Nematic order driven by superconducting correlations
Finn Lasse Buessen, Sopheak Sorn, Ivar Martin, Arun Paramekanti
NNematic order driven by superconducting correlations
Finn Lasse Buessen, Sopheak Sorn, Ivar Martin, and Arun Paramekanti ∗ Department of Physics, University of Toronto, Toronto, Ontario M5S1A7, Canada. Materials Science Division, Argonne National Laboratory, Lemont, IL 60439, USA (Dated: January 14, 2021)Motivated by studying the interplay of nematicity and superconductivity observed in a varietyof quantum materials, we consider a two-dimensional (2D) array of nematogens, local dropletswith Z nematicity, coupled to a network of Josephson junction wires. Using finite temperatureclassical Monte Carlo simulations, we elucidate the phase diagram of this model, showing thatsuperconducting correlations can stabilize long-range nematic order, and we obtain its transportproperties within an effective resistor network model. Our results may be relevant to the 2D electrongas at the (111) KTaO interface and doped topological insulators Nb x Bi Se and Cu x Bi Se . The electron nematic [1], a liquid crystalline state ofelectrons which exhibits spontaneous breaking of lat-tice rotational symmetry, has been extensively exploredin quantum Hall systems [2–9], correlated supercon-ductors [10–20], and the bilayer ruthenate compoundSr Ru O [21–25]. In these systems, nematic orderemerges as a vestige [17, 19, 26] of underlying spin orcharge density wave orders, or due to a density imbalancebetween orbital or valley degrees of freedom. Quantumfluctuations in the nematic order can potentially act as apairing glue for electrons, resulting in a purely electronicmechanism for superconductivity (SC) [27, 28]. This ideahas been substantiated using sign-problem-free quantumMonte Carlo (MC) simulations of electrons coupled to aquantum Ising model of nematic order [29, 30].In this Letter, we consider a model of nematogens,droplets with local Z nematicity, coupled to one-dimensional (1D) Josephson junction wires (JJWs), asshown in Fig. 1. The Cooper pair hopping between ad-jacent sites along a given direction is controlled by theirlocal nematogen orientations. We show that this modelrealizes a converse scenario where superconducting cor-relations – instead of being driven by nematicity – are re-sponsible for establishing nematic order in the first place.Our work is motivated by the discovery of SC and ne-maticity in the doped topological insulators Nb x Bi Se and Cu x Bi Se [31–34] where recent experiments haveobserved Z nematic order below T n ≈ . T c ≈ .
25 K [34],and demonstrated uniaxial strain control of nematic do-mains [35]. A similar interplay of nematicity and SC hasalso been observed in a 2D electron gas (2DEG) formedat (111) KTaO oxide interfaces [36]. At a carrier den-sity n ∼ /cm , SC occurs in the 2DEG with T c ≈ n ∼ × /cm , the nematicity manifests itselfalready in the normal state, as an anisotropic resistivitybelow T n ≈ T c ≈ . FIG. 1. Lattice model of Z nematogens. Nematogens aredepicted with black/gray lobes, with orientations labeled bytheir Z states θ i = 1 , ,
3. They are coupled by three setsof Josephson junction wires (green:1, red:2, blue:3), orientedalong the three lattice directions. The intra-wire nearest-neighbor Josephson coupling ∝ J h is modulated by the ori-entation of the two adjacent nematogens; the three differentpotential nematogen lobes coupled on a bond (black/black,black/gray, gray/gray) are indicated by (solid, dashed, dot-ted) arrows. In addition, the three emanating wires at eachsite are coupled by an onsite Josephson coupling J ‘ . from spin-stripe order due to a nested hexagon-shapedFermi surface (FS) [39], with SC being stabilized on thosefaces of the hexagonal FS which remain ungapped bystripe order. However, transport below T n in the low den-sity 2DEG exhibits a remarkable non-mean-field temper-ature dependence of: the resistivity increases along onedirection [1¯10] as expected for gapping of a Fermi surface,but it decreases by a similar amount along the orthog-onal [11¯2] direction [36]. This behavior is instead remi-niscent of a resistor network made of preformed nemato-gens which act as anisotropic resistance units. In thisscenario, if the resistor network is disordered, it leads toan average isotropic resistivity. Long-range order of thenematogens, on the other hand, naturally results in anincreased resistance along one direction and a decreasedresistance in the transverse direction.We thus propose that these experiments on dopedBi Se and the KTaO a r X i v : . [ c ond - m a t . s up r- c on ] J a n in terms of preformed mesoscopic Z nematogens. Indoped Bi Se , these nematogens may be droplets of abulk nematic pairing state [31–33]. In the KTaO C symmetry, the nematogens may cor-respond to Z domains of unidirectional spin stripe ordercoexisting with SC [39]. Our conceptual work here is ag-nostic about the origin of the nematogens, and the theoryis thus likely to be broadly applicable.In Nb x Bi Se and Cu x Bi Se , there is evidence fordiamagnetism already at T n , hinting at SC fluctuationsbeing important near the nematic transition [34]. InKTaO , the experimental observation [36] that normalstate nematicity does not extend far above T c in zeromagnetic field, or exists only in a small window abovethe critical field for T < T c , suggests that local super-conducting correlations are likely to be important in es-tablishing nematic order. Consistent with this scenario,an in-plane field which is less effective at suppressing SCalso has a smaller impact on the nematic resistivity [36]. Model. —
