Quantum criticality of magnetic catalysis in two-dimensional correlated Dirac fermions

The enhancement of $M(B)$ by the quantum fluctuations can be even more pronounced near the quantum critical point.

Figure 3 shows the CDW order parameter at $V=V_{c}=1.30t$ (blue symbols) together with the mean field result for $V=0.78t$ (red symbols), corresponding to $g=0$. Clearly, the iDMRG result is significantly larger than the mean field result, and the enhancement is much stronger than that in the weak interaction case. There are some deviations between the results for $L_{y}=6$ and $L_{y}=10$ for small magnetic fields, $B\lesssim 0.01B_{0}$, due to a long magnetic length $l_{B}$, and the CDW order gets more strongly stabilized when the system size $L_{y}$ increases from $L_{y}=6$ to $L_{y}=10$. This should be a general tendency since the CDW phase at $B=0$ extends to a smaller interaction region when the system size increases Tada (2019). From this observation, we can discuss scaling behaviors of the CDW order parameter in the thermodynamic limit as a function of $B$ near the quantum critical point. Indeed, as shown in Fig. 3 (b), the calculated $M$ except for the smallest values of $B$ converge for different system sizes $L_{y}=6,10$, and $M(B)$ exhibits a power law behavior for $0.02B_{0}\lesssim B\lesssim 0.1B_{0}$. The finite size effects are negligible in this range of the magnetic field, and furthermore the scaling behavior would hold for smaller magnetic fields down to $B=0$ in a thermodynamic system $L_{y}\rightarrow\infty$, since $M(L_{y}=10)$ shows the scaling behavior in a wider region of $B$ than $M(L_{y}=6)$ does. If we focus on $0.02B_{0}\lesssim B\lesssim 0.1B_{0}$ in Fig. 3, we obtain the anomalous scaling behavior $M(B)\sim B^{0.355(6)}$ by power law fittings for different sets of data points. This is qualitatively different from the mean field (or equivalently large $N_{f}$ limit) result $M_{\textrm{MF}}\sim\sqrt{B}$, which eventually leads to the strong enhancement of $M(B)$ compared to $M_{\textrm{MF}}(B)$.

The calculated magnetic field dependence of the CDW order parameter near $V=V_{c}$ implies a scaling relation characteristic of the quantum criticality. Here, we propose a scaling ansatz for the leading singular part of the ground state energy density of a thermodynamically large $(2+1)$-dimensional system,

$\displaystyle\varepsilon_{\textrm{sing}}(g,h,l_{B}^{-1})=b^{-D}\varepsilon_{\textrm{sing}}(b^{y_{g}}g,b^{y_{h}}h,bl_{B}^{-1}),$

(3)

where $D=2+z=2+1=3$ with $z=1$ being the dynamical critical exponent and $h$ is the conjugate field to the CDW order parameter $M$. The exponents $y_{g,h}$ are corresponding scaling dimensions, and the scaling dimension of the magnetic length is assumed to be one as will be confirmed later. For a thermodynamic system, the magnetic length $l_{B}$ will play a role of a characteristic length scale similarly to a finite system size $L$. Then, a standard argument similar to that for a finite size system at $B=0$ applies, leading to

$\displaystyle M(g=0,l_{B}^{-1})\sim(l_{B}^{-1})^{\beta/\nu}\sim B^{\beta/2\nu},$

(4)

where $\beta$ and $\nu$ are the critical exponents at $B=0$ for the order parameter $M(g,l_{B}^{-1}=0)\sim g^{\beta}$ and the correlation length $\xi(g,l_{B}^{-1}=0)\sim g^{-\nu}$. One sees that this coincides with the familiar finite size scaling if we replace $l_{B}$ with a system size $L$ Cardy (1988). The critical exponents of the CDW quantum phase transition in $(2+1)$-dimensions are $\beta=\nu=1$ in the mean field approximation, and the resulting $M\sim B^{0.5}$ is consistent with our mean field numerical calculations  hyp . The true critical exponents for the present $(2+1)$-dimensional chiral Ising universality class with four Dirac fermion components have been obtained by the quantum Monte Carlo simulations at $B=0$, and are given by $(\beta=0.53,\nu=0.80)$ Wang et al. (2014, 2016), which was further supported by the infinite projected entangled pair state calculation Corboz et al. (2018). Other quantum Monte Carlo studies with different schemes and system sizes give $(\beta=0.63,\nu=0.78)$ Li et al. (2015a, b), $(\beta=0.47,\nu=0.74)$ Hesselmann and Wessel (2016), and $(\beta=0.67,\nu=0.88)$ Huffman and Chandrasekharan (2017). These exponents lead to $\beta/2\nu=0.33,0.40,0.32,0.38$ respectively, and the scaling behavior of $M(B)$ found in our study falls into this range and is consistent with them.

