Universal scrambling in gapless quantum spin chains

Abstract

Information scrambling, characterized by the out-of-time-ordered correlator (OTOC), has attracted much attention, as it sheds new light on chaotic dynamics in quantum many-body systems. The scale invariance, which appears near the quantum critical region in condensed matter physics, is considered to be important for the fast decay of the OTOC. In this paper, we focus on the one-dimensional spin-1/2 XXZ model, which exhibits quantum criticality in a certain parameter region, and investigate the relationship between scrambling and the scale invariance. We quantify scrambling by the averaged OTOC over the Pauli operator basis, which is related to the operator space entanglement entropy (OSEE). Using the infinite time-evolving block decimation (iTEBD) method, we numerically calculate time dependence of the OSEE in the early time region in the thermodynamic limit. We show that the averaged OTOC decays faster in the gapless region than in the gapped region. In the gapless region, the averaged OTOC behaves in the same manner regardless of the anisotropy parameter. This result is consistent with the fact that the low energy excitations of the gapless region belong to the same universality class as the Tomonaga-Luttinger liquid with the central charge c = 1. Furthermore, we estimate c by fitting the numerical data of the OSEE with an analytical result of the two-dimensional conformal field theory, and confirmed that c is close to unity. Thus, our numerical results suggest that the scale invariance leads to a universal behavior of the OTOC that is independent of the anisotropic parameter, which reflects the universality of the two-dimensional conformal field theory at low temperatures.Although the one-dimensional XXZ model is integrable, our results suggest such a universal behavior of generic non-integrable systems, because the Tomonaga-Luttinger liquid serves as a low-energy effective theory for many non-integrable and integrable systems. On the other hand, the OTOC in our numerical result does not exhibit the exponential decay, as our parameter regime is far from the semiclassical limit.

Scrambling in random unitary circuits: Exact results

“…In the two following subsections we will show how (37) can be computed explicitly in the minimal settings discussed in the introduction, namely Haar-random unitary circuits and random dual-unitary circuits. Our approach is based on writing down suitable recurrence relations fulfilled by the tensor networks (35) and (36). These equations will be solved exactly for Haar random circuits while for random dual-unitary circuits they will be truncated to provide strict bounds.…”

“…[61,90] (see also [45][46][47]). While the latter references exploit a mapping between averaged tensor networks like (35) and (36) and classical spin models in 2d here we will work directly with the tensor network.…”

“…The appeal to characterise chaos in terms of the dynamics of local operators lies in the fact that one does not need to work in a semi-classical regime, and that quantities such as OTOCs can be computed (and in principle measured) directly for any many-body quantum system [26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42]. In practice, however, the calculation of OTOCs represents a major challenge, which has provided an increasing motivation to find simplified models for the chaotic dynamics where these objects could be studied in some detail [43][44][45][46][47][48][49][50][51][52][53][54][55][56][57][58][59][60].…”

Out-of-time-order correlations in the quasiperiodic Aubry-André model

“…Recently out of time ordered correlators (OTOCs) have experienced a resurgence of interest from different fields of physics ranging from the black hole information problem [1] to information propagation in condensed matter systems [2][3][4][5][6][7][8][9]. The OTOC is of particular interest due to its role in witnessing the spreading or"scrambling" of locally stored quantum information across all degrees of freedom of the system, something traditional dynamical correlation functions of the form A(t)B cannot.…”

Phase boundary location with information-theoretic entropy in tensor renormalization group flows

“…The first developments explicitly for two-dimensional classical lattices were transfer matrix-based algorithms [5][6][7] , followed by the block spin-like tensor renormalization group (TRG) 8 algorithm. Variations of TRG were then developed [9][10][11][12][13][14][15][16][17][18][19][20][21][22] for better accuracy and enhanced capability, such as applicability in higher dimensions . These tensor network algorithms do not suffer from the notorious sign problem that sometimes arises in Monte Carlo.…”