Correlation function for the one-dimensional extended Hubbard model at quarter filling

# Correlation function for the one-dimensional extended Hubbard model at quarter filling

S. Nishimoto Leibniz-Institut für Festkörper- und Werkstoffforschung Dresden, D-01171 Dresden, Germany    M. Tsuchiizu Department of Physics, Nagoya University, Nagoya 464-8602, Japan
August 1, 2019
###### Abstract

We examine the density-density correlation function in the Tomonaga-Luttinger liquid state for the one-dimensional extended Hubbard model with the on-site Coulomb repulsion and the intersite repulsion at quarter filling. By taking into account the effect of the marginally irrelevant umklapp scattering operator by utilizing the renormalization-group technique based on the bosonization method, we obtain the generalized analytical form of the correlation function. We show that, in the proximity to the gapped charge-ordered phase, the correlation function exhibits anomalous crossover between the pure power-law behavior and the power-law behavior with logarithmic corrections, depending on the length scale. Such a crossover is also confirmed by the highly-accurate numerical density-matrix renormalization group method.

###### pacs:
71.10.Fd, 71.10.Hf, 71.10.Pm

## I Introduction

One-dimensional (1D) electron and spin systems have been attracted much attention since they often exhibit nontrivial quasi-long-range-ordered behavior due to the large low-dimensional quantum fluctuation effects. Gogolin_book (); Giamarchi_book () The critical behavior in the 1D systems, which is called the Tomonaga-Luttinger liquid (TLL) state, has a long history of research and the low-lying modes are known to be described by collective gapless excitations and physical quantities show power-law behavior in the temperature and/or distance dependences. It has also been recognized that, in the systems with spin-rotational symmetry, logarithmic singularities appear in the magnetic-field-dependent corrections to the magnetization YangYang1966 (); Babujian1983 () and the spin susceptibility, Schlottmann1987 () and in the temperature-dependent corrections to the spin susceptibility, Eggert1994 () specific heat, Kluemper1998 () and nuclear magnetic resonance, Barzykin2000 () etc. Motivated by such developments in the field, a number of numerical studies has been successfully performed to examine the logarithmic corrections in spin-chain systems. Kubo1988 (); Liang1990 (); Sandvik1993 (); Hallberg1995 (); Koma1996 (); Eggert1996 (); Nomura1993 (); Hikihara1998 () The spin-spin correlation function of the 1D Heisenberg model has been extensively studied as a most fundamental theoretical model to examine the presence of logarithmic corrections. Finkel'stein1997 (); Affleck1989 (); Giamarchi1989 (); Singh1989 () At this moment, the correlation amplitudes in the asymptotic form of the correlation function can be exactly obtained. Lukyanov1997 (); Lukyanov1999 (); Lukyanov2003 () In contrast, only few efforts have been devoted to those of the Hubbard model due in part to the difficulty of analyzing. Schulz1990 () Therefore, the situation is much less satisfactory as far as the logarithmic corrections in the Hubbard model are concerned.

In the present paper, we focus on the logarithmic corrections in the equal-time density-density correlation function of the quarter-filled Hubbard model including the Coulomb repulsion between electrons on site and the nearest-neighbor sites . So far this model has been analyzed as a minimal model to describe physical phenomena in organic solids. Seo_review () The Hamiltonian of the 1D extended Hubbard model at quarter-filling is given by

 H = −t∑j,s(c†j,scj+1,s+H.c.) (1) +U∑jnj,↑nj,↓+V∑jnjnj+1,

where is the annihilation operator on the th site with spin , and the density operators are and . The hopping energy between the nearest-neighbor sites is represented by . It is known that, at zero temperature, the gapped charge-ordered (CO) state emerges in the large repulsive and region, where the phase boundary is determined numerically by using the exact diagonalization Mila1993 (); Sano1994 () and the highly-accurate density-matrix renormalization group (DMRG) method. Ejima2005 () The mechanism of this quantum phase transition has also been addressed by the bosonization technique and the renormalization-group (RG) method. Giamarchi_book (); Schulz1994 (); Tsuchiizu2001 () In this paper, we perform the detailed analysis on the equal-time correlation function in the TLL phase. The exponent of the correlation functions is characterized by so-called the TLL parameter . Schulz1990 () Especially by focusing on the correlation function near the boundary to the CO insulating state, we show that it exhibits the nontrivial crossover, depending on the length scale, from the power-law behavior with logarithmic correction for short distance, to the pure power-law behavior for large distance.

