Boundary Condition Changing Operators Research Articles

Channeling Quantum Criticality.

We analyze the effect of decoherence, modeled by local quantum channels, on quantum critical states and we find universal properties of the resulting mixed state's entanglement, both between system and environment and within the system. Renyi entropies exhibit volume law scaling with a subleading constant governed by a "g function" in conformal field theory, allowing us to define a notion of renormalization group (RG) flow (or "phase transitions") between quantum channels. We also find that the entropy of a subsystem in the decohered state has a subleading logarithmic scaling with subsystem size, and we relate it to correlation functions of boundary condition changing operators in the conformal field theory. Finally, we find that the subsystem entanglement negativity, a measure of quantum correlations within mixed states, can exhibit log scaling or area law based on the RG flow. When the channel corresponds to a marginal perturbation, the coefficient of the log scaling can change continuously with decoherence strength. We illustrate all these possibilities for the critical ground state of the transverse-field Ising model, in which we identify four RG fixed points of dephasing channels and verify the RG flow numerically. Our results are relevant to quantum critical states realized on noisy quantum simulators, in which our predicted entanglement scaling can be probed via shadow tomography methods.

Ising model in a boundary magnetic field with random discontinuities

We consider a two-dimensional Ising field theory on a space with boundary in the presence of a piecewise constant boundary magnetic field which is allowed to change value discontinuously along the boundary. We assume zero magnetic field in the bulk. The positions of discontinuities are averaged over as in the annealed disorder. This model is described by a boundary field theory in which a superposition of the free spin boundary condition is perturbed by a collection of boundary condition changing operators. The corresponding boundary couplings give the allowed constant values of the magnetic field as well as the fugacities for the transitions between them. We show that when the value of the magnetic field is allowed to take only two different values which are the same in magnitude but have different signs the model can be described by a quadratic Lagrangian. We calculate and analyse the exact reflection matrix for this model. We also calculate the boundary entropy and study in detail the space of RG flows in a three-parameter space and with four different infrared fixed points. We discuss the likely breakdown of integrability in the extended model which allows for two generic values of the boundary magnetic field, backing it by some calculations.

Holographic BCFTs and communicating black holes

We study the AdS/BCFT duality between two-dimensional conformal field theories with two boundaries and three-dimensional anti-de Sitter space with two Karch-Randall branes. We compute the entanglement entropy of a bipartition of the BCFT, on both the gravity side and the field theory side. At finite temperature this entanglement entropy characterizes the communication between two braneworld black holes, coupled to each other through a common bath. We find a Page curve consistent with unitarity. The gravitational result, computed using double-holographically realized quantum extremal surfaces, matches the conformal field theory calculation.At zero temperature, we obtain an interesting extension of the AdS3/BCFT2 correspondence. For a central charge c, we find a gap left(frac{c}{16},frac{c}{12}right) in the spectrum of the scaling dimension ∆bcc of the boundary condition changing operator (which interpolates mismatched boundary conditions on the two boundaries of the BCFT). Depending on the value of ∆bcc, the gravitational dual is either a defect global AdS3 geometry or a single sided black hole, and in both cases there are two Karch-Randall branes.

Properties of RG interfaces for 2D boundary flows

We consider RG interfaces for boundary RG flows in two-dimensional QFTs. Such interfaces are particular boundary condition changing operators linking the UV and IR conformal boundary conditions. We refer to them as RG operators. In this paper we study their general properties putting forward a number of conjectures. We conjecture that an RG operator is always a conformal primary such that the OPE of this operator with its conjugate must contain the perturbing UV operator when taken in one order and the leading irrelevant operator (when it exists) along which the flow enters the IR fixed point, when taken in the other order. We support our conjectures by perturbative calculations for flows between nearby fixed points, by a non-perturbative variational method inspired by the variational method proposed by J. Cardy for massive RG flows, and by numerical results obtained using boundary TCSA. The variational method has a merit of its own as it can be used as a first approximation in charting the global structure of the space of boundary RG flows. We also discuss the role of the RG operators in the transport of states and local operators. Some of our considerations can be generalised to two-dimensional bulk flows, clarifying some conceptual issues related to the RG interface put forward by D. Gaiotto for bulk \U0001d7191,3 flows.

