Quantum phase transitions in a pseudogap Anderson-Holstein model

We study a pseudogap Anderson-Holstein model of a magnetic impurity level that hybridizes with a conduction band whose density of states vanishes in power-law fashion at the Fermi energy, and couples, via its charge, to a nondispersive bosonic mode (e.g., an optical phonon). The model, which we treat using poor-man's scaling and the numerical renormalization group, exhibits quantum phase transitions of different types depending on the strength of the impurity-boson coupling. For weak impurity-boson coupling, the suppression of the density of states near the Fermi energy leads to quantum phase transitions between strong-coupling (Kondo) and local-moment phases. For sufficiently strong impurity-boson coupling, however, the bare repulsion between a pair of electrons in the impurity level becomes an effective attraction, leading to quantum phase transitions between strong-coupling (charge Kondo) and local-charge phases. Even though the Hamiltonian exhibits different symmetries in the spin and charge sectors, the thermodynamic properties near the two types of quantum phase transition are closely related under spin-charge interchange. Moreover, the critical responses to a local magnetic field (for small impurity-boson coupling) and to an electric potential (for large impurity-boson coupling) are characterized by the same exponents, whose values place these quantum-critical points in the universality class of the pseudogap Anderson model. One specific case of the pseudogap Anderson-Holstein model may be realized in a double-quantum-dot device, where the quantum phase transitions manifest themselves in the finite-temperature linear electrical conductance.

Figure 1

Schematic ${\delta }_d-\Gamma$ phase diagrams of the pseudogap Anderson model [Eqs. (1, 2, 3, 4) with $\lambda =0$] for band exponents (a) $0<r<\frac{1}{2}$, (b) $r\ge \frac{1}{2}$. Generically, the system falls into either a local-moment phase (LM) or one of two asymmetric strong-coupling phases (ASC${}_{\pm }$). However, there is also a symmetric strong-coupling phase (the line labeled SSC) that is reached only for $0<r<\frac{1}{2}$ under conditions of strict particle-hole symmetry (${\delta }_d=0$) and for sufficiently large hybridization widths $\Gamma$.

Figure 2

Particle-hole-symmetric pseudogap Anderson-Holstein model near its spin-sector critical point C${}_{\mathrm{S}}$: (a) NRG energy $E_N$ vs even iteration number $N$ of the first excited multiplet having quantum numbers $S=1$, $Q=0$, calculated for $r=0.4$, $U=-2{\varepsilon }_d=0.5$, ${\omega }_0=0.1$, $\lambda =0.05<{\lambda }_0\simeq 0.158$, and seven values of $\Gamma -{\Gamma }_{c1}$ (with ${\Gamma }_{c1}\simeq 0.3166805$) labeled on the plot. (b) Schematic phase diagram on the $\Gamma -T$ plane for $\lambda <{\lambda }_0$, showing the $T=0$ transition between the local-moment ($\Gamma <{\Gamma }_{c1}$) and symmetric strong-coupling ($\Gamma >{\Gamma }_{c1}$) phases. Dashed lines mark the scale $T_1^{*}$ of the crossover from the intermediate-temperature quantum-critical regime to one or other of the stable phases. (c) Crossover scale $T_1^{*}=D{\Lambda }^{-N_1^{*}/2}$ vs $|\Gamma -{\Gamma }_{c1}|$ in the local-moment and symmetric strong-coupling phases, showing the power-law behavior described in Eq. (38). Here, $N_1^{*}$ is the interpolated value of $N$ at which $E_N$ in (a) leaves its quantum-critical range by crossing one or other of the horizontal dashed lines.

