Exact RG flow of point interactions in one dimension (Duality and Scales in Quantum-Theoretical Sciences)

The concept of renormalization group (RG) flow is one of the most novel and broad-reaching aspects of quantum field theory (QFT). The RG flow is implemented by constructing effective descriptions of a QFT at decreasing energy scales. One reason that RG flow is useful is that often one is interested in low-energy properties of theories with complicated short-distance structures. RG flows are subject to C theorems in relativistic QFT. The C theorems order the space of Lorentz-invariant QFTs. RG flows generically begin at scale-invariant fixed points known as conformal field theories (CFTs) and end in trivial massive theories. With tuning, the RG flows may end at non-trivial CFTs. Each CFT has an associated dimensionless C value. The C theorem states that under RG flow from a UV to an IR fixed point, the C value decreases. In this Dissertation I present the F-theorem, which is a C theorem in three spacetime dimensions. I show that the correct quantity to consider is the Euclidean free energy of the CFT conformally mapped to the three-sphere, known as the F value. After motivating the F-theorem, I develop tools for calculating the F value in a variety of CFTs, with and without supersymmetry, including free field theories and gauge theories with large numbers of flavors. I also show that the F value is itself a useful quantity for probing the gauge/gravity duality and understanding other aspects of CFT, such as the scaling dimensions of monopole operators. The F theorem is closely related to quantum entanglement entropy. At conformal fixed points, the F value is equal to minus the renormalized entanglement entropy (REE) in flat Minkowski space across a circle. Away from the fixed points, the REE is a monotonically decreasing function along the RG flow. I compute the REE in a variety of holographic and non-holographic theories. I conclude the Dissertation by discussing a somewhat surprising result: the REE is not stationary at conformal fixed points.The concept of renormalization group (RG) flow is one of the most novel and broad-reaching aspects of quantum field theory (QFT). The RG flow is implemented by constructing effective descriptions of a QFT at decreasing energy scales. One reason that RG flow is useful is that often one is interested in low-energy properties of theories with complicated short-distance structures. RG flows are subject to C theorems in relativistic QFT. The C theorems order the space of Lorentz-invariant QFTs. RG flows generically begin at scale-invariant fixed points known as conformal field theories (CFTs) and end in trivial massive theories. With tuning, the RG flows may end at non-trivial CFTs. Each CFT has an associated dimensionless C value. The C theorem states that under RG flow from a UV to an IR fixed point, the C value decreases. In this Dissertation I present the F-theorem, which is a C theorem in three spacetime dimensions. I show that the correct quantity to consider is the Euclidean free energy of the CFT conformally mapped to the three-sphere, known as the F value. After motivating the F-theorem, I develop tools for calculating the F value in a variety of CFTs, with and without supersymmetry, including free field theories and gauge theories with large numbers of flavors. I also show that the F value is itself a useful quantity for probing the gauge/gravity duality and understanding other aspects of CFT, such as the scaling dimensions of monopole operators. The F theorem is closely related to quantum entanglement entropy. At conformal fixed points, the F value is equal to minus the renormalized entanglement entropy (REE) in flat Minkowski space across a circle. Away from the fixed points, the REE is a monotonically decreasing function along the RG flow. I compute the REE in a variety of holographic and non-holographic theories. I conclude the Dissertation by discussing a somewhat surprising result: the REE is not stationary at conformal fixed points.

Sep 25, 20140 citations

Random geometry at an infinite-randomness fixed point

Akshat Pandey, Aditya Mahadevan, Aditya Cowsik

We study the low-energy physics of the critical ( $2+1$ )-dimensional random transverse-field Ising model. The one-dimensional version of the model is a paradigmatic example of a system governed by an infinite-randomness fixed point, for which many results on the distributions of observables are known via an asymptotically exact renormalization group (RG) approach. In two dimensions, the same RG rules have been implemented numerically and demonstrate a flow to infinite randomness. However, analytical understanding of the structure of this RG has remained elusive due to the development of geometrical structure in the graph of interacting spins. To understand the character of the fixed point, we consider the RG flow acting on a joint ensemble of graphs and couplings. We propose that the RG effectively occurs in two stages: (1) randomization of the interaction graph until it belongs to a certain ensemble of random triangulations of the plane and (2) a flow of the distributions of couplings to infinite randomness while the graph ensemble remains invariant. This picture is substantiated by a numerical RG in which one obtains a steady-state graph degree distribution and subsequently infinite-randomness scaling distributions of the couplings. Both of these aspects of the RG flow can be approximately reproduced in simplified analytical models.We study the low-energy physics of the critical ($2+1$)-dimensional random transverse-field Ising model. The one-dimensional version of the model is a paradigmatic example of a system governed by an infinite-randomness fixed point, for which many results on the distributions of observables are known via an asymptotically exact renormalization group (RG) approach. In two dimensions, the same RG rules have been implemented numerically and demonstrate a flow to infinite randomness. However, analytical understanding of the structure of this RG has remained elusive due to the development of geometrical structure in the graph of interacting spins. To understand the character of the fixed point, we consider the RG flow acting on a joint ensemble of graphs and couplings. We propose that the RG effectively occurs in two stages: (1) randomization of the interaction graph until it belongs to a certain ensemble of random triangulations of the plane and (2) a flow of the distributions of couplings to infinite randomness while the graph ensemble remains invariant. This picture is substantiated by a numerical RG in which one obtains a steady-state graph degree distribution and subsequently infinite-randomness scaling distributions of the couplings. Both of these aspects of the RG flow can be approximately reproduced in simplified analytical models.

arXiv:2304.10564Physical review. B./Physical review. BAug 2, 20232 citations

Symmetry, Duality, and Anholonomy of Point Interactions in One Dimension

Taksu Cheon, Tamás Fülöp, Izumi Tsutsui

arXiv:quant-ph/0008123Annals of PhysicsNov 1, 200175 citations

The generalized point interaction in one dimension

Petr Šeba

Czechoslovak Journal of PhysicsJun 1, 1986151 citations

The Geometry of Quantum Flow

James Lindesay, Harry L. Morrison

Jan 1, 19857 citations

We study the low-energy physics of the critical (

2+1

)-dimensional random transverse-field Ising model. The one-dimensional version of the model is a paradigmatic example of a system governed by an infinite-randomness fixed point, for which many results on the distributions of observables are known via an asymptotically exact renormalization group (RG) approach. In two dimensions, the same RG rules have been implemented numerically and demonstrate a flow to infinite randomness. However, analytical understanding of the structure of this RG has remained elusive due to the development of geometrical structure in the graph of interacting spins. To understand the character of the fixed point, we consider the RG flow acting on a joint ensemble of graphs and couplings. We propose that the RG effectively occurs in two stages: (1) randomization of the interaction graph until it belongs to a certain ensemble of random triangulations of the plane and (2) a flow of the distributions of couplings to infinite randomness while the graph ensemble remains invariant. This picture is substantiated by a numerical RG in which one obtains a steady-state graph degree distribution and subsequently infinite-randomness scaling distributions of the couplings. Both of these aspects of the RG flow can be approximately reproduced in simplified analytical models.