DISSIPATIVE QUANTUM WALKS AND TOPOLOGY M. J. Kastoryano and M. Rudner New Trends in Strongly Entangled Many-Body Systems UCL, November 2015 Wednesday, November 25, 15
TOPOLOGY AND DISSIPATION State engineering Dissipative engineering Computation S. Diehl et. al. , Nat. Phys. 7 (2011), pp. 971–977. S. Diehl et. al. , Nat. Phys. 4.11 (2008): 878-883 S. Diehl et. al. , PRL 105.1 (2010): 015702. MJK et. al. , PRL 106.9 (2011): 090502 ... H. Krauter et. al. , PRL 107 (2011): 080503 F. Verstraete et al.,Nat. phys. 5.9 (2009): 633-636. ... MJK et. al., Phys. Rev. Lett. 110, 110501 Topological systems Wednesday, November 25, 15
TOPOLOGY AND DISSIPATION State engineering Dissipative engineering Computation S. Diehl et. al. , Nat. Phys. 7 (2011), pp. 971–977. S. Diehl et. al. , Nat. Phys. 4.11 (2008): 878-883 We want to find S. Diehl et. al. , PRL 105.1 (2010): 015702. MJK et. al. , PRL 106.9 (2011): 090502 ... H. Krauter et. al. , PRL 107 (2011): 080503 genuinely new F. Verstraete et al.,Nat. phys. 5.9 (2009): 633-636. ... MJK et. al., Phys. Rev. Lett. 110, 110501 open system Topological systems behavior Wednesday, November 25, 15
DISSIPATIVE SYSTEMS m − 1 Master equation X L m ρ L † 2 { L † L ( ρ ) = i [ H, ρ ] + m L m , ρ } + m Stationary states Is the stationary state unique? L ( ρ ss ) = 0 Dark states L m | ψ i = 0 H | ψ i = λ | ψ i all m Eigenvalues of have non- Zero eigenvalue corresponds L Spectral properties positive real part. to the stationary states Relaxation rate is related to Gap of : minimum real part L the inverse of the gap of . of an eigenvalue of . L L Wednesday, November 25, 15
MOTIVATING EXAMPLE Cavity QED | 1 i ν set − ν = δ γ ∆ | 0 i δ H JC = 1 2( ∆ − ν ) σ z + ( g σ + a + Ω σ + + h.c. ) a Ω ρ = i [ H JC , ρ ] + γ ( σ − ρσ + − 1 2 { σ + σ − , ρ } + ) ˙ Assume no cavity decay! ∞ ( � Ω /g ) j | ψ ss i = e − ( Ω /g ) 2 X If Ω > g stationary state is coherent state p j | 0 , j i j =0 independent of . ( ∆ , γ , ν ) Bound topological edge state always pure! Wednesday, November 25, 15
THE MODEL unit cell m A B u m v m periodic bc’s γ J = 2 i tr[ dH s X u m | A, m ih B, m | + v m | A, m + 1 ih B, m | + h.c. Current : H = v → e is v ds ρ ] m m − 1 X L m ρ L † 2 { L † in general currents in open systems are ambiguous, L ( ρ ) = i [ H, ρ ] + m L m , ρ } + as there are usually no conservation laws m L m = p γ | A, m ih B, m | Lindblad (jump) Steady state current: J operators: 0.5 N ∼ even 0.4 N ∼ odd Properties: 0.3 N ∼ ∞ unique steady state 0.2 0.1 can be extracted analytically u ê v 0.5 1.0 1.5 2.0 pure (dark) steady state iff u=v - 0.1 - 0.2 Wednesday, November 25, 15
THE CURRENT unit cell m A B u m v m periodic bc’s γ Steady state current: Current J 0.5 N ∼ even N ∼ odd 0.4 4 v L +1 ( u 2 − v 2 ) γ N ∼ ∞ J L = 8( u 2 − v 2 )( u L − v L ) + ( u L + v L )( γ 2 ) 0.3 0.2 0.1 strictly zero for u>v u ê v 0.5 1.0 1.5 2.0 - 0.1 - 0.2 transition is independent of dissipative strength and on-site What is the origin of energy! this sharp transition? Wednesday, November 25, 15
THE CURRENT unit cell m A B u m v m periodic bc’s γ γ 2 ! i γ Translation invariance, steady state ss = 1 1 + c k = u + e ik v ρ k 4 | c k | 2 c k 2¯ is diagonal in the momentum basis − i γ Z 1 2 c k Can this form be derived in general? k independence of B site population Steady state is reached when probability of entering and exiting momentum k shell is equal. only depends on B population, therefore constant for all k. Steady state can generally be extracted from momentum conserving master equation! Wednesday, November 25, 15
THE CURRENT unit cell m A B u m v m periodic bc’s γ L m = p γ | A, m ih B, m | replace Reduced to solving a 2x2 matrix L k = p γ | A, k ih B, k | equation. with We obtain a solution for each k − 1 L k ρ L † 2 { L † X L ( ρ ) = i [ H, ρ ] + k L k , ρ } + momentum k. k ✓ 0 Steady state of original system is ◆ c k c k = u + e ik v H k = the convex combination that are c k ¯ 0 equal B-site population. γ 2 ! i γ ss = 1 1 + ρ k 4 | c k | 2 c k 2¯ Solution is general! − i γ Z 1 2 c k Wednesday, November 25, 15
THE CURRENT unit cell m A B u m v m periodic bc’s γ ✓ 0 γ 2 ! i γ ◆ ss = 1 1 + c k ρ k 4 | c k | 2 c k 2¯ H k = − i γ ¯ c k 0 Z 1 2 c k Z dk tr[ dH k Steady-state current: J ss = 2 i dk ρ ss ] "✓ ◆ !# i γ dc k Z 0 ∗ 2¯ c k = 2 i dk tr dk − i γ d ¯ c k 0 1 dk c k Winding number! = 2 γ Z dk Im[ d dk log c k ] Z Discontinuity in the current is topological! What about the pre-factor? Wednesday, November 25, 15
