Determination of dynamical quantum phase transition for boson systems using the Loschmidt cumulants method

Nonequlibrium dynamics has draw much attention in recent years Mitra (2018); Langen et al. (2016); Dziarmaga (2010); Will et al. (2010); Wilczek (2012); Goldman and Dalibard (2014); Eisert et al. (2015); Poon and Liu (2016). One major achievement is the investigation of DQPT. DQPTs concern the critical behavior of many-body systems that are driven out of equilibrium via sudden quenches. In fact, There are two types of DQPTs. The first one concerns the nonanalytical change of order parameter along the nonequilibrium evolutionPoon and Liu (2016); Zhang et al. (2017). For instance, in the reference Poon and Liu (2016), it was observed that the evolution dynamics of the spin order parameter undergoes a significant change when the perturbation exceeds a certain threshold value. In this study, we specifically concentrate on the second type of Dynamical Quantum Phase Transitions (DQPT) that has been proposed in recent years Heyl et al. (2013); Heyl (2018). This type of DQPT are inspired by the similarity between the Loschmidt amplitude and the equilibrium thermal phase transitions. It is expected to capture unique features of nonequilibrium dynamics Bhattacharya et al. (2017); Sharma et al. (2016); Heyl and Budich (2017); Budich and Heyl (2016); Karrasch and Schuricht (2017); Wang et al. (2019); Heyl (2015); De Nicola et al. (2021). Due to the absence of a direct connection to local order parameters, experimental verification of this particular type of DQPT remains challenging, particularly for systems with interaction.

The Bose-Hubbard model is a well-known model for investigating the many-body phases, with numerous studies conducted on both equilibrium and nonequilibrium phases. For nonequilibrium dynamics, the Kibble-Zurek mechanism Huang et al. (2021); Braun et al. (2015); Zheng et al. (2023) and DQPTs Lacki and Heyl (2019) have been studied for the Bose Hubbard model. A DQPT occurs in the Bose-Hubbard model when the system is quenched from the Mott-insulator phase to the superfluid phase, as shown in Fig. 1(a).

DQPTs have been investigated by exactly solvable models Heyl et al. (2013); Schmitt and Kehrein (2015); Vajna and Dóra (2015). However, dynamics of the interacting quantum many-body systems have been hard. Traditionally, tensor network methods are used. Recently, a new method Peotta et al. (2021); Brange et al. (2022) has been proposed to determine the dynamical critical points. Different from determining the critical points by identifying the nonanalytics points on the real time, the method determining zeros of the dynamical Loschmidt amplitude analogous to the Lee-Yang zerosYang and Lee (1952); Lee and Yang (1952); Brange et al. (2023) of the thermal phase transition. The critical points are determined from the crossing points of the thermodynamics lines of zeros with the imaginary axis, as shown in Fig. 1(b). Moreover, this method offers a viable approach to determine the dynamical critical points by analyzing the energy fluctuations of the initial state. As a result, it becomes a promising avenue for investigating DQPTs.

Here we focus on DQPTs in the Bose-Hubbard model following a quench. In Sec. II, we introduce the basic ideas for determining the critical points using Loschmit cumulants, as illustrated in Peotta et al. (2021). In Sec. III, we investigate dynamical phase transitions in the Bose-Hubbard model and accurately determine the critical points. In Sec. IV, we consider the nearest-neighbor interaction, and investigate quenches across topologically distinct phases. In Sec. V, we show that how the critical points of DQPT can be determined by analyzing the energy fluctuation of the initial state. In Sec. VI, we present our conclusions and offer an outlook on potential directions for future advancements.

In Reference Heyl et al. (2013), it was initially noted that a formal similarity exists between the canonical partition function

$\displaystyle Z(\beta)=Tre^{-\beta\hat{H}}$

(1)

and the Loschmidt amplitude $G(t)$

$\displaystyle G(t)=\langle\psi_{0}|e^{-i\hat{H}t}|\psi_{0}\rangle.$

(2)

Here, we consider the Loschmidt amplitude on the complex plane

$\displaystyle Z(\tau)=\langle\psi_{0}|e^{-\tau\hat{H}}|\psi_{0}\rangle.$

(3)

According to the factorization theorem Arfken et al. (2013), the Loschmidt amplitude can be witten as

$\displaystyle Z(z)=e^{\alpha\tau}\prod_{k}(1-\frac{\tau}{\tau_{k}})$

(4)

where $\alpha$ is a constant, and $\tau_{k}$ are complex zeros of the Loschmidt amplitude. In the thermodynamics limit, the complex zeros form lines or areas. When such a line intersects or a boundary of a region touches the imaginary axis, a DQPT takes place.

