Dynamics and correlations in Motzkin and Fredkin spin chains

TL;DR

This paper investigates the quantum dynamical behavior of Motzkin and Fredkin spin chains after a quench, analyzing how excitations evolve and correlations decay, revealing unusual properties related to entanglement and correlation functions.

Contribution

It provides a detailed analysis of the time evolution of excitations and correlations in Motzkin and Fredkin chains, highlighting their anomalous dynamical behaviors.

Findings

Unusual correlation decay patterns observed

Violation of cluster decomposition property identified

Distinct dynamical responses compared to typical spin chains

Abstract

The Motzkin and Fredkin quantum spin chains are described by frustration-free Hamiltonians recently introduced and studied because of their anomalous behaviors in the correlation functions and in the entanglement properties. In this paper we analyze their quantum dynamical properties, focusing in particular on the time evolution of the excitations driven by a quantum quench, looking at the correlations functions of spin operators defined along different directions, and discussing the results in relation with the cluster decomposition property.

Equations

$$H=H_0+H_{\partial }$$

$$\displaystyle H_{0}=\frac{1}{2}\sum_{j=1}^{L-1}\left\{{\sf P}\Big{(}\Ket{0_{j}\Uparrow_{j+1}}-\Ket{\Uparrow_{j}0_{j+1}}\Big{)}+{\sf P}\Big{(}\Ket{0_{j}\Downarrow_{j+1}}-\Ket{\Downarrow_{j}0_{j+1}}\Big{)}+{\sf P}\Big{(}\Ket{0_{j}0_{j+1}}-\Ket{\Uparrow_{j}\Downarrow_{j+1}}\Big{)}\right\}$$

$$H_{\partial }=\mathsf{P}\left(|{\Downarrow }_1⟩\right)+\mathsf{P}\left(|{\Uparrow }_L⟩\right),$$

$$\displaystyle H_{0}=\frac{1}{2}\sum_{j=1}^{L-2}\left[{\sf P}\Big{(}\Ket{\downarrow_{j}\uparrow_{j+1}\downarrow_{j+2}}-\Ket{\uparrow_{j}\downarrow_{j+1}\downarrow_{j+2}}\Big{)}+{\sf P}\Big{(}\Ket{\uparrow_{j}\uparrow_{j+1}\downarrow_{j+2}}-\Ket{\uparrow_{j}\downarrow_{j+1}\uparrow_{j+2}}\Big{)}\right]$$

$$H_{\partial }=\mathsf{P}\left(|{\downarrow }_1⟩\right)+\mathsf{P}\left(|{\uparrow }_L⟩\right).$$

$$⟨S_Z(i,t)S_Z(j,t){⟩}_c-⟨S_Z(i,0)S_Z(j,0){⟩}_c\equiv ⟨e^{i{H}_{f}t}{S}_Z(i)S_Z(j)e^{-i{H}_{f}t}{⟩}_c-⟨S_Z(i)S_Z(j){⟩}_c$$

$$⟨S_X(i,t)S_X(j,t){⟩}_c-⟨S_X(i,0)S_X(j,0){⟩}_c\equiv ⟨e^{i{H}_{f}t}{S}_X(i)S_X(j)e^{-i{H}_{f}t}{⟩}_c-⟨S_Z(i)S_Z(j){⟩}_c$$

$$⟨S_Z(i)⟩=\frac{1}{{\mathcal{M}}^{(L)}}\sum _h{\mathcal{M}}_{0h}^{(i-1)}\left({\mathcal{M}}_{h+1,0}^{(L-i)}-{\mathcal{M}}_{h-1,0}^{(L-i)}\right)$$

$$⟨S_Z(i)S_Z(j)⟩=\frac{1}{{\mathcal{M}}^{(L)}}\sum _{h,h^{\prime }}{\mathcal{M}}_{0h}^{(i-1)}\left({\mathcal{M}}_{h+1,h^{\prime }}^{(j-i-1)}-{\mathcal{M}}_{h-1,h^{\prime }}^{(j-i-1)}\right)\left({\mathcal{M}}_{h^{\prime }+1,0}^{(L-j)}-{\mathcal{M}}_{h^{\prime }-1,0}^{(L-j)}\right)$$

