A Holographic Weyl Semimetal in an Effective Gauss–Bonnet Geometry

Introduction

In a three-dimensional Weyl semimetal, two bands touch at isolated momenta called Weyl nodes. Each node carries a chirality that protects the crossing against small perturbations. Nodes of opposite chirality can move together and annihilate, opening a gap. In a system with broken time-reversal symmetry, their separation in momentum space can also give rise to an anomalous Hall current in the absence of an external magnetic field [1]. This raises a question when interactions are strong: does a perturbation that extends the conducting phase also strengthen its electrical response?

Interactions can affect the transition and the current through different mechanisms. Short-range interactions can drive a first-order transition to a band insulator or a continuous transition into a symmetry-broken phase [2]. Other field-theory studies find interaction corrections to anomalous electromagnetic response [3] and changes in both the Weyl nodes and the Hall response in an interacting Weyl–Hubbard model [4].

To identify the two scales that compete in our model, consider first a free Dirac fermion of mass $m$ in a spatial axial source $A^{5}_{z}=b/2$ [5]. Its two positive-energy branches obey

The lower branch has two nodes at $k_{z}=\pm\sqrt{b^{2}/4-m^{2}}$ for $\lvert m/b\rvert<1/2$. Increasing the mass brings the nodes together until a gap opens. The ratio $m/b$ therefore gives a natural parameter for tracking the transition even when a quasiparticle dispersion is unavailable.

In holography one can locate the transition at strong coupling by comparing free energies and compute the current in the selected phase. Bottom-up constructions realize Weyl-semimetal transitions and anomalous transport using axial gauge fields and Chern–Simons interactions [6, 7, 8]. Some holographic models also exhibit a first-order transition into a Chern-insulating phase [9], distinct from the trivial flavour insulator studied here. The axial Hall effect and its anomaly relation have been investigated in related holographic constructions [10]; this axial-current response should not be confused with our vector-current Hall conductivity. Our starting point is the probe-flavour construction of Ref. [11], based on D7-branes in the D3-brane geometry. Its boundary theory is strongly coupled $\mathcal{N}=4$ super-Yang–Mills theory with $N_{f}\ll N_{c}$ fundamental $\mathcal{N}=2$ hypermultiplets [12, 13]. A spatially varying mass phase, supplies the axial source. The D7-branes either reach the horizon or close smoothly above it, giving black-hole (BH) and Minkowski (MN) embeddings, respectively. These topologically distinct probe-brane embeddings and the transitions between them have a well-established history [14]. A first-order transition separates the conducting semimetal and insulating flavour phases.

The Einstein D3/D7 model has several distinctive transport properties. At zero temperature, the Hall conductivity of the semimetal phase is independent of the flavour mass and takes its anomaly-normalized value. At finite temperature, both the longitudinal and Hall conductivities depend on the embedding. In particular, the longitudinal DC conductivity develops a maximum as a function of mass at sufficiently low temperature, while becoming monotonic at higher temperature [11]. A subsequent calculation of finite-frequency conductivities and quasinormal modes identified weakly damped current excitations associated with optical structures near the transition [15]. Optical conductivity has also been studied in bottom-up holographic Weyl semimetals [16]. Those frequency-dependent structures should be distinguished from a maximum of the DC conductivity as the mass is varied.

Deformations of the same flavour-brane construction provide further tests of these properties. The D-instanton extension changes the dilaton and probe dynamics, shifts the phase boundary and suppresses the low-temperature longitudinal DC maximum [17]. We show the same suppression of this maximum by considering higher-curvature corrections. Its interpretation requires both the conductivity formula and the thermodynamic selection of the embedding: an extremum on a stationary branch need not be reached by the preferred phase.

Here we examine a deformation of the thermal geometry through a five-dimensional Gauss–Bonnet (GB) interaction. GB gravity supplies an explicit planar black-brane background and second-order metric equations, making it a useful effective model for investigating geometric corrections. Its thermal response and consistency conditions have been widely studied [18, 19, 20, 21, 22]. Applications to heavy-quark observables and flavour conductivity provide additional motivation for studying the probe sector [23, 24, 25, 26]. Gauss–Bonnet–Maxwell models have also been used to study DC conductivity and charge diffusion [27]; their charged bulk gauge sector differs from the neutral D7 probe flavour sector considered here. This deformation is an effective prescription, not the specific finite-’t Hooft-coupling correction to type IIB theory, whose leading known term involves $R^4$ interactions [28]. A separate Einstein–Gauss–Bonnet holographic construction addresses torsional dislocations [29], not the flavour-brane transition considered here.

We vary the thermal geometry while retaining the D7 action and flavour dictionary of the Einstein model. The resulting construction is a defined effective probe prescription, not a ten-dimensional GB uplift. In particular, the internal $S^5$, RR coupling and relative DBI/WZ normalization are held fixed; changing these choices could alter the response. We therefore focus on what this specific geometric deformation does beyond a shift of the effective thermal scale.

Our central result is the phase diagram in Figure 5. At the sampled finite temperatures, positive GB coupling extends the preferred semimetal region toward larger mass. An on-shell variation predicts the leading shift and separates the thermal-rescaling contribution from an opposing intrinsic fixed-horizon contribution. At $T/b = 0.10$ both conductivity jumps decrease with increasing coupling, whereas at $T/b = 0.07$ the longitudinal jump increases; a finer scan would be needed to determine the temperature at which the response changes sign. Matched-horizon tests show that much of the response at $T/b = 0.10$ follows from thermal rescaling, with smaller residual effects. We also distinguish the kinematic geometric criterion for an interior longitudinal DC maximum from its realization on thermodynamically preferred embeddings.

Section § 2 specifies the effective model, temperature and mass conventions. § 3 compares the competing embeddings and separates the thermal and intrinsic geometric contributions to the coexistence displacement. § 4 treats the DC conductivities. We conclude in § 5, with numerical checks in Appendices § 7 and § 8.

