Continuous and discontinuous topological quantum phase transitions

Bitan Roy, Pallab Goswami, and Jay D. SauCondensed Matter Theory Center and Joint Quantum Institute, Department of Physics,

University of Maryland, College Park, Maryland 20742- 4111 USA(Dated: June 16, 2021)

The continuous quantum phase transition between noninteracting, time-reversal symmetric topo-logical and trivial insulators in three dimensions is described by the massless Dirac fermion. Weaddress the stability of this quantum critical point against short range electronic interactions byusing renormalization group analysis and mean field theory. For sufficiently weak interactions, weshow that the nature of the direct transition remains unchanged. Beyond a critical strength ofinteractions we find that either (i) there is a direct first order transition between two time reversalsymmetric insulators or (ii) the direct transition is eliminated by an intervening time reversal andinversion odd “axionic” insulator. We also demonstrate the existence of an interaction driven firstorder quantum phase transition between topological and trivial gapped states in lower dimensions.

PACS numbers: 64.70.Tg, 71.30.+h, 71.10.Fd, 03.65.Vf

Introduction: The spin-orbit coupled, time reversalsymmetric insulators in two and three dimensions be-long to the Altland-Zirnbauer class AII [1, 2]. Due tothe existence of nontrivial Z2 topological invariants inboth spatial dimensions, a band insulator can be clas-sified either as a strong topological insulator (TI) or atrivial/normal insulator (NI). The four component mas-sive Dirac fermion provides an efficient low energy de-scription of Kramers degenerate conduction and valencebands in such systems [3–6]. The sign of the Dirac massprovides the topological distinction between two insulat-ing states. In a clean, noninteracting system the univer-sality class of a continuous topological quantum phasetransition (QPT) between them (where the Dirac massvanishes) is described by a massless Dirac Hamiltonian.Recently, there has been considerable interest in under-standing the nature of such topological transition in thetwo-dimensional HgCdTe quantum well [7] and three di-mensional materials such as BiTl(S1−δSeδ)2 [8, 9] and(Bi1−xInx)2Se3 [10, 11]. Although the phase diagramof noninteracting, disordered systems has been investi-gated in many analytical and numerical works [12–20]and the interplay of disorder and long range Coulomb in-teractions has also been studied perturbatively for bothtwo [14] and three dimensions [16], the effects of strongshort range electronic interactions on the phase diagramof clean AII insulators is not well understood. The inves-tigation of this fundamental problem is the central themeof this Rapid Communication.

We show that the three-dimensional massless Diracfermion is stable against sufficiently weak, but genericshort range electronic interactions. Consequently, theuniversality class of the continuous QPT between theTI and NI remains unaffected up to a critical strengthof interactions, beyond which the quantum critical point(QCP) becomes destabilized due to the spontaneous gapgeneration for the massless Dirac fermion. There aretwo possibilities for the gap: (i) an inversion (P) and

-0.15 -0.10 -0.05 0.000.0

0.5

1.0

1.5

2.0

2.5

m

gs

TI

NI

(a)

-0.6 -0.4 -0.2 0.0 0.2 0.4 0.60.0

0.5

1.0

1.5

m

gp

AI

TI NI

(b)

FIG. 1: Zero temperature phase diagram of three-dimensionalDirac fermion for (a) gπ = 0 and (b) gσ = 0, for b = 0.2. Allthe parameters are dimensionless (defined in the text). gσand gπ are interaction couplings for scalar and pseudoscalarmass generation, respectively, and m is the band mass forDirac fermions. Dirac semimetal is realized on the solid line.The transition between two insulators is first order (continu-ous) across the dotted (solid and dashed) line(s). The threephases TI, NI and AI meet at a multicritical point as shown in(b). For d=3, similar first order transition can occur betweenstrong and weak topological insulators, and also between weaktopological and normal insulators.

