Dynamical quantum phase transition in quantum spin chains with gapless phases

Quantum phase transitions (QPTs) are one of the most significant phenomena in quantum many-body physicsVojta (2002, 2003); Sachdev (2000). It is characterized by a nonanalytic behavior of some physical observable at a quantum critical point due to the change of the external control parameter. Recently, the dynamical quantum phase transitions (DQPTs) that emerge in the dynamics of an isolated quantum many-body systems have attracted both the theoreticalHeyl et al. (2013); Karrasch and Schuricht (2013); Andraschko and Sirker (2014); Heyl (2014); Hickey et al. (2014); Kriel et al. (2014); Vajna and Dora (2014); Heyl (2015); Schmitt and Kehrein (2015); Vajna and Dora (2015); Budich and Heyl (2016); Divakaran et al. (2016); Huang and Balatsky (2016); Sharma et al. (2016); Zvyagin (2016); Bhattacharya et al. (2017); Halimeh and Zauner-Stauber (2017); Heyl (2017); Heyl and Budich (2017); Homrighausen et al. (2017); Karrasch and Schuricht (2017); Weidinger et al. (2017); Bhattacharjee and Dutta (2018); Cheraghi and Mahdavifar (2018); Heyl (2018); Kennes et al. (2018); Kosior and Sacha (2018, 2018); Lang et al. (2018); Wang and Gao (2018); Sedlmayr et al. (2018); Yin et al. (2018); Zhou et al. (2018); Zunkovic et al. (2018); Abdi (2019); Hagymasi et al. (2019); Huang et al. (2019); Jafari (2019); Khatun and Bhattacharjee (2019); Lacki and Heyl (2019); Lahiri and Bera (2019); Liu and Guo (2019); Mendl and Budich (2019); Piccitto and Silva (2019); Srivastav et al. (2019); Yang et al. (2019); Cao et al. (2020); Hou et al. (2020); Masowski and Sedlmayr (2020); Haldar et al. (2020); Wu (2020); Zamani et al. (2020); Jafari and Akbari (2021) and experimentalJurcevic et al. (2017); Zhang et al. (2017); Flaschner et al. (2018); Wang et al. (2019); Nie et al. (2020); Tian et al. (2020); Xu et al. (2020) interests. Unlike the QPTs driven by the external control parameters, the DQPTs describe the nonanalytic behavior of the Loschmidt echo (LE) during the time evolution, where a common protocol for driven a system out of equilibrium is called quantum quench. In many cases, the DQPTs are found to have a strong connection with the QPTs. The DQPTs are present in the cases of sudden quenches crossing the quantum critical pointsHeyl et al. (2013); Karrasch and Schuricht (2013); Andraschko and Sirker (2014) and the topology changesHickey et al. (2014); Vajna and Dora (2015), although it is also found that the DQPTs occur in the cases of quenches without crossing any quantum critical pointVajna and Dora (2014); Halimeh and Zauner-Stauber (2017); Homrighausen et al. (2017). Therefore, it is still an open and debated issue whether the quench crossing the critical points of QPTs is a necessary condition to induce a DQPTJafari (2019); Haldar et al. (2020).

Up to now, so many works have investigated the quench from gapped phase to gapped phase, but few studies have focused on the case of quench from the gapless phase. The gapless phase is of general existence in the quantum systems, such as the XY chain with the Dzyaloshinskii-Moriya (DM) interactionDzyaloshinsky (1958); Moriya (1960); Derzhko and Moina (1994); Derzhko and Richter (1999), the XY chain with XZY-YZX type of three-site interactionsGottlieb and Rossler (1999); Krokhmalskii et al. (2008) and Kitaev modelKitaev (2006) etc. Unlike the case of the gapped phase, the ground states of the system in the gapless phase corresponds to the configuration where all the states with negative excitation spectra are filled and nonnegative are emptyRodney et al. (2013); Zhong et al. (2013). Recently, Cheraghi and Mahdavifar studied the DQPTs in the quantum Ising chain with DM interaction Cheraghi and Mahdavifar (2018). They only considered the case of quench from gapped phase to gapped phase and did not consider the quench from the gapless phase, so that they concluded that the DM interaction does not affect the DQPT. In a recent paper Haldar et al. (2020), the authors studied the DQPTs in the XY chain with DM interaction in the alternating external transverse field. They also did not discuss the case of quench starting from the gapless phase, because they said that it’s difficult to study the initial condition in the gapless phase in their models Haldar et al. (2020). Therefore, it’s still unknown when the quench is from the gapless phase.