$$⟨S_Z(i)S_Z(j){⟩}_c=⟨S_Z(i)S_Z(j)⟩-⟨S_Z(i)⟩⟨S_Z(j)⟩$$

$$\underset{L\to \infty }{\lim }⟨S_Z(1)S_Z(L){⟩}_c=\underset{L\to \infty }{\lim }\left[{\left(\frac{{\mathcal{M}}_{10}^{(L-1)}}{{\mathcal{M}}^{(L)}}\right)}^2-\frac{{\mathcal{M}}_{11}^{(L-2)}}{{\mathcal{M}}^{(L)}}\right]={\left(\frac{2}{3}\right)}^2-\frac{4}{9}=0$$

$$⟨S_X(i)S_X(j)⟩=\frac{1}{{\mathcal{M}}^{(L)}}\sum _{h,h^{\prime }}{\mathcal{M}}_{0h}^{(i-1)}\left({\mathcal{M}}_{h-1,h^{\prime }}^{(j-i-1)}+{\mathcal{M}}_{h,h^{\prime }}^{(j-i-1)}\right)\left({\mathcal{M}}_{h^{\prime }+1,0}^{(L-j)}+{\mathcal{M}}_{h^{\prime },0}^{(L-j)}\right)$$

$$\underset{L\to \infty }{\lim }⟨S_X(1)S_X(L){⟩}_c=\underset{L\to \infty }{\lim }\left(\frac{{\mathcal{M}}^{(L-2)}}{{\mathcal{M}}^{(L)}}\right)=\frac{1}{9}$$

$$S^{\pm }=\frac{1}{2}\left(S_X\pm i{S}_Y\right)$$

$$\langle S_{X}(i)S_{X}(j)\rangle=\frac{1}{4}\Big{(}\langle S_{+}(i)S_{-}(j)\rangle+\langle S_{-}(i)S_{+}(j)\rangle\Big{)}$$

$$⟨S_{+}(i)S_{-}(j)⟩=⟨S_{-}(i)S_{+}(j)⟩$$

$$⟨S_X(i)S_X(j)⟩=\frac{1}{2}⟨S_{+}(i)S_{-}(j)⟩.$$

$$⟨S_Z(i)⟩=\frac{1}{2{\mathcal{D}}^{(L)}}\sum _h{\mathcal{D}}_{0h}^{(i-1)}\left({\mathcal{D}}_{h+1,0}^{(L-i)}-{\mathcal{D}}_{h-1,0}^{(L-i)}\right).$$

$$⟨S_Z(i)S_Z(j)⟩=\frac{1}{4{\mathcal{D}}^{(L)}}\sum _{h,h^{\prime }}{\mathcal{D}}_{0h}^{(i-1)}\left({\mathcal{D}}_{h+1,h^{\prime }}^{(j-i-1)}-{\mathcal{D}}_{h-1,h^{\prime }}^{(j-i-1)}\right)\left({\mathcal{D}}_{h^{\prime }+1,0}^{(L-j)}-{\mathcal{D}}_{h^{\prime }-1,0}^{(L-j)}\right)$$

$$⟨S_Z(i)S_Z(j){⟩}_c=⟨S_Z(i)S_Z(j)⟩-⟨S_Z(i)⟩⟨S_Z(j)⟩$$

$$\underset{L\to \infty }{\lim }⟨S_Z(2)S_Z(L-1){⟩}_c$$

$$⟨S_X(i)S_X(j)⟩=\frac{1}{2{\mathcal{D}}^{(L)}}\sum _{h,h^{\prime }}{\mathcal{D}}_{0h}^{(i-1)}{\mathcal{D}}_{h-1,h^{\prime }-1}^{(j-i-1)}{\mathcal{D}}_{h^{\prime },0}^{(L-j)}$$

$$\underset{L\to \infty }{\lim }⟨S_X(2)S_X(L-1){⟩}_c=\frac{1}{2}\underset{L\to \infty }{\lim }\frac{{\mathcal{D}}^{(L-4)}}{{\mathcal{D}}^{(L)}}=\underset{L\to \infty }{\lim }\frac{L(L+2)}{32(L^2-4L+3)}=\frac{1}{32}$$

