Adiabatic freezing of entanglement with insertion of defects in a one-dimensional Hubbard model
Sreetama Das, Sudipto Singha Roy, Himadri Shekhar Dhar, Debraj Rakshit, Aditi Sen De, Ujjwal Sen
AAdiabatic freezing of entanglement with insertion of defectsin one-dimensional Hubbard model
Sreetama Das , Sudipto Singha Roy , , Himadri Shekhar Dhar , , Debraj Rakshit , ,Aditi Sen(De) , and Ujjwal Sen Harish-Chandra Research Institute, HBNI, Chhatnag Road, Jhunsi, Allahabad 211 019, India Department of Applied Mathematics, Hanyang University (ERICA),55 Hanyangdaehak-ro, Ansan, Gyeonggi-do, 426-791, Korea Institute for Theoretical Physics, Vienna University of Technology,Wiedner Hauptstraße 8-10/136, 1040 Vienna, Austria Institute of Physics, Polish Academy of Sciences,Aleja Lotnik´ow 32/46, PL-02668 Warsaw, Poland
We report on ground state phases of a doped one-dimensional Hubbard model, which for largeonsite interactions is governed by the t - J Hamiltonian, where the extant entanglement is immutableunder perturbative or sudden changes of system parameters, a phenomenon termed as adiabaticfreezing. We observe that in the metallic Luttinger liquid phase of the model bipartite entanglementdecays polynomially and is adiabatically frozen, in contrast to the variable, exponential decay in thephase-separation and superconducting spin-gap phases. Significantly, at low fixed electron densities,the spin-gap phase shows remarkable affinity to doped resonating valence bond gas, with multipartiteentanglement frozen across all parameter space. We note that entanglement, in general, is sensitiveto external perturbation, as observed in several systems, and hitherto, no such invariance or freezingbehavior has been reported.
I. INTRODUCTION
Over the years, a challenging task has been to ex-plore how entanglement [1] is distributed among the con-stituents of a many-body system and understand its ef-fects on cooperative phenomena [2–4]. For instance, itwas observed that the constituents of the non-criticalphases of many-body systems are, in general, less en-tangled with particles beyond their nearest neighbors(NN), and obey the area law of scaling of entanglemententropy [5, 6], which provides useful information abouttheir ground state properties [2–5] and is closely relatedto its numerical simulability [7, 8]. Hence, study ofquantum correlation may actually provide deeper insightabout the underlying cooperative and critical phenomenain these systems [9–11]. In return, quantum many-bodysystems are also important substrates for quantum com-munication [12, 13] and computation protocols [14, 15],and are thus key enablers for quantum technology.In this work, we report on the entanglement behaviorin the ground state phases of a doped one-dimensional(1D) Hubbard model with large onsite interactions. Thequantum spin-1/2 particles on the lattice doped withholes interact via the t - J Hamiltonian [16], with t repre-senting a typical tunneling strength between two neigh-boring sites and J serving as the spin-spin interactionstrength between particles in filled neighboring sites. The t - J Hamiltonian is widely used to study the physicalproperties of doped quantum spin systems, in particularfor high- T c superconducting phases of strongly-correlatedmatter [17, 18]. The minimum energy configuration ofthe t - J Hamiltonian exhibits a rich phase diagram in the
J/t - n el plane, with n el being the electron concentrationor density, and has already been extensively studied us-ing physical quantities such as ground state energy, spin correlation functions, and spin gap[19–23]. In this re-gard, one of our primary motivations is to investigatehow quantum correlations, especially bipartite entangle-ment (BE) and multipartite entanglement (ME), behavein these different phases, and whether insertion of de-fects play a significant role in altering the entanglementproperties.The key finding of this work is the existence of entan-glement in the ground state of the doped 1D t - J Hamil-tonian, in particular at low electron densities, which re-mains invariant under sudden or perturbative changesto the
J/t ratio, implying potential application in ro-bust quantum technologies [24]. In other words, the en-tanglement remains constant under perturbations of thesystem parameter, a phenomena reminiscent of the adi-abatic freezing of quantum correlations [25] (cf. [26–28]),where the aforementioned quantities are completely in-sensitive or frozen with respect to changes in system pa-rameters [25] or decoherence [26]. We observe that thisadiabatic freezing behavior of entanglement is differentfor bipartite and multipartite cases, and is closely relatedto the relevant ground state phases of this model [19–23].To elaborate, we observe that at low
