Phases of strongly-interacting bosons on a two-leg ladder
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1 Phases of strongly-interacting bosons on a two-leg ladder Marie Piraud Arnold Sommerfeld Center for Theoretical Physics, LMU, Munich April 20, 2015 M. Piraud Phases of strongly-interacting bosons on a two-leg ladder 1/ 32
2 My collaborators LMU, Munich Leibniz U., Hannover F. Heidrich-Meisner U. Schollwöck S. Greschner T. Vekua I. P. McCulloch (Brisbane) M. Piraud Phases of strongly-interacting bosons on a two-leg ladder 2/ 32
3 Electron in a magnetic field Cyclotron orbit M. Piraud Phases of strongly-interacting bosons on a two-leg ladder 3/ 32
4 Electron in a magnetic field Cyclotron orbit Quantum Hall Effect : quantization of conductance Landau levels (highly degenerate) E n = ω c (n+1/2) K. von Klitzing, RMP (1986) M. Piraud Phases of strongly-interacting bosons on a two-leg ladder 3/ 32
5 Electron in a magnetic field Cyclotron orbit Quantum Hall Effect : quantization of conductance Landau levels (highly degenerate) E n = ω c (n+1/2) K. von Klitzing, RMP (1986) Modifies transport properties & creates Topological transitions With interactions : Fractional QHE Unusual excitations, fractional charge and statistics M. Piraud Phases of strongly-interacting bosons on a two-leg ladder 3/ 32
6 Type-II Superconductor in a magnetic field Meissner effect W. Meissner & R. Ochsenfeld, Naturwissenschaften (1933) Meissner state H < H c1 Vortex lattice H c1 < H < H c2 Source : Quantumlevitation.com M. Piraud Phases of strongly-interacting bosons on a two-leg ladder 4/ 32
7 Ultra-cold atoms in synthetic gauge fields Engineer effective B field or SO coupling on lattices : In Synthetic dimensions C. Chin & E.J. Mueller, PRL Viewpoint (2013) Realizations : A. Celi et al., PRL (2014) M. Mancini et al., arxiv1502 :02495 B. Stuhl et al., arxiv1502 :02496 Interplay interactions/ gauge fields/ dimensionality Promising candidate to search for new phases of matter. J. Dalibard & F. Gerbier, RMP (2011) UCA permit to study bosons in magnetic fields Ladders = simplest 2D system M. Piraud Phases of strongly-interacting bosons on a two-leg ladder 5/ 32
8 Weakly-interacting bosons on ladders Vortex/ Meissner Phase Transition (ladders with uniform π/2 magnetic flux) Theo : E. Orignac & T. Giamarchi, PRB (2001), D. Hügel & B. Paredes, PRA (2014) Exp : M. Atala et al., Nat. Phys. (2014) Meissner L R L Vortex R M. Piraud Phases of strongly-interacting bosons on a two-leg ladder 6/ 32
9 Weakly-interacting bosons on ladders Vortex/ Meissner Phase Transition (ladders with uniform π/2 magnetic flux) Theo : E. Orignac & T. Giamarchi, PRB (2001), D. Hügel & B. Paredes, PRA (2014) Exp : M. Atala et al., Nat. Phys. (2014) M. Piraud Phases of strongly-interacting bosons on a two-leg ladder 6/ 32
10 Interacting bosons on ladders in uniform flux A few theoretical results Field-theoretical studies E. Orignac & T. Giamarchi, PRB (2001) A. Tokuno & A. Georges, NJP (2014) Mean-field approach R. Wei & E. J. Mueller, PRA (2014) Chiral Mott Insulator A. Dhar et al., PRA (2012); PRB (2013) Mott Insulator with Meissner Currents Study of the momentum distribution Laughlin-1/2 states A. Petrescu & K. Le Hur, PRL (2013) M.-C. Cha & J.-G. Shin, PRA (2011) A. Petrescu & K. Le Hur, PRB (2015) M. Piraud Phases of strongly-interacting bosons on a two-leg ladder 7/ 32
