Dynamical DMRG study of spin and charge excitations in the four-leg t-t^{\prime}-J ladder

Dynamical DMRG study of spin and charge excitations in the four-leg -- ladder


The ground state of the -- ladder with four legs favors a striped charge distribution for the parameters corresponding to hole-doped cuprate superconductors. We investigate the dynamical spin and charge structure factors of the model by using dynamical density-matrix renormalization group (DMRG) and clarify the influence of the stripe on the structure factors. The dynamical charge structure factor along the momentum direction from to clearly shows low-energy excitations corresponding to the stripe order in hole doping. On the other hand, the stripe order weakens in electron doping, resulting in less low-energy excitations in the charge channel. In spin channel, we find incommensurate spin excitations near forming a hour-glass behavior in hole doping, while in electron doping we find clearly spin-wave-like dispersions starting from . Along the - direction, the spin excitations are strongly influenced by the stripes in hole doping, resulting in two branches that form a discontinuous behavior in the dispersion. In contrast, the electron-doped systems show a downward shift in energy toward . These behaviors along the - direction are qualitatively similar to momentum-dependent spin excitations recently observed by resonant inelastic x-ray scattering experiments in hole- and electron-doped cuprate superconductors.

78.20.Bh, 78.70.Ck, 78.70.Nk, 74.72.-h

I Introduction

In cuprate superconductors, spin excitations near the magnetic zone center in the Brillouin zone change with hole doping from a spin-wave-type excitation in Mott insulating phase to a hour-glass-type excitation as observed by inelastic neutron scattering (INS) Fujita2012 (). The formation of charge stripes in the hole-doped cuprates Tranquada1995 () has been assigned to a possible origin of the hour-glass-type excitation based on a two-dimensional (2D) single-band Hubbard model with nearest-neighbor hopping , next-nearest-neighbor hopping , and on-site Coulomb interaction  Kaneshita2001 (); Seibold2005 (); Seibold2006 () and on a localized spin model Kruger2003 (); Carlson2004 (). Recent quantum Monte Carlo calculations of dynamical spin structure factor for a four-leg three-band Hubbard ladder including oxygen orbitals Huang2017 () and for a -- four-leg ladder Huang2017b () have also indicated the hour-glass-type excitation under the presence of the charge stripes. In contrast to hole doping, the spin-wave-like excitation persists with introducing electron carriers in the -- Hubbard model Huang2017b (), being consistent with INS experiment Fujita2012 ().

Similar to the -- Hubbard model, there is a clear electron-hole asymmetry in a -- model, which is caused by the interplay of spin background and  Tohyama1994 (); Tohyama2004 (). In hole doping, the ground state of a four-leg -- ladder has been studied by the density-matrix renormalization group (DMRG) Tohyama1999 (); White1999 (); Scalapino2012 (); Dodaroo2017 (). The charge stripes are stabilized for negative and with changing the sign of the stripes become weaker White1999 (). The dynamical spin structure factor of the model, however, has not been studied by using DMRG.

Recent development of resonant inelastic x-ray scattering (RIXS) tuned for the Cu edge has provided a lot of new insights about spin excitations in cuprates Ament2011 (); Dean2015 (). Very recently, spin excitations in the so-called 1/8-doped system, LaBaCuO, have been observed by the RIXS and an anomalous change of spin dispersion around has been reported below the stripe-ordered temperature Miao2017 (). The anomaly has been explained by a localized spin model reflecting the effect of the charge stripes Miao2017 (). However, there is no investigation of spin excitations along the momentum perpendicular to the stripes based on microscopic models like the -- model. Furthermore, RIXS can detect momentum-dependent charge excitations Ishii2013 (); Ishii2014 (); Ishii2017 (). Therefore, it is important to study not only spin dynamics but also charge dynamics in such a microscopic model.

In this paper, we investigate both the dynamical spin and charge structure factors in a four-leg -- ladder to give an insight into momentum-dependent spin and charge dynamics in both hole-doped and electron-doped cuprates. We perform large-scale dynamical DMRG calculations. The dynamical charge structure factor along the - direction clearly shows low-energy excitations corresponding to the stripe order in hole doping. On the other hand, the stripe order is weak in electron doping, resulting in less low-energy excitations as expected. In the dynamical spin structure factor, we find incommensurate spin excitations near the magnetic zone center forming a hour-glass behavior in hole doping, while in electron doping we find clearly a spin-wave-like dispersion starting from . Along the - direction, spin excitations are strongly influenced by the stripes in hole doping, resulting in two branches forming a discontinuous dispersion. In contrast, spin excitations in electron doping show a downward shift in energy toward . These behaviors along the - direction are qualitatively similar to RIXS results Miao2017 (); Ishii2014 ().