J/t ratio (
J/t < n el , when the system is known to lie in the metal-lic Luttinger liquid phase [20], two-site BE, as quantifiedby the logarithmic negativity [29, 30], decays polynomi-ally with the increase in lattice distance, r = | i − j | , be-tween the lattice sites i and j , which essentially signalsthe dominating long-range order in the phase. Interest-ingly, within the metallic phase, the BE is invariant tochanges in the J/t ratio and is therefore adiabaticallyfrozen. In contrast, at higher
J/t ratio, superconduct-ing spin-gap phase [21, 22] and electron-hole phase sep-aration (PS) occurs [19], accompanied by an exponen-tial decay of BE. Subsequently, the adiabatic freezing a r X i v : . [ qu a n t - ph ] S e p of BE is lost during the quantum phase transition. Ofgreater significance is the behavior of multipartite en-tanglement, which for low fixed values of n el , remainsadiabatically frozen for all values of the J/t parameterspace. Using generalized geometric measure (GGM) [31](cf. [32]) as the measure of genuine multipartite entan-glement, we show that the variation of GGM across the
J/t - n el phase space, for low n el , remains invariant underadiabatic changes of the J/t ratio. It is important tonote that no such adiabatic freezing of ME is observed inthe undoped anisotropic 1D model [33]. Rather counter-intuitively, it appears that the presence of impurities or defects (as modeled by the holes) in the spin chain acts asa vehicle for phases with frozen ME. The importance ofthe results lie in the fact that many-body systems withrobust ME, which is not sensitive to perturbations insystem parameters or environmental processes, are nec-essary for realizing quantum information-theoretic pro-tocols such as measurement based quantum computation[14] and quantum communication protocols [12, 13]. Thepaper is arranged as follows. In Sec. II we introduce the1D t - J Hamiltonian. We study the decay and adiabaticfreezing of bipartite entanglement in Sec. III. We discussthe low electron density ground states of the model inSec. IV and demonstrate the freezing of genuine multi-partite entanglement in Sec. V. We conclude in Sec. VI.
II. MODEL
In our study, we consider the t - J Hamiltonian as thestructure that governs the interaction between the quan-tum particles in the doped 1D spin lattice, with N sites populated with N el ( < N ) quantum spin-1/2 parti-cles. The rest of the sites are vacant or contain holes .The “electron density” of the lattice is given by n el (= N el /N ). The t - J Hamiltonian can be obtained pertur-batively from the prominent Hubbard model in the limitof large on-site interaction [16], and has been expressedin literature in the form, H = − t (cid:88) (cid:104) i,j (cid:105) ,σ P G ( c † iσ c jσ + h.c.) P G + J (cid:88) (cid:104) i,j (cid:105) (cid:126)S i · (cid:126)S j , (1)where c iσ ( c † iσ ) is the fermionic annihilation (creation)operator of spin σ (= {↑ , ↓} ), acting on site i . P G isthe Gutzwiller projector Π i (1 − n i ↑ n i ↓ ) which enforcesat most single occupancy at each lattice site. S i = σ i ’sare the triad of spin operators { S x , S y , S z } , while t and J correspond to the transfer energy and the spin-exchangeinteraction energy terms, respectively, and each is limitedto nearest-neighbor sites, with periodic boundary con-dition. The ground state phase diagram for the above1D model has received widespread attention in the pastyears [19–23]. In particular, the presence of three primaryphases, namely the repulsive Luttinger liquid or metallic,attractive Luttinger liquid or superconducting, and thephase separation, have been predicted using exact diag-onalization [20]. However, recent results, using density J/t = 0.50
J/t = 0.80
J/t = 1.10
J/t = 1.40
J/t = 1.70
J/t = 2.00
J/t = 3.00
J/t = 3.30
J/t = 3.60 2 4 6 810 -4 -3 -2 E r E r FIG. 1. (Color online.) Decay and adiabatic freezing of bi-partite entanglement in phases of the t - J Hamiltonian. Theplot shows the variation of two-site entanglement ( E ) withincrease in lattice distance r = | i − j | , for the 1D t - J Hamil-tonian, with N = 30 and n el = N . For J/t ≤
2, the groundstate remains in the metallic phase and E decays polynomiallyas 1 / ( A + Br ), with r , exhibiting the presence of a dominatinglong-range order in the ground state. The values of A = 162 . B = 18 .