11 Interacting bosons on ladders in uniform flux A few theoretical results Field-theoretical studies E. Orignac & T. Giamarchi, PRB (2001) A. Tokuno & A. Georges, NJP (2014) Mean-field approach R. Wei & E. J. Mueller, PRA (2014) Chiral Mott Insulator A. Dhar et al., PRA (2012); PRB (2013) Mott Insulator with Meissner Currents Study of the momentum distribution Laughlin-1/2 states A. Petrescu & K. Le Hur, PRL (2013) M.-C. Cha & J.-G. Shin, PRA (2011) A. Petrescu & K. Le Hur, PRB (2015) Explore the phase diagram numerically M. Piraud Phases of strongly-interacting bosons on a two-leg ladder 7/ 32
12 Interacting bosons on a two-leg ladder in a uniform flux : Model H K = J L l=1,2;r=1 ( a l,r+1 a l,r +H.c. ) J L r=1 ( ) e irφ a 1,r a 2,r +H.c. H U = U 2 L l=1,2;r=1 n l,r (n l,r 1) (2,1) (1,1). (2,r) J e irφ. (1,r) φ J.. J e i[r+1]φ U (2,L) (1,L) M. Piraud Phases of strongly-interacting bosons on a two-leg ladder 8/ 32
13 Density Matrix Renormalization Group Ground state and first excited states of 1D many-body systems S. R. White, Phys. Rev. Lett. 69, 2863 (1992) U. Schollwöck, Rev. Mod. Phys. 77, 259 (2005) Successfully applied to (quasi-)2d systems S. Yan et al., Science (2011) S. Depenbrock et al., PRL (2012) E. M. Stoudenmire et al., Annu. Rev. Cond. Mat. Phys. (2012) M. Piraud Phases of strongly-interacting bosons on a two-leg ladder 9/ 32
14 Outline Hard-Core Bosons M. P., F. Heidrich-Meisner, I. P. McCulloch, S. Greschner, T. Vekua and U. Schollwöck, PRB (R) (2015) Phase Diagram : MI and SF, Meissner and vortex phases A few observables : chiral current and vortex density Finite interactions S. Greschner, M. P., F. Heidrich-Meisner, I. P. McCulloch, U. Schollwöck and T. Vekua, submitted (2015) Phase Diagram : vortex lattice phases Chiral current : spontaneous reversal! M. Piraud Phases of strongly-interacting bosons on a two-leg ladder 10/ 32
15 Outline Hard-Core Bosons M. P., F. Heidrich-Meisner, I. P. McCulloch, S. Greschner, T. Vekua and U. Schollwöck, PRB (R) (2015) Phase Diagram A few observables Finite interactions S. Greschner, M. P., F. Heidrich-Meisner, I. P. McCulloch, U. Schollwöck and T. Vekua, submitted (2015) Phase Diagram Chiral current M. Piraud Phases of strongly-interacting bosons on a two-leg ladder 11/ 32
16 Equation of state Filling n = N/2L vs. chemical potential µ 0.5 φ/π = 0.4 φ/π = 0.6 φ/π = 0.8 φ = π n 0.25 increasing φ/π 0 µ [arb. units] Plateau (at n = 0.5, and at n = 0.25 for φ = π) incompressible state Kink change in the number of gapless excitations (+ use S vn ) M. Piraud Phases of strongly-interacting bosons on a two-leg ladder 12/ 32
17 Fractionally filled Mott insulator Hard-core bosons, no flux ( ) Strong rungs (J J) : H K J r a 1,r a 2,r +H.c. s z = 0 triplet ( 1,0 r + 0,1 r )/ 2 E = J s z = ±1 triplets 1,1 r and 0,0 r E = 0 singlet ( 1,0 r 0,1 r )/ 2 E = J rung-triplet phase Mott insulator extends down to J /J = 0 for HCBs M. Piraud Phases of strongly-interacting bosons on a two-leg ladder 13/ 32
