Renormalization of magic and quantum phase transition in spin models

Abstract

The interplay of quantum information theory and condensed matter physics has generated fruitful results, which promote our understanding of quantum phase transition. Magic, as a crucial resource in fault-tolerant quantum computation, may provide new insights into quantum phase transition. Based on quantum renormalization group method, we investigate the role magic plays in detecting quantum phase transition in the one-dimensional anisotropic XXZ model and XY model. The quantifier of magic we employed is defined via the characteristic functions of quantum states; it not only has nice properties, but also can be straightforwardly calculated. As the iteration steps of quantum renormalization group increase in the spin models, the magic quantifier achieves its maximum, while the first-order derivative of the magic is discontinuous around the critical points, which are signatures of quantum phase transition. The scaling behavior of the renormalized magic in terms of the system size is demonstrated, and a comparative study with coherence is made.

Appendices

Appendix A: Renormalization of XXZ spin model under QRG method

The Hamiltonian of the one-dimensional anisotropic XXZ model on a periodic chain of N sites is [10]

$$\begin{aligned} H(J,\Delta )=\frac{J}{4}\sum _{k=1}^{N}(\sigma ^{x}_{k}\sigma ^{x}_{k+1}+\sigma ^{y}_{k}\sigma ^{y}_{k+1} +\Delta \sigma ^{z}_{k}\sigma ^{z}_{k+1}), \end{aligned}$$

where $J,\Delta > 0$, J is the exchange constant, $\Delta$ is the anisotropy parameter, and $\sigma ^{\alpha }_{k}(\alpha =x,y,z)$ denotes Pauli matrices at site k. The spin chain is divided into three-site blocks [40] (by the Kadanoff’s block approach given in Fig. 7) to get a self-similar Hamiltonian

$$\begin{aligned} h^I_l=\frac{J}{4}[\sigma ^{x}_{l,1}\sigma ^{x}_{l,2}+\sigma ^{x}_{l,2}\sigma ^{x}_{l,3}+\sigma ^{y}_{l,1}\sigma ^{y}_{l,2} +\sigma ^{y}_{l,2}\sigma ^{y}_{l,3}+\Delta (\sigma ^{z}_{l,1}\sigma ^{z}_{l,2}+\sigma ^{z}_{l,2}\sigma ^{z}_{l,3})], \end{aligned}$$

where $\sigma ^{\alpha }_{l,j}$ refers to the corresponding Pauli matrix $\alpha$ ($\alpha =x,y,z$) at site j (j=1,2,3) of the l-th block.

The intra-block $H^I$ and the inter-block $H^{II}$ Hamiltonian are

$$\begin{aligned} H^I=\sum _{l}^{N/3}h^I_l,\qquad H^{II}=\frac{J}{4}\sum _{l}^{N/3}(\sigma ^{x}_{l,3}\sigma ^{x}_{l+1,1}+\sigma ^{y}_{l,3}\sigma ^{y}_{l+1,1}+\Delta \sigma ^{z}_{l,3}\sigma ^{z}_{l+1,1}), \end{aligned}$$

respectively.

Fig. 7

Using Kadanoff’s block approach, the Hamiltonian of the system in one-dimensional XXZ chain is divided into intra-block Hamiltonian $H^I$ and inter-block Hamiltonian $H^{II}$. In this way, an effective Hamiltonian $H^\mathrm{{eff}}$ can be obtained by projecting the Hamiltonian H onto the renormalized space

The two degenerate ground states are

$$\begin{aligned} |\phi _0\rangle= & {} \frac{1}{\sqrt{2+q^2}}(|\uparrow \uparrow \downarrow \rangle +q|\uparrow \downarrow \uparrow \rangle +|\downarrow \uparrow \uparrow \rangle ),\\ |\phi ^{'}_0\rangle= & {} \frac{1}{\sqrt{2+q^2}}(|\uparrow \downarrow \downarrow \rangle +q|\downarrow \uparrow \downarrow \rangle +|\downarrow \downarrow \uparrow \rangle ) \end{aligned}$$

where $q=-\frac{1}{2}(\Delta +\sqrt{\Delta ^{2}+8})$, $|\uparrow \rangle$ and $|\downarrow \rangle$ are the eigenstates of $\sigma ^{z}$.

