Impurity-induced environmental quantum phase transitions in the quadratic-coupling spin-boson model

We study the zero temperature properties of the sub-Ohmic spin-boson model with quadratic spin-boson coupling. This model describes experimental setups at the optimal working point where the linear coupling between the qubit (spin) and the environmental noise (bosons) is zero and the leading coupling is quadratic. In the strong coupling regime, we find that the existence of spin induces quantum phase transitions (QPTs) between two states of environment: the normal state and a state with local distortions. The phase diagram contains both continuous and the first-order QPTs, with nontrivial critical properties obtained exactly. At the QPTs, the equilibrium state spin dynamics bears power-law $\omega$ dependence in the small frequency limit and a robust coherent Rabi oscillation at high frequency. We discuss the feasibility of observing such environmental QPTs in the qubit-related experiments.

Figure 1

Ground state NRG phase diagram of $H_{QSB}(\mathrm{\Delta }=0)$ for $s=0.3$. $\langle X\rangle =\langle b_0^{†}+b_0\rangle$ is the order parameter and $\langle S_z\rangle =\langle {\sigma }_z\rangle /2$ is the spin polarization. The phase boundaries are the spin flip (up triangles), continuous QPT (circles), and first-order QPT (squares) lines. Lines are for guiding the eye. NRG parameters are $N_b=8$ (empty symbols), $N_b=12$ (plus-filled symbols), and the extrapolated $N_b=\infty$ (solid squares). $({\alpha }_c=0.0786,{\epsilon }_c=-0.1273)$ (solid dot) is the jointing point of the three transition lines). Inset: details close to the jointing point, with the exact spin flip line (solid line) and the weak-coupling expansion ${\epsilon }_f=-2\alpha /(1+s)$ (dashed line). NRG parameters are $\mathrm{\Lambda }=2.0$ and $M_s=60$.

Figure 2

NRG results for $|\langle X\rangle |$ and $\langle S_z\rangle$ as functions of $\alpha$ for $s=0.3$ and $\mathrm{\Delta }=0.0$, for various $\epsilon$ values. The NRG parameters are $\mathrm{\Lambda }=2.0, M_s=60$, and $N_b=8$.

Figure 3

NRG flow of excitation energies at $s=0.3, \mathrm{\Delta }=0.0$, and $\epsilon =0.1>{\epsilon }_c$. The energy levels flow to three different fixed points: a stable free boson fixed point for $\alpha =0.084<{\alpha }_c^{\left(c\right)}$ (solid lines), an unstable critical fixed point for $\alpha =0.08593945\approx {\alpha }_c^{\left(c\right)}$ (dashed lines), and a strong-coupling fixed point for $\alpha =0.086>{\alpha }_c^{\left(c\right)}$ (dash-dotted lines). The NRG parameters are $\mathrm{\Lambda }=4.0, M_s=100$, and $N_b=8$.

Figure 4

NRG flow of $E_i\left(N\right)$ ($i=1,2,3,...,6$) at $s=0.3, \mathrm{\Delta }=0.0, \epsilon =0.1>{\epsilon }_c$, and $\alpha =0.086>{\alpha }_c^{\left(c\right)}$, obtained using $N_b=8$ (solid lines) and $N_b=12$ (dashed lines). Inset: ground state properties of the fixed point Hamiltonian $H_{N=30}$ as functions of $N_b$, the energy $E_1(N=30)$ (squares) and the boson occupancy number ${\langle b_{30}^{†}{b}_{30}\rangle }_{N=30}$ (up triangles). The lines are for guiding the eye. Other NRG parameters are $\mathrm{\Lambda }=4.0$ and $M_s=100$.

Figure 5

Flow diagrams near the QPTs for $s=0.3$ and $\mathrm{\Delta }=0.0$. (a) Continuous QPT for $\epsilon =0.1, {\alpha }_c\approx 0.085934$, and (b) first-order QPT for $\epsilon =-0.2, {\alpha }_c\approx 0.10147$. From left to right, $\alpha$ increases for $\alpha <{\alpha }_c$ (empty squares) and $\alpha$ decreases for $\alpha >{\alpha }_c$ (solid squares). Lines are for guiding the eye. Inset of (a): power law fitting of $T^{★}={\mathrm{\Lambda }}^{-N^{★}}\propto {|\alpha -{\alpha }_c|}^{z\nu }$ for $\alpha <{\alpha }_c$ (empty squares) and $\alpha >{\alpha }_c$ (solid squares), giving $z\nu =3.333$ and 3.338, respectively. NRG parameters are $\mathrm{\Lambda }=4.0, M_s=100$, and $N_b=12$.

