Multipartite Entanglement in Crossing the Quantum Critical Point

Quantum entanglement plays a crucial role in quantum information processing and quantum computations. For instance, the bipartite entanglement, generally represented by von Neumann or Rényi entropies, is an essential ingredient for supporting quantum advantages in quantum computations. The multipartite entanglement, quantified by spin squeezing [1, 2, 3] or quantum Fisher information (QFI) [4, 5, 6], is a type of indispensable quantum resource to perform quantum metrology beyond the standard quantum limit [7, 8, 9, 10, 11, 12]. In addition, quantum entanglement can also be applied to understand quantum many-body physics [13, 14, 15, 16] both in theory [17] and experiment [18, 19]. The bipartite entanglement can be used to identify different quantum phases and critical phenomena [20, 21]. For example, there exists a universal subleading entanglement entropy, termed topological entanglement entropy [20, 21, 22, 23], in intrinsic topological order systems. In critical systems, conformal field theories (CFT) predict that the entanglement entropy is logarithmically divergent with respect to the system size, where the factor is proportional to the central charge [24, 25, 26]. Recently, the multipartite entanglement is also applied to witness the symmetry-protected topological order  [27, 28, 29, 30].

The Kibble-Zurek mechanism (KZM) describes the universal dynamics of the second-order phase transitions (QPTs). It was first proposed to understand thermodynamic phase transitions in cosmological physics [31, 32], which had been introduced to classical second-order phase transitions [33, 34], and verified by numerical simulations including the density of kinks after a quench [35], vortex in two dimensions [36] and inhomogeneous quenches [37]. In addition, KZM is also valid in quantum phase transitions [38, 39, 40], encompassing symmetry-breaking phase transitions [41, 42], topological phase transitions [43, 44], localization-delocalization transitions [45], and deconfined quantum criticality [46]. Meanwhile, KZM can be applied to understand adiabatic dynamics [47] and quantum annealers [48]. Recently, it is also studied in other aspects of quantum many-body physics [49, 50, 51]. Moreover, KZM has been demonstrated experimentally in a variety of systems, such as solid materials [52, 53], cold atoms [54, 55], Rydberg simulators [56], superconducting circuits [57], and trap-ions [58].

At the critical point, the correlation length and the relaxation time diverge, which is known as critical slowing down. Thus, for a slow quench dynamics crossing a critical point, the adiabatic condition will break down, and the system can excite topological defects in the final state [59]. KZM predicts that the density of topological defects $n_{d}$ has a universal power-law scaling with respect to the quench rate $\tau_{Q}$: $n_{d}=\tau_{Q}^{-d\nu/(z\nu+1)}$ [31], where $d$ is the spatial dimension, $\nu$ and $z$ are the correlation length and dynamical critical exponent, respectively.

In addition to the topological defects, KZM can also be used to predict other physical quantities [60]. It is shown that the diagonal entropy after a slow quantum quench crossing the critical point also satisfies the Kibble-Zurek scaling [61, 62, 63, 64]. For the bipartite entanglement, the von Neumann entropy is found to scale logarithmically with respect to the quench rate, which is closely related to KZM [65, 66].

The multipartite entanglement is more complicated, which has not been depicted uniquely and clearly. Here, the density of topological defects (e.g., domain walls) is a local observable, which generally relates to the short-range correlation function. Thus, topological defects cannot describe the long-range physics. In contrast, the multipartite entanglement is always a nonlocal observable. Therefore, one open question may arise: whether there exists a relation between multipartite entanglement and KZM in crossing a quantum critical point.

In this paper, we focus on the multipartite entanglement after a slow quench crossing a quantum critical point. We discuss the quantum Ising model and the Lipkin-Meshkov-Glick (LMG) model, which are local and full-connected quantum systems, respectively. The multipartite entanglement is witnessed by the QFI density, where the generator is chosen from the symmetries that relate to the phase transitions. For the quantum Ising model, we calculate the QFI density of the final state, based on the matrix product state (MPS) method. We find that the QFI density of the final state exhibits a power-law scaling with respect to the quench rate for both linear and nonlinear quenches. The power-law exponent is close to the KZM scaling with a small correction. We show that this correction is from the long-range physics of the final states, which is beyond KZM. For the LMG model, we calculate the complement of the QFI density. The result conforms to the prediction of KZM, where the correction can be nearly neglected. In addition, based on the above results, we present a phenomenological description of critical slowing down in QPTs from the perspective of multipartite entanglement.

