Favorable phase transitions induced by spinful electron-electron interactions in two-dimensional semimetal with a quadratic band crossing point

The study of semimetal materials is one of the hottest research fields in contemporary condensed matter physics Lee2005Nature ; Neto2009RMP ; Fu2007PRL ; Roy2009PRB ; Moore2010Nature ; Hasan2010RMP ; Qi2011RMP ; Sheng2012book ; Bernevig2013book ; Herbut2018Science . Last two decades have witnessed a phenomenally rapid development on these materials Lee2005Nature ; Neto2009RMP ; Fu2007PRL ; Roy2009PRB ; Moore2010Nature ; Hasan2010RMP ; Qi2011RMP ; Sheng2012book ; Bernevig2013book ; Herbut2018Science ; Armitage2018RMP ; Roy2018RRX , which typically include the Dirac semimetals Wang2012PRB ; Young2012PRL ; Steinberg2014PRL ; Liu2014NM ; Liu2014Science ; Xiong2015Science and Weyl semimetals Neto2009RMP ; Burkov2011PRL ; Yang2011PRB ; Wan2011PRB ; Huang2015PRX ; Xu2015Science ; Xu2015NP ; Lv2015NP ; Weng2015PRX . Such materials are equipped with well-known discrete Dirac points, around which gapless quasiparticles are excited with linear energy dispersions along two or three directions Lee2005Nature ; Neto2009RMP ; Fu2007PRL ; Roy2009PRB ; Moore2010Nature ; Hasan2010RMP ; Qi2011RMP ; Sheng2012book ; Bernevig2013book ; Korshunov2014PRB ; Hung2016PRB ; Nandkishore2013PRB ; Potirniche2014PRB ; Nandkishore2017PRB ; Sarma2016PRB ; Herbut2018Science . Recently, there has been a gradual shift of interest from linear-dispersion toward quadratic-dispersion semimetal-like materials Chong2008PRB ; Fradkin2008PRB ; Fradkin2009PRL ; Cvetkovic2012PRB ; Vafek2014PRB ; Herbut2012PRB ; Mandal2019CMP ; Zhu2016PRL ; Vafek2010PRB ; Yang2010PRB ; Wang2017PRB ; Wang2020-arxiv ; Roy2020-arxiv ; Janssen2020PRB ; Shah2011.00249 ; Nandkishore2017PRB ; Luttinger1956PR ; Murakami2004PRB ; Janssen2015PRB ; Boettcher2016PRB ; Janssen2017PRB ; Boettcher2017PRB ; Mandal2018PRB ; Lin2018PRB ; Savary2014PRX ; Savary2017PRB ; Vojta1810.07695 ; Lai2014arXiv ; Goswami2017PRB ; Szabo2018arXiv ; Foster2019PRB ; Wang1911.09654 ; Wang2303.10163 . In particular, significant attention has been focused on the two-dimensional (2D) electronic system with the upper and lower bands parabolically touching at certain quadratic band crossing point (QBCP) Chong2008PRB ; Fradkin2008PRB ; Fradkin2009PRL ; Cvetkovic2012PRB ; Vafek2014PRB ; Herbut2012PRB ; Mandal2019CMP ; Zhu2016PRL ; Vafek2010PRB ; Yang2010PRB ; Wang2017PRB ; Wang2020-arxiv ; Roy2020-arxiv ; Janssen2020PRB ; Shah2011.00249 . These QBCPs can be established by distinct kinds of models consisting of the kagomé lattice Huse2003PRB ; Fradkin2009PRL ; Janssen2020PRB , checkerboard lattice Fradkin2008PRB ; Vafek2014PRB , collinear spin density wave state Chern2012PRL , and Lieb lattice Tsai2015NJP . Besides, recent studies using the large-scale density-matrix renormalization group have demonstrated a series of essential properties of the QBCP systems Schollwock2005RMP ; Stoudenmire2012ARCMP ; Hubig2015PRB ; Sarma2016PRB ; He2017PRX ; Chatterjee2022PRB ; Nishimoto2010PRL ; Zeng2018npj ; Zhu2016PRL .

