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According to the exact analytical Bethe-ansatz solution of the one-dimensional spin-1/2 XXZ model, we perform a numerical study of one-dimensional spinless fermions in an optical lattice in the presence of harmonic potential by using a local density approximation. We study the density profile, and obtain the -u phase diagram (here =NfV2/t is the characteristic dimensionless density and u=V/t is the interaction strength scaled in units of the hopping parameter t). With the increases of particle density and nearest-neighbour interaction, the system undergoes five different quantum phases. Through calculating the thermodynamic stiffness S, we find that it can be used as an order parameter. Its singular points can measure the quantum phase transitions in such a confined system.
[1] Pierre F, Maldague 1997 Phys. Rev. B 16 2437
[2] Loh E Y, Campbell D K 1998 Synth. Met. 27A 499
[3] Schulz H J 1990 Phys. Rev. Lett. 64 2831
[4] Stafford C A, Millis A J, Shastry B S 1990 Phys. Rev. B 43 13660
[5] Fye R M, Martins M J, Scalapino D J, Wagner J, Hanke W 1991 Phys. Rev. B 44 6909
[6] Giamarchi T, Millis A J 1992 Phys. Rev. B 46 9325
[7] Carmelo J M P, Horsch P 1992 Phys. Rev. Lett. 68 871 Carmelo J M P, Horsch P, Ovchinnikov A A 1992 Phys. Rev. B 46 14728
[8] Wang Y H, Xia Y 2009 Acta Phys. Sin. 58 7479 (in Chinese)[王彦辉, 夏云 2009 物理学报 58 7479]
[9] Song J, Cao Z L 2005 Acta Phys. Sin. 54 696 (in Chinese)[宋军, 曹卓良 2005 物理学报 54 696]
[10] Wei B B, Cao J P, Gu S J, Lin H Q arXiv:0807.2154v1.
[11] Guan L, Chen S, Wang Y, Ma Z Q 2009 Phys. Rev. Lett. 102 160402
[12] Gao X L 2010 Phys. Rev. B 81 104306
[13] Zhang S J, Jiang J J, Liu Y J 2008 Acta Phys. Sin. 57 531 (in Chinese)[张松俊, 蒋建军, 刘拥军 2008 物理学报 57 531]
[14] Jordan P, Wigner E 1928 Z. Phys. 47 631 Yang C N, Yang C P 1966 Phys. Rev. 150 321
[15] Bergkvist S, Henelius P, Rosengren A 2004 Phys. Rev. A 70 053601
[16] Lima N A, Oliveira L N, Cappelle K 2002 Europhys. Lett. 60 601
[17] Lima N A, Silva M F, Oliveira L N, Cappelle K 2003 Phys. Rev. Lett. 90 146402
[18] Gao X L Rizzi M, Polini M, Fazio R, Tosi M P, Campo Jr. V L, Capelle K 2007 Phys. Rev. Lett. 98 030404
[19] Gao X L Polini M, Tosi M P, Campo V L, Capelle K, Rigol M 2007 Phys. Rev. B 73 165120
[20] Alcaraz F C, Capelle K 2007 Phys. Rev. B 76 035109
[21] Schenk S, Dzierzawa M, Schwab P, Eckern U 2008 Phys. Rev. B 78 165102
[22] Gaudin M 1975 Phys. Rev. Lett. 26 1301
[23] Bethe H Z 1931 Phys. 71 205
[24] Kohn W, Sham L J 1965 Phys. Rev. 140 A 1133
[25] Capelle K, Vignale G 2002 Phys. Rev. B 65 113106
[26] Elliott H, Lieb, Wu F Y 1968 Phys. Rev. Lett. 20 1445
[27] Peres N M R, Sacramento P D, Campbell D K, Carmelo J M P 1998 Phys. Rev. B 59 11
[28] Gu S J 2010 Int. J Mod. Phys. B 24 4371 Hu J H, Wang J J, Gao X L, Okumura M, Igarashi R, Yamada S, Machida M 2010 Phys. Rev. B 82 014202
[29] Scarola V W, Pollet L, Oitmaa J, Troyer M 2009 Phys. Rev. Lett. 102 135302, 135305
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[1] Pierre F, Maldague 1997 Phys. Rev. B 16 2437
[2] Loh E Y, Campbell D K 1998 Synth. Met. 27A 499
[3] Schulz H J 1990 Phys. Rev. Lett. 64 2831
[4] Stafford C A, Millis A J, Shastry B S 1990 Phys. Rev. B 43 13660
[5] Fye R M, Martins M J, Scalapino D J, Wagner J, Hanke W 1991 Phys. Rev. B 44 6909
[6] Giamarchi T, Millis A J 1992 Phys. Rev. B 46 9325
[7] Carmelo J M P, Horsch P 1992 Phys. Rev. Lett. 68 871 Carmelo J M P, Horsch P, Ovchinnikov A A 1992 Phys. Rev. B 46 14728
[8] Wang Y H, Xia Y 2009 Acta Phys. Sin. 58 7479 (in Chinese)[王彦辉, 夏云 2009 物理学报 58 7479]
[9] Song J, Cao Z L 2005 Acta Phys. Sin. 54 696 (in Chinese)[宋军, 曹卓良 2005 物理学报 54 696]
[10] Wei B B, Cao J P, Gu S J, Lin H Q arXiv:0807.2154v1.
[11] Guan L, Chen S, Wang Y, Ma Z Q 2009 Phys. Rev. Lett. 102 160402
[12] Gao X L 2010 Phys. Rev. B 81 104306
[13] Zhang S J, Jiang J J, Liu Y J 2008 Acta Phys. Sin. 57 531 (in Chinese)[张松俊, 蒋建军, 刘拥军 2008 物理学报 57 531]
[14] Jordan P, Wigner E 1928 Z. Phys. 47 631 Yang C N, Yang C P 1966 Phys. Rev. 150 321
[15] Bergkvist S, Henelius P, Rosengren A 2004 Phys. Rev. A 70 053601
[16] Lima N A, Oliveira L N, Cappelle K 2002 Europhys. Lett. 60 601
[17] Lima N A, Silva M F, Oliveira L N, Cappelle K 2003 Phys. Rev. Lett. 90 146402
[18] Gao X L Rizzi M, Polini M, Fazio R, Tosi M P, Campo Jr. V L, Capelle K 2007 Phys. Rev. Lett. 98 030404
[19] Gao X L Polini M, Tosi M P, Campo V L, Capelle K, Rigol M 2007 Phys. Rev. B 73 165120
[20] Alcaraz F C, Capelle K 2007 Phys. Rev. B 76 035109
[21] Schenk S, Dzierzawa M, Schwab P, Eckern U 2008 Phys. Rev. B 78 165102
[22] Gaudin M 1975 Phys. Rev. Lett. 26 1301
[23] Bethe H Z 1931 Phys. 71 205
[24] Kohn W, Sham L J 1965 Phys. Rev. 140 A 1133
[25] Capelle K, Vignale G 2002 Phys. Rev. B 65 113106
[26] Elliott H, Lieb, Wu F Y 1968 Phys. Rev. Lett. 20 1445
[27] Peres N M R, Sacramento P D, Campbell D K, Carmelo J M P 1998 Phys. Rev. B 59 11
[28] Gu S J 2010 Int. J Mod. Phys. B 24 4371 Hu J H, Wang J J, Gao X L, Okumura M, Igarashi R, Yamada S, Machida M 2010 Phys. Rev. B 82 014202
[29] Scarola V W, Pollet L, Oitmaa J, Troyer M 2009 Phys. Rev. Lett. 102 135302, 135305
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