We focus on the long- and short-time behaviors of the Loschmidt echo that occur when the system is coupled to a ground-state bath (XXZ) at the initial time. We will consider the case of Ising coupling, see Eq. (3.4), as well as that of Heisenberg coupling, see Eq. (3.5), between the system and the bath. The bath Hamiltonian is that of the XXZ chain, which is
HE =−2J XL
j=1
[SxjSxj+1+SyjSyj+1+ ∆SzjSzj+1]. (3.27)
The ground state of the XXZ chain has three phase with two critical points∆c =−1 and∆c = 1. It is critical for−1 ≤ ∆≤ 1; the critical state, is spanned by| ↑↓↑↓ . . . ↑i,| ↑↑↓↓ . . .↑i, . . ., and | ↓↑↓↑ . . . ↓i. In the antiferromagnetic phase, ∆ < −1, the ground state is doubly degenerate, i.e.,| ↑↓↑↓ . . .↓ior| ↓↑↓↑ . . .↑i. In the ferromagnetic phase,∆> 1, the ground state is| ↑↑↑↑. . .↑↑i.
3.3.1 Ising coupling between S and E
In this section, we consider the case of Ising coupling, Eq. (3.4), between the system S and the bath E. Using adaptive t-DMRG calculations with∆t= 0.001 for the time evolution, discarded weight below 10−13, a typical minimum truncated Hilbert space dimension of D= 100 and a typical maximum truncated Hilbert space dimension of D = 1000, we treat systems with up to 301 sites with OBCs. We choose an odd number of sites with total spin Sz = 1/2 in order to ensure a unique ground state in the antiferromagnetic phase of the XXZ chain. In order to minimize the boundary effects, we consider only the case in which the system is coupled to a spin in the middle of the chain, i.e., j = 25 for L = 49. The error analysis is shown in Appendix B.
The generic behavior of the Loschmidt echo as a function of time for different values of the anisotropy interaction strength∆and lattice sizes L= 49 and 101 with coupling constant ǫ = 0.2 is depicted in Figs. 3.4 and 3.5. Due to the smaller Hilbert space, our calculations for L = 49 can reach a much longer time than those for L = 101. In the antiferromagnetic phase, ∆< −1, the decay of the Loschmidt echo is suppressed, i.e, at∆ = −3 and−2, and it is completely suppressed when∆ → −∞. In the critical region, the decay of the Loschmidt echo is affected by the values of∆only at long times. The oscillation of the curve for∆ =0.5 has a frequency and an amplitude proportional to ǫ that is related to the finite-size effects.
In the ferromagnetic phase, the Loschmidt echo does not decay because the ground state is essentially classical. Near or at the critical points, the decay of the Loschmidt echo is strongly enhanced. Already for L = 49, the asymptotic behavior of the Loschmidt echo is characterized by a decay to zero close to and at the critical point∆c =1. We choose the state with a spin quantum number Sz = 1/2 at the critical point, ∆c = 1, where the ground state is highly degenerate. At long times, the Loschmidt echo revives due to the finite-size effects.
The revival time for this case can be roughly estimated to be t∗ ≈ 2vL, where v is the velocity
0 20 40 60 80 100 t
0.975 0.98 0.985 0.99 0.995 1
LE
∆ = −3
∆ = −2
∆ = −1.5
∆ = −1
∆ = −0.9
∆ = −0.5
∆ = 0
∆ = 0.5
∆ = 0.9
(a)
(a)
0 50 100 150 200
t 0
0.2 0.4 0.6 0.8 1
LE
∆ = 0.985
∆ = 0.99
∆ = 1
(b)
(b)
Figure 3.4: Ising coupling case: Loschmidt echo LE as a function of time t for different values of the anisotropy interaction strength: (a)∆ =−3 (black), -2 (red), -1.5 (green), -1 (blue), -0.9 (yellow), -0.5 (brown), 0 (olive), 0.5 (violet), and 0.9 (cyan); (b)∆ = 0.985 (magenta), 0.99 (orange), and 1 (indigo) for a spin-1/2 XXZ chain with lattice size L =49 and OBCs.
of the elementary excitations in the critical region (XXZ), given by [88, 89]
v= π√ 1−∆2
2 arccos(−∆). (3.28)
For∆→1 and v→0, the revival time becomes very large, see Fig. 3.4.
The behavior of the Loschmidt echo for short time is depicted in Fig. 3.6 (a), which is a magnification of the curve in Fig. 3.5. We have fit the data points of the Loschmidt echo for time t . 0.02 to a Gaussian decay with parameterα. For time t & 0.04, the Loschmidt echo does not, in general, decay as a Gaussian deep in the antiferromagnetic phase. The short-time Gaussian decay parameter αis shown in Fig. 3.6 (b) as a function of∆ for the bath lattice sizes L = 49, 101, and 301. These are no visible finite-size effects in all phases except at the points close to the critical point∆c = −1. For∆ → −∞, αasymptotically tends to zero.
As∆moves close to the critical point ∆c = −1, αincreases to the maximal value 0.01 and remains almost constant in the critical region (−1<∆<1). At the critical point∆c =−1,αis continuous. However, at the critical point∆c = 1,αis completely discontinuous, independent of finite-size effects. In the ferromagnetic phase,αis strictly zero.
0 5 10 15 20 t
0.95 0.96 0.97 0.98 0.99 1
LE
∆ = −3
∆ = −2
∆ = −1.5
∆ = −1
∆ = −0.9
∆ = 0.5
∆ = −0.5
∆ = 0.9
∆ = 0
Figure 3.5: Ising coupling case: the Loschmidt echo LE as a function of time t for different values of the anisotropy interaction strength∆ = −3 (red), -2 (green), 0 (violet), -1.5 (blue), 0.5 (olive), -0.5 (cyan), -0.9 (black), -1 (orange), and 0.9 (magenta) of a spin-1/2 XXZ chain with lattice size L =101 and OBCs.