Motivated by studying the interplay of SCand nematic order beyond Landau theory [34, 40], andin view of the aforementioned experiments, we consider amodel of Z nematogens coupled to a network of Joseph-son junction wires (JJWs), as schematically depicted inFig. 1. The Hamiltonian is given by H = − J h X i,µ cos( ϕ µi − ϕ µi + µ )∆ i,i + µ − J ‘ X i,µ<ν cos( ϕ µi − ϕ νi ) , (1)where ϕ µi denotes the superconducting phase at site i forwire µ = 1 , ,
3. The first term J h > i and i + µ , with the latterbeing the nearest neighbor of site i along wire µ , and thesecond term J ‘ is the local Josephson coupling betweenthe three wires meeting at each site. The informationabout the nematogen configuration is contained in thedirectional Josephson coupling defined as∆ − i,i + µ = g − i,i + µ + g − i + µ,i , (2)where the conductance g i,i + µ depends on the Z nemato-gen orientation θ i = 1 , , g i,i + µ = ( − η if θ i = µ η/ θ i = µ . (3)Following these definitions, for 0 < η <
1, a nematogenwith configuration θ i = µ suppresses the Josephson cou-pling along wire µ relative to the other two directions; onthe other hand, for − < η <
0, the θ i = µ configurationenhances the Josephson coupling along wire µ . Depend-ing on the orientations of the two adjacent nematogens,the nearest-neighbor Josephson coupling ∆ i,i + µ can takeon three values: ∆ bb , ∆ bg , or ∆ gg , where the subscriptsdenote the colors of the lobes (b=black, g=gray) pointingtowards each other as illustrated in Fig. 1. They assumeexplicit values ∆ bb = (2 + η ) /
4, ∆ gg = (1 − η ) /
2, and∆ bg = ( η + 2)(1 − η ) / (4 − η ), respectively. t e m p e r a t u r e T / J h J (cid:30) /J h Josephson coupling
Low density High density
Isotropic normalIsotropic superconductor
Nematicnormal
KTaO (111) 2DEGDoped Bi Se Nematicsuperconductor
FIG. 2. Schematic phase diagram for the model Hamilto-nian as a function of J ‘ /J h and temperature T /J h at fixed η . The high-temperature isotropic normal state is connectedto the low-temperature nematic superconductor either via anintermediate nematic normal phase or an isotropic supercon-ductor. Dashed lines indicate cuts relevant to low-density andhigh-density KTaO (111) 2DEG, and to doped Bi Se . In the limit J ‘ = 0, the different JJWs are decoupledfrom each other, and we can focus on an individual JJWwith periodic boundary conditions which couple chainsof nematogens. For η >
0, we find ∆ bb > ∆ bg > ∆ gg . Min-imizing the Josephson coupling energy on a single JJW(say, green:1) only requires that the nematogen at eachsite is constrained to be θ i = 1, yielding ∆ i,i +1 = ∆ bb for every pair of nearest neighbors. However, if we con-sider adjacent parallel wires, it is easy to check fromFig. 1 that the lowest energy for a system with peri-odic boundary conditions is achieved only when all thenematogens in both wires are globally aligned. Conse-quently, the ground states of the full 2D model will alsoexhibit nematic order. In a similar fashion, one can es-tablish ground state nematic order for η <
0. We em-phasize that, since the nematic order is a discrete order,it can remain stable at finite temperature in the ther-modynamic limit. However, since the individual JJWsremain decoupled 1D XY-type wires, there is no SC forany
T >
0; we thus expect a nematic ordering transitionat T n ∝ J h and a superconducting transition temperature T c = 0. Next, when we switch on weak onsite Josephsoncoupling 0 < J ‘ (cid:28) J h , global 2D SC is established with T c < T n , leading to a window of normal state nematicorder at intermediate temperatures. On the other hand,as J ‘ → ∞ , the phases ϕ µi on different wires get locked ateach site, leading to a single triangular lattice JJ array,for which T c may be larger than T n . These considera-tions lead to the schematic phase diagram for the modelHamiltonian Eq. (1) as a function of J ‘ /J h and tempera-ture (for fixed η ) shown in Fig. 2. Below, we confirm thephase diagram using classical MC simulations. Monte Carlo study. —
We have carried out classi-cal finite-temperature MC simulations of the model de-fined in Eq. (1). We studied finite systems of L × L unitcells with periodic boundary conditions and system sizes (a)(d) (b)(e) (c)(f) .