Also, when a DQPT occurs, the Loschmidt rate function

$\displaystyle\lambda(t)=-\frac{1}{L}\ln|Z(it)|^{2}$

(5)

develops nonanalytic behavior.

The approach used to determine the dynamical critical points involves obtaining the line of zeros in the thermodynamic limit. As illustrated in Peotta et al. (2021), the Loschmidt cumulants and the Loschmidt moments are defined as:

		$\displaystyle\langle\langle\hat{H}^{n}\rangle\rangle_{\tau}=(-1)^{n}\partial^{n}_{\tau}\ln Z(\tau)$		(6)
	$\displaystyle\langle\hat{H}^{n}\rangle=$	$\displaystyle(-1)^{n}\frac{\partial^{n}_{\tau}Z(\tau)}{Z(\tau)}=\frac{\langle\psi_{0}|\hat{H}^{n}e^{-\hat{H}\tau}|\psi_{0}\rangle}{\langle\psi_{0}|e^{-\hat{H}\tau}|\psi_{0}\rangle}$		(7)

At $\tau=0$, the Loschmidt moments reduce to the ordinary moments (7) of the postquench Hamiltonian with respect to the initial state as $\langle\hat{H}^{n}\rangle=\langle\Psi_{0}|\hat{H}^{n}|\Psi_{0}\rangle$.

The Loschmidt cumulants are related to the Loschmidt zeros through

$\displaystyle\langle\langle\hat{H}^{n}\rangle\rangle_{\tau}=(-1)^{n-1}(n-1)!\sum_{k}\frac{1}{(\tau_{k}-\tau)^{n}}$

(8)

According to this expression, the Loschmidt cumulants are primarily influenced by the zeros that are closest to the base point $\tau$. The contribution of each zero to the cumulants decreases rapidly with its inverse distance to the power of the cumulant order $n$. By computing $2m$ high-order Loschmidt cumulants, it becomes possible to determine the $m$ closest zeros to the base point.

We extract the 7 zeros closest to the movable basepoint using Loschmidt cumulants of order n=9 to n=22. In our time evolution of the wavefunction, we select the Krylov subspace dimension to be $N_{\text{vec}}=8$ and the time step for evolution to be $\delta\tau=0.01$.

In this study, we explore DQPTs in the one-dimensional Bose-Hubbard model, characterized by the Hamiltonian:

$\displaystyle\hat{H}=-J\sum_{i}(a_{i}^{\dagger}a_{i+1}+a_{i+1}^{\dagger}a_{i})+\frac{U}{2}\sum_{i}n_{i}(n_{i}-1)$

(9)

Here $a_{i}^{\dagger}$ is the creation operator for a boson on site i, $a_{i}$ is the annihilation operator for a boson on site i, $n_{i}=a_{i}^{\dagger}a_{i}$ is the occupation operator for a boson on site i, $J$ denotes the hopping amplitude between nearest neighbor, $U$ denotes the onsite interaction strength. To minimize boundary effects, we impose the periodic boundary condition.

The properties of the Hamiltonian are determined by the dimensionless ratio $s=J/U$, and a phase transition occurs at the critical value of $s_{c}=0.297$ for unit filling, which separates the system into a Mott-insulator phase for $s<s_{c}$ and a superfluid phase for $s>s_{c}$.

We are now prepared to explore the phenomenon of Dynamical Quantum Phase Transitions (DQPT) in the Bose-Hubbard chain. We initialize the system in the superfluid ground state with a parameter value of $s_{0}=0.36$. The local Hilbert space is truncated to $n=3$ for an balance of efficciency and accuracy. The atom occupation number for $n>3$ is less than 0.03% for the initial state.

We perform a quench into the Mott-insulator phase with $s<0.297$ for later times. The same quench parameters have been explored in Lacki and Heyl (2019) using matrix product method. Here we employ the matrix product state(MPS) method for benchmark.

Figure 2 shows the complex zeros of the Loschmidt amplitude. For the $J=0$ case, the system undergoes periodic evolution with period $T=\frac{2\pi\hbar}{U}$. The first critical time is $t_{1}=\frac{\pi}{U}$. As expected, the zeros forms a line around $t=\frac{\pi\hbar}{U},\frac{3\pi\hbar}{U},\frac{5\pi\hbar}{U}$ on the imaginary axis. As the tunneling amplitude is incrementally increased, it can be observed in Figure 2(b-c) that the Loschmidt zeros gradually shift the critical crossing point with the imaginary axis towards later times. Finally the thermodynamic lines of zeros no longer crosses the imaginary axis.

