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In this work, motivated by the sine-square deformation (SSD) for(1+1)-dimensional quantum critical systems, we study the non-equilibriumquantum dynamics of a conformal field theory (CFT) with SSD, which was recentlyproposed to have continuous energy spectrum and continuous Virasoro algebra. Inparticular, we study the time evolution of entanglement entropy after a quantumquench from a uniform CFT, which is defined on a finite space of length $L$, toa sine-square deformed CFT. We find there is a crossover time $t^{\ast}$ thatdivides the entanglement evolution into two interesting regions. For $t\llt^{\ast}$, the entanglement entropy does not evolve in time; for $t\ggt^{\ast}$, the entanglement entropy grows as $S_A(t)\simeq \frac{c}{3}\log t$,which is independent of the lengths of the subsystem and the total system. This$\log t$ growth with no revival indicates that a sine-square deformed CFTeffectively has an infinite length, in agreement with previous studies based onthe energy spectrum analysis. Furthermore, we study the quench dynamics for aCFT with M$\ddot{\text{o}}$bius deformation, which interpolates between auniform CFT and a sine-square deformed CFT. The entanglement entropy oscillatesin time with period $L_{\text{eff}}=L\cosh(2\theta)$, with $\theta=0$corresponding to the uniform case and $\theta\to \infty$ corresponding to theSSD limit. Our field theory calculation is confirmed by a numerical study on a(1+1)-d critical fermion chain.

Quantum dynamics in sine-square deformed conformal field theory: Quench from uniform to non-uniform CFTs