Complexity in the presence of a boundary

Paolo Braccia; Aldo L. Cotrone; Erik Tonni

Source-linked AI summary

Complexity in the presence of a boundary

Paolo Braccia, Aldo L. Cotrone, Erik Tonni

arXiv:1910.03489v1Oct 8, 2019hep-th

TL;DR

The paper examines how physical boundaries affect complexity in two-dimensional theories, using harmonic chains and holographic boundary CFTs. Across these settings, it finds boundary-related logarithmic divergences except in CA, while CV subregion complexity can jump under continuous configuration changes.

Problem
The paper tests how a physical boundary affects complexity in two-dimensional theories, where definitions and holographic prescriptions remain difficult to settle.
Source: §1 · pp. 2–3
Method
It calculates complexity in Dirichlet harmonic chains and in AdS3/BCFT2 using CV, CA, and CV2.0 prescriptions.
Source: §3 · p. 16, §5 · p. 22
Results
Boundary-dependent logarithmic divergences appear in the harmonic chain and in CV/CV2.0, but not CA, while single-interval CV subregion complexity can jump discontinuously.
Source: §1 · p. 4, §5 · p. 22
Takeaways & Limitations
Within the studied settings, the boundary’s logarithmic contribution distinguishes CV and CV2.0 from CA, whose boundary dependence is finite.
Source: §5 · p. 22
The analysis uses a limited set of models and settings; the effects of boundaries in higher dimensions, time-dependent scenarios, mixed states, and near spacetime singularities remain directions for further study.
Source: §5 · p. 22

Abstract

from arXiv · show

The effects of a boundary on the circuit complexity are studied in two dimensional theories. The analysis is performed in the holographic realization of a conformal field theory with a boundary by employing different proposals for the dual of the complexity, including the "Complexity = Volume" (CV) and "Complexity = Action" (CA) prescriptions, and in the harmonic chain with Dirichlet boundary conditions. In all the cases considered except for CA, the boundary introduces a subleading logarithmic divergence in the expansion of the complexity as the UV cutoff vanishes. Holographic subregion complexity is also explored in the CV case, finding that it can change discontinuously under continuous variations of the configuration of the subregion.

1 Introduction

The paper tests competing complexity prescriptions in two-dimensional theories with boundaries, where circuit and holographic complexity are difficult to define consistently. It finds boundary-dependent logarithmic divergences in the harmonic chain and in CV/CV2.0, but not CA, and reports discontinuous changes in CV subregion complexity.

1 Introduction

Circuit complexity depends on the chosen gates and cost function, while holographic complexity has several competing prescriptions, motivating tests in settings computable in both gravity and discretized field theory.The paper studies two-dimensional theories with a boundary.Source: §1 · p. 2
With Dirichlet boundary conditions, the harmonic chain has leading UV divergences matching the boundary-less case and subleading logarithmic divergences absent in the periodic case.The paper also reports an analytic massless result and numerical support for its lattice analysis.Source: §1 · p. 3
In AdS3/BCFT2, CV and CV2.0 have boundary-data-dependent logarithmic divergences, whereas CA shifts boundary dependence to the finite term.The leading UV divergence is the same as in the boundary-less case for all these prescriptions.Source: §1 · p. 3
CV subregion complexity for a single interval can jump discontinuously as the interval configuration varies continuously.The paper also notes that the logarithmic divergence in CA is absent where it appears in other boundary-complexity analyses.Source: §1 · pp. 3–4

2 Holographic complexity in AdS3/BCFT2

The section reviews the holographic dual of a boundary conformal field theory and calculates ground-state complexity using CV and CA. It also comments on alternative holographic complexity proposals.

2 Holographic complexity in AdS3/BCFT2

The section reviews how to construct the holographic dual of a boundary conformal field theory.Source: §2 · p. 4
It calculates ground-state holographic complexity using the CV prescription.Source: §2 · p. 4
It calculates complexity using CA and closes with comments on other holographic proposals.Source: §2 · p. 4

2.1 AdS3/BCFT2 setup

The setup uses the AdS/BCFT construction for a two-dimensional boundary conformal field theory in its ground state. Its bulk region is bounded by a brane whose angle encodes its tension; alternative BCFT dual constructions are not explored.

2.1 AdS3/BCFT2 setup

The analysis adopts the AdS/BCFT proposal of Takayanagi and focuses on the ground state of a BCFT2 dual to a portion of AdS3.Source: §2.1 · p. 4
The bulk region is delimited by brane Q, anchored at the BCFT boundary and governed by Neumann boundary conditions.The brane delimits the asymptotically AdS region dual to the BCFT2 state.Source: §2.1 · p. 5
For the half-line BCFT2, the static geometry restricts AdS3 to a half-plane, with brane angle α related to tension by T = cos(α)/R.The ansatz z = z(x) yields no solutions other than the stated half-plane.Source: §2.1 · p. 5
The paper does not explore alternative proposals for constructing holographic duals of BCFTs.Source: §2.1 · p. 5

2.2 CV

In CV, complexity is computed from the volume of a maximal bulk surface anchored to the boundary time slice. For the half-line setup, the regulated result has the usual leading UV divergence and an α-dependent subleading logarithm that vanishes at α = π/2.