Holographic model with Gauss–Bonnet corrections

The Einstein model starts with $N_{\mathrm{c}}$ coincident D3-branes extending along the four field-theory directions $(t,x,y,z)$ and $N_{\mathrm{f}}$ D7-branes introduced as probes, with $N_{\mathrm{f}} \ll N_{\mathrm{c}}$ [13, 11]. In the near-horizon limit, the D3-branes generate the $\mathrm{AdS}_5 \times S^5$ geometry. Strings connecting D3- and D7-branes supply the fundamental flavour fields, while the D7 embedding encodes their mass and response. The D7-branes additionally wrap four transverse directions and are embedded in the remaining $(X^8,X^9)$ plane.

These four directions $(X^4,\ldots,X^7)$ have a radial coordinate $r_4$ and the angles of a three-sphere. The remaining plane $(X^8,X^9)$ is described by a radius $r_2$ and an azimuthal angle $\phi$. To define the polar angle on the five-sphere, write

where $0 \le\theta\le\pi/2$ and $\phi$ is periodic with period $2\pi$. The corresponding unit-sphere metric is

Thus $\theta$ determines the radius of the wrapped $S^3$, whereas $\phi$ specifies the direction of the brane in the transverse two-plane [11, 15].

Our bulk radial coordinate is $u$, which increases from the AdS boundary into the interior. In curvature-radius units, the zero-temperature Einstein metric takes the form

Restoring the common Einstein curvature radius gives $u=L_{\mathrm{eff}}^{2}/r_{6}$; Eq. (4) uses $L_{\mathrm{eff}}=1$, so $u=1/r_{6}$ there. Hence $u=0$ is the UV boundary, while larger $u$ probes the infrared. At finite temperature the coordinate $u$ is defined by the metric in Eq. (8), with a horizon at $u=u_{h}$; it should not be identified with the inverse of an isotropic finite-temperature Cartesian radius without the appropriate coordinate transformation.

The D7 wraps the $S^{3}$ and extends along $(t,x,y,z,u)$, so a convenient worldvolume chart and the equilibrium embedding are

Here $\Omega_{3}$ denotes three independent angles. The fields $\theta$ and $\phi$ are embedding functions, not additional independent worldvolume coordinates. The UV coefficient $M$ in $\theta\sim Mu$ specifies the mass source; see Eq. (17). The wrapped $S^{3}$ collapses as $\theta\to\pi/2$, allowing smooth capped solutions; a horizon-reaching brane instead gives a BH embedding. We take $b>0$ and $M\geq0$.

The winding angle supplies the axial source independently of the five-dimensional thermal metric. We therefore keep the internal geometry fixed and deform the black brane by a GB interaction. This specifies an effective probe model. No corrected ten-dimensional type-IIB solution is assumed: the internal $S^{5}$ radius, the RR five-form contribution in the probe action, and the flavour mass and current dictionaries are held fixed by prescription. A genuine finite-’t Hooft-coupling completion would generally change additional bulk and worldvolume terms [28]. To compare thermal states across couplings, we first fix the normalization of boundary time.

Let $L_{0}$ and $L_{\mathrm{eff}}$ denote the bare curvature radius and the asymptotic AdS radius, respectively. The gravitational action is

We use the branch continuously connected to Einstein gravity and define

In units where $L_{\mathrm{eff}}=1$, the black-brane metric is

The Einstein limit is $f(u)=1-u^{4}/u_{h}^{4}$. Since $f(0)=1$, the boundary speed of light is unity, allowing masses, temperatures and response functions to be compared in common boundary units.

Near the horizon, $f(u)=4n(u_{h}-u)/u_{h}+\mathcal{O}((u_{h}-u)^{2})$. Regularity of the Euclidean geometry therefore gives

We impose this relation separately at each coupling when comparing states at fixed physical temperature. In particular, positive $\lambda_{\mathrm{GB}}$ decreases $n$ and hence moves the horizon to smaller $u_{h}$ at fixed $T$. Comparisons at fixed $u_{h}$ instead involve different physical temperatures. The equivalent bare-radius convention is discussed in Appendix § 7.

This normalization also helps interpret the GB dependence: an Einstein background at $T_{\mathrm{eff}} = T/n$ has the same horizon position as the GB background at temperature $T$. Consequently, part of the coupling dependence can reflect a change of the thermal scale. The blackening functions nevertheless remain different. At fixed $u_h$ and $v = (u/u_h)^4$, their difference begins at

Thus matching the horizon position does not make the two embedding problems exactly equivalent.

For reference, the five-dimensional Gauss–Bonnet causality and energy-flux bounds are $-7/36 \leq\lambda_{\mathrm{GB}} \leq9/100$ [21, 22]. All sampled couplings lie within this interval, with $\lambda_{\mathrm{GB}} = 0.09$ at its upper endpoint. These bounds do not by themselves establish a UV completion of the combined gravity–probe construction.

With the thermal metric fixed, the D7 embedding and its current response follow from the DBI and Wess–Zumino actions in the induced geometry,

The equilibrium embedding is given in Eq. (5), with vanishing worldvolume field strength $F = 0$. The profile $\phi= bz$ gives the complex flavour-mass source a spatially varying phase. In the convention of Ref. [11], its axial source is $A^5_z = b/2$, selecting the $z$ direction and distinguishing it from the transverse plane. The equilibrium gauge field vanishes, so the WZ term contributes no background free energy. In the orientation and RR-potential convention of Ref. [11], the part of the four-form pullback that enters the Hall response is

where $\omega(S^3)$ is the unit-volume form of the wrapped sphere. We retain this internal potential when the thermal metric is deformed. Its contribution to the current appears when a transverse electric field is applied. Thus the free-energy crossing and Hall response involve different terms of the probe action, evaluated on the same equilibrium embedding.

After integrating over the wrapped $S^3$ and removing the common normalization, the Euclidean radial density is

The factor $\cos^3\theta$ measures the size of the wrapped three-sphere. The two square roots encode the axial-phase gradient along $z$ and the radial bending of the brane, respectively. Their competition permits horizon-reaching and smoothly capped configurations. Changing $f$ alters the cost of radial bending and the horizon position while keeping the axial source fixed.