Figure 6

Main figure: ground state phase diagram for $s=0.3$ and $\mathrm{\Delta }=0.1$, obtained with $N_b=12$. The continuous QPT, first-order QPT, and spin flip lines are marked by empty circles, empty squares, and empty up triangles with eye-guiding lines, respectively. The jointing point of the continuous and the first-order QPTs is marked by a solid blue dot. The spin flip line obtained with $N_b=20$ (plus-filled up triangles) and $N_b=30$ (cross-filled up triangles) are also plotted. Inset: the change of phase boundaries with $N_b$. $N_b=12$ (empty symbols), $N_b=20$ (plus-filled symbols), $N_b=30$ (cross-filled symbols), and the extrapolated $N_b=\infty$ (solid squares). NRG parameters are $\mathrm{\Lambda }=4.0$ and $M_s=100$.

Figure 7

$|\langle X\rangle |$ and $\langle S_z\rangle$ near the jointing point of the continuous and the first-order QPTs, for $s=0.3$ and $\mathrm{\Delta }=0.1$. (a) $|\langle X\rangle |\left(\alpha \right)$ for various $\epsilon$'s. From right to left, $\epsilon =-0.175, -0.15, -0.125, -0.085, -0.037, -0.012$, and 0.0. In (b) and (c), $|\langle X\rangle |$ and $\langle S_z\rangle |$ values at the upper and lower edge of the transition as functions of $\epsilon$. NRG parameters are $\mathrm{\Lambda }=4.0, M_s=100, N_b=30$.

Figure 8

Dynamical correlations (a) $C_X\left(\omega \right)$ and (b) $C_{Sz}\left(\omega \right)$ for $\alpha \le {\alpha }_c^{\left(c\right)}$ for $s=0.3, \mathrm{\Delta }=0.1$, and $\epsilon =0.0>{\epsilon }_c$. From top to bottom, $\alpha =0.091101\approx {\alpha }_c^{\left(2\right)}$, 0.08, 0.09, and 0.09105. The fitted exponents are for (a): $y_0=0.296$ for $\alpha <{\alpha }_c, y_c=-0.302$ for $\alpha ={\alpha }_c$, and for (b): ${\theta }_0=1.595$ for $\alpha <{\alpha }_c$, and ${\theta }_c=0.396$ for $\alpha ={\alpha }_c$. The zero frequency peak $A\delta \left(\omega \right)$ of $C_{Sz}\left(\omega \right)$ is not shown. NRG parameters are $\mathrm{\Lambda }=4.0, M_s=100, N_b=12$, and $B=1.0$ for broadening.

Figure 9

Dynamical correlation functions (a) $C_X\left(\omega \right)$ and (b) $C_{Sz}\left(\omega \right)$ for different $\epsilon >{\epsilon }_c\approx -0.098$, calculated for $s=0.3, \mathrm{\Delta }=0.1$, and $\alpha ={\alpha }_c^{\left(c\right)}\left(\epsilon \right)$. From top to bottom, $\epsilon =-0.085$, 0.0, 0.1, and 0.2. The corresponding $\langle S_z\rangle$ values are $-0.310, -0.421, -0.465$, and $-0.481$. The fitted exponents in the small $\omega$ regime are $y_c=-0.298$ in (a) and ${\theta }_c=0.396$ in (b). The zero frequency peak $A\delta \left(\omega \right)$ is not shown. In (b), the vertical dashes mark the Rabi frequency ${\omega }_R$ estimated from $\langle S_z\rangle$ and ${\mathrm{\Delta }}_r\approx \mathrm{\Delta }$. NRG parameters are $\mathrm{\Lambda }=4.0, M_s=100, N_b=12$. The broadening parameter $B=1.0$.

