Curved Spacetime Theory of Inhomogeneous Weyl Materials

PHYSICAL REVIEW RESEARCH 1, 032006(R) (2019)

Rapid Communications

Curved spacetime theory of inhomogeneous Weyl materials

Long Liang1,2 and Teemu Ojanen2 1Department of Applied Physics, Aalto University School of Science, FI-00076 Aalto, Finland 2Computational Physics Laboratory, Physics Unit, Faculty of Engineering and Natural Sciences, Tampere University, P.O. Box 692, FI-33014 Tampere, Finland

(Received 19 June 2019; published 16 October 2019)

We show how the universal low-energy properties of Weyl semimetals with spatially varying time-reversal (TR) or inversion (I) symmetry breaking are described in terms of chiral fermions experiencing curved- spacetime geometry and synthetic gauge ﬁelds. By employing Clifford representations and Schrieffer-Wolff transformations, we present a systematic derivation of an effective curved-space Weyl theory with rich geometric and gauge structure. To illustrate the utility of the formalism, we give a concrete prescription of how to fabricate nontrivial curved spacetimes and event horizons in topological insulators with magnetic textures. Our theory can also account for strain-induced effects, providing a powerful uniﬁed framework for studying and designing inhomogeneous Weyl materials.

DOI: 10.1103/PhysRevResearch.1.032006

Introduction. Semimetals and quantum liquids with linear establish a general and fully quantum-mechanical description dispersion near degeneracy points exhibit emergent relativis- of inhomogeneous Weyl semimetals. By employing Clifford tic physics at low energies. Topological Dirac and Weyl representations and Schrieffer-Wolff (SW) transformations semimetals [1–19] have proven to be particularly fertile [42], we systematically consider generic TR- and I-breaking condensed-matter playgrounds to study the interaction of chi- patterns and provide a controlled derivation of the low-energy 0 i i b j a ral fermions with gauge fields. These systems display the rich = σ + − , − ∂ σ . Weyl Hamiltonian HW V ea(ki KW i 2 ei jeb) physics of quantum anomalies discovered originally in the rel- Here σ a denotes the set of Pauli matrices supplemented by ativistic setting [20–22]. In translationally invariant systems, a unit matrix. The effective geometry is encoded in the frame i the twofold band touching in Weyl semimetals can be asso- fields ea while the effective gauge field receives contributions ciated with a conserved Berry charge which is topologically from the spatial variation of the Weyl point KW,i and the frame protected. Moreover, even in the presence of spatially varying fields. Our theory can also account for strain-induced effects perturbations, the low-energy properties can be understood and provides a unified low-energy description of inhomoge- in terms of Weyl particles experiencing artificial gravity and neous Weyl semimetals. gauge fields [23]. The condensed-matter setting allows for The obtained low-energy theory has a number of remark- remarkable opportunities in engineering synthetic gauge fields able consequences. In general, spatially varying TR- and I- and geometries that mimic and generalize the phenomenology breaking textures give rise to frame fields and metric tensors of high-energy physics [16,24–34]. that mix time and space components. In contrast to mere A popular starting point for geometry and gauge-field curved space geometries realized by strain engineering, we engineering in semimetals is a strain-distorted tight-binding obtain nontrivial spacetime geometries. Also, the spectral tilt model [35–41]. This description, case specific to a particular of the Weyl dispersion can be tuned by TR- and I-breaking lattice and orbital structure, can often be regarded as a formal textures. To demonstrate this effect in detail, we consider 3d device to obtain a long-wavelength theory. While being a topological insulators with magnetic textures. Strikingly, vari- powerful method for fabricating synthetic gauge fields, strain ous magnetic textures give rise to Weyl semimetal phases with engineering has the limitation of producing effective geome- a spatial interface between type I and type II regions. We show tries that are small perturbations from flat space. To obtain that the effective geometry near the interface emulates the more general three-dimensional (3d) geometries, Ref. [23] Schwarzschild metric at a black-hole event horizon. Our work proposed a new method of fabricating spatially varying TR- provides powerful tools to analyze and design the properties and I-breaking textures. Semiclassical dynamics of carriers of inhomogeneous Weyl materials. then reflect the interplay of effective curved geometry and Inhomogeneous Weyl systems. The starting point of our Berry curvature effects. The purpose of the present work is to theory of inhomogeneous Weyl systems is a generic four-band parent state with both time-reversal and inversion symmetry intact. By introducing spatially varying TR- or I-breaking fields, the parent state is driven to an inhomogeneous Weyl Published by the American Physical Society under the terms of the semimetal phase. The minimal model for the parent states is Creative Commons Attribution 4.0 International license. Further characterized by the Hamiltonian [23] distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. H0 = n(k)I + κi(k)γi + m(k)γ4, (1)