This paper is organized as follows: In Sec. 2, we review KZM in quantum phase transitions for linear and nonlinear cases. The multipartite entanglement and QFI are introduced in Sec. 3. In Sec. 4, we introduce the quantum Ising model including the spectrum, and the QFI density of the final states after both linear and nonlinear quantum quenches. In Sec. 5, we study QFI of the quenched state in the LMG model. In Sec. 6, we present a phenomenological discussion of our results. Finally, in Sec. 7, we summarize our results and present an outlook.

In this section, we briefly review KZM in quantum phase transitions for both linear and nonlinear quenches [15, 38, 67]. We consider a quantum system driven by a parameter $g$, and the critical point is at $g=g_{c}$. Near the critical point, the correlation length $\xi$ and the energy gap $\Delta$ closely relate to the critical exponents $\nu$ and $z$, i.e.,

Now we consider a slow quench, where the parameter $g$ becomes a time-dependent function $g(t)$. We first consider a linear quench, where we have

When $t=0$, the system approaches the critical point $g(0)=g_{c}$. We note that other functions of $g(t)$ can always be linearized near the critical point once $g^{\prime}(t=0)\neq 0$.

Initially, i.e., $t\rightarrow-\infty$, the system starts with the corresponding ground state, where the energy gap $\Delta$ is large enough. Thus, the adiabatic condition is satisfied, and the system keeps in the ground state. With the increase of $t$, the adiabatic condition will be violated near $t=0$, i.e., in the vicinity of the critical point. The corresponding time range for violating the adiabatic condition is labeled by $[-\hat{t},\hat{t}]$, and excitations can be excited in this regime. When $t>\hat{t}$, the adiabatic condition will recover.

Now we calculate the scale of $\hat{t}$. In the quantum phase transition, the coherent time $\tau$ is determined by the inverse of the energy gap

From Eq. (1), we can find that the energy gap scales as

When $|t|=\hat{t}$, the coherent time satisfies

Thus, we have

and the corresponding energy gap scales as

Therefore, the excitation density scales

where $d$ is the system dimension. The above picture is known as KZM.

Now we also consider a nonlinear slow quench, where we have $g^{\prime}(t=0)=0$. Without loss of generality, we can use the following geometric function to describe the nonlinear slow quench, i.e.,

where $\text{sgn}(t)$ is a sign function. Here, we discuss KZM of a nonlinear quench from the viewpoint of parameter-dependent critical exponents. Now, we assume $g$ is a function of another parameter $\lambda$, i.e., $g=g(\lambda)$. Let $g_{c}=g(\lambda_{c})$, and without loss of generality, we fix $\lambda_{c}=0$. Thus, Eq. (1) can be rewritten as

In addition, if we consider $\lambda$ as the driven parameter of QPT, where $\lambda_{c}$ is the critical point, then the corresponding critical exponents, labeled by $\nu_{\lambda}$ and $z_{\lambda}$, can be defined by

According to Eqs. (1, 11), when $g^{\prime}(\lambda_{c})\neq 0$, we can find that $\nu_{\lambda}=\nu$ and $z_{\lambda}=z$. However, if $g^{\prime}(\lambda_{c})=0$, the situation becomes different. If we let $\lambda=(g-g_{c})^{1/\alpha}$, then $g(\lambda)=\lambda^{\alpha}+g_{c}$. Thus, in the vicinity of $\lambda_{c}$, we can calculate the derivative of $g(\lambda)$ as $g^{\prime}(\lambda)=\alpha\lambda^{\alpha-1}$. Therefore, from Eqs. (1, 10), we can obtain the critical exponents with respective to $\lambda$ as

In addition, we have $\lambda(t)=t/\tau_{Q}$. Thus, this nonlinear quench with respective to $g$ is transformed to a linear quench for $\lambda$, where the results of linear quench can be applied directly. For instance, we can use KZM to obtain the topological defect density of the final state as

which is consistent with the results in Refs. [68, 65].