The homogeneity relation Eq. (3) and the critical exponent can be further confirmed by performing a data collapse. According to Eq. (3), the CDW order parameter for general $g$ is expected to behave as

$\displaystyle M(g,l_{B}^{-1})=l_{B}^{-\beta/\nu}\Phi(gl_{B}^{1/\nu}),$

(5)

where $\Phi(\cdot)$ is a scaling function. This is a variant of the finite size scaling similarly to Eq. (4). When performing a data collapse, we use the results for $0.02B_{0}\lesssim B\lesssim 0.1B_{0}$ so that finite size effects are negligible. As shown in Fig. 4, the calculated data well collapse into a single curve and the critical exponents are evaluated as $\beta=0.54(3),\nu=0.80(2)$ with $V_{c}=1.30(2)t$. This gives $\beta/2\nu=0.34(2)$, which is consistent with $\beta/2\nu=0.36$ obained from $M(V=V_{c},B)$ at the quantum critical point (Fig. 3). Our critical exponents are compatible with those obtained previously by the numerical calculations as mentioned above and roughly with those by the field theoretic calculations  Sorella and Tosatti (1992); Assaad and Herbut (2013); Wang et al. (2014, 2016); Li et al. (2015a, b); Hesselmann and Wessel (2016); Huffman and Chandrasekharan (2017); Hohenadler et al. (2014); Parisen Toldin et al. (2015); Otsuka et al. (2016, 2018); Zhou et al. (2018); Corboz et al. (2018); Braun (2012); Rosenstein et al. (1993); Rosa et al. (2001); Herbut (2006); Herbut et al. (2009); Janssen and Herbut (2014); Ihrig et al. (2018); Dir . Our numerical calculations for the $(2+1)$-dimensional criticality are limited to rather small magnetc lengths $l_{B}$ bounded by the system size $L_{y}$, and we expect that more accurate evaluations of the critical exponents would be possible for larger $L_{y}$ with controlled extrapolations to $\chi\rightarrow\infty$.

The successful evaluation of the critical exponents strongly verifies the scaling ansatz Eq. (3). Although the scaling ansatz may be intuitively clear and similar relations were discussed for the bosonic Ginzburg-Landau-Wilson theory in the context of the cuprate high-$T_{c}$ superconductivity  Fisher et al. (1991); Lawrie (1997); Tes˘anović (1999), its validity is a priori non-trivial and there have been no non-perturbative analyses even for the well-known bosonic criticality. This is in stark contrast to the conventional finite system size scaling at $B=0$ which has been well established for various systems Cardy (1988). The present study is a first non-perturbative analysis of the $l_{B}$-scaling relation, providing a clear insight from a statistical physics point of view for the quantum critical magnetic catalysis. Besides, the scaling ansatz could be used as a theoretical tool for investigating some critical phenomena similarly to the recently developed finite correlation length scaling in tensor network states (see also Appendix B) Corboz et al. (2018); Tada (2019); Rader and Läuchli (2018); Pollmann et al. (2009). Based on this observation, one could evaluate critical behaviors of the magnetic catalysis in other universality classes in $(2+1)$-dimensions, such as $\mathrm{SU}(2)$ and $\mathrm{U}(1)$ symmetry breaking with a general number of Dirac fermion components, by using the critical exponents obtained for the phase transitions at $B=0$  Sorella and Tosatti (1992); Assaad and Herbut (2013); Wang et al. (2014, 2016); Li et al. (2015a, b); Hesselmann and Wessel (2016); Huffman and Chandrasekharan (2017); Hohenadler et al. (2014); Parisen Toldin et al. (2015); Otsuka et al. (2016, 2018); Zhou et al. (2018); Corboz et al. (2018); Braun (2012); Rosenstein et al. (1993); Rosa et al. (2001); Herbut (2006); Herbut et al. (2009); Janssen and Herbut (2014); Ihrig et al. (2018); Dir . It would be a future problem to clarify the exact condition for the $l_{B}$-scaling to hold in general cases.

