Dynamical quantum phase transitions and the Loschmidt echo: A transfer matrix approach

A boundary transfer matrix formulation allows to calculate the Loschmidt echo for one-dimensional quantum systems in the thermodynamic limit. We show that nonanalyticities in the Loschmidt echo and zeros for the Loschmidt amplitude in the complex plane (Fisher zeros) are caused by a crossing of eigenvalues in the spectrum of the transfer matrix. Using a density-matrix renormalization group algorithm applied to these transfer matrices, we numerically investigate the Loschmidt echo and the Fisher zeros for quantum quenches in the XXZ model with a uniform and a staggered magnetic field. We give examples—both in the integrable and the nonintegrable cases—where the Loschmidt echo does not show nonanalyticities although the quench leads across an equilibrium phase transition, and examples where nonanalyticities appear for quenches within the same phase. For a quench to the free fermion point, we analytically show that the Fisher zeros sensitively depend on the initial state and can lie exactly on the real axis already for finite system size. Furthermore, we use bosonization to analyze our numerical results for quenches within the Luttinger liquid phase.

Figure 1

(a) Torus geometry obtained after mapping a one-dimensional quantum system at finite temperatures with periodic boundary conditions to a two-dimensional classical model. (b) Cylinder geometry required to calculate the Loschmidt amplitude. The upper and lower ends of the cylinder are fixed by the initial state $|{\Psi }_0\rangle$.

Figure 2

Lines of Fisher zeros in the complex plane, $z=R+it$, for a quench $g=0.4\to 1.3$ in the transverse Ising model (2.18). The exact results [11] (lines) are compared with numerical scans through the complex plane using the LCRG algorithm (circles).

Figure 3

Schematic phase diagrams for the Hamiltonian (3.1) for (a) the integrable case $h_{\mathrm{st}}=0$, and (b) the nonintegrable case with staggered field but $h=0$. The phase diagrams consist of gapped ferromagnetic (FM) and antiferromagnetic (AFM) phases as well as a gapless Luttinger liquid (LL) phase. In (a), the transition at $\Delta =1$ and $h=0$ is of Berezinsky-Kosterlitz-Thouless (BKT) type, while in (b), the whole line between AFM and LL phase is a BKT transition.

Figure 4

The return rate $l\left(t\right)$, Eq. (4.8), for a system with $N=120$ sites. $l\left(t\right)$ diverges at times $t_c$ given by Eq. (4.9).

Figure 5

The return rate $l\left(t\right)$ in the thermodynamic limit, Eq. (4.8) (dashed line), the two contributions contributions in Eq. (4.13) shown separately (dashed and dotted lines), and the minimum of these two contributions (solid line).

Figure 6

Lines of Fisher zeros obtained by Eq. (4.14) for the quench from the symmetric Néel state (4.11) to the free fermion point (solid lines), and Fisher zeros at times $t_c$, Eq. (4.10), for the quench from the polarized state (symbols).

Figure 7

Comparison between LCRG data (solid lines) and the bosonization result, $l_{\mathrm{Bos}}(t,\Lambda )$ (dashed lines), with fit parameters $\Lambda$ and a velocity shift $v_0$ as given in Table 1.

Figure 8

Return rate $l\left(t\right)$ for a quench from the Luttinger liquid phase, $\Delta =-0.8$, into the ferromagnetic phase, $\Delta =-2.0$.

Figure 9

Return rate $l\left(t\right)$ and next-leading eigenvalues, $-(ln|{\Lambda }_n{|}^2)/2$, for a quench from the AFM to the LL phase. Here ${\Delta }_{\mathrm{ini}}=6$, with the initial state being polarized, and ${\Delta }_{\mathrm{fin}}=-0.6$.

Figure 10

Same quench as shown in Fig. 9 but with a symmetric initial state. The eigenvalues shown are twofold degenerate.

Figure 11

Fisher zeros for a quench from the AFM into the LL phase with ${\Delta }_{\mathrm{ini}}=6$ and ${\Delta }_{\mathrm{fin}}=0.8$. The Fisher zeros cut the $R=0$ axis leading to nonanalyticities in the return rate $l\left(t\right)$. Lines are a guide to the eye.

Figure 12

(a) Quench from the polarized initial state with ${\Delta }_{\mathrm{ini}}=20000$ and ${\Delta }_{\mathrm{fin}}=1.2$, which are both in the AFM phase. Contrary to what might have been expected, $l\left(t\right)$ shows cusps. (b) Same as in (a) for the symmetric initial state. All eigenvalues are twofold degenerate. The inset shows the regime where the first crossings of the leading and a next-leading eigenvalue occur.

Figure 13

Fisher zeros for a quench within the AFM phase with ${\Delta }_{\mathrm{ini}}=6$ (symmetric) and ${\Delta }_{\mathrm{fin}}=1.1$. The Fisher zeros might cut the $R=0$ axis at longer times. Lines are a guide to the eye.

Figure 14

Return rate $l\left(t\right)$ and next-leading eigenvalues, $-(ln|{\Lambda }_n{|}^2)/2$, for a quench across the BKT transition from the AFM to the LL phase. Here, ${\Delta }_{\mathrm{ini}}={\Delta }_{\mathrm{fin}}=-0.8$, $h_{\mathrm{st},\mathrm{ini}}=1.0$, and $h_{\mathrm{st},\mathrm{fin}}=0.0$.

Figure 15

Return rate $l\left(t\right)$ (solid line) and next-leading eigenvalues, $-(ln|{\Lambda }_n{|}^2)/2$, for a quench across the second-order transition between the AFM phases with ${\Delta }_{\mathrm{ini}}={\Delta }_{\mathrm{fin}}=0.1$, $h_{\mathrm{st},\mathrm{ini}}=0.5$, and $h_{\mathrm{st},\mathrm{fin}}=-0.5$.

Figure 16

Return rate $l\left(t\right)$ (solid line) and next-leading eigenvalues, $-(ln|{\Lambda }_n{|}^2)/2$, for a quench across the second-order transition between the AFM phases with ${\Delta }_{\mathrm{ini}}={\Delta }_{\mathrm{fin}}=0.1$, $h_{\mathrm{st},\mathrm{ini}}=0.7$, and $h_{\mathrm{st},\mathrm{fin}}=-0.7$.

Figure 17

Return rate $l\left(t\right)$ for a quench across the BKT transition from the LL to the AFM phase. Here, ${\Delta }_{\mathrm{ini}}={\Delta }_{\mathrm{fin}}=-0.8$, $h_{\mathrm{st}}^{\mathrm{ini}}=0.0$, and $h_{\mathrm{st}}^{\mathrm{fin}}=1.0$.

Figure 18

Return rate $l\left(t\right)$ and next-leading eigenvalues, $-(ln|{\Lambda }_n{|}^2)/2$, for a quench within the AFM phase. Here, ${\Delta }_{\mathrm{ini}}=6$, ${\Delta }_{\mathrm{fin}}=-0.6$ while the staggered field is kept constant, $h_{\mathrm{st}}^{\mathrm{ini}}=h_{\mathrm{st}}^{\mathrm{fin}}=0.05$. Note that $l\left(t\right)$ shows nonanalyticities although no phase transition is crossed. The inset shows the regime where the first crossing occurs.

Figure 19

Fisher zeros for the same quench as in Fig. 18. The Fisher zeros cut the $R=0$ axis leading to the nonanalyticities in $l\left(t\right)$ shown in Fig. 18. Lines are a guide to the eye.