arXiv:1904.08432v2 [cond-mat.quant-gas] 31 Oct 2019 ieeouino est arx(e .. es[,9]): Refs.[8, e.g., the (see governs matrix density that of equation evolution master time Lindblad under the in arises been recently, experiments[100–102]. Very have in observed 40]. 88–99]. been NHSE 58, 37, has 37, NHSE 31, 33, of a 25, 32, investigations[25, zone[24, implications bound- of Brillouin bulk-boundary Broader generalized at non-Bloch eigenstates the based theory the on band localized non-Bloch of and suggests 28] correspondence[24, are majority which aries, operator the non-Hermitian that namely where ytm sthe is systems in- experimental[82–87] vestigations. and in- theoretical[71–81] other among teresting classifications[68–70], and topological semimetals[57–67], novel and topo- 43–56] of insulators[25, generalizations 35–42], logical 31, topological non-Hermitian 27, 25, on new invariants[24, focused correspondence[22–34], been bulk-boundary have Consid- attentions attentions. growing erable topological attracting and been Re- have non-Hermiticity phases of lifetimes[17–21]. interplay finite the systems[6– with cently, quantum quasiparticles open and gain[1–5], 16], sys- and photonic loss as such with phenomena tems of range wide a for work iaosdsrbn unu up u oculn to coupling to and due environment, jumps the quantum describing sipators prtr eoeteocrec fajm,teshort- the jump, a Hamil- tonian non-Hermitian of effective the occurrence follows evolution the time Before operator. ρH o ml,teeeto onaycniini insignifi- commonly is is condition condition boundary boundary periodic such, of As effect cant. the small, too noe unu ytm,nnHriiiynaturally non-Hermiticity systems, quantum open In n ftermral hnmn fnon-Hermitian of phenomena remarkable the of One frame- natural a provide Hamiltonians Non-Hermitian ti eeal eivdta hntesse iei not is size system the when that believed generally is It eff † dρ dt [1 2 103]. 12, )[11, H = H − eff o-emta kneetadcia apn noe quantum open in damping chiral and effect skin Non-Hermitian steHamiltonian, the is i [ ,ρ H, = ad as hrldmigwt hr aern.Teeph These no wavefront. the to sharp dynamics. belong quantum a they with instead, damping Hamiltonians; non-Hermitian chiral un n a algebraic the cause Moreover, is systems bands condition. quantum boundary open shape open of under dramatically class exponential effect un a in its skin we damping uncover non-Hermitian the Letter, and the effect, this that skin In shown non-Hermitian the exhibit evolution. ma can Lindblad time the described generate for well calls superoperators be dynamics often long-time bou while can the tonians, evolution at short-time localized a exponentially systems, are eigenstates the that o-emta kneffect skin non-Hermitian n fteuiu etrso o-emta aitnasis Hamiltonians non-Hermitian of features unique the of One H + ] − X i µ P 1 L nttt o dacdSuy snhaUiest,Beijin University, Tsinghua Study, Advanced for Institute µ 2 scle h iuila super- Liouvillian the called is L L µ † µ L ρL L µ µ µ † as saeteLnba dis- Lindblad the are ’s { − e Song, Fei dρ/dt L µ † L 2,26](NHSE), [24, µ ρ , = 1 } − hnuYao, Shunyu i L ≡ ( H eff ρ, ρ (1) − 1 I.1 a S oe ihsagrdhopping staggered with model SSH (a) 1. FIG. ai change basis a hoyi ae ntefl iuila.Although Liouvillian. the full Crucially, the on damping”. based “chiral is unidirectional, the theory is dubbed damping is the which that implies More- NHSE condition. over, boundary alge- open condition is under boundary exponential damping periodic while time under law) long Notable power the (i.e., braic that Liouvillian. found the the are of from examples NHSE derived the to matrix related damping is periodic-boundary the this a Furthermore, sys- that of system. that open-boundary show from an we differ of dramatically paper, tem dynamics this Lindblad In time rele- long more experiments. is to condition vant open-boundary though adopted, experimentally. feasible more is (b) Hamiltonian nseta ee(.. t aigNS rntde not does not fact in or is NHSE it having role, its matter). important (e.g., an here play inessential to expected be may Hamiltonian hopping The different A model. (b) same h framework. the equation of master realization the in [Eq.