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the surface of a four dimensional topological insulator, the critical exponents have been numerically obtained to high accuracy [17]. For avoiding any misunderstanding, we mention that there is a second quantum phase transi- tion [19] for stronger disorder (Wl Wc than considered in Fig. 1), where the diffusive metal≫ undergoes an An- derson localization. In this work we do not address the E /t F Anderson localization transition at all and exclusively fo- cus on the SM-DM QPT restricting ourselves to disorder strengths below the threshold for the Anderson transi- tion [19]. This itinerant QPT is addressed in our work through numerically exact calculations on sufficiently large sys- tem sizes ( 106 lattice points) and approximate field ∼ FIG. 1: (Color Online) Numerically calculated finite temper- theoretic analysis, which we use to establish: ature (T ) phase diagram as a function of disorder (W ) com- 1. The numerical value of z is universal, being in- puted for a linear system size L = 110 (with t, the band hop- dependent of the disorder type and the details of ping energy, setting the energy scale). The Dirac semimetal (SM), the diffusive metal (DM) and the QCP separating them their probability distribution (box versus a Gaus- only exist at zero temperature and there is no finite temper- sian distribution) and we obtain numerically z = 1.46 0.05 (in good agreement with other numeri- ature phase transition. The squares mark the cross over out ± of the SM phase (green) into the quantum critical (QC) re- cal studies on different models [14, 15, 17]). For the gion (orange), which is anchored by the QCP at the critical correlation length exponent we cannot pinpoint a disorder value Wc/t = 2.55 ± 0.05 (star). This cross over is universal value and we find a value of ν that lies in determined by where the specific heat fails to be described 3 the range 0.9 1.5. (Our numerical technique, by CV ∼ T . The circles mark the cross over out of the DM while providing∼ a− fairly precise value of z, is rela- phase (blue) into the QC region, which is marked by where the tively imprecise in obtaining ν because of the uncer- specific heat fails to be described by CV ∼ T . The dashed tainties in the precise determination of the critical lines connecting the squares/circles are a guide to the eye. disorder strength W which turns out to be crucial The dashed line at Λ/t marks the high energy cut off above c which the low energy continuum description in terms of mass- in calculating ν, but not z.) less Dirac fermions is inapplicable. In a narrow region inside 2. The existence of a broad quantum critical fan at the quantum critical regime the specific heat manifests non- 2 finite temperatures (see Fig. 1), where the fan oc- Fermi-liquid behavior CV ∼ T which gradually crosses over to the power law behaviors of SM and DM. The dashed hor- cupies a large parameter space. izontal line EF /t is a schematic for the (generally unknown) 3. We find considerable agreement between the nu- Fermi energy associated with any residual doping that drives merically exact crossover scaling functions and the system slightly away from the precise Dirac point, indicat- ing the low-energy cut off above which the zero-doping Dirac their approximate analytical forms (Appendix A) point theory is applicable. Note that as long as EF is not for the average density of states and the specific too large, quantum criticality manifests itself in the orange heat. The approximate analytical calculations are region of the phase diagram, even though the QCP itself (the based on one loop renormalization group analysis star at Wc and T = 0) is ‘hidden’ by the residual doping. At which predicts z =3/2 and ν = 1. temperature and energy scales smaller than EF , the system behaves as a conventional metal or diffusive Fermi liquid. In 4. We also construct an order parameter field the- a nominally undoped system, the inequality Λ ≫ EF applies, ory coupled to itinerant fermions, which illustrates and quantum criticality is observable for kB T >EF . how the notion of well defined quasiparticle exci- tations can break down in the quantum critical regime. The microscopic model we solve is non- comes a diffusive metal already for infinitesimal disorder. interacting (with disorder as the tuning parameter Through a d =2+ ǫ expansion of the critical properties, for the QCP), and the sample to sample fluctu- it is found that the leading order results for the critical ations of disorder give rise to effective electronic exponents in d = 3 are z = 3/2 and ν = 1, for both po- interactions of a statistical nature. Even though tential and axial disorder, while there should be no such this does not pertain to correlated electronic sys- transition for mass disorder [8]. Recently, numerical cal- tems per se, the theoretical concepts, regarding the culations have also come to bear on the problem through itinerant quantum critical scaling properties that the study of a three dimensional disordered topological we are able to establish, should be generally appli- insulator [13, 14], a three-dimensional layered Chern in- cable to metallic quantum critical points. In par- sulator [15], a Weyl semimetal [16–18], and the phase ticular, our theoretical finding and the numerical diagram of Dirac [19] and Weyl [20] semimetals. For verification of the non-Fermi-liquid behavior and the case of a single Weyl cone which can be realized on crossover scaling functions in the quantum critical 3
regime should have generic qualitative applicabil- II. LATTICE MODEL AND CONTINUUM ity to itinerant quantum critical systems. There- LIMIT fore, this model may effectively serve as an “Ising model” for itinerant quantum criticality for future We begin by introducing the lattice model and its con- theoretical work. tinuum limit that will be the focus of this work. We consider non-interacting electrons hopping on a simple 5. Our theoretical results are directly pertinent for de- cubic lattice with periodic boundary conditions in the scribing the recently discovered gapless Dirac semi- presence of a random on site potential. The Hamiltonian conductors or semimetals [21–25], where the va- is lence and conduction bands touch linearly at iso- lated points in the Brillioun zone. In particular, we 1 H = itψr† α ψr ˆe + H.c + V (r)ψr†A ψr (1) expect that the quantum critical fan established in 2 j + j W r,j r this work will be experimentally observable in Dirac X X semimetals with a very low carrier concentration. where ψr is a four component spinor. For non- We therefore propose Na3Bi, which has a small superconducting systems with conserved electric charge Fermi energy [24], as one of the most promising T we can choose ψr = (cr,+, ,cr, , ,cr,+, ,cr, , ) which is candidate materials for realizing the predicted crit- ↑ − ↑ ↓ − ↓ composed of an electron annihilation