Abstract
The fate of a hole injected in an antiferromagnet is an outstanding issue of strongly correlated physics. It provides important insights into doped Mott insulators closely related to hightemperature superconductivity. Here, we report a systematic numerical study of tJ ladder systems based on the density matrix renormalization group. It reveals a surprising result for the single hole's motion in an otherwise wellunderstood undoped system. Specifically, we find that the common belief of quasiparticle picture is invalidated by the selflocalization of the doped hole. In contrast to Anderson localization caused by disorders, the charge localization discovered here is an entirely new phenomenon purely of strong correlation origin. It results from destructive quantum interference of novel signs picked up by the hole and since the same effect is of a generic feature of doped Mott physics, our findings unveil a new paradigm which may go beyond the single hole doped system.
Introduction
A critical step in understanding the physics of doped Mott insulators which is closely related to hightemperature superconductivity in cuprates^{1,2,3} is to investigate the motion of a single hole in Mott antiferromagnets^{2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20}. It is easy to imagine that an injected hole in an antiferromagnetically ordered state leaves on its path the trace of spin mismatches in the staggered magnetization direction (to be called “S^{z}string” below). Such S^{z}string was expected^{21} to cause the localization of the hole, but, as was later realized^{8}, it can be selfhealed via quantum spin flips. It was then conjectured^{7,8,10,11} that a quasiparticle picture should be still valid in longdistance, as if the Bloch theorem holds true even for such a manybody system. The quasiparticle picture was further supported by finitesize exact diagonalizations^{12} on lattices up to 32 sites^{13}. However, experimentally, in angleresolved photoemission spectroscopy (ARPES) studies^{22,23,24}, very broad singleparticle spectral features have been observed in parent Mott insulator materials such as Ca_{2}CuO_{2}Cl_{2} and Sr_{2}CuO_{2}Cl_{2}. They cast a serious doubt^{14,17} on the rationale of understanding an individual hole created by the photon within a quasiparticle description. Recently, a strongly localized charge state in the copperoxide layer, induced by an offthelayer dopant defect, was observed in the scanning tunneling microscope experiment on Ca_{2}CuO_{2}Cl_{2}^{25}. In fact, the charge localization is generally found in a lightly doped cuprate before the system becomes a highT_{c} superconductor^{26}. These experiments raise a fundamentally important issue, i.e., whether the observed localization is an intrinsic phenomenon.
Theoretically, the validity of a quasiparticle description has been questioned in the literature as well. It was first argued by Anderson^{9} on general grounds that the quasiparticle spectral weight should vanish due to an unrenomalizable manybody phase shift induced when a hole is injected into a Mott insulator. Later, the discovery of the socalled phase string^{4,5,27} leads to a microscopic justification of this crucial observation. Here, a phase string may be regarded as an S^{±}string as opposed to the aforementioned “S^{z}string”. It is irreparable by quantum spin flips in contrast to the S^{z}string. It was further predicted^{17} that the phase string, instead of the S^{z}string^{21}, is responsible for an intrinsic selflocalization of the injected hole in two dimensions.
In this work, a largesize numerical simulation is used as a powerful machinery to resolve this issue. This approach is of particular importance because the hole's motion is highly sensitive to quantum interference developing at long distance. To this end, we study the ladder systems by the DMRG method^{28}, in which the length scale along one direction can be sufficiently large. As a ladder sample is long enough, we find that the charge generally gets localized so long as the leg number is larger than one. The localization scale decreases monotonically as the leg number increases. Contrary to this, if the sample is short, the injected hole behaves itinerantly, implying that the quasiparticle picture found in earlier numerical studies is likely a smallsize effect. We demonstrate that the charge selflocalization finds its origin in quantum destructive interference of the phase string, a unique property of the tJ model^{4,5,27}. In particular, we show that the novel localization phenomenon disappears once the phase string is turned off artificially in the simulation, where a welldefined quasiparticle description is recovered in consistency with the common wisdom^{8} of spin polaron picture.