$${\cal D}^{(n)}_{hh^{\prime}}=\left[\left(\begin{array}[]{c}n\\ \frac{n+|h-h^{\prime}|}{2}\end{array}\right)-\left(\begin{array}[]{c}n\\ \frac{n+h+h^{\prime}}{2}+1\end{array}\right)\right]p_{n+h+h^{\prime}}$$

$${\mathcal{D}}^{(n)}\equiv {\mathcal{D}}_{00}^{(n)}=C\left(n/2\right)p_n$$

$$C(\ell )=\frac{(2\ell )!}{\ell !(\ell +1)!}$$

$${\cal M}^{(n)}_{hh^{\prime}}=\sum_{\ell=0}^{\lfloor\frac{n-|h^{\prime}-h|}{2}\rfloor}\left(\begin{array}[]{c}n\\ 2\ell+|h^{\prime}-h|\end{array}\right){\cal D}^{(2\ell+|h^{\prime}-h|)}_{hh^{\prime}}$$

$${\cal M}^{(n)}\equiv{\cal M}^{(n)}_{00}=\sum_{\ell=0}^{\lfloor\frac{n}{2}\rfloor}\left(\begin{array}[]{c}n\\ 2\ell\end{array}\right)C(\ell)$$

Full text

I Introduction

The study of quantum systems driven out of equilibrium has attracted a lot of attention in the last few years. Theoretical and experimental interest on how fast the correlations can spread in quantum many-body systems has been renewed polkonikov11 ; eisert15 ; huse15 ; kollath17 after the proof, for critical theories, that the maximum velocity of the spreading of correlations is given by the group velocity in the final gapless system pasquale06 . Actually, the existence of a maximal velocity known as the Lieb-Robinson bound lieb , has been shown to exist theoretically in several interacting many-body systems, due to short-range interactions which limit the propagation of information making finite its spreading speed. Once the system is subject to a sudden change of the parameters of a short-range Hamiltonian, often denoted as a quantum quench, the time evolution of two-point correlation functions shows a well defined light-cone-like propagation defining causally connected regions, up to exponentially small deviations.

This behavior is tightly related to the concept of locality which plays a crucial role in physical theories, with far reaching consequences, the most fundamental being the cluster decomposition property (CDP) hastings ; nachtergaele . The CDP implies that two-point connected correlations functions go to zero when the separation of the points goes to infinity. This is the reason why two parts of a system very far apart, separated by a large distance, behave independently. This property is expected to be verified for non-degenerate states in systems described by local Hamiltonians.

On the other hand, the presence of long-range interaction may cause the violation of the Lieb-Robinson bound and the presence of power-law tails outside the light-cone eisert13 . For these reasons the study of non-local properties and their effects on the light-cone in the time evolution driven by a quantum quench are certainly at the present date a very interesting field of research.

Recently novel quantum spin models have been introduced bravyi12 ; ramis16 ; luca16 ; olof17 . These models, referred to as Motzkin and Fredkin models, in spite of being described by local Hamiltonians and admitting a unique ground state, may exhibit violation of CDP and of the area law for the entanglement entropy, with the presence of anomalous and extremely fast propagation of the excitations after driving the system out-of-equilibrium luca16 . Very recently, also modified versions of these models have been introduced and studied ramis17 ; sugino , which can exhibit a quantum phase transition separating an extensively entangled phase klich ; olof2 from a topological one lucaB . It has been given also a continuum description for the ground-state wavefunctions of those models, which can reproduce quite well some quantities as for example the local magnetization and the entanglement entropy, and whose scaling Hamiltonian is not conformally invariant fradkin . Entanglement properties in these models have also been extensively studied, particularly the Renyi entropy sugino18 , the negativity and the mutual information, revealing a large-distance entanglement behavior luca19 , which can produce intriguing out-of equilibrium properties barbiero18 .