We write the flavour free-energy density as $\mathcal{F} = F/\mathcal{N}$; all free energies and responses below are densities per unit boundary volume. The retained Einstein probe dictionary sets $\mathcal{N} = \lambda_{\mathrm{tH}}N_fN_c/(16\pi^4)$ and $K = \mathcal{N}(2\pi\alpha')^2 = N_fN_c/(4\pi^2)$ in the chosen units. We hold $\lambda_{\mathrm{tH}}$ and the relative DBI and WZ normalizations fixed as $\lambda_{\mathrm{GB}}$ varies. A common positive multiplier of the entire probe action would affect absolute current-response normalizations but not the embedding equations or free-energy crossing. More generally, multiplying the entire probe action by an arbitrary positive coupling-dependent factor $A(\lambda_{\mathrm{GB}})$ leaves the stationary embeddings and the equal-free-energy condition unchanged: $\Delta\mathcal{F} \to A(\lambda_{\mathrm{GB}})\Delta\mathcal{F}$. Such a rescaling changes the absolute mass response and current normalization, but not the coexistence mass. Common finite boundary counterterms also cancel between the two branches at identical sources. These invariances do not cover changes of the internal geometry or of the relative DBI and WZ couplings, which would define different effective probe models.

To compare the phases at a common flavour mass, we need the relation between each regular embedding and its boundary source. Varying the density in Eq. (13) gives the equation for $\theta(u)$.

Its two metric factors are

Variation of Eq. (13) gives

Here primes denote derivatives with respect to $u$. The shrinking three-sphere and the axial winding contribute the two terms on the right-hand side. The GB deformation changes the radial bending cost through $f(u)$ and changes the horizon position through Eq. (9).

The relation between a regular embedding and its field-theory sources is determined near the boundary. Since

the leading embedding equation is $u^2\theta''-3u\theta'+3\theta=0$, with solutions proportional to $u$ and $u^3$. Including the axial contribution gives

Substitution into the nonlinear equation at order $u^3$ gives $2a_{\log}-b^2M=0$, where $a_{\log}$ multiplies $u^3\log(u\Lambda_{\mathrm{UV}})$. The cubic terms present at $b=0$ cancel at this order. Since $f-1=O(u^4)$, its first contribution to the equation multiplied by $u^2$ is of order $u^5$; it cannot modify the displayed powers or logarithmic coefficient. The GB coupling nevertheless changes the relation between $M$ and $c_3$ through the globally regular solution.

We retain the physical mass notation and flavour dictionary of Ref. [11],

Thus $M$ is the UV embedding coefficient, while $m$ is the physical flavour mass. We use $c_3$ for the subleading coefficient of $\theta(u)$ to distinguish it from the coefficient $C$ in the Cartesian embedding expansion of Ref. [11]. Neither coefficient by itself is the renormalized mass response.

Gauss–Bonnet phase structure

The bulk equation and UV dictionary are specified in Section 2. We now construct regular black-hole (BH) and Minkowski (MN) embeddings, compare their free energies at identical sources, and determine how the resulting first-order boundary responds to the GB deformation. These steps must be distinguished: a stationary embedding is not necessarily the thermodynamically preferred phase.

The regular embeddings differ in their infrared boundary conditions. A BH embedding reaches $u_h$ with $0\leq\theta_h<\pi/2$, so the wrapped three-sphere has nonzero radius at the horizon. Regularity fixes

where

The induced horizon permits absorption of flavour-current fluctuations and supplies the dissipative channel relevant to the DC response.

An MN embedding instead reaches $\theta=\pi/2$ at $u_0<u_h$. The wrapped three-sphere shrinks smoothly, and the brane closes before reaching the horizon. A regular local coordinate is $\rho=\cos\theta$, in which

The divergence of $\theta'(u)$ at the cap is therefore a coordinate effect. In the Einstein construction, capped embeddings describe a discrete, gapped flavour spectrum [30, 11, 15]; the retained probe action also gives their vanishing DC response at zero density.

Figure 1 illustrates these two ways of completing the brane in the interior. Within each family, changing the regular infrared datum generates a different UV mass and response. Figure 2 shows how this relation changes with the GB coupling.

For these representative profiles, positive coupling lowers the embedding angle relative to the Einstein solution at fixed $u/u_h$, while negative coupling raises it. This ordering describes a change in the map from infrared data to the UV source. Since the comparison fixes $\theta_h$ or $u_0/u_h$, it does not establish which embedding is preferred at a common physical mass. That question requires the free-energy comparison below.

Free-energy competition and the first-order transition

The embedding equation determines stationary configurations. To select the equilibrium phase, we compare their renormalized free-energy densities at identical sources. We follow probe-brane holographic renormalization [31], with the axial-source counterterms fixed as in the Einstein D3/D7 construction [11]. Denoting the physical flavour free-energy density by $F$, we retain the normalized quantity $\mathcal{F} = F/\mathcal{N}$ introduced above:

where $u_{\mathrm{end}} = u_h$ for BH embeddings and $u_{\mathrm{end}} = u_0$ for MN embeddings. Both families require the same subtraction because their UV divergences are determined by the same sources. We use

Using Eq. (17), the on-shell radial density near the boundary is $u^{-5} - M^{2}u^{-3} + b^{2}M^{2}u^{-1} + \cdots$. In particular, $b^{2}u^{2}\sin^{2}\theta= b^{2}M^{2}u^{4} + \cdots$, so its square is integrable and produces no separate $b^4 \log\epsilon$ divergence. The logarithm from expanding $\theta_\epsilon^2/(2\epsilon^4)$ and the explicit logarithm in Eq. (23) together cancel the logarithm of the regulated integral. Higher positive powers of $\epsilon$ times logarithms vanish in the cutoff limit. This prescription gives $\mathcal{F}(M=0)=-1/(4u_h^4)$ at finite temperature. A common finite counterterm depending only on the sources shifts both branches equally and leaves their crossing unchanged. Further details of the subtraction are given in Appendix § 7.

Following Ref. [11], we express the physical mass response as $\langle\mathcal{O}_m\rangle=\partial\mathcal{F}/\partial m$. In the present coordinates and renormalization scheme,

The same reference scale $\Lambda_{\mathrm{UV}}$ is used in Eq. (17); our numerical units set $\Lambda_{\mathrm{UV}}=1$. The expression at $M=0$ is understood by continuity. For compactness in the analytical relations used later, we also define the auxiliary normalized response

Since this operator is conjugate to an explicitly applied mass, its discontinuity diagnoses the transition without implying spontaneous symmetry breaking.

