79 research outputs found
Critical Ising Model with Boundary Magnetic Field: RG Interface and Effective Hamiltonians
Critical 2D Ising model with a boundary magnetic field is arguably the
simplest QFT that interpolates between two non-trivial fixed points. We use the
diagonalising Bogolyubov transformation for this model to investigate two
quantities. Firstly we explicitly construct an RG interface operator that is a
boundary condition changing operator linking the free boundary condition with
the one with a boundary magnetic field. We investigate its properties and in
particular show that in the limit of large magnetic field this operator becomes
the dimension 1/16 primary field linking the free and fixed boundary
conditions. Secondly we use Schrieffer-Wolff method to construct effective
Hamiltonians both near the UV and IR fixed points.Comment: 38 pages; v.2 minor changes, to appear in JHE
Renormalization group defects for boundary flows
Recently Gaiotto [1] considered conformal defects which produce an expansion
of infrared local fields in terms of the ultraviolet ones for a given
renormalization group flow. In this paper we propose that for a boundary RG
flow in two dimensions there exist boundary condition changing fields (RG
defect fields) linking the UV and the IR conformal boundary conditions which
carry similar information on the expansion of boundary fields at the fixed
points. We propose an expression for a pairing between IR and UV operators in
terms of a four-point function with two insertions of the RG defect fields. For
the boundary flows in minimal models triggered by \psi_{13} perturbation we
make an explicit proposal for the RG defect fields. We check our conjecture by
a number of calculations done for the example of (p,2)--> (p-1,1)+(p+1,1)
flows.Comment: 1+23 pages, 2 Latex figures; v.3: minor corrections throughout the
text, references adde
Gradient formula for the beta-function of 2d quantum field theory
We give a non-perturbative proof of a gradient formula for beta functions of
two-dimensional quantum field theories. The gradient formula has the form
\partial_{i}c = - (g_{ij}+\Delta g_{ij} +b_{ij})\beta^{j} where \beta^{j} are
the beta functions, c and g_{ij} are the Zamolodchikov c-function and metric,
b_{ij} is an antisymmetric tensor introduced by H. Osborn and \Delta g_{ij} is
a certain metric correction. The formula is derived under the assumption of
stress-energy conservation and certain conditions on the infrared behaviour the
most significant of which is the condition that the large distance limit of the
field theory does not exhibit spontaneously broken global conformal symmetry.
Being specialized to non-linear sigma models this formula implies a one-to-one
correspondence between renormalization group fixed points and critical points
of c.Comment: LaTex file, 31 pages, no figures; v.2 referencing corrected in the
introductio
General properties of the boundary renormalization group flow for supersymmetric systems in 1+1 dimensions
We consider the general supersymmetric one-dimensional quantum system with
boundary, critical in the bulk but not at the boundary. The renormalization
group flow on the space of boundary conditions is generated by the boundary
beta functions \beta^{a}(\lambda) for the boundary coupling constants
\lambda^{a}. We prove a gradient formula \partial\ln z/\partial\lambda^{a}
=-g_{ab}^{S}\beta^{b} where z(\lambda) is the boundary partition function at
given temperature T=1/\beta, and g_{ab}^{S}(\lambda) is a certain
positive-definite metric on the space of supersymmetric boundary conditions.
The proof depends on canonical ultraviolet behavior at the boundary. Any system
whose short distance behavior is governed by a fixed point satisfies this
requirement. The gradient formula implies that the boundary energy,
-\partial\ln z/\partial\beta = -T\beta^{a}\partial_{a}\ln z, is nonnegative.
