Genuine multipartite entanglement as the indicator of quantum phase transition in spin systems

Abstract

In this paper, the genuine multipartite entanglement (GME) and quantum criticality property of spin systems with staggered Dzyaloshinskii–Moriya (DM) interaction are investigated by exploiting quantum renormalization group method. The results show that the GME can indicate quantum phase transitions at critical points after several iterations of the renormalization. Moreover, the DM interaction effectively restores the spoiled GME via creation of quantum fluctuations, while it also changes the critical points. At last, the nonanalytic and scaling behaviors of GME are analyzed in detail.

Appendix

In this appendix, we will describe the renormalization of spin models by exploiting QRG method in detail. Followed, we will take the renormalization of one-dimensional Ising model as an example.

The Hamiltonian of Ising model with staggered DM interaction in the z direction on a periodic chain of N sites is

$$\begin{aligned} H=\frac{J}{4}\sum _i^N {\left[ {\sigma _i^z \sigma _{i+1}^z +D\left( {\sigma _i^x \sigma _{i+1}^y -\sigma _i^y \sigma _{i+1}^x } \right) } \right] } . \end{aligned}$$

(8)

By using the Kadanoff’s block method, it is necessary to divide the initial system Hamiltonian shown in Eq. (8) into two parts, namely,

$$\begin{aligned} H=H^\mathrm{B}+H^\mathrm{BB}, \end{aligned}$$

(9)

where $H^\mathrm{B}$ is block Hamiltonian and $H^\mathrm{BB}$ is interblock Hamiltonian. To get a renormalized form for the Hamiltonian, we will choose three sites as a block and the decomposition has been shown in Fig. 7. Note that it is a guarantee of self-similarity after each QRG step.

Fig. 7

(Color line) Decomposition of Ising chain by using Kadanoff’s block approach. The Hamiltonian of the system is divided into two parts, i.e., block Hamiltonian $H^\mathrm{B}$ and interblock Hamiltonian $H^\mathrm{BB}$

Then, the specific forms of Hamiltonian $H^\mathrm{B}$ and $H^\mathrm{BB}$ are

$$\begin{aligned}&H^\mathrm{B}=\frac{J}{4}\sum _L^{N/3} {\left[ {\sigma _{L,1}^z \sigma _{L,2}^z +\sigma _{L,2}^z \sigma _{L,3}^z +D\left( {\sigma _{L,1}^x \sigma _{L,2}^y -\sigma _{L,1}^y \sigma _{L,2}^x +\sigma _{L,2}^x \sigma _{L,3}^y -\sigma _{L,2}^y \sigma _{L,3}^x } \right) } \right] },\nonumber \\ \end{aligned}$$

(10)

$$\begin{aligned}&H^\mathrm{BB}=\frac{J}{4}\sum _{L=1}^{N/3} {\left[ {\sigma _{L,3}^z \sigma _{L+1,1}^z +D\left( {\sigma _{L,3}^x \sigma _{L+1,1}^y -\sigma _{L,3}^y \sigma _{L+1,1}^x } \right) } \right] } , \end{aligned}$$

(11)

where $\sigma _{L,j}^\alpha$ refers to the $\alpha$ component of the Pauli matrix at site j of the $L\hbox {th}$ block Hamiltonian. The exact treatment of this Hamiltonian leads to two distinct eigenvalues which are doubly degenerate with $\left| \uparrow \right\rangle$ and $\left| \downarrow \right\rangle$ defined as the eigenstates of the Pauli matrix $\sigma ^{z}$, both degenerate ground states used to construct the projection operator $P_0$ for the renormalized subspace are given by

$$\begin{aligned} \left| {\varphi _0 } \right\rangle =\frac{1}{\sqrt{2q\left( {1+q} \right) }}\left[ {2D\left| {\downarrow \uparrow \uparrow } \right\rangle +i\left( {1+q} \right) \left| {\uparrow \downarrow \uparrow } \right\rangle -2D\left| {\uparrow \uparrow \downarrow } \right\rangle } \right] , \end{aligned}$$

(12)

$$\begin{aligned} \left| {\varphi _0 ^{\prime }} \right\rangle =\frac{1}{\sqrt{2q\left( {1+q} \right) }}\left[ {2D\left| {\downarrow \downarrow \uparrow } \right\rangle +i\left( {1+q} \right) \left| {\downarrow \uparrow \downarrow } \right\rangle -2D\left| {\uparrow \downarrow \downarrow } \right\rangle } \right] , \end{aligned}$$

(13)

where $q=\sqrt{1+8D^{2}}$. The energy corresponding to the ground states is

$$\begin{aligned} E_0 =-\frac{J}{4}\left( {1+q} \right) . \end{aligned}$$

(14)

