Continuous order-to-order quantum phase transitions from fixed-point annihilation

TL;DR

This work introduces a general mechanism for continuous order-to-order quantum phase transitions that does not rely on deconfined fractionalization. The mechanism hinges on the collision and annihilation of a quantum critical fixed point with an infrared fixed point, which reshapes RG flow to erase the disordered phase and yield a surviving critical point between two distinct orders. The authors demonstrate this in three-dimensional Luttinger fermion systems (pyrochlore iridates) via a 4-ε plus dynamical bosonization RG analysis, showing fixed-point annihilation at a critical flavor number $N_c \,=\, 1.856$, leaving a continuous AIAO Weyl semimetal to nematic topological insulator transition for $N< N_c$, with exact exponents governed by the annihilation-surviving fixed point. They also discuss kagome quantum magnets, where a similar fixed-point collision in a QED$_3$-GN framework yields a continuous Ising-tuned transition between VBS and chiral spin liquid phases, and they highlight distinctive experimental signatures, such as pronounced asymmetry in energy scales across the transition. The results suggest a broad universality class for order-to-order transitions driven by fixed-point annihilation and point to observable consequences in both condensed-matter and high-energy contexts.

Abstract

A central concept in the theory of phase transitions beyond the Landau-Ginzburg-Wilson paradigm is fractionalization: the formation of new quasiparticles that interact via emergent gauge fields. This concept has been extensively explored in the context of continuous quantum phase transitions between distinct orders that break different symmetries. We propose a mechanism for continuous order-to-order quantum phase transitions that operates independently of fractionalization. This mechanism is based on the collision and annihilation of two renormalization group fixed points: a quantum critical fixed point and an infrared stable fixed point. The annihilation of these fixed points rearranges the flow topology, eliminating the disordered phase associated with the infrared stable fixed point and promoting a second critical fixed point, unaffected by the collision, to a quantum critical point between distinct orders. We argue that this mechanism is relevant to a broad spectrum of physical systems. In particular, it can manifest in Luttinger fermion systems in three spatial dimensions, leading to a continuous quantum phase transition between an antiferromagnetic Weyl semimetal state, which breaks time-reversal symmetry, and a nematic topological insulator, characterized by broken lattice rotational symmetry. This continuous antiferromagnetic-Weyl-to-nematic-insulator transition might be observed in rare-earth pyrochlore iridates $R_2$Ir$_2$O$_7$. Other possible realizations include kagome quantum magnets, quantum impurity models, and quantum chromodynamics with supplemental four-fermion interactions.

Figures (14)

• Figure 1: Mechanism for continuous order-to-order transition from fixed-point annihilation. (a) Schematic RG flow in the space spanned by two couplings, $G_1$ and $G_2$, for $N > N_\mathrm{c}$. Arrows indicate flow towards infrared. Beyond an interacting disordered phase (gray) at small couplings, governed by the fixed point D, there are two ordered phases (red and blue) in the strong-coupling regime, governed by the fixed points O$_1$ and O$_2$. The disordered phase is separated from the two ordered phases by continuous phase transition lines (purple), governed by the quantum critical fixed points QCP$_1$ and QCP$_2$. As $N$ decreases, the fixed points D and QCP$_2$ approach each other and eventually collide at a critical value $N_\mathrm{c}$. (b) Same as (a), but for $N < N_\mathrm{c}$. The fixed-point annihilation at $N_\mathrm{c}$ alters the flow topology, eliminating the disordered phase and leaving behind a single continuous phase transition line (purple) between two phases with distinct orders.
• Figure 2: (a) Coefficient $I_{5/2}$ of the nonanalytic term $\propto \chi^{5/2}$ in the mean-field energy, for fixed anisotropy parameter $\delta = -1/2$, as function of polar angle $\alpha$ in $(\phi, \varphi)$ field space. Here, $\alpha \in \{0, \pi\}$ ($\alpha \in \{\pi/2, 3\pi/2\}$) corresponds to the AIAO (nematic) sector. $I_{5/2}$ is positive for all $\alpha \in [0,2\pi)$, justifying the truncation of the mean-field energy after the nonanalytic term. (b) Rescaled coefficient $\overline{I_{5/2}}$ as function of rescaled polar angle $\overline \alpha$, for fixed $\delta = -1/2$, unveiling three global minima at $\overline\alpha = 0, \pi$ (AIAO sector) and $\overline\alpha = \frac{\pi}{2}$ (nematic sector). This excludes the possibility of coexisting AIAO and nematic order.
• Figure 3: (a) Phase diagram of Luttinger model in the low-temperature limit as function of short-range couplings $G_1$ and $G_2$ for fixed representative values of the charge $e^2 = 3 \pi^2 \Lambda/2$ and the anisotropy parameter $\delta = -1/2$ from mean-field theory, which becomes exact in the limit $N \to \infty$. The Luttinger-Abrikosov-Beneslavskii (LAB) phase at small couplings realizes a three-dimensional non-Fermi liquid (gray). For sufficiently large $G_1$, a Weyl semimetal emerges, characterized by all-in-all-out (AIAO) antiferromagnetic order on the pyrochlore lattice. For sizable $G_2$, a nonmagnetic nematic topological insulator is stabilized. Color scale indicates magnitudes of AIAO order parameter $\langle \phi \rangle \propto \langle \psi^\dagger \gamma_{45} \psi \rangle$ (red) and nematic order parameter $\langle \varphi \rangle \propto \langle \psi^\dagger \gamma_{5} \psi \rangle$ (blue). Single (double) lines indicate continuous (discontinuous) phase transitions. (b) Same as (a), but for $N=10$ from RG analysis, qualitatively agreeing with the mean-field result. (c) Same as (b), but for $N=2$. The LAB phase shrinks in favor of the nematic phase as the LAB fixed point and the quantum critical fixed point associated with the nematic instability approach each other. (d) Same as (b), but for $N=1$, corresponding to the case relevant for the pyrochlore iridates $R_2$Ir$_2$O$_7$. The LAB phase is removed as a consequence of the fixed-point annihilation, leaving behind a continuous phase transition line between the AIAO antiferromagnetic Weyl semimetal and the nematic topological insulator for small $G_2$.
• Figure 4: Feynman diagrams at one-loop order contributing to (a)--(c) the fermion anomalous dimensions $\eta_1$, $\eta_\psi$ and the anisotropy parameter renormalization $\Delta\delta$, (d) the Coulomb anomalous dimension $\eta_a$, (e) the AIAO order-parameter anomalous dimension $\eta_\phi$ and the AIAO tuning parameter renormalization $\Delta r_1$, (f,g) the nematic order-parameter anomalous dimension $\eta_\varphi$, the parameter renormalization $\Delta c$, and the nematic tuning parameter renormalization $\Delta r_2$, (h)--(j) the charge renormalizations $\Delta e$, (k)--(m) the AIAO Yukawa vertex renormalizations $\Delta g_1$, (n)--(p) the nematic Yukawa vertex renormalizations $\Delta g_2$, (q)--(s) the self-interaction renormalizations $\Delta \lambda$. The contributions to the flow of $e^2$ from (a) and (h), from (b) and (i), and from (c) and (j), respectively, cancel as a consequence of a Ward identity.
• Figure 5: Complete set of four-fermion box diagrams at the one-loop order. In the dynamical bosonization scheme, diagrams (a)-(f) and (m)-(o) contribute to the flow of the nematic Yukawa coupling $g_2$. In contrast, diagrams (g)-(i) and (j)-(l) do not contribute to the nematic channel when adding the individual contributions.