One-way deficit and quantum phase transitions in XY model and extended Ising model

Abstract

Originating in questions regarding work extraction from quantum systems coupled to a heat bath, the quantum deficit, a kind of quantum correlations in addition to entanglement and quantum discord, links quantum thermodynamics with quantum information theory. In this paper, we investigate the one-way deficit of two adjacent spins in the bulk of the XY model and the extended Ising model. We find that the one-way deficit susceptibility is able to characterize quantum phase transitions in the XY model and even topological phase transitions in the extend Ising model. This study will enlighten extensive studies of quantum phase transitions from the perspective of quantum information processing and quantum computation, including finite-temperature phase transitions, topological phase transitions, and dynamical phase transitions in a variety of quantum many-body systems.

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Acknowledgements

We would like to thank Yu Zeng, Jin-Jun Chen, and Wei Qin for useful discussions. This work was supported by Ministry of Science and Technology of China (Grants Nos. 2016YFA0302104 and 2016YFA0300600), National Natural Science Foundation of China (Grants Nos. 91536108, 11774406, and U1530401), and Strategic Priority Research Program of Chinese Academy of Sciences (Grant No. XDB28000000).

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Correspondence to Yu-Ran Zhang.

Appendices

Appendix A: One-way deficit of X states

In this section, we evaluate the one-way deficit of X states in Eq. (3). Let \(\{\Pi _{k}=|k\rangle \langle k|, k=0, 1\}\) be the local measurement on the party b along the computational base \({|k\rangle }\); then any von Neumann measurement for the party b can be written as

$$\begin{aligned} \{B_{k}=V\Pi _{k}V^{\dag }: k=0, 1\} \end{aligned}$$
(A1)

given some unitary operator \(V\in U(2)\). For any V,

$$\begin{aligned} V=tI+i\vec {y}\cdot \vec {\sigma }=\left( \begin{array}{cc} t+y_{3}i &{} y_{2}+y_{1}i \\ -y_{2}+y_{1}i &{} t-y_{3}i \\ \end{array} \right) , \end{aligned}$$
(A2)

with \(t\in \mathbb {R}\), \(\vec {y}=(y_{1}, y_{2}, y_{3})\in \mathbb {R}^{3}\), and

$$\begin{aligned} t^{2}+y_{1}^{2}+y_{2}^{2}+y_{3}^{2}=1, \end{aligned}$$
(A3)

after the measurement \({B_{k}}\), the state \(\rho ^{ab}\) will be changed into the ensemble \(\{{\rho _{k}, p_{k}}\}\) with

$$\begin{aligned} \rho _{k}=\frac{1}{p_{k}}(I\otimes B_{k})\rho (I\otimes B_{k}), \end{aligned}$$
(A4)
$$\begin{aligned} p_{k}=\text {Tr}(I\otimes B_{k})\rho (I\otimes B_{k}). \end{aligned}$$
(A5)

To evaluate \(\rho _{k}\) and \(p_{k}\), we write

$$\begin{aligned} p_{k}\rho _{k}= & {} (I\otimes B_{k})\rho (I\otimes B_{k})\nonumber \\= & {} \frac{1}{4}(I\otimes V)(I\otimes \Pi _{k})[I+r\sigma _{3}\otimes I+sI\otimes V^{\dag }\sigma _{3}V^{\dag }\nonumber \\&+\sum _{j=1}^3 c_{j}\sigma _{j}\otimes (V^{\dag } \sigma _{j} V)](I\otimes \Pi _{k})(I\otimes V^{\dag }). \end{aligned}$$
(A6)

Using the relations [46]

$$\begin{aligned} V^{\dag }\sigma _{1}V= & {} (t^{2}+y_{1}^{2}-y_{2}^{2}-y_{3}^{2})\sigma _{1}+2(ty_{3}+y_{1}y_{2})\sigma _{2}\nonumber \\&+2(-ty_{2}+y_{1}y_{3})\sigma _{3}, \end{aligned}$$
(A7)
$$\begin{aligned} V^{\dag }\sigma _{2}V= & {} 2(-ty_{3}+y_{1}y_{2})\sigma _{1}+(t^{2}+y_{2}^{2}-y_{1}^{2}-y_{3}^{2})\sigma _{2}\nonumber \\&+2(ty_{1}+y_{2}y_{3})\sigma _{3}, \end{aligned}$$
(A8)
$$\begin{aligned} V^{\dag }\sigma _{3}V= & {} 2(ty_{2}+y_{1}y_{3})\sigma _{1}+2(-ty_{1}+y_{2}y_{3})\sigma _{2}\nonumber \\&+(t^{2}+y_{3}^{2}-y_{1}^{2}-y_{2}^{2})\sigma _{3}, \end{aligned}$$
(A9)