16 0 .
18 0 .
20 0 .
22 0 . temperature T/J h . . . ph a s e c o rr e l a t i o n S µ L J (cid:28) /J h = 0 . L = 24 L = 48 L = 72 L = 960 .
05 0 .
10 0 .
15 0 .
20 0 . temperature T/J h . . . n e m a t i c i t y N J (cid:28) /J h = 0 . L = 24 L = 48 L = 72 L = 960 .
05 0 .
10 0 .
15 0 .
20 0 . temperature T/J h s p e c i fi c h e a t C J (cid:28) /J h = 0 . L = 24 L = 48 L = 72 L = 96 0 .
06 0 .
08 0 .
10 0 .
12 0 . temperature T/J h . . . ph a s e c o rr e l a t i o n S µ L J (cid:28) /J h = 0 . L = 24 L = 48 L = 72 L = 960 .
05 0 .
10 0 .
15 0 .
20 0 . temperature T/J h . . . n e m a t i c i t y N J (cid:28) /J h = 0 . L = 24 L = 48 L = 72 L = 960 .
05 0 .
10 0 .
15 0 .
20 0 . temperature T/J h s p e c i fi c h e a t C J (cid:28) /J h = 0 . L = 24 L = 48 L = 72 L = 96 FIG. 3. Monte Carlo simulation results for the model Hamiltonian in Eq. (1), for fixed η = 2 / J ‘ /J h = 0 .
005 and (d-f) J ‘ /J h = 0 .
05, respectively. (a) Specific heat per site showing a sharp peak at the nematic transition at T n ≈ . J h (dashed line),and a weak bump near the BKT superconducting transition T c ≈ . J h (dotted line). (b) The nematic order parameter N nearthe nematic transition point. (c) The scaled superconducting phase correlator S µL shows a system-size independent crossingpoint at the superconducting BKT transition. Panels (d-f): Similar results, but for J ‘ /J h = 0 .
05, showing a nearly unchanged T n but a significantly higher T c ≈ . J h > T n . up to L = 96. We equilibrated the system for 5 × MC sweeps, where a single sweep is defined as one at-tempted update per degree of freedom on average, beforetaking measurements for up to 5 × sweeps. For animproved sampling of the configuration space – in partic-ular when resolving the Berezinskii Kosterlitz Thouless(BKT) transition [41, 42] in a nematic background – weimplemented a parallel tempering scheme across 196 tem-perature points in an optimized temperature ensemble inthe range 0 . < T /J h < .
25 [43, 44].We explore the phase diagram as a function of J ‘ /J h and temperature, keeping η = 2 / N =(1 /L ) h (cid:12)(cid:12) P i e i πθ i / (cid:12)(cid:12) i , where θ i = 1 , , S µ = (1 /L ) h (cid:12)(cid:12) P i e iϕ µi (cid:12)(cid:12) i . Thelatter quantity is useful for the following reason. When J ‘ >
0, we expect the superconducting transition to bea BKT transition, which implies a universal r − / powerlaw decay of phase correlations at the critical point. Thispower law manifests in the finite-size dependence of thephase correlations, S µ ∼ L − / . As a result, the scaledsuperconducting phase correlations S µL = L / S µ are ex-pected to be system-size independent at the BKT criticalpoint, and the S µL curves for different L should cross at the BKT transition temperature [45].The results of our MC analysis are summarized inFig. 3. Panels 3a and 3d show the specific heat for J ‘ /J h = 0 .
