In this paper we argue that boundary condition may run with energy scale. As an illustrative example, we consider one-dimensional quantum mechanics for a spinless particle that freely propagates in the bulk yet interacts only at the origin. In this setting we find the renormalization group flow of U(2) family of boundary conditions exactly. We show that the well-known scale-independent subfamily of boundary conditions are realized as fixed points. We also discuss the duality between two distinct boundary conditions from the renormalization group point of view. Generalizations to conformal mechanics and quantum graph are also discussed.In this paper we argue that boundary condition may run with energy scale. As an illustrative example, we consider one-dimensional quantum mechanics for a spinless particle that freely propagates in the bulk yet interacts only at the origin. In this setting we find the renormalization group flow of U(2) family of boundary conditions exactly. We show that the well-known scale-independent subfamily of boundary conditions are realized as fixed points. We also discuss the duality between two distinct boundary conditions from the renormalization group point of view. Generalizations to conformal mechanics and quantum graph are also discussed.

Jan 1, 20101 citation

Vacuum energy and renormalization on the edge

M. Asorey, D García-Álvarez, J. M. Munõz-Castañeda

The vacuum dependence on boundary conditions in quantum field theories is analysed from a very general viewpoint. From this perspective, the renormalization prescriptions not only imply the renormalization of the couplings of the theory in the bulk but also the appearance of a flow in the space of boundary conditions. For regular boundaries, this flow has a large variety of fixed points and no cyclic orbit. The family of fixed points includes Neumann and Dirichlet boundary conditions. In one-dimensional field theories, pseudo-periodic and quasi-periodic boundary conditions are also RG fixed points. Under these conditions massless bosonic free field theories are conformally invariant. Among all fixed points only Neumann boundary conditions are infrared stable fixed points. All other conformal invariant boundary conditions become unstable under some relevant perturbations. In finite volumes, we analyse the dependence of the vacuum energy along the trajectories of the renormalization group flow providing an interesting framework for dark energy evolution. In contrast, the renormalization group flow on the boundary does not affect the leading behaviour of the entanglement entropy of the vacuum in one-dimensional conformally invariant bosonic theories.The vacuum dependence on boundary conditions in quantum field theories is analysed from a very general viewpoint. From this perspective, the renormalization prescriptions not only imply the renormalization of the couplings of the theory in the bulk but also the appearance of a flow in the space of boundary conditions. For regular boundaries, this flow has a large variety of fixed points and no cyclic orbit. The family of fixed points includes Neumann and Dirichlet boundary conditions. In one-dimensional field theories, pseudo-periodic and quasi-periodic boundary conditions are also RG fixed points. Under these conditions massless bosonic free field theories are conformally invariant. Among all fixed points only Neumann boundary conditions are infrared stable fixed points. All other conformal invariant boundary conditions become unstable under some relevant perturbations. In finite volumes, we analyse the dependence of the vacuum energy along the trajectories of the renormalization group flow providing an interesting framework for dark energy evolution. In contrast, the renormalization group flow on the boundary does not affect the leading behaviour of the entanglement entropy of the vacuum in one-dimensional conformally invariant bosonic theories.

arXiv:0704.1084Journal of Physics A Mathematical and TheoreticalJun 6, 200723 citations

Teaching renormalization, scaling, and universality with an example from quantum mechanics

Steve Paik

We discuss the quantum mechanics of a particle restricted to the half-line >x 0 with potential energy a = V x 2 for a -< < 1 4 0. It is known that two scale-invariant theories may be defined.By regularizing the near-origin behavior of the potential by a finite square well with variable width b and depth g, it is shown how these two scale-invariant theories occupy fixed points in the resulting (b, g)space of Hamiltonians.A renormalization group (RG) flow exists in this space and scaling variables are shown to exist in a neighborhood of the fixed points.Consequently, the propagator of the regulated theory enjoys homogeneous scaling laws close to the fixed points.Using RG arguments it is possible to discern the functional form of the propagator for long distances and long imaginary times, thus demonstrating the extent to which fixed points control the behavior of the cut-off theory.By keeping the width fixed and varying only the well depth, we show how the mean position of a bound state diverges as g approaches a critical value.It is proven that the exponent characterizing the divergence is universal in the sense that its value is independent of the choice of regulator.Two classical interpretations of the results are discussed: standard Brownian motion on the real line, and the free energy of a certain one-dimensional chain of particles with prescribed boundary conditions.In the former example, V appears as part of an expectation value in the Feynman-Kac formula.In the latter example, V appears as the background potential for the chain, and the loss of extensivity is dictated by a universal power law.We discuss the quantum mechanics of a particle restricted to the half-line >x 0 with potential energy a = V x 2 for a -< < 1 4 0. It is known that two scale-invariant theories may be defined.By regularizing the near-origin behavior of the potential by a finite square well with variable width b and depth g, it is shown how these two scale-invariant theories occupy fixed points in the resulting (b, g)space of Hamiltonians.A renormalization group (RG) flow exists in this space and scaling variables are shown to exist in a neighborhood of the fixed points.Consequently, the propagator of the regulated theory enjoys homogeneous scaling laws close to the fixed points.Using RG arguments it is possible to discern the functional form of the propagator for long distances and long imaginary times, thus demonstrating the extent to which fixed points control the behavior of the cut-off theory.By keeping the width fixed and varying only the well depth, we show how the mean position of a bound state diverges as g approaches a critical value.It is proven that the exponent characterizing the divergence is universal in the sense that its value is independent of the choice of regulator.Two classical interpretations of the results are discussed: standard Brownian motion on the real line, and the free energy of a certain one-dimensional chain of particles with prescribed boundary conditions.In the former example, V appears as part of an expectation value in the Feynman-Kac formula.In the latter example, V appears as the background potential for the chain, and the loss of extensivity is dictated by a universal power law.