Results
Model Hamiltonians
A prototypical doped Mott insulator is described by the tJ Hamiltonian, H_{t}_{J} = H_{t} + H_{J}, with
Here, is the electron creation operator at site i, S_{i} the spin operator and n_{i} the number operator. The summation is over all the nearestneighbors, 〈ij〉. The Hilbert space is constrained by the nodoubleoccupancy condition, i.e., n_{i} ≤ 1. At halffilling, n_{i} = 1, the system reduces to Mott insulators (antiferromagnets) with a superexchange coupling, J. Upon doping a hole into this system, n_{i} = N − 1 (N the number of the lattice sites) and the hopping process is triggered as described by the hopping term, H_{t}, with t the hopping integral.
For the single hole doped tJ model in a bipartite lattice, there exists an exact theorem, which states^{4,5,27} that the propagation of the hole is a superposition of quantum amplitudes of all the paths, with each path carrying a unique sign sequence known as the phase string, i.e.,
Here, the sign ± on the righthand side keeps track of an ↑ or ↓spin exchanged with the hole at each step of hopping as illustrated in Fig. 1 (A).
The quantum interference of phase strings from different paths is strong and play the fundamental role in dictating the longdistance behavior of the hole. In order to uniquely pin down the effect of phase strings, we also introduce the socalled σ · tJ model with the hopping term H_{t} in Eq. (1) replaced by
(i.e., an extra sign σ = ±1 is added). It is easy to show, following Refs. 4, 5, 27, that the phase string (2) disappears, whereas the positive weight for each path remains unchanged [as illustrated in Fig. 1 (B)]. In other words, one can artificially switch on and off the phase string effect between the tJ and σ · tJ models to study its novel consequences.
Previously the DMRG algorithm has already been used to study the tJ model in ladder systems at low hole doping^{16,29}. Below we shall focus on the one hole case on bipartite lattices of N = N_{x} × N_{y}, where N_{x} and N_{y} are the site numbers in the x and y directions, respectively. By using the DMRG method, we shall study the ladders with small N_{y} (from 1 to 5) and sufficiently large N_{x}. Here we set J as the unit of energy and focus on the t/J = 3 case unless otherwise specifically stated.
Selflocalization of the charge
Typical examples of the hole density distribution, , are shown in Fig. 2 (A) and (B) for N_{y} = 3 and 4, respectively. Here the charge profiles are plotted along the x direction for a middle leg of the ladders. They are localized in the central region of the ladders with open boundaries. Upon summing up the distribution at all the sites of different legs, the sum rule: is satisfied. Examples of the contour plot of in the xy plane can be found in Fig. S3 in Supplementary Information.
We systematically change the ladder length N_{x} to study the full width at halfmaximum (FWHM) of the charge profile. Fig. 2 (C) shows that for ladders with different N_{y}, the FWHM increases linearly for small sample lengths, while saturates for sufficiently large sample lengths (except the N_{y} = 1 case). The former indicates that the doped hole still remains itinerant at small N_{x}. The latter suggests that a selflocalization takes place for the doped hole, when the ladder length is long enough where the FWHM is no longer sensitive to the boundaries along the xdirection. Later on, by a more precise analysis, it will be shown that such localization only involves the charge of the doped hole, not its spin. Furthermore, Fig. 2 (C) clearly shows that the saturated FWHM at N_{y} > 1 monotonically decreases with the increase of the leg number N_{y}, implying that the localization effect becomes stronger and stronger. By contrast, there is no indication of saturation in FWHM for long onedimensional chains (N_{y} = 1, with N_{x} up to 500), which is consistent with the fact that the doped hole in strictly one dimension exhibits the Luttinger liquid behavior^{2}.
The destructive quantum interference effect of phase strings in the tJ model provides a natural explanation for the charge “selflocalization” observed in the above DMRG results. To further confirm the phase string origin of selflocalization, we switch off the phase string interference by two methods and then repeat the previous procedure of numerical simulations. In the first method, we study the σ · tJ model introduced above. In the second method, we set the interchain hopping coefficient, t_{⊥}, to be zero such that the hole can only move in the xdirection as if in a onedimensional chain. In both cases, the interference picture shown in Fig. 1 (A) breaks down. The numerical results for these two models are presented in Fig. 2 (D) for twoand threeleg cases, in which the FWHM of the hole density distribution restores the behavior of linear increase with the sample length. It indicates that the charge becomes delocalized in both cases, in sharp contrast to the selflocalization in the tJ model.
