HMF模型中的量子涨落抑制对称性破缺
Quantum fluctuations inhibit symmetry breaking in the HMF model
摘要：
人们普遍认为，平均场论对许多经典的长程相互作用系统都是精确的。一旦考虑了量子涨落，这也是真的吗？作为一个测试案例，我们研究了一个不可分辨玻色子系统的哈密顿平均场模型，该系统（根据平均场论）被预测在零温度下经历二阶量子相变。有序相的特征在于自发破缺的$O(2)$对称性，尽管发生在一维模型中，但由于长程相互作用的存在，Mermin-Wagner定理并不排除这种对称性。然而，一个自发破缺的对称性意味着无隙的戈德斯通模式，其大的波动可以恢复破缺的对称性。在这项工作中，我们研究了量子涨落的影响，通过投影到对称性破缺平均场态的连续子空间的哈密顿量。我们发现，能量成本的梯度质心波函数抑制打破的$O(2)$对称性，但能量成本是非常小的-缩放为$\mathcal{O}(1/N^2)$。然而，对于任何有限的$N$，不管有多大，这意味着基态具有恢复的$O(2)$对称性。有限温度相的影响，和经典的限制，HMF模型进行了讨论。
英文摘要：
It is widely believed that mean-field theory is exact for a wide-range of classical long-range interacting systems. Is this also true once quantum fluctuations have been accounted for? As a test case we study the Hamiltonian Mean Field (HMF) model for a system of indistinguishable bosons which is predicted (according to mean-field theory) to undergo a second-order quantum phase transition at zero temperature. The ordered phase is characterized by a spontaneously broken $O(2)$ symmetry, which, despite occurring in a one-dimensional model, is not ruled out by the Mermin-Wagner theorem due to the presence of long-range interactions. Nevertheless, a spontaneously broken symmetry implies gapless Goldstone modes whose large fluctuations can restore broken symmetries. In this work, we study the influence of quantum fluctuations by projecting the Hamiltonian onto the continuous subspace of symmetry breaking mean-field states. We find that the energetic cost of gradients in the center of mass wavefunction inhibit the breaking of the $O(2)$ symmetry, but that the energetic cost is very small --- scaling as $\mathcal{O}(1/N^2)$. Nevertheless, for any finite $N$, no matter how large, this implies that the ground state has a restored $O(2)$ symmetry. Implications for the finite temperature phases, and classical limit, of the HMF model are discussed.
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