operator cr,τ,s at ical properties. Our results can also be relevant for site r, with parity τ = , and spin-projections s = / . the three dimensional topological quantum phase We work in the Dirac± representation and therefore↑ the↓ transitions between a topological and band insula- matrices are tor in the presence of disorder [26–30]. Although our quantum critical results strictly apply only for 0 σj 0 1 1 0 αj = , γ5 = , β = , (2) the intrinsic (i.e. undoped) Dirac semimetal, the σj 0 1 0 0 1 results should remain valid even for finite doping − as long as the doping density is low enough so that where σj denotes the Pauli matrices and 1 denotes the the associated chemical potential is smaller than 2 2 identity matrix. Each site is labeled by the vector × the temperature [31]. r and the nearest neighbors are connected by the unit ˆe In our numerically calculated phase diagram vectors j , with j =1, 2, 3. The strength of hopping be- tween neighboring sites is determined by t, and the ran- (Fig. 1) we depict the SM, DM, and the critical fan r regions with green, blue, and orange colors respec- dom potential is V ( ). (We often use t = 1 in the rest of tively whereas the (bottom) dashed horizontal line this paper using the hopping as the unit of energy in the problem.) To study the universality of the transition we schematically indicates the Fermi level (or chemi- r cal potential) associated with any residual (unin- consider two types of random distributions for V ( ): (i) tentional and hence unknown) doping in the sys- a box distribution (BD), i.e. a randomly distributed vari- able between [ W/2,W/2], (ii) a Gaussian distribution tem providing the low-energy cut off for the quan- − 2 tum critical theory. The high-energy cut off is in- (GD) with zero mean and variance W . The type of dis- dicated by the top horizontal line which is associ- order is specified by the AW matrix, we consider three ated with the usual ultraviolet cut off for studying different types of disorder: (1) axial disorder A5 = γ5, Dirac physics on a lattice. We note that the invari- (2) mass disorder Am = β, and (3) potential disorder able presence of some residual doping in the sys- Ap = I4 4 (where I4 4 denotes the four by four identity matrix).× Physically,× each type of disorder can naturally tem producing the low-energy cut-off EF in Fig. 1 acts to ‘hide’ the QCP by introducing an incipient occur in experimental systems without any control over (unintentional) metallic phase. Although the QCP which one may appear or produce the dominant effect, itself is inaccessible and hidden, its effect persists and it is not unreasonable to assume that the QPT tuned in the quantum critical regime (the orange region by each type of disorder belongs to a different universal- ity class (which turns out not to be the case here as we above the EF line in Fig. 1) provided that the dop- ing is not too high. This situation is not dissimilar will see). to many examples of itinerant quantum criticality In addition to the global U(1) symmetry for total in correlated metallic systems. number conservation, the tight binding model also has a continuous U(1) chiral symmetry, as the Hamiltonian The remainder of the paper is organized as follows: In commutes with r ψr†γ5ψr, where γ5 = iα1α2α3. This section II we introduce the lattice model and its contin- Hamiltonian serves as a model for a topological Dirac uum limit. In section III we present detailed numerical semimetal withP eight Dirac cones at the high symme- results, which is followed by the construction of the or- try points of the cubic Brillouin zone, and the excita- der parameter field theory in section IV, and we conclude tions around different cones may be coupled by inter- in section V. In Appendix A, we theoretically determine valley scattering due to disorder. As a result of the the general scaling formulas used to analyze the numer- U(1) chiral symmetry we expect the axial and poten- ical data. Finally additional numerical details for our tial disorders to have identical critical properties, man- calculations are provided in Appendix B. ifesting universality. An important question is whether 4 the universality of the SM-DM QCP remains unaffected For simplicity of the analysis, the analytic calcula- when the U(1) chiral symmetry is absent. We address tions are performed with the independent Dirac cone ap- this question by studying the effects of mass disorder, proximation, while neglecting intervalley scattering (i.e. which reduces the U(1) chiral symmetry down to a Z B I ). We choose a Gaussian white noise disorder 2 λσ → λσ chiral or particle-hole symmetry, described by the rela- distribution with a variance ∆,˜ and define the dimension- d 2 2 2 tion H, r ψr†iβγ5ψr = 0. While potential disorder is less disorder strength ∆ = ∆Λ˜ − Ωd/[(2π) v ] where Λ { } d/2 only relevant for non-superconducting systems, axial and is the large momentum cut off and Ωd = 2π /Γ(d/2) mass disorderP can naturally appear for superconducting is the area of the unit sphere in d dimensions, and Γ(x) systems. Due to the discrete particle-hole symmetry, the is the gamma function. Note that for analytic calcula- mass disorder problem can appear for class AIII or DIII tions we refer to the reduced distance from the QCP as (depending on the representation of the spinor). By con- δ = ∆ ∆c /∆c, while for the numerical work we de- trast, the axial disorder displays the discrete particle hole note| the− reduced| distance as δ = W W /W . To avoid | − c| c symmetry with respect to both β, iβγ5 and any arbitrary confusion we explicitly mention our definitions when we linear combination of them. Thus, the axial disorder use δ. The one loop calculation (or to the lowest order model is shared by both AIII and DIII Altland-Zirnbauer in ǫ) for potential or axial disorder yields the same beta classes. One of our completely unanticipated interesting function [8] and the SM-DM QPT is controlled by the new findings is that the average density of states at zero QCP located at ∆c = (d 2)/2 = ǫ/2 with z =1+ ǫ/2 energy (ρ(0)) across the SM-DM QCP is independent of and ν = 1/(d 2). For d−= 3, this leads to ν = 1, and the type of disorder (potential, mass, or axial) driving z = 3/2. The− implications of one loop calculations for the transition, see Fig. 2 (a). As ρ(0) acts as the order determining quantum critical scaling properties are dis- parameter of the SM-DM transition, this implies same cussed in detail in Appendix A. By contrast, the one-loop power law dependence of ρ(0) for potential, axial, and beta function shows intravalley mass disorder to be an mass disorder. Consequently, the SM-DM QPTs driven irrelevant perturbation and does not predict any QPT. by potential, mass, and axial disorders belong to the same However, through our nonperturbative numerical calcu- universality class. We establish this universality of the lations