Non-Perturbative Approaches to Condensed-Matter Physics

将对具有强相关性的凝聚态系统进行理论研究。 该研究是在NSF范围内的数学科学优先领域的保护伞下授予的。 该研究的优点包括理解具有定性新行为的系统，例如在具有令人沮丧的相互作用的磁体，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，特别是非微扰的方法有用的理解新的阶段，不能接近传统的方法。 研究将包括：继续导出拓扑相的精确结果，其中准粒子激发不具有电子的量子数，即，被细分;发现和描述量子临界点，这些临界点将新相与传统有序相分离;更深入地探索强关联电子的一维和二维模型的对称结构;在量子杂质问题中使用可积场论。 这些技术在当前研究中的成功应用包括：对广泛研究的经典和量子二聚体模型导出了许多精确结果;深入探索了具有拓扑相的模型的相图;利用超对称性描述强关联电子系统的基态;在具有竞争相互作用的模型中描述不同种类的密度波序，例如在一维光学晶格中的玻色子模型中;导出无序模型的精确结果;导出Haldane带隙自旋链的相关函数和其他精确结果;理解量子点和原子系统中的Fano共振。 在更广泛的背景下，这种类型的研究正在将凝聚态理论推向全新的领域。 在这些系统中，传统的将准粒子视为类电子的朗道范式甚至不能给出一个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，该领域扩大和发展新的思想和方法来解决这些问题。 一个更广泛的影响将来自一本正在写的书，题为《对抗无限》。 这本书将向普通观众解释用于在极端条件下探测物理的新理论，例如，超低温，密码破解量子计算机，早期宇宙，以及重正化群如何在不同的长度尺度上连接物理学。将对具有强相关性的凝聚态系统进行理论研究。 该研究是在NSF范围内的数学科学优先领域的保护伞下授予的。 该研究的优点包括理解具有定性新行为的系统，例如在具有令人沮丧的相互作用的磁体，高温超导体和拓扑量子计算的研究中出现的系统。 一个更广泛的影响，赠款将来自一本书正在写的题为反对无限。 这本书将向普通观众解释用于在极端条件下探测物理的新理论，例如，超低温，密码破解量子计算机，早期宇宙，以及重正化群如何在不同的长度尺度上联系物理学。

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.***