time reversal (T ) preserving scalar mass, which doesnot break any bonafide microscopic symmetry and (ii)a pseudoscalar mass that spontaneously breaks genuinediscrete P and T symmetries and corresponds to an “ax-ionic insulator” (AI), which has recently been discussedin the context of magnetic insulators [21, 22], a Kondosinglet phase [23], and p + is superconductor [24]. Thescalar mass generation leads to a direct first order QPT(without band gap closer) between the TI and NI [seeFig. 1(a)]. This is a fluctuation driven first order transi-tion and is distinct from the generic first order transitionbetween two different broken symmetry phases [25, 26].On the other hand, the nucleation of pseudoscalar massseparates two T symmetric insulators, while eliminatinga direct transition between them [see Fig. 1(b)], and three

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phases (TI, NI and AI) meet at a multi-critical point(MCP). In contrast to the TI and NI, which are respec-tively characterized by the quantized magnetoelectric co-efficients π and 0, the AI phase supports a nonquantizedmagnetoelectric coefficient. Consequently, the AI phasedoes not possess gapless surface states, and the massivefluctuations of the magnetoelectric coefficient cause dy-namic magnetoelectric effects. We also reach similar con-clusions regarding the fate of the QPT between the TI(quantum spin Hall insulator) and NI in two dimensions.However, the nature of the possible T breaking phases intwo and three dimensions are considerably different. Wealso show a similar first order transition between topo-logical and trivial superconductors in one dimension.

Model : We will mostly be interested in the three-dimensional materials with P and T symmetries. Forconcreteness, we choose the following tight bindingHamiltonian on a cubic lattice [4]

H =∑k

Ψ†k [t1

3∑j=1

Γj sin(kja) + Γ4 M ]Ψk

+t2∑k

3∑j=1

Ψ†k Γ4 [1− cos(kja)] Ψk, (1)

as a minimal model for describing the insulating statesin class AII, where a is the lattice constant. The fourcomponent spinor Ψ>k = (c+,↑,k, c+,↓,k, c−,↑,k, c−,↓,k) iscomprised of the electron annihilation operator cr,s,k forstates with wavevector k, the orbital parity r = ± andthe spin projections s =↑ / ↓. The relevant discrete sym-metry operations are defined according to P: k → −kand Ψk → Γ4Ψ−k, and T : k→ −k and Ψk → Γ1Γ3Ψ−k.We have used only four mutually anticommuting matri-ces Γj = σj ⊗ τ1 (with j = 1, 2, 3), Γ4 = σ0 ⊗ τ3 inEq. (1) to enforce simultaneous P and T symmetries.The bilinear Ψ†Γ5Ψ, where Γ5 = σ0 ⊗ τ2, maintains twofold degeneracy of the valence and conduction bands, butseparately breaks P and T . By contrast, fermion bilin-ears Ψ†ΣabΨ, with Σab = [Γa,Γb]/(2i), either break Por T , and lift the Kramers degeneracy. Consequently,Ψ†ΣabΨ and Ψ†Γ5Ψ are absent in H. Here, σµ and τµare two sets of Pauli matrices respectively operating onthe spin and the parity indices. The above model de-scribes both strong (for −2 < M

t2< 0) and weak (for

−4 < Mt2< −2) TIs. The strong TI and NI phases re-

spectively correspond to sgn(Mt2) < 0 and sgn(Mt2) > 0and M = 0 describes the QPT between them. In con-trast, the transition between weak TI and strong TI (NI)takes place when M

t2= −2(−4). For simplicity, we focus

on the strong TI-NI QCP. But, our results are equallyapplicable near the weak TI-strong TI or NI QCPs.

Since the minimal gap in the spectrum near the strongTI-NI QPT occurs at the Γ point k = (0, 0, 0) [27], thelow energy quasiparticles in its vicinity are described by

the following continuum Hamiltonian

H3d =

∫ ′ d3k

(2π)3Ψ†k

v 3∑j=1

Γjkj + Γ4(M +Bk2)

Ψk,

(2)where v = t1a and B = t2a

2. The integral over the mo-mentum is restricted to an ultraviolet cutoff Λ. The TIand NI phases respectively correspond to sgn(MB) < 0and sgn(MB) > 0. At the QCP (M = 0) between thesetwo insulators the critical excitations are described bymassless Dirac fermions and Bk2 act as a momentum de-pendent Wilson mass. Only the fixed point Hamiltonianof massless Dirac fermion (M = B = 0) possesses a globalchiral U(1) symmetry under which Ψ→ ΨeiθΣ45 [28]. Atthe microscopic level B 6= 0 and H3d enjoys only a re-duced Z2 particle-hole symmetry, captured by the anti-commutation relation {H3d,Γ5} = 0. Additionally, af-ter an appropriate modification of the spinor, the sameHamiltonian operator can also describe the topologicalQPT (BEC-BCS transition) for superconducting systemsin Altland-Zirnbauer classes AIII and DIII.