In a previous work luca16 we showed that, looking at the connected correlation functions of spins along $z$-directions ($S_{Z}$-$S_{Z}$ correlators) the ground states exhibit a violation of the cluster decomposition for the Motzkin and the Fredkin models with spins higher than $1$ (the so-called colorful versions of these models). This behavior goes with a square-root violation of the area law for the entanglement entropy and an extremely fast spreading of the excitations after a quantum quench. For spin $1/2$ in the Fredkin model and spin $1$ in the Motzkin model (the so-called colorless cases), instead, the $S_{Z}$-$S_{Z}$ connected correlation functions vanish for long distances and their dynamics is characterized by a clear light-cone-like evolution after switching a local perturbation along the $z$-direction. The question we address in this paper is whether one can have absence of cone and extremely fast excitation spreading in the colorless cases. What we will show is that, in the same situation where there is a conventional light-cone dynamics for the $S_{Z}$-$S_{Z}$ correlators, one can still observe anomalous dynamics looking at the transverse spin correlations, e.g. the time evolution of the $S_{X}$-$S_{X}$ correlation functions, which also at equilibrium does not fulfill the cluster decomposition property. This analysis shows that also in the colorless version of the Motzkin and Fredkin models (for spin $1$ and $1/2$, respectively) there is a violation of the cluster decomposition along the $x$-direction and the absence of a light-cone behavior of the propagation of excitations looking at the correlations of the spins in the transverse directions. This result is in agreement with what found for the mutual information, which, at least for conformal theories, represents an upper bound for any connected correlation functions pasquale11 , and which in our cases remains finite even when measured between infinitely distant separated regions luca19 .

II The models

In this section we will briefly review the spin models under study, considering only the case with spin $1$ for the Motzkin model and and spin $1/2$ for the Fredkin model. In both cases the full Hamiltonian can be written as

[TABLE]

where $H_{0}$ is the bulk Hamiltonian and $H_{\partial}$ is the boundary term which remove the huge degeneracy of the ground state of $H_{0}$, making it uniquely defined.

II.1 Motzkin model

The Motzkin Hamiltonian bravyi12 ; ramis16 can be written as a local Hamiltonian made by a bulk contribution,

[TABLE]

where ${\sf P}(|.\rangle)$ denotes the projector $|.\rangle\langle.|$, and $\Ket{\Uparrow}$ ($\Ket{\Downarrow}$) the $1$-spin up (down). $L$ is the size of the chain, namely the total number of sites. The Hamiltonian is constructed in terms of projection operators which commute with the total spin $S^{tot}_{Z}=\sum_{j=1}^{L}S_{Z}(j)$ where $S_{Z}(j)\Ket{\Uparrow_{j}}=\Ket{\Uparrow_{j}}$, $S_{Z}(j)\Ket{\Uparrow_{j}}=0$, and $S_{Z}(j)\Ket{\Downarrow_{j}}=-\Ket{\Downarrow_{j}}$. The unique ground-state, $\ket{M}$, is known exactly and corresponds to an equal-weight superposition of states defined through Motzkin paths bravyi12 ; ramis16 , so that $S^{tot}_{Z}\ket{M}=0$, namely it belongs to the zero-total spin sector. These states are such that, denoting the spins up, ${\Uparrow}$, by /, the spins down, ${\Downarrow}$, by ${\backslash}$ and spins zero, ${0}$, by $-$, one can construct a Motzkin path. A Motzkin path is any path on a $x$-$y$ plan connecting the origin $(0,0)$ to the point $(0,L)$ with steps $(1,0)$, $(1,1)$, $(1,-1)$, where $L$ is an integer number. Any point $(x,y)$ of the path is such that $x$ and $y$ are not negative.

II.2 Fredkin model

For spin $1/2$ we will consider the following Fredkin model luca16 ; olof17

[TABLE]

Also in this case the Hamiltonian is constructed in terms of projection operators which commute with the total spin $S^{tot}_{Z}=\sum_{j=1}^{L}S_{Z}(j)$ where $S_{Z}(j)\Ket{\uparrow_{j}}=\frac{1}{2}\Ket{\uparrow_{j}}$ and $S_{Z}(j)\Ket{\downarrow_{j}}=-\frac{1}{2}\Ket{\downarrow_{j}}$. The unique ground-state, $\ket{D}$, is known exactly and corresponds to an equal-weight superposition of states defined through Dyck paths luca16 ; olof17 , such that it belongs to the zero-total spin sector, $S^{tot}_{Z}\ket{D}=0$. These states are such that, denoting the spins up, ${\uparrow}$, by / and the spins down, ${\downarrow}$, by ${\backslash}$, one can construct a Dyck path. A Dick path is any path on a $x$-$y$ plan connecting the origin $(0,0)$ to the point $(0,L)$ with steps $(1,1)$, $(1,-1)$, where $L$ is an even integer number. Any point $(x,y)$ of the path is such that $x$ and $y$ are not negative.

