Exponential Wormhole Length Operator

• The paper’s main contribution is the formulation of an operator that connects wormhole length in JT gravity to the spread complexity of quantum information in the DSSYK framework.
• It employs a Krylov basis and Lanczos process to precisely define and extend the wormhole length operator in both infinite and finite-dimensional Hilbert spaces.
• The analysis reveals universal dynamical features, including exponential growth, ramp, and plateau phenomena, with influences from random matrix theory and orthogonal polynomial identities.

The Exponential Wormhole Length Operator, also referred to as the chord-number operator or the spread-complexity operator, is a quantum mechanical observable that equates the geometric size of Einstein-Rosen bridges in Jackiw-Teitelboim (JT) gravity to the spreading of quantum information in its dual description via double-scaled Sachdev-Ye-Kitaev (DSSYK) theory. The operator acts as a “position” operator along the Krylov chain generated by repeated applications of the system Hamiltonian on a thermofield double (TFD) state. Its non-perturbative extension enables exact analysis across the full finite-dimensional Hilbert space, elucidating dynamics such as exponential growth, ramp and plateau phenomena, and universality across random-matrix classes (Balasubramanian et al., 2 Dec 2024).

1. Definition and Construction of the Length Operator

The wormhole length operator is defined in terms of the Krylov basis, which is constructed via the Lanczos or Gram–Schmidt process iteratively applied to the set $\{H^n |0\rangle\}$, where $|0\rangle \equiv |\psi_\beta\rangle$ is the initial (possibly infinite-temperature) TFD state. Explicitly, the Krylov chain is

$$|K_0\rangle = |0\rangle, \quad |K_1\rangle \propto H|0\rangle - \langle 0|H|0\rangle|0\rangle, \quad ... \quad |K_n\rangle.$$

In the Krylov basis, the system Hamiltonian $H$ reduces to a tridiagonal (Hessenberg) form: $$(H_K)_{mn} = \langle K_m|H|K_n \rangle = a_n \delta_{mn} + b_n (\delta_{m,n+1} + \delta_{m+1,n}),$$ with real coefficients $a_n$, $b_n$.

The wormhole length (chord-number) operator is

$$\hat{L} = \sum_{n=0}^{L} n |K_n\rangle \langle K_n|$$

where $L$ is the length of the Krylov chain, bounded in the finite-$N$ system. Its expectation value in an arbitrary state $|\psi(t)\rangle = \sum_n \psi_n(t) |K_n\rangle$ measures the spread complexity,

$$C(t) = \langle \psi(t) | \hat{L} | \psi(t) \rangle = \sum_n n |\psi_n(t)|^2.$$

In the strict double-scaling or DSSYK limit ($N \to \infty$), the Krylov chain becomes infinite, and the tridiagonal coefficients are $a_n \approx 0$, $b_n = \mathcal{J}\sqrt{1 - q^n}$, where $q = e^{-\lambda}$, $\lambda = 2p^2/N$. The infinite-chain transfer matrix $T_\infty$ coincides with the chord diagram solution, with $T_\infty$ exhibiting the same bandwidth structure as $H_K$, and $\hat{L}$ takes the form $\sum_n n |n\rangle \langle n|$.

2. Non-Perturbative Finite-Dimensional Extension

The operator’s non-perturbative characterization involves extending the definition to finite-dimensional Hilbert spaces—specifically, the SYK model with Hilbert space dimension $L = 2^{N/2}$. For sub-exponential Krylov index $n \ll L$, the DSSYK form for the Lanczos coefficients ($b_n = \mathcal{J}\sqrt{1 - q^n}$) remains accurate. For $n \sim O(L)$, corrections derived from integral or saddle-point equations become dominant, and $b_n$ “descends” rapidly to zero as $n \to L$, reflecting the finite support of the spectrum.

For large Hilbert space dimension $L$, the coarse-grained variables $x = n/L$ are employed, and the density of states $\rho(E)$ must satisfy the integral constraint: $$\int_0^1 dx\, \Theta [4b(x)^2 - (E - a(x))^2] \frac{1}{\pi} \sqrt{4b(x)^2 - (E - a(x))^2} = \rho(E).$$ This is equivalent to a pair of saddle-point equations involving the random-matrix potential $V(E)$ for $\rho(E)$. This formalism allows identification of the bulk Krylov spectrum, yielding the non-perturbative extension of the length operator on the full chain: $\hat{L} = \sum_{n=0}^L n |K_n\rangle \langle K_n|,$ with $H_K$ set by the bulk $a_n, b_n$.

3. Time Evolution and Dynamical Behavior

The time-dependent expectation value $$C(t) = \langle \text{TFD}(t) | \hat{L} | \text{TFD}(t) \rangle$$ quantifies the growth and saturation of spread complexity, and hence the wormhole length, in the TFD state under time evolution. The analysis leverages the connection to spectral quantities: the survival amplitude $$Z(\beta - it) = \langle \psi_\beta | \psi_\beta(t) \rangle$$ and the spectral form factor ${S}(t) = |Z(\beta-it)|^2$, with the Lanczos coefficients being direct functionals of the time-evolved moments $$M_n = \operatorname{Tr}[e^{-\beta H} H^n]/Z_{\beta}$$.

