Hyperscaling Violation Geometry

• Hyperscaling violation geometry is a gravitational background characterized by a dynamical exponent z and a hyperscaling violation exponent θ that alter the scaling behavior of thermodynamic and transport quantities.
• The metric, defined by a scale-dependent factor and blackening function, results in an effective spatial dimension (d - θ) which influences entanglement entropy and phase transition properties.
• Holographic models using Einstein-Maxwell-dilaton actions reveal instabilities and emergent orders such as holographic superconductivity, spatially modulated, and crystalline phases.

A hyperscaling violation geometry is a gravitational background exhibiting scaling transformations characterized not only by a dynamical exponent $z$ (as in Lifshitz geometries) but also by a nontrivial hyperscaling violation exponent $\theta$. These exponents govern the anisotropic scaling between time and space and break the naive hyperscaling property, resulting in nontrivial scaling of thermodynamic and transport quantities in the dual field theory. Such geometries and their associated holographic duals have been extensively studied in the context of strongly correlated systems, quantum criticality, non-Fermi liquids, and exotic universality classes.

1. Definition and Metric Structure

A canonical hyperscaling violation (hvLif) metric in $d+2$ bulk dimensions takes the form

$$ds^2 = r^{-2\theta/d} \left( - r^{-2(z-1)} f(r)\,dt^2 + \frac{dr^2}{f(r)} + \sum_{i=1}^d dx_i^2 \right),$$

where:

• $z$ is the dynamical critical exponent controlling $t\to\lambda^z t$ vs. $x_i\to\lambda x_i$ scaling,
• $\theta$ is the hyperscaling violation exponent; under $t\to\lambda^z t$, $x_i\to\lambda x_i$, $\theta$0, the metric transforms as $\theta$1,
• $\theta$2 is a blackening factor introducing a horizon, $\theta$3, with $\theta$4 the horizon location.

This scaling property is covariant rather than invariant, expressing the “hyperscaling violation” that modifies the effective spatial dimensionality perceived by the dual field theory: thermodynamic quantities and operator dimensions transform as if the theory lives in $\theta$5 spatial dimensions (Fan, 2013, Karch, 2014). The special case $\theta$6 reduces to standard Lifshitz scaling.

2. Null Energy Condition and Admissible Exponents

Physically allowed (realizable by stress–energy tensors satisfying the null energy condition, NEC) exponents $\theta$7 are highly constrained. The explicit inequalities in $\theta$8 spatial dimensions are

$\theta$9

Generally, for $d+2$0 this yields $d+2$1 and $d+2$2 (Fan, 2013, Fonda et al., 2014, Chen et al., 2015). The NEC ensures positive specific heat and the absence of superluminal propagation; violation generally signals unphysical stress–energy.

Allowed regions in the $d+2$3-plane also affect matter couplings: the existence and regularity of static or time-dependent solutions, as well as potential stability to spatially modulated (striped or crystalline) phases, all depend on where these exponents lie relative to NEC windows (Cremonini et al., 2012).

3. Hyperscaling Violation and Effective Dimension

The defining feature of hyperscaling violation is the scaling of the line element such that extensive quantities exhibit non-standard dependence on temperature, length, or chemical potential. Specifically, the entropy and free energy density scale as

$d+2$4

and the entanglement entropy of a region of characteristic size $d+2$5 scales as $d+2$6 for $d+2$7 (Fan, 2013, Shaghoulian, 2013). Hyperscaling violation with $d+2$8 leads to logarithmic violations of the area law, linked to “hidden Fermi surface” or non-Fermi liquid behavior.

The operator spectrum is also modified: the effective scaling dimension for scalar operators becomes $d+2$9, and the Breitenlohner-Freedman (BF) bound is correspondingly shifted, affecting possible phase transitions (Fan, 2013, Cremonini et al., 2016).