This paper is organized as follows. The four-leg -- ladder and dynamical DMRG method are introduced in Sec. II. In Sec. III, we calculate the charge distribution in the ground state. The dynamical charge structure factors obtained by the dynamical DMRG are compared between hole and electron dopings. The dynamical spin structure factors in both hole and electron dopings are shown in Sec. IV. Finally, a summary is given in Sec. IV.

Ii Model and method

The Hamiltonian of the -- model in two dimensions reads


where , , and are the nearest-neighbor hopping, the next-neighbor hopping, and the antiferromagnetic (AF) exchange interaction, respectively; , and , , and being the unit vectors in the and directions, respectively; the operator with annihilates a localized particle with spin at site with the constraint of no double occupancy; is the spin operator at site ; and .

In the model (1), the difference between hole and electron dopings is taken into account by the sign difference of the hopping parameters Tohyama1994 (): For hole doping, the particle is an electron with and , while the particle is a hole with and for electron doping. We take and for both the hole- and electron-doped cases, which are typical values appropriate for cuprates with  eV.

We use a -site lattice with cylindrical geometry where the direction is of open boundary condition while the direction is of periodic boundary condition. This lattice is called the four-leg -- ladder. The carrier density for holes ( electrons) in the ladder is defined by (). In the ladder ( and ), the component of momentum is determined by using a standard translational symmetry, i.e., (), but the component is given by () because of the open boundary condition. Defining () as the () component of site , respectively, we can write the Fourier component of charge operator and that of the component of spin operator as





The dynamical charge and spin structure factors, and , are defined as


where represents the ground state with energy ; ; and is a small positive number.

We calculate Eqs. (4) and (5) for the -- ladder by dynamical DMRG, where we use three kinds of target states: For , (i) , (ii) , and (iii) . The target state (iii) is evaluated by using a kernel-polynomial expansion method Sota2010 (), where the Lorentzian broadening in Eqs. (4) and (5) is replaced by a Gaussian broadening with half width at half maximum . In our numerical calculations, we divide the energy interval by 100-mesh points and have targeted all of the points at once. To perform DMRG, we construct a snakelike one-dimensional chain, and use the maximum truncation number , and resulting truncation error is less than .

Iii Dynamical charge structure factor

We first examine the carrier distribution in the ground state of the four-leg -- ladder to confirm the nature of the charge stripes reported previously by DMRG Tohyama1999 (); White1999 (); Scalapino2012 (); Dodaroo2017 (). Figure 1 shows the carrier number along the leg position . Note that there is no carrier number dependence on the rung position . As expected, there is an oscillation of in the middle of the ladder, depending on the charge density in both hole and electron dopings. The period is 6-, 4-, and 3-lattice spacing for , 1/8, and 1/6 as shown in Figs. 1(a), 1(b), and 1(c), respectively, implying that the period is given by . The amplitude of the oscillation is smaller in electron doping than in hole doping. This has been pointed out in the context of sign change of , where positive suppresses the stripes White1999 (). The standard deviation for is plotted as a function of the carrier number in Fig. 1(d). In hole doping, has a maximum at , implying the strongest stripe order near the doping as observed in hole-doped cuprates Tranquada1995 (). In electron doping, is small and decreases above .

Figure 1: Carrier distribution in the -- ladder. , , and for hole (electron) doping. The carrier number along the leg position for (a) , (b) , and (c) . The black squares (red circles) represent for hole (electron) doping. The horizontal dotted line denotes averaged number . (d) The () dependence of the standard deviation for in hole (electron) doping.

In contrast to carrier density, spin density is zero in all cases because of no spin-symmetry breaking in the calculations. If one introduces an external magnetic field at the edges, the symmetry is broken and thus local spin density becomes finite as discussed in the previous DMRG calculations White1999 (). In hole doping, there is a ferromagnetic spin arrangement across the charge stripes, resulting in anti-phase spin structures White1999 ().

Figure 2 shows along the - direction for both hole and electron dopings. As expected from the stripe ground state, strong low-energy excitations whose energy minimum is located around emerge in the hole-doped case [see Figs. 2(a), 2(b), and 2(c)]. There are broad but weak excitations at . In contrast to the hole doping, low-energy excitations near for electron doping [see Figs. 2(d), 2(e), and 2(f)] are very weak reflecting weak stripe ordering as discussed above. High-energy excitations above show broad dispersive features that are steeper than those in hole doping. Such dispersive high-energy excitations have been observed in the Cu -edge RIXS for electron-doped cuprates Ishii2014 (); Lee2014 ().

Figure 2: along the - direction in the -- ladder with . (a) , (b) , and (c) for hole doping ( and ). (d) , (e) , and (f) for electron doping ( and ).