Figure 3

Particle-hole-symmetric pseudogap Anderson-Holstein model near its charge-sector critical point C${}_{\mathrm{C}}$: (a) NRG energy $E_N$ vs even iteration number $N$ of the first excited state having quantum numbers $S=Q=0$, calculated for $r=0.4$, $U=-2{\varepsilon }_d=0.5$, ${\omega }_0=0.1$, $\lambda =0.2>{\lambda }_0\simeq 0.158$, and seven values of $\Gamma -{\Gamma }_{c2}$ (with ${\Gamma }_{c2}\simeq 0.6878956$) labeled on the plot. (b) Schematic $\Gamma -T$ phase diagram for $\lambda >{\lambda }_0$, showing the $T=0$ transition between the local-charge and symmetric strong-coupling phases and the scale $T_2^{*}$ of the crossover from the quantum-critical regime to a stable phase. (c) Crossover scale $T_2^{*}=D{\Lambda }^{-N_2^{*}/2}$ vs $|\Gamma -{\Gamma }_{c2}|$ in the local-charge and symmetric strong-coupling phases, showing the power-law behavior described in Eq. (39). Here, $N_2^{*}$ is the interpolated value of $N$ at which $E_N$ in (a) leaves its quantum-critical range by crossing one or other of the horizontal dashed lines.

Figure 4

Phase boundaries of the particle-hole-symmetric pseudogap Anderson-Holstein model in zero magnetic field: variation with bosonic coupling $\lambda$ of the critical hybridization widths ${\Gamma }_{c1}$ and ${\Gamma }_{c2}$. Data for $U=-2{\varepsilon }_d=0.5$, ${\omega }_0=0.1$, and band exponents $r=0.1$, 0.2, 0.3, and 0.4 are scaled and offset for clarity: the quantities plotted are ${\Gamma }_{c1}\left(\lambda \right)/{\Gamma }_{c1}\left(0\right)+10r-1$ (empty symbols) and ${\Gamma }_{c2}\left(\lambda \right)/20{\Gamma }_{c1}\left(0\right)+10r-1$ (filled symbols). Solid lines show the prediction of Eq. (34), while dashed lines plot the form $A\left(r\right){\lambda }^{2(2-r)}$ suggested by Eq. (37) with a prefactor $A\left(r\right)$ determined by fitting values of ${\Gamma }_{c2}$ for $0.3\le \lambda \le 0.4$.

Figure 5

Thermodynamic properties of the particle-hole-symmetric pseudogap Anderson-Holstein model near its spin-sector critical point C${}_{\mathrm{S}}$: Temperature dependence of the impurity contribution to (a) the static spin susceptibility ${\chi }_{s,\mathrm{imp}}$ multiplied by temperature, (b) the static charge susceptibility ${\chi }_{c,\mathrm{imp}}$ multiplied by temperature, and (c) the entropy $S_{\mathrm{imp}}$, for $r=0.4$, $U=-2{\varepsilon }_d=0.5$, ${\omega }_0=0.1$, $\lambda =0.05<{\lambda }_0\simeq 0.158$, and the seven values of $\Gamma -{\Gamma }_{c1}$ labeled in the legend. Filled (open) symbols connected by guiding lines represent data in the local-moment (symmetric strong-coupling) phase, while thick lines without symbols show the critical properties at C${}_{\mathrm{S}}$. $N_s=3000$ states were retained after each NRG iteration.

Figure 6

Thermodynamic properties of the particle-hole-symmetric pseudogap Anderson-Holstein model near its charge-sector critical point C${}_{\mathrm{C}}$: Temperature dependence of the impurity contribution to (a) the static spin susceptibility ${\chi }_{s,\mathrm{imp}}$ multiplied by temperature, (b) the static charge susceptibility $T{\chi }_{c,\mathrm{imp}}$ multiplied by temperature, and (c) the entropy $S_{\mathrm{imp}}$, for $r=0.4$, $U=-2{\varepsilon }_d=0.5$, ${\omega }_0=0.1$, $\lambda =0.2>{\lambda }_0\simeq 0.158$, and the seven values of $\Gamma -{\Gamma }_{c2}$ labeled in the legend. Filled (open) symbols connected by guiding lines represent data in the local-charge (symmetric strong-coupling) phase, while thick lines without symbols show the critical properties at C${}_{\mathrm{C}}$. $N_s=3000$ states were retained after each NRG iteration.