The present paper is organized as follows. In Sec. II, the analytical form of the correlation function is obtained by utilizing the RG technique based on the bosonization method. In Sec. III, the analytical results are confirmed by using the highly-accurate DMRG method. The summary is given in Sec. IV. Detailed derivation of the analytical form of the correlation function is given in the Appendix.

## Ii Bosonization approach

In this section, we derive the generalized analytical form of the correlation function in the TLL state. We analyze the limit case and the finite case separately, since the picture of the can become transparent with the analogy of the spin-chain system which properties are well understood.

### ii.1 The U→∞ limit

In the limit, since the double occupancy of electrons is excluded, the extended Hubbard model [Eq. (1)] reduces to the spinless half-filled model:

 HU→∞=−t∑j(d†jdj+1+H.c.)+V∑jndjndj+1, (2)

where . It is well known that this model can be mapped onto the XXZ spin-chain model by using the Jordan-Wigner transformation Giamarchi_book (); Affleck () and the physical properties have been extensively studied with both the exact treatment based on the Bethe ansatz and numerical approaches. Here we examine the analytical form of the correlation function by using the exact results obtained in the context of spin-chain problems.

The density operator is expressed in terms of the bosonic field operator as Giamarchi_book (); Affleck (); Hikihara1998 ()

 ρ(x)=ndja=12πηdφdx−(−1)jcπasin√2πηφ, (3)

where is the lattice constant and we will set in the following. The parameter can be related to the TLL parameter by . The nonuniversal parameter will be shown later. The model Hamiltonian (2) can be expressed in terms of the bosonic field .

It is well-known that the TLL phase is realized for small , while the gapped CO state appears for . In the TLL phase, the parameter is known exactly from the Bethe ansatz as , and the TLL parameter varies within the range of for . The TLL parameter approaches to an universal value when . The mechanism of this quantum phase transition has also been addressed by the bosonization technique and the RG method Giamarchi_book (); Schulz1994 (); Tsuchiizu2001 () and it has been clarified that the 1/4-filled umklapp scattering has a crucial role in making the TLL state into the gapped CO state. It has also been shown that the universality class of this TLL-to-CO phase transition is in the Kosterlitz-Thouless transition, where the umklapp scattering is irrelevant in the TLL phase while it becomes relevant in the CO phase. It is worthwhile to note that, on the phase boundary between the TLL and CO states, the umklapp scattering term becomes marginally irrelevant, where it shows very slow scaling-parameter dependence and can give rise to anomalous corrections to physical quantities. We analyze this effect on the density-density correlation function by using the RG method developed in Ref. Giamarchi1989, . The resultant form of the correlation function for in the TLL state is given by

 N(x)=−Kρπ2x2+A2cos4kFxx4Kρ[1−(α/x)2−8Kρ]1/2[1+(α/x)2−8Kρ]3/2, (4)

where is the Fermi wave vector, . The derivation of this form is given in the Appendix. The parameter is the short-distance cutoff which appears in the RG method. The coefficient is given by by using the parameter in Eq. (3). In Refs. Lukyanov1997, and Lukyanov1999, , the exact form of the parameter has been proposed, where it is controlled by the TLL parameter, i.e., , and its exact form is given by

 c(η) = 2[Γ\biglb(η/(2−2η)\bigrb)2√πΓ\biglb(1/(2−2η)\bigrb)]1/2ηexp[12∫∞0dtt(sinh[(2η−1)t]sinh(ηt)cosh[(1−η)t]−2η−1ηe−2t)]. (5)

The numerical values of are shown in Table. I. For , the quantity becomes . We note here that the quantity in Eq. (4) is the only unknown parameter, which is to be determined numerically.

Here we find that equation (4) has two different asymptotics: (i) In the short-range region, the power-law behavior with logarithmic correction is obtained, while (ii) the logarithmic correction disappears in the long-range region. By noting , the length scale which characterizes crossover between these two regions is given by

 xcross=αexp[1/(8Kρ−2)]. (6)

We note that, for , the logarithmic correction appears in the whole length scale. By noting for , the explicit form of the correlation function (4) at is given by

 N(x) = −14π2x2+1(2π)3/2cos4kFxxln1/2(x/α) (7) for (U,V)=(∞,2t).

This formula was reported in Ref. Affleck1998, for the antiferromagnetic Heisenberg spin chain.

### ii.2 For finite U

Next we examine the generic case. In this case, there appears the conventional oscillation term in addition to the one. We show that the additional logarithmic correction appears near the phase boundary to the CO phase, not only in the oscillation term but in the oscillation term.

