Quantum Lifshitz criticality in a frustrated two-dimensional XY model

Antiferromagnetic quantum spin systems can exhibit a transition between collinear and spiral ground states, driven by frustration. Classically this is a smooth crossover and the crossover point is termed a Lifshitz point. Quantum fluctuations change the nature of the transition. In particular, it has been argued previously that in the two-dimensional (2D) case a spin liquid (SL) state is developed in the vicinity of the Lifshitz point, termed a Lifshitz SL. In the present work, using a field theory approach, we solve the Lifshitz quantum phase transition problem for the 2D frustrated XY model. Specifically, we show that, unlike the $SU\left(2\right)$ symmetric Lifshitz case, in the XY model, the SL exists only at the critical point. At zero temperature we calculate nonuniversal critical exponents in the Néel and in the spin spiral state and relate these to properties of the SL. We also solve the transition problem at a finite temperature and discuss the role of topological excitations.

Figure 1

The $O\left(2\right)J_1\text{−}{J}_3$ model. At $J_3/J_1<0.4.$ the ground state is the Néel state, and at $J_3/J_1>0.4$ the ground state is the spin spiral state. The spin liquid is realized at one point, $J_3/J_1\approx 0.4$. (a) Energy of the ground state vs $J_3/J_1$. (b) The spin spiral wave vector squared vs $J_3/J_1$. (c) Static magnetization $M$ vs $J_3/J_1$.

Figure 2

The $O\left(3\right)J_1\text{−}{J}_3$ model, results of Ref. [6] At $J_3/J_1<0.33,$ the ground state is the Néel state, and at $J_3/J_1>0.55$ the ground state is the spin spiral state. The quantum disordered (spin liquid) phase is realized within the range $0.33<J_3/J_1<0.55$. (a) Energy of the ground state vs $J_3/J_1$. (b) The spin spiral wave vector squared vs $J_3/J_1$. (c) Static magnetization vs $J_3/J_1$.

Figure 3

Single loop self-energy diagram for $\theta$-field generated by the quartic term in Eq. (5).

Figure 4

Sketch of the $T\text{−}\rho$ phase diagrams of the quantum Lifshitz transition in XY model for a finite system size $L$. Neel and Spin spiral phases at $T>0$ possess a quasi-long-range order and algebraic spin-spin correlations.

Figure 5

Two-loop self-energy diagram that generates width of the magnon.