2.2 CV

CV complexity is proportional to the volume V of the maximal codimension-one bulk surface anchored to the boundary time slice.The calculation leaves the arbitrary length scale l unspecified rather than setting it to the AdS radius.Source: §2.2 · p. 5
For the symmetric setup, the extremal surface is the bulk time slice, and its volume integral uses UV and IR regulators ϵ and zIR.The depicted volume is bounded by the UV cutoff, the IR cutoff, and brane Q.Source: §2.2 · p. 6, Figure 1 · p. 6
The CV result has the boundary-less leading UV linear divergence plus an α-dependent subleading logarithmic divergence, which vanishes at α = π/2.At α = π/2, the infrared logarithmic divergence also disappears.Source: §2.2 · p. 6

2.3 CA

The CA calculation evaluates gravitational action on a regularized WDW patch by summing bulk, boundary, joint, and brane contributions. Its result retains the leading UV divergence but has no subleading logarithmic divergence; instead, the boundary modifies the finite term through α.

2.3 CA

The CA prescription evaluates gravitational action in the WDW patch, with contributions split into bulk, boundary, and joint terms, plus a contribution from brane Q.The patch is regulated near the boundary at z = ϵ and deep in the bulk at z = zIR.Source: §2.3 · pp. 7–8

2.3.1 Bulk terms

The bulk calculation separates the WDW patch into central and left regions, evaluates each future half using time symmetry, and sums the contributions.The central region spans x ∈ [0, L]; the left region extends to x = −z cot α.Source: §2.3.1 · p. 9

2.3.2 Timelike boundary and joint terms

Timelike boundaries include the UV and IR regulators and the intersection of Q with the WDW patch; their boundary and joint contributions are evaluated and summed.The regulators are constant-z surfaces, and the IR regulator contribution differs in sign because its normal points oppositely.Source: §2.3.2 · pp. 9–10

2.3.3 Null boundary and joint terms

For the null boundaries, κ vanishes on the affinely parameterized central surface, leaving the counterterm contribution; the outermost null-brane joint contributes zero.That joint has a vanishing line element, and the limiting procedure is applicable because the null surface is stationary (Θ = 0).Source: §2.3.3 · pp. 11, 13

2.3.4 Brane term

The brane contribution comes from Q ∩ WDW; possible corner contributions are discarded because no local codimension-three counterterms exist and regulated limits have yielded zero.The authors note that regulated corner contributions vanish in the cases in the literature where they could be calculated.Source: §2.3.4 · p. 13

2.3.5 Result for CA

CA retains the boundary-less leading 1/ϵ UV divergence but has no subleading logarithmic divergence; the boundary modifies the finite term through α.The finite term comes entirely from time-like joints and is also present with the alternative regularization scheme.Source: §2.3 · p. 9, §2.3.5 · p. 14
The surface-and-joint proposal [12] has a CV-like boundary-dependent logarithmic divergence and would also include a boundary-dependent finite term.The logarithmic functions differ between that result and CV.Source: §2.3.5 · p. 14

3 Complexity in the harmonic chain with Dirichlet boundary conditions

The harmonic-chain analysis adapts circuit-space methods to a segment with Dirichlet boundary conditions and finds logarithmic divergences absent in periodic systems. The results differ between the two cost functions and depend on mass in the κ = 2 case.

Method

The calculation adapts circuit-space geodesic methods to a Dirichlet harmonic chain, using a factorized Gaussian reference state and two cost functions.The generalized κ cost function is evaluated for κ = 1 and κ = 2.Source: §3 · pp. 15, 16, 18

C1 complexity

C1 has a logarithmic divergence absent with periodic boundary conditions, and its massive regime retains the massless coefficient −1/4.For ω = 0, the large-N leading term matches the periodic-chain result; for ω ≠ 0, the O(1) terms depend nontrivially on ω.Source: §3 · pp. 17–18
For a massless field, C1 has a logarithmic divergence that remains in the large-L regime used for comparison with holography.Its leading large-N term matches the periodic-chain result, which also has a linear divergence in N.Source: §3 · p. 17

Cκ=2 complexity

The massless Cκ=2 expansion contains a logarithmic divergence whose coefficient is a nontrivial function of LωR, determined numerically.The large-N expansion was checked numerically; the corresponding term does not occur for periodic boundary conditions.Source: §3 · p. 19

4 Holographic subregion complexity

In the BCFT2 setup, CV subregion complexity is computed from the bulk volume bounded by the interval’s RT surface. Changes in the minimal RT-surface configuration can produce a finite discontinuity as the interval moves away from the boundary.

