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 func- tion 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) 8U > 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 y 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 transi- tion line can be verified numerically by extracting, e.g., wherec ˆy (ˆc ) creates (annihilates) an electron with jσ jσ the central charge from the the correlation length (ξχ) y 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.