The above discussions can be summarized into a global phase diagram near the quantum critical point in the $V$-$B$ plane at zero temperature as shown in Fig. 4. Here, we mainly focus on the critical behaviors of the order parameter but not on phase boundaries. In this phase diagram, there are two length scales; one is the correlation length of the CDW order parameter $\xi$ at $B=0$, and the other is the magnetic length $l_{B}$. One can compare it with the familiar finite temperature phase diagram near a quantum critical point  Sondhi et al. (1997); Moriya and Ueda (2000); Löhneysen et al. (2007). The length scale $l_{B}$ in our case corresponds to a system size along the imaginary time, $L_{\tau}=1/T$, in a standard quantum critical system at finite temperature $T$. In a finite temperature system, anomalous finite temperature behaviors are seen when the dynamical correlation length $\xi_{\tau}\sim\xi^{z}$ becomes longer than the temporal system size, $\xi_{\tau}\gg L_{\tau}$, so that the critical singularity is cut off by $L_{\tau}$ in the imaginary time direction Sondhi et al. (1997); Moriya and Ueda (2000); Löhneysen et al. (2007). Similarly in the present system at $T=0$, physical quantities will exhibit anomalous $B$-dependence when the spatial correlation length $\xi$ is longer than the magnetic length, $\xi\gg l_{B}$, and the critical singularity is cut off by $l_{B}$ in the spatial direction. In this way, we can understand the scaling behavior $M\sim l_{B}^{-\beta/\nu}\sim B^{\beta/2\nu}$ in close analogy with the finite temperature scaling behaviors associated with a quantum critical point at $B=0$. On the other hand, the order parameter shows conventional $B$-dependence, $M(B)\sim B$ or $M(B)-M(0)\sim B^{2}$, when the system is away from the quantum critical point, $\xi\ll l_{B}$. We note that our phase diagram would be qualitatively applicable to an interacting Dirac system with a general flavor number $N_{f}$ including $N_{f}\rightarrow\infty$ with $\beta=\nu=1$ Shovkovy (2013); Miransky and Shovkovy (2015). It is also noted that the Dirac semimetal phase will be extended to a $B\neq 0$ region at finite low temperature  Shovkovy (2013); Miransky and Shovkovy (2015); Buividovich et al. (2009); D’Elia et al. (2010); Ilgenfritz et al. (2012); Bali et al. (2012); Boyda et al. (2014); DeTar et al. (2016, 2017), and the critical behaviors can be modified as will be briefly discussed later.

We would also expect that a similar phase diagram could be seen even in a system with long-range interactions such as QED-like theories in the massless limit, because it is considered that criticality of a quantum phase transition in a $(2+1)$-dimensional Dirac system driven by a short-range interaction is not affected by the long-range Coulomb interaction Hohenadler et al. (2014); Herbut et al. (2009). It is noted that, while the Coulomb interaction is (marginally) irrelevant at the transition point, it will play an important role at a weak coupling regime and an order parameter could behave as $M\sim\sqrt{B}$ even for any small coupling Shovkovy (2013); Miransky and Shovkovy (2015).

In this section, we discuss several issues in the magnetic catalysis including possible future studies.

