Unraveling Deconfined Quantum Criticality in Non-Hermitian Easy-Plane $J$-$Q$ Model

TL;DR

The study addresses whether a genuine deconfined quantum critical point (DQCP) exists between antiferromagnetic and valence-bond-solid phases in SU(2) spins by constructing a sign-problem-free non-Hermitian easy-plane $J$-$Q$ model (NHJQ) with parameters $Δ$, $δ$, and $g=Q/J$, and analyzing it with large-scale, unbiased quantum Monte Carlo using AFM and VBS order parameters and RG-invariant ratios. The results show that increasing the non-Hermitian strength $δ$ shifts the AFM–VBS transition to larger $g$ and weakens the first-order discontinuity, with critical exponents $ u$ converging and $ar{η}$ increasing, consistent with a continuous or pseudo-continuous transition near a complex fixed point described by a non-unitary CFT. This provides numerical evidence that the DQCP may be governed by fixed points residing in the complex plane and demonstrates a viable route to study non-unitary CFTs in microscopic, non-Hermitian spin models. The approach opens avenues to generalize to SU($N$) spins and further explore deconfined criticality in non-Hermitian quantum systems with potential experimental relevance.

Abstract

Deconfined quantum critical point (DQCP) characterizes the continuous transition beyond Landau-Ginzburg-Wilson paradigm, occurring between two phases that exhibit distinct symmetry breaking. The debate over whether genuine DQCP exists in physical SU(2) spin systems or the transition is weakly first-order has persisted for many years. In this letter, we construct a non-Hermitian easy-plane $J$-$Q$ model and perform sign-problem-free quantum Monte Carlo (QMC) simulation to explore the impact of non-Hermitian microscopic interactions on the transition that potentially features a DQCP. Our results demonstrate that the intensity of the first-order transitions significantly diminishes with the amplification of non-Hermitian interactions, serving as numerical evidence to support the notion that the transition in $J$-$Q$ model is quasi-critical, possibly in the vicinity of the fixed point governing DQCP in the complex plane, described by a non-unitary conformal field theory (CFT). The non-Hermitian interaction facilitates the approach towards such a complex fixed point in the parameter regime. Furthermore, our QMC study on the non-Hermitian J-Q model opens a new route to numerically investigating the nature of complex CFT in the microscopic model.

Figures (9)

• Figure 1: A sketch of the phase diagram of the NHJQ model is presented, depicting the transition lines with (solid green line) and without (dashed green line) non-Hermitian interaction. Here, $\Delta$ represents the strength of the easy-plane term, $\delta$ signifies the strength of non-Hermitian interaction, and $g = \frac{Q}{J}$ corresponds to the ratio between the strengths of two interaction terms in J-Q model. The green dot marks the transition point in the original $J$-$Q$ model. For clarity and comparison, the transition line from the Hermitian case $\delta = 0$ is also presented in the non-Hermitian $\delta > 0$ regime, and vice versa. The arrows indicate that the strength of the first-order transition becomes weaker. The green star denotes the transition point under non-Hermitian interaction at SU(2) limit, signifying the weakest first-order transition in the phase diagram, which potentially could be interpreted as a proximate continuous transition.
• Figure 2: At $\Delta=0.6$, the RG-invariant correlation ratios for the AFM order $R(L)$ vary as a function of the coupling ratio $g = Q/J$ for different system sizes, under different strengths of non-Hermitian coupling: (a) $\delta = 0.0$ and (b) $\delta = 0.6$, respectively. The intersection points in these plots demarcate the phase transition points. Panels (c) and (d) display the RG-invariant correlation ratios for the VBS order. The phase transition points from AFM to VBS states shift towards higher values of $g$ with an increase in the strength of the non-Hermitian coupling parameter $\delta$.
• Figure 3: (a,c) The order parameters for the AFM order are examined as functions of the system size at the critical transition point, for various strengths of non-Hermitian interactions at (a) $\Delta=0.6$ and at (c) $\Delta=0.0$. The analysis in (a) and (c) employs fitting curves based on the power-law function as $m_a^2(L)=m^2_0+b/L^c$. (b,d) The fitting results $m_0^2$ of order parameters with a series of minimum system sizes $L_{\text{min}}$ utilizing at (b) $\Delta=0.6$ and at (d) $\Delta=0.0$. The discontinuities $m_0^2$ in the order parameters are interpreted as indicators of the strength of first-order phase transitions. A decrease in these discontinuities suggests that the first-order transitions become less pronounced as the strength of the non-Hermitian interaction increases.
• Figure 4: At $\Delta=0.0$, the critical exponent (a) $\nu$ and (b) $\eta$ are examined as functions of the system size at the critical transition point for both the Hermitian case (red) and the non-Hermitian case (blue). (a) The analysis utilizes fitting curves based on the power-law function as $\nu(L)=\nu+b/L^c$, employing a minimum system size of $L_{\text{min}}=40$. The empty data points are excluded from the fitting. The fitting results is $\nu_0(\delta=0.6)\approx0.49(4)$. (b) The analysis utilizes fitting curves based on the power-law function as $\eta(L)=\eta+b/L^c$, employing a minimum system size of $L_{\text{min}}=24$. The fitting results are $\eta_0(\delta=0.0)\approx0.38(1)$ and $\eta_0(\delta=0.6)\approx0.41(1)$.
• Figure S1: At $\Delta=0.6$, the RG-invariant correlation ratio for (a-d) the AFM order and (e-h) the VBS order, $R(L)$, measured across various system sizes, vary as a function of the coupling ratio $g = Q/J$, under different strengths of non-Hermitian coupling $\delta=0.0,0.2,0.6$ and 1.0. The intersection points in these plots demarcate the phase transition points. The phase transition points shift towards higher values with an increase in the strength of the non-Hermitian coupling parameter $\delta$.