Emergence of localized magnetic moments near antiferromagnetic quantum criticality

We revisit an antiferromagnetic quantum phase transition with $\mathit{Q}=2{\mathit{k}}_F$, where $\mathit{Q}$ is an ordering wave vector and ${\mathit{k}}_F$ is a Fermi momentum. Reformulating the Hertz-Moriya-Millis theory within the strong-coupling approach to diagonalize the spin-fermion coupling term and performing the scaling analysis for an effective-field theory with quantum corrections in the Eliashberg approximation, we propose an interacting fixed point for this antiferromagnetic quantum phase transition, where antiferromagnetic spin fluctuations become locally critical to interact with renormalized electrons. The emergence of local quantum criticality suggests a mechanism of $\omega /T$ scaling for antiferromagnetic quantum criticality, generally forbidden in the context of the Hertz-Moriya-Millis theory.

Figure 1

A schematic phase diagram for antiferromagnetic quantum phase transitions. The $\mathit{x}$ axis is the inverse of a spin-fermion coupling constant, and the $\mathit{y}$ axis is temperature, constituting a conventional phase diagram. Here, we suggest to add an additional axis into this phase diagram, which corresponds to an ordering wave vector, but appropriately changeable to be related to Fermi-surface nesting. SDW and FL denote spin-density wave and Fermi liquid, respectively. U1SSR FP represents a U(1) slave spin-rotor fixed point, and HMM and LQC express Hertz-Moriya-Millis quantum criticality and local quantum criticality, respectively. It is a conventional view that the fixed point of an antiferromagnetic quantum critical point with $\mathit{Q}\ne 2{\mathit{k}}_F$ is described by the HMM theory. Applying the U(1) slave spin-rotor formulation to this antiferromagnetic quantum phase transition, regarded to be a strong-coupling approach, we find the possibility for the emergence of another fixed point, where the HMM theory becomes modified to contain relevant interactions between transverse spin fluctuations and renormalized electrons (Sec. 3b). It is not possible to clarify the condition for which the fixed point will be realized between the HMM and U(1) slave spin-rotor fixed points at present. An essential aspect of the present study is that the structure of the HMM theory breaks down at an antiferromagnetic quantum critical point with $\mathit{Q}=2{\mathit{k}}_F$, where interactions between longitudinal spin fluctuations and renormalized electrons, usually referred to as the spin-fermion coupling, turn out to be irrelevant (Secs. 3c and 3d). This situation differs from the case with $\mathit{Q}\ne 2{\mathit{k}}_F$, where the structure of the HMM theory is preserved at least partially for the U(1) slave spin-rotor fixed point. The role of transverse spin fluctuations becomes more dominant in the antiferromagnetic quantum criticality with $\mathit{Q}=2{\mathit{k}}_F$, giving rise to a critical field theory within the U(1) slave spin-rotor formulation, where renormalized electrons interact with localized transverse spin fluctuations (Sec. 3d).

Figure 2

A schematic diagram of a Fermi surface in the double-patch construction. Red curved lines denote a pair of Fermi surfaces, connected by $2{\mathit{k}}_F$, which are filled with electrons. A coordinate system is defined as the figure for each patch of $s=\pm$.

Figure 3

Feynmann diagrams in the Eliashberg approximation. Renormalized electrons $f_{s\sigma }$ and longitudinal antiferromagnetic fluctuations ${\varphi }_2$ consist of the Hertz-Moriya-Millis theory while transverse spin fluctuations $z_{\sigma }$ and U(1) gauge fields $a$ appear in the U(1) slave spin-rotor formulation. Longitudinal ferromagnetic fluctuations ${\varphi }_1$ give rise to consistency for dynamics of transverse spin fluctuations $z_{\sigma }$. All interaction vertices are given by diagrams with three lines, which characterize the U(1) slave spin-rotor theory. Self-energy corrections of three bosonic fields, ${\varphi }_2$, ${\varphi }_1$, and $a$ are given by polarization bubbles of renormalized electrons and transverse spin fluctuations. Self-energy corrections of renormalized electrons and transverse spin fluctuations are given by typical one-loop diagrams with a bosonic line. Since longitudinal ferromagnetic fluctuations are gapped, one may neglect all diagrams involved with ${\varphi }_1$ except for the spinon self-energy, where it makes dynamics of spinons consistent. In the lattice-model construction, dynamics of gauge fluctuations are neglected, not incorporated into the free energy.

Figure 4

A schematic diagram for renormalization-group flows in an antiferromagnetic quantum phase transition with $\mathit{Q}\ne 2{\mathit{k}}_F$. The U(1) slave spin-rotor theory suggests the possible existence of another fixed point beyond the HMM theory on the HMM critical surface. When spinons are forced to condense, the HMM fixed point would be realized. On the other hand, if the dynamics of such transverse spin fluctuations becomes localized, we expect to reach the U(1) slave spin-rotor fixed point, differentiated from the HMM. In this study, the nature of the U(1) slave spin-rotor fixed point is not clarified.

Figure 5

A schematic diagram for renormalization-group flows in an antiferromagnetic quantum phase transition with $\mathit{Q}=2{\mathit{k}}_F$. The U(1) slave spin-rotor formulation allows three possible fixed points. The first fixed point is realized by the condensation of spinons, described by the HMM scenario, but the dynamics of longitudinal antiferromagnetic fluctuations becomes localized at this fixed point. Since the spin-fermion coupling turns out to be irrelevant (not shown here), the local quantum criticality given by longitudinal antiferromagnetic fluctuations is decoupled from the dynamics of renormalized electrons in the low-energy limit. The second is described by Eq. (42), but not a critical field theory, where the dynamics of all bosonic excitations of gauge fields, longitudinal antiferromagnetic fluctuations, and transverse spin fluctuations becomes localized and $g_2$ is irrelevant, but $\frac{g_1{g}_z}{N_{\sigma }{m}_1^2}$ is relevant while $e_f$ is marginal (left). The last fixed point is well defined, described by the critical field theory of Eq. (46) in terms of renormalized electrons and localized transverse spin fluctuations. All interaction vertices turn out to be irrelevant, except for the holon-spinon coupling $\frac{g_1{g}_z}{N_{\sigma }{m}_1^2}$, marginal in the one-loop level (right).