Generally, for an operator $\hat{O}$ and a quantum state with density matrix $\hat{\rho}=\sum_{i}p_{i}\ket{\psi_{i}}\bra{\psi_{i}}$, where $\langle\psi_{i}|\psi_{j}\rangle=\delta_{ij}$, the corresponding QFI can be defined as

For a pure state $\hat{\rho}=\ket{\psi}\bra{\psi}$, QFI can be simplified as

In addition, we can define the QFI density of a system with $N$ particles as

Here, to understand the connection between the QFI density and the multipartite entanglement, we introduce the concepts of $k$-producible and $k$-entangled states. If a state is $k$-producible, it can be decomposed as a product of states of a series of subsystems, where each subsystem contains no more than $k$ particles. In addition, a state is $k$-entangled, if it is $k$-producible but not $(k-1)$-producible, i.e., it contains at least one decomposed state with $k$ particles entangled. For a $k$-producibel state, QFI has an upper bound corresponding to the $k$ entangled particles [1]

where $[\frac{N}{k}]$ represents the maximum integer that is smaller than $\frac{N}{k}$. Therefore, if the QFI density $f_{Q}$ of a $k$-producible state violates bound, the entangled particles of the state must be larger than $k$, i.e., at least $k+1$. Thus, according to Eq. (17), the physical significance of the QFI density is that there are at least $[f_{Q}]+1$ particles entangled.

Thus, QFI is one of the quantifications of the multipartite entanglement. Generally, QFI is an important quantum resource in quantum metrology, which quantifies the maximal precision of the parameter estimation. Recently, the multipartite entanglement is also applied to understand quantum many-body physics, e.g., as a witness of topological phases [27, 30]. Naturally, one may wonder whether the scaling behaviors of the multipartite entanglement can also be described as KZM. Next, we will study QFI of the final state quenched from the initial ground state crossing the critical point with the rate $\tau_{Q}$.

To further demonstrate our results, we also investigate the scaling behaviors in the LMG model, which is a full-connected model. The time-dependent Hamiltonian reads

where $N$ is the number of spins, and $g(t)$ is an external field. The LMG model has a second-order phase transition at the critical point $g=g_{c}=1$. When $g<1$, the system is in the ferromagnetic phase, where the parity symmetry is spontaneously breaking. When $g>1$, the system is in a symmetry-unbreaking phase with non-degenerate ground states.

Introducing the total spin operator

we can rewrite the Hamiltonian as

which is convenient for numerical simulations.

The LMG model has no spatial dimensions. Compared with the quantum Ising model, the scaling behaviors of QFI after a slow quench may be different. In the LMG model, the corresponding parity operator associated to QPT reads

which is the same as the quantum Ising model. When $g=0$, the final ground state is also a GHZ state, which has the maximum QFI density $N$, with respect to the operator $\hat{\sigma}_{z}$. Thus, we can also choose $\hat{\sigma}_{z}$ as the operator to calculate the QFI density.

The LMG model is full-connected, so the QFI density of the final state scales as$f_{Q}\approx N$, i.e., almost all spins are entangled. According to the results of the quantum Ising model, we can find that crossing a critical point can decrease the multipartite entanglement. Thus, for the LMG model, we can consider how many spins that are not entangled. Here, we introduce the complement of the QFI density $f_{Q}$ to the maximum entanglement $N$, which is defined as

In the following discussion, we mainly focus on the scaling of $r_{Q}$ with respect to the quench rate.

For convenience, we only discuss the results of the linear quench. The transverse field satisfies

We drive the external field from $g=5$ to $0$. Here, the number of spins $N$ varies from $2^{3}$ to $2^{11}$, and $\tau_{Q}$ varies from $10^{1}$ to $10^{5}$. Our numerical simulations are based on the Hamiltonian Eq. (46), which is a $(N+1)\times(N+1)$ matrix. We obtain the ground state at $t=-4\tau_{Q}$ by the exact diagonalization as the initial state. The time evolution is obtained by solving an ordinary differential equation.

