From Doubled Chern-Simons-Maxwell Lattice Gauge Theory to Extensions of the Toric Code

We regularize compact and non-compact Abelian Chern-Simons-Maxwell theories on a spatial lattice using the Hamiltonian formulation. We consider a doubled theory with gauge fields living on a lattice and its dual lattice. The Hilbert space of the theory is a product of local Hilbert spaces, each associated with a link and the corresponding dual link. The two electric field operators associated with the link-pair do not commute. In the non-compact case with gauge group $\mathbb{R}$, each local Hilbert space is analogous to the one of a charged "particle" moving in the link-pair group space $\mathbb{R}^2$ in a constant "magnetic" background field. In the compact case, the link-pair group space is a torus $U(1)^2$ threaded by $k$ units of quantized "magnetic" flux, with $k$ being the level of the Chern-Simons theory. The holonomies of the torus $U(1)^2$ give rise to two self-adjoint extension parameters, which form two non-dynamical background lattice gauge fields that explicitly break the manifest gauge symmetry from $U(1)$ to $\mathbb{Z}(k)$. The local Hilbert space of a link-pair then decomposes into representations of a magnetic translation group. In the pure Chern-Simons limit of a large "photon" mass, this results in a $\mathbb{Z}(k)$-symmetric variant of Kitaev's toric code, self-adjointly extended by the two non-dynamical background lattice gauge fields. Electric charges on the original lattice and on the dual lattice obey mutually anyonic statistics with the statistics angle $\frac{2 \pi}{k}$. Non-Abelian $U(k)$ Berry gauge fields that arise from the self-adjoint extension parameters may be interesting in the context of quantum information processing.Comment: 38 pages, 4 figure

Graphical Tensor Product Reduction Scheme for the Lie Algebras so(5) = sp(2), su(3), and g(2)

We develop in detail a graphical tensor product reduction scheme, first described by Antoine and Speiser, for the simple rank 2 Lie algebras so(5) = sp(2), su(3), and g(2). This leads to an efficient practical method to reduce tensor products of irreducible representations into sums of such representations. For this purpose, the 2-dimensional weight diagram of a given representation is placed in a "landscape" of irreducible representations. We provide both the landscapes and the weight diagrams for a large number of representations for the three simple rank 2 Lie algebras. We also apply the algebraic "girdle" method, which is much less efficient for calculations by hand for moderately large representations. Computer code for reducing tensor products, based on the graphical method, has been developed as well and is available from the authors upon request.Comment: 43 pages, 18 figure

The (2+1)-d U(1) Quantum Link Model Masquerading as Deconfined Criticality

The $(2+1)$-d U(1) quantum link model is a gauge theory, amenable to quantum simulation, with a spontaneously broken SO(2) symmetry emerging at a quantum phase transition. Its low-energy physics is described by a $(2+1)$-d \RP(1) effective field theory, perturbed by a dangerously irrelevant SO(2) breaking operator, which prevents the interpretation of the emergent pseudo-Goldstone boson as a dual photon. At the quantum phase transition, the model mimics some features of deconfined quantum criticality, but remains linearly confining. Deconfinement only sets in at high temperature.Comment: 4.5 pages, 6 figure

Crystalline Confinement

We show that exotic phases arise in generalized lattice gauge theories known as quantum link models in which classical gauge fields are replaced by quantum operators. While these quantum models with discrete variables have a finite-dimensional Hilbert space per link, the continuous gauge symmetry is still exact. An efficient cluster algorithm is used to study these exotic phases. The $(2+1)$-d system is confining at zero temperature with a spontaneously broken translation symmetry. A crystalline phase exhibits confinement via multi-stranded strings between charge-anti-charge pairs. A phase transition between two distinct confined phases is weakly first order and has an emergent spontaneously broken approximate $SO(2)$ global symmetry. The low-energy physics is described by a $(2+1)$-d $\mathbb{R}P(1)$ effective field theory, perturbed by a dangerously irrelevant $SO(2)$ breaking operator, which prevents the interpretation of the emergent pseudo-Goldstone boson as a dual photon. This model is an ideal candidate to be implemented in quantum simulators to study phenomena that are not accessible using Monte Carlo simulations such as the real-time evolution of the confining string and the real-time dynamics of the pseudo-Goldstone boson.Comment: Proceedings of the 31st International Symposium on Lattice Field Theory - LATTICE 201