18 Fractionally filled Mott insulator Hard-core bosons, no flux ( ) Strong rungs (J J) : H K J r a 1,r a 2,r +H.c. s z = 0 triplet ( 1,0 r + 0,1 r )/ 2 E = J s z = ±1 triplets 1,1 r and 0,0 r E = 0 singlet ( 1,0 r 0,1 r )/ 2 E = J 1 boson per rung rung-triplet phase Mott insulator extends down to J /J = 0 for HCBs T. Vekua et al., PRB (2003); F. Crépin et al., PRB (2011). M. Piraud Phases of strongly-interacting bosons on a two-leg ladder 13/ 32
19 Phase diagram for Hard-Core Bosons Parameter space : rung-tunneling J /J, flux φ, filling n = N/2L. n MI, gapped SF, c=1 MI, c=1 SF, c=2 J /J = 1 SF, c=2 J /J = φ/π Upper half 0.5 < n < 1 related by particle-hole symmetry. Phase diagram consistent with field-theory. M. Piraud Phases of strongly-interacting bosons on a two-leg ladder 14/ 32
20 Current patterns Local ) currents ) j l,r (a = ij l+1,r a l,r H.c. and jr = ij (e irφ a 1,r a 2,r H.c. n MI, gapped SF, c=1 MI, c=1 SF, c=2 0 J /J = 1 J /J = 1.5 SF, c= φ/π Meissner-SF φ/π =0.5, J /J =1.5, n=0.25 Vortex-SF φ/π =0.9, J /J =1.5, n=0.1 cf E. Orignac & T. Giamarchi, PRB (2001), A. Tokuno & A. Georges, NJP (2014), M. Atala et al., Nat. Phys. (2014) M. Piraud Phases of strongly-interacting bosons on a two-leg ladder 15/ 32
21 Current patterns Meissner-MI φ/π =0.5, J /J =2, n = 0.5 Vortex-MI φ/π =0.5, J /J =0.5, n = 0.5 cf A. Petrescu & K. Le Hur, PRL (2013) n M-MI, gapped M-SF, c=1 V-MI, c=1 V-SF 0 J /J = 1 J /J = 1.5 V-SF, c= φ/π Meissner-SF φ/π =0.5, J /J =1.5, n=0.25 Vortex-SF φ/π =0.9, J /J =1.5, n=0.1 M. Piraud Phases of strongly-interacting bosons on a two-leg ladder 16/ 32
22 Tractable limit : around φ π For strong rung-tunneling J J and n < 0.5 : effective spin H Pseudo-spin-1/2 on each rung r : ( 1,0 r +e irφ 0,1 r )/ 2 r and 0,0 r r. H eff (φ = π) = J2 2 J ] (2S [S ) zr Szr+1 +r (12 Szr+1 )S r+2 +H.c. r M. Piraud Phases of strongly-interacting bosons on a two-leg ladder 17/ 32
23 Tractable limit : around φ π For strong rung-tunneling J J and n < 0.5 : effective spin H H eff (φ = π) = J2 2 J r ( 2Sr z Sr+1 z [S r + ( 1 ] ) 2 Sz r+1)sr+2 +H.c. (n = 0.25,φ = π) becomes gapped V-SF splitted in 2 lobes (for J 1.3J) V-MI and upper lobe of V-SF disappears n M-MI, gapped M-SF, c=1 V-MI, c=1 V-SF J /J = 1 V-SF, c=2 J /J = φ/π M. Piraud Phases of strongly-interacting bosons on a two-leg ladder 17/ 32
24 Tractable limit : around φ π For strong rung-tunneling J J and n < 0.5 : effective spin H H eff (φ = π) = J2 2 J r ( 2Sr z Sr+1 z [S r + ( 1 ] ) 2 Sz r+1)sr+2 +H.c. (n = 0.25,φ = π) becomes gapped V-SF splitted in 2 lobes (for J 1.3J) V-MI and upper lobe of V-SF disappears (for J 1.7J) n M-MI, gapped M-SF, c=1 V-MI, c=1 V-SF J /J = 1 V-SF, c=2 J /J = φ/π M. Piraud Phases of strongly-interacting bosons on a two-leg ladder 17/ 32
25 Tractable limit : around φ π For strong rung-tunneling J J and n < 0.5 : effective spin H H eff (φ = π) = J2 ( 2Sr z Sr+1 z [S r + ( 1 ] ) 2 J 2 Sz r+1)sr+2 +H.c. r Meta-magnetic transition at φ = π : jump in density from n > 0.25 to n = 0.5 (for J 1.5J) 0.5 J /J = 2 J /J = 4 J /J = 8 J /J = n 0.25 increasing J /J 0 µ [arb. units] M. Piraud Phases of strongly-interacting bosons on a two-leg ladder 18/ 32
