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Universal Behavior of Multipartite Entanglement in Crossing the Quantum Critical Point

We investigate multipartite entanglement of the quantum Ising model with transverse fields for a slow quantum quench crossing a critical point. The multipartite entanglement is quantified by quantum Fisher information with the generator defined as the operator of the ferromagnetic order parameter due to the system symmetry. The quench dynamics begin with the ground state of the large transverse field in the paramagnetic phase, and then the transverse field is driven slowly to cross a quantum critical point and ends with a zero transverse field. Based on methods of matrix product state, we calculate the quantum Fisher information density of the final state. Numerical results of both linear and nonlinear quench show that the quantum Fisher information density of the final state scales as a power law of the quench rate, which has an opposite exponent of the Kibble-Zurek scaling. Our results reveal that the multipartite entanglement provides a new viewpoint to understand the dynamics of quantum phase transition. Based on the observation of entanglement in the quenched state, we also expect that our result would inspire approaches for preparing multipartite entangled states in quantum information processing.

In addition to the topological defects or energy production, KZM can be used to predict other physical quantities [51]. It is shown that the diagonal entropy after a slow quantum quench crossing the critical point also satisfies Kibble-Zurek scaling [52][53][54][55]. For the bipartite entanglement, von Neumann entropy is found to scale logarithmically with the quench rate, which is closely related to KZM [41,56]. However, it is not clear the behaviors of multipartite entanglement relating to KZM in crossing the quantum critical point.
In this paper, we focus on the multipartite entanglement of the transverse field Ising model after a arXiv:2202.07892v1 [quant-ph] 16 Feb 2022 slow quench crossing a quantum critical point. The multipartite entanglement is witnessed by QFI density, where the generator is the ferromagnetic order parameter. We calculate numerically the time evolution of the QFI density after a slow quench by matrix product state (MPS) based method. We find that the QFI density of the final state has a power-law scaling of the quench rate after both linear and nonlinear quench, where the exponent is opposite to the KZM scaling. 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. II, we review the quantum Ising model with transverse fields including its spectrum and KZM. The multipartite entanglement and QFI are introduced in Sec. III. In Sec. IV, we study QFI of the final states after both linear and nonlinear quench. In Sec. V, we present a phenomenological discussion of our results. Finally, in Sec. VI, we summarize our results and present our outlook on further research. Additional details of the derivation are shown in Appendix A.

II. QUANTUM ISING MODEL AND KIBBLE-ZUREK MECHANISM
Here, we consider the one-dimensional spin− 1 2 quantum Ising model with transverse fields, of which Hamiltonian reads where N is system size,σ x z is Pauli matrix, J > 0 is the nearest-neighbor interaction strength and is fixed to 1 in the following discussion, and g is the strength of transverse field. By Jordan-Wigner transformation, the Hamiltonian (1) can be written as a spinless fermionic form, whereĉ n (ĉ † n ) is the fermionic annihilate (create) operator at site n. If we choose periodic boundary condition for spin system, i.e.,σ N +1 =σ 1 , then it is periodic/antiperiodic boundary condition in fermionic system for odd/even fermion parity. Now we introduce a Fourier transform,ĉ where Thus Hamiltonian (2) can be rewritten in momentum space as (where the constant term is neglected) By Bogoliubov transformation the Hamiltonian (5) can be diagonalized as The mode (u k , v k ) T is the eigenstate of the Bogoliubovde Gennes (B-dG ) equation with eigenvalue When g = g c = 1, the system is gapless and thus locates at the critical point. When g > g c , the spins tend to polarize in x-direction at the ground state, so the system is in the paramagnetic phase. When g < g c , the system emerges long-range order, i.e., in ferromagnetic phase. Now, we consider a slow quench dynamics of the quantum Ising model. Here, the transverse field is timedependent and satisfying where −∞ ≤ t ≤ τ Q . The initial state is the ground state of g = ∞, i.e. all spins are polarized to the ground state of −σ x n . During the slow quench to zero transverse field, the system needs to cross the critical point and the adiabaticity breaks. We label the instantaneous eigenstate at time t as ψ(t)⟩. Thus, the final state can be denoted as ψ(τ Q )⟩, which can be regarded as the state annihilated by operators followingγ k ψ(τ Q )⟩ = 0. Hereγ k is defined by the time dependent Bogoliubov transformationĉ where Since k-mod is decoupled to each other, the dynamical Bd-G equation can be mapped into a Landau-Zener (LZ) problem [39][40][41]. Thus, we have the excitation probability of a k-mode as [38] The topological density of the final state is the summation of all k, i.e., n d = ∑ k p k . In the thermodynamic limit N → ∞, it is obtained by a Gaussian integral as This result reveals that the topological defect density has a power law scaling with respect to the quench rate [38], To understand this result, we need to introduce KZM [18,37]. The excitation can be only excited around the critical point, where the adiabatic approximation breaks down. The coherence time is determined by the inverse of the energy gap, i.e., τ ∼ ∆ −1 . The adiabatic condition fails at τ ∼t, which is determined bŷ where and the corresponding energy gap scales aŝ Therefore, the excitation density scales where d is the system dimension. This is known as KZM.

