Loschmidt echo, emerging dual unitarity and scaling of generalized temporal entropies after quenches to the critical point

TL;DR

The paper studies the Loschmidt echo after quenches to a conformal critical point and shows that its leading decay and finite-time corrections are governed by universal CFT data, notably the central charge $c$ and the operator content. By mapping the problem to a boundary CFT on a strip and analyzing the spectrum of the spatial transfer matrix ${\mathcal{T}}$, the authors predict and verify, via tensor-network simulations, that $c$ and the boundary exponents can be extracted from the dynamics, and that an emergent dual-unitarity appears at late times as ${\mathcal{T}}$ becomes unitary in the large-$T$ limit. They also introduce generalized temporal entropies, which grow logarithmically with time, and show their behavior matches CFT/holographic predictions while remaining numerically tractable with temporal MPS-based TN methods. The results are confirmed numerically for Ising and Potts minimal models, with implications for efficient classical simulations of out-of-equilibrium critical dynamics and potential experimental probes of universal temporal entropies.

Abstract

We show how the Loschmidt echo of a product state after a quench to a conformal invariant critical point and its leading finite time corrections can be predicted by using conformal field theories (CFT). We check such predictions with tensor networks, finding excellent agreement. As a result, we can use the Loschmidt echo to extract the universal information of the underlying CFT including the central charge, the operator content, and its generalized temporal entropies. We are also able to predict and confirm an emerging dual-unitarity of the evolution at late times, since the spatial transfer matrix operator that evolves the system in space becomes unitary in such limit. Our results on the growth of temporal entropies also imply that, using state-of-the art tensor networks algorithms, such calculations only require resources that increase polynomially with the duration of the quench, thus providing an example of numerically efficiently solvable out-of-equilibrium scenario.

Figures (8)

• Figure 1: The Loschmidt echo can be studied numerically by defining the tensor network made by a) the initial product state b) the MPO of the evolution operator and c) its matrix element. It can also be characterized using field theories, by analytically continuing the results obtained by mapping the CFT on the plane to the finite geometries shown in d). The field theory infrared divergence is handled by studying a finite spatial extent $L$ that is then sent to infinity, while the UV divergencies are cured by introducing a finite $\beta_0$ close to the conformally invariant boundary states $\ket{b}$.
• Figure 2: The transverse transfer matrix ${\mathcal{T}}$ involves the contraction of a column of elementary tensors a). Its leading left and right eigenvectors b) allow to characterize the Loschmidt echo \ref{['eq:Lochmidt']} in the thermodynamic limit. Generalized temporal entropies are extracted from the reduced transition matrices defined in Eq. \ref{['eq:rtm']}.
• Figure 3: Real vs imaginary parts of $\tau_0$, the dominant eigenvalue of ${\mathcal{T}}$ for various values of $T$ and $\beta_0$ for the Ising (left) and Potts (right) models. For a fixed $\beta_0$, as we vary $T$ they distribute on an almost constant radius circle.
• Figure 4: Top row: Imaginary parts of $\lambda_0$ for different values of $\beta_0$. The points are data from our TN calculation, solid lines are fits using the expected CFT form, which we use to extract the value of the central charge (see main text). The fits show an excellent agreement with data for larger $T$, as expected. Bottom row: Imaginary part of the first gap $\lambda_1 - \lambda_0$ for different $\beta_0$ and BC, and corresponding fits to the CFT formulas to extract $x_1$. Left: Results for the Ising model, Right: Potts model.
• Figure 5: Numerical results for the real (left) and imaginary (right) parts of the generalized entropies for the critical Ising (top row) and critical Potts (bottom row) with free BC. Solid lines denote the CFT predictions \ref{['eq:Sgen_cft']}. For Ising, we use $c=1/2$ central charge and $s_0 = 0.3$ the constant providing the best match for the real part, whereas for Potts we use the central charge $c=4/5$ and $s_0= 0.46$. The agreement with the CFT predictions is excellent for both real and imaginary parts, and it improves as we go to longer chains, as expected.