Scaling of ground-state fidelity in the thermodynamic limit: $XY$ model and beyond

Scaling of ground-state fidelity in the thermodynamic limit: $X Y$ model and beyond

Marek M. Rams and Bogdan Damski

Phys. Rev. A 84, 032324 – Published 16 September 2011

Abstract

We study ground-state fidelity defined as the overlap between two ground states of the same quantum system obtained for slightly different values of the parameters of its Hamiltonian. We focus on the thermodynamic regime of the $X Y$ model and the neighborhood of its critical points. We describe extensively fidelity when it is dominated by the universal contribution reflecting the quantum criticality of the phase transition. We show that proximity to the multicritical point leads to anomalous scaling of fidelity. We also discuss fidelity in a regime characterized by pronounced oscillations resulting from the change in either the system size or the parameters of the Hamiltonian. Moreover, we show when fidelity is dominated by non-universal contributions, study fidelity in the extended Ising model, and illustrate how our results provide additional insight into dynamics of quantum phase transitions. Special attention is given to studies of fidelity from the momentum space perspective. All our main results are obtained analytically. They are in excellent agreement with numerics.

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Received 2 May 2011

DOI:https://doi.org/10.1103/PhysRevA.84.032324

Authors & Affiliations

Marek M. Rams^1,2 and Bogdan Damski¹

¹Los Alamos National Laboratory, Theoretical Division, Mail Stop B213, Los Alamos, New Mexico, 87545, USA
²Institute of Physics, Jagiellonian University, Reymonta 4, PL-30059 Kraków, Poland

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Vol. 84, Iss. 3 — September 2011