Based on the conventional bosonization for electron systems, the density operator is given by Giamarchi_book (); Schulz1994 ()

 ρ(x) = 1πdϕρdx−2c1πasin(2kFx+ϕρ)cosϕσ (8) +2c2πacos(4kFx+2ϕρ),

where and are the charge and spin phase fields. The and are nonuniversal numerical quantities satisfying and in the noninteracting case. In the similar way to the case, the most general form of the density-density correlation function is derived as (see Appendix)

 N(x) = −Kρπ2x2+A1cos2kFxxKρ+1ln−3/2(x/ασ)[1−(α/x)4−16Kρ]1/8 (9) +A2cos4kFxx4Kρ[1−(α/x)2−8Kρ]1/2[1+(α/x)2−8Kρ]3/2,

where and are the short-distance cutoff parameters for the charge and spin sectors, respectively. In the case of , the coefficients and , which are proportional to and respectively, are to be determined numerically. The logarithmic correction in the oscillating term appears due to the marginally irrelevant coupling of the spin channel. Giamarchi1989 () In the noninteracting limit , the quantity vanishes and the logarithmic correction is replaced by a constant, and then the correlation function reproduces the trivial result .

From Eq. (9), we find that an anomalous logarithmic correction also appears in the oscillating term near the boundary of the CO phase. On the phase boundary, the correlation function reads

 N(x) = −14π2x2 (10) +~A1cos2kFxx5/4ln−3/2(x/ασ)ln−1/8(x/α) +~A2cos4kFx(2π)3/2ln1/2(x/α) for (U,V)=(Uc,Vc)

where and for .

## Iii Numerical results

For numerical confirmation of the logarithmic corrections, we employ the DMRG technique which provides very accurate data for the ground-state correlation functions of 1D correlated electron systems. White1992 () We consider electrons on a chain with sites and calculate the equal-time density-density correlation function

 N(x)=⟨njnj+x⟩−⟨nj⟩⟨nj+x⟩, (11)

under the open-end boundary conditions (OBC). Here, the distance is centered at the middle of the system. The application of OBC enables us to obtain the correlation function (11) quite accurately for very large finite-size systems up to sites. However, the real-space DMRG method works with finite number of sites, so that we have to pay special attention to the finite-size effects for a precise comparison with the RG results. In the present calculations, the most problematic finite-size effect is the Friedel oscillation starting from the open edges. To eliminate it, we simply add on-site potential energy on both edge sites. It corresponds to a compensation of the “missing correlation” caused by the absence of their neighboring site. Hereby the Friedel oscillation is fairly suppressed.

On that basis, the remaining finite-size effects are investigated. We now choose some parameters in the vicinity of the CO phase where the finite-size effect is relatively large due to strong charge fluctuations. For these parameters, we calculate for several chains with length up to sites and then obtain the extrapolated values to the thermodynamic limit () using the finite-size-scaling analysis. By comparing the extrapolated values and the finite-size data, we find that in the thermodynamic limit can be reproduced with extracted central sites of a chain with sites within a few percent error. The relative error in the ground-state energy, , is below , where is the ground-state energy per site for a chain with sites. Consequently, we will study the equal-time correlation function for the central sites of a chain with sites without the finite-size-scaling analysis. We keep up to density-matrix eigenstates in the DMRG procedure and all the calculated quantities are extrapolated to the limit. For your information, in this way we obtain [the exact are ] for the coefficients of Eq. (9) in the non-interacting case which poses a non-trivial problem to the DMRG method.

### iii.1 The U→∞ limit

Let us first consider the correlation function in the limit. We now attempt to fit the DMRG results of into the analytical form of Eq. (4). Since the exact solutions of and are available, the quantity is the only fitting parameter in Eq. (4). Figure 1 (a) shows the DMRG results of for . An excellent agreement of the DMRG data with the fitted line is found. We then obtain , which leads to . It means that the logarithmic correction appears at for . We note that the central sites out of are used to carry out the fitting procedure and, however, the fitting results are almost unchanged for any choice of the length from to . In the same way, we can also estimate the values of for the other values. In Fig. 1 (b), the estimated values of are plotted as a function of . We find that the logarithmic correction is hardly present at and the length scale increases rapidly in the vicinity of the CO insulating phase. Note that at .

