Symmetry resolved out-of-time-order correlators of Heisenberg spin chains using projected matrix product operators

Abstract

We extend the concept of operator charge in the context of an abelian U (1) symmetry and apply this framework to symmetry-preserving matrix product operators (MPOs), enabling the description of operators projected onto specific sectors of the corresponding symmetry. Leveraging this representation, we study the effect of interactions on the scrambling of information in an integrable Heisenberg spin chain, by controlling the number of particles. Our focus lies on out-of-time order correlators (OTOCs) which we project on sectors with a fixed number of particles. This allows us to link the non-interacting system to the fully-interacting one by allowing more and more particle to interact with each other, keeping the interaction parameter fixed. While at short times, the OTOCs are almost not affected by interactions, the spreading of the information front becomes gradually faster and the OTOC saturate at larger values as the number of particle increases. We also study the behavior of finite-size systems by considering the OTOCs at times beyond the point where the front hits the boundary of the system. We find that in every sector with more than one particle, the OTOCs behave as if the local operator was rotated by a random unitary matrix, indicating that the presence of boundaries contributes to the maximal scrambling of local operators.

Figures (21)

• Figure 1: Sketch of the meaning of the generalized operator $\alpha$-charge defined by different values of $\alpha$, for $L=6$. Each parametrization corresponds to a different block structure defined by $\alpha$ and the corresponding $\alpha$-charge $q^{\alpha}_{O}$ of the operator $\hat{O}$, whose eigenvalues are indicated by a single color in each panel. Left: $\alpha=-1$ corresponds to the standard definition of the operator charge. Middle: $\alpha=L+1$ is the smallest integer value allowing a single-block $(n,n')$ resolution. Right: $\alpha=+1$, allows to define blocks together on the anti-diagonal.
• Figure 2: a) Diagrammatic representation of a symmetric MPO, in which each vector space is decorated by a charge. $m$ denotes the charge of physical vector spaces, and $q$ the charge of virtual ones. b) Charge-preserving MPOs ($\alpha=-1$): allowed charges of virtual indices $a_i$, encoding the difference of accumulated local charges $q^{{\space}}_i = \sum^{i}_{j=1} m^{\prime}_j- m_j$. c) Projected MPOs ($\alpha=L+1$): the corresponding charge becomes $q_i=\sum^{L}_{j=1} m^{\prime}_i+(L+1)m^{{\space}}_i$. The charge of the operator in the generalized definition is imposed at the boundary: $q^{}_L = n' + \alpha n$.
• Figure 3: Left: Matrix representation of a charge-preserving operator. Only the non-zero elements are highlighted, and every sector (in the main diagonal) is spanned. Right. Projected representation of an operator, equivalent to the projection of the operator within a given symmetry sector. Every element outside the highlighted sector are projected out.
• Figure 4: OTOC in different sectors for $L=15$, $dt=0.05$, without truncation. The OTOC are computed up to times far beyond the boundary reflection, in order to identify the different time regions. The normalization $C_{\mathrm{max}}^{[n]}$ allows the scale in each subplot to be similar, the amplitude being larger in bigger sectors. The horizontal blue line indicates the time $t=L$, at which the reflection hits back the site in the middle.
• Figure 5: Time dependence of the OTOC projected on different sectors for $L=20$ obtained with typical states evolution (see text). The log scale allows to dissociate time regions, and red circles are indicators for the reflection on the boundary hitting the site in the middle. We indicate the number of particles corresponding to each curve by a number next to it with the corresponding color. The blue color labelled Full corresponds to the full symmetric operator, defined as the sum over all sectors.