III. QUANTUM FISHER INFORMATION AND MULTIPARTITE ENTANGLEMENT
Generally, given an operatorÔ and a quantum state with density matrixρ = ∑ i p i ψ i ⟩ ⟨ψ i , where ⟨ψ i ψ j ⟩ = δ ij and p i is the probability, the corresponding QFI can be defined as For a pure stateρ = ψ⟩ ⟨ψ , QFI can be simplified as In addition, we can define the QFI density of a system with N particles as The physical significance of QFI density is that there are at least [f Q ] + 1 particles entangled [57], where [f Q ] represents the maximum integer that is smaller than f Q . That is, QFI is a quantification of multipartite entanglement. Generally, QFI or multipartite entanglement is an important quantum resource in quantum metrology, which quantifies the maximal precision of parameter estimation. Recently, multipartite entanglement is also applied to understand quantum many-body physics, e.g., as a witness of topological phase [26,29]. It is known that bipartite entanglement, generally represented by von Neumann or Rényi entropy, is closely related to QPTs [9]. One may wonder whether the scaling behaviors of multipartite entanglement quantified by QFI can be described as KZM. Next, we will study the QFI of the final state quenched from the polarized state crossing the critical point with the rate τ Q .

IV. QUANTUM FISHER INFORMATION AFTER A SLOW QUANTUM QUENCH
Now we start to investigate multipartite entanglement of transverse Ising model after a slow quantum quench, i.e., considering the dynamics of following timedependent Hamiltonian We begin with a ground state of large g in the paramagnetic phase and mainly concern the relation between QFI density and quench rate. Generally, QFI depends on the corresponding generatorÔ, so we first need to choose a proper operator in Eq. (19). For H(t), there is a global Z 2 spin-flip symmetry, of which transformation operator has the form Here, the initial state is a ground state of the paramagnetic phase, which is Z 2 symmetry unbroken. In addition, we consider an isolated system, so the final state after the quench dynamics is also symmetry unbroken. For the final Hamiltonian, i.e., g = 0, the corresponding symmetry unbroken ground state is a GHZ state which has the maximum QFI density N with respect to the operatorÔ = ∑ N n=1σ z n . Assuming that the quench is slow enough, we expect that the final state will be very close to the GHZ state. Based on these considerations, we can choose the operator asÔ = ∑ N n=1σ z n to calculate QFI.
According to Eq. (19) and since ⟨σ z n ⟩ = 0 (due to the existence of spin-flip symmetry.), we can calculate QFI density as where ⟨⋅⟩ means the expectation value with respect to the final state. Now, one can find that f Q is the summation of two-site spin-spin correlation function of each pair. In the thermodynamic limit, i.e., N → ∞, f Q of the final state is where C z (r) = ⟨σ z iσ z i+r ⟩ is the correlation function with distance r.
Here, we adopt MPS based method to study the slow quench dynamics with open boundary conditions. Firstly, we apply the DMRG algorithm to obtain the ground state for g = 5 as the initial state. Then, utilizing time-evolving block decimation (TEBD) algorithm [58][59][60][61][62], we can calculate the wave function of the final state. Thus, according to Eq. (24), we can obtain the QFI density of the final state. For DMRG calculation, the maximum bond dimension is 100 with truncation error smaller than 10 −12 . For TEBD calculation, we choose second-order Suzuki-Trotter decomposition. Furthermore, we also where the intercept f ∞ Q (red circle) can be considered as the QFI density in the thermodynamic limit. Here, we obtain the interception value of about 7.6358. (b) The relation between QFI density f Q and quench rate τ Q in the thermodynamic limit. Each scatters is obtained by the method shown in (a). The dashed line is the linear fitting curve. It is shown that f Q exhibits a power law scaling with respect to τ Q , where the fitting result is f Q ∼ τ 0.474±0.002 enlarge the maximum bond dimension and decrease the time of a single step to increase the precision of results, where the truncation error is smaller than 10 −11 and the minimum time step of Suzuki-Trotter decomposition is δt = 0.02.