Strong low-energy excitations for small at the low doping region in have been proposed in the electron-doped side of the 2D Hubbard model with  Tohyama2015 (). The origin of the strong intensity has been attributed to the proximity to the phase separation from the study of the -- model Greco2017 (). Such strong intensity is also seen at in Fig. 2(d). The intensity is, in fact, reduced with introducing the next-nearest neighbor Coulomb interaction (not shown here) as demonstrated in Greco2017 ().

Iv Dynamical spin structure factor

At half filling (), the four-leg -- ladder exhibits a spin gap whose magnitude is close to  White1994 (). In Fig. 3(a), the gap is identified as the peak position of at , which is close to . A spin-wave-like dispersion exists toward from and along the - direction [Fig. 3(b)], whose energy is slightly higher than the energy of dispersion obtained by the linear-spin wave theory for a 2D Heisenberg model (purple lines in Figs. 3(a) and 3(b)).

Figure 3: in the -- ladder with . (a) Half filling () and . (b) Half filling and . (c), (d), and (e): Hole doping ( and ) with , , and , respectively, along the - direction. The purple dots in (c), (d), and (e) represent the peak position of INS experiments for LaSrCuO Lipscombe2009 (), LaBaCuO Tranquada2004 (), and LaSrCuO Vignolle2007 (), respectively, where we assume  meV Hayden1991 (). (f), (g), and (h): Electron doping ( and ) with , , and , respectively, along the - direction. The purple dots in (f) and (h) represent the peak position of INS experiments for PrLaCeCuO Asano2017 () with and , respectively. The purple lines in (a), (b), and (f) represent a single magnon dispersion at half filling obtained by the linear spin-wave theory for the 2D Heisenberg model.

With hole doping, the excitation at half filling splits into two low-energy excitations along the - direction as shown in Figs. 3(c), 3(d), and 3(e). The wave vector measured from is approximately given by , which is consistent with incommensurate vectors reported in hole-doped cuprate superconductors LaSrCuO Yamada1998 (). Linear dispersive branches emerge from the position toward both the (inward) and (outward) directions in all three densities, , 1/8, and 1/6.

In INS experiment Fujita2012 (), the outward dispersion has not been observed. Furthermore, in other calculations of under the stripe order for the 2D extended Hubbard model based on the random phase approximation Kaneshita2001 () and time-dependent Gutzwiller approximation (TDGA) Seibold2006 (), the outward dispersion looses its intensity quickly for small . However, the outward dispersion is clearly seen in the -- ladder. This inconsistency may arise from ladder geometry in our model, which is different from 2D geometry in the experiment and other calculations.

The inward dispersive structure merges with that from the opposite side and forms an intense structure at , whose energy position is clearly lower than , i.e., at and increases with increasing up to at . Above the structure there is no gap in contrast to the TDGA results for the 2D extended Hubbard model Seibold2006 (). Rather there is a linear dispersive structure with small intensity extending, for example, up to at as seen Fig. 3(c). The linear dispersive feature looks continuously connected to the linear dispersions starting from the incommensurate position. Such a behavior has been reported in a linear spin-wave theory for a spin model assuming bond-centered vertical stripe, where AF exchange interaction is assumed for every nearest-neighbor bonds except for the ferromagnetic bonds across the stripe Carlson2004 (). In this view, the upward-energy shift of the strong-intensity position at with increasing can be partly related to the outward shift of the incommensurate wave vectors, where we assume that the velocity of spin-wave-like dispersion starting from the incommensurate points does not change significantly with .

In Figs. 3(c), 3(d), and 3(e), the calculated spectra are compared with the experimental peak positions Lipscombe2009 (); Tranquada2004 (); Vignolle2007 (), assuming  meV Hayden1991 (). The neck position of the hour-glass dispersions is lower in energy than the position of the calculated spectrum. The difference would be attributed to the ladder geometry whose spectral weight tends to shift to higher-energy position as compared with 2D systems, as demonstrated by the comparison with the linear spin-wave theory at half filling [see Figs. 3(a) and 3(b)].

In contrast to hole doping, the lowest-energy excitation of in electron doping remains at for all as shown in Figs. 3(f), 3(g), and 3(h). Such low-energy excitations at are expected from strong AF correlation in electron doping due to the effect of  Tohyama1994 (); Tohyama2004 (). The dispersive behavior near is similar to that at half filling, but away from the spectral distribution becomes broader as compared with that in Fig. 3(a). This is qualitatively consistent with INS experiments for electron-doped cuprates Fujita2012 (); Asano2017 (); Wilson2006 (); Fujita2006 (). The experimental peak positions for PrLaCeCuO Asano2017 () are plotted in Figs. 3(f) and 3(h). We find a rough agreement with our calculated results.

Figure 4: Integrated weight of with respect to along the - direction for the -- ladder with and . (a) Hole doping and (b) electron doping.