Comparison with conventional finite size effects— In the previous section, we have discussed the effects of a finite $l_{B}$ in analogy with the temporal size $L_{\tau}$. Here, we make a comparison of the magnetic catalysis as a finite size effect in spatial directions with the conventional finite size effects. In a finite size Dirac system with an isotropic linear system size $L$ in absence of a magnetic field, an order parameter $M$ (more precisely, a long range order $M=\sqrt{\langle\hat{M}^{2}\rangle}$) is usually overestimated when compared with the thermodynamic value, and it shows smooth crossover for a wide range of interaction strength when the system size is fixed  Sorella and Tosatti (1992); Assaad and Herbut (2013); Wang et al. (2014, 2016); Li et al. (2015a, b); Hesselmann and Wessel (2016); Huffman and Chandrasekharan (2017); Hohenadler et al. (2014); Parisen Toldin et al. (2015); Otsuka et al. (2016, 2018); Zhou et al. (2018). For different system sizes, it behaves as $M\sim L^{-\beta/\nu}$ at the critical point based on the conventional finite size scaling ansatz. Similar scaling relations hold also for an infinite system within a framework of tensor network states where the system size $L$ is replaced by the correlation length due to a finite bond dimension (see also Appendix B) Corboz et al. (2018); Tada (2019). In this sense, at least formally, the enhanced $M$ by the magnetic field in the present study is analogous to the overestimated $M$ in a conventional finite size system without a magnetic field. Furthermore, these two phenomena share a physical origin in common, i.e. the dimensional reduction. As mentioned in Sec. I, a magnetic field reduces the spatial dimensionality $d\rightarrow d-2$ via the Landau quantization. Similarly, a small system size quantizes the spatial degrees of freedom and possible wavenumbers are discretized. Consequently, the density of states at low energy can become larger than that in the thermodynamic limit and correlation effects can be amplified, which would lead to enhanced/overestimated $M$. Therefore, the magnetic catalysis can be regarded as a finite size effect and is expected to be a quite universal phenomenon. However, there is a crucial difference that the finite $l_{B}$ effect can be observed in an experiment as an anomalous $B$-dependence $M(B)\sim l_{B}^{-\beta/\nu}\sim B^{\beta/2\nu}$, in contrast to the familiar finite size scaling, $M\sim L^{-\beta/\nu}$.

Ground state energy density— Although we have been focusing on the CDW order parameter, scaling behaviors will also be seen in other quantities such as the ground state energy density $\varepsilon$ itself. According to Eq. (3), $\varepsilon$ of a thermodynamically large system is expected to behave as

$\displaystyle\varepsilon(g,l_{B}^{-1})=\varepsilon(g,0)+\frac{\varepsilon_{\textrm{sing}}(gl_{B}^{1/\nu})}{l_{B}^{3}}+\cdots.$

(6)

At the quantum critical point $g=0$ (i.e. $V=V_{c}$), the prefactor in front of $l_{B}^{-3}$ might be factorized as $\varepsilon_{\textrm{sing}}(0)=C_{0}v$ with a constant $C_{0}$ and the “speed of light” $v$ characterizing the underlying field theory with the Lorentz invariance  Rader and Läuchli (2018). Away from the quantum critical point, the mean field behaviors will be qualitatively correct as we have seen in the CDW order parameter $M$ (Sec. III). Indeed, our iDMRG calculation and mean field calculation suggest for a small magnetic field $l_{B}^{-1}\rightarrow 0$, $\varepsilon_{\textrm{sing}}(gl_{B}^{1/\nu}\ll-1)\sim$ const $>0$ in the Dirac semimetal regime $g<0$ (i.e. $V<V_{c}$), while $\varepsilon_{\textrm{sing}}(gl_{B}^{1/\nu}\gg 1)\sim l_{B}^{-1}>0$ in the ordered phase $g>0$ (i.e. $V>V_{c}$), which is in agreement with the large $N_{f}$ field theory Shovkovy (2013); Miransky and Shovkovy (2015). Consequently, the orbital magnetic moment $m_{\textrm{orb}}=-\partial\varepsilon/\partial B$ will be $m_{\textrm{orb}}\sim-\sqrt{B}$ for the former (and also at the critical point), and $m_{\textrm{orb}}\sim-B$ for the latter. Details of the ground state energy density and the diamagnetic orbital magentic moment will be discussed elsewhere.

Finite temperature correction— Finally, we briefly touch on finite temperature effects around $T=0$. At finite temperature, the new length scale $L_{\tau}$ is introduced and we expect an anomalous $T/\sqrt{B}$ scaling in our system, by following a scaling hypothesis for the singular part of the free energy density, $f_{\textrm{sing}}(g,h,l_{B}^{-1},L_{\tau}^{-1})=b^{-D}f_{\textrm{sing}}(b^{y_{g}}g,b^{y_{h}}h,bl_{B}^{-1},b^{z}L_{\tau}^{-1})$ with $z=1$. For example, the CDW order parameter would have a finite temperature correction given by $M(B,T)=B^{\beta/2\nu}\Psi(T/\sqrt{B})$ at the critical point $g=0$, where $\Psi(\cdot)$ is a scaling function with the property $\Psi(x\rightarrow 0)=$ const. Since finite temperature effects are important in experiments, detailed investigations of them would be an interesting future problem.