Define the differences between the two regular branches by

The coexistence mass is determined by

A negative $\Delta\mathcal{F}$ favours the BH branch, whereas a positive $\Delta\mathcal{F}$ favours the MN branch among the compared homogeneous solutions. Near a simple crossing,

For the positive $\Delta Q_t$ found here, the BH phase is preferred below $M_t$ and the MN phase above it. The equilibrium free energy is continuous at the crossing, but its mass derivative is discontinuous.

The swallow tails in Figure 3 display several stationary embeddings at the same mass. Equilibrium follows the branch with the lowest free energy and switches between two distinct embeddings at coexistence. The turning points of the families and the limiting topology-changing embedding are therefore different from the thermodynamic transition: neither determines $M_t$ without the free-energy comparison.

The mass-response curves provide the complementary signature of this same crossing. At coexistence the source-dependent terms in Eq. (24) agree on the two branches, giving

Although individual expectation values depend on the finite renormalization scheme, their difference at common sources is unchanged by a common finite counterterm. The unequal responses therefore give the discontinuity in the slope of the equilibrium free energy.

Gauss–Bonnet phase boundary

Having selected the lower-free-energy embedding at each common set of physical sources, we obtain the phase boundaries in Figure 5. Positive GB coupling shifts the transition toward larger flavour mass at fixed physical temperature; negative coupling produces the opposite trend in the sampled range. The boundary is expressed using the mass convention in Eq. (18), with $M_t$ the UV coexistence coefficient.

At fixed temperature, increasing the mass eventually replaces the horizon-reaching embedding by a capped embedding. Positive GB coupling delays this change to a larger mass in the sampled parameter range. Equivalently, on the monotonically rising parts of the phase boundaries, increasing the coupling lowers the temperature required to enter the BH phase at fixed mass. These are two descriptions of the same displacement in Figure 5.

Since the two coexisting branches have different mass responses [Eq. (29)], this shifted boundary remains first order at the sampled couplings. Its displacement is a change in the competition between optimized free energies, rather than a criterion read off from an individual embedding profile. A larger conducting region need not imply larger conductivities, as we show in § 4.

The ordering of the boundaries is supported at the sampled crossings: the interpolation does not resolve the interval $0<T/b<0.05$ or establish a critical endpoint. Numerical matching residuals and selected refinement checks are reported in Appendices § 7 and § 8; they are not uniform convergence bounds for the full phase diagram.

Analytical displacement and temperature normalization

We can predict the leading boundary shift directly from the coexistence condition (27) and the response (25). Differentiating $\Delta\mathcal{F}(M_t,T,b,\lambda_{\mathrm{GB}})=0$ gives

The derivative is taken at fixed physical temperature and axial scale, with the retained flavour dictionary held fixed. For a nonzero mass-response jump the crossing is simple, so this derivative is well defined. When the jump is positive, lowering the BH free energy relative to the MN free energy at the old crossing gives a positive displacement of $M_t$.

An independent prediction follows by evaluating the explicit variation of the probe action on the coexisting Einstein embeddings. Define

At fixed $(u,T)$, the blackening function and horizon position satisfy

For either coexisting Einstein embedding $\theta_0(u)$, introduce

Stationarity removes the bulk variation of the embedding at fixed source. Including the displacement of the BH endpoint gives

The horizon angle and cap position in these expressions belong to the unperturbed embeddings. The integrals are UV finite. The smooth MN cap contributes no endpoint term, while the UV contributions cancel against the counterterm variation at fixed mass.

The coexistence mass therefore obeys

where all quantities on the right are evaluated at Einstein coexistence. The initial shift can thus be predicted from the unperturbed embeddings without solving their first-order deformations. Independently computed crossings at small positive and negative coupling reproduce this prediction at $T/b=0.10$, $1/\pi$, $0.50$, with a largest relative discrepancy of 0.017% across the tested settings.

The physical-temperature convention is essential to interpreting this response. Since $u_h=n/(\pi T)$, varying $\lambda_{\mathrm{GB}}$ at fixed $T$ also changes the horizon position. Treat $M_t$ as a function of $(u_h,\lambda_{\mathrm{GB}},b)$. Since $n^{(0)}=-1$, at the Einstein point $(\partial_{\lambda_{\mathrm{GB}}}u_h)_T=-u_h$, while $(\partial_Tu_h)_{\lambda_{\mathrm{GB}}=0}=-u_h/T$. The chain rule at unchanged axial source $b$ therefore gives

The first term measures the change at fixed horizon position; the second accounts for the change of thermal scale needed to maintain the same physical temperature. At the three tested temperatures, these contributions have opposite signs: the fixed-horizon contribution is negative, while the positive thermal contribution is larger, giving the positive fixed-temperature slope.

The intrinsic term can also be expressed directly as a difference of radial integrals over the coexisting Einstein embeddings. At fixed $u_h$, with $v=(u/u_h)^4$ and $W$ defined in Eq. (33),

The fixed-horizon contribution is therefore a genuine bulk-shape response, even within this effective prescription. The archived Einstein-point coexistence checks in Table 3 find fixed-horizon slopes $-0.026917$, $-0.199205$ and $-0.339305$ at $T/b=0.10$, $1/\pi$ and $0.50$, opposed by thermal contributions $+0.129795$, $+0.852981$ and $+1.404065$, respectively. These are first-order slopes at $\lambda_{\rm GB}=0$, not estimates of global monotonicity. The corresponding analytical and direct finite-difference checks are tabulated in Appendix § 8.