we show that the mass disorder with intervalley disorder-tuned SM-DM transitions through nonperturba- scattering actually drives a SM-DM QPT (see Fig. 2 (a)), tive exact numerical calculations. in contrast to the one loop perturbative result with only In the absence of disorder, the model reduces to the intravalley scattering. well known Dirac Hamiltonian on a lattice, with a dis- The important question is whether the results derived persion at the one loop level for potential and axial disorders in the single-cone approximation remains valid in the pres- E (k)= t sin(k )2 . (3) ence of intervalley scattering, i.e., how robust the analyti- 0 ± j s j cally obtained values of the critical exponents and scaling X functions in the general situation are where the precise Expanding the dispersion in the low energy limit gives theoretical assumptions necessary for obtaining the ana- rise to eight Dirac cones with linearly dispersing exci- lytical results may no longer apply. To answer these ques- tations around eight high symmetry points of the cubic tions non-perturbatively, we have performed detailed nu- Brillouin zone. Considering only the long wavelength de- merical calculations on the tight binding model in Eq. (1) grees of freedom (which are the most dominant at a sec- using large system sizes, which we present in the next ond order phase transition) in the presence of disorder section. we arrive at the following continuum Hamiltonian as the effective theory III. NUMERICAL RESULTS = ψ† ( ivα ∂ I + V (x)A B ) ψ , (4) HD λ − j j ⊗ λσ W ⊗ λσ σ We are primarily interested in obtaining the disorder where Iλσ is an eight by eight identity matrix in the val- averaged DOS. As shown below the DOS at zero energy ley space, Bλγ is the intervalley mixing matrix, and now the Dirac spinor carries a valley or flavor index λ. In serves as the order parameter for the SM-DM QPT. The the absence of intervalley scattering there is an SU(8) DOS is defined as flavor symmetry. The intervalley scattering due to disor- 1 D der breaks this flavor symmetry. We mention, however, ρ(E,L)= δ(E E ), (5) D − i that for long-ranged disorder, e.g., Coulomb impurities, i=1 such a disorder-induced valley symmetry breaking may X be weak and thus operational only at very low temper- where L is the linear size of the system with a vol- atures, which appears to be the situation in (the two- ume V = L3, D = 4V is the total number of states dimensional Dirac material) graphene. In such a situa- of the model, and Ei corresponds to each eigenstate. tion, the independent Dirac cone approximation that ig- We calculate the DOS using the numerically exact ker- nores intervalley scattering effects is valid down to very nel polynomial method (KPM) (see Ref. 32 for explicit low temperatures. details), which allows us to reach sufficiently large sys- 5
0.35 (a) 0.1 (b) 0.2 0.09 0.3 0.08 0.15 2d 0.25 0.07 0.1 0.06 0.2 0.05
0.05 (E) (0)
ρ 0.15 ρ 0 0.04 0 0.5 1 1.5 2 2.5 3 0.03 0.1 0.02 axial 0.05 0.01 mass potential 0 0 2 2.2 2.4 2.6 2.8 3 3.2 3.4 -1 -0.5 0 0.5 1 W/t E/t (c) (d) 0.07 0.14 W/t=1.5 W/t=2.6 1.6 2.7 1.7 0.06 2.8 0.12 1.8 2.9 1.9 0.05 3.0 0.1 2.0 3.1 2.1 0.04 3.2 0.08 (T) 2.2 (T) 3.3
V 2.3 V 3.4 2.4 0.03 3.5 C 0.06 C 2.5 0.04 0.02
0.02 0.01
0 0 0 0.05 0.1 0.15 0.2 0 0.02 0.04 0.06 0.08 0.1 0.12 T/t T/t
FIG. 2: (Color Online) Numerically (KPM)-calculated DOS as a function of disorder strength for a linear system size L = 110. (a) DOS at zero energy ρ(0) as a function of disorder strength for each type of disorder matrix AW , which shows that axial, mass, and potential disorder yield identical results. (Inset) Corresponding calculation for axial disorder in two dimensions [i.e. setting the hopping in the z-direction to zero in Eq. (1)] clearly establishing that there is no stable SM phase and ρ(0) is finite even for an infinitesimal amount of disorder. (b) DOS for axial disorder as a function of energy passing through the QCP with the arrow indicating increasing disorder strength. Specific heat (solid lines) with axial disorder for W
tem sizes (also see Appendix B). We calculate the spe- As Chebyshev polynomials are only defined on an cific heat CV (T ) = ∂ E /∂T by numerically integrating interval [ 1, 1], we have to rescale the Hamiltonian to the density of states usingh i the following formula for non- ensure that− its eigenvalues fall within the corresponding interacting fermions range. This can be done from a simple transformation
2 H = aH′ + b where a and b are related to the maximum 2 ∞ ρ(E)E CV = β dE , (6) and minimum eigenvalues, which we estimate using the 4 cosh(βE/2)2 Lanczos method. In short, the KPM reduces the prob- Z−∞ lem of diagonalizing the Hamiltonian into calculating where β = 1/T is the inverse temperature and we are the coefficients of the expansion µ = Tr[T (H )], which working in units where k = 1. n n ′ B can be done using only matrix-vector multiplication, and the trace can be evaluated stochastically. For the A. Method sparse Hamiltonian matrix considered here, this can be done very efficiently, and therefore, KPM is capable of handling system sizes much larger than what can be Central to KPM [32], we expand the density of states done by direct diagonalization for our purpose. As is well in terms of Chebyshev polynomials T (x) and truncate n known from the analysis of Fourier series, truncating a the expansion at order n = N . (The dependence of c series expansion can lead to artificial oscillations (Gibb’s the numerical convergence on the parameter N must c oscillations) in the calculation. We filter out these be checked explicitly- see Appendix B for details.) 6
(a) 0.12 (b) 0.18 0.16 0.1 E1.05 0.14 T2.03 0.08 0.12 0.1 0.06 (T) (E) V 0.08 ρ C 0.04 0.06 0.04 0.02 0.02 0 0 0.05 0.1 0.15 0.2 0.25 0 0.05 0.1 0.15 0.2 E/t T/t
FIG. 3: (Color Online) Determination of the critical exponent z for potential disorder with the box distribution and L = 130. (a) DOS as a function of energy, and (b) the specific heat as a function of temperature, both obtained at the QCP, Wc = 2.55±0.05. spurious effects by using the Jackson Kernel [32]. In refers to the universality of the critical exponents (and Appendix B we discuss in detail the convergence of the the scaling functions in dimensionless units), but not to KPM and how to choose appropriate values for Nc. For the critical coupling strength (or the critical tempera- the results presented in Figs. 1 and 2, we have calculated ture Tc in thermodynamic phase transitions). The fact Nc = 1028 moments such that the DOS is smooth, while that the mass disorder leads to the QPT highlights the considering a lattice size L = 110 (i.e. a system size importance of intervalley scattering. At the same time V = 1103, which would be unthinkable from the exact the equal strength of microscopic critical couplings for all diagonalization perspective). As the density of states three types of