26 Outline Hard-Core Bosons M. P., F. Heidrich-Meisner, I. P. McCulloch, S. Greschner, T. Vekua and U. Schollwöck, PRB (R) (2015) Phase Diagram A few observables Finite interactions S. Greschner, M. P., F. Heidrich-Meisner, I. P. McCulloch, U. Schollwöck and T. Vekua, submitted (2015) Phase Diagram Chiral current M. Piraud Phases of strongly-interacting bosons on a two-leg ladder 19/ 32
27 Chiral current, HCB at n = 0.5 j c = 1 j 1,r 2L j 2,r r J /J M-MI V-MI j c /J cf non-interacting phases : φ/π 0 Vortex phase is disfavored M. Atala et al., Nat. Phys. (2014) M. Piraud Phases of strongly-interacting bosons on a two-leg ladder 20/ 32
28 Chiral current, HCB at n = 0.5 J /J M-MI V-MI φ/π j c /J At J J and small φ : j c (J /φ) 2 At J J : j c = J2 J sin(φ) no saturation for HCB in gapped phases 0.3 (a) (b) J /J = 1 J /J = 2 J /J = j c /J j c /J φ/π = 0.2 φ/π = 0.5 φ/π = J /J φ/π M. Piraud Phases of strongly-interacting bosons on a two-leg ladder 21/ 32
29 Vortex density, HCB at n = 0.5 (a1) φ/π =0.5, J /J =0.05 (a2) φ/π =0.5, J /J =0.5 (a3) φ/π =0.5, J /J =2 0.4 l 1 V 0.2 φ/π = 0.2 φ/π = 0.5 φ/π = J /J Inverse spatial extension of the vortices : ρ V = l 1 V Fourier transform of the rung current j r ). (obtained from the M. Piraud Phases of strongly-interacting bosons on a two-leg ladder 22/ 32
30 Outline Hard-Core Bosons M. P., F. Heidrich-Meisner, I. P. McCulloch, S. Greschner, T. Vekua and U. Schollwöck, PRB (R) (2015) Phase Diagram A few observables Finite interactions S. Greschner, M. P., F. Heidrich-Meisner, I. P. McCulloch, U. Schollwöck and T. Vekua, submitted (2015) Phase Diagram Chiral current M. Piraud Phases of strongly-interacting bosons on a two-leg ladder 23/ 32
31 Phase Diagram, at U = 2J and n = 0.8 a 1 VL 1/2 -SF φ / π 0.8 V-SF VL 1/3 -SF BLP M-SF J / J Vortex Lattices E. Orignac & T. Giamarchi, PRB (2001) at ρ V = 1/2 and ρ V = 1/3 Biased Leg Phase R. Wei & E. J. Mueller, PRA (2014) M. Piraud Phases of strongly-interacting bosons on a two-leg ladder 24/ 32
32 Vortex Lattices Commensurate/ incommensurate transitions E. Orignac & T. Giamarchi, PRB (2001) Vortex (incommensurate) Vortex Lattice Meissner The size of the plateaux reduces with interactions M. Piraud Phases of strongly-interacting bosons on a two-leg ladder 25/ 32
33 Phase Diagram, at U = 2J and n = 0.8 a cut at J = 1.6J 1 VL 1/2 -SF φ / π 0.8 V-SF VL 1/3 -SF BLP M-SF J / J M. Piraud Phases of strongly-interacting bosons on a two-leg ladder 26/ 32
34 Phase Diagram : Mott Insulating phases φ / π (J = 1.6J, φ = 0.9π) a 1 VL 1/2 -SF BLP VL 0.8 1/3 -SF V-SF M-SF J / J b density ρ MI phase at fractional filling V-SF VL 1/2 - SF VL 1/2 - MI V-SF U / J M. Piraud Phases of strongly-interacting bosons on a two-leg ladder 27/ 32
35 Outline Hard-Core Bosons M. P., F. Heidrich-Meisner, I. P. McCulloch, S. Greschner, T. Vekua and U. Schollwöck, PRB (R) (2015) Phase Diagram A few observables Finite interactions S. Greschner, M. P., F. Heidrich-Meisner, I. P. McCulloch, U. Schollwöck and T. Vekua, submitted (2015) Phase Diagram Chiral current M. Piraud Phases of strongly-interacting bosons on a two-leg ladder 28/ 32
