Spectra and Eigenstates of Spin Chain Hamiltonians
Spectra and Eigenstates of Spin Chain Hamiltonians
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自旋链哈密顿量的谱和本征态
DOI：
10.1007/s00220-015-2366-0
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发表时间：
2014
影响因子：
2.4
通讯作者：
H. J. Wells
中科院分区：
文献类型：
--
作者：
J. Keating;N. Linden;H. J. Wells
我们证明了具有最近邻相互作用的n个量子比特链的平移不变哈密顿量有两个看似矛盾的特征。首先，在极限→∞中，我们证明了n个量子比特链的任何平移不变的哈密顿量都有一个本征基，使得几乎所有本征态在固定大小的量子比特子块与系统的其余部分之间具有最大纠缠；在这个意义上，这些本征态类似于完全一般的哈密顿量(即，任意量子比特组之间存在所有阶相互作用的哈密顿量)。其次，在极限→∞中，我们证明了n个量子比特链的任何最近邻哈密顿量都具有高斯态密度；因此，就本征值而言，系统就像一个非相互作用的系统。这种比较适用于具有平移不变近邻相互作用的量子比特链，但我们表明它可以扩展到更一般的系统(无论是在局域维度还是相互作用的几何方面)。数值证据也表明，在最近邻链的情况下，平移不变性条件可以被放弃。
We prove that translationally invariant Hamiltonians of a chain of n qubits with nearest-neighbour interactions have two seemingly contradictory features. Firstly in the limit $${n \rightarrow \infty}$$n→∞ we show that any translationally invariant Hamiltonian of a chain of n qubits has an eigenbasis such that almost all eigenstates have maximal entanglement between fixed-size sub-blocks of qubits and the rest of the system; in this sense these eigenstates are like those of completely general Hamiltonians (i.e., Hamiltonians with interactions of all orders between arbitrary groups of qubits). Secondly, in the limit $${n \rightarrow \infty}$$n→∞ we show that any nearest-neighbour Hamiltonian of a chain of n qubits has a Gaussian density of states; thus as far as the eigenvalues are concerned the system is like a non-interacting one. The comparison applies to chains of qubits with translationally invariant nearest-neighbour interactions, but we show that it is extendible to much more general systems (both in terms of the local dimension and the geometry of interaction). Numerical evidence is also presented that suggests that the translational invariance condition may be dropped in the case of nearest-neighbour chains.