disorder can not be addressed within any is self averaging (which we have explicitly checked), effective low energy theory and we have no theoretical we only have to average over different realizations of explanation for this finding, since the critical coupling the disorder to reduce the noise in the calculation and strength is not known to be a universal quantity, in con- obtain a smooth DOS. For the critical exponents, we trast to critical exponents and scaling functions, in any consider Nc = 4096 and focus on a two component quantum (or classical) critical phenomenon. More work model (i.e. replacing all the αj matrices in Eq. (1) would be necessary, which is well beyond the scope of the with Pauli matrices σj ) after isolating one of the two current work, to understand this unexpected ‘universality degenerate eigenvalues due to the axial symmetry. In in critical coupling strength’ which we have numerically this case, we consider system sizes ranging from L = 60 discovered in this problem. up to L = 130, and in order to completely eliminate By contrast, the DOS for a two dimensional Dirac any statistical fluctuations we perform 100 disorder SM becomes finite for an infinitesimally weak disorder averages. The scaling formulas used for the analysis of strength as shown in the inset of Fig. 2(a), which agrees the numerical data are described in Appendix A. with the field theoretic predictions that d = 2 is the lower critical dimension for the disorder-tuned SM-DM QPT. The behavior of the three dimensional SM is also clearly distinct from the effects of disorder in compress- B. Potential, Axial, and Mass Disorder ible normal metals, which for an infinitesimal amount of disorder converts into a DM (from a ballistic metal As shown in Fig. 2(b), the DOS inside the SM phase with perfect conductivity). We have explicitly shown indeed scales as ρ(E) E 2, making the SM an in- in Ref. 19 that the DM phase (in the present model) compressible, gapless state,∝ | which| remains stable up to is not Anderson localized, and only for a large disorder a critical strength of disorder. For all three types of dis- strength Wl/t 8.8 (for the box distribution) does the order (potential, axial, and mass), ρ(0) becomes finite DM undergo Anderson≈ localization, similar to what hap- after passing through a QCP at a finite disorder strength pens in three dimensional conventional disordered met- W 2.55t, the putative SM-DM transition driven by als [34–37]. Thus, within the present calculation we find c ≈ increasing disorder. Remarkably, the microscopic criti- the SM (for W < Wc) and the DM (for W > Wc) phases cal coupling for all three types of disorder are identical to be stable over finite ranges of disorder in the undoped within our numerical accuracy, as clearly demonstrated three dimensional Dirac system and we use the specific in Fig. 2(a). This is a rather amazing unanticipated heat to determine the cross over energy scales for each result since universality in critical phenomena usually phase as shown in Figs. 2 (c) and (d). 7
(b) (a) L=70 L=70 90 0.1 90 0.1 110 110 130 130
0.01 0.01 (0) (0) ρ ρ
0.001 0.001
0.0001 0.0001 0.01 0.1 0.01 0.1 δ δ
120 120 (c) 400 (d) 200 300 100 ) 100 ) ν 200 ν 100 S( L=60 S( 80 100 80 d-z 0 70 d-z 0 1 1.5 2 80 1 1.5 2 60 ν 60 ν 90 100 (0,L)L 40 (0,L)L 40
ρ 110 ρ 120 20 130 20 0 0 0 2 4 6 8 10 0 5 10 15 20 ν ν δL1/ δL1/
FIG. 4: (Color Online) Determination of the critical exponent ν for potential disorder with the box distribution [(a) and (c)] and the Gaussian distribution [(b) and (d)]. (a) and (b) show the numerical DOS as a function of the distance to the QCP. The dashed lines are power law fits to extract ν, the thick (thin) dashed lines are fits closer to (further from) the QCP. (c) and (d) show the numerical finite size scaling and data collapse of ρ(0, L), the labels for L are shared between (c) and (d). (Insets) The local-linearity function [33], which yields the best scaling collapse at its minimum value. Here δ = |W − Wc|/Wc is the usual dimensionless tuning parameter defining the QPT.
C. Universal scaling and critical exponents scaling functions, which are related through
νz + νz 1/ν Tδ− ∞ 2 2 (Tδ− ,L δ) = du u cosh− (u/2) From our numerical results we now show that the den- C 4 sity of states and specific heat obey universal scaling Z−∞ νz 1/ν forms in the vicinity of the QCP, which enables us to de- ( u Tδ− ,L δ). (9) ×R | | termine the critical exponents characterizing the univer- sality class (see Appendix A for the details regarding the Based on the numerical calculations presented in Fig. 2 scaling ansatz). Assuming that the QCP is non-Gaussian (a), we have established that the QCP driven by axial (i.e. non-mean field), we apply the scaling hypothesis to or mass or potential disorder between the SM and the the density of states and specific heat implying that they DM falls within the same universality class (within our must obey the scaling forms numerical accuracy). Therefore, we conclude that irre- spective of the underlying chiral symmetry being present or absent, the disorder driven SM-DM QPT of three- (d z)ν νz 1/ν ρ(E,L) = δ − ( E δ− ,L δ) (7) dimensional massless Dirac fermions exhibits a manifest R | | νd νz 1/ν universality, which is one of our main results. CV (T ) = δ (Tδ− ,L δ) (8) C We determine the location of the critical point by ex- trapolating ρ(0) to zero from the DM phase, which yields where we have introduced the reduced distance to the Wc/t =2.55 0.05 for box disorder (and for all three dif- QCP, δ W W /W , and and are two unknown ferent types of± disorder) and W /t =0.60 0.03 for Gaus- ≡ | − c| c R C c ± 8
(a) 105 (b) 104
104 103 (E) (E) 103 ρ ρ ν ν 102 102 -(d-z) -(d-z)
δ 1 δ 10 101
0 100 10 1 2 3 4 5 101 102 103 104 10 10 ν10 10 10 ν δ- z|E| δ- z|E|
(c) 1012 (d) 1011 1010 1010 109 108 108 (T) (T) 7 V V 10 106 C C 106 ν ν 4 5 -d 10 -d 10 δ δ 104 102 103 100 102 100 101 102 103 104 105 101 102 103 104 105 ν ν δ- zT δ- zT
FIG. 5: (Color Online) Scaling collapse of the DOS and the specific heat in the quantum critical fan δνz ≪ E/t,T/t ≪ Λ/t (with δ = |W − Wc|/Wc) establishing that both exhibit single parameter scaling as a function of the distance to the QCP given by Eqs. (7) and (8). The dashed line is the approximate crossover functions analytically determined from the one loop beta functions (see Appendix A); (a) and (c) are for W
