Systematic study of d-wave superconductivity in the 2D repulsive Hubbard model

The cluster size dependence of superconductivity in the conventional two-dimensional Hubbard model, commonly believed to describe high-temperature superconductors, is systematically studied using the Dynamical Cluster Approximation and Quantum Monte Carlo simulations as cluster solver. Due to the non-locality of the d-wave superconducting order parameter, the results on small clusters show large size and geometry effects. In large enough clusters, the results are independent of the cluster size and display a finite temperature instability to d-wave superconductivity.Comment: 4 pages, 3 figures; updated with version published in PRL; added values of Tc obtained from fit

Signature Characters for A_2 and B_2

The signatures of the inner product matrices on a Lie algebra's highest weight representation are encoded in the representation's signature character. We show that the signature characters of a finite-dimensional Lie algebra's highest weight representations obey simple difference equations that have a unique solution once appropriate boundary conditions are imposed. We use these results to derive the signature characters of all $A_2$ and $B_2$ highest weight representations. Our results extend, and explain, signature patterns analogous to those observed by Friedan, Qiu and Shenker in the Virasoro algebra's representation theory.Comment: 22 p

Anti-Evasion Doctrines in Constitutional Law

Recent constitutional scholarship has focused on how courts - the Supreme Court in particular - implement constitutional meaning through the use of doctrinal constructs that enable judges to decide cases. Judges first fix constitutional meaning, what Mitchell Berman terms the constitutional operative proposition, but must then design decision rules that render the operative proposition suitable to use in the third step, the resolution of the case before the court. These decision rules produce the familiar apparatus of constitutional decision-making - strict scrutiny, rational basis review, and the like. For the most part, writers have adopted a binary view of doctrine. Doctrinal tests can defer or not to other actors; implementing doctrines can be fashioned as rules or standards; doctrines can over-enforce or under-enforce constitutional commands. In this Article, though, we unsettle this dialectical view of doctrinal design by identifying and describing anti-evasion doctrines (AEDs) in constitutional law: doctrines developed by courts - usually designed as standards, as opposed to rules - that supplement other doctrines (designed as rules) to implement particular constitutional principles. AEDs touch all areas of constitutional law. In addition to being ubiquitous, AEDs have a long pedigree. Early examples appear in famous Marshall Court opinions; thus, they are not some modern innovation. In addition to naming AEDs, describing the forms they take, and discussing the characteristics the forms share, this Article also seeks to describe the benefits and costs to constitutional law resulting from AEDs, as well as their implications for doctrinal formation more generally. We back our claim about the omnipresence and pedigree of AEDs in Part II with examples from a broad swath of constitutional law. AEDs supplement rule-like decision rules with decision rules that tend to resemble standards. In Part III we discuss the benefits of AEDs and the tradeoffs for doctrine in their widespread use. AEDs are designed to help optimize enforcement of constitutional principles - by addressing problems with rules, for example. This gap-filling function comes at a cost, however. Not only does the addition of AEDs tend to increase doctrinal complexity, but that complexity can also increase decision costs for courts and dilute the benefits of using rules in the first place. The tradeoffs are almost mirror images of the benefits. We discuss the implications of AEDs for constitutional doctrine generally in Part IV. That they seem to be everywhere in constitutional law suggests that doctrinal complexity should be seen as a feature of our system, not a bug, because it attempts to ensure form will not trump constitutional substance. If a certain amount of complexity is inevitable, then that suggests one should be skeptical about claims that constitutional law could be rationalized by abandoning the \u27formulaic Constitution in favor of simple, predictable, and easy-to-apply rules. Further, the presence of AEDs furnishes strong evidence for Frederick Schauer\u27s convergence hypothesis, which holds that [w]hen authorised to act in accordance with rules, rule-subjects will tend to convert rules into standards by employing a battery of rule-avoiding devices that serve to soften the hard edges of rules, and vice-versa. Finally, highlighting the role AEDs play in constitutional doctrine, for good or ill, is another reason to take doctrine, its formation, and application by courts, seriously