and

$$\begin{aligned} \Pi _{0}\sigma _{3}\Pi _{0}=\Pi _{0}, \Pi _{1}\sigma _{3}\Pi _{1}=-\Pi _{1},\Pi _{j}\sigma _{k}\Pi _{j}=0, \end{aligned}$$
(A10)

for \(j=0, 1, k=1, 2\), we obtain

$$\begin{aligned} p_{0}\rho _{0}= & {} \frac{1}{4}[I+sz_{3}I+c_{1}z_{1}\sigma _{1}+c_{2}z_{2}\sigma _{2}+(r+c_{3}z_{3})\sigma _{3}]\nonumber \\&\otimes (V\Pi _{0}V^{\dag }), \end{aligned}$$
(A11)
$$\begin{aligned} p_{1}\rho _{1}= & {} \frac{1}{4}[I-sz_{3}I-c_{1}z_{1}\sigma _{1}-c_{2}z_{2}\sigma _{2}+(r-c_{3}z_{3})\sigma _{3}]\nonumber \\&\otimes (V\Pi _{1}V^{\dag }), \end{aligned}$$
(A12)

where

$$\begin{aligned} z_{1}= & {} 2(-ty_{2}+y_{1}y_{3}),\ z_{2}=2(ty_{1}+y_{2}y_{3}), \end{aligned}$$
(A13)
$$\begin{aligned} z_{3}= & {} t^{2}+y_{3}^{2}-y_{1}^{2}-y_{2}^{2}.\ \ \ \ \ \ \ \ \ \ \ \ \ \end{aligned}$$
(A14)

Then, we will evaluate the eigenvalues of \(\sum _{k}\Pi _{k}\rho ^{ab}\Pi _{k}\) by \(\sum _{k}\Pi _{k}\rho ^{ab}\Pi _{k}=p_{0}\rho _{0}+p_{1}\rho _{1}\), and

$$\begin{aligned} p_{0}\rho _{0}+p_{1}\rho _{1}&=\frac{1}{4}(I+r\sigma _{3})\otimes I\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \nonumber \\&\quad +\frac{1}{4}(sz_{3}I+c_{1}z_{1}\sigma _{1}+c_{2}z_{2}\sigma _{2}+c_{3}z_{3}\sigma _{3}) \otimes V\sigma _{3}V^{\dag }. \end{aligned}$$
(A15)

The eigenvalues of \(p_{0}\rho _{0}+p_{1}\rho _{1}\) are the same with the eigenvalues of the states \((I\otimes V^{\dag })(p_{0}\rho _{0}+p_{1}\rho _{1})(I\otimes V)\), and

$$\begin{aligned}&(I\otimes V^{\dag })(p_{0}\rho _{0}+p_{1}\rho _{1})(I\otimes V)\nonumber \\&\quad =\frac{1}{4}(sz_{3}I+c_{1}z_{1}\sigma _{1}+c_{2}z_{2}\sigma _{2}+c_{3}z_{3}\sigma _{3})\otimes \sigma _{3}\nonumber \\&\qquad +\frac{1}{4}(I+r\sigma _{3})\otimes I. \end{aligned}$$
(A16)

The eigenvalues of the states in Eq. (A16) are given in Eqs. (8,9). Thus, the entropy of \(\sum _{k}\Pi _{k}\rho ^{ab}\Pi _{k}\) is

$$\begin{aligned} S\left( \sum _{k}\Pi _{k}\rho ^{ab}\Pi _{k}\right) =-\sum _{i=1}^{4}w_{i}\log w_{i}. \end{aligned}$$
(A17)

When \(\gamma \) and h are fixed, \(r, s, c_{1}, c_{2}\), and \(c_{3}\) are constant. By using \(z_{1}^{2}+z_{2}^{2}+z_{3}^{2}=1\), it converts the problem about \(\min _{\{\Pi _{k}\}} S(\sum _{k}\Pi _{k}\rho ^{ab}\Pi _{k})\) to the problem about the function of three variables \(z_{1},z_{2},z_{3}\) for minimum, that is,

$$\begin{aligned} \min _{\{\Pi _{k}\}} S\left( \sum \limits _{k}\Pi _{k}\rho ^{ab}\Pi _{k}\right) =\!\!\!\!\min \limits _{\{z_{1}^{2}+z_{2}^{2}+z_{3}^{2}=1\}}\!\!\!\!S\left( \sum \limits _{k}\Pi _{k}\rho ^{ab}\Pi _{k}\right) . \end{aligned}$$
(A18)

Therefore, by Eqs. (1), (6), and (A18), the one-way deficit of X states in Eq. (3) is obtained as shown in Eq. (7).

