Calculation of spin hamiltonians of rank zero in a permutation model. II

H -U Desgranges, John B. Parkinson

The authors present calculations of the specific heat of 1D quantum spin chains for a number of integrable and non-integrable Hamiltonians, in the presence of a magnetic field. For S=1 the Hamiltonian has both bilinear and biquadratic isotropic exchange and includes ferromagnetic and antiferromagnetic versions of the Heisenberg, Takhtajan-Babujian (TB), Lai-Sutherland (LS) (permutation) and biquadratic models. The two methods used are (i) direct numerical diagonalisation for chains of up to nine atoms and (ii) numerical solution of the Bethe ansatz equations for the integrable models (TB and LS). Generalisations of the latter method to S>1 are also given. For the integrable models the two methods give results in good agreement. Various types of behaviour, particularly at low T, are found and these are related to known phase changes at T=0. The results of the first method for the non-integrable Hamiltonians are discussed with reference to the existence of the Haldane gap in the S=1 Heisenberg model and a possible gap in the biquadratic model.

秀稔 西森, Hidetoshi Nishimori, Yoshihiro Taguchi

Indrani Bose, S. Chatterjee

Ramis Movassagh

standard spin-basis representation. We prove that at zero temperature, the magnetization is along the z direction i.e.,hs x i = hs y i = 0 (everywhere on the chain). We then analytically calculatehs z i and the two-point correlation functions of s z . By analytically diagonalizing the reduced density matrices, we calculate the Schmidt rank, von Neumann and Renyi entanglement entropies for: 1. Any partition of the chain into two pieces (not necessarily in the middle) and 2. L consecutive spins centered in the middle. Further, we identify entanglement Hamiltonians (Eqs. 40 and 50). We prove a small lemma (Lemma 1) on the combinatorics of lattice paths using the reflection principle to relate and calculate the Motzkin walk ’height’ to spin expected values. We also calculate the, closely related, (scaled) correlation functions of Brownian excursions. The known features of this model are summarized in a table in Sec. I.

Manju C, Uma Divakaran

Permutation symmetry plays a central role in the understanding of collective quantum dynamics. By introducing power law couplings that algebraically decay with the distance between the spins $r$ as $1/r^α$, we break this symmetry with a non-zero $α$. This allows us to probe the emergence of new dynamical behaviors, including thermalization in an otherwise permutation symmetric Hamiltonian with all-to-all spin interactions along $x$ direction subjected to periodic kicks in transverse direction. As we increase $α$, the system interpolates from an infinite range spin system at $α=0$ exhibiting permutation symmetry, to a short range integrable model as $α\rightarrow \infty$ where this permutation symmetry is absent. We focus on this change in the behavior of the system as $α$ is tuned, using dynamical quantities like total angular momentum and von Neumann entropy. Starting from the chaotic limit of the permutation symmetric Hamiltonian at $α=0$, for the finite system sizes considered, we find that for small $α$, the steady state values of these quantities remain close to the permutation symmetric subspace values corresponding to $α=0$. At intermediate $α$ values, these show signatures of thermalization exhibiting values corresponding to that of random states in full Hilbert space. On the other hand, the large $α$ limit approaches the values corresponding to integrable kicked Ising model. In addition, we also study the dependence of thermalization on the driving period $τ$, with results indicating the onset of thermalization for smaller values of $α$ when $τ$ is large, thereby extending the thermalizing window in the intermediate range of $α$. We further confirm these results using effective dimension and spectral statistics.