Bose-Einstein condensation of deconfined spinons in two dimensions

The transition between the Néel antiferromagnet and the valence-bond solid state in two dimensions has become a paradigmatic example of deconfined quantum criticality, a non-Landau transition characterized by fractionalized excitations (spinons). We consider an extension of this scenario whereby the deconfined spinons are subject to a magnetic field. The primary purpose is to identify the exotic scenario of a Bose-Einstein condensate of spinons. We employ quantum Monte Carlo simulations of the $J-Q$ model with a magnetic field, and we perform a quantum field theoretic analysis of the magnetic field and temperature dependence of thermodynamic quantities. The combined analysis provides evidence for Bose-Einstein condensation of spinons and also demonstrates an extended temperature regime in which the system is best described as a gas of spinons interacting with an emergent gauge field.

Figure 1

Finite-size scaling of the stiffness ${\rho }_s(T,h)$ from QMC simulations with $j=j_c=0.045$ for (a) $h=0$, (b) $h=0.3$, (c) $h=0.6$, and (d) $h=1.0$. Error bars are omitted for clarity, but they are smaller than or equal to the markers. The black lines show the Nelson-Kosterlitz criterion, Eq. (2). Note the nonmonotonic size dependence at low $T$ in (b) and (c).

Figure 2

Phase diagram of the $J-Q$ model in the $h-T$ plane with $j=j_c=0.045$. Triangles represent the QMC values of $T_{\mathrm{BEC}}\left(h\right)$ extracted from ${\rho }_s$. Fine lines represent phase boundaries based on the field theoretic solution to ${\mathrm{\Delta }}_T\left(h_c\right)-\mu h_c=\delta$, as defined by Eq. (14), with each curve based on a different $\delta \in \{{10}^{-6},{10}^{-4}\}$; the predicted transition temperature decreases as $\delta$ decreases (see Sec. 4b).

Figure 3

BEC of spinons for $T<T_{\mathrm{BEC}}\left(h\right)$. Colored $•$ are QMC results for $E(T,h)+$offset compared to the field theory predictions for a BEC of spinons (solid line) and a BEC of magnons (broken line). The points $E(T_{\mathrm{BEC}},h), E(2T_{\mathrm{BEC}},h)$ are marked with $\square$ and $\circ$, respectively. Theory lines are numerically exact; QMC results' error bars are smaller than the markers.

Figure 4

Spinon gas for $T>T_{\mathrm{BEC}}\left(h\right)$. Colored $•$ are QMC results for $E(T,h)+$offset (same data as Fig. 3) compared to the field theory predictions for a gas of deconfined spinons (solid line) and a gas of magnons (broken line). The points $E(T_{\mathrm{BEC}},h), E(2T_{\mathrm{BEC}},h)$ are marked with $\square$ and $\circ$, respectively. Theory lines are numerically exact; QMC results' error bars are smaller than the markers.

Figure 5

Fitting phenomenological parameters. (a) Magnetic susceptibility over temperature, $\chi /T$: Points are QMC data from Ref. [25]; the line is Eq. (A1), taking $\mathrm{\Delta }(h=0,T)=\mathrm{\Theta }T$, with $\mathrm{\Theta }=0.59$ (which produces a constant). (b) Condensate energy vs field at $T=0, {\mathcal{E}}_0\left(h\right)$: Points are our QMC results; the line is a QFT prediction from Eq. (4) using parameters $\{\tilde{\alpha },c\}=\{\frac{2}{3}\pi c^2\left(0.32\right),2.42\}$. Everywhere we take $Q=1$.

Figure 6

Loop corrections to (a) the spinon Green's function Eq. (B1), and (b) the gauge field Green's functions Eqs. (B3) and (B4). Labeling of external momenta is used to help define the various self-energy components. Terms ${\mathrm{\Pi }}_2^{0i}$ are intentionally omitted. In both figures, we have used $B\equiv \mu h$.

Figure 7

Loop corrections to the partition function $lnZ$ [Eq. (C6)] due to the interaction terms in $ln{Z}_I$. (a) The standard partition function loop corrections with interactions taken from Eq. (C2). (b) The additional corrections due to the reorganization of the perturbation theory, Eq. (C3). Here the loops traced by red lines are the self-energy loops appearing in the reorganized interaction Lagrangian, Eq. (C5).