The projection operator $P_{0}$ is expressed as

$$\begin{aligned} P_0=\prod \limits _{l}^{N/3}P^l_0, \end{aligned}$$

where $P^l_0$ is the projection operator $P_{0}$ for the l-th block, which is defined as

$$\begin{aligned} P^l_0=|\Uparrow \rangle _l\langle \phi _0|+|\Downarrow \rangle _l\langle \phi ^{'}_0|, \end{aligned}$$

with $|\Uparrow \rangle _l$ and $|\Downarrow \rangle _l$ being the renamed states of each block to represent the effective site degrees of freedom. The renormalization of Pauli matrices can be calculated as

$$\begin{aligned} P^l_0\sigma ^{\alpha }_{l,i}P^l_0=\xi ^{\alpha }_l\sigma ^{'\alpha }_l\quad \quad ( i=1,2,3; \alpha =x,y,z) \end{aligned}$$

with

$$\begin{aligned} \xi ^{x,y}_1=\xi ^{x,y}_3=\frac{2q}{2+q^2},\quad \xi ^{x,y}_2=\frac{2}{2+q^2},\quad \xi ^{z}_1=\xi ^{z}_3=\frac{q^2}{2+q^2},\quad \xi ^{z}_2=\frac{2-q^2}{2+q^2}, \end{aligned}$$

where the indices 1,2,3 refer to the first, second and third site of each block.

The effective Hamiltonian $H^\mathrm{{eff}}$ can be written as

$$\begin{aligned} H^\mathrm{{eff}}= H^\mathrm{{eff}}_0+H^\mathrm{{eff}}_1=P_0H^IP_0+P_0H^{II}P_0, \end{aligned}$$

from which we obtain

$$\begin{aligned} H^\mathrm{{eff}}= \frac{J^{'}}{4}\sum ^{N/3}_{k=1}(\sigma ^{x}_{k}\sigma ^{x}_{k+1} +\sigma ^{y}_{k}\sigma ^{y}_{k+1}+\Delta ^{'}\sigma ^{z}_{k}\sigma ^{z}_{k+1}) \end{aligned}$$

with

$$\begin{aligned} J^{'}=J{(\frac{2q}{2+q^2})}^2,\qquad \Delta ^{'}=\Delta \frac{q^2}{4}. \end{aligned}$$

Appendix B: Calculation of the magic $M(\rho )$ on the ground state in XXZ model

The density matrix of the ground state in XXZ model is expressed as

$$\begin{aligned} \begin{aligned} \rho =|\phi _0\rangle \langle \phi _0|=&\tfrac{1}{2+q^2}(|\uparrow \rangle \langle \uparrow |\otimes |\uparrow \rangle \langle \uparrow |\otimes |\downarrow \rangle \langle \downarrow |\\&+q|\uparrow \rangle \langle \uparrow |\otimes |\uparrow \rangle \langle \downarrow |\otimes |\downarrow \rangle \langle \uparrow |+|\uparrow \rangle \langle \downarrow |\otimes |\uparrow \rangle \langle \uparrow |\otimes |\downarrow \rangle \langle \uparrow |\\&+q|\uparrow \rangle \langle \uparrow |\otimes |\downarrow \rangle \langle \uparrow |\otimes |\uparrow \rangle \langle \downarrow |+q^2|\uparrow \rangle \langle \uparrow |\otimes |\downarrow \rangle \langle \downarrow |\otimes |\uparrow \rangle \langle \uparrow |\\&+q|\uparrow \rangle \langle \downarrow |\otimes |\downarrow \rangle \langle \uparrow |\otimes |\uparrow \rangle \langle \uparrow |+|\downarrow \rangle \langle \uparrow |\otimes |\uparrow \rangle \langle \uparrow |\otimes |\uparrow \rangle \langle \downarrow |\\&+q|\downarrow \rangle \langle \uparrow |\otimes |\uparrow \rangle \langle \downarrow |\otimes |\uparrow \rangle \langle \uparrow |+|\downarrow \rangle \langle \downarrow |\otimes |\uparrow \rangle \langle \uparrow |\otimes |\uparrow \rangle \langle \uparrow |). \end{aligned} \end{aligned}$$

(B1)

