15 FIG. 11: Boson correlation 〈 b † $_{r}$b$_{r}$ ′ 〉 as a function of real space distance | r − r $^{′}$| . Top row: r belongs to sublattice A (where n$_{A}$ > n$_{B}$ = n$_{C}$ ) while r ′ includes sites on all three sublattices. Bottom row: r belongs to sublattice B while r ′ includes sites on sublattices B and C . Hexagonal clusters with 84, 144, and 48 sites are respectively used for Y (left column), uud (middle), and V perm (right) calculations. Acknowledgments We would like to thank J. Alicea and O. Starykh for many stimulating discussions and for reading and com- menting on the manuscript. This research is supported by the A. P. Sloan Foundation and the National Science Foundation through grant DMR-0907145. -0.15 -0.10 -0.05 0.00 0.05  1  2  3  4  5 Correlation Y: n=0.38 -0.15 -0.10 -0.05 0.00 0.05  1  2  3  4  5 uud: n=1/3 -0.24 -0.16 -0.08 0.00 0.08 0.16  1  2  3  4 Vperm: n=0.29 -0.10 -0.05 0.00 0.05 0.10  1  2  3  4  5 Correlation |r-r’| 0.00 0.01 0.02 0.03 0.04 0.05  1  2  3  4  5 |r-r’| 0.00 0.02 0.04 0.06 0.08  1  2  3  4 |r-r’| 1 T. Ono, H. Tanaka, H. Aruga Katori, F. Ishikawa, H. Mi- tamura, and T. Goto, Phys. Rev. B 67 , 104431 (2003). 2 T. Ono, H. Tanaka, O. Kolomiyets, H. Mitamura, T. Goto, K. Nakajima, A. Oosawa, Y. Koike, K. Kakurai, J. Klenke, et al., J. Phys. Condens. Matter 16 , S773 (2004). 3 T. Ono, H. Tanaka, O. Kolomiyets, H. Mitamura, F. Ishikawa, T. Goto, K. Nakajima, A. Oosawa, Y. Koike, J. Klenke, et al., Prog. Thoer. Phys. Supp 159 , 217 (2005). 4 Y. Fujii, T. Nakamura, H. Kikuchi, M. Chiba, T. Goto, S. Matsubara, K. Kodama, and M. Takigawa, Physica B 346-347 , 45 (2004). 5 Y. Fujii, H. Hashimoto, Y. Yasuda, H. Kikuchi, M. Chiba, S. Matsubara, and M. Takigawa, J. Phys. Condens. Matter 19 , 145237 (2007). 6 H. Tsujii, C. R. Rotundu, T. Ono, H. Tanaka, B. Andraka, K. Ingersent, and Y. Takano, Phys. Rev. B 76 , 060406(R) (2007). 7 N. A. Fortune, S. T. Hannahs, Y. Yoshida, T. E. Sherline, T. Ono, H. Tanaka, and Y. Takano, Phys. Rev. Lett. 102 , 257201 (2009). 8 A. V. Chubukov and D. I. Golosov, J. Phys. Condens. Mat- ter 3 , 69 (1991). 9 J. Alicea, A. V. Chubukov, and O. A. Starykh, Phys. Rev. Lett. 102 , 137201 (2009). 10 O. A. Starykh and L. Balents, Phys. Rev. Lett 98 , 077205 (2007). 11 T. Nikuni and H. Shiba, J. Phys. Soc. Jpn 64 , 3471 (1995). 12 M. Y. Veillette and J. T. Chalker, Phys. Rev. B 74 , 052402 (2006). 13 B. Bernu, P. Lecheminant, C. Lhuillier, and L. Pierre, Phys. Rev. B 50 , 10048 (1994). 14 A. Honecker, J. Schulenburg, and J. Richter, J. Phys. Con- dens. Matter 16 , S749 (2004). 15 S. Yoshikawa, K. Okunishi, M. Senda, and S. Miyahara, J. Phys. Soc. Jpn 73 , 1798 (2004). 16 S. Miyahara, K. Ogino, and N. Furukawa, Physica. B 378 , 587 (2006). 17 A. Honecker, J. Phys. Condens. Matter 11 , 4697 (1999). 18 T. Pardini and R. R. P. Singh, Phys. Rev. B 77 , 214433 (2008). 19 D. J. J. Farnell, R. Zinke, J. Schulenburg and J. Richter, J. Phys. Condens. Matter 21 , 406002 (2009). 20 M. Q. Weng, D. N. Sheng, Z. Y. Weng, and R. J. Bursill, Phys. Rev. B 74 , 012407 (2006). 21 H. C. Jiang, M. Q. Weng, Z. Y. Weng, D. N. Sheng, and L. Balents, Phys. Rev. B 79 , 020409(R) (2009). 22 D. A. Huse and V. Elser, Phys. Rev. Lett. 60 , 2531 (1988). 23 D. Heidarian, S. Sorella, and F. Becca, Phys. Rev. B 80 , 012404 (2009). 24 D. Heidarian and A. Paramekanti, Phys. Rev. Lett. 104 , 015301 (2010). 25 S. Yunoki and S. Sorella, Phys. Rev. B 74 , 014408 (2006). 26 F. Wang, F. Pollmann, and A. Vishwanath, Phys. Rev. Lett. 102 , 017203 (2009). 27 D. Ceperley, G. V. Chester, and M. H. Kalos, Phys. Rev. B 17 , 1070 (1978). 28 I. M. Wanless, Permanents , Chapter 31 in Handbook of Linear Algebra (ed. L. Hogben), Chapman & Hall/CRC