$T/b$	$M_t^{(0)}/b$	Action $m_1$	Direct $m_1$	Fixed $u_h$	Thermal
0.10	0.492932	0.102878	0.102879	$-0.026917$	0.129795
$1/\pi$	0.996177	0.653776	0.653788	$-0.199205$	0.852981
0.50	1.497160	1.064760	1.064804	$-0.339305$	1.404065

This decomposition connects the phase-boundary displacement to the effective-temperature comparison discussed in § 2. An Einstein background at $T/\eta$ has the same horizon position as the GB background at $T$. Part of the observed displacement can therefore be understood through the temperature dependence of the Einstein phase boundary. The remaining difference reflects the deformation of the blackening function at the matched horizon position. Equation (36) makes this distinction explicit at first order; it does not establish an exact equivalence between the two geometries.

The common zero-temperature intercept of Figure 5 has a stronger, exact explanation. At $T=0$, the normalized vacuum geometry satisfies

Because the internal metric, probe action, counterterms and flavour dictionary are retained, the zero-temperature embedding equations and normalized free-energy differences coincide with those of the Einstein model. Consequently,

This coupling independence follows from the vacuum equations, rather than from extrapolating the finite-temperature curves. It is a property of the present geometric prescription: a deformation that also changes the probe action can modify the vacuum problem, as illustrated by the D-instanton construction of Ref. [17].

A complementary local check explains why the topology-changing embedding retains its usual finite-temperature scaling. Writing

the near-horizon geometry is Rindler and the shrinking three-sphere contributes a factor $y^3$. The leading local action and equation are

where derivatives are with respect to $\zeta$. Linearizing about $y_c=\sqrt{3}\,\zeta$ gives

These are the usual finite-temperature probe-brane exponents [32, 11]. The GB coupling changes the connection between the local near-horizon region and the UV boundary, while leaving this leading local equation unchanged. The exponents explain the turning of nearby embedding families; they do not locate the free-energy crossing or constitute critical exponents of the equilibrium first-order transition.

The phase boundary fixes the preferred embedding. We next use its horizon data to determine the electrical response.

Gauss–Bonnet corrections to DC conductivity

The phase diagram determines where the conducting BH embedding is thermodynamically preferred. We now examine how the GB deformation changes its electrical response. Two results will be emphasized. At the sampled low temperature, positive coupling reduces both conductivity jumps at coexistence, despite extending the preferred BH region. Near the temperature at which the longitudinal conductivity becomes monotonic, it also suppresses and eventually removes the mass-dependent maximum. The horizon formulas explain these effects through the thermal scale and the selected embedding.

The conductivities $\sigma_{xx}$ and $\sigma_{xy}$ describe the response in the transverse $(x,y)$ plane. Our convention is

so an electric field along $x$ gives $\sigma_{xx}=J_x/E_x$ and $\sigma_{xy}=-J_y/E_x$. All conductivities refer to the probe flavour current and use the same normalization $K=N_fN_c/(4\pi^2)$. The physical temperature relation $u_h=n/(\pi T)$ is imposed separately at every coupling.

Conductivity jumps and their GB dependence

The DC response follows from the radially conserved currents of the quadratic DBI and WZ actions, using the standard probe-brane current and horizon-regularity method [33, 34]. The Hall current in our model is induced by the axial-source WZ term, rather than by the external magnetic field used in some of those constructions. Evaluating these currents with regular ingoing conditions at the induced horizon gives

These are the horizon expressions of the retained flavour-brane construction [11], evaluated in the prescribed GB geometry. For a uniform DC field along $x$, the dissipative radial flux has magnitude $K\cos^3\theta\sqrt{D/H}\,f|A'_x|/u$, with $H=1+u^2f\theta'^2$. Regularity in ingoing coordinates gives $f|A'_x|\to|E_x|$ and $H\to1$ at the horizon. Consequently, there is no additional explicit factor of $n$ multiplying Eq. (44): the coupling enters through $u_h$ and the regular horizon angle $\theta_h$.

The factors in the two conductivities have distinct origins. The wrapped-sphere volume contributes $\cos^3\theta_h$ to the longitudinal response, while $\sqrt{1+b^2u_h^2\sin^2\theta_h}$ accounts for the axial stretching of the induced metric. The Hall response follows from the retained WZ coupling and has the angular dependence $\cos^4\theta_h$. Smooth MN embeddings have $\sigma_{xx}=\sigma_{xy}=0$ in this zero-density probe prescription.

We evaluate these expressions on the embeddings selected by the free-energy comparison. In the sampled region, the equilibrium conductivities therefore satisfy

The discontinuities compare the two pure-phase limits:

They are evaluated at each coupling’s own coexistence mass. They do not describe the effective response of a spatial mixture at coexistence.

Figure 6 shows that both jumps decrease as $\lambda_{\mathrm{GB}}$ increases over the sampled interval at $T/b = 0.10$. Between $\lambda_{\mathrm{GB}} = -0.05$ and $0.05$, the coexistence mass increases by approximately $2.13\%$, whereas the longitudinal and Hall jumps decrease by approximately $5.77\%$ and $22.85\%$, respectively. Thus the Hall discontinuity is more sensitive over this interval. Negative coupling gives larger jumps than the Einstein result, while positive coupling gives smaller ones. This ordering is established for the sampled low-temperature comparison and is not assumed at every temperature.

The massless embedding provides a useful exact reference at $T > 0$. For $\theta_h = 0$, Eq. (44) becomes

Positive coupling therefore increases the massless longitudinal conductivity at fixed physical temperature, while leaving the massless Hall value unchanged. The decrease of the conductivity jumps must consequently involve the changing embedding at coexistence.

This interpretation can be made precise without assuming the sign of the embedding response. Let $\theta_{h,t}=\theta_h(M_t(T,b,\lambda_{\mathrm{GB}}),T,b,\lambda_{\mathrm{GB}})$ and define $D_t=1+b^2u_h^2\sin^2\theta_{h,t}$. Differentiating Eq. (44) along the coexistence line at fixed physical $(T,b)$ gives

These identities are exact consequences of the horizon formulas. Their interpretation requires the coexistence embeddings obtained numerically.

For a compact sign test define

When $G_t > 0$, suppression of the Ohmic jump requires

The horizon-angle derivative is a total derivative along coexistence:

Only the sum in Eq. (52) is evaluated in the present numerical table; we do not claim separately converged values for its two terms.