Figure 7

Static local-spin response of the particle-hole-symmetric pseudogap Anderson-Holstein model near its spin-sector critical point C${}_{\mathrm{S}}$. Circles are NRG data for $r=0.4$, $U=-2{\varepsilon }_d=0.5$, ${\omega }_0=0.1$, and $\lambda =0.05$, at or near the critical hybridization width ${\Gamma }_{c1}\simeq 0.3166805$. Straight lines represent power-law fits. (a) Static local-spin susceptibility ${\chi }_{s,\mathrm{loc}}(T=0)$ vs $\Gamma -{\Gamma }_{c1}$ in the symmetric strong-coupling phase. (b) Local magnetization $M_{\mathrm{loc}}(h\to 0,T=0)$ vs ${\Gamma }_{c1}-\Gamma$ in the local-moment phase. Inset: Continuous vanishing of $M_{\mathrm{loc}}(h\to 0,T=0)$ as $\Gamma$ approaches ${\Gamma }_{c1}$ from below. (c) Local magnetization $M_{\mathrm{loc}}(\Gamma ={\Gamma }_{c1},T=0)$ vs local magnetic field $h$. (d) Static local-spin susceptibility ${\chi }_{s,\mathrm{loc}}(\Gamma ={\Gamma }_c)$ vs temperature $T$.

Figure 8

Static local charge response of the particle-hole-symmetric pseudogap Anderson-Holstein model near its charge-sector critical point C${}_{\mathrm{C}}$. Circles are NRG data for $r=0.4$, $U=-2{\varepsilon }_d=0.5$, ${\omega }_0=0.1$, and $\lambda =0.2$, at or near the critical hybridization width ${\Gamma }_{c2}\simeq 0.6878956$. Straight lines represent power-law fits. (a) Static local charge susceptibility ${\chi }_{c,\mathrm{loc}}(T=0)$ vs $\Gamma -{\Gamma }_{c2}$ in the symmetric strong-coupling phase. (b) Local charge $Q_{\mathrm{loc}}(\varphi \to 0,T=0)$ vs ${\Gamma }_{c2}-\Gamma$ in the local-charge phase. Inset: Continuous vanishing of $Q_{\mathrm{loc}}(\varphi \to 0,T=0)$ as $\Gamma$ approaches ${\Gamma }_{c2}$ from below. (c) Local charge $Q_{\mathrm{loc}}(\Gamma ={\Gamma }_{c2},T=0)$ vs local electric potential $\varphi$. (d) Static local charge susceptibility ${\chi }_{c,\mathrm{loc}}(\Gamma ={\Gamma }_{c2})$ vs temperature $T$.

Figure 9

Comparison of the static local responses of the particle-hole-symmetric pseudogap Anderson-Holstein model near its critical points C${}_{\mathrm{S}}$ and C${}_{\mathrm{C}}$: (a) Dependence of the local magnetization $M_{\mathrm{loc}}(h\to 0,T=0)$ on ${\Gamma }_{c1}-\Gamma$ and of the local charge $Q_{\mathrm{loc}}(\varphi \to 0,T=0)$ on ${\Gamma }_{c2}-\Gamma$, for $U=-2{\varepsilon }_d=0.5$, ${\omega }_0=0.1$, $\lambda =0.05$ (magnetic response) or 0.2 (charge response), and band exponents $r=0.2$ and $0.4$. (b) Static local-spin susceptibility ${\chi }_{s,\mathrm{loc}}(T;\Gamma ={\Gamma }_{c1})$ and static local charge susceptibility ${\chi }_{c,\mathrm{loc}}(T;\Gamma ={\Gamma }_{c2})$ vs temperature $T$. All parameters other than $\Gamma$ take the same values as in (a). Straight lines represent power-law fits.

Figure 10

Schematic renormalization-group flows on the $U_{\mathrm{eff}}-\Gamma$ plane for the particle-hole-symmetric pseudogap Anderson-Holstein model with a band exponent $0<r<\frac{1}{2}$. Trajectories with arrows represent the flow of the couplings with decreasing temperature. Dashed lines connecting unstable fixed points (open circles) separate the basins of attraction of stable fixed points (filled circles). The right and left dashed lines represent the boundary values ${\Gamma }_{c1}$ and ${\Gamma }_{c2}$ defined in Eqs. (34) and (37), respectively. The asymmetry of the flows about the line $U_{\mathrm{eff}}=0$ stems from differing symmetries of the model in the spin and charge sectors. See the text for a discussion of each fixed point.