For simplicity, we here restrict ourselves to class AIIand consider the effects of repulsive, short-range inter-actions. For a P, T symmetric system with rotationalsymmetry, generic model of short-range interaction

Hint =

∫d3x

[λ1

2

(Ψ†Γ0Ψ

)2+λ2

2

(Ψ†Σ45Ψ

)2+λσ2

(Ψ†Γ4Ψ

)2+λπ2

(Ψ†Γ5Ψ

)2 ](3)

is described by only four linearly independent couplingsdue to the Fierz identity [29]. For λ1 = λ2 = 0 andM = B = 0, H3d + Hint corresponds to the celebratedNambu-Jona-Lasinio (NJL) model for mass generationthrough spontaneous U(1) chiral symmetry breaking [30].

RG analysis: Since the dynamic scaling exponent formassless Dirac fermion is z = 1, the scaling dimensionsof the quartic interactions are [λj ] = (z − d) = (1 − d).Therefore, any sufficiently weak short range interactionis an irrelevant perturbation at the massless Dirac QCPfor d > 1, and leaves the universality class of the directTI-NI transition unchanged. In the weak coupling limit,λis only modify the phase boundary in a non-universalmanner. These features and the potential breakdown ofthe massless Dirac theory for strong interactions can becaptured through a renormalization group (RG) analysis,controlled via simultaneous 1/Nf - and ε-expansion aboutthe lower critical dimension d = 1, where Nf is the flavornumber of four component fermions and ε = d−1. Withinthe Wilsonian momentum shell method, we integrate outthe degrees of freedom inside −∞ < ω <∞ and Λe−l <k < Λ, and subsequently rescale according to x → xel,τ → τel, Ψ→ e−dl/2Ψ. In the Nf →∞ limit, we obtain

3

the following flow equations

dm

dl= m (1 + 2gσ) + 2gσb,

db

dl= −b, dg1

dl= −εg1,

dg2

dl= −εg2,

dgσdl

= −εgσ + 2g2σ,

dgπdl

= −εgπ + 2g2π, (4)

where we have introduced the dimensionless couplingsgi = λiSdNfΛd−1/[v(2π)d], m = M/(vΛ), b = BΛ/v,and Sd is the surface area of a d-dimensional unit sphere.

For sufficiently weak interactions, Dirac mass m is theonly relevant variable. In this regime, gi(l) ∼ gi,0e

−εl,b(l) ∼ b0e

−l, where gi,0 and b0 are the bare values ofthe corresponding couplings and the solution of Eq. (4)yields

m(l) +2

d+ 1b(l)gσ(l) = el

(m0 +

2

d+ 1b0gσ,0

). (5)

Therefore, m∗ = m0 + 2 b0d+1gσ,0 acts as an effective mass

and the TI-NI phase boundary is determined by m∗ = 0.This agrees well with the phase boundary determinedthrough minimizing the free energy for weak interactions,as shown in Fig. 1(a) for d = 3 [31]. The flow equationsalso show that g1 and g2 are always irrelevant pertur-bations, and can be ignored. Thus, we can restrict our-selves to the NJL model that supports three unstablefixed points: (i) gcσ = ε/2, gcπ = 0; (ii) gcσ = 0, gcπ = ε/2;and (iii) gcσ = gcπ = ε/2. The first two are QCPs, re-spectively describing the generation of scalar and pseu-doscalar masses for massless Dirac fermion at strong cou-plings, whereas the third one is a bicritical point. Ford > 1 and at strong coupling (gi > gci ), the appropri-ate correlation length ξi ∼ Λ−1(gi − gci )−νi provides theinfra-red cutoff for the RG flow, with a correlation lengthexponent νi = 1/ε. Therefore, for d = 2 and d = 3 we re-spectively have non-Gaussian and Gaussian (mean-field)itinerant QCPs [32]. As b(l) = b0e

−l, we can ignore it inthe RG sense in the weak-coupling regime. However, theeffects of b cannot be neglected for gi > gci s. For eluci-dating the dramatic effects of b in determining the natureof the strong coupling phases and the associated transi-tions, next we adopt the method of large Nf mean-fieldtheory.

