Particle-hole Symmetry And The Dirty Boson Problem

Peter B. Weichman; Ranjan Mukhopadhyay

摘要

We study the role of particle-hole symmetry on the universality class of various quantum phase transitions corresponding to the onset of superfluidity at zero temperature of bosons in a quenched random medium. To obtain a model with an exact particle-hole symmetry it is necessary to use the Josephson junction array, or quantum rotor, Hamiltonian, which may include disorder in both the site energies and the Josephson couplings between wave function phase operators at different sites. The functional integral formulation of this problem in d spatial dimensions yields a (d + 1)-dimensional classical XY model with extended disorder, constant along the extra imaginary time dimension-the so-called random rod problem. Particle-hole symmetry may then be broken by adding nonzero site energies, which may be uniform or site dependent. We may distinguish three cases: (ⅰ) exact particle-hole symmetry, in which the site energies all vanish; (ⅱ) statistical particle-hole symmetry, in which the site energy distribution is symmetric about zero, vanishing on average; and (ⅲ) complete absence of particle-hole symmetry in which the distribution is generic. We explore in each case the nature of the excitations in the nonsuperfluid Mott insulating and Bose glass phases. We show, in particular, that, since the boundary of the Mott phase can be derived exactly in terms of that for the pure, nondisordered system, there can be no direct Mott-superfluid transition. Recent Monte Carlo data to the contrary can be explained in terms of rare region effects that are inaccessible to finite systems. We find also that the Bose glass compressibility, which has the interpretation of a temporal spin stiffness or superfluid density, is positive in cases (ⅱ) and (ⅲ), but that it vanishes with an essential singularity as full particle-hole symmetry is restored. We then focus on the critical point and discuss the relevance of type (ⅱ) particle-hole symmetry-breaking perturbations to the random rod critical behavior, identifying a nontrivial crossover exponent. This exponent cannot be calculated exactly but is argued to be positive and the perturbation therefore relevant. We argue next that a perturbation of type (ⅲ) is irrelevant to the resulting type (ⅱ) critical behavior: The statistical symmetry is restored on large scales close to the critical point, and case (ⅱ) therefore describes the dirty boson fixed point. Using various duality transformations we verify all of these ideas in one dimension. To study higher dimensions, we attempt, with partial success, to generalize the Dorogovtsev-Cardy-Boyanovsky double-epsilon expansion technique to this problem. We find that when the dimension of time ε_τ ＜ ε_τ~c approx= 8/29 is sufficiently small a type (ⅱ) symmetry-breaking perturbation is irrelevant, but that for sufficiently large ε_τ ＞ ε_τ~c particle-hole asymmetry is a relevant perturbation and a new stable fixed point appears. Furthermore, for ε_τ ＞ ε_τ~(c2) ≈ 2/3, this fixed point is stable also to perturbations of type (ⅲ): at ε = ε_τ~(c2) the generic type (ⅲ) fixed point merges with the new fixed point. We speculate, therefore, that this new fixed point becomes the dirty boson fixed point when ε_τ= 1. We point out, however, that ε_τ= 1 may be quite special. Thus, although the qualitative renormalization group flow picture the double-epsilon expansion technique provides is quite compelling, one should remain wary of applying it quantitatively to the dirty boson problem.

机译：我们研究了在淬灭的随机介质中，与玻色子在零温度下的超流体的开始相对应，粒子-孔对称性在各种量子相变的普遍性类中的作用。为了获得具有精确的粒子-孔对称性的模型，必须使用约瑟夫逊结阵列或量子转子哈密顿量，这可能包括位能的无序以及不同位置处的波函数相位算子之间的约瑟夫森耦合。该问题在d空间维度上的函数积分公式产生了（d +1）维度的经典XY模型，该模型具有扩展的无序性，沿着额外的虚假时间维度恒定，即所谓的随机杆问题。然后可以通过添加非零位能量来破坏粒子-孔的对称性，该能量可以是均匀的或与位有关的。我们可以区分三种情况：（ⅰ）精确的粒子-孔对称性，其中该位能全部消失；（ⅱ）统计的粒子-孔对称性，其中位能量分布对称于零，平均消失；（ⅲ）完全不存在一般分布的粒子-孔对称性。我们分别探讨了非超流体Mott绝缘和Bose玻璃相中激发的性质。我们特别表明，由于可以精确地根据纯无序系统的边界得出Mott相的边界，因此不会有直接的Mott-超流体跃迁。相反，可以用有限系统无法访问的稀有区域效应来解释最近的蒙特卡洛数据。我们还发现，玻色玻璃可压缩性可以解释时间自旋刚度或超流体密度，在情况（ⅱ）和（ⅲ）中为正，但是随着完全的粒子-孔对称性的恢复，它以本质的奇点消失。。然后，我们将重点放在临界点上，并讨论类型（ⅱ）破坏粒子孔对称性的扰动与随机杆的临界行为的相关性，从而确定非平凡的交叉指数。该指数无法精确计算，但被认为是正数，因此扰动是相关的。接下来，我们认为类型（ⅲ）的扰动与所得类型（ⅱ）的临界行为无关：统计对称性在接近临界点的范围内大规模恢复，因此情况（ⅱ）描述了肮脏的玻色子固定点。使用各种对偶转换，我们在一维中验证了所有这些想法。为了研究更高的维度，我们尝试部分成功地将Dorogovtsev-Cardy-Boyanovsky双ε展开技术推广到该问题。我们发现，当时间ε_τ＜ε_τ〜c大约= 8/29的维数足够小时，类型（ⅱ）的对称破坏扰动是不相关的，而对于足够大的ε_τ＞ε_τ〜c则粒子孔不对称是相关的扰动和新的稳定不动点出现。此外，对于ε_τ＞ε_τ〜（c2）≈2/3，该不动点对（ⅲ）类型的扰动也是稳定的：在ε=ε_τ〜（c2）时，泛型（ⅲ）不动点与新的不动点合并点。因此，我们推测，当ε_τ= 1时，这个新的固定点变为脏玻色子固定点。但是，我们指出，ε_τ= 1可能很特殊。因此，尽管双ε展开技术提供的定性重归一化组流程图非常引人注目，但应警惕将其定量地应用于脏玻色子问题。

Particle-hole Symmetry And The Dirty Boson Problem

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