(2)] ihteoasidctn h ntcls h lc Hamil- Bloch The cells. unit the is indicating tonian ovals the with L osadgi r ecie ytedissipators the by described are gain and loss rainadanhlto prtr tsite at operators annihilation and creation n hn Wang Zhong and ( x l k Model.– ( = ) = √ dr ftesse.Froe quantum open For system. the of ndary -emta hsc ffl-egdopen full-fledged of physics n-Hermitian γ treutos nwihteLiouvillian the which in equations, ster nHriinsi ffc n non-Bloch and effect skin on-Hermitian t yteeetv o-emta Hamil- non-Hermitian effective the by l 1 c h xetdpyia osqecs tis It consequences. physical expected h o-emta kneet namely effect, skin non-Hermitian the h ogtm yais uhthat such dynamics, long-time the s n htLovlinsuperoperators Liouvillian that find xA h ytmi lutae nFg1a.Our Fig.1(a). in illustrated is system The ( e eidcbudr odto but condition boundary periodic der + k ( = ) nmn r eodteeffective the beyond are enomena t and H 2 σ cos y = ↔ L t ,108,China 100084, g, k 1 x g 1, ) P σ + σ = z ∗ x eas h anadls son-site, is loss and gain the Because . ij t 2 + √ h cos γ t ij g 2 c c sin k xA † i † ) c σ j b seuvln o()via (a) to equivalent is (b) . kσ x where , + z n h isptr are dissipators the and , t 2 sin systems c kσ i † c , y h fermion The . i i r fermion are L (including t l 1 and and H L t eff 2 g , 2 additional degrees of freedom such as spin is straight- (a) L=150 forward). We will consider single particle loss and gain, ˜n(t) l l -2 with loss dissipators Lµ = i Dµici and gain dissipa- 10 g g † tors Lµ = i Dµici , respectively.P For concreteness, we take h to be the Su-Schrieffer-Heeger (SSH) Hamiltonian,
P 10-4
namely hij = t1 and t2 on adjacent links. A site is also A t1=t2=1,¡ g= l=0.4 £ labelled as i = xs, where x refers to the unit cell, and B t1=0.7,t2=1,¢ g= l=0.6
s = A, B refers to the sublattice. For simplicity, let each 10-6 ¥ C t1=1.2,t2=1,¤ g= l=0.05
unit cell contain a single loss and gain dissipator (namely, § D t1=1.2,t2=1,¦ g= l=0.1 µ is just x): 10-8 1 10 100 t 1000 l g † † Lx = γl/2(cxA − icxB),Lx = γg/2(cxA + icxB); p q (2) l l g in other words, Dx,xA = iDx,xB = γl/2,Dx,xA = g p −iDx,xB = γg/2. We recognized in Eq.(2) that the σy = +1 statesp are lost to or gained from the bath. A seemingly different but essentially equivalent realization of the same model is shown in Fig.1(b), which can be obtained from the initial model [Fig.1(a)] after a basis change σy ↔ σz. Accordingly, the dissipators in Fig.1(b) are σz = 1 states. As the gain and loss are on-site, its experimental implementation is easier. Keeping in mind that Fig.1(b) shares the same physics, hereafter we focus on the setup in Fig.1(a). To see the evolution of density matrix, it is convenient to monitor the single-particle correlation † FIG. 2. (a) The damping of fermion number towards the ∆ij (t) = Tr[ci cj ρ(t)], whose time evolution is d∆ij /dt = † steady state of a periodic boundary chain with length L = 150 Tr[c cj dρ/dt]. It follows from Eq.(1) that (see the Sup- i (unit cell). The damping is algebraic for cases A, B with t1 plemental Material) ≤ t2, while exponential for C, D with t1 >t2. The initial state is † d∆(t) the completely filled state Q cxs 0 . (b) The eigenvalues of = i[hT , ∆(t)] −{M T + M , ∆(t)} +2M , (3) x,s | i dt l g g the damping matrix X. Blue: periodic-boundary; Red: open- boundary. The Liouvillian gap of the periodic-boundary chain g∗ g l∗ l where (Mg)ij = µ Dµi Dµj and (Ml)ij = µ DµiDµj , vanishes for A and B, while it is nonzero for C and D. For and both Ml andPMg are Hermitian matrices.P Majorana the open-boundary chain, the Liouvillian gap is nonzero in versions of Eq.(3) appeared in Refs.[8, 16, 104]. We can all four cases. This drastic spectral distinction between open define the damping matrix and periodic boundary comes from the NHSE (see text). T T X = ih − (Ml + Mg), (4) which recasts Eq.(3) as and express Eq.(6) as
