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

We study quantum criticality of the magnetic field induced charge density wave (CDW) order in correlated spinless Dirac fermions on the $\pi$-flux square lattice at zero temperature as a prototypical example of the magnetic catalysis, by using the infinite density matrix renormalization group. It is found that the CDW order parameter $M(B)$ exhibits an anomalous magnetic field $(B)$ scaling behavior characteristic of the $(2+1)$-dimensional chiral Ising universality class near the quantum critical point, which leads to a strong enhancement of $M(B)$ compared with a mean field result. We also establish a global phase diagram in the interaction-magnetic field plane for the fermionic quantum criticality.

Correlation effects in a Dirac system become even more significant in presence of an applied magnetic field.It is known that an infinitesimally small magnetic field induces a magnetic/charge order for any non-zero interaction V , which is called the "magnetic catalysis" [21][22][23][24][25][26][27][28][29][30][31] .A uniform magnetic field B will effectively reduce spatial dimensionality d of the system via the Landau quantization, d → d − 2. Therefore, the system becomes susceptible to formation of a bound state by interactions.For example in the (2 + 1)-dimensional Gross-Neveu-Yukawa type models, it is shown that in the limit of the large number of fermion flavors N f corresponding to a mean field approximation, the order parameter behaves as Although the magnetic catalysis was first studied in high energy physics, it was also discussed in condensed matter physics, especially for graphene and related materials [29][30][31] .Recently, there are a variety of candidate Dirac materials with strong elec-tron correlations [32][33][34][35][36] which could provide a platform for an experimental realization of the magnetic catalysis, and a detailed understanding of this phenomenon is an important issue.
In this work, we study quantum criticality of the field induced charge density wave (CDW) order in spinless Dirac fermions on the two-dimensional π-flux square lattice, which is one of the simplest realizations of the magnetic catalysis.We use a non-perturbative numerical technique, infinite density matrix renormalization group (iDMRG) which can directly describe spontaneous Z 2 symmetry breaking of the CDW order [37][38][39][40][41][42] .It is found that the order parameter exhibits an anomalous critical behavior, which characterizes the fermionic criticality as clarified by a scaling argument with respect to the magnetic length.Based on these observations, we establish a global phase diagram for the ground state near the quantum critical point.

II. MODEL
We consider spinless fermions on a π-flux square lattice at half-filling under a uniform magnetic field.There are two Dirac cones in the Brillouin zone and each Dirac fermion has two (sublattice) components, which corresponds to a case where the total number of Dirac fermion components is four, similarly to the honeycomb lattice model [3][4][5][6][7][8] .The Hamiltonian is given by arXiv:2005.01990v1 [cond-mat.str-el]5 May 2020 where i, j is a pair of the nearest neibghbor sites and the energy unit is t = 1.The hopping is t ij = te iπyi exp(iA ij ) along the x-direction on the y = y i bond and t ij = t exp(iA ij ) along the y-direction.The vector potential is given in the string gauge 43 with the period L x ×L y where L x is the superlattice unit period used in the iDMRG calculations for the system size L x × L y = ∞ × L y .Typically, we use L x = 20 for L y = 6 and L x = 10 for L y = 10.A ij = 0 corresponds to the conventional π-flux square lattice without an applied field, while A ij = 0 describes an applied magnetic field for a plaquette p, B p = ij ∈p A ij .The magnetic field is spatially uniform and an integer multiple of a unit value allowed by the superlattice size, where δB = 2π/L x L y .The lattice constant a as a length unit and the electric charge e have been set as a = 1, e = 1, and the magnetic field is measured in the unit of B 0 = 2π.
The V -term is a repulsive nearest neighbor interaction leading to the CDW state and the quantum phase transition with B = 0 takes place at V = V c 1.30t according to the quantum Monte Carlo calculations for the bulk two dimensional system, where the criticality belongs to the (2 + 1)-dimensional chiral Ising universality class [3][4][5][6][7][8] .On the other hand, our cylinder system is anisotropic and the CDW quantum phase transition at B = 0 is simply (1+1)-dimensional Ising transition 44 .However, the system can be essentially two-dimensional in space under a magnetic field when the magnetic length l B = 1/ √ B becomes shorter than the system size L y .We will use this property to discuss the (2 + 1)-dimensional criticality.Note that the critical interaction strength V c 1.30t will be confirmed later within the present framework.
In the following, we focus on the CDW order parameter, where the summation runs over the superlattice unit cell.
In the iDMRG calculation, we introduce a finite bond dimension χ up to χ = 1600 and the true ground state is achieved when χ → ∞ [37][38][39][40][41][42] .As we will show, an extrapolation to χ → ∞ works well, because our system has a gap in presence of a non-zero B due to the magnetic catalysis of the broken discrete symmetry Z 2 where there is no gapless Nambu-Goldstone mode.