Scaling of Loschmidt echo in a boundary-driven critical Z3 Potts model

Low-frequency perturbations at the boundary of critical quantum chains can be understood in terms of the sequence of boundary conditions imposed by them, as has been previously demonstrated in the Ising and related fermion models. Using extensive numerical simulations, we explore the scaling behavior of the Loschmidt echo under longitudinal field perturbations at the boundary of a critical ${\mathbb{Z}}_{3}$ Potts model. We show that at times much larger than the relaxation time after a boundary quench, the Loschmidt echo has a power-law scaling with time as expected from interpreting the quench as insertion of boundary condition changing operators. Similar scaling is observed as a function of time period under a low-frequency square-wave pulse. We present numerical evidence for one cycle that indicates that under a sinusoidal or triangular pulse, scaling with time period is modified by the Kibble-Zurek mechanism, again similar to the case of the Ising model. Results confirm the validity, beyond the Ising model, of the treatment of the boundary perturbations in terms of the effect on boundary conditions.

Conformally invariant boundary conditions in the antiferromagnetic Potts model and the SL(2, \u211d)/U(1) sigma model

We initiate a study of the boundary version of the square-lattice Q-state Potts antiferromagnet, with Q ∈ [0, 4] real, motivated by the fact that the continuum limit of the corresponding bulk model is a non-compact CFT, closely related with the SL(2, ℝ)k/U(1) Euclidian black-hole coset model. While various types of conformal boundary conditions (discrete and continuous branes) have been formally identified for the the SL(2, ℝ)k/U(1) coset CFT, we are only able in this work to identify conformal boundary conditions (CBC) leading to a discrete boundary spectrum. The CBC we find are of two types. The first is free boundary Potts spins, for which we confirm an old conjecture for the generating functions of conformal levels, and show them to be related to characters in a non-linear deformation of the W∞ algebra. The second type of CBC — which corresponds to restricting the values of the Potts spins to a subset of size Q1, or its complement of size Q − Q1, at alternating sites along the boundary — is new, and turns out to be conformal in the antiferromagnetic case only. In the following, we refer to these new boundary conditions as “alt” boundary conditions. Using algebraic and numerical techniques, we show that the corresponding spectrum generating functions produce all the characters of discrete representations for the coset CFT. The normalizability bounds of the associated discrete states in the coset CFT are found to have a simple interpretation in terms of boundary phase transitions in the lattice model. In the two-boundary case, with two distinct alt conditions, we obtain similar results, at least in the case when the corresponding boundary condition changing operator also inserts a number of defect lines. For sqrt{Q} = 2 cos frac{pi }{k} , with k ≥ 3 integer, we show also how our boundary conditions can be reformulated in terms of a RSOS height model. The spectrum generating functions are then identified with string functions of the compact SU(2)k−2/U(1) parafermion theory (with symmetry Zk−2). The new alt conditions are needed to cover all the string functions. We provide an algebraic proof that the two-boundary alt conditions correctly produce the fusion rules of string functions. We expose in detail the special case of Q = 3 and its link with three-colourings of the square lattice and a corresponding boundary six-vertex model. Finally, we discuss the case of an odd number of sites (in the loop model) and the relation with wired boundary conditions (in the spin model). In this case the RSOS restriction produces the disorder operators of the parafermion theory.

Taming boundary condition changing operator anomalies with the tachyon vacuum

Motivated by the appearance of associativity anomalies in the context of superstring field theory, we give a generalized solution built from boundary condition changing operators which can be associated to a generic tachyon vacuum in the KBc subalgebra of the Okawa form. We articulate sufficient conditions on the choice of tachyon vacuum to ensure that ambiguous products do not appear in the equations of motion.