Although the observed selflocalization is insensitive to the parity (evenodd) of the leg number, the hole distribution is, as shown in Fig. 2 (A) and (B) and Supplementary Information: for the evenleg ladders (N_{y} = 2, 4), there are always small spatial oscillations on top of the Gaussian density profile, while they are absent for the oddleg ladders (N_{y} = 3, 5). This can be attributed to distinct decaying behavior of spinspin correlations for the odd and evenleg ladders at halffilling: the former (latter) follows a power (an exponential) law reflecting the absence (presence) of spin gap, see Supplementary Information for further explanations. Furthermore, the distinction of the evenodd effect disappears with the increase of t/J ratio (see Fig. S4), indicating that the detailed spin dynamics, governed by J, is not essential to the charge selflocalization effect.
Momentum distribution
Now let us examine the momentum distribution of the hole by studying . At halffilling, one finds n(k) = 1 and for the one hole case, 1 − n(k) is shown in Fig. 3.
The insets of Fig. 3 (A) and (B) present the hole momentum distribution 1 − n(k) as a function of k_{x} for fixed k_{y} = 2π/3 in the 3leg ladder and k_{y} = π/2 in the 4leg ladder, respectively. The value of k_{y} is chosen in a way that the “sudden change” in 1 − n(k) can reach maxima by varying k_{x}, according to the contour plots in the k_{x}k_{y} plane (see Supplementary Information). A very interesting feature is that after the rescaling: 1 − n(k) → [1 − n(k)]N, all the curves in the inset of Fig. 3(A) [or (B)] collapse onto a universal curve shown in the corresponding main panel. If one defines the Fermi surface by the sudden jump in the momentum distribution function, then the two universal curves suggest that the quasiparticle spectral weight is upperbounded by 1/N for large N and vanishes in the thermodynamic limit.
To satisfy the sum rule, , the width of the jump in 1 − n(k) must remain finite in the limit N → ∞. This is clearly shown in Fig. 3 (A) and (B), consistent with a finite localization length in the real space. We have also calculated the hole momentum distribution at different ratios of t/J. As shown in Supplementary Information, for a given sample size, the jump near the Fermi point is continuously reduced with increasing t/J, which is qualitatively consistent with earlier work^{12}.
Again a sharp contrast arises once the phase string is turned off in the σ ·tJ model, as shown in Fig. 3(C) for the 4leg ladder. Here the wide spread of 1 − n(k) in the momentum space seen in (A) and (B) collapses into a sharp peak at k_{x},k_{y} = 0. In particular, a universal curve (the inset) is obtained by the rescaling: k_{x} → N_{x}k_{x} instead of by 1 − n(k) → [1 − n(k)]N as done in the main panel. This clearly indicates that the quasiparticle spectral weight is finite and a Blochlike state is restored with a definite momentum at k = 0 (a peak at the equivalent of k_{x}, k_{y} = π is also found).
Spin degree of freedom
Complementary to the charge localization illustrated in Fig. 2, we further explore the fate of the spin1/2 associated with the hole injected into the halffilled spinsinglet background.
For the 3leg ladder, the spin1/2 is explicitly separated from the charge as demonstrated by Fig. 4(A). Here the single hole is created by removing a ↓ spin electron. While the charge of the doped hole is well localized at the sample center, the extra spin of S^{z} = 1/2 spreads over the sample. It is indicated by spin density as well as the coarse grained , which is approximately uniform across the ladder. This is probably the most direct display of electron fractionalization for a doped Mott insulator, thanks to the charge localization.
By contrast, for an evenleg ladder, because of the presence of a spin gap in the chargefree spin background, the extra spin of S^{z} = 1/2 tends to stay in the hole region in order to reduce its superexchange energy cost, as illustrated in Fig. 4(B) for the 2leg ladder case. In order to examine properly how the charge and spin associated with the doped hole behave, one has to resort to a different method.
Note that, first, if the charge part – namely the holon – is localized, the localization center can be anywhere in a translationally invariant system. In particular, an “extended” profile as a linear superposition of the localized wave packets at different positions should be also energetically degenerate. Second, if Fig. 4(B) really corresponds to spincharge separation, the extra spin1/2 – namely the spinon – should be more or less “free” within the regime expanded by the charge distribution. Then the spinon will prefer a more extended distribution of the hole to further lower its energy. In other words, the degeneracy of localized states should be lifted in favor of an extended profile in order to gain additional spinon energy.