Figure 11

Phase boundaries of the zero-field pseudogap Anderson-Holstein model on the $\Gamma -{\delta }_d$ plane for weak bosonic couplings $\lambda <{\lambda }_0=0.15812\left(1\right)$ and band exponent $r=0.4$ (left), $r=0.6$ (right). Data calculated for $U=0.5$, $h=0$, ${\omega }_0=0.1$, and the three values of $\lambda$ listed in the legend.

Figure 12

Phase boundaries of the particle-hole-symmetric pseudogap Anderson-Holstein model on the $\Gamma -h$ plane for strong bosonic couplings $\lambda >{\lambda }_0=0.15812\left(1\right)$ and band exponent $r=0.4$ (left), $r=0.6$ (right). Data calculated for $U=-2{\epsilon }_d=0.5$, ${\omega }_0=0.1$, and the three values of $\lambda$ listed in the legend.

Figure 13

Thermodynamic properties of the particle-hole-asymmetric pseudogap Anderson-Holstein model in zero magnetic field near the spin-sector critical point C${}_{-}$: Temperature dependence of the impurity contribution to (a) the static spin susceptibility ${\chi }_{s,\mathrm{imp}}$ multiplied by temperature, (b) the static charge susceptibility ${\chi }_{c,\mathrm{imp}}$ multiplied by temperature, and (c) the entropy $S_{\mathrm{imp}}$, for $r=0.4$, ${\varepsilon }_d=-0.2$, $U=0.5$, ${\omega }_0=0.1$, $\lambda =0.05<{\lambda }_0\simeq 0.158$, and the seven values of $\Gamma -{\Gamma }_{c1}$ labeled in the legend. Filled (open) symbols connected by guiding lines represent data in the local-moment (ASC${}_{-}$) phase, while thick lines without symbols show the critical properties at C${}_{-}$. $N_s=3000$ states were retained after each NRG iteration.

Figure 14

Thermodynamic properties of the particle-hole-symmetric pseudogap Anderson-Holstein model in nonzero local magnetic field near the charge-sector critical point C${}_{\downarrow }$: Temperature dependence of the impurity contribution to (a) the static spin susceptibility ${\chi }_{s,\mathrm{imp}}$ multiplied by temperature, (b) the static charge susceptibility $T{\chi }_{c,\mathrm{imp}}$ multiplied by temperature, and (c) the entropy $S_{\mathrm{imp}}$, for $r=0.4$, $U=-2{\varepsilon }_d=0.5$, $h=0.05$, ${\omega }_0=0.1$, $\lambda =0.2>{\lambda }_0\simeq 0.158$, and the seven values of $\Gamma -{\Gamma }_{c2}$ labeled in the legend. Filled (open) symbols connected by guiding lines represent data in the local-charge (ASC${}_{\downarrow }$) phase, while thick lines without symbols show the critical properties at C${}_{\downarrow }$. $N_s=3000$ states were retained after each NRG iteration.

Figure 15

Phase diagrams of a $U_2=0$ double-quantum-dot device with $U_1=0.5$, ${\Gamma }_1=0.05$, ${\varepsilon }_2=0$, ${\Gamma }_2=0.02$, and ${\omega }_0=0.1$: (a) The ${\lambda }^2-{\varepsilon }_1$ plane for $h=0$. (b) The ${\lambda }^2-h$ plane for ${\varepsilon }_1=-\frac{1}{2}{U}_1$. Note the near linearity of the phase boundaries in each panel when plotted against the square of the bosonic coupling.