Appendix B: Diagonalization, winding numbers and characteristic functions of extended Ising model

The Hamiltonian of the extended Ising model (10) can be mapped to a spinless fermion Hamiltonian by the Jordan-Wigner transformation \(-c_1=\sigma _1^+=(\sigma ^x_1+i\sigma ^y_1)/2\), \(-c_j=\sigma _j^+\prod _{i=1}^{j-1}\sigma ^z_i\) [30]. In the thermodynamic limit \(L\gg 1\), we can use the Bogoliubov–Fourier transformation to obtain a Bogoliubov–de Gennes (BdG) Hamiltonian as [30]

$$\begin{aligned} H=\sum _{\phi }(c_\phi ^\dag \ c_{-\phi })\mathcal {H}_\phi \left( \begin{array}{c}{c_\phi }\\ {c_{-\phi }^\dag }\end{array}\right) , \end{aligned}$$
(B1)

where the complete set of wavevectors is \(\phi =2\pi m/L\) with \(m=-\frac{L-1}{2},-\frac{L-3}{2},\cdots ,\frac{L-3}{2},\frac{L-1}{2}\). Here, we can write [30]

$$\begin{aligned} \mathcal {H}_\phi =\varvec{r}(\phi )\cdot \varvec{\sigma }, \end{aligned}$$
(B2)

with the vector \(\varvec{r}(\phi )=(0~Y(\phi )~Z(\phi ))\) in the auxiliary two-dimensional \(y-z\) space,

$$\begin{aligned} Y(\phi )=\lambda \delta \sin (2\phi )+\gamma \sin \phi , \end{aligned}$$
(B3)
$$\begin{aligned} Z(\phi )=\lambda \cos (2\phi )+\cos \phi -h, \end{aligned}$$
(B4)

and \(\varvec{\sigma }=(\sigma _1~\sigma _2~\sigma _3)\). The winding number of the closed loop in auxiliary \(y-z\) plane around the origin point can be written as

$$\begin{aligned} \nu =\frac{1}{2\pi }\oint (YdZ-ZdY)/|\varvec{r}|^2, \end{aligned}$$
(B5)

which is used to identify different topological phases in the BDI class one-dimensional fermion systems [47].

Using the Bogoliubov transformation \(c_\phi =\cos \frac{\Theta }{2}\eta _\phi +i\sin \frac{\Theta }{2}\eta _{-\phi }^\dag \) with \(\tan \Theta \equiv Y(\phi )/Z(\phi )\), we can diagonalize the Hamiltonian as

$$\begin{aligned} {H}=\sum _\phi \omega _\phi (\eta _\phi ^\dag \eta _\phi -{1}/{2}) \end{aligned}$$
(B6)

and obtain the ground state as

$$\begin{aligned} |\mathcal {G}\rangle =\prod _{\phi }\left[ \cos \frac{\Theta }{2}+i\sin \frac{\Theta }{2}\eta _\phi ^{\dag }\eta _{-\phi }^\dag \right] |0\rangle , \end{aligned}$$
(B7)

where \(|0\rangle \) is the vacuum state and the energy spectra are

$$\begin{aligned} \omega _\phi =\sqrt{Y(\phi )^2+Z(\phi )^2}. \end{aligned}$$
(B8)

Via a substitute \(\zeta (\phi )\equiv \exp (i\phi )\), the characteristic function is defined as [29]

$$\begin{aligned}&g(\zeta )\equiv Z(\phi )+iY(\phi ), \end{aligned}$$
(B9)
$$\begin{aligned}&=\lambda [\zeta ^2+(1-\delta )\zeta ^{-2}/2]+\zeta +(1-\gamma )\zeta ^{-1}/2-h, \end{aligned}$$
(B10)

with which we can calculate the critical points for the quantum topological phase transitions by the characteristic equation \(g(\zeta )=0\) with \(|\zeta |=1\) required.

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Wang, Y., Zhang, Y. & Fan, H. One-way deficit and quantum phase transitions in XY model and extended Ising model. Quantum Inf Process 18, 19 (2019). https://doi.org/10.1007/s11128-018-2132-2

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Keywords

  • One-way deficit
  • Quantum phase transition
  • Symmetry-protected topological order