### iii.2 For finite U

We next turn to the case of . In this case, the numerical results of the correlation function can be fitted with the analytical form of Eq. (9). Differently from the case of , there are five fitting parameters; namely, , , , , and . Of them, may be obtained very accurately with the DMRG method via the derivative of charge structure factor at ,

 Kρ=12limq→0⟨n(q)n(−q)⟩, (12)

with and . Thus, we can reduce the fitting parameters from five to four. In Fig. 2, we show the fitting results of the correlation function near the boundary of the CO phase for and [the critical boundary has been estimated as () for () in Ref. Ejima2005, ]. We can see that the DMRG data is in good agreement with the fitted line for all the parameter sets. From the obtained results of , we find that the logarithmic correction appears for and . Especially at , the length scale is extremely large ; it allows us to crossly notice that this point is very close to the boundary of the CO phase. Meanwhile, the logarithmic correction is hardly present for and . As a result, we confirm that the logarithmic correction is present also for and its length scale grows rapidly near the CO phase boundary.

Finally, we discuss the correlation amplitudes, and , of Eqs. (4) and (9). Figure 3 shows the DMRG results of the amplitude as a function of for several values of . In the limit, we can see an excellent agreement between the DMRG and exact results. We also find a very sharp increase near . For , the behavior of seems to be quite similar to that for ; while, the amplitude is rapidly decreased near , e.g., at . Thus, the oscillating term would be negligible in the vicinity of the CO phase. When , the amplitude decreases with increasing in reflecting that the fluctuation is not enhanced by . We note that the two amplitudes and are rather small with the same order of magnitude for small and larger values.

## Iv Summary

We study the density-density correlation function in the TLL state for the 1D extended Hubbard model at quarter filling. Based on the bosonization and RG techniques, we obtain the generalized analytical form of the correlation function which exhibits anomalous power-law behavior with logarithmic corrections near the phase boundary to the CO insulating state. Using the DMRG method, we confirm the appearance of the logarithmic corrections not only in the limit but also for finite . Moreover, we find that the length scale of the corrections grows rapidly near the CO phase boundary.

###### Acknowledgements.
The authors thank A. Furusaki and E. Orignac for valuable discussions.

## Appendix A Derivation of the analytical form of the correlation function

In this appendix, we derive the generalized form of the correlation function [Eqs. (4) and (9)] based on the RG approach. In our derivation, we follow the formalism of the RG method developed in Ref. Giamarchi1989, .

The RG equations for the TLL parameter and the 1/4-filled umklapp scattering are given by Giamarchi_book (); Tsuchiizu2001 ()

 ddlKρ(l) = −2G2u(l)K2ρ(l), (13a) ddlGu(l) = [2−8Kρ(l)]Gu(l), (13b)

where the initial values are estimated based on the perturbative treatment in Ref. Tsuchiizu2001, . The TLL parameter in the low-energy effective theory can be evaluated from the fixed point value of , i.e., . The correlation functions for the and oscillation parts, defined as and , respectively, are given in the RG scheme by Giamarchi_book (); Giamarchi1989 ()

 C2kF(x) = exp[−∫ln(x/α0)0dlKρ(l)], (14a) C4kF(x) = exp[−∫ln(x/α0)0dl(4Kρ(l)−2Gu(l))],

where is the short-distance cutoff. The couplings and are determined by solving Eq. (13).

From Eq. (13), we find that the fixed point values are given by on the phase boundary between the TLL and CO states. Near this phase boundary, the TLL parameter can be expanded as and we can treat perturbatively. Up to the second order in and , the RG equations (13) are rewritten as

 ddlGρ(l)=−2G2u(l),ddlGu(l)=−2Gρ(l)Gu(l). (15)

In the case of , the umklapp scattering flows to zero, i.e., is irrelevant, and has a finite fixed point . Thus the TLL parameter in the low-energy limit is given by . The explicit solutions of Eq. (15) are given by

 Gρ(l) = θ2coth[θl+tanh−1(θ/2Gρ(0))], (16a) Gu(l) = θ2cosech[θl+tanh−1(θ/2Gρ(0))], (16b)

where is a scaling invariant quantity. Near the phase boundary, i.e., for small , the umklapp scattering approaches to zero very slowly as increasing . By substituting Eq. (16) into Eq. (14), we obtain the analytical form of the correlation functions:

 C2kF(x) = (α0x)1/4+θ/8(1−d2θ1−(dα0/x)2θ)1/8, (17) C4kF(x) = (α0x)1+θ/2(1+dθ1+(dα0/x)θ)3/2 (18) ×(1−(dα0/x)θ1−dθ)1/2,

where is the nonuniversal quantity depending on the initial values of RG equations, defined by . In terms of , the parameter is given by . By defining we can derive Eqs. (4) and (9).

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