Recently, Jafari has studied the DQPTs in the extended XY chain with XZX+YZY type of three-site interaction in the staggered transverse field Jafari (2019). He found that the DQPTs may not occur when the quench is from the gapped phase to gapless phase. The model has the gapless phase due to the staggered external field and the quasiparticle excitation spectra are symmetrical Jafari (2019). In this paper, we investigate the DQPTs in more general quantum spin chains with asymmetrical quasiparticle excitation spectra. We study all possible situations of ground states of pre-quench Hamiltonian and obtain the general expression of LE for the systems with gapless phases [see Sec. II]. For the homogeneous system, the LE can be given by $\mathcal{L}(t)=\prod_{k>0}\mathcal{L}_{k}(t)$, where $\mathcal{L}_{k}(t)$ equal unity corresponding to the quasiparticle excitation spectra of pre-quench Hamiltonian satisfying $\varepsilon_{k}\cdot\varepsilon_{-k}<0$ in the momentum subspace $k>0$. The general conditions for the occurrence of DQPTs are also obtained in Sec. II. In Secs. III and IV, we discuss the behaviors of DQPTs in two typical models—the XY chains with DM and XZY-YZX type of three-site interactions, respectively. It’s found that the DQPTs may not occur in the quench across the quantum phase transitions regardless of whether the quench is from the gapless phase to gapped phase or from gapped phase to gapless phase. This is different from that in the quantum spin chain with symmetrical excitation spectra. A brief conclusion is given in Sec. V.

We consider general quantum spin chains, whose Hamiltonian can be written in a quadratic form

where $c_{n}$ and $c^{{\dagger}}_{n}$ are fermion annihilation and creation operators respectively. The Hermiticity of $H$ demands $A$ to be a Hermitian matrix and anti-commutation of fermion operators demands $B$ to be an antisymmetric matrix. In the uniform case, the Hamiltonian (1) can be written in a diagonal form in the momentum space via the Fourier transformation ($c_{k}=\frac{1}{\sqrt{N}}\sum_{n}e^{ikn}c_{n}$, $c^{{\dagger}}_{k}=\frac{1}{\sqrt{N}}\sum_{n}e^{-ikn}c^{{\dagger}}_{n}$) and Bogoliubov transformation Sachdev (2000); Suzuki et al. (2013) $\eta_{k}=u_{k}c_{k}+iv_{k}c^{{\dagger}}_{-k}$:

where $k$ are the waves vectors, $\varepsilon_{k}$ are the quasiparticle excitation spectra, $\eta_{k}$ and $\eta^{{\dagger}}_{k}$ are fermion annihilation and creation operators. It should be noticed that we consider the more general cases in which the quasiparticle excitation spectra may not be asymmetrical ($\varepsilon_{k}\neq\varepsilon_{-k}$) and positive ($\varepsilon_{k}<0$).

The ground state of the system can be constructed as

Here $|GS\rangle_{k}\in\mathbb{C}^{2}\bigotimes\mathbb{C}^{2}$ in the subspace $k>0$ is of the form

where $n_{k}(n_{-k})=0$ or $1$ are the eigenvalues of spinless fermion number operator $\hat{n}_{k}=\eta^{{\dagger}}_{k}\eta_{k}(\hat{n}_{-k}=\eta^{{\dagger}}_{-k}\eta_{-k})$. To be specific, the ground states are given by

In a quantum quench, the system is prepared in the ground states $|GS\rangle=\otimes_{k>0}|n_{k}n_{-k}\rangle$ for the Hamiltonian $H$. At time $t=0$, the system undergoes a sudden quench with its Hamiltonian switched from $H$ to $\widetilde{H}$, where the ground states of the post-quench Hamiltonian $\widetilde{H}$ is $|\widetilde{GS}\rangle=\otimes_{k>0}|\widetilde{n}_{k}\widetilde{n}_{-k}\rangle$ accordingly. The quantities associated with the post-Hamiltonian are labeled by sign “$\sim$”. The Loschmidt amplitude is defined as (we take $\hbar=1$)

where ${\cal G}_{k}(t)=\langle n_{k}n_{-k}|e^{-i\widetilde{H}_{k}t}|n_{k}n_{-k}\rangle$, $E_{0}(\widetilde{E}_{0})$ is the ground state energy. To calculate the Loschmidt amplitude, we should expand the states $|n_{k}n_{-k}\rangle$ of the pre-quench Hamiltonian $H_{k}$ by the eigenstates of the post-quench $\widetilde{H}_{k}$.