36 Chiral current : Swimming against the tide! b density ρ (J = 1.6J, φ = 0.9π) cut at n = VL 1/2 - SF V-SF VL 1/2 - MI V-SF U / J edge current j c / j c VL 1/2 -SF U/J V-SF ρ= In and near the VL 1/2, the current circulates in the clock-wise direction, on average : M. Piraud Phases of strongly-interacting bosons on a two-leg ladder 29/ 32
37 Chiral current : Swimming against the tide! In the VL 1/2 phase : spontaneous symmetry breaking doubling of the unit cell j c is odd and 2π-periodic with the flux j c -π π 2π 0 φ, φ e If we have B on with φ [π/2,π]; the effective flux on (2φ) corresponds to B. spontaneous current reversal M. Piraud Phases of strongly-interacting bosons on a two-leg ladder 30/ 32
38 Chiral current : Swimming against the tide! In the VL 1/2 phase : spontaneous symmetry breaking doubling of the unit cell j c is odd and 2π-periodic with the flux j c -π π 2π 0 φ, φ If we have B on with φ [π/2,π]; the effective flux on (2φ) corresponds to B. spontaneous current reversal Genuine many-body effect M. Piraud Phases of strongly-interacting bosons on a two-leg ladder 30/ 32
39 Current reversal : Remarks edge current j c / j c 0 a φ / π b VL 1/2 -SF U/J V-SF ρ= V-SF VL 1/3 -SF VL 1/2 -SF BLP M-SF J / J V-SF The amplitude of the reversal is 3j 0 c The VL 1/3 does not lead to a current reversal (because φ > 2π/3) The current reversal is robust against the presence of a mass gap Study at n 1 : the current reversal occurs at arbitrary small U/J density ρ VL 1/2 - SF VL 1/2 - MI It is stable against a finite number of defects, and at finite T V-SF U / J M. Piraud Phases of strongly-interacting bosons on a two-leg ladder 31/ 32
40 a φ / π Conclusion Strongly-interacting bosons on a ladder with uniform flux : Phase diagram and observables (DMRG and field theory) 1 V-SF VL 1/3 -SF VL 1/2 -SF BLP M-SF J / J Plethora of phases : Mott insulating and superfluid, Meissner and Vortex phases Vortex lattices (ρ V = 1/2 and 1/3) Biased Leg Phase Vortex lattice phases, Spontaneous current reversal M. P., F. Heidrich-Meisner, I. P. McCulloch, S. Greschner, T. Vekua and U. Schollwöck, PRB (R) (2015) S. Greschner, M. P., F. Heidrich-Meisner, I. P. McCulloch, U. Schollwöck and T. Vekua, submitted (2015) M. Piraud Phases of strongly-interacting bosons on a two-leg ladder 32/ 32
41 a φ / π Conclusion Strongly-interacting bosons on a ladder with uniform flux : Phase diagram and observables (DMRG and field theory) 1 V-SF VL 1/3 -SF VL 1/2 -SF BLP M-SF J / J Plethora of phases : Mott insulating and superfluid, Meissner and Vortex phases Vortex lattices (ρ V = 1/2 and 1/3) Biased Leg Phase Vortex lattice phases, Spontaneous current reversal M. P., F. Heidrich-Meisner, I. P. McCulloch, S. Greschner, T. Vekua and U. Schollwöck, PRB (R) (2015) S. Greschner, M. P., F. Heidrich-Meisner, I. P. McCulloch, U. Schollwöck and T. Vekua, submitted (2015) Thank you for your attention! M. Piraud Phases of strongly-interacting bosons on a two-leg ladder 32/ 32
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