The magic in the three-site block can be written as

$$\begin{aligned} M(\rho )=\sum _{\varvec{k} }|\textrm{tr} (\rho \sigma ^{\varvec{k}} )|=\sum _{k_1,k_2,k_3} |\textrm{tr} (\rho \sigma ^{k_1}\otimes \sigma ^{k_2}\otimes \sigma ^{k_3})| \end{aligned}$$

(B2)

where $k_j=0,x,y,z (j=1,2,3)$. Substituting the quantum state in Eq. (B1) into Eq. (B2), we can straightforwardly calculate the trace of the product between $\rho$ and each of the 64 combinations $\sigma ^{k_1}\otimes \sigma ^{k_2}\otimes \sigma ^{k_3}$, respectively. For example,

$$\begin{aligned} {\begin{matrix} |\textrm{tr} (\rho \sigma ^{0}\otimes \sigma ^{0}\otimes \sigma ^{0})|&{} =|\textrm{tr} (\rho \textbf{1}\otimes \textbf{1}\otimes \textbf{1})|=\tfrac{1}{2+q^2}|\langle \uparrow |\textbf{1}|\uparrow \rangle \times \langle \uparrow |\textbf{1}|\uparrow \rangle \times \langle \downarrow |\textbf{1}|\downarrow \rangle \\ &{}\quad +q\langle \uparrow |\textbf{1}|\uparrow \rangle \times \langle \uparrow |\textbf{1}|\downarrow \rangle \times \langle \downarrow |\textbf{1}|\uparrow \rangle +\langle \uparrow |\textbf{1}|\downarrow \rangle \langle \uparrow |\textbf{1}|\uparrow \rangle \times \langle \downarrow |\textbf{1}|\uparrow \rangle \\ &{}\quad +q\langle \uparrow |\textbf{1}|\uparrow \rangle \times \langle \downarrow |\textbf{1}|\uparrow \rangle \times \langle \uparrow |\textbf{1}|\downarrow \rangle +q^2\langle \uparrow |\textbf{1}|\uparrow \rangle \times \langle \downarrow |\textbf{1}|\downarrow \rangle \times \langle \uparrow |\textbf{1}|\uparrow \rangle \\ &{}\quad +q\langle \uparrow |\textbf{1}|\downarrow \rangle \times \langle \downarrow |\textbf{1}|\uparrow \rangle \times \langle \uparrow |\textbf{1}|\uparrow \rangle +\langle \downarrow |\textbf{1}|\uparrow \rangle \times \langle \uparrow |\textbf{1}|\uparrow \rangle \times \langle \uparrow |\textbf{1}|\downarrow \rangle \\ &{}\quad +q\langle \downarrow |\textbf{1}|\uparrow \rangle \times \langle \uparrow |\textbf{1}|\downarrow \rangle \times \langle \uparrow |\textbf{1}|\uparrow \rangle +\langle \downarrow |\textbf{1}|\downarrow \rangle \times \langle \uparrow |\textbf{1}|\uparrow \rangle \times \langle \uparrow |\textbf{1}|\uparrow \rangle |\\ &{}=\tfrac{1}{2+q^2}|1+q^2+1|=1. \end{matrix}} \end{aligned}$$

Similarly, another combination $|\textrm{tr} (\rho \sigma ^{z}\otimes \sigma ^{z}\otimes \sigma ^{z})|$ has the same result 1. By analogous method, we can calculate all 64 traces; adding these values together, we get the value of the magic as

$$\begin{aligned} M(\rho )=\frac{2|q^2-2|+6q^2+16|q|+12}{2+q^2}. \end{aligned}$$

Appendix C: Renormalization of XY spin model under QRG method

The Hamiltonian of the XY model on a periodic chain with N sites reads [11]

$$\begin{aligned} H(J,\gamma )=\frac{J}{4}\sum _{k=1}^{N}[(1+\gamma )\sigma ^{x}_{k}\sigma ^{x}_{k+1}+(1-\gamma )\sigma ^{y}_{k}\sigma ^{y}_{k+1}], \end{aligned}$$