By considering both $\eta^{{\dagger}}_{k}$ and $\widetilde{\eta}^{{\dagger}}_{k}$ are related by the Bogoliubov transformation, we can obtain the relations between the eigenstates of $H_{k}$ and $\widetilde{H}_{k}$:

where $\alpha_{k}=\theta_{k}-\tilde{\theta}_{k}$, $\theta_{k}$ ($\tilde{\theta}_{k}$) are the Bogoliubov angles of the pre- and post-quench Hamiltonian defined by $u_{k}=\cos{\theta_{k}}$ and $v_{k}=\sin{\theta_{k}}$ Sachdev (2000); Suzuki et al. (2013). It’s noticed that $|1_{k}0_{-k}\rangle$ and $|0_{k}1_{-k}\rangle$ of the pre-quench Hamiltonian are also the eigenstates of the post-quench Hamiltonian $\widetilde{H}_{k}$.

By substituting Eq. (7) into Eq. (6), the Loschmidt amplitudes ${\cal G}_{k}(t)$ are given by

Then, the LE is

with

In order to show the DQPT more directly, one can use the dynamical free energy density, which is defined as the rate function $\lambda(t)=-\lim_{N\rightarrow\infty}\ln{{\cal L}(t)}/N$ of the LE. Analogous to the equilibrium phase transitions, the rate function will show the sharp nonanalyticities at the critical times of the DQPTs Karrasch and Schuricht (2013). According to Eq. (10), the rate function of LE is

Obviously, for the cases of $\varepsilon_{k}\cdot\varepsilon_{-k}\leq 0$, $\ln{{\cal L}_{k}(t)}=0$.

Another way to show the DQPTs is using the Fisher zeros of the Loschmidt amplitude in the complex time plane Heyl et al. (2013); Heyl (2018). According to Eq. (8), for $\varepsilon_{k}<0,\varepsilon_{-k}\geq 0$ and $\varepsilon_{k}\geq 0,\varepsilon_{-k}<0$, ${\cal G}_{k}(t)=e^{-it(\widetilde{\varepsilon}_{k}+\widetilde{\varepsilon}_{-k})}=0$ have no solutions in the complex time plane. While for $\varepsilon_{k},\varepsilon_{-k}\geq 0$, we obtain the Fisher zeros $z_{n}$ of the Loschmidt amplitude located on the lines ($n\in\mathbb{Z}$)

and for $\varepsilon_{k},\varepsilon_{-k}<0$

The DQPTs occur if the Fisher zeros lines interacting with the imaginary axis. The imaginary parts of the Fisher zeros on the imaginary axis denote the critical times of the DQPTs. Therefore, the conditions for the occurrence of DQPTs are given by

If the above conditions (14) hold, the corresponding critical times of the DQPTs are given by

As the first example, we begin our study on the anisotropic XY chain with the DM interaction in the transverse field Rodney et al. (2013); Zhong et al. (2013)

where $\sigma^{x,y,z}_{n}$ are the Pauli matrices, $\gamma$ is the anisotropic parameter, $h$ is the external field, and $D$ is the strength of the DM interaction, respectively.

The Hamiltonian (16) can be expressed in the quadratic form (1) via the Jordan-Wigner transformation, written by

The quasiparticle excitation spectra $\varepsilon_{k}$ of the XY chain with DM interaction are given by

According to Eq. (18), the quasiparticle excitation spectra $\varepsilon_{k}$ are not symmetrical in the momentum space and can only have negative values for $k>0$.

Fig. 1 (a) shows a typical phase diagram of the system with $D=0.2$ in the plane $(h,\gamma)$ Zhong et al. (2013), where the region CP is the gapless chiral phase, region PM is the gapped paramagnetic phase, region FM_x is the gapped ferromagnetic phase along $x$ direction, FM_y is the gapped ferromagnetic phase along $y$ direction, respectively. The critical line separating the gapless chiral phase and the PM phase is given by $h_{c}=\sqrt{4D^{2}-\gamma^{2}+1}$, and those separating the gapless chiral phase and the FM phases are $\gamma_{c}=\pm 2D$. In Fig. 1 (b), we show three examples of quasiparticle excitation spectra, in which the Hamiltonian parameters $(h=0.5,\gamma=0.2)$, $(h=0.5,\gamma=-0.5)$ and $(h=1.5,\gamma=0.2)$ are marked by black dots in Fig. 1 (a). The excitation spectra $\varepsilon_{k}$ are negative for $1.7<k<2.4$ when $h=0.5,\gamma=0.2$, while they are positive for all $k$ when $h=0.5,\gamma=-0.5$ and $h=1.5,\gamma=0.2$.