$T/b$	$\lambda_{\mathrm{GB}}=-0.05$	$\lambda_{\mathrm{GB}}=0$	$\lambda_{\mathrm{GB}}=+0.05$
0.050	0.150854	0.158606	0.168464
0.070	0.242838	0.264987	0.293493
0.075	0.280121	0.305729	0.334463
0.085	0.351400	0.367867	0.378432
0.100	0.375739	0.367854	0.354067
0.150	0.238217	0.223511	0.208212

The increase at $T/b = 0.070$ confirms that suppression is not universal over the sampled coupling interval. The finite-interval ordering of the three couplings reverses between $T/b = 0.085$ and 0.10; the six-temperature table does not locate the precise crossover or establish a sign change of the infinitesimal derivative at every coupling. A finer scan is needed for those stronger claims.

Since $d\ln n/d\lambda_{\mathrm{GB}} = -1/[n\sqrt{1-4\lambda_{\mathrm{GB}}}] < 0$, the explicit thermal-scale contribution in Eq. (48) is positive. At the sampled crossings, $\theta_{h,t}$ increases with coupling, which suppresses the Hall jump. The angular part of the Ohmic identity can have either sign: its competition with the positive explicit thermal-scale term changes with temperature. The identity therefore organizes the computed response but does not by itself predict suppression.

The angular change includes both the deformation of the BH solution at fixed mass and the displacement of the coexistence mass itself. The identities therefore explain the numerical trend but do not independently establish its sign. Likewise, the comparison with Einstein gravity at $T/n$ can account for part of this response through its effect on the coexistence embedding; the horizon formulas alone do not quantify the remaining GB contribution.

The central physical distinction is between the extent of the conducting phase and its response at the boundary of that phase. The former follows from a difference of optimized free energies, while the latter depends on local horizon data. Positive coupling can consequently extend the preferred BH region while decreasing or increasing the Ohmic jump, depending on temperature. With the WZ normalization fixed, this finite-temperature Hall suppression does not represent a correction to the axial-anomaly coefficient.

Matched-horizon comparison and residual GB response

To quantify the part of the response not explained by the thermal scale alone, we compare GB coexistence at physical $T$ with Einstein coexistence at $T_{\mathrm{eff}} = T/n(\lambda_{\mathrm{GB}})$, so that $u_h$ is the same. Both points lie on their own coexistence lines: this is not a comparison at equal flavour mass. For an observable $\mathcal{O}$ define the relative residual

The residual combines the explicit difference of blackening functions and the resulting shift of the optimized embedding. It does not isolate a unique local higher-curvature operator. The residuals

in Table 2 are small on these coexistence states (at most 1.5% in magnitude for the listed observables). Thus thermal rescaling reproduces most of the response for the listed quantities at $T/b = 0.10$, but not exactly. These percentages quantify differences between two coexistence solutions, not the fraction of a coupling derivative due to the intrinsic term. For the latter, the analytical phase-boundary decomposition in Eq. (37) gives a more direct interpretation. The available matched-horizon data do not establish a similar bound at other physical temperatures or at fixed mass.

Observable	$\lambda_{\mathrm{GB}}=-0.05$	$\lambda_{\mathrm{GB}}=+0.05$
$M_t/b$	+0.210%	-0.365%
$\Delta\sigma_{xx}/(Kb)$	-0.257%	+0.603%
$\Delta\sigma_{xy}/(Kb)$	-1.142%	+1.499%

Suppression and disappearance of the longitudinal maximum

The longitudinal response contains a second feature beyond its discontinuity at coexistence. The original Einstein model exhibits a maximum of the DC conductivity as the mass is varied at sufficiently low temperature, followed by monotonic behavior at higher temperature [11]. The D-instanton extension also reports suppression of this low-temperature maximum when the instanton density is introduced [17]. The disappearance of such a maximum is therefore not unique to GB corrections. Here the retained horizon formula allows its geometric condition to be determined analytically and tested on the preferred embeddings.

At fixed $(T,b,\lambda_{\mathrm{GB}})$, introduce $B = bu_h = bn/(\pi T)$ and $x = \sin^2\theta_h$. Dividing the longitudinal conductivity by its massless value at the same temperature and coupling gives

The first factor decreases as the wrapped sphere shrinks, whereas the second increases because of axial stretching. Their competition is expressed by

For $B \le\sqrt{3}$, the conductivity decreases monotonically with the horizon angle. For $B > \sqrt{3}$, there is exactly one interior maximum:

The geometric existence condition is therefore

Positive coupling decreases $n$ and lowers this threshold. It also reduces the maximum relative to the massless reference while $B>\sqrt{3}$. In terms of $T_{\rm eff}=T/n$, Eq. (57) is precisely the Einstein condition $T_{\rm eff}/b<1/(\pi\sqrt{3})$. The GB dependence of this geometric criterion thus follows directly from the thermal-scale relation.

An interior maximum of the horizon formula need not appear in the equilibrium conductivity against mass. The preferred BH family must reach $\theta_h^*$ before its free-energy crossing. Moreover, the local source map must be regular, $\partial M/\partial\theta_h\ne0$, so the angular extremum corresponds to an extremum against mass. A monotonic map along the preferred branch makes this correspondence unambiguous. Phase selection can therefore exclude the maximum even when Eq. (57) is satisfied.

We test this distinction at $T/b=0.18$, where the threshold passes through the sampled coupling range. The preferred BH branches have a monotonic mass–angle map up to coexistence. For $\lambda_{\rm GB}=-0.05$ and $0$, the maximum is reached before the first-order transition, with relative enhancements of approximately $0.333\%$ and $0.032\%$ above each coupling’s own massless conductivity. For $\lambda_{\rm GB}=0.05$, the geometric threshold is $T/b\simeq0.17408$, below the chosen temperature, and the conductivity decreases monotonically with mass.