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Images

Figure 1
Phase diagram of the $X Y$ model (6). Critical points lie on green (thick) lines. (A)–(D) show different characteristic lines along which we calculate fidelity. The dashed “circle” separates two regions of the ferromagnetic phase differing by the structure of the excitation gap. The minimum of the energy gap is reached for $k_{c} = 0$ outside the “circle” ( $k_{c} = π$ for $g < 0$ ) and for $k_{c} = \arccos [g / (1 - γ^{2})]$ inside the “circle” (see, e.g., Ref. [35]).Reuse & Permissions
Figure 2
The transition from the “small system” limit (left part of the plot) to the thermodynamic limit (right part of the plot) resulting from a variation in the anisotropy parameter $γ$ . The solid black line is a numerical result showing $F = | 〈 g_{c} - 2 δ, γ | g_{c}, γ 〉 |$ , red dots depict the “small system” prediction (21), and blue crosses represent the thermodynamic result (27). The figure is prepared for $δ = 3 \times 10^{- 7}$ and $N = 10^{6}$ .Reuse & Permissions
Figure 3
Numerical study of the crossover condition (20): see text for details. The crosses come from numerics, while the straight line is a linear fit, $\ln γ_{3 / 2} = - 15.51 \pm 0.05 + (1.007 \pm 0.004) \ln N$ [39]. The fit shows that the crossover takes place when $γ \sim N$ as predicted by Eq. (20). Here, we used $δ = 3 \times 10^{- 7}$ as in Fig. 2 and extracted $γ_{3 / 2}$ from $F = | 〈 g_{c} - 2 δ, γ | g_{c}, γ 〉 |$ .Reuse & Permissions
Figure 4
The derivative of the scaling parameter $\frac{\partial}{\partial g_{2}} \tilde{d} (g_{1}, g_{2})$ . The solid black line is the analytical result (32). The dashed lines, from top to bottom, correspond to numerical results for $N = 100, 300, 1000, 3000$ , respectively. The field $g_{1} = 1.1$ and $γ = 1$ .Reuse & Permissions
Figure 5
The thermodynamic vs the “small system” limit near the Ising critical point. The solid lines correspond to $f_{k}$ calculated for $γ = 1$ , $δ = 0.01$ , and $c = 0$ , 1, and 1.5 (bottom to top). The small red crosses mark $f_{k}$ ’s contributing to fidelity calculated for $N = 3000$ . The large black crosses show the same but when the system size is $N = 50$ . For $N = 3000$ the thermodynamic limit is reached. $N = 50$ represents the “small system” limit. In the “small system” limit the resolution in momentum space is insufficient to account for abrupt changes in $f_{k}$ taking place at small momenta.Reuse & Permissions
Figure 6
A discontinuous change in $f_{k}$ around $c = 1$ . The curves from bottom to top correspond to $c$ equal to 0.98, 1, and 1.02, respectively. The red crosses mark $f_{k}$ ’s contributing to fidelity calculated for $N = 3000$ . The parameter shift is $δ = 0.01$ . The lowest order Taylor expansions of $f_{k}$ are listed (but are not plotted) for all three curves.Reuse & Permissions
Figure 7
Panels in the left and right columns illustrate $\ln F$ and $\ln F / N | δ |$ as a function of $N$ . From top to bottom, rows show results for $c$ equal to 0.9, 1, and 1.1. The black solid lines show numerics. The red dashed lines present $- A (c) N | δ |$ in the left column and $- A (c)$ in the right column. The green dashed-dotted line in the top row shows $- A (c) N | δ | + \ln 2 / 2$ in the left panel and $- A (c) + \ln 2 / 2 N | δ |$ in the right one. The parameter shift is $δ = 10^{- 3}$ and $γ = 1$ .Reuse & Permissions
Figure 8
The plot of $f_{k}$ for the transition across the $γ = 0$ line. We choose $g = 0.5$ and $δ = 0.1$ . The curves from bottom to top correspond to $c$ equal to 0, 1, and 1.5, respectively. The dominating contribution to fidelity is centered around $k_{c} = \arccos (g) = π / 3$ .Reuse & Permissions
Figure 9
Comparison between the smooth part of fidelity (44), given by the red dashed line, and the exact numerical result (solid black line). In the large $N$ limit and for $| c | > 1$ , oscillations of fidelity are absent, which is presented here for $c = 2$ (top panel). At $c = 1$ , some oscillations of fidelity are visible for not-too-large systems (middle panel), but they gradually disappear for large $N$ . For $| c | < 1$ pronounced oscillations of fidelity surviving in the thermodynamic limit exist: bottom panel prepared for $c = 1 / 2$ . The plots are created for $δ = 0.002$ and $g = 0.99$ .Reuse & Permissions
Figure 10
Plot of $f_{k}$ for the transition across the $γ = 0$ line for $δ = 0.01$ , $c = 0$ , and several values of $g$ such that (a) $k_{c} = π \frac{1002}{4000}$ , (b) $k_{c} = π / 4$ , and (c) $k_{c} = π \frac{1001}{4000}$ [note that $k_{c}$ depends on $g$ (39)]. Pluses mark the discretization in momentum space for the size of the system $N = 2000$ . We see a systematic shift in $f_{k}$ ’s contributing to fidelity as we change $g$ slightly. This results in oscillations of fidelity from the lowest panel of Fig. 9. Parameter $φ$ from Eq. (45) equals 1, 0, and 1/2 for panels (a), (b), and (c), respectively.Reuse & Permissions
Figure 11
A schematic illustration of the shift $φ$ from Eq. (45). The crosses show discretized momenta, the dot denotes the position of $k_{c}$ (39), and the dashed line goes through the midpoint between the two discretized momenta being closest to $k_{c} = (2 m + φ) π / N$ . Note that $φ > 0$ ( $φ < 0$ ) when $k_{c}$ is to the right (left) of the dashed line. Naturally, when $φ$ approaches $\pm 1$ , $k_{c}$ collapses onto one of the discretized $k$ ’s.Reuse & Permissions
Figure 12