arXiv:1707.04388Journal of Physics CommunicationsDec 21, 20175 citations

Entanglement entropy and boundary renormalization group flow: Exact results in the Ising universality class

Eyal Cornfeld, Eran Sela

The entanglement entropy in one-dimensional critical systems with boundaries has been associated with the noninteger ground-state degeneracy. This quantity, being a characteristic of boundary fixed points, decreases under renormalization group flow, as predicted by the g theorem. Here, using conformal field theory methods, we exactly calculate the entanglement entropy in the boundary Ising universality class. Our expression can be separated into the well-known bulk term and a boundary entanglement term, displaying a universal flow between two boundary conditions, in accordance with the g theorem. These results are obtained within the replica trick approach, where we show that the associated twist field, a central object generating the geometry of an $n$ -sheeted Riemann surface, can be bosonized, giving simple analytic access to multiple quantities of interest. We argue that our result applies to other models falling into the same universality class. This includes the vicinity of the quantum critical point of the two-channel Kondo model, allowing one to track in real space the presence of a region containing one-half of a qubit with entropy $\frac{1}{2}log(2)$ , associated with a free local Majorana fermion.The entanglement entropy in one-dimensional critical systems with boundaries has been associated with the noninteger ground-state degeneracy. This quantity, being a characteristic of boundary fixed points, decreases under renormalization group flow, as predicted by the g theorem. Here, using conformal field theory methods, we exactly calculate the entanglement entropy in the boundary Ising universality class. Our expression can be separated into the well-known bulk term and a boundary entanglement term, displaying a universal flow between two boundary conditions, in accordance with the g theorem. These results are obtained within the replica trick approach, where we show that the associated twist field, a central object generating the geometry of an $n$-sheeted Riemann surface, can be bosonized, giving simple analytic access to multiple quantities of interest. We argue that our result applies to other models falling into the same universality class. This includes the vicinity of the quantum critical point of the two-channel Kondo model, allowing one to track in real space the presence of a region containing one-half of a qubit with entropy $\frac{1}{2}log(2)$, associated with a free local Majorana fermion.

arXiv:1705.04181Physical review. B./Physical review. BAug 24, 201715 citations

Boundary Entropy Can Increase Under Bulk RG Flow

Green, Daniel; Mulligan, Michael; Starr, David;

The boundary entropy log(g) of a critical one-dimensional quantum system (or two-dimensional conformal field theory) is known to decrease under renormalization group (RG) flow of the boundary theory. We study instead the behavior of the boundary entropy as the bulk theory flows between two nearby critical points. We use conformal perturbation theory to calculate the change in g due to a slightly relevant bulk perturbation and find that it has no preferred sign. The boundary entropy log(g) can therefore increase during appropriate bulk flows. This is demonstrated explicitly in flows between minimal models. We discuss the applications of this result to D-branes in string theory and to impurity problems in condensed matter.The boundary entropy log(g) of a critical one-dimensional quantum system (or two-dimensional conformal field theory) is known to decrease under renormalization group (RG) flow of the boundary theory. We study instead the behavior of the boundary entropy as the bulk theory flows between two nearby critical points. We use conformal perturbation theory to calculate the change in g due to a slightly relevant bulk perturbation and find that it has no preferred sign. The boundary entropy log(g) can therefore increase during appropriate bulk flows. This is demonstrated explicitly in flows between minimal models. We discuss the applications of this result to D-branes in string theory and to impurity problems in condensed matter.

Aug 14, 20180 citations

Boundary Entropy Can Increase Under Bulk RG Flow

Green, Daniel, Mulligan, Michael, Starr, David

Oct 30, 20070 citations

n

-sheeted Riemann surface, can be bosonized, giving simple analytic access to multiple quantities of interest. We argue that our result applies to other models falling into the same universality class. This includes the vicinity of the quantum critical point of the two-channel Kondo model, allowing one to track in real space the presence of a region containing one-half of a qubit with entropy

\frac{1}{2}log(2)

, associated with a free local Majorana fermion.