In marked contrast to their 2D Dirac/Weyl counterparts, in which the density of state (DOS) vanishes at Dirac points, the 2D QBCP materials possess a finite DOS exactly at their reduced Fermi surfaces (i.e., QBCP) Fradkin2009PRL ; Vafek2014PRB . This unique feature together with the gapless quasiparticles (QPs) from discrete QBCPs plays an essential role in activating a plethora of critical behavior in the low-energy regime Fradkin2009PRL ; Vafek2010PRB ; Yang2010PRB ; Vafek2014PRB ; Venderbos2016PRB ; Wu2016PRL ; Zhu2016PRL ; Wang2017PRB . It is of remarkable significance to highlight that the 2D QBCP systems are unstable under the electron-electron interactions, giving rise to the possibility of weak-coupling interaction-driven phase transitions Fradkin2009PRL ; Venderbos2016PRB ; Wu2016PRL ; Vafek2014PRB ; Vafek2010PRB ; Yang2010PRB ; Wang2020-arxiv . As delivered recently Fradkin2009PRL ; Vafek2014PRB ; Vafek2010PRB ; Yang2010PRB , one can expect the development of the quantum anomalous Hall (QAH) with breaking time-reversal symmetry and quantum spin Hall (QSH) protected by time-reversal symmetry states in the presence of electron-electron repulsions in the checkerboard lattice Fradkin2009PRL ; Vafek2014PRB or two-valley bilayer graphene with QBCPs Vafek2010PRB ; Yang2010PRB .

However, the spinful effects can play a critical role in modifying the low-energy behavior as well. The authors in Refs. Vafek2010PRB ; Cvetkovic2012PRB carefully investigated the spinful effects on the honeycomb lattice and showed several interesting results. In addition, an investigation of such effects on the instability was examined for a checkerboard lattice Fradkin2009PRL ; Vafek2014PRB , which principally considered the contributions from spin up and down are equivalent and hence employed the two-component spinor to describe the low-energy excitations. Motivated by these works and given the important role of spinful effects, we will explicitly take into account the spinful ingredients by adopting a four-component spinor to establish the low-energy theory as presented in Sec. II. This expansion can involve more kinds of electron-electron interactions and enable us to capture all the potential instabilities induced by the spinful degrees of freedom. In consequence, this implies that it is inadequate to capture all the potential instabilities generated by the electron-electron interactions without considering the spinful effects. Therefore, it is possible that other phase transitions from 2D QBCP semimetals to certain fascinating states, aside from QAH and QSH states, may occur once the spinful contributions are taken into account. Unambiguously clarifying this issue would be particularly helpful to improve our understandings on the low-energy properties of 2D QBCP and analogous materials.

To this goal, we explicitly consider all the marginal spinful electron-electron interactions in this work. This involves sixteen different types, which are distinguished by the coupling vertexes (matrices) shown in Sec. II, compared to only four spinless sorts of interactions in earlier studies Vafek2014PRB ; Wang2017PRB . In order to unbiasedly treat all these kinds of interactions and their intimate interplay, it is suitable to adopt the momentum-shell renormalization-group (RG) approach Shankar1994RMP ; Wilson1975RMP ; Polchinski9210046 , which is a powerful tool to unravel the energy-dependent hierarchical properties in the presence of various types of physical ingredients. Performing the RG analysis yields a set of coupled energy-dependent evolutions of all fermion-fermion interaction parameters, from which the several interesting results are obtained in the low-energy regime.

At first, we realize the electron-electron interactions are closely coupled to exhibit various energy-dependent tendencies, which are broken down into three categories including the Limit case, Special case, and General case as designated in Sec. IV. With variations of the initial values of interaction parameters and sign of structure parameter, the 2D QBCP systems flow towards several distinct sorts of fixed points in the low-energy regime. In the Limit and Special cases, the system can be driven to the fixed points $\mathrm{FP}_{1}^{+}$/$\mathrm{FP}_{2}^{-}$ and $\mathrm{FP}_{1}^{\pm}$/ $\mathrm{FP}_{2}^{\pm}$/$\mathrm{FP}_{3}^{\pm}$, respectively. The General case, in addition to $\mathrm{FP}_{1}^{\pm}$ and $\mathrm{FP}_{3}^{\pm}$, also harbors the $\mathrm{FP}_{41,42,43}^{\pm}$ (all these fixed points will be designated and explained in Sec. IV).

