We develop a general method to compute correlation functions of fractional quantum Hall (FQH) states on a curved space. In a curved space, local transformation properties of FQH states are examined through local geometric variations, which are essentially governed by the gravitational anomaly. Furthermore, we show that the electromagnetic response of FQH states is related to the gravitational response (a response to curvature). Thus, the gravitational anomaly is also seen in the structure factor and the Hall conductance in flat space. The method is based on iteration of a Ward identity obtained for FQH states.Introduction Important universal properties of fractional quantum Hall (FQH) states are evident in the quantization of kinetic coefficients in terms of the filling fraction. The most well-known kinetic coefficient is the Hall conductance [1], a transversal response to the electromagnetic field. Beside this, FQH states possess a richer structure evident through their response to changes in spatial geometry and topology, both captured by the gravitational response.A kinetic coefficient which reflects a transversal response to the gravitational field is the odd viscosity (also referred as anomalous viscosity, Hall viscosity or Lorentz shear modulus) [2][3][4]. This coefficient also exhibits a quantization and reveals universal features of FQH states as much as the Hall conductance. While the Hall conductance is seen in an adiabatic response to homogeneous flux deformation [1], the anomalous viscosity is seen as an adiabatic response to homogeneous metric deformations [2]. However, even more universal features become apparent when one considers the adiabatic response to inhomogeneous deformations of the flux and the metric. This is the subject of this paper.In this paper, we compute the response of the FQH states to local curvature and show that this response reveals corrections to physical quantities in a flat space that remain hidden otherwise. We compute the particle density through a gradient expansion in local curvature, explain the relation of the leading terms to the gravitational anomaly, and show that they are geometrical in nature. For this reason, we expect these terms to be universal (i.e. insensitive to the details of the underlying electronic interaction as long as the interaction gives rise to the FQH state). We develop a general method to compute these terms. Additionally, we show that the dependence on curvature determines the long wavelength expansion of the static structure factor in a flat background, linking the electromagnetic response to the gravitational anomaly. Furthermore, correlation functions computed on arbitrary surfaces provide information about the properties of FQH states under general covariant and, in particular, conformal transformations.We consider only Laughlin states for which the filling
We show that universal transport coefficients of the fractional quantum Hall effect (FQHE) can be understood as a response to variations of spatial geometry. Some transport properties are essentially governed by the gravitational anomaly. We develop a general method to compute correlation functions of FQH states in a curved space, where local transformation properties of these states are examined through local geometric variations. We introduce the notion of a generating functional and relate it to geometric invariant functionals recently studied in geometry. We develop two complementary methods to study the geometry of the FQHE. One method is based on iterating a Ward identity, while the other is based on a field theoretical formulation of the FQHE through a path integral formalism.
We introduce a framework that abstracts Reinforcement Learning (RL) as a sequence modeling problem. This allows us to draw upon the simplicity and scalability of the Transformer architecture, and associated advances in language modeling such as GPT-x and BERT. In particular, we present Decision Transformer, an architecture that casts the problem of RL as conditional sequence modeling. Unlike prior approaches to RL that fit value functions or compute policy gradients, Decision Transformer simply outputs the optimal actions by leveraging a causally masked Transformer. By conditioning an autoregressive model on the desired return (reward), past states, and actions, our Decision Transformer model can generate future actions that achieve the desired return. Despite its simplicity, Decision Transformer matches or exceeds the performance of state-of-the-art model-free offline RL baselines on Atari, OpenAI Gym, and Key-to-Door tasks.
We study quantum Hall states on surfaces with conical singularities. We show that a small parcel of electronic fluid at the cone tip gyrates with an intrinsic angular momentum whose value is quantized in units of the Planck constant and exists solely due the gravitational anomaly. We show that quantum Hall states behave as conformal primaries near singular points, with a conformal dimension equal to the angular momentum. We argue that the gravitational anomaly and conformal dimension determine the fine structure of electronic density at the tip. The singularities emerge as quasi-particles with spin and exchange statistics arising from adiabatically braiding conical singularities. Thus, the gravitational anomaly, which appears as a finite-size correction on smooth surfaces, dominates geometric transport on singular surfaces.
We develop a collective field theory for fractional quantum Hall (FQH) states. We show that in the leading approximation for a large number of particles, the properties of Laughlin states are captured by a Gaussian free field theory with a background charge. Gradient corrections to the Gaussian field theory arise from the covariant ultraviolet regularization of the theory, which produces the gravitational anomaly. These corrections are described by a theory closely related to the Liouville theory of quantum gravity. The field theory simplifies the computation of correlation functions in FQH states and makes manifest the effect of quantum anomalies.PACS numbers: 73.43. Cd, 73.43.Lp, Introduction Since the work of Laughlin [1], a common approach to analyzing the physics of the fractional quantum Hall effect (FQHE) starts with a trial ground state wave function for N electrons. Despite its success, this approach is an impractical framework for studying the collective behavior of a large number of electrons ( N ∼ 10 6 , in samples exhibiting the QHE). As a result, some subtle properties of QHE states, such as the gravitational anomaly [2-10], were computed only recently.The effects of quantum anomalies are essential in the physics of the QHE. Although anomalies originate at short distances on the order of the magnetic length, they control the large-scale properties of the state, such as transport. It was recently shown in [10] that, like the Hall conductance, transport coefficients determined by the gravitational anomaly are expected to be quantized on QH plateaus. For this reason it is important to formulate the theory of QH effect in a fashion which makes the quantum anomalies manifest. The field theory approach seems the most appropriate for this purpose.In this paper, we develop a field theory for Laughlin states. This approach naturally captures universal features of the QHE, and emphasizes geometric aspects of QH-states. We demonstrate how the field theory encompasses recent developments in the field [2-10] and obtain some properties of quasi-hole excitations. Preliminary treatment of this approach appears in [3].The field theory framework uncovers a connection between the QHE and random geometry, specifically 2D Liouville quantum gravity. Since its introduction, the Laughlin wave function has been a practical model wave function mainly because of the plasma analogy. This analogy to a 2D statistical mechanical system allowed the most salient features of the state -uniform density and fractional quasi-hole charge -to be easily captured by a saddle point approach to the partition function of the equivalent plasma.Every analysis to date has stopped at the saddle point. as a result subtle features of the theory such as the gravitational anomaly were missed. We show how the Laughlin wave function maps to a full quantum field theory. This approach allows to go beyond the saddle point and
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