Critical Behavior of the Extended Hubbard Model with Bond Dimerization

Critical behavior of the extended Hubbard model with bond dimerization

Satoshi Ejimaa, Florian Langea,b, Fabian H. L. Esslerc, Holger Fehskea

a Institut f¨urPhysik, Ernst-Moritz-Arndt-Universit¨atGreifswald, 17487 Greifswald, Germany b Computational Condensed Matter Physics Laboratory, RIKEN, Wako, Saitama 351-0198, Japan c The Rudolf Peierls Centre for Theoretical Physics, Oxford University, Oxford OX1 3NP, United Kingdom

Abstract Exploiting the matrix-product-state based density-matrix renormalization group (DMRG) technique we study the one-dimensional extended (U-V) Hubbard model with explicit bond dimerization in the half-ﬁlled band sector. In particular we investigate the nature of the quantum phase transition, taking place with growing ratio V/U between the symmetry-protected-topological and charge-density-wave insulating states. The (weak-coupling) critical line of continuous Ising transitions with central charge c = 1/2 terminates at a tricritical point belonging to the universality class of the dilute Ising model with c = 7/10. We demonstrate that our DMRG data perfectly match with (tricritical) Ising exponents, e.g., for the order parameter β = 1/8 (1/24) and correlation length ν = 1 (5/9). Beyond the tricritical Ising point, in the strong-coupling regime, the quantum phase transition becomes ﬁrst order. Keywords: Extended Hubbard model, (tricritical) Ising universality class

1. Introduction methods.

Half a century has passed since it was proposed, yet At present, quantum phase transitions between topo- the Hubbard model [1] is still a key Hamiltonian for logically trivial and nontrivial states arouse great in- the investigation of strongly correlated electron sys- terest [10, 11, 12]. In this context, extensions of tems. Originally designed to describe the ferromag- the half-ﬁlled EHM also attracted attention, mainly netism of transition metals, in successive studies the with regard to the formation of symmetry-protected- Hubbard model has also been used for heavy fermions topological (SPT) states [11]. Including an alternating and high-temperature superconductors. The physics of ferromagnetic spin interaction [13] or an explicit dimer- the model is governed by the competition between the ization [14] in the EHM, the SDW and BOW phases itinerancy of the charge carriers and their local Coulomb are completely replaced by an SPT insulator, whereby a interaction. In one dimension (1D), seen from a theoret- quantum phase transition occurs between the SPT and ical point of view, the Hubbard model is a good starting the CDW, the area of which shrinks. Most interestingly, point to explore, for example, Tomonaga-Luttinger liq- the SPT-CDW continuous Ising transition with central uid behavior (including spin-charge separation). charge c = 1/2 ends at a tricritical point, belonging to While the 1D Hubbard model is exactly solvable by the universality class of the tricritical Ising model, a sec- Bethe Ansatz [2], most of its extensions are no longer ond minimal model with c = 7/10 [15, 16]. Above this integrable. This is even true if only the Coulomb in- point, the quantum phase transition becomes ﬁrst order. teraction between electrons on nearest-neighbor lattice In Ref. [14] it has been demonstrated that the transi- sites is added. The ground-state phase diagram of this tion region of the EHM with bond dimerization can be so-called extended Hubbard model (EHM) is still a described by the triple sine-Gordon model by extend-

arXiv:1706.07486v1 [cond-mat.str-el] 22 Jun 2017 hotly debated issue. At half filling, this relates in partic- ing the former bosonization analysis [17]. The predic- ular to the recently discovered bond-order-wave (BOW) tions of field theory regarding power-law (exponential) state located in between spin-density-wave (SDW) and decay of the density-density (spin-spin) and bond-order charge-density-wave (CDW) phases [3, 4]. To charac- correlation functions are shown to be in excellent ac- terize the BOW state and determine its phase boundaries cordance with the numerical data obtained by a matrix- considerable efforts were undertaken in the last few product-states (MPS) based density-matrix renormal- years, using both analytical [5, 6] and numerical [7, 8, 9] ization group (DMRG) technique [18, 19].

