A Magnetic Model with a Possible Chern-Simons Phase
A Magnetic Model with a Possible Chern-Simons Phase
复制标题
具有可能的 Chern-Simons 相的磁模型
DOI：
10.1007/s00220-002-0785-1
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发表时间：
2001
影响因子：
2.4
通讯作者：
Michael H. Freedman
中科院分区：
文献类型：
--
作者：
Michael H. Freedman
翻译后摘要：一个基本家庭的本地哈密顿，描述了一个2维量子力学系统的自旋粒子。 在环面上，基态空间Gynn，Gynn是（log）广泛退化的，但在"扰动“下应该坍缩成一个具有完整数学描述的任意子系统：SO（3）-Chern-Simons模函子在q=e2πi/π i +2处的量子二重，我们称之为DE π i。哈密尔顿算符Hlr，Hlr定义了一个量子圈气体。我们认为，当G =1和2时，G○，G ε是不稳定的，通过微扰可以真正地坍缩到Gε，G ε DE ε.对于ε ≥3，G○，ε是稳定的，在这种情况下，找到Gε，ε ∈ DE ε必须要求ε>ε ε ∈>0，这得益于有限的系统尺寸，表面粗糙化（见第3节）或其他技巧，因此最初使用引号“”。一个假设的相图包括在介绍。用代数的方法研究了扰动的影响：H○，H的基态空间G○，G ε被描述为一个表面代数，我们的推论是扰动应尊重这个结构，得到一个由商代数描述的扰动基态Gε，G ε.通过分类，这意味着Gε，DE。基本的一点是，非线性结构可能存在于初始扰动的退化特征空间上，这限制了扰动可能的有效作用。没有理由期望Gε，ε_DE_ε作为一个任意子系统的物理实现需要分数量子霍尔效应（Fractional Quantum Hall Effect）系统或旋转的玻色-爱因斯坦凝聚体（Bosé-Einstein condensates）所固有的低温和时间不对称性-目前已知的物理系统由拓扑模函子建模。DE 3的固态实现，甚至在室温下，可能通过建立和研究系统，“量子回路气体”，其主要项是H 0，3。这是本世纪固体物理学家面临的一个挑战。对于ε ≥3，ε ε 2 mod 4，DE ε的物理实现将产生固有容错的通用量子计算机。但必须提出一个警告，在λ =2处的理论在计算上并不普适，而在λ =3处的第一个普适理论似乎更难定位，因为相应的环圈气体的稳定性。大自然憎恶量子计算机吗？
Abstract: An elementary family of local Hamiltonians , is described for a 2-dimensional quantum mechanical system of spin particles. On the torus, the ground state space Gŝ,ℓ is (log) extensively degenerate but should collapse under ``perturbation'' to an anyonic system with a complete mathematical description: the quantum double of the SO(3)-Chern-Simons modular functor at q=e2πi/ℓ+2 which we call DEℓ. The Hamiltonian Hŝ,ℓ defines a quantum loop gas. We argue that for ℓ=1 and 2, G○,ℓ is unstable and the collapse to Gε,ℓ≌DEℓ can occur truly by perturbation. For ℓ≥3, G○,ℓ is stable and in this case finding Gε,ℓ≌DEℓ must require either ε>εℓ>0, help from finite system size, surface roughening (see Sect. 3), or some other trick, hence the initial use of quotes `` ''. A hypothetical phase diagram is included in the introduction. The effect of perturbation is studied algebraically: the ground state space G○,ℓ of H○,ℓ is described as a surface algebra and our ansatz is that perturbation should respect this structure yielding a perturbed ground state Gε,ℓ described by a quotient algebra. By classification, this implies Gε,ℓ≌DEℓ. The fundamental point is that nonlinear structures may be present on degenerate eigenspaces of an initial Hŝ which constrain the possible effective action of a perturbation. There is no reason to expect that a physical implementation of Gε,ℓ≌DEℓ as an anyonic system would require the low temperatures and time asymmetry intrinsic to Fractional Quantum Hall Effect (FQHE) systems or rotating Bosé-Einstein condensates − the currently known physical systems modelled by topological modular functors. A solid state realization of DE3, perhaps even one at a room temperature, might be found by building and studying systems, ``quantum loop gases'', whose main term is H○,3. This is a challenge for solid state physicists of the present decade. For ℓ≥3,ℓ≠2 mod 4, a physical implementation of DEℓ would yield an inherently fault-tolerant universal quantum computer. But a warning must be posted, the theory at ℓ=2 is not computationally universal and the first universal theory at ℓ=3 seems somewhat harder to locate because of the stability of the corresponding loop gas. Does nature abhor a quantum computer?