9, obtained from the average best-fitted curve, re-mains almost unchanged for all the curves in this phase, andBE is adiabatically frozen. This freezing behavior of bipartiteentanglement is shown more clearly in Fig. 2. In contrast,for
J/t ≥
3, the superconducting and PS phases leads to ex-ponential decay of BE, given by
E ∼ C exp ( − rξ ), where ξ isthe characteristic length and the constant C can be obtainedfrom the best-fitted curve. ξ and C are dependent on J/t andthe adiabatic freezing of BE is lost in this phase. The verticalaxis are in ebits and the horizontal axes are dimensionless.
J/t is also dimensionless. In the inset, we set the vertical axisin the logarithmic scale and plot E for J/t ≥ matrix renormalization group techniques, have also re-ported the presence of a superconducting spin-gap phaseat low n el [22, 23]. These phases play a significant role inthe entanglement properties of the doped quantum spinmodel. III. DECAY OF BIPARTITE ENTANGLEMENTAND ADIABATIC FREEZING IN METALLICPHASE
We now focus on the behavior of bipartite entangle-ment in the ground state of the t - J Hamiltonian. Inparticular, we look at the logarithmic negativity ( E ) inthe state, ρ ij , shared between two-sites i and j , and itsdecay with increase in lattice distance, r = | i − j | , fordifferent phases of the model in the J/t - n el plane. For abipartite state ρ ij , shared between two sites i and j , itslogarithmic negativity is defined as E ( ρ ij ) = log (2 N ( ρ ij ) + 1) , (2)where N is the negativity [29, 30], defined as the absolute J/t r = 1 r = 3 r = 5 E FIG. 2. (Color online.) Adiabatic freezing of bipartite entan-glement. Variation of two-site entanglement ( E ) with J/t fordifferent lattice distances, r = | i − j | =1 (black-circle), r = 3(blue-diamond), r = 5 (red-triangle), for the 1D t - J Hamilto-nian, with N = 30 and n el = N . From the figure, one can seethat for J/t ≤
2, the ground state BE remains adiabaticallyfrozen. The vertical axis is in ebits and the horizontal axis isdimensionless. Although the region considered in the figureis 0 . ≤ J/t ≤
2, the freezing behaviour extends all the wayto
J/t = 0. value of the sum of the negative eigenvalues of ρ T i ij , sothat N ( ρ ij ) = || ρ Tiij || − , where ρ T i ij denotes the partialtranspose of ρ ij with respect to the subsystem i .The decay of spin correlation functions with inter-sitedistance r , often signals the nature of correlation presentin the system [9, 10, 34]. In general, for non-critical statesof strongly-correlated 1D spin systems, quantum correla-tions are short-ranged and decay exponentially with theincrease of lattice distance [35], giving rise to featuressuch as the area law [5, 6]. As discussed earlier, forall n el in the J/t - n el phase space, at low values of J/t ( ≈ E ( ρ ij ), with thelattice distance r , for different values of the J/t ratio,using exact diagonalization to obtain the ground statefor N = 30 and n el = 2 /N [36]. In the metallic phase( J/t ≤ . r can be encapsu-lated as E ∼ / ( Ar + B ), where the numerically obtainedvalues of A and B , from the best-fit curve, are given by A = 162 . B = 18 .