Now, we consider the quench from $H=H(h_{0},\gamma_{0})$ to $\widetilde{H}=H(h_{1},\gamma_{1})$, where we keep the same strength of DM interaction before and after the quench. By considering the Bogoliubov angle satisfying $\tan{2\theta_{k}}=\gamma\sin{k}/(h+\cos{k})$, the condition $|\tan{\alpha_{k}}|=1$ of (14) is equivalent to the following quadratic equation

Therefore, for a given quench process, the conditions for the occurrence of DQPTs are given by

where $\Delta=(h_{0}+h_{1})^{2}-4(1-\gamma_{0}\gamma_{1})(h_{0}h_{1}+\gamma_{0}\gamma_{1})$.

In Fig. 2, we use the domains ${\cal R}(h_{0},\gamma_{0})$ to cover all the post-quench Hamiltonian parameters where the Eqs. (20) are satisfied in the case of quench from the pre-quench Hamiltonian $H(h_{0},\gamma_{0})$. In other words, the DQPTs occur in the quench from $H(h_{0},\gamma_{0})$ to the post-quench Hamiltonian parameters covered by the domain ${\cal R}(h_{0},\gamma_{0})$.

As another example, we investigate the anisotropic XY chain with the XZY-YZX type of three-site interaction defined by the Hamiltonian Krokhmalskii et al. (2008); Liu et al. (2012)

where $\xi$ is the strength of the three-site interaction. By using the Jordan-Wigner transformation, the Hamiltonian (21) can be written in the quadratic form (1) with

The quasiparticle excitation spectra of the XY chain with three-site interaction are given by

In Fig. 6 (a), we display a typical phase diagram of the system with three-site interaction $\xi=0.5$ Liu et al. (2012). The phase diagram contains four parts: the region CP is the gapless chiral phase, region PM is the gapped paramagnetic phase, region FM_x is the gapped ferromagnetic phase along $x$ direction, FM_y is the gapped ferromagnetic phase along $y$ direction, respectively. The critical lines separate the gapless phase and FM phase are given by $\gamma=\pm\xi h$, and that separates the gapless phase and PM phase is $h=\sqrt{\xi^{2}-\gamma^{2}+1}$. In Fig. 6 (b), we show two examples of quasiparticle excitation spectra, in which the Hamiltonian parameters $(h=0.5,\gamma=0.1)$ and $(h=1.5,\gamma=0.1)$ are marked by dots in Fig. 6 (a). The excitation spectra $\varepsilon_{k}$ are negative for $1.93<k<2.38$ when $h=0.5,\gamma=0.1$, while they are positive for all $k$ when $h=1.5,\gamma=0.1$.

Now, we consider the quantum quench from $H=H(h_{0},\gamma_{0})$ to $\tilde{H}=H(h_{1},\gamma_{1})$, where we keep the same strength of three-site interaction before and after the quench ($\xi_{0}=\xi_{1}$).

We study the properties of DQPTs in general quantum spin chains, in which the systems have gapless phases and asymmetrical quasiparticle excitation spectra. By considering the quench starting from different initial states, we find that the factors $\mathcal{L}_{k}(t)$ of LE equal unity where the quasiparticle excitation spectra of pre-quench Hamiltonian satisfy $\varepsilon_{k}\cdot\varepsilon_{-k}<0$. Therefore, we obtain the general conditions for the occurrence of DQPT, that is the occurrence of DQPT not only requires the Bogoliubov angle to satisfy $|\tan{\alpha_{k}}|=1$, but also requires the quasiparticle excitation spectra of the pre-quench Hamiltonian to satisfy $\varepsilon_{k}\cdot\varepsilon_{-k}\geq 0$. We have confirmed our conclusions in two typical models—the XY chains with DM interaction and XZY-YZX type of three-site interaction.

From the discussion above and previous results Jafari (2019); Haldar et al. (2020), we summarize the connection between DQPT and QPT as following. For the system only with gapped phase, the DQPTs will occur when the quench crosses the critical lines of QPTs. For the system with gapless phase and symmetrical quasiparticle excitation spectra, the DQPTs may not occur in the quench from the gapped phase to gapless phase but occur in the quench from the gapless phase to gapped phase. For the system with gapless phase and asymmetrical quasiparticle excitation spectra, the DQPTs may not occur in the quench across the quantum phase transitions regardless of whether the quench is from the gapless phase to gapped phase or from the gapped phase to gapless phase.