The plot of fidelity for the transition across the $γ = 0$ line for $δ = 0.002$ and various parameters $g$ and $| c | < 1$ . The black line and crosses show numerical results. The red dashed line and circles show the theoretical prediction (48).Reuse & Permissions
Figure 13
$f_{k}$ along the Ising critical line (49), for $ε = 1$ and $δ = 0.05, 0.1, 0.2$ (top to bottom). The minimum of $f_{k}$ is reached for $k = \arccos (\frac{1 - ε^{2} + δ^{2}}{1 + ε^{2} - δ^{2}}) ≃ \arccos (\frac{1 - ε^{2}}{1 + ε^{2}})$ .Reuse & Permissions
Figure 14
Transition from the “small system” limit to the thermodynamic limit near the multicritical point located at $g = 1$ and $γ = 0$ . (a) The transition as a function of the system size $N$ at fixed $δ = 10^{- 8}$ . (b) The transition as a function of parameter difference $δ$ at fixed $N = 3000$ . In both plots, the solid (black) line shows fidelity for $c = 1$ , i.e., $F = | 〈 1, 0 | 1 + 2 δ, 2 δ 〉 |$ , while the dashed (red) line shows the $c = 5$ case, i.e., $F = | 〈 1 + 4 δ, 4 δ | 1 + 6 δ, 6 δ 〉 |$ , Eq. (54). Note that $α = 1$ is assumed here.Reuse & Permissions
Figure 15
Numerical study of the crossover condition (55). The crosses come from numerics (see text for details), while the straight lines are linear fits [39]. In both panels, upper (lower) data sets correspond to $c = 1$ ( $c = 5$ ). Panel (a), we fit $a$ and $b$ coefficients of $\ln N_{3 / 2} = a + b \ln δ$ . For $c = 1$ ( $c = 5$ ), we obtain $b = - 0.499 \pm 0.0003$ ( $b = - 0.4992 \pm 0.0005$ ). Panel (b), we fit $a$ and $b$ coefficients of $\ln δ_{7 / 4} = a + b \ln N$ . For $c = 1$ ( $c = 5$ ), we obtain $b = - 1.989 \pm 0.005$ ( $b = - 1.9987 \pm 0.0008$ ). The numerical simulations for these plots are done with $α = 1$ . The fitting coefficient $a$ is not listed as it is of minor interest.Reuse & Permissions
Figure 16
Comparison between numerics (solid lines) and analytics [pluses given by Eq. (57)]. In all panels $N = 10^{5}$ and $α = 1$ . The parameter shift $δ$ is chosen to be small enough to keep the $O (| δ |^{5 / 2})$ corrections to Eq. (57) negligible.Reuse & Permissions
Figure 17
We show from the momentum space angle that $\ln F \sim - {| δ |}^{3 / 2}$ (57). We plot the rescaled $f_{k}$ functions for $c = 1$ and $α = 1$ : one for $δ = 0.001$ (solid black line) and the other for $δ = 0.01$ (dashed blue line). Because the two curves overlap very well, we see that $(1 - f_{k}) / | δ | = h (k / \sqrt{| δ |})$ , where $h$ is some nonsingular function. This implies that $\ln F \sim \int d k \ln f_{k} \sim - {| δ |}^{3 / 2}$ ( $f_{k} \approx 1$ so we approximate $\ln f_{k} ≃ f_{k} - 1$ ). This confirms the leading scaling of Eq. (57) with $δ$ and independently confirms the condition (55) with $ν = 1 / 2$ . We also note that these rescalings work in the limit of $δ \to 0$ .Reuse & Permissions
Figure 18
The lines correspond to $f_{k}$ plotted for $δ = 0.01$ (69). From bottom to top, $c$ equals 0, 1, and 1.5, respectively. For $c = 0$ the ground states entering fidelity are calculated on the opposite sides of the critical point. For $c = 1$ one of them is obtained at the critical point. Finally, for $c = 1.5$ they are both obtained on the same side of the critical point.Reuse & Permissions
Figure 19
Discontinuous change in $f_{k}$ around $c = 1$ . The curves from bottom to top correspond to $c$ equal to 0.98, 1, and 1.02, respectively. The parameter shift is $δ = 0.01$ . Lowest order Taylor expansions of $f_{k}$ around $k_{c} = 0$ are listed (but are not plotted) for all three curves. When $| c | < 1$ , then $f_{k}$ crosses zero for $k_{0} \neq 0$ .Reuse & Permissions
Figure 20
Scaling function (77) and its derivative (78). Panel (a), the black line shows the analytical expression (77), while red crosses show numerically evaluated $n_{ex} / | δ |$ from Eq. (76) for $N = 10^{5}$ and $δ = 10^{- 3}$ . Panel (b), logarithmic divergence of $d B (c) / d c$ around $c = 1$ (78). The solid black line shows the analytical result (78). The dashed lines present numerics done for $δ = 10^{- 3}$ and $N$ equal to $10^{5}$ (red), $2 \times 10^{5}$ (blue), and $4 \times 10^{5}$ (green) (top to bottom). The logarithmic singularity is reached in the limit of $N \to \infty$ (compare with Fig. 4 and the discussion in Sec. 3a).Reuse & Permissions
Figure 21
A schematic illustration of the dynamics of a generic quantum phase transition (see text for details).Reuse & Permissions
Figure 22
Numerical calculation of the error (A1) for different $γ$ , $δ$ , and $c = ε / | δ |$ . All the curves show that $| E | \times γ^{3} / δ^{2}$ is bounded from above by about 0.2. They also illustrate that $E$ scales as $δ^{2} / γ^{3}$ . Upper panel: $δ = 10^{- 3}$ (dashed red line) and $δ = 10^{- 5}$ (solid black line); $γ = 1$ for both curves. Middle panel: $δ = 10^{- 4}$ (dashed red line) and $δ = 10^{- 6}$ (solid black line); $γ = 0.5$ for both curves. Bottom panel: $δ = 10^{- 5}$ (dashed red line) and $δ = 10^{- 7}$ (solid black line); $γ = 0.1$ for both curves. We keep $| ε |, | δ | ≪ γ^{2}$ in these calculations.Reuse & Permissions
Figure 23
Integration error (A10). The curves from bottom to top correspond to $δ = 10^{- 3}$ , $10^{- 4}$ , and $10^{- 5}$ , respectively. For each $δ$ , we choose $g = 0, 0.9, 1$ . The curves for different $g$ ’s, but the same $δ$ , perfectly collapse onto each other. This plot supports $E \approx 0.25 δ^{2}$ .Reuse & Permissions

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