S this is not a fit, and the only parameter that is adjusted e− , where the effective action is is the non-universal coefficient of the analytical scaling d ω functions. Finally, the consistency between our numeri- S = d x Ψ†[E + iη τ3 H]Ψ. (15) cal calculations and the scaling hypothesis (contained in − 2 − Z Eqs. 7) has led us to conclude that the QCP is indeed Here Ψ† = i(ψ† , ψ† ), and τ = diag(1, 1) is a Pauli non-Gaussian and the disorder-tuned SM-DM QPT in − R A 3 − Dirac systems cannot be described by a Gaussian theory. matrix that operates on the retarded and advanced la- bels. All physical quantities such as the average DOS and Motivated by the universality of the non-Gaussian the density-density correlation functions can be obtained QCP, in the following section we construct the effective from the generating functional after taking derivatives field theory of the long wavelength fluctuations, while ac- with respect to the appropriate source terms. For sim- counting for the strong coupling between the underlying plicity we again consider only a single Dirac cone and the order parameter and the itinerant fermions. case of a random scalar potential. After averaging over disorder with the replica method we arrive at
IV. ORDER PARAMETER FIELD THEORY d ω S = d x Ψ† [E + iη τ + ivα ∂ ]Ψ a − 2 3 j j a Z Within the replica field theory formulation of disor- ∆ d d xΨ† Ψ Ψ†Ψ . (16) dered systems, the sample to sample fluctuations are cap- − 2 a a b b tured in terms of a disorder induced effective statistical Z interaction between the fermions although the starting When ω = η = 0, the retarded and the advanced sectors bare Hamiltonian is noninteracting [cf. Eqs. (1) and appear in a symmetric manner, implying there is a flavor (4)]. The collective modes of the density fluctuations symmetry between these two sectors. Therefore, η acts as an infinitesimal external field for the bilinear iΨ τ Ψ, and are captured by a bosonic matrix field Q ψ ψ† and † 3 ∼ h a b i for ∆ > ∆ (ρ(0) = 0) this symmetry is spontaneously T r(Q) ρ(0) acts as the order parameter. The devia- c 6 tion of ∝T r(Q) from its vacuum expectation value consti- broken. We perform the following Hubbard-Stratonovich tutes the amplitude or the DOS fluctuations, while the transformation transverse part of the matrix field captures the slow diffu- ∆ d exp d x Ψ† Ψ Ψ†Ψ = sons or Goldstone modes. At the SM-DM QCP, both the 2 a a b b Z amplitude T r(Q) and the transverse diffuson modes are 2 d T r(Q ) gapless and remain strongly coupled with the underlying D[Q]exp d x + iΨ† Ψ Q (17), − 2∆ a b ab fermions. As we show below, the average DOS controls Z Z the residue of the diffusion pole, and it therefore vanishes where Q constitutes a matrix order parameter, and as W W +. Similarly, the quasiparticle residue of the c Q Ψ Ψ† . By computing the susceptibilities for Dirac→ excitations vanishes as W W . Consequently, ab a b c− differenth i ∼ orderingh i channels, we find that the most favor- inside the critical fan the entire notion→ of weakly coupled able choice for Q inside the metallic phase is Q = iQ τ , quasiparticle excitations becomes invalid and an emer- 0 3 for which η acts as the external field. For 2
δνηψ [Ψa,Qab] = D[Ψa]+ LB[Qab]+ LDB[Qab, Ψa],(20) G (E = 0) . (27) L L R (z 1)ν k Γ ∼ [ηΓ0 + vδ − ] d [Ψ ] = iv d x Ψ† α Ψ , (21) · LD a a · ∇ a Z Therefore the quasiparticle residue and the effective νη d 1 r Fermi velocity are respectively given by δ ψ and B[Qab] = d x Tr( Q† Q)+ Tr(Q†Q) ν(z 1) L 2 ∇ ∇ 2 vδ − . At the one loop order ηψ = (z 1) = ǫ/2 and Z ν = 1/ǫ. Consequently, both the quasiparticle− residue 2 2 1/2 +u1Tr[(Q†Q) ]+ u2[Tr(Q†Q)] , (22) and the Fermi velocity behave as δ . This is an arti- fact of the one loop analysis and at higher loop orders d 2 [Q , Ψ ] = iλ d x Ψ† Ψ Q , (23) (beginning at ǫ ) they will follow different power laws. LDB ab a a b ab Z Hence, we conclude that the quasiparticle residue and the effective Fermi velocity of the Dirac fermion vanish, where the summation over repeated replica indices is im- as we approach the QCP from the semimetal side. plied. We are only considering Q in the regular and ab Consequently, the residue of the two different types of the axial density channels. When an external frequency quasiparticles vanish while approaching the QCP from is considered, it couples to T r[Qτ ] in the regular density 3 either side. Thus, the critical region of this system can channel as an external field. not have any simple quasiparticle description, qualita- At the critical point r = 0, both regular and axial tively similar to what is envisaged for QC regions in cor- density fluctuations become gapless. If we scale x xel, related materials. Thus, the orange region in our nu- Ψ Ψe(1 d)l/2, Q Qe(2 d)l/2, we find λ λe(4→d)l/2, − − − merically obtained quantum phase diagram of Fig. 1 is u → u e(4 d)l. Therefore,→ d = 4 serves→ as the upper j j − a finite-temperature ‘non-Fermi liquid’ quantum critical critical→ dimension, and the interaction effects can be ad- crossover regime arising from the critical fluctuations of dressed through a d =4 ǫ expansion, which leads to a the non-Gaussian QCP underlying the SM-DM QPT in critical point at λ = O(ǫ−), u = O(ǫ). c j the three dimensional Dirac materials. In the DM phase the density-density correlation func- To summarize this section, the finite expectation value tion (involving both retarded and advanced sectors) dis- of the DOS [as shown numerically in Fig. 2(a)] on the plays a diffusion pole corresponding to the gapless Gold- metallic side and its critical fluctuations in the vicinity stone modes or the diffusons, which behaves as of the QCP can be described by an order parameter in the 2 zν form of the lagrangian B for the matrix field Q. While Q0 δ L Π (ω, k) 2 (24) d Q 2 approaching the QCP from the SM phase, the quasiparti- νηQ 0 2 iω Dk ∼ iQ ω δ 2 k ∼ 0 − − Λ − cle residue of the underlying Dirac fermions (as described by D) vanishes continuously as δ 0. Thus, the gapless where ηQ is the anomalous dimension of the gapless col- fermionicL and bosonic degrees of freedom→ must be treated lective mode. If the order parameter inside the DM phase on an equal footing and this has led us to construct the varies as Q δβ, the hyperscaling relation leads to 0 ∼ effective action in Eqs. (20)-(23), which governs all the ν long wavelength degrees of freedom. This effective the- β = (d 2+ η ) = zν, (25) − Q 2 ory retains both longitudinal and transverse fluctuations 12 of the Q matrix, as required by the vanishing of the av- phase transitions [40–42]. erage DOS at the QCP. This should be contrasted with the familiar nonlinear sigma model description of the An- derson localization at stronger disorder [19], where the V. DISCUSSION AND CONCLUSION average DOS remains noncritical across the transition. The nonlinear sigma model is obtained by integrating The experimentally observed non-Fermi liquid scaling out the gapped ballistic fermions and the gapped longi- behavior of many correlated metals [43–46] over a sig- tudinal fluctuations of the Q matrix inside the metallic 1 nificant range of temperatures is usually reconciled with phase (below the energy scale of τ − Q ). ∼ 0 the existence of a large quantum critical fan driven by In our earlier work [19] on the phase diagram of a dis- the putative QCP, whose intrinsic properties are often ordered DSM, we have noticed a significant difference