Additionally, accessing the fixed points is always accompanied by certain instabilities that result in breaking some symmetries Cvetkovic2012PRB ; Vafek2014PRB ; Wang2017PRB ; Maiti2010PRB ; Altland2006Book ; Vojta2003RPP ; Halboth2000RPL ; Halboth2000RPB ; Eberlein2014PRB ; Chubukov2012ARCMP ; Nandkishore2012NP ; Chubukov2016PRX ; Roy2018RRX ; Wang2020NPB . This motives us to examine and carefully select the favorable phase transitions from the candidate states shown in Table 1 nearby these distinct kinds of fixed points. Detailed analysis reveals that the spinful fermion-fermion interactions can induce a number of leading and subleading instabilities as collected in Table 2. Notably, the QAH, QSH, and Nem.site(bond) states are dominant nearby $\mathrm{FP}_{1}^{\pm}$, $\mathrm{FP}_{2}^{\pm}$, and $\mathrm{FP}_{3}^{\pm}$, respectively. Instead, around $\mathrm{FP}_{41,42,43}^{\pm}$, QSH becomes anisotropic, with one component being the leading instability and the others being subleading. Besides, Nem.site(bond), Chiral SC-I, and NSN.site(bond) are subleading candidates for these fixed points. It is worth highlighting that the spinful fermion-fermion interactions, compared to the spinless case Vafek2014PRB ; Wang2017PRB , generate more fixed points and induce more favorable phase transitions, and henceforth play an essential role in reshaping the low-energy properties of 2D QBCP materials.

The rest of this paper is organized as follows. In Sec. II, we introduce the microscopic model and construct the effective theory and then carry out the RG analysis in Sec. III to derive the coupled RG equations of all spinful interaction parameters. In Sec. IV, we present the tendencies of interaction parameters and all potential sorts of fixed points in the interaction-parameter space that dictate the low-energy behavior of 2D QBCP materials. Sec. V is followed to address the leading and subleading instabilities around all these fixed points that are induced by the spinful fermion-fermion interactions. At last, we exhibit a brief summary of the basic results in Sec. VI.

The microscopic noninteracting model for a 2D QBCP semimetal with spin one-half electrons on a checkerboard lattice in the low-energy regime can be expressed by the following Hamiltonian Fradkin2008PRB ; Fradkin2009PRL ; Vafek2014PRB ,

where $\Lambda$ serves as the momentum cutoff that is associated with the lattice constant and the Hamiltonian density takes the form of

with $t_{I}$, $t_{x}$ and $t_{z}$ being the microscopic structure parameters. Hereby, $\Psi_{\mathbf{k}}$ characterizes the low-energy quasi-particles excitations coming from two energy bands, which is a four-component spinor and designated as $\Psi_{\mathbf{k}}^{T}\equiv(c_{A\uparrow},c_{A\downarrow},c_{B\uparrow},c_{B\downarrow})$ with $A$ and $B$ denoting two sublattices in the checkerboard lattice Fradkin2008PRB ; Fradkin2009PRL . In addition, the $4\times 4$ matrix is introduced by $\Sigma_{\mu\nu}\equiv\tau_{\mu}\otimes\sigma_{\nu}$, where $\tau_{\mu}$ and $\sigma_{\nu}$ are Pauli matrices and identity matrix, which act on the lattice space and spin space, respectively.

After diagonalizing the free Hamiltonian density (2), we are left with the parabolical energy eigenvalues Fradkin2008PRB ; Fradkin2009PRL ; Vafek2014PRB

where the related coefficients are defined as

with $\theta_{k}$ specifying the direction of momentum. There exist one upward and one downward dispersing band at $|t_{I}|<\mathrm{min}(|t_{x}|,|t_{z}|)$, which touch parabolically at $\mathbf{k}=0$ and are invariant under $C_{4v}$ point group and time-reversal symmetry Fradkin2008PRB ; Fradkin2009PRL ; Vafek2014PRB .

Without loss of generality, we will consider in the remainder a particle-hole and rotational symmetric QBCP semimetal, which requires $t_{I}=0$ and $t_{x}=t_{z}\equiv t$. To proceed, the interacting part which includes all the marginal short-range electron-electron interactions can be introduced as follows Fradkin2009PRL ; Vafek2014PRB ; Vafek2010PRB ; Yang2010PRB ; Wang2020-arxiv

where the $\lambda_{\mu\nu}$ with $\mu,\nu=0,1,2,3$, which are positive and represent the repulsive interactions between electrons, are adopted to measure the coupling strengths that are related to the interactions distinguished by the matrices $\Sigma_{\mu\nu}$. Given that the fermionic couplings are marginal at the tree level due to the unique features of the 2D QBCP semimetals and become relevant at the one-loop level, it is worth highlighting that the fermion-fermion interactions are much more important than the other interactions and play an essential role in determining the low-energy properties of 2D QBCP materials. Accordingly, we obtain our effective action by taking into account both the free part (1) and the interacting part (4) as follows

The free electron propagator can be extracted from the noninteracting terms and written as

With these in hand, we are in a suitable position to make the RG analysis.