Preprint submitted to Physica B June 26, 2017 6 ﬁrst order χ = 400 5 100 CDW (∆c > 0, ∆s > 0) 5 U/t = 4 χ ξ 4 tricritical point critical end point 50

4 (a) V/t 3 χ = 200 0 2 continuous (c = 1) BOW (∆c > 0, ∆s > 0) 1.9 2.0 2.1 2.2 V/t SDW (∆c > 0, ∆s = 0) 1 2.1 (b) V = 2.160t (≈ Vc) 0 0 2 4 6 8 10 12 U/t χ 2 ∗ S c ' 0.996 Figure 1: DMRG ground-state phase diagram of the 1D EHM (1) at half filling [9]. The red dotted line gives the continuous SDW-BOW y = a · x + b transition. The bold (thin) blue dashed line marks the continuous 1.9 (first-order) BOW-CDW transition and the green dashed-dotted line 5.6 5.8 6 6.2 6.4 denotes the first-order SDW-CDW transition. ln ξχ

The Ising criticality of the EHM with explicit dimer- Figure 2: (a): Correlation length ξχ of the EHM as a function of V/t ization was established in early work [17] that also spec- for U/t = 4 obtained from iDMRG. The dashed line indicates the ifies the critical exponents. The critical exponents at BOW-CDW transition point. (b): von Neumann entropy S χ as a function of logarithm of ξχ at V ≈ Vc for U/t = 4. The iDMRG data the tricritical point should differ from those at the ordi- for ln ξχ > 6 (χ ≥ 1800) provide us the numerically obtained central nary Ising transition because the tricritical Ising quan- charge c∗ ' 0.996 by fitting to Eq. (2). tum phase transition belongs to a different universality class. Simulating the neutral gap and the CDW order pa- system shows fluctuating SDW order. The spin (charge) rameter by DMRG, in this paper we will determine the excitations are gapless (gapped) ∀U > 0 [2]. At fi- critical exponents at both Ising and tricritical Ising tran- nite V, for V/U . 1/2, the ground state is still a sitions. The paper is structured as follows. Section 2 SDW. When V/U becomes larger than 1/2 a 2kF-CDW introduces the model Hamiltonians under consideration is formed. As pointed out first by Nakamura [3, 4] and discusses their ground-state properties. The critical and confirmed later by various analytical and numeri- exponents will be derived in Sect. 3. Section 4 summa- cal studies [8, 9, 20, 21], the SDW and CDW phases are rizes our main results. separated by a narrow BOW phase below the critical EHM EHM end point, (Uce ,Vce ) ≈ (9.25t, 4.76t). In the BOW phase translational symmetry is spontaneously broken, 2. Model which implies that the spin gap opens passing the SDW- EHM 2.1. Extended Hubbard model BOW phase boundary at fixed U < Uce . Increasing V further, the system enters the CDW phase with finite The Hamiltonian of the EHM is defined as spin and charge gaps. The BOW-CDW Gaussian tran- X ˆ † sition line with central charge c = 1 terminates at the HEHM = −t (ˆc jσcˆ + H.c.) j+1σ EHM EHM jσ tricritical point, (Utr , Vtr ) ≈ (5.89t, 3.10t) [9]. For ! ! EHM EHM X 1 1 Utr < U < Uce , the BOW-CDW transition becomes +U nˆ j↑ − nˆ j↓ − first order, characterized by a jump in the spin gap (see, 2 2 j Fig. 3 in Ref. [9]). Figure 1 summarizes the rich physics X +V (ˆn j − 1)(ˆn j+1 − 1) , (1) of the half-filled EHM. j The criticality at the continuous BOW-CDW transition line can be verified numerically by extracting, e.g., wherec ˆ† (ˆc ) creates (annihilates) an electron with jσ jσ the central charge from the the correlation length (ξχ) † spin projection σ =↑, ↓ at Wannier site j,n ˆ jσ = cˆ jσcˆ jσ, and von Neumann entropy (S χ), where ξχ can be ob- andn ˆ j = nˆ j↑ + nˆ j↓. In the Hubbard model limit (V = 0), tained from the second largest eigenvalue of the trans- at half-filling, no long-range order exists. Instead the fer matrix for some bond dimension χ used in a infinite 2 8 Previous studies of this model have shown that the low δ/t = 0.2 first order lying excitations in the large-U limit are chargeless spin-triplet and spin-singlet excitations [24, 25, 26, 27, 6 28, 29, 30], whereby the dynamics is described by an ef- tricritical Ising fective spin-Peierls Hamiltonian. Moreover, at finite U, point (c = 7/10) the Tomonaga-Luttinger parameters have been explored 4 CDW V/t at and near commensurate fillings by DMRG [31]. Par- continuous Ising ticularly for half filling, it has been proven by pertur- (c = 1/2) PI 2 bative [32, 33] and renormalization group [6, 34, 35] approaches that the system realizes Peierls insulator +δ −δ (PI) and CDW phases in the weak-coupling regime. 