2643-1564/2019/1(3)/032006(6) 032006-1 Published by the American Physical Society LONG LIANG AND TEEMU OJANEN PHYSICAL REVIEW RESEARCH 1, 032006(R) (2019) where the repeated indices are implicitly summed over, I is after the transformation becomes the identity matrix, and γμ with μ = 1, 2, 3, 4 denotes the four = 1 i γ a,κ − ω + − ω γ + γ . γ matrices satisfying anticommutation relations {γμ,γν }= H 2 Ea i(k ) m(k ) 4 u 12 (4) 2δμν . The parameters n(k) and m(k) are even functions of the The anticommutator structure results from the noncommuta- momentum while κ (k) is odd. Therefore, γ , , are odd under i 1 2 3 tivity of position and momentum. Here E i with i and a running both TR and I, while γ is even. The fifth γ matrix is defined a 4 from 1 to 3 are frame fields defined through W †γ W = E i γ a. as γ = γ γ γ γ , which is odd under both TR and I. i a 5 1 2 3 4 ij = i jηab × Using the frame fields, we can define a metric g EaEb To write down the most general 4 4 Hamiltonian, η η = − , , , we introduce ten additional matrices, γ =−i[γ ,γ ]/2. It with being the Minkowski metric, diag( 1 1 1 1). ij i j , , ij is convenient [13,23] to separate the ten matrices into The indices i j k can be raised or lowered by g or its , , ηab η three vectors b = (γ ,γ ,γ ), b = (γ ,γ ,γ ), and p = inverse gij, and a b c are raised or lowered by or ab. 23 31 12 15 25 35 It is easy to verify that E i E b = δb and E i E a = δi .Thespin (γ14,γ24,γ34 ) and one scalar ε = γ45. The transformation a i a a j j ω = †∂ properties of the four groups can be easily deduced from their connection iW rW can be written in terms of the frame ω = ωabγ =− a∇ jbγ / ∇ constituent γ matrices, and it turns out that b and b break fields as i i ab E j iE ab 4, with being the TR symmetry, while p and ε break I symmetry. The general covariant derivative on the manifold. Alternatively, ω can also ω = ω j 4 × 4 Hamiltonian with TR- and I-breaking terms can thus be be viewed as a SU(2) gauge connection i i b j. Equation written as (4) describes a Dirac electron moving in a curved space [43] u = + · + · + · + ε, in the presence of time-reversal symmetry-breaking field . H H0 u b w p u b f (2) The elegant gauge structure of Eq. (2) is discussed in detail in where the functions u, w, u, and f characterize the symmetry- the Supplemental Material (SM) [44]. While the Dirac equa- breaking fields which are position dependent. Because r is tion still describes flat space gij = δij, it contains redundant even under TR but odd under I, w and f should be even high-energy bands. The curved space geometry and gauge functions of r to break the inversion symmetry. If u and u fields emerge as we project the four-band Dirac Hamiltonian, are not even functions of r, the inversion symmetry is also Eq. (4), to the two-band low-energy Weyl Hamiltonian. If broken. But since u and u fields always break time-reversal the symmetry-breaking field u is constant, this can be done symmetry irrespective of their r dependence, we label them as exactly by a momentum-dependent unitary transformation. time-reversal symmetry-breaking terms. In general, we regard However, for the position-dependent fields we are interested TR- and I-breaking terms as smooth functions of position r in, this method is no longer exact since the momentum and and assume that r and k are conjugate variables [ri, k j] = position do not commute with each other. There will be iδij. Elastic deformations and strain typically induce spatial additional terms related to the derivatives of the fields that dependence in H0 [35–40]. While we are mainly considering mix the high- and low-energy degrees of freedom, and in inhomogeneous TR and I breaking, we will discuss below how general it is impossible to perform the block diagonalization. strain is included in our formalism in the continuum limit. However, in the large-mass limit and slowly varying fields, we TR-breaking case. Here we derive an effective curved- can derive an effective low-energy theory in a controlled way space Weyl equation for inhomogeneous TR-breaking sys- by employing the SW transformation. tems in two opposite limits, vanishing mass (m = 0) and large First, we expand Eq. (4) to the first order in derivatives of mass. Physically these correspond to metallic and insulating the symmetry-breaking field. This yields u r · b parent states. We consider the ( ) term which corresponds ≈ κ , γ + γ + γ + + γ , to 3d magnetization or any field which transforms as mag- H ˜i(k r) i m(k) 4 u 12 uibi f 45 (5) netization under TR and spatial rotations. The unit matrix i i whereκ ˜i ={κi(k), E (r)}/2, u = ∂k mω , and f = term n(k) can be set as zero since it only shifts the energy. a i j j ∂ κ ωaE i . Choosing a particular representation γ = τ ⊗ σ , For the m(k) = 0 case, the Hamiltonian can be readily block k j i j a 1 0 x γ2 = τ0 ⊗ σy, γ3 = τx ⊗ σz, and γ4 = τz ⊗ σz,theterms diagonalized. Defining two sets of Pauli matrices σi and τi (σ0 proportional to γ , , , b , and b are block diagonal. and τ are the 2 × 2 unit matrix) and working in the chiral 1 2 4 3 3 0 Employing the SW transformation, we seek matrix S such representation γ = τ ⊗ σ , γ = τ ⊗ σ , the Hamiltonian i 3 i 4 1 0 that the unitary equivalent Hamiltonian eSH e−S is block splits to two blocks diagonal. In the SM, we explicitly write S in the lowest order ± = ± κ σ = ± , σ . / ,κ / HW [ui(r) i(k)] i di (k r) i (3) in the large-mass limit ui m 3 m 1. The transformed ± Hamiltonian is block diagonal, yielding an effective Weyl The local Weyl point are determined by d = u (r) ± i i Hamiltonian κi(KW ) = 0. Thus, u(r) gives rise to an axial gauge field while the frame fields can be straightforwardly obtained as indicated a HW = da(k, r)σ , (6) [by Eq. (8)] below. = κ − / − / =− + − The complementary regime of nonzero mass gives rise with di=1,2 ˜i fui (2m) fui (2u), d3 m u κ2 + 2 / + 2 + 2 / = + κ / to rich geometric and gauge structure. It is convenient (˜3 f ) (2m) (u1 u2 ) (2u), and d0 u3 f ˜3 m. to parametrize the symmetry-breaking fields as u(r) = The Weyl points are determined by di=1,2,3 = 0, and due to u(r)[sin θ(r) cos φ(r), sin θ(r)sinφ(r), cos θ(r)]. To derive the inversion symmetry, if KW is a Weyl point, −KW will an effective Weyl Hamiltonian, we first rotate the fields also be a Weyl point. The higher order corrections to Eq. (6) ∂2κ ∂ 2 ∂2 ∂ 2 along the z direction by applying a unitary transforma- are proportional to k i( ru) , k m( ru) , which are always † tion, W (r)u · bW (r) = u(r)b3. This is achieved by choos- small for smooth u and vanish completely for H0 with a linear ing W = exp (−iφb3/2)exp (−iθb2/2). The Hamiltonian dispersion. Expanding the Hamiltonian around KW ,wearrive