In Fig. 5, we show the numerical results of $r_{Q}$ versus $\tau_{Q}$, where there are two scaling behaviors. For instance, when $N=2^{11}$, for small $\tau_{Q}$, $r_{Q}$ scales as

For large $\tau_{Q}$, $r_{Q}$ scales as

Now, we try to understand the two scaling behaviors. The complement $r_{Q}$ can be considered as the residual energy [76], defined as

where $E_{\text{final}}$ is the energy of the final state, i.e., the expectation value of $\hat{H}(\tau_{Q})$ with respect to the final state, and $E_{\text{gs}}$ is the ground state energy of $\hat{H}(\tau_{Q})$. The final Hamiltonian is

When $g=0$, due to the symmetry breaking, all spins are polarized in the same direction. Thus, the ground state energy $E_{\text{gs}}=-N$. Here, the residual energy can be written as

which can also be regarded as the excitation energy. In the LMG model, at the critical point $g_{c}=1$, the energy gap decays with the system size as

i.e., the dynamical exponent is $z=1/3$ [76, 77]. For small $\tau_{Q}$, we can use the LZ formula [69, 70] to estimate the scaling of $r_{Q}$, see Eq. (28). The LZ formula gives the probability that a system driven through an energy gap is

According to Eq. (55, 56), we have $r_{Q}$, also the residual energy scales as

which is consistent with the scaling of small $\tau_{Q}$ [39, 67]. For a larger $\tau_{Q}$, i.e., $\tau_{Q}>\Delta$, we know that the LZ formula becomes invalid, so the scaling of QFI cannot conform to the predictions of KZM [76, 77]. We note that, for a local system, if $\tau_{Q}>\Delta$, the adiabatic condition recovers, where the topological defect density will exhibit an exponential decay with respect to $\tau_{Q}$.

According to Eqs. (50, 57), we can find that the scaling of QFI in the LMG model can be well predicted by KZM, where there are hardly corrections. Here, as a full-connected system, the LMG model has no spatial dimensions, where the long-range physics cannot be defined.

Generally, near the critical point, the relaxation time of the system is divergent, where the divergent exponent depends on critical exponents. This phenomenon is known as critical slowing down, which can be regarded as the origin of KZM. Here, based on the above results, we try to understand the phenomenon of critical slowing down in QPTs from the viewpoint of multipartite entanglement.

For the quantum Ising model, due to the Lieb-Robinson bounds [78, 79], the minimum time for creating a state with extensive multipartite entanglement from a short-range entangled state scales linearly in the size of the system. That is, if the local system size is $N$, and the QFI density of the initial state is $O(1)$, the time cost for creating a state with $O(N)$ QFI density is at least $O(N)$. In the quantum Ising model, for the linear quench, if we want $f_{Q}\sim N$ for the final state, then the quench rate $\tau_{Q}\sim N^{2}$. Thus, we need time $t\sim N^{2}$ to create a state with the extended QFI density via a linear quench crossing a QPT, which is slower than the adiabatic case. For the LMG model, we can also find that the crossing of the critical point can decrease the multipartite entanglement, where the scaling of QFI is closely related to the critical exponents.

Therefore, compared with the adiabatic process, the rate of generating the multipartite entanglement near the critical point becomes slow and is closely related to the critical exponents. This behavior of the multipartite entanglement crossing a QPT can be considered as a reflection of critical slowing down.

In summary, we have investigated the multipartite entanglement after slow quantum quenches crossing QPTs. For the quantum Ising model, by numerical calculations via MPS-based methods, we find that the QFI density of the final state follows the scaling behavior nearly opposite to the one of KZM with a small correction. We demonstrate that this correction originates from the long-range physics after a slow quench. We have also studied the LMG model, where the results conform better to KZM, since the long-range physics is absent in this full-connected system. Our results reveal that KZM may not describe the long-range behaviors after a slow quantum quench crossing a critical point, where the multipartite entanglement is a suitable observable to demonstrate these nontrivial long-range physics. Moreover, our results can also enhance the understanding of QPTs and the behavior of critical slowing down from another viewpoint, i.e., from the perspective of multipartite entanglement. In addition, since QFI is accessible in quantum simulation experiments, we expect that our results can be realized experimentally in state-of-art quantum simulators.

There also remain many interesting open questions, which are deserved to be further studied. In this work, we only consider 1D and full-connected systems, and it is meaningful to generalize our results to higher-dimensional systems. Moreover, when calculating the QFI density, we choose the corresponding local operators based on the symmetries of the Hamiltonians. In these systems, the QPTs originate from spontaneous symmetry breaking, so we can use the order parameter to define QFI. Therefore, it will be an interesting issue to study the multipartite entanglement after slow quantum quenches crossing a topological QPT, which is absent of spontaneous symmetry breaking. Furthermore, maybe other quantifications of multipartite entanglement could be adopted to testify to the generality of the scaling behavior.