0 According to weak-coupling renormalization-group re- 0 2 4 6 8 10 12 U/t sults [6], any finite bond dimerization δ will change the universality class of the continuous BOW-CDW transi- Figure 3: Ground-state phase diagram of the 1D EHM with bond tion (realized in the pure EHM) from Gaussian to Ising dimerization in the half-filled band sector [14]. The red solid line type. Thereby the PI-CDW transition in the weak-to- marks the PI-CDW phase boundaries for δ/t = 0.2. The tricritical intermediate coupling regime belongs to the universality Ising point [Utr, Vtr] separates continuous Ising and first-order phase transitions. For comparisons, the phase boundaries of the pure EHM class of the two-dimensional (2D) Ising model [6, 17]. (δ = 0) were included. Even more interesting physics appears analyzing the intermediate-to-strong-coupling regime [14] by analogy with an effective spin-1 (EHM) system with alternating DMRG (iDMRG) simulation [19, 22]. Conformal field ferromagnetic spin interaction [13]: Here the continu- theory tells us that the von Neumann entropy for a sys- ous PI-CDW Ising transition line with central charge tem between two semi-infinite chains is [23] c = 1/2 terminates at a tricritical point that belongs c to the universality class of the 2D dilute Ising model S = ln ξ + s (2) χ 6 χ 0 with c = 7/10. Above the tricritical Ising point the quantum phase transition becomes first order. Display- with a non-universal constant s . 0 ing the ground-state phase diagram, Fig. 3 summarizes Figure 2(a) shows iDMRG results of ξχ as a func- these results. A field theoretical description of the tri- tion of V/t for fixed U/t = 4. Since the system is criti- critical transition region has been performed in terms of cal in the SDW phase and at the BOW-CDW transition a triple sine-Gordon model [14], based on the bosoniza- point, we find a rapid increase of ξχ in the SDW phase tion analysis in Ref. [17], providing results for the de- and a distinct peak at the BOW-CDW critical point cay of various correlation functions, such as the density- ≈ (Vc/t 2.160) when we increase χ from 200 to 400. density, bond-order or spin-spin two-points functions. This indicates the divergence of the correlation length The predictions of field theory are in excellent agree- → ∞ → ∞ ξχ as χ . Now, plotting the von Neumann ment with iDMRG data. entropy S χ as a function of ln ξχ and fitting the graph to Eq. (2), the criticality at V = Vc can be proved, as demonstrated by Fig. 2(b). The obtained c∗ ' 0.996 for 3. Critical exponents iDMRG data with χ ≥ 1800 corroborates the Gaussian transition resulting from a bosonization analysis [5, 6]. In the following, we give further evidence for Note that for the confirmation of the SDW-BOW tran- the Ising respectively the tricritical Ising universality sition much larger bond dimensions χ are required in classes of the quantum phase transitions in the EHM order to make clear the convergence of ξχ in the BOW with bond dimerization by calculating the critical ex- phase of Fig. 2. ponents of various physical quantities. When approaching a continuous phase transition by varying a parame- 2.2. EHM with explicit bond dimerization ter (e.g., a coupling strength) g of the Hamiltonian, the correlation length diverges as Let us now add a staggered bond dimerization to the ∝ | − |−ν EHM, Hˆ = Hˆ EHM + Hˆδ, where ξ g gc . (4) X ˆ j † Here, gc denotes the (critical) value of g at the transition Hδ = −t δ(−1) (ˆc jσcˆ j+1σ + H.c.) . (3) jσ point and ν is the corresponding critical exponent. Other 3 tricritical quantity exponent Ising 2 Ising U/t = 4 magnetization β 1/8 1/24 correlation length ν 1 5/9 i| 3 CDW 2