005 and J ‘ /J h = 0 .
05, respectively. In bothcases, we observe a sharp peak at T n ≈ . J h which weassociate with the onset of nematicity. This is supportedby the nematic order parameter becoming finite at thesame temperature scale (Figs. 3b and 3e). The BKTtransition is detected as a crossing point of the curves of S µL for different system sizes, as shown in Figs. 3c and 3f.We point out that at a BKT transition, it is well knownthat there is an undetectable essential singularity in thespecific heat; however, a rough indication of its locationis given by a weak bump in the specific heat associatedwith the quenching of entropy tied to phase fluctuations.We distinguish two qualitatively different cases. For J ‘ /J h = 0 . T c ≈ . J h lies belowthe nematic transition temperature T n . For J ‘ /J h = 0 . T c ≈ . J h , so that T c > T n . The ex-istence of these two different sequences of phase transi-tions, with an intermediate phase which is either nematicwith finite resistance or isotropic and superconducting,confirms the schematic phase diagram in Fig. 2. Transport. —
In the normal state, far above T c , super-conducting correlations are short ranged, and the inter-site Josephson links act as normal resistances ∝ ∆ − i,i ± µ .We can thus approximately compute transport proper-ties in the normal state by translating nematogen config- .
05 0 .
15 0 . temperature T /J h . . . . . . . . . r e s i s t i v i t y ρ / ρ η = 2 / ρ xx ρ yy .
05 0 .
15 0 . temperature T /J h . . . . . . . . . r e s i s t i v i t y ρ / ρ η = − ρ xx ρ yy (a) (b) FIG. 4. Resistivity tensor eigenvalues for classical resistancenetwork model with (a) η = 2 / x ) and(b) η = − x ). The network is obtainedby using conductances as given in Eq. (2), with nematogenconfigurations drawn from the MC simulations of the modelin Eq. (1) with J ‘ = 0 . L × L system with L = 20. Weaverage the resistivity over 100 configurations and normalizeit by the high temperature isotropic value ρ . urations into configurations of a corresponding resistornetwork [40, 46, 47]. We average the network resistivityover 100 nematogen configurations drawn from our MCsimulations at each temperature. Fig. 4 shows the eigen-values of the resistivity tensor, which correspond to ρ xx and ρ yy if we choose the unique hard (easy) axis in thenematic phase to be along the x -direction, correspond-ing to the parameter choice η > η < T n , the resistivity increases along one directionand decreases along the other direction, consistent witha symmetry analysis [40]. This anisotropic behavior ofthe resistivity in the nematic normal state is in qualita-tive agreement with the experiments on the KTaO (111)2DEG. For nonzero J ‘ (cid:28) J h , the linear resistivity willvanish at the 2D BKT transition temperature T c < T n ,and thus occurring below the displayed onset of resistiveanisotropy. The resistor network calculation does notaccount for the growth of superconducting correlationswhich will eventually lead to vanishing resistivity alongboth directions at T c , but we present a phenomenologi-cal account of this in the Supplemental Material [40]. Inthe opposite regime, when T c > T n , SC develops beforethe onset of nematicity. The superconductor should thenexhibit an anisotropic critical current. Impact of a magnetic field. —
A perpendicular mag-netic field will suppress the SC gap on individual grains,leading to a decrease in the Josephson couplings J h , J ‘ .Within Landau theory, we expect J h ( B ) and J ‘ ( B ) todecrease ∝ (1 − B/B c ), where B c is the bulk upper criti-cal field. Since T n ∝ J h , with J h → T n ( B ) ∝ (1 − B/B c ). Here B c ∼ Φ /ξ , with Φ and ξ being the superconducting flux quantum and co-herence length respectively. Experiments on the (111)KTaO B c ∼ ξ ∼
10 nm [36].However, the perpendicular magnetic field may alsolead to Josephson frustration, if we recognize that ourmodel of Josephson coupling between superconductinggrains being mediated by 1D wires reflects a convenientidealization of the real system. In reality, the Joseph-son coupling between grains of size ξ g (cid:29) ξ will oc-cur via the entire inter-grain region. When the field isstrong enough to insert a vortex in this region, it caneffectively suppress the Josephson coupling between twoadjacent grains. This interference effect, which leads tothe familiar Fraunhofer-like pattern in Josephson junc-tions [48, 49], would manifest itself at a much smallerfield scale B g ∼ Φ /ξ g set by the grain size. Assuming ξ g ∼ ξ yields B g ∼
