ESQPT in systems with long-range interactions Lea F. Santos Yeshiva - PowerPoint PPT Presentation

ESQPT in systems with long-range interactions Lea F. Santos Yeshiva University, New York, NY, USA Francisco Pérez-Bernal Universidad de Huelva, Spain G. Luca Celardo and Fausto Borgonovi Universita Cattolica del Sacro Cuore, Italy Ø Consequences of the presence of an ESQPT (static and dynamics). Ø Lipkin model [U(2)]: experiments with ion traps, BEC, NMR but valid also for U(n+1) PRA 94, 012113 (2016) arXiv:1604.06851 (Fort. Physik) PRA 92, 050101R (2015) Lea F. Santos, Yeshiva University Long-range, Trieste, 2016

Trapped ions: long-range interaction J ! z ! x ! m x H = B ! n + | n " m | " ! n 0 ~ ! ! 3 n n < m P. Richerme et al, Nature 511 , 198 (2014) P. Jurcevi et al, Nature 511 , 202 (2014) Ion Traps Lea F. Santos, Yeshiva University Long-range, Trieste, 2016

Trapped ions: long-range interaction J ! z ! x ! m x H = B ! n + | n " m | " ! n 0 ~ ! ! 3 n n < m P. Richerme et al, Nature 511 , 198 (2014) P. Jurcevi et al, Nature 511 , 202 (2014) P. Hauke and L. Tagliacozzo, PRL 111 , 207202 (2013) Magnetization ! = 3 ! = 0.7 in z of each site Faster than Lieb-Robinson bound N=100, excitation on 50 !! .. !!"!! .. !! Z Lea F. Santos, Yeshiva University Long-range, Trieste, 2016

Lipkin Model: infinite-range interaction J ! z ! x ! m x H = B ! n + | n " m | " ! n dimension = 2 N n n < m ! z ! x ! m x H = B + J ! n ! n to ! = 0 dimension = N 2 + 1 n n < m N = number of sites S z = S x = ! z ! x ! n ! n n n Lea F. Santos, Yeshiva University Long-range, Trieste, 2016

Lipkin Model: infinite-range interaction J ! z ! x ! m x H = B ! n + | n " m | " ! n dimension = 2 N n n < m ! z ! x ! m x H = B + J ! n ! n to ! = 0 dimension = N 2 + 1 n n < m N = number of sites S z = S x = ! z ! x ! n ! n n n " % H = (1 ! ! ) N '! 4 ! 2 + S z N S x 2 Lipkin-Meshkov-Glick $ model # & Ground state quantum phase transition Control parameter ! c = 0.2 Lea F. Santos, Yeshiva University Long-range, Trieste, 2016

Lipkin Model: U(2) algebraic structure U(2) U(1) SO(2) " % H = (1 ! ! ) N '! 4 ! 2 + S z N S x 2 $ # & S z = 1 S + = 2 t + s 2 ( t + t ! s + s ) Schwinger representation: U(2) U(1) SO(2) Ground state H = (1 ! ! ) n t + ! Two species of N ( t + s + s + t ) 2 QPT scalar bosons ! c = 0.2 n t = t + t In general: H U ( n + 1) = (1 ! ! ) H U ( n ) + ! N H SO ( n + 1) PRA 92, 050101R (2015) Lea F. Santos, Yeshiva University Long-range, Trieste, 2016

Excited State Quantum Phase Transition QPT ESQPT ESQPT ESQPT ! = 0.4 ! = 0.6 ! = 0.8 ! c = 0.2 " % H = (1 ! ! ) N '! 4 ! 2 + S z N S x 2 $ # & Separatrix that marks the ESQPT E ESQPT = (1 ! 5 ! ) 2 16 ! H U ( n + 1) = (1 ! ! ) H U ( n ) + ! N H SO ( n + 1) Lea F. Santos, Yeshiva University Long-range, Trieste, 2016