III Dynamics after a quantum quench

We study, by means of t-DMRG t_dmrg ; note_numerics , the time evolution of the correlation functions after a local quench. We will consider the time evolution of the $\langle S_{Z}S_{Z}\rangle_{c}$ and $\langle S_{X}S_{X}\rangle_{c}$ correlation functions in both the models, after suddenly switching on a local perturbation $h_{z}S_{Z}(j_{0})$, so that the final Hamiltonia is $H_{f}=H+h_{z}S_{Z}(j_{0})$, while the initial state is the ground-state of $H$. We will calculate the time evolution of the excitations after the quench looking at the following quantities

[TABLE]

the difference between the connected corelation function at $t=0$ and at later time $t$ along the quantization axis, and

[TABLE]

the correlators orthogonal to the quantization axis. Notice that the perturbation we choose preserve the $U(1)$ symmetry of the model, so that $\langle S_{X}\rangle=\langle S_{y}\rangle=0$ and $\langle S_{X}(i,t)S_{X}(j,t)\rangle=\langle S_{Y}(i,t)S_{Y}(j,t)\rangle$.

As shown in Fig. 1 for both cases, clear light-cones are visible in the dynamics of $\langle S_{Z}S_{Z}\rangle_{c}$, driven by switching-on a local field at one edge of the chain. On the other hand the same quench does not produce a light-cone profile in the transverse correlation functions, $\langle S_{X}S_{X}\rangle_{c}$, as one can see from Fig 2. Notice that, for the Fredkin model, we plotted the correlation functions between any points with the second one, since the first site is totally uncorrelated.

As we can clearly see, the two correlation functions have very different behaviors: $\langle S_{Z}S_{Z}\rangle$ behaves as expected by the theory of quantum quenches, namely, the perturbation takes some time in order to reach distant regions, while $\langle S_{X}S_{X}\rangle$ shows that the whole system reacts immediately to the perturbation, as being fully causally connected. Because of the arguments reported in the introduction, as we will be seeing in the next section, this anomalous dynamical behavior goes with a violation of the cluster decomposition property for the transverse spin components already at equilibrium.

IV Correlation functions

The anomalous behavior described above can be explained by calculating the correlation functions at equilibrium. Since the ground state is a uniform superposition of states which can be mapped to some random walks, we can calculate many ground-state properties resorting to combinatorics luca16 , see Appendix. In particular we can calculate the magnetization and the correlation functions. We remind that the $S_{Z}$-$S_{Z}$ connected correlation functions vanish when measured on two points very far apart luca16 . Proceeding as in Ref. luca16 , one can calculate also the $S_{X}$-$S_{X}$ connected correlation functions exactly, showing that, unlike $S_{Z}$-$S_{Z}$, they do not vanish even at infinitely large distances, exhibiting therefore a violation of the cluster decomposition property, which is at the origin of the unconventional dynamical properties observed numerically and reported in the previous section.

IV.1 Motzkin model

$S_{Z}$-$S_{Z}$ correlation functions.

Defining ${\cal M}^{(n)}_{hh^{\prime}}$ as the number of Motzkin-like paths connecting two points at heights $h$ and $h^{\prime}$ by $n$ steps, from combinatorics the magnetization as a function of the position is given by luca16

[TABLE]

where ${\cal M}^{(L)}\equiv{\cal M}^{(L)}_{00}$ is the Motzkin number (explicit expressions are given in Appendix). The two-point correlation function can be also calculated exactly luca16 and reads as it follows

[TABLE]

The quantity we are interested in is the connected correlation function

[TABLE]

Examples of this quantity are given in Fig. 3 and Fig. 4 (left plots, blue curves). As shown in those figures, the connected correlation functions go to zero at large distances, for instance

[TABLE]

$S_{X}$-$S_{X}$ correlation functions.