4. Holographic Realizations and Matter Sectors

Hyperscaling violation backgrounds are constructed as solutions to Einstein-Maxwell-dilaton (EMD) or more general gravitational actions including appropriate matter couplings: $$ds^2 = r^{-2\theta/d} \left( - r^{-2(z-1)} f(r)\,dt^2 + \frac{dr^2}{f(r)} + \sum_{i=1}^d dx_i^2 \right),$$0 with $$ds^2 = r^{-2\theta/d} \left( - r^{-2(z-1)} f(r)\,dt^2 + \frac{dr^2}{f(r)} + \sum_{i=1}^d dx_i^2 \right),$$1, $$ds^2 = r^{-2\theta/d} \left( - r^{-2(z-1)} f(r)\,dt^2 + \frac{dr^2}{f(r)} + \sum_{i=1}^d dx_i^2 \right),$$2 exponential potentials engineered to support the desired $$ds^2 = r^{-2\theta/d} \left( - r^{-2(z-1)} f(r)\,dt^2 + \frac{dr^2}{f(r)} + \sum_{i=1}^d dx_i^2 \right),$$3 scaling (Chen et al., 2015, Cremonini et al., 2012, Cremonini et al., 2016). A running dilaton is essential for sustaining nontrivial hyperscaling violation, with scalar and gauge kinetic couplings dictating the hyperscaling structure in both the UV and IR.

Magnetically charged black brane solutions and probe sectors (scalars, fermions) are routinely analyzed in these backgrounds. Intrinsically, these setups admit nontrivial IR flows and possible instabilities towards inhomogeneous or crystalline phases, especially under strong magnetic fields or when the coupling functions cross criticality (Bao et al., 2013, Chen et al., 2015, Cremonini et al., 2012).

5. Phase Structure, Instabilities, and Emergent Orders

A salient theme in recent research is the instability of hyperscaling violation geometries towards novel ordered phases:

• Holographic superconductors: The critical temperature $$ds^2 = r^{-2\theta/d} \left( - r^{-2(z-1)} f(r)\,dt^2 + \frac{dr^2}{f(r)} + \sum_{i=1}^d dx_i^2 \right),$$4 and spectral gap $$ds^2 = r^{-2\theta/d} \left( - r^{-2(z-1)} f(r)\,dt^2 + \frac{dr^2}{f(r)} + \sum_{i=1}^d dx_i^2 \right),$$5 depend nontrivially on both $$ds^2 = r^{-2\theta/d} \left( - r^{-2(z-1)} f(r)\,dt^2 + \frac{dr^2}{f(r)} + \sum_{i=1}^d dx_i^2 \right),$$6 and $$ds^2 = r^{-2\theta/d} \left( - r^{-2(z-1)} f(r)\,dt^2 + \frac{dr^2}{f(r)} + \sum_{i=1}^d dx_i^2 \right),$$7. Coupling the probe gauge field to the background dilaton ($$ds^2 = r^{-2\theta/d} \left( - r^{-2(z-1)} f(r)\,dt^2 + \frac{dr^2}{f(r)} + \sum_{i=1}^d dx_i^2 \right),$$8) can suppress $$ds^2 = r^{-2\theta/d} \left( - r^{-2(z-1)} f(r)\,dt^2 + \frac{dr^2}{f(r)} + \sum_{i=1}^d dx_i^2 \right),$$9 as $z$0, with characteristic deviations from universal conductivity gap ratios known in AdS$z$1 (Zhang et al., 2015, Fan, 2013). The phase transition remains mean-field, but the interplay between $z$2, $z$3, and probe mass can invert the monotonicity of $z$4 for fixed $z$5.
• Spatially modulated/striped phases: Analysis of linearized perturbations about AdS$z$6 IR regions reveals windows of finite-momentum instabilities, controlled by the effective curvature of the scalar and gauge kinetic functions at the IR fixed point. These instabilities induce ground states with spontaneously broken translation symmetry (“striped” black branes), which remove the extensive IR entropy and modify thermodynamics (Cremonini et al., 2012, Bhattacharya et al., 2012).
• Crystalline (vortex-lattice) orders: Charged scalar and fermion probes in strong magnetic backgrounds experience instabilities towards crystalline vortex lattice formation. The emergence and thermodynamic stability of lattices are generically favored—triangular configurations minimize free energy regardless of $z$7, and larger $z$8 or smaller $z$9 enhance lattice stability. The backreacted metric acquires a further hyperscaling violation exponent shift, representing a flow toward lower effective IR dimension (Chen et al., 2015, Bao et al., 2013).

