Non-Perturbative Approaches to Condensed-Matter Physics

将对具有强关联的凝聚态系统进行理论研究。这项研究是在NSF全数学科学优先领域的保护伞下授予的。这项研究的优点包括理解具有新行为的系统，例如在具有受阻相互作用的磁体、高温超导体和拓扑量子计算的研究中出现的系统。这项工作将利用场论技术，特别是有助于理解新相的非微扰方法，这是传统方法无法接近的。这项研究将包括：继续得到拓扑相的精确结果，其中准粒子激发不具有电子的量子数，即被细分；发现并描述将新相与常规有序相分开的量子临界点；更深入地探索强关联电子的一维和二维模型的对称结构；将可积场理论应用于量子杂质问题。该原理研究人员在开发和利用现代技术方面具有丰富的经验。这些技术在当前研究中的成功应用包括：推导出被广泛研究的经典和量子调光模型的许多精确结果；深入探索具有拓扑相的模型的相图；利用超对称性来描述强关联电子系统的基态；描述具有竞争相互作用的模型中不同类型的密度波序，例如在一维光学晶格中的玻色子模型；推导具有无序的模型的精确结果；推导关联函数和其他关于Haldane-Gap自旋链的精确结果；理解量子点和原子系统中的Fano共振。在更广泛的背景下，这类研究正在将凝聚态理论推向全新的领域。在这些系统中，将准粒子视为类电子的传统朗道范式甚至无法给出很好的物理定性描述，更不用说准确的定量预测了。重要的是，该领域必须拓宽和发展新的想法和方法来解决这些问题。更广泛的影响将来自一本正在撰写的名为《对抗无限》的书。这本书将向普通读者解释用于在极端条件下探测物理的新类型的理论，例如，超低温、破译密码的量子计算机、早期宇宙，以及重整化群如何在巨大不同的长度尺度上连接物理。%理论研究将在具有强关联的凝聚态系统上进行。这项研究是在NSF全数学科学优先领域的保护伞下授予的。这项研究的优点包括理解具有新行为的系统，例如在具有受阻相互作用的磁体、高温超导体和拓扑量子计算的研究中出现的系统。这笔拨款的更广泛影响将来自一本名为《对抗无限》的书。这本书将向普通读者解释用于在极端条件下探测物理的新类型的理论，例如超低温、破译密码的量子计算机、早期宇宙，以及重整化小组如何以非常不同的长度尺度连接物理。*

Theoretical research will be conducted on condensed matter systems with strong correlations. The research is awarded under the umbrella of the NSF-wide Mathematical Sciences Priority Area. Merits of the research include understanding systems with qualitatively new behavior, such as have arisen in studies of magnets with frustrating interactions, high-temperature superconductors, and topological quantum computation. This work will utilize field-theoretic techniques, especially non-perturbative methods useful for understanding novel phases which cannot be approached by conventional methods. The research will include: continuing to derive exact results for topological phases, where quasiparticle excitations do not have the quantum numbers of the electron, i.e., are fractionalized; discovering and describing quantum critical points which separate novel phases from phases with conventional order; probing more deeply the symmetry structure of one-and two-dimensional models of strongly correlated electrons; using integrable field theory in quantum impurity problems.The principle investigator has extensive experience in developing and utilizing modern techniques. Successful applications of these techniques in current research include: deriving many exact results for the widely-studied classical and quantum dimmer models; exploring in depth the phase diagrams of models with topological phases; utilizing supersymmetry to describe the ground state of strongly correlated electron systems; describing different kinds of density-wave order in models with competing interactions, such as in a model of bosons in a one-dimensional optical lattice; deriving exact results for models with disorder; deriving correlation functions and other exact results for Haldane-gap spin chains; understanding Fano resonances in quantum dots and atomic systems. In a broader context, this type of research is pushing condensed matter theory into brand new arenas. In these systems, the traditional Landau paradigm of treating quasiparticles as electron-like does not give even a good qualitative description of the physics, much less an accurate quantitative prediction. It is important that the field broaden and develop new ideas and methods to attack these problems. An even broader impact will come from a book being written entitled Up Against the Infinite. This book will explain to general audiences the new sorts of theories used to probe physics in extreme conditions, e.g., ultra-low temperature, code-cracking quantum computers, the early universe, and how the renormalization group links physics at vastly different length scales.%%%Theoretical research will be conducted on condensed matter systems with strong correlations. The research is awarded under the umbrella of the NSF-wide Mathematical Sciences Priority Area. Merits of the research include understanding systems with qualitatively new behavior, such as have arisen in studies of magnets with frustrating interactions, high-temperature superconductors, and topological quantum computation. A broader impact of the grant will come from a book being written entitled Up Against the Infinite. This book will explain to general audiences the new sorts of theories used to probe physics in extreme conditions, e.g., ultra-low temperature, code-cracking quantum computers, the early universe, and how the renormalization group links physics at vastly different length scales.***