032006-2 CURVED SPACETIME THEORY OF INHOMOGENEOUS … PHYSICAL REVIEW RESEARCH 1, 032006(R) (2019) at our main result, Application I: Engineering tilts and horizons. Here, we i illustrate the power and utility of the developed formal- ≈ σ 0 + i − − b∂ j σ a, HW V ea ki KW,i ei jeb (7) ism by proposing concrete systems with spatial interface 2 between type I and type II Weyl fermions. We consider where i = 1, 2, 3 and a = 0, 1, 2, 3. The Weyl point depends a simple parent model with κi = ki and m(k) = m which on the position and can be interpreted as U (1) vector poten- could be realized in 3d TR-invariant insulators with magne- = − i b∂ j tization texture [23] or in the topological insulator–magnet tial, and V d0(KW ) 2 e0 jeb acts as an effective scalar potential. The potentials acquire corrections from the frame heterostructures [12]. The TR-breaking field, representing, fields, ensuring the Hamiltonian Hermitian. The frame fields for example, 3d magnetization, is parametrized as u = are defined through u(r)(sin θ cos φ,sin θ sin φ,cos θ ) with spatially varying an- gles. ∂ i da e (r) = , (8) Following the procedure discussed above, we obtain a ∂k × = θ φ + i KW an effective 2 2 Hamiltonian with d1 kx cos cos k cos θ sin φ − k sin θ, d =−k sin φ + k cos φ, d = u − with additions e0 =−1 and e0 = 0. We obtain from the y z 2 x y 3 0 1,2,3 m − (˜κ2 + f 2 )/(2m), and d =−κ˜ f /m, whereκ ˜ = μν = μ ν ηab 3 0 3 3 frame fields the emergent metric g ea e . Explicitly, φ θ + φ θ + θ = ∂ φ + b kx cos sin ky sin sin kz cos and f ( rz j cos φ∂ θ − sin φ∂ θ )/2. The products of k- and r-dependent g00 =−1, g0i = ei , gij =−ei e + ei e j . (9) ry rx 0 0 0 a a terms should be understood as symmetrized. For the It should be stressed that, as indicated by the effective metric lowest-order of the SW transformation to be a good tensor with space and time mixing terms g0i, the emergent approximation around the Weyl point, the condition geometry of Eq. (7) is fundamentally different from the in- u − m m should be satisfied. The Weyl points are homogeneous strain-induced metrics that do not contain the determined by ±KW =±KW (sin θ cos φ,sin θ sin φ,cos θ ) mixing terms. The dispersion relation of the Weyl fermion can with K = 2m(u − m) − f 2. For simplicity, we assume μν W be determined by g pμ pν = 0 with p = (ω, k), which gives that 2m(u − m) − f 2 > 0 such that there always exist two well-separated Weyl points. Expanding around KW , we obtain ω = i ± i j . e0ki el el kik j (10) the linearized Weyl Hamiltonian with the frame fields being e0 =−1, ei =−f /m(sin θ cos φ,sin θ sin φ,cos θ ), ei = | i l | < | i l | > 0 0 1 If e0ei 1, it is a type I Weyl semimetal, and if e0ei 1 θ φ, θ φ,− θ i = − φ, φ, (cos cos cos sin sin ), e2 ( sin cos 0), we obtain a type II Weyl semimetal with overtilted Weyl cone. i =− / i and e3 KW fe0. After straightforward calculations, Remarkably, as shown below, this fact can be employed in we find ei ex = ei ey = 0 and ei ez = f /K . Therefore, if engineering a spatial interface between type I and type II Weyl 0 i 0 i 0 i W | f /K | < 1, the node is of type I, if | f /K | > 1, it is of type semimetals. The interface between type I and type II Weyl W W II, and the interface between the type I and type II regions, semimetals, which could simulate properties of the black i.e., the event horizon, is determined by | f /K |=1. In the hole horizon, may be designed experimentally by controlling W SM, it is shown that in a suitable local basis the metric is the magnetic texture. This could open a method to study analogous to the Schwarzschild metric near the horizon in the Hawking radiation [45–47] and quantum chaos [48–51]in Gullstrand-Painlevé coordinates [45,52,53]. Weyl semimetals. Having worked out the general case, we now study a To conclude this section, we outline how things would special case with u as a constant, φ = 0, and θ = r/ξ with r = have changed had we considered the other TR-breaking triplet 2 + 2 + 2 = / ξ u · b instead of u · b. In this case, we can use essentially x y z . Correspondingly, we find f y (2r ). This the same method to derive a two-band model, but in general magnetic texture is slowly varying in the length scale much ξ | | / ξ this leads to a nodal line semimetal [13] instead of a Weyl smaller than . Since 0 f 1 (2 ), there always exists semimetal. While interesting, this case is not relevant for a type I region, and to have a type II region, the condition − ξ 2 < Weyl-type behavior and is postponed to the SM. 4m(u m) 1 should be satisfied. The event horizon is 2 2 2 I-breaking case. The derivation of the curved-space Weyl determined by y = 4ξ m(u − m)r , which defines a conical Hamiltonian can be extended to I-breaking systems. Sev- surface; see Figs. 1(a) and 1(b). Clearly, the interface can be eral Weyl materials that break inversion (but preserve TR) tuned when it is possible to manipulate ξ. Thus, the horizon symmetry have be observed [15–17]. In general, a spatially may be tuned experimentally, providing a way to simulate varying I-breaking term w(r) · p can be treated similarly as the Hawking radiation using Weyl semimetals [45–47]. As the TR-breaking case. This has been carried out in the SI depicted by Figs. 1(c) and 1(d), different shapes of the event where we obtain an I-breaking variant of Eq. (6). The essential horizon can also be realized; see the SM for more details. difference from the TR-breaking case is that the role of the Application II: Inclusion of strain effects. As noted above, mass term m is played byκ ˜1 orκ ˜2. Thus, the controlling there exists a significant body of literature on strain-induced parameter in this situation isκ ˜1 (orκ ˜2) instead of m. Assuming artificial gauge fields and synthetic geometry [35–41]. In these κ˜1 sufficiently large, which is also a required to obtain Weyl treatments, the strain effects manifest at the level of the TR- points in this case, we can find a SW transformation to block and I-preserving parent state Hamiltonian H0 which becomes diagonalize H and obtain a two-band Hamiltonian. Finally, position dependent. In this respect, the strain engineering can analogously to the u · b term, the f γ45 term as a sole I- be viewed as complementary to the studied case with spatially breaking term leads to a nodal line semimetal as shown in the dependent TR- and I-breaking fields. However, here we show SM. how to incorporate the strain effects to a unified theory of