not well-understood. It is generally believed that the between the critical exponents for the typical DOS at generic behavior of the ‘strange metal’ within the finite- the SM-DM QCP and the QCP describing an Anderson temperature critical fan regime is non-Fermi liquid like localization transition (at stronger disorder). We believe because of strong quantum fluctuations arising from the that the differences between the multifractal properties QCP (which may often be ‘hidden’ or ‘inaccessible’ ex- at these two QCPs are inherited from (i) the presence or perimentally), but no general proof exists. For describ- absence of itinerant, ballistic fermions, and (ii) the pres- ing such a metallic or itinerant QCP, it is necessary to ence or absence of gapless longitudinal mode of the Q address both the gapless fermionic degrees of freedom matrix. A detailed analysis of different correlation func- and the bosonic order parameter fluctuations on an equal tions and the multifractal properties associated with the footing, which is a highly challenging task for analyti- field theory in Eqs. (20)-(23) is beyond the scope of this cal calculations, particularly for strongly correlated sys- work and remains an open problem for the future. The tems, where even the minimal model is intractable with Dirac SM and the DM phases preserve time reversal and or without the QCP. In this work even though we have inversion symmetries, and do not display any topologi- focused on non-interacting massless Dirac fermions we cal transport properties such as anomalous Hall effect. have been able to make various connections to the general For this reason, there is no Chern Simons term in the notion of scaling phenomena of itinerant QCPs. First, effective field theory. By contrast, a time reversal sym- we find the appearance of a broad quantum critical fan metry breaking Weyl semimetal phase and the resulting at finite temperatures. Second, we have succeeded in diffusive metal possess an anomalous Hall effect. In the constructing an effective field theory of bosonic fluctua- effective field theory this arises due to a topological Chern tions (captured here by the Q matrix for diffusive modes) Simons term [39]. that are strongly coupled to itinerant electron degrees of One direct implication of our effective field theory is freedom (i.e. massless Dirac fermions). Third, we have the existence of an upper critical dimension of du = 4 shown how the general notion of quasiparticle excitations for the disorder-tuned SM-DM transition in Dirac-Weyl breaks down (via their residue) in the quantum critical systems (in addition to the known lower critical dimen- fan region even for our noninteracting model, explicitly sionality of dl = 2) . For dimensions d du, the correla- demonstrating that a QCP by itself, independent of the ≥ tion length exponent ν should be given by the mean-field underlying model being interacting or not, leads to a value ν = 1/2 and hyperscaling relations will be vio- generic non-Fermi liquid behavior. However, as we are lated. It will be interesting to check this observation in dealing with non-interacting fermions, it is important to future numerical studies on higher dimensional models contrast the version of itinerant quantum criticality we of a Dirac SM. Finally, the existence of an upper critical obtain here with the form that is applicable to strongly dimension for the SM to DM QCP will further distin- correlated electronic systems. We are dealing with free guish this transition from Anderson localization, which fermions and the interaction is statistical and only due does not possess an upper critical dimension. to disorder (only elastic scattering). Consequently, the We mention here that we do not, however, have an ex- effective field theory will not be a dynamical one (with planation for our numerical finding of the exact z( 1.5) inelastic scattering) as we have found. This is in sharp being equal within the numerical accuracy to the≈ one- contrast to that of the strongly correlated systems where loop theoretical value (z = d/2). Whether this is a mere the dynamics of the bosonic field is also important and coincidence or a deeper truth (i.e. somehow all higher- must be incorporated in the theory on an equal footing loop corrections cancel out) is unknown at this stage. Of with the fermionic field. course, our numerical error in estimating z, while being Through a precise numerically exact calculation using small, is not zero, and thus small corrections in the ex- the KPM technique and an approximate field theoretical act theoretical value of z (well less than 10%) away from analysis, we have established that the three-dimensional z =1.5 cannot be ruled out without more numerical work Dirac SM phase undergoes a disorder-tuned QPT into a with much larger systems. There are well-known exam- DM state for different types of disorder (potential, axial, ples in the literature for numerical and one-loop theoret- and mass), which belong to the same universality class ical z values being very close to each other purely for- (and surprisingly, even with the identical critical coupling tuitously in completely different contexts of dynamical for different disorder types) of an itinerant QCP. We find 13 the critical exponents for box and Gaussian distributions which necessitates a detailed focused numerical study on agree to within numerical accuracy (with z =1.46 0.05 its own. Although we cannot rule out the possibility of and ν [0.9, 1.5]), which points to the universal nature± rare regions based on our study, we can assert that the of the∈ QCP. For this model of an itinerant QCP, the disorder-tuned SM-DM QPT is well-captured by our the- universal scaling functions obtained from numerical and ory and numerics once any contribution from rare fluctu- analytical methods show a remarkable agreement, and ations are subtracted out. It is important in this context thus can serve as a conceptual framework for address- to emphasize that the real Dirac systems are likely to ing other itinerant critical phenomena in diverse corre- have random impurity-induced ‘puddles’ or spatial den- lated metallic systems. Experimentally, a dilute three- sity inhomogeneities (i.e. random spatially nonuniform dimensional Dirac material such as Na3Bi with a small doping) around zero energy (i.e. the Dirac point) any value of the Fermi energy EF Λ associated with resid- way, as is ubiquitous in graphene [48], leading to a lo- ual doping, can be a promising≪ material for verifying our cally fluctuating chemical potential masking the QCP in theoretical results. In a real system, the residual uninten- a way similar to what rare regions would do. Any exper- tional doping always produces a finite Fermi energy shift- imental study of the QPT itself must find a reasonable ing the system from the zero chemical potential Dirac way of subtracting out these