The enlarged panel in Figure 7 resolves the shallow Einstein maximum. Its small height follows from the analytical behavior near the threshold. Writing $\epsilon=1-T/T_{\rm geom}^*$ at fixed coupling gives

The relative enhancement therefore vanishes quadratically as the maximum approaches the massless endpoint. At fixed $T/b=0.18$, the geometric threshold corresponds to $\lambda_{\rm GB}^*\simeq0.02013$. The same expansion applies when $\lambda_{\rm GB}$ is varied at fixed $T$, since $\epsilon=1-\pi\sqrt{3}T/(bn)$ measures distance from the threshold in either scan. At $\lambda_{\mathrm{GB}}^{*}-0.002$ the interior peak lies at $M_{*}/b=0.04670$, with $R_{\max}-1=3.38\times10^{-6}$, whereas $M_{t}/b=0.65202$. At $\lambda_{\mathrm{GB}}^{*}+0.002$ there is no interior peak. These embedding and free-energy checks support a smooth disappearance through zero mass while coexistence remains at finite mass. In this region the maximum is lost through the shape of the conductivity curve, rather than being bypassed by the first-order transition.

This conclusion concerns the DC conductivity as a function of mass. It does not identify the appearance or disappearance of a pole in the finite-frequency current correlator. The optical structures and long-lived current modes studied in the Einstein model [15] require a separate GB fluctuation calculation.

The two transport results thus probe different aspects of the same equilibrium family. The conductivity jumps compare the horizon data selected at coexistence, whereas the interior maximum measures the competition between axial stretching and the shrinking wrapped sphere along the BH branch. Their coupling dependence is constrained by exact horizon relations, with the physical realization determined by regular embeddings and free-energy selection.

Conclusions and outlook

We have examined how an effective Gauss–Bonnet deformation changes the phase structure and DC transport of the D3/D7 holographic Weyl semimetal. The background is an effective five-dimensional GB deformation combined with the unchanged internal geometry and probe DBI/WZ prescription. We do not claim a corrected ten-dimensional type-IIB solution or systematic finite-’t Hooft-coupling expansion, and the probe approximation does not assess all possible inhomogeneous or dynamical instabilities. Comparing phases at common physical sources leads to the following results:

• The first-order boundary moves while its character persists. Positive GB coupling extends the preferred conducting BH phase toward larger flavour mass at finite temperature in the sampled range; negative coupling produces the opposite shift. The unequal mass responses at coexistence establish the first-order character of the transition. The normalized zero-temperature boundary remains unchanged because the vacuum probe problem is unchanged.

• The leading boundary shift follows from the Einstein embeddings. An on-shell action variation predicts the initial displacement without solving the first-order embedding deformation. Its agreement with independently computed crossings supports the interpretation of the shift as a competition between the two optimized free energies. The decomposition into fixed-horizon and thermal-scale contributions identifies the origin of the positive shift at the tested temperatures.

• An enlarged conducting region need not have larger conductivity jumps. At $T/b=0.10$, increasing the coupling suppresses both longitudinal and anomalous Hall discontinuities at their respective coexistence masses. The changing horizon embedding reduces the wrapped-sphere factors in the conductivities. For the longitudinal jump, this suppression outweighs the increase associated explicitly with the thermal scale. The Hall change follows from the embedding with the WZ normalization held fixed.

• The longitudinal maximum can disappear within the preferred phase. The horizon formula admits an interior maximum when $T/b<n(\lambda_{\mathrm{GB}})/(\pi\sqrt{3})$. This geometric criterion is the Einstein condition expressed at the matched-horizon temperature. Free-energy selection supplies the additional requirement that the preferred BH branch reaches the maximum. The targeted scan resolves its smooth merger with the massless endpoint, with a quadratically vanishing relative enhancement, while coexistence remains at finite mass.

The conductivity-jump suppression is not universal: at $T/b = 0.070$ the Ohmic jump increases over the same coupling interval. The matched-horizon residuals in Table 2 show that thermal rescaling accounts for most of the listed responses at $T/b = 0.10$, while nonzero residuals test deformation-dependent embedding and blackening-function effects. The intrinsic coexistence slope is nonzero and of opposite sign to the thermal contribution at the three temperatures checked analytically.

An important extension is to determine how the GB deformation modifies the frequency-and momentum-dependent flavour-current response. Optical conductivities, spectral functions and quasinormal modes in the Einstein D3/D7 Weyl semimetal provide a direct baseline for this question [15]. Tracking resonance positions, widths and spectral weight on the thermodynamically preferred BH and MN branches would test whether the GB background produces changes beyond those reproduced by the matched-horizon effective temperature. It would also clarify whether the discontinuities in DC transport across coexistence have distinctive finite-frequency signatures.

Fermionic flavour excitations offer a complementary probe of the same transition. The quadratic D7-brane fermion equations, with their induced geometry, axial embedding and background Ramond–Ramond flux, would need to be derived and solved on the competing embeddings. Polarization-resolved retarded fermionic correlators could then be used to follow spectral weight and pole structure across the insulating–Weyl-semimetal transition. A useful objective would be to identify any GB-dependent features that survive a matched-horizon comparison and go beyond the expected distinction between normal modes on capped embeddings and dissipative excitations on horizon-reaching embeddings. These questions lie outside the present bosonic embedding and DC-conductivity analysis.

Acknowledgements

We thank Ronnie Rodgers for helpful discussions. The initial embedding and free-energy analysis builds on the code developed by the authors. The authors used OpenAI’s ChatGPT to assist with analytical derivations, additional numerical calculations and consistency checks, and manuscript editing.

Conventions and numerical checks

• Radial coordinates and physical temperature.

The temperature relation depends on the radial convention, although the physical Hawking temperature is invariant. In the commonly used bare-radius convention [19], we denote the radial coordinate by $\bar{u}$ to distinguish it from the boundary direction $z$. The metric is

Here

with the Einstein limit understood at $\lambda_{\mathrm{GB}} = 0$. Using $L_{\mathrm{eff}}^{2} = nL_{0}^{2} = 1$ and defining $u = \sqrt{n}\,\bar{u}$ gives the metric of Eq. (8), with $f = nf_{0}$ and $u_{h} = \sqrt{n}\,\bar{u}_{h}$. Consequently,

The two expressions therefore describe the same temperature. The horizon coordinates $\bar{u}_{h}$ and $u_{h}$ must not be interchanged when comparing different couplings.