Figure 10

Exponents of $C_{S_z}\left(\omega \right)$: ${\theta }_0$ and ${\theta }_c$. The solid lines are ${\theta }_0=1+2s$ and ${\theta }_c=1-2s$.

Figure 11

The critical behavior of the order parameter $|\langle X\rangle |$ near the continuous QPT $\alpha ={\alpha }_c^{\left(c\right)}$, for $s=0.3, \mathrm{\Delta }=0.0$, and $\epsilon =0.1>{\epsilon }_c$. (a) $|\langle X\rangle |\left(\alpha \right)$ curves for different $N_b$ values. From bottom to top, $N_b=8,12,20,30,50$. (b) Power law fitting of $\left|\langle X\rangle \right|=c{(\alpha -{\alpha }_c)}^{\beta }$ with the fitted value $\beta =1.16$ and ${\alpha }_c^{\left(c\right)}=0.09918$, being independent of $N_b$. (c) Log-log plot of the prefactor $c$ versus $N_b$. The dashed line gives the fitting $c\left(N_b\right)\propto N_b^{0.56}$ in the large $N_b$ limit. The NRG parameters are $\mathrm{\Lambda }=9.0, M_s=100$.

Figure 12

Extrapolation of ${\alpha }_c^{\left(c\right)}\left(\mathrm{\Lambda }\right)$ to $\mathrm{\Lambda }=1.0$ for $s=0.3, \mathrm{\Delta }=0.0$, and $\epsilon =0.1>{\epsilon }_c$. The squares and circles represent data obtained using $M_s=100$ and $Ms=200$, respectively, with $N_b=12$. The four-point Lagrangian extrapolation using data of $\mathrm{\Lambda }=1.6$, 1.8, 2.0, and 2.2 gives the dashed line and the extrapolated value ${\alpha }_c^{\left(c\right)}(\mathrm{\Lambda }=1)=0.0749$, in good agreement with the exact value ${\alpha }_c=s/\left(4g_2{\omega }_c\right)=0.075$.

Figure 13

The curves $|\langle X\rangle |\left(\alpha \right)$ near the first-order QPT for $s=0.3, \mathrm{\Delta }=0.0$, and $\epsilon =-0.2<{\epsilon }_c$. In (a), the empty circles with eye-guiding lines are NRG data using various $N_b$'s. From bottom to top, $N_b=8,12,20,30,50$. The dashed lines are corresponding data for $\epsilon =0.1>{\epsilon }_c$. (b) Distance of the first-order QPT point ${\alpha }_c^{\left(1\right)}$ to the continuous one ${\alpha }_c^{\left(c\right)}$ as a function of $1/N_b$. The solid squares are NRG data and the solid line is a power law fit with the slope $-0.4$. (c) $|\langle X\rangle |$ value at the upper edge of ${\alpha }_c^{\left(1\right)}$ as a function of $1/N_b$. The NRG parameters are $\mathrm{\Lambda }=9.0$ and $M_s=100$.

Figure 14

$C_X\left(\omega \right)$ for $s=0.3, \mathrm{\Delta }=0.0$, and $\epsilon =0.1>{\epsilon }_c$. The solid lines are NRG results and the dashed lines are exact solution Eq. (9) of the main text. From bottom to top, $\alpha ={\alpha }_c-\delta \alpha$ with $\delta \alpha =1.0\times {10}^{-2}, 2.0\times {10}^{-3}, 4.0\times {10}^{-4}, 8.0\times {10}^{-5}$, and 0.0. For NRG and the exact solution, we use, respectively, ${\alpha }_c^{NRG}=0.085934484$ and ${\alpha }_c^{exc}=0.075$. The NRG parameters are $\mathrm{\Lambda }=4.0, M_s=100, N_b=12$. The broadening parameter is $B=1.0$.

Figure 15

Comparison of the universal exponents calculated from NRG and the exact solution. (a) The energy scale exponent $z\nu$. Symbols are NRG data and the solid line $z\nu =1/s$ is the exact solution; (b) exponents of $C_X\left(\omega \right)$: $y_0$ and $y_c$. Symbols are NRG data and the solid lines are exact solution $y_0=s$ and $y_c=-s$.