The transverse correlation functions, $\langle S_{X}(i)S_{X}(j)\rangle=\langle S_{Y}(i)S_{Y}(j)\rangle$, are given by

[TABLE]

which is also the connected correlation function, $\langle S_{X}(i)S_{X}(j)\rangle_{c}=\langle S_{X}(i)S_{X}(j)\rangle$ since $\langle S_{X}(i)\rangle=0$ $\forall i$. In this case, as shown in Figs. 3, 4 (left plots, red curves), this correlation function does not vanish at infinity distances. For example

[TABLE]

Equation (12) can be derived introducing

[TABLE]

such that $S^{+}\Ket{\Uparrow}=0$, $S^{+}\Ket{0}=\sqrt{2}\Ket{\Uparrow}$ and $S^{+}\Ket{\Downarrow}=\sqrt{2}\Ket{0}$, and $S^{-}\Ket{\Uparrow}=\sqrt{2}\Ket{0}$, $S^{-}\Ket{0}=\sqrt{2}\Ket{\Downarrow}$ and $S^{-}\Ket{\Downarrow}={0}$, and writing

[TABLE]

since $\langle S_{+}(i)S_{+}(j)\rangle=\langle S_{-}(i)S_{-}(j)\rangle=0$, because the operators $S^{+}S^{+}$ ans $S^{-}S^{-}$ project onto finite-total spin sectors, ortogonal to the ground-state. As already pointed out in Ref. fradkin , the quantity $\langle S_{+}(i)S_{-}(j)\rangle$ is the probability of finding $\Ket{0}$ or $\Ket{\Downarrow}$ at site $i$ and $\Ket{\Uparrow}$ or $\Ket{0}$ at site $j$, times $\sqrt{2}$. This gives half of the expression in Eq. (12). Now we can prove that

[TABLE]

since the action of the operator $S_{-}(i)S_{+}(j)$ onto the ground-state is non-zero if, at site $i$ the spin is not $\ket{\Downarrow}$ and at site $j$ the spin is not $\ket{\Uparrow}$, and, in addition, if all the paths beewen the two spins never cross level $1$ (or the horizon $z=1$, according to the definition in Ref. luca19 ). This means that the string of spins between sites $i$ and $j$ should be the Motzkin-like paths with heights lowered by one, namely the same paths of those allowed for the $S_{+}(i)S_{-}(j)$ operator. As a result, we have simply

[TABLE]

Our exact result, Eq. (12), perfectly agrees with the numerical result for $\langle S_{+}(i)S_{-}(j)\rangle$ reported in Ref. fradkin , (cfr. Fig. 4, left plot) and with DMRG calculation we performed for $\langle S_{X}(i)S_{X}(j)\rangle$. We found therefore that, for $L\rightarrow\infty$, $\langle S_{X}(i)S_{X}(j)\rangle$ goes to $\frac{4}{9}$ deeply in the bulk and to $\frac{1}{9}$ at the edges.

IV.2 Fredkin model

$S_{Z}$-$S_{Z}$ correlation functions.

We define ${\cal D}^{(n)}_{hh^{\prime}}$ the number of paths connecting two points at heights $h$ and $h^{\prime}$ with $n$ steps, which never cross the ground, such that ${\cal D}^{(2n)}\equiv{\cal D}^{(2n)}_{00}=C(n)$, where $C(n)$ are the Catalan numbers (see Appendix for details). The magnetizations is, then, given by luca16

[TABLE]

We can also calculate analytically the correlation functions getting luca16

[TABLE]

so that the connected correlation function is, once again,

[TABLE]

Some plots of this quantity are reported in Fig. 3 and Fig. 4 (right plots, blue curves). Also in this case, as for the Motzkin model, the $S_{Z}$-$S_{Z}$ correlation functions go to zero for large distances, for instance

[TABLE]

$S_{X}$-$S_{X}$ correlation functions.