A. Linear quench
Since the adiabatic condition breaks only in the vicinity of the critical point, only the quench rate at t c (where g(t c ) = g c = 1), i.e., g ′ (t c ), dominates the transition from ground state to excited states. That is, if the derivative of g(t) at t c is a finite value, g(t) can be replaced by a linear function with slope g ′ (t c ). Thus, following the progress of conventional KZM in quantum Ising model, we first consider a linear slow quench. Thus, the time-dependent transverse field has the form where τ Q represents the quench rate and t c = 0. The whole quench dynamics begins with g = 5 and ends with g = 0.
In Fig. 1, we present the correlation function C z (r) of the final state. It is shown that C z (r) decays very quickly as the increase of distance r, even faster than the exponential law [41]. Thus, the final state has no long-range correlation. Now we start to consider the QFI density f Q of the final state.
To obtain more precise f Q in the thermodynamic limit, we first calculate f Q for a fixed τ Q with different system sizes, and then we use finite-size scaling to fit out f Q for N → 0, see Fig. 2(a). Finally, we study the relation between f Q and quench rate τ Q . From Fig. 2(b), we can find that f Q ∼ τ 0.47 Q . Therefore, the number of entangled spins after a slow quench has a power-law scaling with respect to the quench rate.
As mentioned in Sec. II, the conventional KZM predicts that the topological defect density of final state satisfying the scaling n d ∼ τ Q . Therefore, after a linear quench, the QFI density f Q has the inverse scaling to one of the topological defects. Now we try to understand the above result. For QFI density, according to the behavior of correlation function C z (r), we can find the main contributions to f Q originate from C z (r < ξ), where ξ can be regarded as the correlation or coherence length of the final state. That is, QFI density is the reflection of the correlation length. Generally, after a linear quench crossing critical point, the correlation length is the inverse of topological defect density, i.e., ξ ∼ τ ν (zν+1) Q [41]. Therefore, the scaling of QFI density should satisfy which is consistent with the numerical results.

B. Nonlinear quench
To further illustrate the results in Eq. (27), we study the QFI density after a nonlinear quench. Generally, if the derivative of g(t) is zero at t c , then the approximation of the linear quench is invalid. For this type of nonlinear quench, we can apply Taylor expansion and use the leading term to describe this nonlinear quench dynamics. Hence, we can use the following power law function to study the general nonlinear quench where α > 0, see Fig. 3. In addition, similar to the linear case, the nonlinear quench dynamics begins with g = 5 and ends with g = 0.
Here, the nonlinear quench of Eq. (28) can be considered as a linear quench with respect to the driven parameter λ = (g − 1) 1 α with the critical exponents The detailed analysis is presented in Appendix A. Therefore, according to Eqs (27) and (29), the QFI density of the final state after a nonlinear quench, labeled f nl Q , should have the form We note that the scaling behavior of topological defects after a nonlinear quench has been studied in Refs [41,42], where the results are consistent with ours. In Fig. 4, we present the numerical results of QFI density. According to Fig. 4(a), we can find that f nl Q indeed exhibits a power law relation with τ Q . By linear fittings, we can obtain the corresponding powerlaw indexes for different α, which is consistent with Eq (30), see Fig. 4(b). These results further demonstrate the validity of Eq. (27).

V. DISCUSSION
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 a local quantum many-body system, due to the Lieb-Robinson bounds [63,64], the minimum time it takes to create a state with extended 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), then the time cost for creating a state with QFI density proportional to N is at least ∝ N . In our system, for the linear quench, if we want f Q ∝ N for the final state, then the quench rate τ Q ∝ N 2 . Thus, we need time t ∝ N 2 to create a state with extended QFI density via a linear quench crossing a QPT, which is slower than the adiabatic case. Furthermore, in the case of nonlinear quench, the corresponding time cost is N α+1 α > N . Only if α → ∞, which is equivalent to an adiabatic process, the rate of increasing QFI is identical to the adiabatic case.
Therefore, compared with the adiabatic process, the rate of generating multipartite entanglement near the critical point becomes slow and is closely related to the critical exponents. This behavior of multipartite entanglement crossing a QPT can be considered as a reflection of critical slowing down.

VI. SUMMARY AND OUTLOOK
In summary, we have studied the multipartite entanglement after slow quantum quenches crossing a QPT. The QFI density f nl Q after nonlinear quenches crossing a quantum critical point. (a) The relations between QFI density f nl Q and quench rate τ Q in the thermodynamic limit for different α. Each scatters is obtained by the finite-size scaling in Fig. 2(a) By numerical calculations via MPS-based methods, we find that the QFI density of the final state follows the scaling behavior opposite to the one in KZM. Our results reveal that KZM can not only predict the topological defect density but also the multipartite entanglement after crossing a QPT. Moreover, these results can also help us understand QPTs and the behavior of critical slowing down in 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-ofart quantum simulators. There also remain many interesting open questions, which are deserved to be further studied. In this work, we only consider the 1D system, and it is meaningful to generalize our results to higher-dimensional systems to verify whether QFI density still satisfies the conventional KZM f Q ∼ τ dν zν+1 Q . Moreover, when calculating QFI density, we choose the corresponding local operator based on the symmetry of the Hamiltonian.
In this system, the QPT originates from a spontaneous symmetry breaking, so we can use the order parameter to define QFI. Therefore, it will be an interesting issue to study multipartite entanglement after slow quantum quenches crossing a topological QPT, which is absent of spontaneous symmetry breaking.