additional effects (from rare point, but as long as this doping-induced Fermi energy is regions and/or puddles) in order to capture the underly- lower than the temperature, one expects intrinsic Dirac ing critical behavior. point behavior to prevail, leading to the manifestation of the QPT studied in this work. One experiment could be the measurement of the specific heat in the quantum Acknowledgements critical regime which, according to our theory, should manifest a non-Fermi liquid like quadratic temperature We have benefited from insightful discussions with dependence associated with the critical quantum fluctu- Sudip Chakravarty, Qimiao Si, Steven Kivelson, Subir ations. One key aspect of this QCP is its fundamental Sachdev, David Huse, A. Peter Young, Matthew Foster, non-Gaussian nature. Our KPM-based estimate of the Bj¨orn Sbierski, and Piet Brouwer. This work is supported correlation length exponent ν, however, has rather large by JQI-NSF-PFC, LPS-MPO-CMTC, and Microsoft Q. error bars, and much more work would be necessary (well The authors acknowledge the University of Maryland su- beyond the scope of the current work) to obtain a pre- percomputing resources (http://www.it.umd.edu/hpcc) cisely numerically accurate value of ν in the SM-DM QPT made available in conducting the research reported in in three dimensions. this paper. Finally, we comment on the possible role that ‘Grif- fiths physics’ (associated with rare fluctuations) might Appendix A: Scaling formulas play in the SM-DM QPT studied in the current work. The physics of rare regions (or the so-called “Griffiths In the vicinity of the QCP, we have two divergent physics”) is not captured by the long wavelength field ν scales, namely the spatial correlation length ξl δ− theory developed in the present manuscript. Recently it z νz∝ and the temporal correlation length ξ ξ δ− . The has been suggested that rare regions affecting the SM- t ∼ l ∝ DM QCP can induce a finite DOS at the Dirac point universal scaling functions are determined by these two divergent scales. Quite generally for an observable we for an infinitesimally weak disorder [47], thus converting O the SM phase into effectively a DM phase. Compared to have the scaling formula, the box distribution, the unbounded tails of the Gaussian ∆O 1 z z (L,E,T ) ξ F (Lξ− , Eξl , T ξl ), (A1) disorder distribution are expected to enhance these rare O ∼ l O l fluctuations, but we get the same universal QCP prop- where L, E and T respectively denote the system size, erties for both box and Gaussian distributions without the energy and the temperature, and ∆ is the scaling di- O seeing any obvious effects of rare regions. Within our mension of . For example, the scaling dimensions ∆ O O numerical calculations we have not found any evidence for the average density of states, the free energy den- of rare region effects, as the results for both distribu- sity, the specific heat and the longitudinal conductivity tions agree with the field theoretical predictions for the are respectively given by (d z), (d + z), d, and − − − − existence of the transition (e.g. considering the problem (d 2). The leading scaling behavior is determined by − − 1 1 in two versus three dimensions). However, the detec- max L− ,E,T,ξ− . Since there are two fixed points { l } tion of rare region effects can be quite subtle (due to a respectively with z = 1 and z = 3/2 (at one loop), the possibly exponentially small DOS contribution from the crossover scaling functions are quite nontrivial. rare regions) and therefore the possible existence of rare For example, first consider the average density of states regions still cannot be ruled out completely (e.g. the re- (d z) 1 z ρ(L, E) ξ− − F (Lξ− , Eξ ). (A2) gions are rarer than our numerical system size or the rare ∼ l ρ l l regions being independent of the QCP itself or the expo- At zero energy this becomes nentially small DOS introduced by the rare regions are ν(d z) ν (d z) 1/ν simply too small to show up in our KPM simulations), ρ(L, 0) δ − f(Lδ− ) L− − g(L δ). (A3) ∼ ∼ 14
ν When L ξ δ− , we are in the quantum critical where ∆ = ∆(l = 0) is the bare value of the coupling ≪ l ∼ 0 regime and constant. On the SM side ∆0 < ∆c, the renormalized disorder coupling monotonically decreases and satisfies 1 the asymptotic form ρ(L, 0) d z , (A4) ∼ L − ∆0 (d 2)l ∆(l) e− − . (A14) reflects the scale invariance. By contrast, inside the ≈ δ metallic phase (∆ > ∆c) In contrast, for the metallic side ∆0 > ∆c the renormal- ν(d z) ρ(L, 0) δ − . (A5) ized disorder coupling diverges at the RG scale ∼ l1 ν On the semimetal side we rather expect e = ξlΛ δ − , (A15) ∼ | | (d 1) ρ(L, 0) L− − . (A6) which defines the correlation length ξl and the scaling ∼ exponent ν =1/(d 2). By contrast, the divergent time − These finite size scaling properties constitute the basis scale has to be found from the solution of the differential for the analysis of our numerical results. In addition, equation for energy we consider a large enough system size, L (ξl,v/E). ≫ ∆(l) ∆ zν ∆ ν In this case, the scaling behavior is determined by the ǫ(l)= ǫ(0) c 0 z − interplay between v/E and ξ . The scaling function now ∆(0) ∆c ∆(l) l − behaves as (z 1)ν ∆ − ǫ(0)ezl = c . (A16) ν(d z) νz ∆ (z 1)ν ρ(E) δ − h(Eδ− ). (A7) 0 1+ ∆c 1 el/ν − ∼ ∆0 − When, E δνz, we are in the quantum critical regime h i For that we need to find the RG scale l as a function and obtain≫ the following scale invariant answer 2 of the bare energy ǫ(0) and δ by imposing the condition d/z 1 ǫ(l2) 1. This results into the following equation for l2 ρ(E) E − . (A8) ∼ | | ∼ (z 1)ν (z 1)ν E ∆ − ∆ − Inside the metallic phase (∆ > ∆c), the leading behavior ezl2 c = 1+ c 1 el2/ν . is given by vΛ ∆ ∆ − (A17) ν(d z) ρ(E) δ − . (A9) When ∆ = ∆ , l , and the infrared cutoff is solely ∼ c 1 → ∞ determined by l2 where By contrast, inside the Dirac semimetal phase we obtain l2 1/z e E − . d 1 d(z 1)ν d 1 ∼ | | ρ(E) c(δ)E − δ− − E − . (A10) ∼ ∼ | | Inside the semimetal phase ∆ 0, and again the infrared →l2 1 (z 1)ν We can gain considerable analytical insight regarding cutoff is determined by l2 with e E− δ − . Since ρ ∼ (d/z 1) the crossover functions by integrating the one-loop renor- has the scaling dimension (d z), we find ρ E − − − (d 1)∼ | | malization group equations. The one loop RG equations inside the critical fan, and ρ c(δ) E − inside the for potential and axial disorder are [8] semimetal phase, with the c(δ∼) quoted| | in the previous paragraph. d∆ l2 z(l)=1+∆, = (d 2)∆ + 2∆2. (A11) The explicit solution for e in d = 3 inside the quan- dl − − tum critical fan is obtained by finding the roots of a cubic equation. When ∆ > ∆ , we have the following cubic We simultaneously solve the beta function for the disor- 0 c equation der coupling ∆ and δ 1+ δ dǫ e3l2 + el2 =0, (A18) = z(l)∆ = (1 + ∆)ǫ, (A12) ǫ2(0) − ǫ2(0) dl which has the only one real root given by where ǫ = E/(vΛ) is the dimensionless quantity de- fined from the energy E. The dimensionless tempera- √ l2 δ 1 1 (1 + δ)3 3ǫ(0) ture t = kBT/(vΛ) also satisfies the same flow equation − e = 2 2 sinh sinh 3/2 as ǫ. From these solutions we can identify two RG flow s3ǫ (0) " 3 ( 2δ )# times associated with two divergent scales. The disorder 3ξlΛ 1 1 coupling ∆(l) is given by sinh sinh− x , for δ << 1, (A19) ≈ x 3 ∆ ∆(l)= c , (A13) with the anticipated dimensionless variable x = −3 2 ∆c (d 2)l 3√3ǫ(0)δ / 1+ 0 1 e − . When x >> 1 we have the quantum critical ∆ − 2 15