9, respectively. Significantly, thecurves of E ( ρ ij ) with respect to r for different values of J/t are almost invariant in the metallic phase, i.e., thedecay is not only polynomial, but it is the same poly-nomial for all
J/t (see Fig. 2 for a more clear illustra-tion). The entanglement therefore remains adiabaticallyfrozen under perturbations of
J/t . It is known that inthe Luttinger-liquid phase, the NN spin correlation func-tions are independent of
J/t and the electron density[23].Therefore, one can infer that the freezing of bipartite en-tanglement is characteristic of the ground state phasediagram of the 1D t - J model. However, for non-NN spin correlation functions there is a very slow variation withthe system parameters. Therefore, the behavior of E in Fig. 1 not only expectedly follows the properties ofspin correlation functions but also provides more insightabout the ground state in the metallic phase. The freez-ing of bipartite entanglement with respect to system pa-rameters can be advantageous for implementing quantumtechnologies that is robust to fluctuations in the systemparameters, potentially due to errors in the preparationprocedure [24].In Fig. 1, for higher values of J/t ( ≥ J/t , an exponential decay ofspin correlation functions is expected. From Fig. 1, it isquite prominent that as the
J/t ratio increases, the BEmeasure E ( ρ ij ) exhibits an exponential decay with theincrease of r , given by E ∼ C exp( − r/ξ ), where ξ is thecharacteristic length of the decay. Again from the best-fit data, one can estimate the value of the constant C .As an example, for J/t = 3.6, the best-fitted plot yields C = 0 . ξ = 0 . J/t in the superconducting and PS phase. It isobserved that the decay becomes steeper, with increase in
J/t , such that entanglement vanishes quicker with r , andthe freezing behavior is completely lost in these regions.Moreover, if we introduce additional next-nearestneighbor interactions in the t - J Hamiltonian, the subse-quent spin model is known to have a rich phase diagramin the
J/t - n el plane [21], which is qualitatively similar tothat of the Hamiltonian in Eq. 1, apart from the fact that,in this case, the intermediate spin-gap phase is spreadover a larger area in the phase plane. The boundaries be-tween the metallic, superconducting, and PS phases arealtered. Interestingly, the freezing of BE, or lack thereof,in the different phases remains unaltered. IV. GROUND STATE PHASE AT LOWELECTRON DENSITIES
To understand the behavior of bipartite entanglementin the different phases of the 1D t - J Hamiltonian, wenow discuss the ground state properties of the model atlow electron densities. In the superconducting phase ofthe model, at low n el , a finite spin gap opens up, whichis in contrast to the behavior at the high density regionwhere the system remains gapless [23]. Interestingly, wefind that in this spin-gap phase, the ground state of thesystem is essentially a long-range resonating valence bond N = 12 N = 16 N = 20 N = 24 F J/t
FIG. 3. (Color online.) RVB gas as the spin-gap phase of the t - J Hamiltonian at low electron densities. We plot the fidelity( F ) of the ground state of the 1D t - J Hamiltonian, obtainedvia exact diagonalization, and the variational long-range RVBstate, at electron density n el = 2 /N . The curves shown inthe figure pertain to 1D lattices with N = 12 , , ,
24 sites.Note that the curves corresponding to N ≥
24 coincide withreasonable numerical accuracy. We note that the RVB gasstate considered for different values of
J/t and N are not thesame, as the set { r C } that maximizes the fidelity are different.All quantities used are dimensionless. (RVB) state or the RVB gas [37]. Thus, the ground statecan be expressed as | ψ (cid:105) RVB = (cid:88) C r C (cid:89) i (cid:54) = j | A i B j (cid:105) ⊗ (cid:89) k | k (cid:105) , (3)where | A i B j (cid:105) = √ ( | (cid:105) i | (cid:105) j − | (cid:105) i | (cid:105) j ) is the spin singletformed between two spin-1/2 particles at spin-occupiedsites ‘ i ’ and ‘ j ’, corresponding to the sublattices A and B , respectively. The product is over all such non-overlapping dimers between N el / { i, j } . The state (cid:81) k | k (cid:105) represents the k