To eliminate the higher energy of the system and retain the lower, the projection operator $P_0$ is composed of its lowest-energy eigenstates. Then, the effective Hamiltonian $H^\mathrm{eff}$ and original Hamiltonian H have in common the low-lying spectrum, which can be given by the projection operator, i.e., $H^\mathrm{eff}=P_0^\dagger HP_0$ wherein $P_0^\dagger$ is the Hermitian operator of $P_0$. In the effective Hamiltonian, we consider only the first-order correction in the perturbation theory, which is

$$\begin{aligned} H^\mathrm{eff}=H_0^\mathrm{eff} +H_1^\mathrm{eff} =P_0^\dagger H^\mathrm{B}P_0 +P_0^\dagger H^\mathrm{BB}P_0 . \end{aligned}$$

(15)

At the same time, the projection operator $P_0$ can be put in a factorized form

$$\begin{aligned} P_0 =\prod _{i=1}^{N/3} {P_0^L } , \end{aligned}$$

(16)

where the specific form of $P_0^L$ is

$$\begin{aligned} P_0^L =\left| \Uparrow \right\rangle _L \left\langle {\varphi _0 } \right| +\left| \Downarrow \right\rangle _L \left\langle {\varphi _0 } \right| , \end{aligned}$$

(17)

where $\left| \Uparrow \right\rangle _L$ and $\left| \Downarrow \right\rangle _L$ are the renamed states of the $L\hbox {th}$ block to represent the effective site degrees of freedom. Then, the Pauli matrices in the effective Hilbert space have the following transformations:

$$\begin{aligned} P_0^L \sigma _{L,1}^x P_0^L= & {} \frac{2D}{q}{\sigma }_L^{'y} ,\quad P_0^L \sigma _{L,2}^x P_0^L =\frac{4D^{2}}{q\left( {q+1} \right) }{\sigma }_L^{'x} ,\quad P_0^L \sigma _{L,3}^x P_0^L =-\frac{2D}{q}{\sigma }_L^{'y} ,\nonumber \\ P_0^L \sigma _{L,1}^y P_0^L= & {} -\frac{2D}{q}{\sigma }_L^{'x} ,\quad P_0^L \sigma _{L,2}^y P_0^L =\frac{4D^{2}}{q\left( {q+1} \right) }{\sigma }_L^{'y} ,\quad P_0^L \sigma _{L,3}^y P_0^L =\frac{2D}{q}{\sigma }_L^{'x} ,\nonumber \\ P_0^L \sigma _{L,1}^z P_0^L= & {} \frac{1+q}{2q}{\sigma }_L^{'z} ,\quad P_0^L \sigma _{L,2}^z P_0^L =-\frac{1}{q}{\sigma }_L^{'z} ,\quad P_0^L \sigma _{L,3}^z P_0^L =\frac{1+q}{2q}{\sigma }_L^{'z} , \end{aligned}$$

(18)

Then, we can obtain the effective Hamiltonian of renormalized Ising model, which is similar to the initial one with the sign of DM interaction changed.

Similarly, to renormalize the one-dimensional anisotropic XXZ model, we still choose three sites as a block. The block Hamiltonian has two degenerate ground states as follows

$$\begin{aligned} \left| {\phi _0 } \right\rangle= & {} \frac{1}{\sqrt{2q\left( {q+\Delta } \right) \left( {1+D^{2}} \right) }}\left\{ 2\left( {D^{2}+1} \right) \left| {\downarrow \downarrow \uparrow } \right\rangle -\left( {1-iD} \right) \left( {q+\Delta } \right) \left| {\downarrow \uparrow \downarrow } \right\rangle \right. \nonumber \\&\left. -2\left[ {2iD+\left( {D^{2}-1} \right) } \right] \left| {\uparrow \downarrow \downarrow } \right\rangle \right\} \end{aligned}$$

(19)

$$\begin{aligned} \left| {\phi _0^{\prime } } \right\rangle= & {} \frac{1}{\sqrt{2q\left( {q+\Delta } \right) \left( {1+D^{2}} \right) }}\left\{ 2\left( {D^{2}+1} \right) \left| {\downarrow \uparrow \uparrow } \right\rangle -\left( {1-iD} \right) \left( {q+\Delta } \right) \left| {\uparrow \downarrow \uparrow } \right\rangle \right. \nonumber \\&\left. -2\left[ {2iD+\left( {D^{2}-1} \right) } \right] \left| {\uparrow \uparrow \downarrow } \right\rangle \right\} \end{aligned}$$

(20)

with $q=\sqrt{\Delta ^{2}+8\left( {D^{2}+1} \right) }$. Using the similar processing implemented in the Ising model, we can also achieve the renormalization of XXZ model.