Real-Time Simulation of Large Open Quantum Spin Systems driven by Measurements

We consider a large quantum system with spins $\frac{1}{2}$ whose dynamics is driven entirely by measurements of the total spin of spin pairs. This gives rise to a dissipative coupling to the environment. When one averages over the measurement results, the corresponding real-time path integral does not suffer from a sign problem. Using an efficient cluster algorithm, we study the real-time evolution of a 2-d Heisenberg antiferromagnet, which is driven to a disordered phase, either by sporadic measurements or by continuous monitoring described by Lindblad evolution.Comment: 5 pages, 7 figure

What's All the Drama About?: The Development of Tragedy in Ancient Greece

The art of storytelling is only a step away from the art of performance. Yet it took centuries to develop into the form we easily recognize today. Theatre, a gift of ancient Greece, pulled threads from many facets of Greek society - religion, festival, poetry, and competition. And yet, as much as historians know about the Greek theatre, they are still theorizing about its origins and how it suddenly developed and flourished within the confines of the 5th century B.C. Fascination concerning the origin of the theatre developed as quickly as the 4th century B.C. and remains a topic of speculation even today

A Comprehensive Bison Management and Research Plan for the Crane Trust

The Great Plains were once a vast grassland ecosystem, but, due to agricultural and human development, are one of the most endangered ecosystems in North America. What remains is generally fragmented, threatened by invasive species, and lacks the natural ecosystem processes that shaped these grasslands such as periodic wildfire and bison grazing. Since 1978, the Platte River Whooping Crane Maintenance Trust, Inc. (dba “Crane Trust”) has worked to maintain the function of grassland and riparian habitats to benefit endangered Whooping Cranes, Sandhill Cranes, and other migratory bird species. They protect ~8,100 acres, including the largest contiguous portion of lowland tallgrass prairie and wet meadow remaining along the Central Platte River in southcentral Nebraska. To manage their prairie ecosystems, the Crane Trust mimics natural disturbances to create a diverse mosaic of habitat structure on the landscape, supporting hundreds of grassland-obligate species. The Crane Trust piloted bison reintroduction with a small bison herd loaned throughout 2013- 2014. After a successful pilot period, the Crane Trust purchased and reintroduced forty-one American bison (Bison bison) in 2015 within a portion of their protected land. Their primary goal was to restore the functional services of bison as “ecosystem engineers”. They sought to allow bison grazing patterns to create structural heterogeneity on the landscape for the betterment migratory bird species and other grassland taxa. Beyond using bison as a management tool, the Crane Trust also sought to contribute to the continental effort to recover and research bison, while developing ways to make the herd economically self-sustaining. Though the Crane Trust has made great contributions to these goals, the various components of the bison program had yet to be synthesized into one cohesive plan, direction, and vision. The Crane Trust bison program has reached a point of relative stability, creating an opportunity to develop a reasonable long-term outlook for the bison program. This Bison Management and Research Plan (the plan) was created to document the current status, vision, goals, and practices of the Crane Trust’s bison program using conservation literature, internal records and research, and coordinated planning meetings with members of the Natural Resource Team. The creation of habitat structure can be facilitated using methods such as patch-burn-grazing and encouraging bison movement and herbivory throughout the landscape. However, invasive species and woody encroachment need to be addressed through more intensive management practices before bison are allowed to freely graze throughout their range. Like many small, conservation-oriented herds, the Crane Trust bison are spatially limited, contained within a fenced boundary. Likewise, several species that once interacted ecologically with bison, like wolves and prairie dogs, are no longer present on the landscape. These limitations to ecological function have precluded small herds from several bison recovery conversations. However, there are several advantages and opportunities for bison herds like the Crane Trust’s. This plan demonstrates the adaptability of small conservation herds. These herds provide unique opportunities for research and serve as a laboratory of bison practice. The Crane Trust has already advanced the scientific understanding of bison behavior, health, and ecology through published research conducted on their lands. Long-term monitoring, record keeping, and cooperative partnerships have and will continue to be pivotal for the Crane Trust’s