032006-3 LONG LIANG AND TEEMU OJANEN PHYSICAL REVIEW RESEARCH 1, 032006(R) (2019)

becomes = 1 ˜ i γ a, − ω + γ + , H 2 Ea ki ˜ i m 4 u(r)b3 (12)

˜ i = ¯ i b ω = †ω + ω where Ea EbEa and ˜ W ¯ W are the new frame fields and the modified spin connection containing the com- bined effects of strain and inhomogeneous TR breaking. In the SM, we show that the spin connection transforms exactly in agreement with the modification of the frame fields. Thus, after inclusion of strain effects, Eq. (12) takes mathematically the same form as Eq. (4) without strain. We can proceed with the projection to the low-energy space precisely as before to obtain an effective Weyl Hamiltonian of form Eq. (7), now accounting for the strain and inhomogeneous TR-breaking texture. Summary and outlook. By employing Clifford representations and Schrieffer-Wolff transformations, we carried out a controlled derivation of quantum-mechanical low-energy theory for chiral fermions in Weyl semimetals with smooth TR- and I-breaking textures. The resulting effective Weyl Hamil- tonian, describing carriers experiencing an effective curved FIG. 1. Magnetic textures and the corresponding event hori- spacetime, provides a unified approach also in the presence zons. [(a),√ (b)] u = u(sin r/ξ, 0, cos r/ξ ) with u/m = 11/10 and of strain. To illustrate the utility of the developed formalism, 2 2 mξ = 2. The horizon is a conical surface defined by x + z = we proposed a concrete prescription to realize a spatial type 2/ = − / 2 + 2, / 2 + 2, y 4. [(c), (d)] u ( u||y x y u||x x y u3 ), where I–type II interface in magnetic topological insulators. This = − 2 + 2/ξ / = / / = / ξ = u|| u0 exp ( x y ), u3 m 10 9, u0 u3 1 9, and m interface is mathematically analogous to an event horizon / . ξ 9 70. The horizon is a cylinder surface with the radius about 0 232 . of a black hole and may provide an experimental access to The values of the coordinates in both plots are given in the units exotic high-energy phenomena. The developed low-energy ξ of . theory is applicable to a wide variety of inhomogeneous Weyl semimetals and provides a powerful framework for analyzing inhomogeneous Weyl systems in the continuum limit. In the and designing these systems. presence of small strain, the spatial metric is related to the An interesting avenue for future work is the generalization strain tensor uij as gij = δij + 2uij, and the frame fields can be of our theory to time-dependent TR- and I-breaking textures ¯ i = δi − δ ij chosen as Ea a aju [35]. Employing these frame fields that give rise to nonstationary geometries and gauge fields. ¯ i ω Ea and the corresponding spin connection ¯ , the four-band Another intriguing problem concerns the connections between Dirac Hamiltonian in the presence of strain- and TR-breaking different geometric responses in Weyl semimetals. As high- field u(r) · b can be written in a Hermitian form as (we assume lighted in the present work, low-energy carriers respond to κi = ki and m(k) = m) magnetization through the change of effective geometry. This is analogous to elastic deformations in response to stress. H = 1 E¯ i γ a, k − ω¯ + mγ + u(r) · b. (11) 0 2 a i i 4 Furthermore, thermal transport coefficients are also related This prescription can be understood as a momentum- to effective geometry through gravitational response [31,56]. dependent minimal substitution [26] accounting for strain. These facts lead us to speculate on possible novel connections The strain-induced gauge fields [37,54] are encoded in the between seemingly distinct (magnetic, elastic, and thermal) frame fields and the spin connection [35,55]. Using the same response properties. unitary transformation W as previously, we can rotate u · b Acknowledgment. The authors acknowledge Aalto Center along the z direction, and the Hamiltonian after rotation for Quantum Engineering for financial support.

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