ESQPT: participation ratio in U(1) basis " % H = (1 ! ! ) N '! 4 ! 2 + S z N S x 2 $ # & U(1)-basis N /2 ( k ) S m z ( k ) " C s z ! U (2) = s z = ! N /2 Participation Ratio 1 Large PR: delocalized state PR ( k ) ! N /2 Small PR: localized state ( k ) | 4 # | C s z s z = " N /2 LFS & Pérez-Bernal PRA 92 , 050101R (2015). Lea F. Santos, Yeshiva University Long-range, Trieste, 2016

Eigenstate at ESQPT localized at U(1) basis with mz=-N/2 ! = 0.4 " % H = (1 ! ! ) N '! 4 ! 2 + S z N S x 2 $ # & E ESQPT U(1)-basis N /2 ( k ) S m z ( k ) " C s z ! U (2) = s z = ! N /2 Participation Ratio 1 PR ( k ) ! N /2 ( k ) | 4 # | C s z s z = " N /2 N=600, 2000 PRA 92 , 050101R (2015). Lea F. Santos, Yeshiva University Long-range, Trieste, 2016

Eigenstate at ESQPT localized at U(1) basis with mz=-N/2 N /2 ( k ) s m z " % H = (1 ! ! ) N '! 4 ! 2 + S z N S x ( k ) " 2 C s z U(1) basis s m z ! U (2) = $ # & s z = ! N /2 E ESQPT ! = 0.6 ( k ) 2 C s z N=600 e z '/ N e z '/ N e z '/ N e z '/ N e z = s m z H U (2) s m z Lea F. Santos, Yeshiva University Long-range, Trieste, 2016

Eigenstate at ESQPT localized at U(1) basis with mz=-N/2 N /2 ( k ) s m z " % H = (1 ! ! ) N '! 4 ! 2 + S z N S x ( k ) " 2 C s z U(1) basis s m z ! U (2) = $ # & s z = ! N /2 E ESQPT ! = 0.6 ( k ) 2 C s z N=600 e z '/ N e z '/ N e z '/ N e z '/ N e z = s m z H U (2) s m z Energy of the U(1) basis vectors e z = s m z H U (2) s m z !! .. !!!! .. !! Z m z = ! N / 2 Lea F. Santos, Yeshiva University Long-range, Trieste, 2016

Eigenstate at ESQPT localized at U(1) basis with mz=-N/2 N /2 ( k ) s m z " % H = (1 ! ! ) N '! 4 ! 2 + S z N S x ( k ) " 2 C s z U(1) basis s m z ! U (2) = $ # & s z = ! N /2 E ESQPT ! = 0.6 ( k ) 2 C s z N=600 e z '/ N e z '/ N e z '/ N e z '/ N Energy of the U(1): z U(1) basis vectors e z = s m z H U (2) s m z SO(2): x !! .. !!!! .. !! Z m z = ! N / 2 ! c Lea F. Santos, Yeshiva University Long-range, Trieste, 2016 Ground state QPT

Quench from U(n) to U(n+1) " % H U (2) = (1 ! ! ) N '! 4 ! 2 + S z N S x 2 !! .. !!!! .. !! Z $ U(1) ground state # & Initial state U(1)-basis " % H U (2) = (1 ! ! ) N '! 4 ! 2 + S z N S x 2 ! (0) = S m z $ # & Survival Probability 2 F ( t ) = ! (0) | ! ( t ) " Lea F. Santos, Yeshiva University Long-range, Trieste, 2016

Initial state: U(1)-basis vector Slow decay ( k ) ! U (2) " ( k ) Initial state ! (0) = s m z = C s z k 2 2 ( k ) 2 e # iE k t % 2 = ! ini ( E ) e # iEt dE $ & Survival Probability F ( t ) = ! (0) | ! ( t ) " C m z = k #% !! .. !!!! .. !! Z U(1) ground state 0 10 F(t) !! .. !""! .. !! Z -2 10 -4 10 0 0 0.2 0.4 0.8 1 0.6 Time 0 ! = 0.6 N=1000 10 Lea F. Santos, Yeshiva University Long-range, Trieste, 2016 PRA 94, 012113 (2016)