RT surfaces and volume

The CV subregion complexity is the volume enclosed by the RT surface on the t = 0 slice, whose configuration depends on the interval’s distance from the boundary.An attached interval has an arc centered at the boundary; at small nonzero distance, the RT surface comprises two arcs connecting the interval endpoints to the brane.Source: §4 · p. 20

RT-surface transition

At the critical distance dc, the RT surface changes configuration and the subregion complexity has a finite discontinuous jump.For d > dc, the RT surface is a semicircle; below dc, the near-boundary configuration uses two arcs ending on the brane.Source: §4 · pp. 20–22

Semicircle configuration

For the semicircle configuration, the subregion complexity is independent of both boundary data and distance from the boundary.It has the usual UV divergence of holographic complexity plus an extra finite term.Source: §4 · p. 21

Boundary-attached interval

For an interval attached to the boundary, the boundary adds a subleading UV divergence and finite terms depending on α; the subleading divergence vanishes at α = π/2.At α = π/2, the result is half the boundaryless result with ℓ replaced by 2ℓ.Source: §4 · p. 21

5 Conclusions

The paper studies boundary effects on complexity in a Dirichlet harmonic chain and in AdS3/BCFT2 using CV, CA, and CV2.0. It finds boundary-dependent logarithmic divergences for CV and CV2.0, but not CA, and discontinuous jumps in CV subregion complexity.

Approach and results

The study evaluates Dirichlet harmonic-chain complexity and applies CV, CA, and CV2.0 to the AdS3/BCFT2 holographic setup.The paper reports the corresponding holographic results and large-N expansions for the chain’s C1 and massless Cκ=2 complexities.Source: §5 · p. 22

Holographic complexity

In AdS3/BCFT2, the boundary introduces a boundary-data-dependent logarithmic divergence in CV and CV2.0, but not CA, where boundary dependence appears in the finite term.This distinguishes the CV and CA prescriptions.Source: §5 · p. 22

Harmonic-chain complexity

The harmonic chain has divergences growing as O(log N) for C1 and O in the number of sites for Cκ=2, unlike periodic systems.The paper notes that the models are not directly comparable, while C1’s logarithmic divergence resembles CV and CV2.0.Source: §5 · p. 22

Future directions

Further study is needed on boundary effects in higher dimensions, time-dependent settings, mixed states, spacetime singularities, and links with thermodynamics.These are identified as directions for further analysis.Source: §5 · p. 22

A A different regularization scheme for the WDW patch

The appendix tests the CA result with a different WDW-patch regularization scheme. Although some boundary contributions change, cancellations leave the final complexity unchanged.

A A different regularization scheme for the WDW patch

The alternative scheme uses z = const surfaces centrally and r = const surfaces to connect the patch to the boundary brane.The central contributions are unchanged, so the calculation evaluates only the boundary-region terms.Source: A A different regularization scheme… · p. 23
The evaluated bulk, brane, null-boundary, and joint terms reproduce the corresponding results from the original regularization.The passages report matching results for the left bulk contribution, brane contribution, null-boundary terms, and timelike joints.Source: A A different regularization scheme… · pp. 23–25
The alternative regulator produces only the logarithmic part of f(α), but its difference cancels against contributions from the other regulator.The timelike-regulator contributions cancel when both r = ϵ and r = zIR are included.Source: A A different regularization scheme… · p. 23
The alternative regularization leaves the final CA complexity unchanged despite modifications to some action contributions.The null joint and corners are taken to vanish, as in the original scheme.Source: A A different regularization scheme… · p. 25

B Cκ=2 for large N

The appendix studies the large-N behavior of S2 using the Euler–Maclaurin formula. A bound and numerically fitted constants supply an approximation for R2 that yields the reported result for Cκ=2.

B Cκ=2 for large N

The large-N analysis approximates S2 by an integral and applies the Euler–Maclaurin formula with p = 2.The appendix introduces a = π/(2N) and θ = ak, then applies the formula to the resulting function.Source: B Cκ=2 for large N · pp. 25–26
The Euler–Maclaurin approach can also explore the large-N behavior of C1, justifying (3.9).Source: B Cκ=2 for large N · p. 26
For large N, R2 is bounded but cannot be evaluated analytically; replacing it with 4ν log N + 4µ using numerically fitted constants yields (3.18).The approximation is combined with (B.1) and (B.5).Source: B Cκ=2 for large N · p. 26