Figure 16

Crossover temperature scales near QPTs in a $U_2=0$ double-quantum-dot device with $U_1=0.5$, ${\Gamma }_1=0.05$, ${\varepsilon }_2=0$, ${\Gamma }_2=0.02$, and ${\omega }_0=0.1$: (a) $T_1^{*}$ vs $\Delta {\varepsilon }_1={\varepsilon }_1-{\varepsilon }_{1,c}^{+}$ in the ASC${}_{-}$ phase for $h=0$ and $\lambda =0.1$. (b) $T_2^{*}$ vs $\Delta h=h-h_c$ in the ASC${}_{\downarrow }$ phase for ${\varepsilon }_1=-0.25$ and $\lambda =0.2$. Straight lines represent power-law fits, which yield the correlation-length exponents ${\nu }_1={\nu }_2=1$. $N_s=1000$ states were retained after each NRG iteration.

Figure 17

Thermodynamic properties of a $U_2=0$ double-quantum-dot device near a spin-sector QPT: Temperature dependence of the impurity contribution to (a) the static spin susceptibility ${\chi }_{s,\mathrm{imp}}$ multiplied by temperature, (b) the static charge susceptibility ${\chi }_{c,\mathrm{imp}}$ multiplied by temperature, and (c) the entropy $S_{\mathrm{imp}}$, for $U_1=0.5$, ${\Gamma }_1=0.05$, ${\varepsilon }_2=0$, ${\Gamma }_2=0.02$, $h=0$, ${\omega }_0=0.1$, $\lambda =0.1$, and nine values of $|{\varepsilon }_1-{\varepsilon }_{1,c}^{+}|$ where ${\varepsilon }_{1,c}^{+}\simeq -0.124985$. Properties at the transition (thick lines without symbols) are those expected at a level crossing between the local-moment phase (filled symbols) and the ASC${}_{-}$ phase. $N_s=3000$ states were retained after each NRG iteration.

Figure 18

Thermodynamic properties of a $U_2=0$ double-quantum-dot device near a charge-sector QPT: Temperature dependence of the impurity contribution to (a) the static spin susceptibility ${\chi }_{s,\mathrm{imp}}$ multiplied by temperature, (b) the static charge susceptibility ${\chi }_{c,\mathrm{imp}}$ multiplied by temperature, and (c) the entropy $S_{\mathrm{imp}}$, for $U_1=-2{\varepsilon }_1=0.5$, ${\Gamma }_1=0.05$, ${\varepsilon }_2=0$, ${\Gamma }_2=0.02$, ${\omega }_0=0.1$, $\lambda =0.2$, and nine values of $|h-h_c|$ where $h_c\simeq 0.284959$. Properties at the transition (thick lines without symbols) are those expected at a level crossing between the local-charge phase (filled symbols) and the ASC${}_{\downarrow }$ phase. $N_s=3000$ states were retained after each NRG iteration.

Figure 19

Linear conductance of a $U_2=0$ double-quantum-dot device near a spin-sector QPT: (a) Linear conductance $g$ vs $\Delta {\varepsilon }_1={\varepsilon }_1-{\varepsilon }_{1,c}^{+}$ for $U_1=0.5$, ${\Gamma }_1=0.05$, ${\varepsilon }_2=0$, ${\Gamma }_2=0.02$, $h=0$, ${\omega }_0=0.1$, $\lambda =0.1$, and different temperatures $T$ specified in the legend as multiples of $T_{K0}=7\times {10}^{-4}$. The retention of $N_s=1000$ states after each NRG iteration accounts for the small discrepancy between ${\varepsilon }_{1,c}^{+}\simeq -0.1249871$ and its value in the case $N_s=3000$ shown in Fig. 17. (b) The same data scaled as $(g-g_c)T_{K0}/T$ vs $\Delta {\varepsilon }_1/T$, where $g_c\left(T\right)$ is the conductance at ${\varepsilon }_1={\varepsilon }_{1,c}^{+}$. The solid line was obtained from Eq. (62) by approximating $A_q\left(x\right)=a_q$ and using values of $a_q$ and ${\omega }_q/({\varepsilon }_1-{\varepsilon }_1^{+})$ fitted from $A_{11}^{\sigma }\left(\omega \right)$.