III. AWAY FROM QUANTUM CRITICAL POINT
Before discussing quantum criticality, we investigate the magnetic catalysis when the system is away from the critical point.Firstly, we consider a weak interaction V = 0.50t < V c = 1.30t for which the system at B = 0 is a Dirac semimetal renormalized by the interaction.As exemplified in Fig. 1, dependence of M (B) on the bond dimension χ used in the calculation is negligibly small for L y = 6, and it can be safely extrapolated to χ → ∞ even for L y = 10.Standard deviations of the extrapolations are less than 1% and within the symbols.Such an extrapolation can be done also for other values of V as mentioned before, and all results shown in this study are extrapolated ones.In Fig. 2 (a), we show the CDW order parameter M extrapolated to χ → ∞ for the system sizes L y = 6 and L y = 10 at V = 0.50t.The calculated results almost converge for L y = 6, 10 and are independent of the system size, except for B = 0 where there is a finite size effect due to l B = ∞, although there is some accidental deviation around B 0.1B 0 .Therefore, these results give the CDW order parameter essentially in the thermodynamic limit L y → ∞.In order to understand impacts of quantum fluctuations, we also performed a mean field calculation for a comparison 45 .The critical interaction within the mean field approximation is found to be V c = 0.78t and the interaction V = 0.30t corresponds to the same coupling strength in terms of the normalized interaction g = (V − V c )/V c = 0.38.The iDMRG reuslts of M (blue symbols) are larger than the corresponding mean field results (red symbols), M > M MF , which suggests that quantum fluctuations enhance the order parameter even for the present weak V .It is noted that the order parameter behaves roughly as M (B) ∼ B as seen for small B, which is consistent with the large N f field theory [21][22][23][24][25][26][27][28] .
Similarly to the weak interaction case, the CDW order parameter M calculated by iDMRG (blue symbols) is enhanced at a strong interaction V = 2.0t > V c compared to the mean field result M MF (red symbols) at the corresponding interaction V = 1.20t (or equivalently g = 1.5) as shown in Fig. 2 (b).However, this is governed by the B = 0 values and increase of M (B) by the magnetic field is roughly comparable to that of M MF (B).The result that M > M MF already at B = 0 is because they behave as M (V, B = 0) ∼ g β with β 0.5 ∼ 0.6 < 1 3-8 while M MF (V, B = 0) ∼ g βMF with β MF = 1 near the quantum critical point, and these critical behaviors essentially determine magnitudes of the CDW order parameters away from the critical points.For B = 0, the order parameter behaves roughly as M (B) − M (0) ∼ B 2 in agreement with the large N f field theory [21][22][23][24][25][26][27][28] .

IV. NEAR QUANTUM CRITICAL POINT
In this section, we discuss quantum criticality of the magnetic catalysis based on a variant of finite size scaling ansatzes.Then, we establish a global phase diagram around the quantum critical point in the interactionmagnetic field plane, in close analogy with the wellknown finite temperature phase diagram near a quantum critical point.

A. Scaling argument
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 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 44 .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 B 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 → ∞, 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 B 0.1B 0 in Fig. 3, we obtain the anomalous scaling behavior M (B) ∼ 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 MF ∼ √ B, which eventually leads to the strong enhancement of M (B) compared to M 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, 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 where β and ν are the critical exponents at B = 0 for the order parameter M (g, l −1 B = 0) ∼ g β and the correlation length ξ(g, l −1 B = 0) ∼ g −ν .One sees that this coincides with the familiar finite size scaling if we replace l B with a system size L 46 .The critical exponents of the CDW quantum phase transition in (2 + 1)-dimensions are β = ν = 1 in the mean field approximation, and the resulting M ∼ B 0.5 is consistent with our mean field numerical calculations 47 .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 (β = 0.53, ν = 0.80) 3,4 , which was further supported by the infinite projected entangled pair state calculation 13 .Other quantum Monte Carlo studies with different schemes and system sizes give (β = 0.63, ν = 0.78) 5,6 , (β = 0.47, ν = 0.74) 7 , and (β = 0.67, ν = 0.88) 8 .These exponents lead to β/2ν = 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 where Φ(•) 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 B 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 β = 0.54(3), ν = 0.80(2) with V c = 1.30(2)t.This gives β/2ν = 0.34 (2), which is consistent with β/2ν = 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 with those by the field theoretic calculations [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20] .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 χ → ∞.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 [48][49][50] , 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 46 .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 13,44,51,52 .Based on this observation, one could evaluate critical behaviors of the magnetic catalysis in other universality classes in (2 + 1)-dimensions, such as SU(2) and 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 [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20] .It would be a future problem to clarify the exact condition for the l B -scaling to hold in general cases.