Free energy : We perform the Hubbard-Stratonovichdecoupling of the quartic terms proportional to gσ andgπ, with bosonic fields Σ and Π, respectively, that cou-ple to the fermion bilinears as Σ Ψ†Γ4Ψ and Π Ψ†Γ5Ψ.When the corresponding coupling constants exceed theircritical strengths, these bosonic fields acquire expectationvalues, while giving rise to the gap in the spectrum. With〈Π〉 = 0, 〈Σ〉 6= 0, 〈Σ〉 + M acts as the effective scalarmass, with TI and NI being the only possible phases. Incontrast, when 〈Π〉 6= 0, P and T are spontaneously bro-ken, giving rise to the AI phase. Since the free energydensity (F ) has the dimension EL−d ∼ L−(d+1), we de-fine a dimensionless quantity f = F (2π)d/(NfSdvΛd+1),

which takes the form

f =σ2

2gσ+π2

2gπ−2

∫ 1

0

dxxd−1√x2 + (σ +m+ bx2)2 + π2.

(6)We have also introduced the following dimensionless vari-ables σ = Σ/(vΛ), π = Π/(vΛ), x = k/Λ.

Note σ = −m, π = 0 corresponds to the masslessDirac fermion describing the continuous QPT betweenthe TI and NI, and remains as the global minimum of fup to critical strengths of gis. Consequently, the phaseboundary between the TI and NI is determined by

m =2gσ3b3

[(2− b2)

√1 + b2 − 2

], (7)

in d = 3, which is equivalent to m∗ = 0 for small b, upto the critical strengths of the quartic interactions

g∗σ =1

2

[1 + b2 +

√1 + b2

], g∗π =

1

2

[1 +

√1 + b2

]. (8)

Notice that g∗σ > g∗π for an arbitrary value of b. This isrelated to the fact {Γ4,Γ5} = 0. Thus a finite b favorsthe nucleation of the pseudoscalar mass. In a continuummodel without any momentum dependent mass (b = 0)the critical couplings g∗σ = g∗π = ε/2 = 1 become identicalto the ones found from the RG analysis of the NJL model.

We numerically minimize the free energy to obtain thephase diagrams in Figs. 1(a) and 1(b). However, all thesalient features can be understood by expanding f forsmall σ + m and π, and retaining only the lowest orderb-linear contributions. The free energy is then given by

f3d ≈[m2

2gσ− 1

2

]− σ

[m

gσ+b

2

]+σ2

2

[1

gσ− 1

gcσ

]+π2

2[1

gπ− 1

gcπ

]+b

2σ(σ2 + π2) +

(σ2 + π2)2

4log

[2e−1/4

σ2 + π2

],

(9)

for d = 3, after shifting σ → σ −m. When we considerthe nucleation of the scalar mass (for sufficiently stronggσ), the following subtleties need to be addressed: (i)(mgσ

+ b2

)acts as the external field for σ and (ii) the

momentum dependent mass ∝ b gives rise to all oddpowers (cubic and higher) of σ in the expression for thefree energy density. Due to the presence of all the oddpowers of σ, the nucleation of the scalar mass proceedsthrough a first order QPT between the TI and NI [seeFig. 1(a)]. By contrast, the pseudoscalar mass genera-tion (for sufficiently strong gπ) occurs through continu-ous QPT. Three phases (TI, NI and AI) meet at a MCP(located at m∗ = 0 and gπ = g∗π in the gσ = 0 plane),as shown in Fig. 1(b). The MCP displays mean-field ex-ponents, but the hyperscaling is violated by logarithmiccorrections [due to the π4 log(π2) term in Eq. (9)].