10 mT. The published data on lowerdensity (111) KTaO Impact of disorder and strain. —
In our model, it isclear that short-range superconducting correlations aresufficient to establish 2D nematic order. Disorder whichthus limits the range of superconducting correlations, sayby cutting the 1D JJWs to remove the Josephson cou-pling on a small fraction of bonds, may suppress T c but isnaively not expected to significantly impact the nematicorder. However, an Imry-Ma argument [17, 19, 50] sug-gests that local random fields arising from impurities caneventually kill long-range nematic order on sufficientlylong length scales. Experiments on the (111) KTaO Discussion. —
We have proposed a model of meso-scopic nematogens which are coupled to each other viaJJWs. SC correlations in the JJWs have been shownto drive nematogen ordering and a spontaneous breakingof lattice rotational symmetry. Our theory is a bosonicanalogue of the Kugel-Khomskii model [51, 52] which de-scribes spin and orbital order of electrons in solids. In thisanalogy, SC and Z nematicity act respectively as ‘spin’and ‘orbital’ degrees of freedom at a site, while inter-siteand local Josephson couplings on the JJWs play the roleof orbital-dependent exchange and Hund’s coupling re-spectively. Our results explain various observations onthe (111) 2DEG in KTaO , and may also be relevant toultrathin films of Nb x Bi Se and Cu x Bi Se . An equi-librium manifestation of nematicity in the superconduc-tor would be a spontaneous ellipticity in the shape ofsuperconducting vortices, which could be probed usinga scanning superconducting quantum interference device(SQUID) [53]. Such vortices may themselves exhibit un-conventional crystal orders [54]. Formulating and study-ing a quantum version of our model, and variants whichsupport time-reversal breaking SC [55, 56], are interest-ing future research directions. Finally, we note that ifthe JJWs in our model represent dislocation lines withenhanced 1D pairing [57], and the nematogens representthe local orientation of dislocation lines, a similar modelbut with randomness may be relevant for recent experi-ments on plastically deformed SrTiO crystals [58].We thank Anand Bhattacharya, Changjiang Liu, MikeNorman, Peter Littlewood, Pablo Villar Arribi, GauravChaudhary, and Daniel Podolsky for fruitful discussions.This work was supported by NSERC of Canada. The nu-merical simulations were performed on the Cedar cluster,hosted by WestGrid and Compute Canada. ∗ [email protected][1] E. Fradkin, S. A. Kivelson, M. J. Lawler, J. P. Eisen-stein, and A. P. 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Nematic order driven by superconducting correlations
Finn Lasse Buessen, Sopheak Sorn, Ivar Martin, and Arun Paramekanti Department of Physics, University of Toronto,Toronto, Ontario M5S 1A7, Canada Materials Science Division, Argonne National Laboratory, Lemont, IL 60439, USA (Dated: January 14, 2021) S1 a r X i v : . [ c ond - m a t . s up r- c on ] J a n
1. LANDAU THEORY
In the limit when J ‘ >
0, and all the wires get coupled into a single 2D superconductingstate, we can define complex order parameters: Φ for SC, and Ψ for the Z nematic. Theindependent order parameters have Landau free energies S Φ = Z d r h r s | Φ | + u s | Φ | + κ s | ~ ∇ Φ | + . . . i (S1) S Ψ = Z d r h r n | Ψ | + w n (Ψ n + Ψ ∗ n ) + u n | Ψ | + κ n | ~ ∇ Ψ | + . . . i (S2)where the cubic term is permitted by C rotation symmetry under which Ψ → e i π/ Ψ.This reduces the nematic theory to a 3-state clock/Potts model. The coupling between thenematic order and the SC takes the form S Φ , Ψ = Z d r (cid:2) u sn | Φ | | Ψ | + κ sn { Ψ Φ ∗ ∂ Φ + c . c . } + . . . (cid:3) (S3)where “c.c” refers to complex conjugate and ∂ ± ≡ ∂ x ± i∂ y . The gradient coupling is chosento be invariant under a C rotation which leads to Ψ → Ψ e i π/ and ∂ ± → ∂ ± e ∓ i π/ . Wehave omitted additional gradient terms, and higher order terms in this action. The gradientcoupling κ sn causes the superconducting stiffness to become anisotropic in the presence ofnematic order h Ψ i 6 = 0. At the same time, integrating out the fluctuating superconductingorder parameter Φ from the gradient terms can renormalize the nematic mass r n and stiffness κ n , thus helping to stabilize nematic order.As an illustration, we compute the mass renormalization to Gaussian order by dropping u s and keeping r s > u sn in Eq. S3, and integrating out Φ leads to˜ r n = r n − κ sn Z Λ0 d q q ( r s + κ s q ) (S4)As r s decreases, the superconducting correlation length grows, and the renormalized nematicmass can change sign, ˜ r n <