B. Phase diagram
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. In this phase diagram, there are two length scales; one is the correlation length of the CDW order parameter ξ at B = 0, and the other is the magnetic length l B which corresponds to a system size along the imaginary time L τ = 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 ξ τ ∼ ξ z becomes longer than the temporal system size, ξ τ L τ , so that the critical singularity is cut off by L τ in the imaginary time direction [53][54][55] .Similarly in the present system at T = 0, physical quantities will exhibit anomalous B-dependence when the spatial correlation length ξ is longer than the magnetic length, ξ 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 ∼ B β/2ν 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 Bdependence, M (B) ∼ B or M (B)−M (0) ∼ B 2 , when the system is away from the quantum critical point, ξ l B .
Scaling behaviors will also be seen in other quantities such as the ground state energy density ε itself.According to Eq. ( 3), ε of a thermodynamically large system is expected to behave as the Lorentz invariance 51 .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 −1 B → 0, ε sing (gl 1/ν B −1) ∼ const > 0 in the Dirac semimetal regime g < 0 (i.e.V < V c ), while ε sing (gl B > 0 in the ordered phase g > 0 (i.e.V > V c ), which is in agreement with the large N f field theory 21,22 .Consequently, the orbital magnetic moment m orb = −∂ε/∂B will be m orb ∼ − √ B for the former (and also at the critical point), and m orb ∼ −B 2 for the latter.Details of the ground state energy density and the diamagnetic orbital magentic moment will be discussed elsewhere.
Finally, we briefly touch on finite temperature effects.At finite temperature, the new length scale L τ is introduced and we expect an anomalous T / √ B scaling in our system, by following a scaling hypothesis for the singu-lar part of the free energy density, f sing (g, h, l −1 B , L −1 τ ) = b −D f sing (b yg g, b y h h, bl −1 B , b z L −1 τ ) with z = 1.For example, the CDW order parameter would have a finite temperature correction given by M (B, T ) = B β/2ν Ψ(T / √ B) at the critical point g = 0, where Ψ(•) is a scaling function with the property Ψ(x → 0) = const.Since finite temperature effects are important in experiments, detailed investigations of them would be an interesting future problem.

V. SUMMARY
We have discussed quantum criticality of the magnetic catalysis in spinless fermions on the π-flux square lattice by non-perturbative calculations with iDMRG.We found the scaling behavior of the CDW order parameter M (B) characteristic of the (2 + 1)-dimensional chiral Ising universality class, and established a global phase diagram near the quantum critical point.The present study is a first non-perturbative investigation of fermionic quantum criticality under a magnetic field, and could provide a firm basis for deeper understandings of other related systems.

FIG. 1 .
FIG.1.Extrapolation of the CDW order parameter M (B) for the χ → ∞ limit at V = 0.50t.The blue (red) symbols are for Ly = 6(Ly = 10) and the curves are power law fittings.Each curve corresponds to a magnetic field in the range 0 ≤ B ≤ 0.06B0.

FIG. 2 .
FIG. 2. (a)The CDW order parameter M at a weak coupling.The blue symbols are the iDMRG results at V = 0.50t < Vc for Ly = 6 (squares) and Ly = 10 (circles), while the red symbols are the mean field results (V = 0.30t) for the same system sizes.(b) M at a strong coupling V = 2.0t > Vc calculated by iDMRG (blue) and V = 1.20t by the mean field approximation (red).The interactions for iDMRG and the mean field approximation correspond to the same value of the normalized coupling constant g.

FIG. 3 .
FIG. 3. (a) The CDW order parameter M at the quantum critical point V = Vc = 1.30t caculated by iDMRG together with the mean field result at V = 0.78t corresponding to g = 0. Definitions of the symbols are the same as in Fig. 2. (b) M in the log-log plot.The black solid line is the power law fitting M ∼ B 0.355 , while the black dashed line is the large N f result M ∼ √ B shown for the eyes.

FIG. 4 .
FIG. 4. Scaling plot of the CDW order parameter M (V, B) in terms of g = (V − Vc)/Vc and lB = 1/ √ B. The blue squares are for Ly = 6 and red circles for Ly = 10.

) 3 BFIG. 5 .
FIG. 5. Schematic phase diagram in the V -B plane at zero temperature and the B-dependence of M (V, B) for fixed V in each region.The CDW state at B = 0 is denoted as CDW0 and M0(V ) = M (V, B = 0) ∼ (V − Vc) β .The crossover boundaries (dashed lines) are characterized by lB ξ.