The transverse correlation functions, $\langle S_{X}(i)S_{X}(j)\rangle=\langle S_{Y}(i)S_{Y}(j)\rangle$, are given by

[TABLE]

which is also the connected correlation function, $\langle S_{X}(i)S_{X}(j)\rangle_{c}=\langle S_{X}(i)S_{X}(j)\rangle$ since $\langle S_{X}(i)\rangle=0$ $\forall i$. As shown in Figs. 3, 4 (right plots, red curves), it does not vanishes at infinity distances. For instance,

[TABLE]

Notice that the first and the last spin are completely uncorrelated with the rest of the chain, so that we considered the second and second-last spins. In order to derive Eq. (22), one can introduce the operators as in Eq. (14), such that $S^{+}\Ket{\uparrow}=0$, $S^{+}\Ket{\downarrow}=\Ket{\uparrow}$, and $S^{-}\Ket{\downarrow}={0}$, $S^{-}\Ket{\uparrow}=\Ket{\downarrow}$. One can write $\langle S_{X}(i)S_{X}(j)\rangle$ as in Eq. (15) and realize that $\langle S^{+}(i)S^{-}(j)\rangle$ is the probability $P$ of finding spin down at site $i$ and spin up at site $j$. Repeating the argument described before one gets that $\langle S^{-}(i)S^{+}(j)\rangle=\langle S^{+}(i)S^{-}(j)\rangle$ so that Eq. (17) is valid. One obtain, therefore, Eq. (22) which is half of $P$, the probability of having spin down and spin up at $i$ and $j$ respectively. Also in this case our exact result, Eq. (22), is in perfect agreement with the numerical result for $\langle S_{+}(i)S_{-}(j)\rangle$ reported in Ref. fradkin , (cfr. Fig. 4, right plot) and with DMRG calculation we performed for $\langle S_{X}(i)S_{X}(j)\rangle$. We find that, for $L\rightarrow\infty$, $\langle S_{X}(i)S_{X}(j)\rangle$ goes to $\frac{1}{8}$ deeply inside the bulk and to $\frac{1}{32}$ at the edges, for second and second-last spins.

V Conclusions

We performed a quantum quench by applying a perturbation which preserve the $U(1)$ symmetry of the model and observe a light-cone propagation of the excitations looking at the correlation functions of spins along $z$-directions, consistently with the vanishing behavior of the connected correlation functions at equilibrium. On the contrary, looking at the correlation functions of spins along the transverse directions we do not observe a light-cone because the system is already long-range correlated, as shown by exact results for the $S_{X}$-$S_{X}$ correlation functions. This work extends significantly the analysis done in a previous paper luca16 , where violation of the cluster decomposition has been observed for the colorful versions of the models looking at the correlation functions of spins along $z$-directions. The results reported here are in perfect agreement with numerical results for transverse spin correlation functions fradkin and with what found for the mutual information, which remains finite even when measured between infinitely distant separated regions luca19 . In this paper we considered the colorless versions of the Motzkin and Fredkin spin chains, with spins $1$ and $1/2$. It would be very interesting to study the transverse spin correlations and the related dynamics also in the corresponding colorful models, namely with higher values of the spins.

Acknowledgements.

We thank X. Chen, E. Fradkin, I. Klich, V. Korepin, O. Salberger, W. Witczak-Krempe for useful discussions. L.D. thanks also SISSA for kind hospitality. L.B. acknowledges ERC Starting Grant TopoCold for financial support.

Appendix A Combinatorics

Let us define $p_{n}=(1-\textrm{mod}(n,2))$ such that $p_{2n+1}=0$ and $p_{2n}=1$, so that it selects only even integer numbers, and

[TABLE]

where ${\cal D}^{(n)}_{hh^{\prime}}$ are the number of Dyck-like paths between two points at distance $n$ and heights $h$ and $h^{\prime}$. In particular

[TABLE]

with

[TABLE]

the Catalan numbers. Useful relations are {\cal D}^{(n)}_{h0}=\frac{h+1}{\frac{n+h}{2}+1}\left(\begin{array}[]{c}n\\ \frac{n+h}{2}\end{array}\right)p_{n+h} and ${\cal D}^{(n)}_{0h}={\cal D}^{(n)}_{h0}$. Let us also define

[TABLE]

the number of Motzkin-like paths between two points at heights $h$ and $h^{\prime}$ connected by $n$ steps. In particular

[TABLE]

are called Motzkin numbers.