d∆(t) † = X∆(t) + ∆(t)X +2Mg. (5) ˜ ∗ ˜ dt ∆(t)= exp[(λn + λn′ )t]|uRnihuLn|∆(0)|uLn′ ihuRn′ |. n,nX′ The steady state correlation ∆s = ∆(∞), to which long time evolution of any initial state converges, is deter- (8) † mined by d∆s/dt = 0, or X∆s + ∆sX +2Mg = 0. In this paper, we are concerned mainly about the dynam- By the dissipative nature, Re(λn) ≤ 0 always holds true. ics, especially the speed of converging to the steady state, The Liouvillian gap Λ = min[2Re(−λn)] is decisive for therefore, we shall focus on the deviation ∆(˜ t) = ∆(t) − the long-time dynamics. A finite gap implies exponential † converging rate towards the steady state, while a vanish- ∆s, whose evolution is d∆(˜ t)/dt = X∆(˜ t) + ∆(˜ t)X , which is readily integrated to ing gap implies algebraic convergence[106]. Periodic chain.–Let us study the periodic boundary ˜ Xt ˜ X†t ∆(t)= e ∆(0)e . (6) chain, for which going to momentum space is more con- We can write X in terms of right and left venient. It can be readily found that h(k) = (t1 + eigenvectors[105], t2 cos k)σx + t2 sin kσy and
X = λn|uRnihuLn|, (7) γl γg Ml(k)= (1 + σy),Mg(k)= (1 − σy). (9) Xn 2 2 3
(a) γg=γl=0.2 also give the steady state. In fact, our Ml and Mg ˜ satisfy M T + M = M γ/γ , which guarantees that n(t) l g g g ∆s = (γg/γ)I2L×2L is the steady state solution. It is -4 10 independent of boundary conditions. periodic Now we show the direct relation between the algebraic open L=30 damping and the vanishing gap of X. The eigenvalues of 10-8 open L=40 X(k) are open L=50 open L=60 -12 2 2 2 10 λ±(k)= −γ/2 ± i (t1 + t2 +2t1t2 cos k − γ /4) − it2γ sin k. (11) q 20 40 60 t 80 Let us consider t = t ≡ t for concreteness (case A in (b) γg=γl=0.2 1 2 0 ˜ L=60 nx(t) Fig.2), then λ−(π) = 0 and the expansion in δk ≡ k − π reads 10-3 t2 λ (π + δk) ≈−it δk − 0 (δk)4. (12) − 0 4γ periodic 10-7 open x=1 Now Eq.(7) becomes X = k,α=± λα(k)|uRkαihuLkα|, open x=10 and Eq.(8) reads P open x=20 -11 ∗ ′ 10 open x=30 λα(k)t+λ ′ (k )t ∆(˜ t)= e α |uRkαihuLkα|∆(0)˜ |uLk′α′ ihuRk′ α′ |. 20 40 t 60 kkX′ ,αα′ FIG. 3. (a) The particle number damping of a periodic bound- (13) ary chain (solid curve) and open-boundary chains for several L For the initial state with translational symmetry, chain length . The long time damping of a periodic chain ˜ ˜ follows a power law, while the open boundary chain follows we have huLkα|∆(0)|uLk′α′ i = δkk′ huLkα|∆(0)|uLkα′ i. an exponential law after an initial power law stage. (b) The The long-time behavior of ∆(˜ t) is dominated by the site-resolved damping. The left end (x = 1) enters the expo- α = α′ = − sector, which provides a decay factor 2 nential stage from the very beginning, followed sequentially t0 4 exp (2Re[λ−(π + δk)]t) ≈ d(δk) exp[− (δk) t] ∼ by other sites. For both (a) and (b), the initial state is the δk 2γ † −1/4 ˜ −1/2 completely filled state Q c 0 , therefore, ∆(0) the identity tP . Similarly, for t1
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l i Dµici are The corresponding terms in Eq.(17) sum to T P −{Ml , ∆(t)}ij , l† † l l∗ † † l 2 Lµ [ci cj ,Lµ]=2 Dµmcm[ci cj , Dµncn] Similarly, for the commutators from the gain dissipa- Xµ Xµm Xn tors, we have l∗ l † † =2 DµmDµncm[ci cj ,cn] µmnX 2 Lg†[c†c ,Lg ]=2 (M ) c c†, l∗ l † µ i j µ g mj m i (21) =2 DµmDµn(−δincmcj ) Xµ Xm µmnX † = −2 (Ml)micmcj , (19) and Xm and g† g † † † [Lµ Lµ,ci cj ]= −(Mg)nj cnci + (Mg)incj cn . l† l † l∗ l † † Xµ Xn [Lµ Lµ,ci cj ]= DµmDµn[cmcn,ci cj ] Xµ µmnX (22) † † l∗ l † † Writing cnci = δni − ci cn, we see that the corresponding = DµmDµn(−δmj ci cn + δincmcj ) terms in Eq.(17) sum to 2(Mg)ij Tr(ρ) −{Mg, ∆(t)}ij = µmnX 2(Mg)ij −{Mg, ∆(t)}ij . Therefore, all terms at the right † † = −(Ml)jnci cn + (Ml)nicncj (20). hand side of Eq.(17) sum to that of Eq.(16). Xn