pair correlation η 1/4 3/20 m 1 |h iDMRG 1 1/8 10−3 10−2 ∝ (V − Vc) Table 1: Critical exponents belonging to the Ising and tricritical Ising (V − Vc)/t universality classes in 2D [36, 37, 38]. The critical exponent η for the 0 pair correlation function has been confirmed in Ref. [14]. 2.49 2.50 2.51 2.52 V/t quantities such as the order parameters or energy gaps Figure 4: Absolute value of the CDW order parameter in the vicinity also show power-law behavior. In this way the system of the Ising transition at fixed U/t = 4. Symbols are iDMRG data; the β is characterized by a set of universal exponents near the dashed line displays the fitting function |hmCDWi| ∝ (V − Vtr) with critical exponent β = 1/8 (Ising universality class). Inset: Log-log continuous phase transitions. The exact values of the plot of the order parameter for V > Vtr demonstrating the power-law most common exponents for the 2D Ising and tricriti- decay with exponent β. cal Ising universality classes are listed in Table 1. The exponents satisfy the following scaling relation Extrapolating the values of the neutral gap ∆ to the ν n (η + d − 2) = β , (5) thermodynamic limit, the critical exponent ν = 1 is ver- 2 ified, as demonstrated by Fig. 5. Increasing V at fixed where d is the spatial dimension (in our case d = 2). U/t = 4, the neutral gap decreases linearly and closes at For the EHM with bond dimerization, β and ν can be the Ising transition point. If V grows further, ∆n opens extracted from the CDW order parameter and the neutral again with linear slope. This is clearly visible in the log- gap, respectively. The CDW order parameter is defined log plots representation, both for V > Vc and V < Vc; as see Fig. 5(b). 1 X m = (−1) j(ˆn − 1) . (6) CDW L j 3.2. Perturbed tricritical Ising model j As quoted above and demonstrated in Ref. [14], the The neutral gap is obtained from tricritical point in the EHM with bond dimerization belongs to the universality class of the 2D tricritical Ising ∆n(L) = E1(N) − E0(N) , (7) model with the critical exponents given in Table 1. Let where E0(N)[E1(N)] denotes the energy of the ground us emphasize that it is exceptionally challenging to ver- state [first excited state] of a system with L sites, N elec- ify the critical exponents at the tricritical Ising point nu- trons, and vanishing total spin z component. merically, not least because one first has to determine the tricritical point itself, with high precision, varying 3.1. Ising transition U and V simultaneously [14]. The exponent η characterizes the power-law decay of We now show that the critical exponents β = 1/8 and the CDW order-parameter two-point function at the crit- ν = 1 follow from (i)DMRG simulations by varying V at ical point. As shown in Ref. [14] one has fixed U and δ, just as the corresponding phase transition line was obtained in Fig. 3. Note that β = 1/8 and ν = h(−1)`(ˆn − 1)(ˆn − 1)i ∝ `−3/20 , ` 1 . (8) 1 were extracted in Ref. [17] by means of the DMRG j+` j method, varying δ for fixed U and V. This establishes that η = 3/20. In order to determine Figure 4 gives the CDW order parameter as a func- the exponents β and ν one needs to consider the off- tion of V/t, fixing U/t = 4 and δ/t = 0.2, calculated by critical regime. We therefore consider the perturbation iDMRG technique with bond dimensions χ = 800. Ob- of the tricritical Ising conformal field theory by the “en- viously, in the CDW (PI) realized for V > Vc (V < Vc), ergy operator” (x), which has conformal dimensions |m | is finite (zero). Using V /t ≈ 2.5035, the iDMRG 1 1 CDW c ∆ , ∆¯ = , [36, 37, 38] β 10 10 data are well fitted by (V − Vc) near the transition, where the critical exponent β = 1/8 can be easily read Z off from a log-log plot; see inset of Fig. 4. H = HCFT + h dx (x) . (9) 4 1 1 U/t = 4 U/t = 10.56 0.8 i| /t n 0.6 1 CDW ∆ 0.5 m

