Quantum tricritical point emerging in the spin-boson model with two dissipative spins in staggered biases

We study the spin-boson model (SBM) with two spins in staggered biases by a numerically exact method based on variational matrix product states. Several observables such as the magnetization, the entanglement entropy between the two spins and the bosonic environment, the ground-state energy, as well as the correlation function for two spins are calculated exactly. The characteristics of these observables suggest that the staggered biases can drive the second-order quantum phase transition (QPT) to the first-order QPT in the sub-Ohmic SBM, while the Kosterlitz-Thouless QPT in the Ohmic SBM goes directly to the first-order one. A quantum tricritical point, where the continuous QPT meets the first-order one, can then be detected. It is found that the staggered biases would not change the universality of the phase transition in this model below the quantum tricritical point.

Figure 1

Illustration of the two-spin-boson model with staggered biases $\pm \epsilon$ along the $x$ direction. The two spins denoted by red spheres interact with a common continuous bosonic reservoir represented by the big blue region. No direct interaction between spins is considered.

Figure 2

(Upper panels) Phase diagram in the $\epsilon -\alpha$ plane for 2SBM drawn from the magnetization $M$: delocalized phases ($M=0$) and the localized phase ($M\ne 0$). (Lower panels) Entanglement entropy $S_E$. The power of the spectral function is (left) $s=0.3$ and (right) 0.7. $\mathrm{\Delta }=0.1, {\omega }_c=1$. The QuTP is marked by a red dot, which separates the intersection of the second- and first-order phase transition. The parameters used in the VMPS approach are $\mathrm{\Lambda }=2, L=50, d_{\text{opt}}=12$, and $D=20$ for $s=0.3,0.7$.

Figure 3

Magnetization $M$, entanglement entropy $S_E$ (upper panels), the first- and second-order derivatives of the ground-state energy (lower panels) as a function of $\alpha$ for $\epsilon =0.1$ (left) and $\epsilon =0.2$ (right) for the sub-Ohmic bath with $s=0.3$ by VMPS approach. $\mathrm{\Delta }=0.1, {\omega }_c=1, \mathrm{\Lambda }=2, L=50, d_{\text{opt}}=12$, and $D=20$.

Figure 4

Magnetization $M$, entanglement entropy $S_E$ (upper panels), the first- and second-order derivatives of the ground-state energy (lower panels) as a function of $\alpha$ for $\epsilon =0.5$ (left) and $\epsilon =0.7$ (right) for the sub-Ohmic bath with $s=0.7$ by VMPS approach. $\mathrm{\Delta }=0.1, {\omega }_c=1, \mathrm{\Lambda }=2, L=50, d_{\text{opt}}=12$, and $D=20$.

Figure 5

The correlation function $〈{\sigma }_1^x{\sigma }_2^x〉$ as a function of the coupling strength for the sub-Ohmic bath by VMPS approaches at $\epsilon =0.1,0.2$ for $s=0.3$ (left) and $\epsilon =0.5,0.7$ for $s=0.7$ (right). Other parameters: $\mathrm{\Delta }=0.1, {\omega }_c=1, \mathrm{\Lambda }=2, L=50, d_{\text{opt}}=12, D=20$.

Figure 6

The log-log plot of the magnetization $M$ as a function of $\alpha -{\alpha }_c$ at $\epsilon =0.1,0.2$ for $s=0.3$ (left panels) and $\epsilon =0.5,0.7$ for $s=0.7$ (right panels). The numerical results by VMPS are denoted by black circles and the power-law fitting curves are denoted by the red dashed lines, which indicates the second-order QPT takes place in the smaller staggered biases and gives similar critical behaviors compared to the standard spin-boson model, while the larger staggered biases induce the first-order QPT and vanishing of the critical exponent $\beta$. $\mathrm{\Delta }=0.1, {\omega }_c=1, \mathrm{\Lambda }=2, L=50, d_{\text{opt}}=12$, and $D=20,40$ for $s=0.3,0.7$, respectively.

Figure 7

Entanglement entropy $S_E$ as a function of $\alpha$ in the ground state for the Ohmic bath ($s=1$) at $\epsilon =0,0.1,0.5,0.9,1.2,1.5$ for $s=1$ by VMPS approach. $\mathrm{\Delta }=0.1, {\omega }_c=1, \mathrm{\Lambda }=2, L=50, d_{\text{opt}}=12$, and $D=20$.

Figure 8

Magnetization $M$, entanglement entropy $S_E$ (upper panels), the first- and second-order derivatives of the ground-state energy (lower panels) as a function of $\alpha$ in the Ohmic bath ($s=1$) for a weak bias $\epsilon =0.5$ (left panels) and strong bias $\epsilon =0.9$ (right panels), representative of the KT transition, by VMPS approach. $\mathrm{\Delta }=0.1, {\omega }_c=1, \mathrm{\Lambda }=2, L=50, d_{\text{opt}}=12$, and $D=20$ for $s=1$.

Figure 9

Magnetization $M$, entanglement entropy $S_E$ (upper panels), the first- and second-order derivatives of the ground-state energy (lower panels) as a function of $\alpha$ in the Ohmic bath ($s=1$) for two strong biases $\epsilon =1.5$ (left panels) and $\epsilon =3$ (right panels), representative of the first-order phase transition, by VMPS approach. $\mathrm{\Delta }=0.1, {\omega }_c=1, \mathrm{\Lambda }=2, L=50, d_{\text{opt}}=12$, and $D=20$ for $s=1$.