Asymptotics of Toeplitz determinants and the emptiness formation probability for the XY spin chain
Asymptotics of Toeplitz determinants and the emptiness formation probability for the XY spin chain
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Toeplitz 行列式的渐近性和 XY 自旋链的空性形成概率
DOI：
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发表时间：
2005
期刊：
影响因子：
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通讯作者：
A. Abanov
中科院分区：
文献类型：
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作者：
F. Franchini;A. Abanov
本文研究了横向磁场中一维各向异性XY自旋-1/2链的空性形成概率（EFP）的渐近行为。这个相关器本质上是在反铁磁基态的链中形成长度为n的铁磁串的概率，在可积模型理论中起着重要的作用。对于XY自旋链，相关函数可以表示为Toeplitz矩阵的行列式，利用已知的Toeplitz行列式定理和公式，得到了它在相图中n → ∞的渐近行为.我们发现，在XY模型的相图中，除了在临界线上，即在谱是无隙的地方，衰减是指数的。在这些情况下，除了指数或高斯衰减之外，还出现了具有普适指数的幂律前因子。后一种高斯行为在对应于各向同性XY模型的临界线上保持，而在磁场的临界值处，EFP指数衰减。在小的各向异性有一个交叉从高斯指数的行为。我们使用玻色化方法研究这种交叉。
We study an asymptotic behaviour of a special correlator known as the emptiness formation probability (EFP) for the one-dimensional anisotropic XY spin-1/2 chain in a transverse magnetic field. This correlator is essentially the probability of formation of a ferromagnetic string of length n in the antiferromagnetic ground state of the chain and plays an important role in the theory of integrable models. For the XY spin chain, the correlator can be expressed as the determinant of a Toeplitz matrix and its asymptotical behaviours for n → ∞ throughout the phase diagram are obtained using known theorems and conjectures on Toeplitz determinants. We find that the decay is exponential everywhere in the phase diagram of the XY model except on the critical lines, i.e. where the spectrum is gapless. In these cases, a power-law prefactor with a universal exponent arises in addition to an exponential or Gaussian decay. The latter Gaussian behaviour holds on the critical line corresponding to the isotropic XY model, while at the critical value of the magnetic field the EFP decays exponentially. At small anisotropy one has a crossover from the Gaussian to the exponential behaviour. We study this crossover using the bosonization approach.