scientific capacity for bison research. The Crane Trust is in an opportune position to contribute to the genetic, health, and cultural recovery of bison as a species. Through their genetic monitoring and management strategies, the herd is producing bison calves with high genetic diversity. These gains in diversity are largely due to a strategy of introducing young bulls that are genetically dissimilar to the current Crane Trust bison herd. The Crane Trust needs to consider retaining some of their female bison calves to preserve the genetic heritage of the herd and maintain a relatively young cow herd poised for high annual production. To maximize the Crane Trust’s role in genetic recovery of the species, efforts need to be made to distribute genetically diverse bison calves born at the Crane Trust to other conservation herds. There are several diseases that threaten bison conservation and recovery. Diseases such as Brucellosis and Mycoplasmosis threaten bison at a continental scale, while disease like Anthrax and Pink Eye are more localized concerns for the Crane Trust herd. Despite these concerns, the bison herd has not experienced an outbreak of any fatal diseases and remains relatively healthy. In an effort to preserve their “wild” nature, the Crane Trust limits human intervention in the health of bison to maintain the processes of natural selection. However, some intervention may be warranted, particularly if health concerns exceed that of the individual and threaten the herd as a whole. Standard practices for body condition scoring, record keeping, necropsy, and quarantine procedures within the plan will be used to monitor the health and productivity of the bison. The accessibility of the Crane Trust bison provides an opportunity to build cultural and community connections. The Crane Trust’s Nature and Visitor Center attracts thousands of visitors each spring to witness the Sandhill Crane migration and has been used as the interface between the public and the landscape, raising awareness for conservation needs and educational engagement. Reintroduction of bison on the landscape has attracted visitors outside of the spring migration, and they have become “prairie ambassadors” for the Crane Trust. Programing and educational curricula need to be developed around bison, their recovery, and their relationship to grassland and human dimensions. Bison also play an important role in many indigenous cultures and we hope to support the cultural recovery of the Bison through friendships and partnerships with regional Tribal Communities. The reintroduction of bison to the Crane Trust has provided a diverse grassland structure and suitable habitat for a wide-range of grassland species. As long-term ecological data is evaluated, the picture of bison’s ecological role within the grasslands of Nebraska and along the Platte River will become clearer. Likewise, the value of small, conservation herds to the recovery of the bison species has yet to be fully recognized. The Crane Trust’s bison plan is a testament to the organization’s commitment to realizing the potential of bison reintroduction on small to medium scales. Goal 1: Improve ecosystem structure and function by reintroducing bison as a keystone species to enhance the diversity of the native prairie and wet meadow ecosystem along the Platte River. Goal 2: Support the genetic recovery of bison in North America and provide a model of genetic diversity management. Goal 3: Maintain bison well-being with limited human intervention and develop standard operating procedures to monitor bison health while maintaining the safety of bison and bison handlers. Goal 4: Improve outreach and education efforts, contributing to the cultural significance of bison by impressing the story of their extinction and recovery, and the need to conserve native habitats similarly on to visitors and the community. Goal 5: Develop strategies of long-term economic sustainability for the bison program using ecologically sound culling decisions

Death of a Myth: The Patenting of Internet Business Models after State Street Bank

The case of State Street Bank & Trust Co. v. Signature Financial Group, Inc., which extended patent protection to a computerized financial method, was regarded by many as a revolutionary expansion of patentable subject matter. The author, however, argues that this notion is overstated. The author explains that the State Street Bank decision will be of little consequence because the business exception was a myth in the first place. The author reasons that courts often cited other bars to patentability when denying business methods protection. Furthermore, the author argues that the recent relaxation of the patentability requirements of computer related inventions would likely include the business method exception, rendering any change created by State Street Bank meaningless. Finally, the author asserts that claims for Internet business models will still be rejected by the PTO and courts under the patentability requirements of the Patent Act