holes at N − N el vacant sites. The summation corresponds to thesuperposition of all possible dimer coverings ( C ) on thelattice, each with relative weight r C .The RVB gas description of the superconducting spin-gap phase of the 1D t - J Hamiltonian, at low electrondensity, has a remarkable significance, since it allows forthe study related to the phase properties of this modeland beyond, using the RVB ansatz [21, 38, 39] undersuitable doping. Hence, even for moderate-sized sys-tems, where exact diagonalization is not possible, thedoped RVB ansatz opens up the possibility of investi-gating different properties of the t - J Hamiltonian [40]using tensor network [41] or other approximate ap-proaches [42]. Fig. 3 depicts the behavior of the fidelity, F = max { r C } |(cid:104) φ g | ψ (cid:105) RV B | , between the ground state | φ g (cid:105) as estimated by exact diagonalization and the RVB state | ψ (cid:105) RV B , for low electron density, n el = 2 /N . One ob-serves that after a certain J/t ( ≈ . J/t ratio to a large value, the ground state at low n el still exhibitsRVB behavior but the probability of formation of nearest-neighbor singlet pairing increases as compared to distantpairs due to the formation of electron-hole phase separa-tion. In principle, this may lead to the formation of anRVB liquid state or NN dimer phase for high J/t , whichhas a decisive bearing on the exponential decay patternof the two-site entanglement of the system as the quan-tum correlation of the NN RVB states are known to beshort-ranged.
V. FREEZING OF MULTIPARTITEENTANGLEMENT
A significant outcome of our analysis of the entangle-ment properties of ground state phases of the 1D t - J Hamiltonian, is the existent characteristics of genuinemultipartite entanglement. To measure the genuine MEin the different regions of the
J/t - n el plane, we use thegeneralized geometric measure (GGM)[31] (cf. [32]).For an N -party pure quantum state | φ (cid:105) , the GGM is acomputable measure of genuine multisite entanglement,which is formally defined as the optimized fidelity-baseddistance of the state from the set of all states that arenot genuinely multiparty entangled. Mathematically, theGGM can be evaluated as G ( | φ (cid:105) ) = 1 − λ ( | ξ N (cid:105) ) , where λ max = max |(cid:104) ξ N | φ (cid:105)| , and | ξ N (cid:105) is an N -partynon-genuinely multisite entangled quantum state and themaximization is performed over the set of all such states.The GGM can be effectively computed using the relation G ( | φ (cid:105) ) = 1 − max { λ A : B | A ∪ B = A , . . . , A N , A ∩ B = φ } , where λ A : B is the maximum Schmidt coefficient in allpossible bipartite splits A : B of the given state | φ (cid:105) .A complexity in computation of the multiparty entan-glement measure G lies in the fact that the number ofpossible bipartitions increases exponentially with an in-crease of the lattice size. Therefore, we need to restrictourselves to moderate-sized systems only, which in ourcase restricts us to N = 16. We observe that at low elec-tron concentrations the GGM is adiabatically frozen oversignificant regions of the phase space.We study the variation of GGM in the ground state ofthe 1D t - J Hamiltonian, with respect to system param-eters
J/t and n el , as depicted in Fig. 4. For conveniencein representation, we look at higher values of J/t ( ≥ . G increases linearly with n el ,at low values of n el , for fixed J/t . It reaches a maximumat n el ≈ n el . This behavior is similar to the ground state prop-erties of spin liquid phases in doped Heisenberg ladders[40]. Significantly, in the low electron density regime, i.e., n el (cid:46) G ) is in-sensitive to the parameter J/t , and is thus adiabatically n el J/t = 10
J/t = 5.0
J/t = 3.3
J/t = 2.5
J/t n el = 0.12 n el = 0.25 n el = 0.37 n el = 0.5 G G FIG. 4. (Color online) Adiabatic freezing of genuine mul-tipartite entanglement. The plot shows the variation of thegeneralized geometric measure, G , with n el for different valuesof J/t . The number of lattice sites in the 1D model is fixed at N = 16. At low electron density, viz. n el (cid:46) . G increaseslinearly, along the same line , with n el , and reaches its max-imum value at n el ≈ .