• Dimensionless shooting variables and the mass response. For finite-temperature shooting, we use $x = u/u_h$ and $\hat{b} = b u_h$. Define the dimensionless UV fit by

Comparing with Eq. (17) gives

Here $\widehat{Q}$ and $\widehat{\mathcal{F}}$ use the same subtraction as Eqs. (23) and (25), with dimensionless reference scale equal to one. The factors of $u_h\Lambda_{\mathrm{UV}}$ cancel in $Q$; Eq. (63) reduces to the numerical convention $\Lambda_{\mathrm{UV}} = 1$ used in the data. The logarithmic conversion of $c_3$ is essential: omitting it changes the extracted mass response.

In the Cartesian embedding notation of Ref. [11], the corresponding subleading coefficient is

with the same logarithmic reference convention. The physical expectation value can therefore be written equivalently as

This reproduces the notation of the original construction while distinguishing $C$ from the coefficient $c_3$ of the angular embedding.

Changing the reference scale at fixed embedding gives

This change cancels the explicit scale dependence in Eq. (24). It is a reparametrization of the same UV solution, distinct from adding a finite counterterm to change the renormalization scheme.

• Initialization and free-energy checks. For BH embeddings, the near-horizon expansion uses Eq. (19). For MN embeddings, the inverse-coordinate series in Eq. (21) is initialized near the smooth cap and converted to the $u$ coordinate in an overlap region. The ODE solver is DOP853. The UV fit includes the logarithm in Eq. (17).

Solutions are continued in their infrared parameter rather than sorted only by mass. This retains folds and multivalued portions of the embedding families. Free energies are compared only after matching the physical UV mass, using the same $(T,b,\lambda_{\mathrm{GB}})$ and subtraction prescription.

The leading $u^{-5}$ divergence is subtracted analytically before numerical integration. The renormalized actions are evaluated at five UV cutoffs and extrapolated using the basis

As an additional check, direct action differences are compared with differences obtained by integrating $Q\,dM$ along each continued branch, with integration constants fixed by direct action values, where $Q$ is the auxiliary normalized mass response defined in Eq. (25).

For the four coexistence points at $T/b = 1/\pi$, refinement reduced the UV windows and cutoffs by a factor of 0.75, the infrared offsets by a factor of 0.20, and the relative ODE tolerance from $2 \times10^{-12}$ to $5 \times10^{-13}$. The largest reported change in $M_t/b$ was $1.3 \times10^{-7}$. Matched-mass action comparisons on either side of coexistence checked the sign change of $\Delta F$. A further refinement of the transport coexistence points at $T/b = 0.10$ changed the mass by less than $1.7 \times10^{-8}$.

For the 33 finite-temperature phase-boundary pairs, mass-matching residuals were below $1.08 \times10^{-7}$ and direct free-energy residuals below $9.4 \times10^{-9}$. These residuals test the matching equations; they are not uniform convergence error bars for the entire phase diagram. The sensitivity tests support the reported results within the sampled homogeneous families and do not determine fluctuation stability.

The flat-boundary subtraction used here and the subtraction using the induced boundary volume element differ by

At common sources and coupling, this offset is identical for BH and MN embeddings and cancels in $\Delta F$. Its temperature dependence must nevertheless be retained when comparing absolute free energies or their thermal derivatives. Because it contains $u_h^{-4}$, it should be distinguished from a finite local counterterm depending only on the fixed boundary sources.

• Resolution of the shallow conductivity maximum.

The peak analysis includes eighteen coexistence calculations for $\lambda_{\mathrm{GB}}=-0.05$, 0, 0.05, covering $T/b=0.17$, 0.18, 0.19, 0.20 and temperatures 0.0005 below and above each geometric threshold. The map $M(\theta_h)$ was sampled below coexistence and was monotonic in these tests.

Six points were repeated with the refined settings described above. The largest change in $M_t/b$ was $6.3 \times10^{-8}$. The baseline mass and normalized free-energy matching residuals were below $7.6 \times10^{-10}$ and $8.9 \times10^{-11}$, respectively.

At $T/b=0.18$, broader scans of both embedding families were followed by direct matched-mass action comparisons. None of the sampled competitors had lower free energy than the selected BH branch. At the two nonzero peak masses, the broad scan found one matching BH solution and no MN solution. These checks support the equilibrium interpretation within the sampled families without proving that all stationary solutions have been found or establishing dynamical stability.

Additional coexistence calculations were performed at $\lambda_{\mathrm{GB}}^*$ and $\lambda_{\mathrm{GB}}^* \pm0.002$, where $\lambda_{\mathrm{GB}}^*$ denotes the geometric threshold at $T/b=0.18$. The peak heights are evaluated from the exact horizon formula at $\theta_h^*$; they are not extracted from differences between nearly equal free energies.

The two jumps also obey an algebraic relation because they depend on the same horizon angle. Defining

eliminating $\theta_{h,t}$ gives

This is an exact consistency relation within the retained DBI/WZ prescription when the MN conductivities vanish. It checks the common normalization and implementation, with a maximum absolute residual in $X$ of $3.1 \times10^{-16}$ over the 33 stored finite-temperature coexistence states. Both sides use the same horizon data.

The angular dependence used in this check and the threshold of the DC maximum are shown in Figure 8.

Independent coexistence-response checks

The following perturbative slope values were recovered from the archived analytical and numerical verification accompanying an earlier manuscript draft; they are retained here as a comparison rather than presented as a newly repeated high-precision run. The finite-difference estimate uses $[M_t(+\delta) - M_t(-\delta)]/(2b\delta)$ with $\delta= 0.002$. The largest relative discrepancy in the original

Table 3: Einstein-point coexistence displacement $m_1 = \partial(M_t/b)/\partial\lambda_{\mathrm{GB}}|_{T,b,0}$. The temperature-compensation column equals the difference between fixed-temperature and fixed-$u_h$ slopes in the stated radial gauge. The displayed digits are archived numerical estimates, not certified error bounds.

multi-step validation was reported as 0.017%. That statement covers the archived numerical settings and is not a uniform error bound for all parameters. At these temperatures the fixed-$u_h$ contribution is negative while the positive thermal term is larger; the decomposition is given analytically in Eq. (36) and discussed in the main text.

References