Two-dimensions: Our study can be generalized to ad-dress the transition between the TI (quantum spin-Hall

4

insulator) and NI in two dimensions. In the continuumlimit, these two insulators and the transition betweenthem can be described by the Bernevig-Hughes-Zhangmodel [33]

H2d =

∫ ′ d2k

(2π)2Ψ†k[v (Γ3kx + Γ5ky) + Γ4(M +Bk2)

]Ψk.

(10)

The TI and NI phases in this model are realized forsgn(MB) < 0 and sgn(MB) > 0, and the continu-ous QPT between them (M = 0) is described by two-dimensional massless Dirac fermion. Strong interac-tions can give rise to the following four Dirac masses:Σ = 〈Ψ†Γ4Ψ〉, Π1,2 = 〈Ψ†Γ1,2Ψ〉, and Ξ = 〈Ψ†Σ35Ψ〉.As in d = 3, the nucleation of the scalar mass Σ (for suf-ficiently strong gσ) proceeds through a first order QPT,which can be understood from the following free energydensity

f2d =

(m2

2gσ− 2

3

)− σ

(m

gσ+

2

3b

)+σ2

2

(1

gσ− 1

gcσ

)+ |σ|3

(2

3+ b sgn(σ)

)+O(b2, σ4), (11)

where gcσ = 1/2. The corresponding phase diagram isqualitatively similar to Fig. 1(a). In contrast, conden-sation of T -odd magnetic masses Π1,2 or the anomalouscharge Hall mass Ξ [34] takes place through a continuousQPT as in Fig. 1(b) (see also Ref. [35]), giving rise toa MCP in the (m, gπ) plane (with non-mean-field expo-nents [36, 37]). Our proposed first order transition be-tween TI and NI has recently been observed in numericalworks in both d = 2 [38] and d = 3 [39].

One dimension: In one dimension, the quartic interac-tion (gσ) is marginally relevant and destabilizes the mass-less Dirac fixed point for infinitesimal strength [settingε = 0 in Eq. (4)] [40]. In the presence of a momen-tum dependent mass, the QPT between topological andtrivial gapped states can be first order for a sufficientlylarge number of flavors. This is the reason behind theexistence of a fluctuation driven first order QPT for theN -color Ashkin-Teller chain when N ≥ 3 [41–43]. Thepertinent continuum model is described by N species oftwo component Majorana/Jordan-Wigner fermions witha k dependent mass in the vicinity of the decoupled IsingQCP, and microscopic four-spin coupling gives rise toO(N) invariant quartic interaction. Such a first orderQPT can be germane for describing the direct QPT be-tween topological and trivial superconductors in differentclasses.

Conclusions: Our results can also shed light onto thefinite temperature phase diagram of strongly interactingDirac fermion. At finite temperatures and for sufficientlystrong interactions, the massless Dirac fermion can un-dergo either (i) a first-order classical phase transition andenter into the TI or NI phase, or (ii) a continuous classical

phase transition, giving rise to a T -breaking insulator. Ingeneral, the semiconductors (where most of the TIs havebeen identified) are weakly correlated materials. How-ever sufficiently strong electronic interactions (local andnonretarded) can be mediated by optical phonons, belowthe scale of optical frequency [44–46].

Finally, we comment on the effects of disorder on var-ious QPTs, discussed for clean systems. Harris criteriondictates that any continuous QPT is stable against weakdisorder if ν > 2/d [47] and this has been explicitly shownfor the TI-NI QPT at d = 3 [16]. Since ν = 1/2 < 2/3at the clean MCP in d = 3, the universality class will bechanged by an infinitesimal amount of randomness intoa disorder controlled class satisfying ν > 2/3 [48]. Atpresent it is not well established whether the exact valueof ν for MCP in d = 2 is bigger or less than one (= 2/d),and we cannot properly assess its stability against weakdisorder. On the other hand, in the thermodynamiclimit, the first order transition (both classical and quan-tum) will be rounded by disorder into a continuous onein d = 1, 2, while it can survive in d = 3 for sufficientlyweak randomness [49–55]. The analysis of the universal-ity class at a putative disorder and interaction controlledcritical point is beyond the scope of the present RapidCommunication.

Acknowledgements: B. R. and J. D. S. were supportedby the start-up grant of J. D. S. from the University ofMaryland. P. G. was supported by NSF-JQI-PFC andand LPS-CMTC.

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