6. This feature remains invariant forany value of the
J/t ratio. However at large n el , G becomes afunction of system parameters and the feature – of increasingalong the same line – obtained earlier, disappears. The insetshows that G is frozen with respect to change in J/t , for low n el . The axes dimensions are the same as in Fig. 1. frozen. We have numerically observed that at low n el this phenomenon extends to lower values of J/t . How-ever, this freezing of GGM completely vanishes as theelectron density is increased. We note that such adia-batic freezing of ME is not observed in other models, forinstance in the undoped anisotropic 1D model [33].This highlights a set of very unique features of theground state phases of the 1D t - J Hamiltonian. In par-ticular, in the metallic Luttinger liquid phase, at low
J/t and n el , bipartite entanglement is long-ranged and adi-abatically frozen, in stark contrast to the exponentiallydecaying BE in superconducting and PS phases. How-ever, at low n el but all J/t , including the latter phases,multipartite entanglement is frozen and completely in-variant to system parameters. This provides an inter-esting interplay between the behavior of BE and ME indifferent phases of the doped Hubbard model.
VI. CONCLUSION
Entanglement is an important resource in quantum in-formation protocols [1–3]. However, in general, both bi- partite and multipartite entanglement are fragile to deco-herence [43], and this is one of the main obstacles in real-ization of these protocols. Moreover, entanglement mayalso be highly sensitive to perturbative or sudden changesin system parameters and may fluctuate close to criticalpoints, as observed during collapse and revival [44] anddynamical transitions of entanglement [45]. It was ob-served that certain information-theoretic quantum corre-lations, such as quantum discord, could exhibit freezingin the face of decoherence [28], espousing a strong be-lief that this could lead to robust information protocols.However, entanglement, the workhorse of key quantuminformation protocols, rarely freezes under system pa-rameter or temporal changes, including under decoher-ence (cf. [46]). Our results show that doped quantumspin chains described by the 1D t - J Hamiltonian con-tain ground state phases that exhibit adiabatic freezingof both bipartite and genuine multisite entanglement. In-terestingly, the same model without the insertion of de-fects – in the form of doping – does not exhibit a similarfreezing phenomenon [33]. It is the presence of defects inthe quantum spin system that gives rise to the nascentphenomenon of adiabatic freezing of entanglement. Animportant observation in this regard is that no freezingphenomenon of multiparty entanglement (or other multi-party quantum correlations) has hitherto been observedin any quantum system. For applications in quantuminformation protocols, such as fault-tolerant [15] or one-way computation [14], robustness of multisite entangle-ment over fluctuating system parameters can be a signif-icant resource in achieving desired levels of stability.
ACKNOWLEDGMENTS
The research of SSR was supported in part by the IN-FOSYS scholarship for senior students. HSD acknowl-edges funding by the Austrian Science Fund (FWF),project no. M 2022-N27, under the Lise Meitner pro-gramme. DR acknowledges support from the EU Horizon2020-FET QUIC 641122. [1] R. Horodecki, P. Horodecki, M. Horodecki, and K.Horodecki,
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