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.

Universal Entanglement Dynamics following a Local Quench

We study the time dependence of the entanglement between two quantum wires after suddenly connecting them via tunneling through an impurity. The result at large times is given by the well known formula S(t) \approx {1\over 3}\ln {t}. We show that the intermediate time regime can be described by a universal cross-over formula S=F(tT_K)S=F(tTK), where T_KTK is the crossover (Kondo) temperature: the function FF describes the dynamical “healing” of the system at large times. We discuss how to obtain analytic information about FF in the case of an integrable quantum impurity problem using the massless Form-Factors formalism for twist and boundary condition changing operators. Our results are confirmed by density matrix renormalization group calculations and exact free fermion numerics.

String field theory solution corresponding to constant background magnetic field

Following the method recently proposed by Erler and Maccaferri, we construct solutions to the equation of motion of Witten's cubic string field theory, which describe constant magnetic field background. We study the boundary condition changing operators relevant to such background and calculate the operator product expansions of them. We obtain solutions whose classical action coincide with the Born-Infeld action.

On form factors of boundary changing operators

We develop a form factor bootstrap program to determine the matrix elements of local, boundary condition changing operators. We propose axioms for these form factors and determine their solutions in the free boson and Lee–Yang models. The sudden change in the boundary condition, caused by an operator insertion, can be interpreted as a local quench and the form factors provide the overlap of any state before the quench with any outgoing state after the quench.

String field theory solution for any open string background

We present an exact solution of open bosonic string field theory which can be used to describe any time-independent open string background. The solution generalizes an earlier construction of Kiermaier, Okawa, and Soler, and assumes the existence of boundary condition changing operators with nonsingular OPEs and vanishing conformal dimension. Our main observation is that boundary condition changing operators of this kind can describe nearly any open string background provided the background shift is accompanied by a timelike Wilson line of sufficient strength. As an application we analyze the tachyon lump describing the formation of a D$(p-1)$-brane in the string field theory of a D$p$-brane, for generic compactification radius. This not only provides a proof of Sen's second conjecture, but also gives explicit examples of higher energy solutions, confirming analytically that string field theory can "reverse" the direction of the worldsheet RG flow. We also find multiple D-brane solutions, demonstrating that string field theory can add Chan-Paton factors and change the rank of the gauge group. Finally, we show how the solution provides a remarkably simple and nonperturbative proof of the background independence of open bosonic string field theory.

Phase separation in a wedge: exact results.

The exact theory of phase separation in a two-dimensional wedge is derived from the properties of the order parameter and boundary condition changing operators in field theory. For a shallow wedge we determine the passage probability for an interface with endpoints on the boundary. For generic opening angles we exhibit the fundamental origin of the filling transition condition and of the property known as wedge covariance.

Logarithmic operator intervals in the boundary theory of critical percolation

We consider the sub-sector of the c = 0 logarithmic conformal field theory (LCFT) generated by the boundary condition changing (bcc) operator in two dimensional critical percolation. This operator is the zero weight Kac operator ϕ1, 2 ≕ ϕ, identified with the growing hull of the SLE6 process. We identify percolation configurations with the significant operators in the theory. We consider operators from the first four bcc operator fusions: the identity and ϕ; the stress-tensor and its logarithmic partner; ∂ϕ and its logarithmic partner; and the pre-logarithmic operator ϕ1, 3. We construct several intervals in the percolation model, each associated to one of the LCFT operators we consider, allowing us to calculate crossing probabilities and expectation values of crossing cluster numbers. We review the CG, which we use as a method of calculating these quantities when the number of bcc operator makes a direct solution to the system of differential equations intractable. Finally we discuss the case of the six-point correlation function, which applies to crossing probabilities between the sides of a conformal hexagon. Specifically we introduce an integral result that allows one to identify the probability that a single percolation cluster touches three alternating sides a hexagon with free boundaries. We give results of the numerical integration for the case of a regular hexagon.