Dynamically, $C(t)$ displays three principal regimes:

• Early times ($t \ll e^{S}$): Linear growth, $C(t) \approx v_E t$, with velocity $v_E = 2\mathcal{J} \sqrt{1 - q}$ for DSSYK at infinite temperature.
• Intermediate times: Emergence of a “ramp” (semi-linear intermediate growth), and “peak-overshoot” (local maximum), governed by random-matrix coherence effects.
• Late, exponential times ($t \sim e^S$): Downward slope (“white-hole shrinking”) followed by saturation to a plateau.

At low temperature, the late-time plateau (saturated wormhole length) is determined by a stationary distribution,

$$\omega(x) = \frac{L}{Z_\beta} \int_{E_L(x)}^{E_H(x)} dE\, e^{-\beta E} [\pi \sqrt{-(E-E_L(x))(E-E_H(x))}]^{-1},$$

where $E_L(x), E_H(x)$ correspond to spectral band-edges, and $\langle x \rangle = C_{\text{plateau}}/L$ can be obtained from the associated integral formula. The plateau value is numerically $\mathcal{O}(e^S)$ but parametrically has polynomial-in-$S$ corrections.

4. Influence of Random Matrix Universality

The dynamics toward, and the structure of, the late-time saturation of $C(t)$ and related observables depend on the random-matrix universality class of the SYK model or its generalizations. In particular:

• The Dyson $\beta$-index (GOE, GUE, GSE) strongly affects the approach to the plateau: increasing $\beta$ implies stronger level repulsion, producing a more pronounced post-ramp slope.
• The plateau height is $\beta$-independent to leading order, with late-time fine structure only mildly sensitive to the Altland–Zirnbauer $\alpha$-index (extra $|\lambda|^\alpha$ factors in the joint PDF).
• Numerical analysis (see Figs. 8–9 in the source) confirms qualitative consistency across all seven AZ classes sharing the DSSYK density of states $\rho(E)$.

A plausible implication is that certain universal features of wormhole length saturation are robust across a broad family of random Hamiltonians, while intermediate-time details depend sensitively on the underlying symmetry class.

5. Orthogonal Polynomial Identities and Operator Algebra

The underlying structure of the Krylov chain and the spectral properties are governed by orthogonal polynomials and associated identities:

• Scaled Chebyshev polynomials $C_n(x)$, defined recursively by $C_0 = 1$, $C_1 = x$, $C_n = xC_{n-1} - C_{n-2}$, with $C_n(2\cos \theta) = \sin[(n+1)\theta]/\sin \theta$.
• Polynomials $P_n(\omega)$ entering the random-matrix potential construction, satisfying the same three-term recursion as Chebyshev polynomials.
• The exact DSSYK density of states $\rho(E)$ can be written via the Jacobi $\theta_1$-function:

$$\rho(2 \cos \theta | q) = \frac{\theta_1(\theta, q^{1/2})}{2\pi q^{1/8}},$$

and expanded in a Fourier–Chebyshev sum.

• The generating function for ensemble-averaged moments $M_{2k}$ is given in terms of sums over chord diagrams weighted by $q^{\#\,\text{intersections}}$:

$$M_{2k} = \langle \mathrm{Tr}\, H^{2k} \rangle = (1-q)^k \sum_{\mathrm{chord\, diag}} q^{(\#\,\text{intersections})},$$

with an explicit combinatorial expression.

For finite $L$, the prescription is to generate a large Hermitian matrix of the desired universality class, stretch its spectrum to match $\rho(E)$, use the Lanczos process to produce tridiagonal $H_K$, and read off the sequence $\{a_n, b_n\}$ for operator construction.

6. Physical and Conceptual Significance

The exponential wormhole length operator provides a direct quantitative bridge between quantum gravitational observables in JT gravity (wormhole length) and operator-theoretic notions of quantum information spreading (spread complexity) in DSSYK and its generalizations. Its non-perturbative formulation enables precise tracking of black hole interior growth and saturation, revealing regimes reminiscent of "white hole" physics—where the wormhole shrinks from maximum size onto a finite plateau at late times.

The operator encapsulates and formalizes proposals equating wormhole geometric size with quantum mechanical spread complexity, thus unifying geometric growth and quantum complexity in a precise operator-theoretic framework. Its behavior elucidates late-time quantum gravity phenomena—such as information "leakage" or evaporation signaled by plateauing length—which remain central to ongoing research in holography and quantum chaos.

A plausible implication is that investigation of further extensions beyond the strictly 1D chain, as well as systematic paper of higher-dimensional generalizations and various symmetry classes, may reveal even deeper connections between complexity, geometry, and universality in quantum gravity models.