6. Thermodynamic and Transport Properties

Thermodynamics of hyperscaling violation geometries display signatures unavailable in scale-invariant (relativistic or Lifshitz) settings:

• Black brane entropy and temperature: $t\to\lambda^z t$0, $t\to\lambda^z t$1. The scaling of specific heat, free energy, and other quantities follow directly (Pourhassan et al., 2017, Alishahiha et al., 2018).
• Quantum corrections and stability: Quantum (thermal) fluctuations produce entropy corrections of the form $t\to\lambda^z t$2, where $t\to\lambda^z t$3 is the classical entropy. These corrections can stabilize or destabilize the black brane, and in sufficiently high $t\to\lambda^z t$4 induce van der Waals-type phase transitions (Pourhassan et al., 2017).
• Transport coefficients and strange metal phenomenology: Holographic conductivities exhibit hyperscaling violation-dependent power laws. An additional scaling exponent $t\to\lambda^z t$5, associated with the gauge sector, is necessary to reproduce experimental scalings such as linear-in-$t\to\lambda^z t$6 resistivity and quadratic Hall angle found in strange metals (Karch, 2014). The scaling of the DC conductivity is $t\to\lambda^z t$7, where $t\to\lambda^z t$8 is the number of spatial directions probed, and $t\to\lambda^z t$9 is the matter hyperscaling exponent.

7. Singularities, IR Completions, and Information-Theoretic Aspects

Most hyperscaling violation geometries generically possess IR tidal-force singularities at $x_i\to\lambda x_i$0, even when all curvature invariants are finite. There exists a special class (with exponents satisfying $x_i\to\lambda x_i$1 in the generalized metric parameterization) allowing a smooth analytic extension with two disconnected boundaries, analogous to AdS$x_i\to\lambda x_i$2, but this regularity is destroyed by any generic finite-energy bulk excitation (Lei et al., 2013).

To remedy IR incompleteness due to running dilaton, one introduces quantum corrections or modified potentials to stabilize the scalar and induce a flow to an $x_i\to\lambda x_i$3 throat in the deep IR, yielding a consistent global geometry interpolating between $x_i\to\lambda x_i$4 (UV), an intermediate hyperscaling violation regime, and $x_i\to\lambda x_i$5 (IR) (Bhattacharya et al., 2012).

Recent developments connect hyperscaling violation geometry to quantum information measures:

• Entanglement entropy: The Ryu-Takayanagi prescription yields entropies scaling as $x_i\to\lambda x_i$6, and in the presence of a cutoff, all entanglement quantities (entropies, mutual information, entanglement wedge cross sections) become finite and sensitive to $x_i\to\lambda x_i$7, the cutoff scale. Hyperscaling violation modifies universal entanglement structures and the phase transitions in mutual information (Khoeini-Moghaddam et al., 2020, Fonda et al., 2014, Shaghoulian, 2013).
• Complexity growth: The violation of Lloyd's bound in complexity=action computations is generic in hvLif black brane backgrounds. The late-time complexity growth rate saturates to $x_i\to\lambda x_i$8 (with $x_i\to\lambda x_i$9 the energy), which can exceed $\theta$0 (the naive Lloyd bound with $\theta$1 the mass) (Alishahiha et al., 2018).

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References (by arXiv id):

• (Fan, 2013) Holographic superconductors with hyperscaling violation
• (Zhang et al., 2015) Holographic superconductor in hyperscaling violation geometry with Maxwell-dilaton coupling
• (Chen et al., 2015) Crystalline geometries from fermionic vortex lattice with hyperscaling violation
• (Cremonini et al., 2012) Spatially Modulated Instabilities of Geometries with Hyperscaling Violation
• (Bao et al., 2013) Crystalline Scaling Geometries from Vortex Lattices
• (Karch, 2014) Conductivities for Hyperscaling Violating Geometries
• (Cremonini et al., 2016) Criteria For Superfluid Instabilities of Geometries with Hyperscaling Violation
• (Pourhassan et al., 2017) Thermal Fluctuations in a Hyperscaling Violation Background
• (Bhattacharya et al., 2012) On the IR completion of geometries with hyperscaling violation
• (Lei et al., 2013) Extending the nonsingular hyperscaling violating spacetimes
• (Alishahiha et al., 2018) Complexity Growth with Lifshitz Scaling and Hyperscaling Violation
• (Khoeini-Moghaddam et al., 2020) Aspects of Hyperscaling Violating Geometries at Finite Cutoff
• (Fonda et al., 2014) Holographic thermalization with Lifshitz scaling and hyperscaling violation
• (Shaghoulian, 2013) FRW cosmologies and hyperscaling-violating geometries: higher curvature corrections, ultrametricity...
• (Hassaine, 2015) New black holes of vacuum Einstein equations with hyperscaling violation and Nil geometry horizons
• (Andrade et al., 2024) Spatially localized scalar structures on hyperscaling violating geometries