To proceed, we within this section perform the RG analysis to construct the coupled energy-dependent flows of all spinful electron-electron couplings, which contain the low-energy behaviors of 2D QBCP materials. Following the spirit of RG framework Shankar1994RMP ; Wilson1975RMP ; Polchinski9210046 , we separate the fermionic fields into the fast and slow modes within the momentum shell $b\Lambda<k<\Lambda$ and $0<k<b\Lambda$, respectively. Hereby, we utilize $\Lambda$ to characterize the energy scale and a variable parameter $b$ with $b=e^{-l}<1$ to serve as a running energy scale Vafek2014PRB ; Altland2006Book ; Cvetkovic2012PRB ; Wang2011PRB ; Wang2013PRB ; Wang2017PRB ; Huh2008PRB ; Kim2008PRB ; Maiti2010PRB ; She2010PRB ; She2015PRB ; Cvetkovic2012PRB ; Vafek2014PRB ; Roy2016PRB . On the basis of these, the noninteracting parts of the effective field action (5) consequently can be selected as a free fixed point. Keeping such fixed point invariant under RG transformations gives rise to the RG rescaling transformations of fields and momenta as follows Vafek2014PRB ; Wang2011PRB ; Wang2013PRB ; Wang2017PRB ; Huh2008PRB ; Wang2020-arxiv ,

Here, the parameter $\eta_{f}$ is so-called anomalous dimension of fermionic spinor which is equivalent to zero owing to the marginal fermion-fermion interactions for 2D QBCP systems Vafek2014PRB .

In order to include the higher-level contributions, we endeavor to carry out the analytical calculations of one-loop electron-electron corrections to interaction parameters as depicted in Fig. 11 of Appendix A. For convenience, the cutoff $\Lambda_{0}$ which is linked to the lattice constant can be adopted to measure the momenta and energy with rescaling $k\rightarrow k/\Lambda_{0}$ and $\omega\rightarrow\omega/\Lambda_{0}$ Vafek2014PRB ; Wang2011PRB ; Huh2008PRB ; She2010PRB . Subsequently, we obtain the one-loop corrections in Appendix A after paralleling the tedious but straightforward evaluations Vafek2014PRB ; Wang2017PRB ; Wang2018JPCM ; Wang2019JPCM ; Wang2020-arxiv . At current stage, we are able to derive the coupled RG flow equations by combining the RG scalings (7)-(10) and the one-loop corrections in Appendix A Wang2011PRB ; Wang2013PRB ; Wang2017PRB ; Huh2008PRB ; Kim2008PRB ; Maiti2010PRB ; She2010PRB ; She2015PRB ; Cvetkovic2012PRB ; Vafek2014PRB ; Roy2016PRB , which take the form of

These RG equations are closely coupled and ferociously compete with each other, which give rise to the energy-dependent interaction parameters and govern the physical behavior in the low-energy regime Shankar1994RMP ; Altland2006Book . In order to unveil the underlying physical information of 2D QBCP system, we are going to investigate the potential fixed points of such interaction parameters in the following Sec. IV, and defer the study of accompanying instabilities and phase transitions induced by fermionic interactions to Sec. V, respectively.

As aforementioned, the low-energy fate of 2D QBCP system is dictated by the coupled RG equations (11)-(26), which capture the interplay among all electron-electron interactions. In this section, we examine the behavior of the interaction parameters as the energy scales decrease, aiming to reveal their tendencies and identify potential fixed points at the lowest-energy regime. After carrying out the numerical analysis of RG equations (11)-(26), we figure out that the energy-dependent interaction parameters exhibit a series of interesting evolutions and are attracted to distinct kinds of fixed points that are of close dependence upon the initial conditions. As the symmetries of the free Hamiltonian do not impose strict constraints on the independence of fermion-fermion interactions, the initial values of fermion-fermion interactions can be taken independently. To simplify our analysis, we cluster the starting conditions into three distinct cases: (i) Limit case in which all 16 interaction parameters are assigned the same value at the beginning, (ii) Special case for which only parts of interaction parameters share certain initial value, and (iii) General case where all 16 interaction parameters are independent and hence randomly take their own starting values. Hereby, it is necessary to highlight that such three distinct cases are denominated only based on the initial conditions of fermion-fermion interactions. As the parameter $t$ in our model (1) is an energy-independent constant to the one-loop level, the QBCP semimetal owns both the particle-hole and rotational symmetry for all these three cases unless certain instabilities are induced at the lowest-energy limit. In the following, we are going to consider these three cases one by one.