Spread complexity for the planar limit of holography

TL;DR

This work generalizes spread complexity from bosonic systems to fermionic and supercoherent states, enabling analytic access in semiclassical and holographic regimes. By introducing a Krylov path on higher-dimensional weight lattices generated by Lie (super)algebras, the authors connect complexity to the geometry of the underlying space, including AdS/CFT planar strings and their spin-chain duals. They derive explicit Lanczos data and complexity formulas for fermionic HW, $OSp(2|1)$, and superstring subsectors, showing how complexity encodes worldsheet and target-space geometry and remains meaningful despite fermionic finite or infinite Krylov spaces. The results establish a versatile framework for diagnosing holographic dynamics and supersymmetric systems through complexity, with broad potential for exploring topology, integrability, and higher-rank algebras in semiclassical regimes.

Abstract

Complexity is a fundamental characteristic of states within a quantum system. Its use is however mostly limited to bosonic systems, inhibiting its present applicability to supersymmetric theories. This is also relevant to its application to the AdS/CFT correspondence. To address this limitation, we extend the framework of spread complexity beyond bosonic systems to include fermionic and supercoherent states. This offers a gateway to compute spread complexity analytically for any semiclassical system governed by a Hamiltonian associated with a Lie (super)algebra. This requires extending the Krylov chain to a Krylov path in a higher-dimensional lattice. A detailed analysis of supercoherent states within the super Heisenberg-Weyl and OSp$(2|1)$ algebras elucidates distinct contributions from bosonic and fermionic degrees of freedom to the complexity. This generalisation allows us to access the semiclassical regime of the planar limit of the holographic correspondence. We then compute the spread complexity of large charge superstring states on the gravity side, which are equivalent to the dual gauge states. The resulting complexity leads to Krylov paths capturing the geometry in which the string propagates.

Figures (10)

• Figure 1: The Lanczos algorithm constructs a Krylov basis which leads to an auxiliary semi-infinite chain reflecting the gradual spread of the initial state over the Hilbert space.
• Figure 2: The time evolution in the spread complexity is fixed by solving the semi-classical path in phase space of the coherent state. On the left the path in phase space for the linear $SL(2,\mathbb R)$ Hamiltonian. Choosing a different Hamiltonian, for the same set of coherent states, leads to a different path in phase space. Such a hypothetical path is sketched on the right. Although described by the same set of coherent states, the resulting complexity is distinct due to a different time evolution.
• Figure 3: Spread complexity for bosonic states, for example the $\mathfrak{sl}(2,\mathbb R)$ coherent state, leads to an effective semi-infinite chain. For purely fermionic states, the chain becomes finites, reflecting the statistic of those states.
• Figure 4: The left panel shows the plot of complexity contributions of spreading in the Fock basis for fixed total fermion number states normalised by the dimension of the Hilbert space. In the left panel, warmer colours indicate contributions from states with lower fermion numbers, while colder colours correspond to contributions from states with higher fermion numbers. At early times, the dynamics is dominated by states with smaller fermion numbers, whereas at later times, states with larger fermion numbers become dominant. The total complexity, shown on the right, is the envelope obtained by summing all contributions. On the right, the total normalised complexity of spreading in the Fock basis for the overall system for a 15-mode fermionic coherent state where the coefficients $\{\alpha_{k}\}$ are randomly chosen in the range [0, 1]. This illustrates how the total complexity of the system in the Fock basis arises as a cumulative contribution from individual fermion number states.
• Figure 5: The lattice points represent a basis of the Hilbert space $\mathcal{H}$, and a locus is read out by position operators $\vec{X}$. An initial vector $\ket{\psi_0}$ can be any superposition of these lattice points. Picking $\ket{\psi_0}$ to be the basis vector in the lower left corner, say, the evolution driven by a displacement operator $D$ leads through superpositions of basis vectors, i.e the Krylov path (dashed line). This path is an embedding of the Krylov chain into the total Hilbert space $\mathcal{K}\hookrightarrow\mathcal{H}$.