Indeed, by increasing DMRG sweep number with the truncation error reaching much lower than 10^{−8}, the localized profile in Fig. 4(B) is found to slowly converge to an extended one as shown in (C) with the total energy lowered by ~ 10^{−4} J. To identify the nature of this small energy gain, one may vary the sample size from N_{x} = 16 to 800 and repeat the same procedure to obtain the onehole energy at each N_{x} [with the same full convergence to the extended charge profile as in Fig. 4(C)]. As shown in the inset of Fig. 5(A), with a constant term excluded, is well fitted by α(π/N_{x})^{2} (blue triangles) with the coefficient α = 0.87 J, as if it is contributed by some “free particle” in a box of length N_{x}^{30}.
Next, we show that such (finitesize) contribution to is not from the charge sector. For this purpose, we make the 2leg ladder as a ribbon by connecting the boundaries on the two edges of the ladder in the xdirection. Then we calculate the energy difference with threading through a flux π into the ribbon. It corresponds to the change of boundary condition from the periodic to antiperiodic one for the hopping term and is equivalent to coupling a small electric field to the charge sector. As shown in Fig. 5(A), oscillates strongly and falls off exponentially as with ξ ~ 14.5. It behaves completely different from behavior exhibited in , indicating that the charge indeed remains localized. The “free particlelike” behavior in should therefore solely originate from the chargeneutral part of the doped hole, i.e., the spinon.
By sharp contrast, upon turning off the phasestring in the σ · tJ model, both and follow the same behavior as clearly shown in the inset of Fig. 5(A). It is consistent with the previous conclusion that in this case the doped hole is restored as a quasiparticle, which carries both charge and spin and thus responds very similarly to the boundary condition (i.e., the finitesize effect) as a single mobile object.
The degenerate charge localized states can be superposed to form a more “extended” distribution to minimize the spinon energy due to the residual spincharge interaction,. It has been shown^{31} that generally the DMRG method tends to pick up a minimally entangled state among quasidegenerate states. Thus, spatially localized ones such as those in Fig. 2 are naturally obtained because a further spinon energy gain for a bigger FWHM would be too tiny to be detected with a given small DMRG truncation error. It also explains why the FWHM of charge distribution shown in Figs. 2 and 4 is usually bigger than the intrinsic localization length ξ as exhibited by . But the former is still valid to characterize the general trend of the localization, say, as a function of the leg number in Fig. 2. On the other hand, by adding some weak local chemical potentials, the quasidegeneracy of the hole states can be lifted such that the hole density profile is truly localized around the impurities at different locations [as indicated by different colored curves in Fig. 4(D) for a N = 400 × 2 ladder]. Here the increase of the total energy of the original tJ Hamiltonian is weak (about 10^{−4} J), presumably from the spinon part as the FWHM is still larger than ξ.
Finally, the spincharge separation is wellknown for the onedimensional chain^{2}. Figure 5(B) presents the corresponding 1leg results. As shown in the inset, the finitesize effect of is indistinguishable from the phase string free case (with open boundary conditions). But the charge response of is again totally different [the main panel of Fig. 5(B)]. It oscillates strongly as a function of N_{x}, similar to the 2leg case in Fig. 5(A). As a matter of fact, it can be directly attributed to the phase string modulation in the expression for the ground state energy given in Ref. 17. The main difference from the 2leg case is that the envelop of decays with increasing N_{x} in a power law fashion () instead of an exponential one. This is consistent with the fact that the holon is not localized in the onedimensional case. It is interesting to notice that even without the phase string effect, and behave distinctly for the σ · tJ model. The former , while the latter , also indicating a spincharge separation although the distinction is much less dramatic as compared to the tJ model.
Discussion
Charge selflocalization is a striking result of a singleholedoped Mott insulator. However, for this novel effect to get fully unveiled, a large sample is needed. Instead of tackling directly a twodimensional square lattice, in this work, we have chosen ladders that the DMRG method can access. As it turns out, very surprising results show up so long as the ladders are long enough, and, there are more than one paths to realize quantum interference, e.g., in a twoleg system.