CRITICALITY unit cell m A B u m v m periodic bc’s γ J ss = 2 γ Z dk Im[ d Time to equilibrium: # of jumps dk log c k ] Z required to reach equilibrium times the probability for a jump to occur. Probability of a jump = population at B sites. gives the probability for a jump 1 /Z to occur γ 2 ! i γ ss = 1 1 + ρ k 4 | c k | 2 c k 2¯ − i γ Z 1 2 c k J ss = ν τ is the average time between jumps τ Wednesday, November 25, 15
GENERALITY Negative winding number = current t 3 unit cell m t − 1 t 2 in opposite direction. A B periodic bc’s t 1 t 0 γ Z dk Im[ d X t l e ilk c k = ν = dk log c k ] J ss = ν l τ γ 2 τ = 1 Z dk (2 + 4 | c k | 2 ) 2 γ diverges quadratically at the Is this really topology? τ transition. Wednesday, November 25, 15
NEXT NEAREST NEIGHBOR w A B v u γ c k = we − ik + u + ve ik w 0 , if u > v + w − 1 ν = − 1 , if u < v + w and v > w , 1 1 , if u < v + w and v < w u = 1 ( 1 / | u 2 − ( v + w ) 2 | , around u = v + w 1 τ ∼ 1 / | v 2 − w 2 | , around v = w 0 v 1 topological sectors ν Wednesday, November 25, 15
EDGE STATES Back to nearest neighbor chain Boundary acts as a sink! Recall, in TI system, stationary state dark iff u=v. Stationary state must be As particle crosses between the dark! two sectors, it gets caught in a dark state. current no current ν = 0 ν = 1 u > v u < v Wednesday, November 25, 15
EDGE STATES X X u m | A, m ih B, m | + v m | A, m + 1 ih B, m | + h.c. Assume: H = | ψ ss i = p m | A, m i m m p m u m + p m +1 v m = 0 H | ψ ss i = 0 ) p − m ∝ ( − v u ) m p m ∝ ( − u v ) m current no current ν = 0 ν = 1 u > v u < v Wednesday, November 25, 15
EDGE STATES X X u m | A, m ih B, m | + v m | A, m + 1 ih B, m | + h.c. Assume: H = | ψ ss i = p m | A, m i m m p m u m + p m +1 v m = 0 H | ψ ss i = 0 ) General p − m ∝ ( − v u ) m p m ∝ ( − u v ) m phenomenon? current no current ν = 0 ν = 1 u > v u < v Wednesday, November 25, 15
NEXT NEAREST NEIGHBOR ν w A B v u γ w − 1 p 2 p 1 p 1 1 1 − 1 ν u = 1 0 1 p 2 v 1 0 − 1 ν 1 topological sectors ν Wednesday, November 25, 15
NEXT NEAREST NEIGHBOR w − 1 p 2 p 1 p 1 1 1 ν u = 1 0 1 p 2 v − 1 1 0 ν 1 topological sectors ν Each boundary can support What are the general rules for the bulk a pure edge state! edge correspondence? But all currents have to How is it actually related to the end in an edge! topology? Wednesday, November 25, 15
TRANSFER MATRIX L X X General model t l | A, m ih B, m + l | + h.c. H = m l = − L X | ψ ss i = p m | A, m i Assume dark stationary states m L X t l p m + l = 0 Recurrence relation l = − L p m + L − 1 p m + L Transfer Matrix T ... ... = p m − L p m − L +1 Physical states correspond to eigenvectors with eigenvalues of magnitude less than one. Wednesday, November 25, 15
TRANSFER MATRIX However not all eigenvalues in the Physical states correspond to unit circle correspond to legitimate eigenvectors with eigenvalues of edge states. magnitude less than one. Left and right moving transfer matrices must be bounded from any initial point! p 2 − 1 2 1 ν Wednesday, November 25, 15
TRANSFER MATRIX However not all eigenvalues in the Physical states correspond to unit circle correspond to legitimate eigenvectors with eigenvalues of edge states. magnitude less than one. Left and right moving transfer matrices must be bounded from any initial point! p 2 − 1 2 1 ν Wednesday, November 25, 15
TRANSFER MATRIX However not all eigenvalues in the Physical states correspond to unit circle correspond to legitimate eigenvectors with eigenvalues of edge states. magnitude less than one. Left and right moving transfer matrices must be bounded from any initial point! p 2 − 1 2 1 ν Wednesday, November 25, 15
TRANSFER MATRIX However not all eigenvalues in the Physical states correspond to unit circle correspond to legitimate eigenvectors with eigenvalues of edge states. magnitude less than one. Left and right moving transfer matrices must be bounded from any initial point! p 2 − 1 2 1 ν Wednesday, November 25, 15
TRANSFER MATRIX However not all eigenvalues in the Physical states correspond to unit circle correspond to legitimate eigenvectors with eigenvalues of edge states. magnitude less than one. Left and right moving transfer matrices must be bounded from any initial point! p 2 − 1 2 1 ν Wednesday, November 25, 15
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