Figure 4 shows the integrated weight of with respect to along the - direction for both hole and electron dopings. In hole doping, the lowest-energy peak at reduces its weight with and broadens with extending the weight higher energy. The energy region higher than looses the weight significantly. In contrast, the weight at the high-energy region in electron doping remains less dependent as shown in Fig. 4(b). We also notice that a high-energy peak at at half filling shifts to the lower-energy side around with increasing . This corresponds to the broadening of the spin-wave-like dispersion as discussed above.

Figure 5: along the - direction in the hole-doped -- ladder with . (a) , (b) , and (c) .

Since the charge stripe has a charge modulation along the - direction, is expected to show spectral features associated with the stripe. In fact, for hole doping clearly exhibits such a feature as shown in Fig. 5. For (1/6), discontinuous spectral intensity appears at close to the stripe wave vector as seen in Figs. 5(b) [5(c)]. More precisely, there are two branches, one of which has low-energy excitations with maximum energy near and the other of which exhibits high-energy excitations around .

The anomaly at in spin excitation has been reported in RIXS for LaBaCuO Miao2017 (), although clear discontinuous spectral intensity has not been identified. In the interpretation of the experimental data, a localized spin model has been introduced, where AF magnetic exchange interaction across disordered charge stripes is replaced by a ferromagnetic one Miao2017 (). A simple view for the presence of two branches in our results is also given by a one-dimensional spin model where a ferromagnetic exchange interaction is periodically introduced onto one of two bonds. In this simple model, the two branches show an anti-crossing, leading to a gap at the middle of the magnetic Brillouin zone and a clear separation of the two branches. Therefore, the separated two branches obtained by our dynamical DMRG calculations indicate the presence of ferromagnetic effective interaction along the perpendicular direction of the charge stripes.

Figure 6: along the - direction in the electron-doped -- ladder with . (a) , (b) , and (c) .

In electron doping, such a discontinuous behavior of spectral weights is invisible in as shown in Fig. 6. This is consistent with weak charge stripe moderation in electron doping as discussed in Sec. III. Alternatively one can find a peculiar spectral behavior in contrast to hole doping, which is a downward shift of the peak position of spectral weight beyond for all three cases. This is a counterintuitive behavior in the sense that spin excitations similar to the Heisenberg model might be expected as evidenced from the spin-wave-like dispersions near . This downward behavior is, thus, due to the presence of electron carriers, suggesting the influence of itinerant nature in the electron-dopes system. The signature of such a downward shift has not clearly been seen in the experimental data of RIXS for NdCeCuO Ishii2014 (); Lee2014 (). However, it is clear that the dispersion along the - direction in electron-doped cuprates Ishii2014 () becomes flat above , in contrast to hole-doped cuprates with monotonically increasing dispersion LeTacon2011 (); LeTacon2013 (); Dean2013 ().

The spin-excitation energy at in Fig. 6 remains almost same at half filling in Fig. 3(b). This is different from the experimental data where the energy increases with increasing  Ishii2014 (); Lee2014 (). This difference will disappear if one introduces the so-called three-site terms into the -- model Jia2014 ().

V Summary

In summary, we have investigated the dynamical spin and charge structure factors, and , in the four-leg -- ladder by using dynamical DMRG. along the - direction clearly shows the low-energy excitations corresponding to the stripe order in hole doping, while the stripe order weaken in electron doping, resulting in less low-energy excitations. In , we find incommensurate spin excitations near the magnetic zone center forming a hour-glass behavior in hole doping. However, the outward dispersion from the incommensurate position is strong in intensity, inconsistent with INS experiments. In electron doping, clear spin-wave-like dispersions starting from are seen, being similar to INS experiments. Along the - direction, the spin excitations are strongly influenced by the stripes in hole doping, resulting in two branches that form a jump in the dispersion. In contrast, the spin excitations show a downward shift in energy toward . These behaviors along the - direction are also qualitatively consistent with RIXS results. For more quantitative descriptions, we need to treat 2D systems rather than ladder systems. This remains as a future problem.

We thank M. Fujita for providing us experimental data and for fruitful discussions. We also thank K. Ishii for useful discussions. This work was supported by MEXT, Japan, as a social and scientific priority issue (creation of new functional devices and high-performance materials to support next-generation industries) to be tackled by using a post-K computer, by MEXT HPCI Strategic Programs for Innovative Research (SPIRE) (hp170114), and by the inter-university cooperative research program of IMR, Tohoku University. The numerical calculation was carried out at the K Computer and HOKUSAI, RIKEN Advanced Institute for Computational Science, and the facilities of the Supercomputer Center, the Institute for Solid State Physics, the University of Tokyo, This work was also supported by the Japan Society for the Promotion of Science, KAKENHI (Grants No. 26287079 and 15K05192).


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