1 (z 1)ν behavior, which eventually gives away to the metallic be- When E >> L− δ − , these analytically obtained havior for x< 1 when the infrared scale is determined by approximate| | crossover scaling functions have been com- l1 e or the correlation length. When ∆0 < ∆c, the cubic pared with the numerically determined exact scaling equation becomes function in section III. In this comparison, we have only adjustedR an overall numerical prefactor for the an- 3l2 δ l2 1 δ e 2 e 2− =0. (A20) alytical formula. A similar approximate expression for − ǫ (0) − ǫ (0) is also found by substituting the explicit expressions of Cρ (1 δ)3√3ǫ(0) (in terms of g(x)) in Eq. 6. − When 2δ3/2 > 1 the only real root is given by
√ 0.1 l2 δ 1 1 (1 δ)3 3ǫ(0) (a) − e = 2 2 cosh cosh − 3/2 s3ǫ (0) "3 ( 2δ )#
3ξlΛ 1 1 0.01 cosh cosh− x , for δ << 1. (A21) ≈ x 3 (1 δ)3√3ǫ(0) − 3 2 0.001 For 2δ / < 1, there are three real roots, and only (0) one of them is positive. This root is given by ρ
√ l2 δ 1 1 (1 δ)3 3ǫ(0) 0.0001 e = 2 2 cos cos− − 3/2 s3ǫ (0) "3 ( 2δ )#
3ξlΛ 1 1 cos cos− x , for δ << 1, (A22) 1e-05 ≈ x 3 0.005 0.01 0.02 0.03 where the inverse trigonometric function is restricted 1/L to the first quadrant. For small δ, δel2 is again a (b) 1 3/2 scaling function of ǫ(0)δ− . For energies satisfying 3/2 ǫ(0)δ− >> 1 or x >> 1 inside the quantum critical fan cosh and sinh terms determine the z = 3/2 quan- 0.1 tum critical behavior, while inside the SM phase the cos term leads to the z = 1 critical behavior. Therefore, the 0.01 crossover from the z = 3/2 to z = 1 scaling is appropri- (0) ρ ately captured by how the cosh term transforms into a cos term. 0.001 In the thermodynamic limit the total number of states per unit volume can be estimated to be 0.0001 N(ǫ) d dl2 d d 100 1000 10000 100000 Λ e− = ξ− f − (x). (A23) L3 ∼ l Nc Since the average DOS can be obtained by differentiating d N(ǫ)L− with respect to ǫ, we can write FIG. 6: (a) (Color Online) Finite size scaling of ρ(0, L, Nc) Λ2δ3/2 on a log-log scale, for Nc = 1028 in the clean limit (W = 0) ρ(ǫ) g(x), (A24) yields a power law decay in system size 1/L2.85 consistent ∼ √ 3 2 3v with 1/L . (b) Scaling of ρ(0, L, Nc) on a log-log scale with where the crossover scaling function g(x) is determined Nc for L = 60 yielding a linear in Nc scaling. Here Nc is by the number of terms kept in the Chebyshev expansion in the KPM numerics. 1 1 x2 x coth sinh− x g(x) = 1 3 , 3 1 1 − √ 2 sinh 3 sinh− x " 3 1+ x # for ∆ > ∆c,x> 1, (A25) Appendix B: Numerical details 2 1 1 x x tanh 3 cosh− x g(x) = 3 1 1 1 , In this appendix we discuss the scaling and conver- cosh cosh− x " − 3√x2 1 # 3 − gence of the zero energy DOS as a function of various for ∆ < ∆c,x> 1, (A26) KPM parameters. We begin with the clean limit W =0 2 1 1 and compute the zero energy density of states as a func- x x tan cos− x g(x) = 1 3 , 3 1 1 2 tion of system size ranging from L = 60 to 140 in steps cos cos− x " − 3√1 x # 3 − of 10. Generally in the thermodynamic limit we expect for ∆ < ∆ ,x< 1. (A27) ρ(0,L) 1/L2. However, this behavior cannot be ob- c ∼ 16
(a) 0.005 gree of accuracy as shown in Fig. 6(b). These results are obtained by considering the asymptotic behavior of the 0.0045 Jacobi theta function 0.004 ∞ 2 0.0035 xπn πx e− = θ3(0,e− )= ϑ(0, ix). (B2) 0.003 n= −∞ (0) X 0.0025 ρ 0.002 0.039 0.0015 (a) 0.001 0.038 0.0005 0.037 0 3000 6000 9000 12000 15000 18000 0.036 Nc 0.035 (0) 0.0025 ρ (b) 11 0.034 Nc=2 0.033 212 0.002 213 0.032 14 2 0.031 0.0015 0 3000 6000 9000 12000 15000 18000
(E) Nc ρ 0.001 (b) 0.04 0.0005
0.035 0 -0.02 -0.01 0 0.01 0.02
E/t 0.03 (E) ρ FIG. 7: (Color Online) DOS for L = 60 and W = 2.0t FIG. 8: (Color Online) DOS for L = 60 and W = 3.0t>Wc, tained within the KPM. Rather within the KPM we (a) Nc dependence of ρ(0, L, Nc) converges in Nc for Nc ≥ find ρ(0,L) 1/L2.85 consistent with a 1/L3 scaling 4096. (b) Low energy dependence of ρ(E, L, Nc) for various [see Fig. 6(a)].∼ This can be understood by considering Nc showing that there is a very weak dependence in the DM the density of states with a Gaussian broadening fac- phase. tor σ = π/Nc, where Nc is the number of terms kept in the Chebyshev expansion (as described in section III Moving away from the clean limit by introducing dis- A), as this is an excellent approximation to the effect of order provides a natural broadening of the energy eigen- the Jackson kernel [32]. Since there is only an intensive values and thus changes the scaling with Nc. Focusing on the SM phase, we find that ρ(0,L,N ) 1/L3 is number (8 corresponding to each cone) of states at zero c ∼ energy the KPM yields essentially satisfied for weak disorder strengths. Even though we find a large deviation in the zero energy DOS 2 exp 2π (n2 + n2 + n2) as a function of Nc, it does converge for increasing Nc as − Lσ x y z ρ(0,L,Nc) = shown in Figs. 7 (a) and (b). The deviations in ρ(0,L,Nc) h 3√ 2 i n ,n ,n L 2πσ are due to the KPM beginning to resolve the energy xXy z 1 N eigenstates. = c (B1) Inside the DM phase, ρ(0,L,N ) has a very weak de- ∼ σL3 πL3 c pendence on Nc [as shown in Figs. 8 (a) and (b)] and for where we have used ki =2πni/L. We find ρ(0,L,Nc) sufficiently large system sizes, converges in L [see Figs. 9 1.02 ∼ Nc reproducing the linear in Nc scaling to a good de- (a) and (b)]. We conclude that in each phase choosing 17 Nc = 4096 gives a converged estimate of ρ(0), and there- fore we use this value of Nc to determine the critical exponents. 0.01 (a) W/t=2.0 2.1 2.2 2.3 2.4 2.5 2.55 0.001 2.6 (0) 2.65 ρ 2.70 2.75 0.0001 0.01 1/L 0.01 (b) W/t=0.55 0.56 0.57 0.58 0.59 0.60 0.61 0.001 0.62 (0) 0.625 ρ 0.63 0.64 0.65 0.675 0.0001 0.01 1/L FIG. 9: (Color Online) DOS at zero energy ρ(0) as a function of of the inverse system size 1/L on a log-log scale in both the SM and DM phases for potential disorder up to L = 110. Results for the box distribution (a) and the Gaussian distribution (b). In the SM phase ρ(0) goes to zero as the system size is increased, while it saturates in the DM phase, and at the QCP ρ(0) is still decreasing for increasing L. In the DM phase the results are saturating in 1/L. [1] N. Goldenfeld, Lectures on phase transitions and the [7] E. Fradkin, Phys. Rev. B 33, 3263 (1986). renormalization group, vol. 5 (Addison-Wesley Reading, [8] P. 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