. We also use our approach to utilize results from spectral graph theory to solve new spin models: the $XY$ model and Heisenberg model in a magnetic field on the complete graph.We consider the statics and dynamics of distinguishable spin-$\frac{1}{2}$ systems on an arbitrary graph $G$ with $N$ vertices. In particular, we consider systems of quantum spins evolving according to one of two Hamiltonians: (i) the $XY$ Hamiltonian ${H}_{XY}$, which contains an $XY$ interaction for every pair of spins connected by an edge in $G$; and (ii) the Heisenberg Hamiltonian ${H}_{\mathrm{Heis}}$, which contains a Heisenberg interaction term for every pair of spins connected by an edge in $G$. We find that the action of the $XY$ (respectively, Heisenberg) Hamiltonian on state space is equivalent to the action of the adjacency matrix (respectively, combinatorial Laplacian) of a sequence ${G}_{k}$, $k=0,\dots{},N$ of graphs derived from $G$ (with ${G}_{1}=G$). This equivalence of actions demonstrates that the dynamics of these two models is the same as the evolution of a free particle hopping on the graphs ${G}_{k}$. Thus we show how to replace the complicated dynamics of the original spin model with simpler dynamics on a more complicated geometry. A simple corollary of our approach allows us to write an explicit spectral decomposition of the $XY$ model in a magnetic field on the path consisting of $N$ vertices. We also use our approach to utilize results from spectral graph theory to solve new spin models: the $XY$ model and Heisenberg model in a magnetic field on the complete graph.

arXiv:quant-ph/0312126Physical Review BSep 12, 200624 citations

We consider the statics and dynamics of distinguishable spin-

\frac{1}{2}

systems on an arbitrary graph

G

with

N

vertices. In particular, we consider systems of quantum spins evolving according to one of two Hamiltonians: (i) the

XY

Hamiltonian

{H}_{XY}

, which contains an

XY

interaction for every pair of spins connected by an edge in

G

; and (ii) the Heisenberg Hamiltonian

{H}_{\mathrm{Heis}}

, which contains a Heisenberg interaction term for every pair of spins connected by an edge in

G

. We find that the action of the

XY

(respectively, Heisenberg) Hamiltonian on state space is equivalent to the action of the adjacency matrix (respectively, combinatorial Laplacian) of a sequence

{G}_{k}

k=0,\dots{},N

of graphs derived from

G

(with

{G}_{1}=G

). This equivalence of actions demonstrates that the dynamics of these two models is the same as the evolution of a free particle hopping on the graphs

{G}_{k}

. Thus we show how to replace the complicated dynamics of the original spin model with simpler dynamics on a more complicated geometry. A simple corollary of our approach allows us to write an explicit spectral decomposition of the

XY

model in a magnetic field on the path consisting of

N

vertices. We also use our approach to utilize results from spectral graph theory to solve new spin models: the

XY

model and Heisenberg model in a magnetic field on the complete graph.