Universal scaling for a quantum discontinuity critical point and quantum quenches

We introduce the notion of a quantum version of a discontinuity critical point (DCP) in the context of a first-order quantum phase transition referred to as a quantum DCP (QDCP). We evaluate the critical exponents associated with such a QDCP using the appropriate scaling relations focusing mainly on a QDCP that is located at the boundary between a gapped phase and a gapless critical line as observed in a spin-$\frac{1}{2}$ XXZ chain while we also touch upon the conventional Ising case. We then study temporal and spatial quenches in the vicinity of the QDCP associated with the XXZ chain and propose the scaling relation for the defect density and the residual energy characterized by associated critical exponents. These predictions are numerically verified for quenches of the XXZ chain to the QDCP (or across it), thereby establishing the existence of a Kibble-Zurek scaling for such quenches given in terms of appropriate critical exponents. Finally, in the Appendix we consider a generic situation which reduces to the case when the QDCP separates two gapped phases and also to that occurring in the XXZ chain, and present detail analysis of the universality and scaling relations concerning quantum quenches.

Figure 1

Schematic representation of two possible situations associated with a discontinuous transition. For a classical phase transition, temperature ($T$) is varied; for a quantum phase transition (at $T=0$), $\mathrm{\Gamma }$ tunes the strength of quantum fluctuations. The parameter $h$ denotes the strength of the symmetry-breaking field in both classical and quantum cases. The bottom figure corresponds to the situation that represents a DCP.

Figure 2

The phase diagram of the Hamiltonian (9) with $h_i=0, \forall i$; the gapless region lying between $-1<\mathrm{\Delta }\le 1$ is the Tomonaga-Luttinger liquid (TLL) phase. The transition between the gapless phase and the antiferromagnetically (AFM) ordered phase at $\mathrm{\Delta }=1$ is Berezinski-Kosterlitz-Thouless (BKT) transition point while the critical point separating the gapless phase from the ferromagnetic phase at $\mathrm{\Delta }=-1$ is a quantum discontinuity critical point (QDCP).

Figure 3

Residual energy per spin of the XXZ chain after a temporal quench $\mathrm{\Delta }\left(t\right)=-t/t_Q$ with time $t$ from 0 to $t_Q$ and rate $1/t_Q$ in the absence of a magnetic field. Results of DMRG for sizes $L=100$ and 200 and a fitting line are shown. Fitting is done by assuming a function $a{t}_Q^{-b}$ with the data of $L=100$ and $t_Q$'s from 10 to 100 and yields $a=0.0276$ and $b=0.779$.

Figure 4

The scaling of ${\varepsilon }_{\mathrm{res}}$ for a sudden quench of small magnitude $\delta$ starting from the gapless phase and ending at the QDCP, obtained by the static DMRG of the XXZ chain. We have ${\varepsilon }_{\mathrm{res}}\sim {\delta }^{(d+z){\nu }_{+}}$ with $\nu =\frac{1}{2}$ and $z=2$.

Figure 5

Local magnetization $\langle {\sigma }_i^z\rangle$ obtained by the static DMRG of the XXZ chain with $\mathrm{\Delta }=-1$ and with a sloped magnetic field $h=(i-1/2-L/2)/x_Q$. Only the results for $L=200$ are shown. We note that the difference between the results for $L=100$ and 200 are invisibly small. The abscissa indicates the position scaled by $x_Q^{1/3}$. Results for several $x_Q$ collapse into a single curve thereby confirming the predicted scaling.