Periodic quenching modulated quantum phase transitions in transverse XY spin-chains

Abstract

The XY spin-chain in transverse fields under the modulation of periodic quenching has been investigated in this paper. We have obtained the exact expression of the Floquet effective Hamiltonian and analyzed the phase diagram, the corresponding critical properties, as well as geometric phase of the system. Our results show that the critical properties and phase diagram of the spin-chain can be modulated by the periodic quenching. We also considered the central spin system’s Loschmidt echo coupling to such spin-chain and showed that the Loschmidt echo can be used to detect the new critical points of the effective system.

Graphical abstract

6 Appendix

In order to obtain the effective Hamiltonian (9) in the main text, which have the following form,

$$\begin{aligned} e^{i a_1 ({\mathbf{n_1}}\cdot \sigma )}e^{i a_2({\mathbf{n_2}}\cdot \sigma )}=e^{i a (\mathbf{n}\cdot \sigma )}, \end{aligned}$$

(31)

where ${\mathbf{n_1}}, {\mathbf{n_2}}, \mathbf{n}$ are unit vectors, we use the following formulas

$$\begin{aligned}&e^{i a(\mathbf{n}\cdot \sigma )}= \cos a + i (\mathbf{n}\cdot \sigma ) \sin a, \\&({\mathbf{n_1}}\cdot \sigma )({\mathbf{n_2}}\cdot \sigma )={\mathbf{n_1}}\cdot {\mathbf{n_2}}+i ({\mathbf{n_1}}\times {\mathbf{n_2}})\cdot \sigma , \end{aligned}$$

and obtain

$$\begin{aligned}&e^{i a_1({\mathbf{n_1}}\cdot \sigma )}e^{i a_2({\mathbf{n_2}}\cdot \sigma )} \nonumber \\= & {} [\cos a_1 + i ({\mathbf{n_1}}\cdot \sigma ) \sin a_1 ][\cos a_2 + i ({\mathbf{n_2}}\cdot \sigma ) \sin a_2] \nonumber \\= & {} \cos a_1\cos a_2-({\mathbf{n_1}}\cdot \sigma )({\mathbf{n_2}}\cdot \sigma )\sin a_1\sin a_2 \nonumber \\&+i({\mathbf{n_1}}\cdot \sigma )\sin a_1\cos a_2+i({\mathbf{n_2}}\cdot \sigma )\cos a_1\sin a_2 \nonumber \\= & {} \cos a_1\cos a_2-{\mathbf{n_1}}\cdot {\mathbf{n_2}}\sin a_1\sin a_2 \nonumber \\&-i \sin a_1\sin a_2({\mathbf{n_1}}\times {\mathbf{n_2}})\cdot \sigma \nonumber \\&+ i({\mathbf{n_1}}\cdot \sigma )\sin a_1\cos a_2+i({\mathbf{n_2}}\cdot \sigma )\cos a_1\sin a_2. \end{aligned}$$

(32)

By comparing the real part and the image part with the equation $e^{i a(\mathbf{n}\cdot \sigma )}= \cos a + i (\mathbf{n}\cdot \sigma ) \sin a$, we will obtain the relations between the parameters, i.e.,

$$\begin{aligned}&\cos a=\cos a_1 \cos a_2- {\mathbf{n_1}}\cdot {\mathbf{n_2}}\sin a_1\sin a_2, \nonumber \\&{\mathbf{n}}=\frac{1}{\sin a} ({\mathbf{n_1}}\sin a_1\cos a_2 +{\mathbf{n_2}}\sin a_2\cos a_1\nonumber \\&\quad - {\mathbf{n_1}}\times {\mathbf{n_2}}\sin a_1\sin a_2 ). \end{aligned}$$

(33)

In the main text, we can re-express the Hamiltonian as

$$\begin{aligned} {{{\mathcal {H}}}}_{0}(\lambda _{1})=\varepsilon _{k,1}\mathbf{\sigma }\cdot \mathbf{n_{1}}.\nonumber \\ {{{\mathcal {H}}}}_{0}(\lambda _{2})=\varepsilon _{k,2}\mathbf{\sigma }\cdot \mathbf{n_{2}}. \end{aligned}$$

(34)

with

$$\begin{aligned} \varepsilon _{k,1} = \sqrt{(\lambda _1-\cos k)^2+\gamma ^2\sin ^2k } , \nonumber \\ \varepsilon _{k,2} = \sqrt{(\lambda _2-\cos k)^2+\gamma ^2\sin ^2k } . \end{aligned}$$

(35)

and

$$\begin{aligned} {\mathbf{n_{1}}}=\left( 0, \frac{-\gamma \sin k}{ \varepsilon _{k,1}}, \frac{\lambda _1-\cos k}{ \varepsilon _{k,1}}\right) , \nonumber \\ {\mathbf{n_{2}}}=\left( 0, \frac{-\gamma \sin k}{ \varepsilon _{k,2}}, \frac{\lambda _2-\cos k}{ \varepsilon _{k,2}}\right) , \end{aligned}$$

(36)

then, substitute this expression into formula (33), one can obtain the values of parameters $\varepsilon _k$ and $\mathbf {n}$ in the effective Hamiltonian $\mathcal{H}_{\text {eff}}=\xi _{k}{\varvec{\sigma }}\cdot \mathbf{n}=\mathcal{H}_{\text {eff},x}\sigma _x+{{{\mathcal {H}}}}_{\text {eff},y}\sigma _y+\mathcal{H}_{\text {eff},z}\sigma _z$, i.e.,

$$\begin{aligned} \xi _{k}= & {} \frac{1}{T} \arccos \Big [ \cos \varepsilon _{k,1}T_1\cos \varepsilon _{k,2}T_2 \nonumber \\&- \sin \varepsilon _{k,1}T_1\sin \varepsilon _{k,2}T_2 \nonumber \\&\times \frac{\gamma ^2\sin ^2k+(\lambda _{1}-\cos k)(\lambda _{2}-\cos k)}{\varepsilon _{k,1}\varepsilon _{k,2}} \Big ] , \nonumber \\ {{{\mathcal {H}}}}_{\text {eff},x}= & {} \frac{\xi _k}{\sin {\xi _{k}T}}\Big [ {-}\sin \varepsilon _{k,1}T_1 \sin \varepsilon _{k,2}T_2 \nonumber \\&\frac{\gamma \sin k (\lambda _1-\lambda _2)}{\varepsilon _{k,1}\varepsilon _{k,2}}\Big ], \nonumber \\ {{{\mathcal {H}}}}_{\text {eff},y}= & {} \frac{\xi _k}{\sin {\xi _{k}T}}\Big [\sin \varepsilon _{k,1}T_1 \cos \varepsilon _{k,2}T_2\frac{\gamma \sin k}{\varepsilon _{k,1}}\nonumber \\&+\sin \varepsilon _{k,2}T_2 \cos \varepsilon _{k,1}T_1\frac{\gamma \sin k}{\varepsilon _{k,2}}\Big ],\nonumber \\ {{{\mathcal {H}}}}_{ \text {eff},z}= & {} \frac{\xi _k}{\sin {\xi _{k}T}}\Big [{-} \sin \varepsilon _{k,1}T_1 \cos \varepsilon _{k,2}T_2 \frac{\lambda _1-\cos k}{\varepsilon _{k,1}}\nonumber \\&- \cos \varepsilon _{k,1}T_1 \sin \varepsilon _{k,2}T_2 \frac{\lambda _2-\cos k}{\varepsilon _{k,2}} \Big ] . \nonumber \\ \end{aligned}$$

(37)