|h 0.4 (a) iDMRG ∝ (V − V )1/24 0.5 0 0.2 tr 10−4 10−3 10−2 10−1 2.4 2.5 2.6 (V − Vtr)/t 0 V/t 5.46 5.48 5.5 5.52 5.54 V/t 100 V > Vc V < Vc Figure 6: Absolute value of the CDW order parameter in the vicinity /t

n of the tricritical Ising point at fixed U/t = 10.56. Symbols are iDMRG (b) β ∆ data; the dashed line displays the fitting function |hm i| ∝ (V−V ) −1 CDW tr 10 with critical exponent β = 1/24 (tricritical Ising universality class). ∝ |V − Vc| Inset: Log-log plot of the order parameter for V > Vtr demonstrating the power-law decay with exponent β. 10−2 10−1 |V − Vc| /t gap ∆n. Increasing V(< Vtr) at fixed U/t = 10.56, ∆n is

Figure 5: (a): Neutral gap ∆n near the Ising transition at fixed U/t = 4 reduced but not linearly as in the Ising case (cf. Fig. 4), (symbols are DMRG data taken from Ref. [14]). (b): Log-log plots and closes at V ≈ V before it becomes finite again for ν tr of ∆n as a function of |V − Vc|, fitted by |V − Vc| with ν = 1 (Ising V > Vtr. Again the log-log representation can be used universality class). ν to extract the critical exponent for |V − Vtr| , ν = 5/9, for both V < Vtr and V > Vtr, in conformity with the The perturbing operator has scaling dimension d = 1/5 tricritical Ising universality class. and is therefore relevant in the renormalization group (RG) sense. It generates a spectral gap M that scales as M ∼ Ch1/(2−d) = Ch5/9, (10) 4. Summary where C is a constant. This identifies the critical exponent ν = 5/9. The magnetization operator σ(x) in the To conclude, we have investigated the criticality of tricritical Ising model has scaling dimension ∆σ, ∆¯ σ = the 1D half-filled extended Hubbard model (EHM) with explicit dimerization δ. The BOW-CDW Gaussian tran- 3 , 3 . In the perturbed theory (9) it acquires a non- 80 80 sition with central charge c = 1 of the pure EHM gives zero expectation value that scales as way to an Ising transition with c = 1/2 at any finite δ. hσ(x)i ∼ Dh∆σ/(1−∆ ) = Dh1/24 , (11) The Ising transition line terminates at a tricritical point, which belongs to the universality class of the tricritical where D is a constant. This identifies the critical expo- Ising model in two dimensions. The change of the uni- nent β = 1/24. versality class is verified numerically by (i)DMRG (see The predictions of perturbed conformal field theory also [14]). Furthermore, we demonstrate that not only for β and ν can be checked against numerical computa- the Ising but also the tricritical Ising critical exponents β tions as follows. Fixing U = 10.56t (' U ), we first tr and ν can be obtained with high accuracy by simulating give the iDMRG results for the CDW order parameter the CDW order parameter and the neutral gap. |hmCDWi| as a function of V, cf. Fig. 6. Just as in the case of the Ising universality class, |hmCDWi| is finite (zero) for V > Vtr (V < Vtr). The order parameter |hmCDWi| now vanishes much more abruptly approaching the quantum We thank M. Tsuchiizu for fruitful discussions. The phase transition point from above. Fitting the iDMRG DMRG simulations were performed using the ITensor β data for V > Vtr to (V − Vtr) with Vtr/t ≈ 5.497 and library [39]. This work was supported by Deutsche β = 1/24 works perfectly, see the log-log representa- Forschungsgemeinschaft (Germany), SFB 652, project tion. B5 (SE and HF), and by the EPSRC under grant In order to verify the field theory prediction for ν we EP/N01930X/1 (FHLE). FL thanks RIKEN for the hos- examine the L → ∞ extrapolated values of the neutral pitality sponsored by the IPA program. 5 1.5 [21] K.-M. Tam, S.-W. Tsai, and D. K. Campbell, Phys. Rev. Lett. U/t = 10.56 96, 036408 (2006). 1 [22] I. P. McCulloch, arXiv:0804.2509. [23] P. Calabrese and J. Cardy, J. Stat. Mech. (2004), P06002. /t

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