The critical time obtained from the crossings of the thermodynamic lines of zeros with the imaginary axis are in excellent agreement with the critical times abtained using the MPS method. To quantify the accuracy of the Loschmidt cumulants method, we determine the critical points and compare it with those obtained using MPS method. We find that for all typical cases in Fig. 2 the discrepancy is lower than 2%. It is worth noting that these results are obtained for rather short length from L=4 to L=10. The use of such system sizes makes the approach very attractive for strongly intracting systems.

We will now explore DQPTs in the extended Bose-Hubbard model, which includes nearest-neighbor interactions and exhibits rich phase diagrams such as the Haldane insulator(HI) and the charge density wave (CDW). The Hamiltonian for the extended Bose-Hubbard model is given by:

	$\displaystyle\hat{H}=$	$\displaystyle-J\sum_{i}(a_{i}^{\dagger}a_{i+1}+a_{i+1}^{\dagger}a_{i})+\frac{U}{2}\sum_{i}n_{i}(n_{i}-1)$		(10)
		$\displaystyle+V\sum_{i}n_{i}n_{i+1}$

Here V denotes the nearest neighbor interaction strength. For a ratio of interaction to hopping strength of $U/J=5$, the equilibrium phase transition point is $p=V/J=2.95$ for the Mott insulator-Haldane insulator transition and $p=3.53$ for the Haldane insulator-density wave transition Rossini and Fazio (2012). We adopt periodic boundary conditions in this study Stumper and Okamoto (2020); Ejima et al. (2014).

We quench from the Mott phase $p_{0}=1.0$ to larger nearest neighbor interaction Stumper et al. (2022). The results are shown in Fig. 3. Dynamical quantum phase transition occurs at about $p=3.5$. As we increase the ratio $p$, dynamical phase transition happens at ealier times. The discrepancy is larger near the equilibrium phase transition points and decreases for cases further away from these points. This may be due to the finite-size effect. For further improvement, other boundary conditions may be introduced Peotta et al. (2021). Although our study focuses on a narrow range of parameters, it is feasible to investigate DQPTs for other parameter ranges in the extended Bose-Hubbard model as well.

Finally, we demonstrate the ability to predict DQPT solely through measuring the energy fluctuations in the initial state. At $\tau=0$, the Loschmidt moments reduce to the ordinary moments (7) of the postquench Hamiltonian with respect to the initial state as $\langle\hat{H}^{n}\rangle=\langle\Psi_{0}|\hat{H}^{n}|\Psi_{0}\rangle$. If we expand the wavefunction with respect to the post-quench Hamiltonian $|\Psi_{0}\rangle=\sum_{m}a_{m}|\tilde{\Psi}_{m}\rangle$, we get $\langle\hat{H}^{n}\rangle=\sum_{m}P(E_{m})E_{m}^{n}$, where $P(E_{m})=|a_{m}|^{2}$. Thus, by repeatedly preparing the system in the state $|\psi_{0}\rangle$ and measuring the energy with respect to the postquench Hamiltonian, we can construct the energy distribution and extract the corresponding moments and cumulants. Therefore, by following the same procedure presented in the method section, we can obtain the Loschmidt zeros.

In Fig. 4, we present the procedure of determination the critical points from $10^{6}$ energy measurements. This predicts the critical time to be around $t\;U$/$\hbar=3.75$. This demonstrate the ability to predict the first dynamical phase transition using initial energy fluctuations. The determination of a dynamical Lee-Yang zeros was accomplished in Ref. Brandner et al. (2017) through the measurement of high cumulants. It should be noted that obtaining precise energy measurements of a many-body quantum system is challenging. Thus, the presented method offers an approach to establish a connection between DQPTs and measurable quantities i.e energy, at least in principle.

In summary, we have investigated the Loschmidt zeros associated with the dynamical quantum phase transitions in the Bose-Hubbard model and the extended Bose-Hubbard model. This analysis was conducted through the utilization of the Loschmidt cumulants method. We have determined the locations of the zeros of the Loschmidt amplitude in the complex plane of time. And by identifying the crossing point with the imaginary axis we get the dynamical phase transition point to an discrepancy lower than 2% for the Bose-Hubbard model. Also, we show an avenue for determining the dynamical phase transition by measuring the initial state energy fluctuation.

Acknowledgements: This work is supported by the National Natural Science Foundation of China (Grants No. 11920101004, 11934002, 12174006), and the National Key Research and Development Program of China (Grant No. 2021YFA1400900, 2021YFA0718300).