Magnetization in z: slow dynamics " % H U (2) = (1 ! ! ) N '! 4 ! ( k ) / N = ! k S z ! k / N 2 + S z N S x 2 U(1): z m z $ # & SO(2): x ! (0) = "" .. "##" .. "" Z Same initial states studied in ion traps ! (0) = "" .. """" .. "" Z Lea F. Santos, Yeshiva University Long-range, Trieste, 2016 PRA 94, 012113 (2016)

Magnetization in z: dip " % H U (2) = (1 ! ! ) N '! 4 ! ( k ) / N = ! k S z ! k / N 2 + S z N S x 2 U(1): z m z $ # & SO(2): x At the separatrix is localized at m z = ! N / 2 ! k !! .. !!!! .. !! Z E ESQPT E ' k / N PRA 94, 012113 (2016) Lea F. Santos, Yeshiva University Long-range, Trieste, 2016

Magnetization in x: bifurcation U(1): z " % H U (2) = (1 ! ! ) N '! 4 ! 2 + S z N S x 2 $ # & SO(2): x ( k ) / N = ! k S z ! k / N ( k ) / N = ! k S x ! k / N m z m x E ESQPT E ESQPT E ' k / N PRA 94, 012113 (2016) Lea F. Santos, Yeshiva University Long-range, Trieste, 2016

QPT with parity-symmetry breaking H = ! 2 " S x # ! 2 z 2 " 1 " z 2 cos ! 2 S z Imbalance Trenkwalder … Inguscio, Fattori arXiv: 1603.02979 Tuning g to large negative values, the ground state of the system goes from a gapped symmetric state (z = 0) to two degenerate asymmetric states ( | z | > 0). The system undergoes a second-order QPT where the spatial parity symmetry is broken. Lea F. Santos, Yeshiva University Long-range, Trieste, 2016

Magnetization in x: bifurcation H U (2) = (1 ! ! ) N " '! 4 ! % U(1): z 2 + S z N S x 2 $ # & SO(2): x ( k ) / N = ! k S x ! k / N ! c = 0.2 m x Ground state QPT Bifurcation of m x for the ! < ! c ! > ! c ground state as increases ! Lea F. Santos, Yeshiva University Long-range, Trieste, 2016

Magnetization in x: bifurcation " % H U (2) = (1 ! ! ) N '! 4 ! 2 + S z N S x 1 2 $ # & 0.8 U(1): z 0.6 k /N E’ 0.4 0.2 SO(2): x 0 0 0.2 0.4 0.6 0.8 1 ξ Lea F. Santos, Yeshiva University Long-range, Trieste, 2016

Magnetization in x: bifurcation " % H U (2) = (1 ! ! ) N '! 4 ! 2 + S z N S x 2 U(1): z $ # & ( k ) / N = ! k S x ! k / N SO(2): x m x E ESQPT E ESQPT Bifurcation of m x at the ESQPT Bifurcation of m x for the ground state as increases ! Lea F. Santos, Yeshiva University Long-range, Trieste, 2016

Classical bifurcation H = ! 2 " J x # ! " % 2 z 2 " 1 " z 2 cos ! H = (1 ! ! ) N '! 4 ! 2 J z 2 + S z N S x 2 $ # & Oberthaler’s group PRL 105 (2010) V ( ! ) = 1 ! z 2 cos ! BEC z = ( N a ! N b ) / N V ( ! ) Oliveira’s group PRA 87 (2013) NMR z = Temporal mean magnetization ! Lea F. Santos, Yeshiva University Long-range, Trieste, 2016

Self-trapping: depending on ! " % H = (1 ! ! ) N '! 4 ! 2 + S z N S x H = ! 2 " J x # ! 2 z 2 " 1 " z 2 cos ! 2 J z 2 $ # & ! = 0.15 < ! c ! < 1 SO(2) basis ! (0) = s m x ++ .. + ! ++ .. ! + X ! = 0.6 > ! c ! > 1 Lea F. Santos, Yeshiva University Long-range, Trieste, 2016

Self-trapping: depending on energy Superposition of eigenstates only below or only above the ! (0) separatrix ( k ) / N = ! k S x ! k / N m x E ini < E ESQPT E ini > E ESQPT Bifurcation of m x at the ESQPT Lea F. Santos, Yeshiva University Long-range, Trieste, 2016