Equivalently, the quantity \ln z(\lambda) decreases under the renormalization
group flow.Comment: 21 pages, Late
On asymptotic behaviour in truncated conformal space approach
The Truncated conformal space approach (TCSA) is a numerical technique for
finding finite size spectrum of Hamiltonians in quantum field theory described
as perturbations of conformal field theories. The truncation errors of the
method have been systematically studied near the UV fixed point (when the
characteristic energy related to the coupling is less than the truncation
cutoff) where a good theoretical understanding has been achieved. However
numerically the method demonstrated a good agreement with other methods for
much larger values of the coupling when the RG flow approaches a new fixed
point in the infrared. In the present paper we investigate this regime for a
number of boundary RG flows testing the leading exponent and truncation errors.
We also study the flows beyond the first fixed point which have been observed
numerically but yet lack a theoretical understanding. We show that while in
some models such flows approximate reversed physical RG flows, in other models
the spectrum approaches a stable regime that does not correspond to any local
boundary condition. Furthermore we find that in general the flows beyond the
first fixed point are very sensitive to modifications of the truncation scheme.Comment: v2: presentation restructured, general considerations are put forward
into section 2, section on bulk flows removed, quality of all pictures and
referencing improved; 40 pages, 22 figures, 11 tables; to appear in JHE
Open topological defects and boundary RG flows
In the context of two-dimensional rational conformal field theories we
consider topological junctions of topological defect lines with boundary
conditions. We refer to such junctions as open topological defects. For a
relevant boundary operator on a conformal boundary condition we consider a
commutation relation with an open defect obtained by passing the junction point
through the boundary operator. We show that when there is an open defect that
commutes or anti-commutes with the boundary operator there are interesting
implications for the boundary RG flows triggered by this operator. The end
points of the flow must satisfy certain constraints which, in essence, require
the end points to admit junctions with the same open defects. Furthermore, the
open defects in the infrared must generate a subring under fusion that is
isomorphic to the analogous subring of the original boundary condition. We
illustrate these constraints by a number of explicit examples in Virasoro
minimal models.Comment: 26 pages; v.2: section 3 rewritten and now includes a detailed
discussion of RG counterterms, new example added at the end of section 4.2,
extended discussion of the \psi_1,2 boundary flow in the Pentacritical model,
minor improvements throughout the tex
Entropy of conformal perturbation defects
We consider perturbation defects obtained by perturbing a 2D conformal field
theory (CFT) by a relevant operator on a half-plane. If the perturbed bulk
theory flows to an infrared fixed point described by another CFT, the defect
flows to a conformal defect between the ultraviolet and infrared fixed point
CFTs. For short bulk renormalization group flows connecting two fixed points
which are close in theory space we find a universal perturbative formula for
the boundary entropy of the corresponding conformal perturbation defect. We
compare the value of the boundary entropy that our formula gives for the flows
between nearby Virasoro minimal models Mm with the boundary entropy of the
defect constructed by Gaiotto in [1] and find a match at the first two orders
in the 1/m expansion.Comment: 24 pages, 2 figure
Open string radiation from decaying FZZT branes
In this paper we continue studying the decay of unstable FZZT branes
initiated in [1],[2]. The mass of tachyonic mode in this model can be chosen
arbitrarily small and we use it as a perturbative parameter. In [2] a
time-dependent boundary conformal field theory (BCFT) describing the decay
process was studied and it was shown that in a certain sense this BCFT
interpolates between two stationary BCFT's corresponding to the UV and IR fixed
points of the associated RG flow. In the present work we find in the leading
order vertex operators of the time-dependent BCFT. We identify the "in" and
"out" vertex operators assigned to the UV and IR fixed points and compute the
related Bogolyubov coefficients. We show that there is a codimension one
subspace of the out-going states for which pair creation amplitudes are
independent of the initial wave function of the tachyonic mode. We demonstrate
that such amplitudes can be computed within the framework of first quantized
open string theory via suitably defined string two-point functions. We also
evaluate a three point function which we interpret as an amplitude for string
triplet creation due to interaction. Some peculiarities of scattering
amplitudes in the presence of tachyonic modes in the far past are discussed.Comment: 1+39 pages, Latex; v.2: minor improvements all over the tex
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