Numerical determination of the boundary condition changing operators

A new numerical method to determine the boundary condition changing (bcc) operators in the boundary conformal field theory (BCFT) is introduced. This method is based on a variant of Schramm–Loewner Evolution (SLE), namely SLE(κ,ρ). As a prototype, the Abelian Sandpile Model (ASM) with a sink point on the boundary is considered. Using this method we study the bcc operator corresponding to the sink in the boundary of c=−2 BCFT. It is numerically shown that the conformal dimension of this operator is nearly 0. The most appropriate candidate for this operator is the logarithmic partner of the unity operator, i. e. Ĩ≡:θθ̄: as it has been conjectured theoretically.

Conformal boundary state for the rectangular geometry

We discuss conformal field theories (CFTs) in rectangular geometries, and develop a formalism that involves a conformal boundary state for the 1+1d open system. We focus on the case of homogeneous boundary conditions (no insertion of a boundary condition changing operator), for which we derive an explicit expression of the associated boundary state, valid for any arbitrary CFT. We check the validity of our solution, comparing it with known results for partition functions, numerical simulations of lattice discretizations, and coherent state expressions for free theories.

Solutions from boundary condition changing operators in open superstring field theory

We construct analytic solutions of open superstring field theory in the Berkovits formulation using boundary condition changing operators under some regularity conditions, extending the previous construction in the bosonic string. We also consider the gauge-invariant observables corresponding to closed string one-point functions on the disk. We analytically calculate the gauge-invariant observables for the solutions both in the bosonic string and in the superstring and find the expected change of boundary conditions of the disk.

Solutions from boundary condition changing operators in open string field theory

We construct analytic solutions of open string field theory using boundary condition changing (bcc) operators. We focus on bcc operators with vanishing conformal weight such as those for regular marginal deformations of the background. For any Fock space state phi, the component string field <phi,Psi> of the solution Psi exhibits a remarkable factorization property: it is given by the matter three-point function of phi with a pair of bcc operators, multiplied by a universal function that only depends on the conformal weight of phi. This universal function is given by a simple integral expression that can be computed once and for all. The three-point functions with bcc operators are thus the only needed physical input of the particular open string background described by the solution. We illustrate our solution with the example of the rolling tachyon profile, for which we prove convergence analytically. The form of our solution, which involves bcc operators instead of explicit insertions of the marginal operator, can be a natural starting point for the construction of analytic solutions for arbitrary backgrounds.

Constructing Open String Field Theory Solutions from Boundary Condition Changing Operators

This is a short review of the results of the work [arXiv:1009.6185] with Michael Kiermaier and Yuji Okawa, which was presented at the String Field Theory and Related Aspects 2010 conference. We construct an analytic solution of open string field theory which is given in terms of boundary condition changing (bcc) operator. The solution is valid for bcc operators with vanishing conformal weight such as those corresponding to regular marginal deformations of the background, but looks like a good starting point to look for solutions corresponding to more general deformations. Applications to the rolling tachyon profile are also presented.

Conformal boundary conditions in the critical [formula omitted] model and dilute loop models

We study the conformal boundary conditions of the dilute O ( n ) model in two dimensions. A pair of mutually dual solutions to the boundary Yang–Baxter equations are found. They describe anisotropic special transitions, and can be interpreted in terms of symmetry breaking interactions in the O ( n ) model. We identify the corresponding boundary condition changing operators, Virasoro characters, and conformally invariant partition functions. We compute the entropies of the conformal boundary states, and organize the flows between the various boundary critical points in a consistent phase diagram. The operators responsible for the various flows are identified. Finally, we discuss the relation to open boundary conditions in the O ( n ) model, and present new crossing probabilities for Ising domain walls.