Random Matrix Theory and Entanglement in Quantum Spin Chains
Random Matrix Theory and Entanglement in Quantum Spin Chains
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随机矩阵理论和量子自旋链中的纠缠
DOI：
10.1007/s00220-004-1188-2
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发表时间：
2004
影响因子：
2.4
通讯作者：
F. Mezzadri
中科院分区：
文献类型：
--
作者：
J. Keating;F. Mezzadri
我们计算纠缠的熵在基态的一般类的量子自旋链哈密顿-那些相关的费米算子的二次形式-之间的第一个N自旋和其余的系统在无限总链长度的限制。我们表明，熵可以表示在古典紧群的平均数，并建立一个明确的对应关系之间的对称性的一个给定的哈密尔顿和那些特征的Haar措施的相关组。这些平均值是Toeplitz行列式或Toeplitz和Hankel矩阵组合的行列式。最近推广的Fisher-Hartwig猜想被用来计算N→∞时熵的首阶渐近。这表明，增长与N。比例常数是明确确定的，因为是下一个（常数）项的渐近展开。熵的对数增长以前预测的数值计算和共形场论计算的基础上。在这些计算中，比例常数是根据Virasoro代数的中心电荷确定的。因此，我们的结果导致一个明确的公式，这种电荷。我们还表明，熵与Painlevé型常微分方程的解有关。在某些情况下，这些解决方案可以评估的所有订单使用递归关系。
We compute the entropy of entanglement in the ground states of a general class of quantum spin-chain Hamiltonians — those that are related to quadratic forms of Fermi operators — between the first N spins and the rest of the system in the limit of infinite total chain length. We show that the entropy can be expressed in terms of averages over the classical compact groups and establish an explicit correspondence between the symmetries of a given Hamiltonian and those characterizing the Haar measure of the associated group. These averages are either Toeplitz determinants or determinants of combinations of Toeplitz and Hankel matrices. Recent generalizations of the Fisher-Hartwig conjecture are used to compute the leading order asymptotics of the entropy as N→∞. This is shown to grow logarithmically with N. The constant of proportionality is determined explicitly, as is the next (constant) term in the asymptotic expansion. The logarithmic growth of the entropy was previously predicted on the basis of numerical computations and conformal-field-theoretic calculations. In these calculations the constant of proportionality was determined in terms of the central charge of the Virasoro algebra. Our results therefore lead to an explicit formula for this charge. We also show that the entropy is related to solutions of ordinary differential equations of Painlevé type. In some cases these solutions can be evaluated to all orders using recurrence relations.