0 0.1 0.2 0.3 0.4 0.5
t 0.9995
0.9996 0.9997 0.9998 0.9999 1
LE
L = 101
(a)
(a)
-8 -6 -4 -2 0 2 4
∆ 0
0.002 0.004 0.006 0.008 0.01
α
L = 49 L = 101 L = 301
(b)
(b)
Figure 3.6: Ising coupling case: (a) short-time behavior of the Loschmidt echo LE with bath lattice sizes L = 101 and t ≤ 0.5, where the LE is fit to a Gaussian decay, LE(t) ∼ e−αt2. The plot corresponds to an enlargement of Fig. 3.5 at small times. The various curves are for different values of∆, from top to bottom: ∆ =−3 (red), -2 (green), -1.5 (blue), 0 (violet), 0.5 (olive), -0.5 (cyan), . . . , 0.9 (magenta), where the curves overlap for−1< ∆≤ 1, i.e., in the critical phase of the XXZ chain. (b) Scaling analysis of the decay parameterαas a function of∆for different bath lattice sizes L= 49 (black), L = 101 (red), and L=301 (green). Here the coupling constant between the system and bath is kept fixed,ǫ =0.2.
3.3.2 Heisenberg coupling between S and E
In this section, we consider the case of Heisenberg coupling, Eq. (3.5), between the system S and the bath E. Using the adaptive t-DMRG calculations with a time step of∆t = 0.001, and keeping the discarded weight below 10−13 for t <15 and 10−10for 15< t< 100, and the size of the truncated Hilbert space above D = 100 and below D = 1000, we treat systems with up to 101 sites, OBCs, and an odd number of sites with a spin quantum number Sz = 1/2.
In order to minimize the boundary effects, we consider only the case in which the system is coupled to a spin in the middle of the chain, i.e., j= 25 for L=49.
In Fig. 3.7, we plot the decay of the Loschmidt echo as a function of time for different values of the anisotropy interaction strength∆for a bath lattice size L = 49, with isotropic coupling constantεx = εy = εz = 0.2. In the antiferromagnetic phase,∆< −1, the long-time decay of the Loschmidt echo is strongly suppressed (i.e.,∆ =−3 and−2) and is completely suppressed as ∆ → −∞. In the critical region (−1 < ∆ < 1), the decay of the Loschmidt echo is affected by the values of ∆ only at long times. For L = 49, the Loschmidt echo already decays to zero for all parameters. In the ferromagnetic phase, unlike the case of Ising coupling, decay of the Loschmidt echo takes place because the effects ofεx andεydestroy the fully polarization of the ground state. The decay is also strongly suppressed, with complete suppression occurring when∆→ ∞. Near or at the critical points, the short-time decay of the Loschmidt echo is strongly enhanced. However, the long-time behavior is more complicated to interpret. We can still see that the Loschmidt echo revives at a long time scale. The revival time is approximately proportional to L. For∆ =−1, -0.9, -0.5, and 0, a kink in the curve of the Loschmidt echo at about t = 20 is related to the odd number of the bath lattice size. We cannot find such kinks for the case of even bath lattice size.
The short-time Gaussian decay parameterαis shown in Fig. 3.8 as a function of∆for the bath lattice size L = 25,49, and 101. We have fit the data points of the Loschmidt echo for times t . 0.02 to obtain the Gaussian decay parameterα. For time t > 0.04, the Loschmidt echo does not, in general, decay strictly as a Gaussian deep in the antiferromagnetic phase.
As shown in the figure, as for the case of Ising coupling, the finite-size effects are only obvi-ous near the critical point∆c = −1. In other regions, the finite-size effects are not noticeable.
For∆→ −∞, it asymptotically tends to 0 (not shown in the figure) because the effects of the small coupling between the qubit and the bath are negligible for an infinite value of∆. From
∆ = −8 to ∆ = −1, αincreases from 0.01 to the maximal value 0.02. In the critical region (−1 < ∆ < 1) on finite lattices, the decay parameterαdepends on ∆. However, as the bath size approaches the thermodynamic limit, the dependence ofαon ∆is reduced. In Fig. 3.8,
0 20 40 60 80 100 t
0.000 0.200 0.400 0.600 0.800 1.000
LE
0.992 0.996 1.000
∆ = −3
∆ = −2
∆ = 1.5
∆ = 2
∆ = −1.5
∆ = −1
∆ = −0.9
∆ = −0.5
∆ = 0
∆ = 0.9 ∆ = 0.5
Figure 3.7: Heisenberg coupling case: Loschmidt echo LE as a function of time t for different values of the anisotropy interaction strength:∆ =−3 (indigo), -2 (maroon), 1.5 (turquoise), 2 (olive), -1.5 (black), -1 (red), -0.9 (green), -0.5 (blue), 0 (orange), 0.5 (brown), and 0.9 (violet) of a spin-1/2 XXZ chain with lattice size L= 49 and OBCs.
the decay parameter α retains an almost constant value 0.02, as expected in the critical re-gion, already for a lattice size L = 101. As in the case of Ising coupling, the curves ofαare continuous at the critical point ∆c = −1. At the critical point∆c = 1, they are completely discontinuous, and are also not dependent on L. In the ferromagnetic phase, in contrast to the case of Ising coupling, α remains a constant finite value 0.01 and asymptotically tends to 0 when ∆ → ∞. Since the ferromagnetic ground state is not an eigenvector of the com-plete Hamiltonian HS E, the effects of the small coupling between the qubit and the bath are negligible for an infinite value of∆.