(C3)

where $J>0$ and $-1<\gamma <1$, J is the exchange coupling constant, $\gamma$ is the anisotropy parameter, and $\sigma ^{\alpha }_{k}$ are Pauli matrices at site k. After one renormalization, the Hamiltonian H is divided into the intra-block $H^I$ and the inter-block $H^{II}$ Hamiltonian as

$$\begin{aligned} H^I= & {} \frac{J}{4}\sum _{l}^{N/3}[(1+\gamma )(\sigma ^{x}_{l,1}\sigma ^{x}_{l,2} +\sigma ^{x}_{l,2}\sigma ^{x}_{l,3})+(1-\gamma )(\sigma ^{y}_{l,1}\sigma ^{y}_{l,2}+\sigma ^{y}_{l,2}\sigma ^{y}_{l,3})],\\ H^{II}= & {} \frac{J}{4}\sum _{l}^{N/3}[(1+\gamma )\sigma ^{x}_{l,3}\sigma ^{x}_{l+1,1}+(1-\gamma )\sigma ^{y}_{l,3}\sigma ^{y}_{l+1,1}], \end{aligned}$$

where $\sigma ^{\alpha }_{l,j}$ refers to the $\alpha$ ($\alpha =x,y,z$) of the Pauli matrix at site j (j=1,2,3) of the l-th block Hamiltonian.

The two degenerate ground states of XY Hamiltonian are

$$\begin{aligned} |\psi _0\rangle= & {} \frac{1}{2\sqrt{1+{\gamma }^2}}(-\sqrt{1+{\gamma }^2}|\uparrow \uparrow \downarrow \rangle +\sqrt{2}|\uparrow \downarrow \uparrow \rangle -\sqrt{1+{\gamma }^2}|\downarrow \uparrow \uparrow \rangle +\sqrt{2}\gamma |\downarrow \downarrow \downarrow \rangle ),\\ |\psi ^{'}_0\rangle= & {} \frac{1}{2\sqrt{1+{\gamma }^2}}(\sqrt{1+{\gamma }^2}|\downarrow \downarrow \uparrow \rangle -\sqrt{2}|\downarrow \uparrow \downarrow \rangle +\sqrt{1+{\gamma }^2}|\uparrow \downarrow \downarrow \rangle -\sqrt{2}\gamma |\uparrow \uparrow \uparrow \rangle ). \end{aligned}$$

The projection operator $T_{0}$ can be expressed as

$$\begin{aligned} T_0=\prod \limits _{l}^{N/3}(|\Uparrow \rangle _l\langle \phi _0|+|\Downarrow \rangle _l\langle \phi ^{'}_0|), \end{aligned}$$

where $|\Uparrow \rangle _l$ and $|\Downarrow \rangle _l$ are renamed states of each block to represent the effective site degrees of freedom.

The renormalization of Pauli matrices is defined by

$$\begin{aligned} T^l_0\sigma ^{\alpha }_{l,i}T^l_0=\zeta ^{\alpha }_l\sigma ^{'\alpha }_l,\quad \quad (i=1,2,3; \alpha =x,y) \end{aligned}$$

where

$$\begin{aligned} \zeta ^{x}_1=\xi ^{x}_3=\frac{1+\gamma }{\sqrt{2(1+{\gamma }^2)}},\quad \zeta ^{x}_2=-\frac{(1+\gamma )^2}{2(1+{\gamma }^2)},\\ \quad \zeta ^{y}_1=\xi ^{y}_3=\frac{1-\gamma }{\sqrt{2(1+{\gamma }^2)}},\quad \zeta ^{y}_2=-\frac{(1-\gamma )^2}{2(1+{\gamma }^2)}. \end{aligned}$$

The effective Hamiltonian of the renormalized chain can be expressed as

$$\begin{aligned} H^\mathrm{{eff}}= \frac{J^{'}}{4}\sum ^{N/3}_{k=1}[(1+\gamma ^{'})\sigma ^{x}_{k}\sigma ^{x}_{k+1}+(1-\gamma ^{'})\sigma ^{y}_{k}\sigma ^{y}_{k+1}], \end{aligned}$$

where the parameters are

$$\begin{aligned} J^{'}=J{\frac{3\gamma ^{2}+1}{2(\gamma ^{2}+1)}},\quad \gamma ^{'}=\frac{\gamma ^{3}+3\gamma }{3\gamma ^{2}+1}. \end{aligned}$$