0, thus favoring nematic order, even when the bare r n > S2. RESISTIVITY CALCULATIONSA. Resistor network model
We consider a triangular mesh of sites { i } with a resistor on each nearest-neighbour bond[1] whose resistance is proportional to ∆ − i,i + µ from Monte-Carlo simulations. To obtain theS2ffective resistivity [1, 2] for a given configuration { ∆ i,i + µ } , we apply a potential differencebetween the two edges, as illustrated in Fig. S1 with a periodic boundary condition alongy, and solve equations from Kirchhoff’s laws to obtain current I i,i + µ . The current density j x and j y can be computed and then used to determine the conductivity σ xx and σ yx . σ yy and σ xy can be obtained similarly by instead having periodic and open boundaries along x-and y-direction respectively. The conductivity tensor can then be inverted and diagonalizedto arrive at the principal eigenvalues of the resistivity tensor. The results, averaged overMonte-Carlo configurations of { ∆ i,i + µ } , are reported in the main text. R i , i +m i i +m I i , i +m xy FIG. S1. Triangular lattice of resistors residing on each bond whose resistance R i,i + µ is proportionalto ∆ − i,i + µ . Sites on the left edge are held at the same potential, and so are those on the right. Apotential difference between the edges induces current I i,i + µ which is a vector quantity. The y-direction is periodic, and the dashed lines represent wires with zero resistance. B. Resistivity tensor and nematic order parameter
Let us write the nematic order parameter as Ψ = | Ψ | e iθ . In terms of this, the resistivitytensor due to nematicity is expected to take the following form on symmetry grounds:∆ ρ = | Ψ | cos θ sin θ sin θ − cos θ (S5)so that a C rotation which sends θ → θ + 4 π/ ρ → R T ∆ ρR . The tensor ∆ ρ has eigenvalues ±| Ψ | . This eigenvalue splitting reveals itselfin Fig. 4 of the manuscript as we go below the nematic transition. For θ = 0, this leadsS3 c T / J h R e s i s t i v i t y / (a) = 2/3, A = 0.1 xxyy T c T / J h (b) = 2/3, A = 1.0 xxyy FIG. S2. Illustration of the impact of growing superconducting correlations on the resistivity. to ∆ ρ xx = − ∆ ρ yy and ∆ ρ xy = ∆ ρ yx = 0, while for θ = 2 π/ , π/
3, there is a symmetricoff-diagonal component to the resistivity tensor.
C. Phenomenological theory of resistivity with superconducting correlations
The classical resistor network model is a valid approach to compute the resistance of theJosephson junction array when superconducting correlations are very short-ranged. How-ever, as we approach the superconducting transition T c , these superconducting correlationsgrow and must be taken into account. A phenomenological route to incorporating thesecorrelations is to view patches of linear dimension ξ ( T ), where ξ ( T ) is the temperature de-pendent correlation length, as zero resistance ‘short’ regions. We thus expect the resultingnetwork to have a renormalized resistivity˜ ρ αβ ( T ) = ρ αβ ( T ) /ξ ( T ) , (S6)where α, β = x, y and ρ ( T ) is shown in Fig. 4 of the manuscript for J ‘ /J h = 0 . T c , where we haveused ρ ( T ) obtained from the resistor network model, using nematogen configurations fromthe Monte Carlo simulations, and the following ansatz for the BKT correlation length, ξ ( T ) = exp ( A p T /T c − ) . (S7)Here, T c = 0 . J h as obtained from our Monte Carlo simulations shown in the manuscriptfor J ‘ /J h = 0 . A = 0 . , . ρ ( T ). These plots are in qualitative agreement with experimental data on the (111) KTaO [1] S. Kirkpatrick, Percolation and conduction, Rev. Mod. Phys. , 574 (1973).[2] M. Parish and P. Littlewood, Non-saturating magnetoresistance in heavily disordered semicon-ductors, Nature , 162 (2003)., 162 (2003).