Anti-Evasion Doctrines in Constitutional Law

Recent constitutional scholarship has focused on how courts - the Supreme Court in particular - implement constitutional meaning through the use of doctrinal constructs that enable judges to decide cases. Judges first fix constitutional meaning, what Mitchell Berman terms the constitutional operative proposition, but must then design decision rules that render the operative proposition suitable to use in the third step, the resolution of the case before the court. These decision rules produce the familiar apparatus of constitutional decision-making - strict scrutiny, rational basis review, and the like. For the most part, writers have adopted a binary view of doctrine. Doctrinal tests can defer or not to other actors; implementing doctrines can be fashioned as rules or standards; doctrines can over-enforce or under-enforce constitutional commands. In this Article, though, we unsettle this dialectical view of doctrinal design by identifying and describing anti-evasion doctrines (AEDs) in constitutional law: doctrines developed by courts - usually designed as standards, as opposed to rules - that supplement other doctrines (designed as rules) to implement particular constitutional principles. AEDs touch all areas of constitutional law. In addition to being ubiquitous, AEDs have a long pedigree. Early examples appear in famous Marshall Court opinions; thus, they are not some modern innovation. In addition to naming AEDs, describing the forms they take, and discussing the characteristics the forms share, this Article also seeks to describe the benefits and costs to constitutional law resulting from AEDs, as well as their implications for doctrinal formation more generally. We back our claim about the omnipresence and pedigree of AEDs in Part II with examples from a broad swath of constitutional law. AEDs supplement rule-like decision rules with decision rules that tend to resemble standards. In Part III we discuss the benefits of AEDs and the tradeoffs for doctrine in their widespread use. AEDs are designed to help optimize enforcement of constitutional principles - by addressing problems with rules, for example. This gap-filling function comes at a cost, however. Not only does the addition of AEDs tend to increase doctrinal complexity, but that complexity can also increase decision costs for courts and dilute the benefits of using rules in the first place. The tradeoffs are almost mirror images of the benefits. We discuss the implications of AEDs for constitutional doctrine generally in Part IV. That they seem to be everywhere in constitutional law suggests that doctrinal complexity should be seen as a feature of our system, not a bug, because it attempts to ensure form will not trump constitutional substance. If a certain amount of complexity is inevitable, then that suggests one should be skeptical about claims that constitutional law could be rationalized by abandoning the \u27formulaic Constitution in favor of simple, predictable, and easy-to-apply rules. Further, the presence of AEDs furnishes strong evidence for Frederick Schauer\u27s convergence hypothesis, which holds that [w]hen authorised to act in accordance with rules, rule-subjects will tend to convert rules into standards by employing a battery of rule-avoiding devices that serve to soften the hard edges of rules, and vice-versa. Finally, highlighting the role AEDs play in constitutional doctrine, for good or ill, is another reason to take doctrine, its formation, and application by courts, seriously

Judicial Doctrine as Risk Regulation

Much of the literature on risk regulation concerns first-order risks--e.g., those addressed by environmental law or workplace safety rules. But scholars recently have suggested that risk regulation can provide a helpful framework for thinking about second-order, or political, risks arising from allocations of power and institutional design. Although a few commentators have utilized this perspective to suggest connections between risk regulation and particular areas of constitutional law, in this essay we take a broader view. Building on the existing literature, we argue that the selection of constitutional decision rules is a judicial effort to regulate the political risk that government officials will violate constitutional principles. After making the case that it is helpful to view judicial doctrinal formation as a species of political risk regulation, we discuss some implications of this risk regulation model and pose some questions for future research. We conclude that the risk regulation model reinforces the notion that the formation of doctrine is a temporally extended process, rather than a one-time event, and it provides a metric by which that doctrine can be evaluated. Additionally, the risk regulation model helps explain some of the more commonly critiqued features of constitutional law. Finally, the risk regulation model raises important questions that merit further investigation: (1) Should we trust judicial perceptions of and responses to political risk? (2) What influences risk assessment among judges, and are those influences (and resulting assessments) normatively defensible? (3) What connection, if any, exists between judicial risk assessment and the myriad doctrinal formulae employed by the Supreme Court