The most essential physics underlying the observed exotic phenomena is that the motion of the single hole acquires a sequence of signs that precisely and permanently record each microscopic holehoppingspinbackflow event in the quantum spin background. These irreparable phase strings cause singular and destructive interference of quantum waves of the charge. As the distance of wave propagation increases, effects of destructive interference become stronger and stronger. The present results show unequivocally that the selflocalization of the charge, irrespective of even or odd ladders, is a combined effect of quantum interference and multiple scattering between the doped charge and spin background, in analogy to Anderson localization^{32}. This analogy notwithstanding, their nature is fundamentally different: The charge selflocalization arises purely from strong electron correlation in clean systems; whereas Anderson localization from disorders in noninteracting electron systems. Consistent with the picture of charge selflocalization above, in the single chain case, since quantum interference involving different paths is absent, the doped charge is delocalized.
The irreparable phase string effect is fundamentally different from the premise for a spin polaron picture – a conventional wisdom in which the disturbance of the hole hopping to the spin background is described by virtual spin excitations that eventually relax back after the hole moves away. In fact, the spinpolaron or quasiparticle behavior has been shown to be recovered once the phase string is “switched off” in the socalled σ · tJ model, where everything seems “normal” in a conventional sense. Here, a spincharge recombination with a good kinetic energy for the quasiparticle can be realized since there is no more singular phase string effect to strongly suppress the kinetic energy of the charge.
Phase strings have been proved^{27} rigorously as the sole sign structure in the tJ model on bipartite lattices, regardless of doping concentration, temperature and dimensions. As such, the present work may provide significant insights into the longstanding issue of high temperature superconductivity in the cuprates at finite doping. One may point out that for the realistic materials, the tJ model needs modified. For example, a nonbipartite lattice with a finite nextnearestneighboring hopping integral t′ or the Hubbard model with an intermediate onsite repulsion U is probably more relevant than the simpliest tJ model. We have numerically checked that the singlehole localization remains robust so long as t′ is small (e.g., t′/t < 0.2 for the 3leg ladder at t = 3 J), or in a Hubbard model, at least, with U/t > 10. It suggests that the generic localization found^{26} in the lightly doped cuprates may intrinsically belong to the new paradigm revealed in this work.
Methods
The numerical simulations in this work are performed by using the standard DMRG method. For Fig. 5, because the systems involved are small, we performed the simulation for both the open and periodic boundary conditions. For other results, the systems involved are large and we adopted the open boundary condition due to the numerical difficulty. In DMRG simulations, we keep up to m = 5000 states in the DMRG block with up to 40 sweeps to get converged results. For the measurement of the groundstate energy and other observables, the total truncation error is of the order or less than 10^{−8} ~ 10^{−12}.
To examine the validity of our numerical method, we calculated the groundstate energy for small systems and compared it with that obtained by the exact diagonalization method. The results are in excellent agreement. We also calculated the spinspin correlation functions of the Heisenberg ladders (halffilled tJ ladders). The results are consistent with those reported in earlier works (see Supplementary Information for details).
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Acknowledgements
Stimulating discussions with L. Balents, D. N. Sheng, S. R. White, J. Zaanen and especially P. A. Lee are acknowledged. This work was partially supported by the KITP (NSF no. PHY0551164) and the Center for Scientific Computing at the CNSI and MRL: the NSF MRSEC (DMR 1121053) and NSF CNS0960316; by the NBRPC (nos. 2009CB929402, 2010CB923003 and 2011CBA00108); by the NSFC (no. 11174174) and the Tsinghua University ISRP (no. 2011Z02151).
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H.C.J. developed the simulation codes, Z.Z. conducted the numerical experiments, Y.Q., C.T. and Z.Y.W. designed the project. All authors equally participated in discussing the results and writing the manuscript.
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Supplementary Information for: Strong correlation induced charge localization in antiferromagnets
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Zhu, Z., Jiang, HC., Qi, Y. et al. Strong correlation induced charge localization in antiferromagnets. Sci Rep 3, 2586 (2013). https://doi.org/10.1038/srep02586
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