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Analytic approaches to anisotropic holographic superfluids in asymptotically hyperscaling violation geometry

T0 review · 3 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read The paper constructs the first analytic anisotropic superfluid solutions in asymptotically hyperscaling-violating geometries, in three and four bulk dimensions, and shows the superfluid phase is thermodynamically preferred at the critical…

desk verdict A careful but modest extension of the AdS5 anisotropic superfluid to hyperscaling-violating backgrounds, with the phase-transition claim outrunning the calculation. read the letter →

arxiv 2504.13635 v2 pith:TBEHTIBO submitted 2025-04-18 hep-th

Gitae Kim , Yeong Seok Choi , Jae-Hyuk Oh This is my paper · ORCID
classification hep-th
keywords holographicsuperfluidityp-wavesuperconductorhyperscalingviolationLifshitzgeometrySU(2)Yang-Millsentanglemententropyanalyticblackbranesolutionsanisotropicphasetransition
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper extends a known exact construction of anisotropic holographic superfluids, previously available only in five-dimensional anti-de Sitter spacetime, to three- and four-dimensional bulk geometries that are asymptotically hyperscaling violating. Working with Einstein-scalar–U(1)×SU(2) Yang-Mills theory, the authors find analytic vector-order solutions $w_1(u)=u^2/(1+u^2)^2$ at rescaled chemical potential $\mu=4/\sqrt{3}$, together with the leading-order backreaction on the metric. They then compute holographic entanglement entropies for small boundary subsystems and find positive leading corrections, which indicates that the anisotropic superfluid phase is preferred over the isotropic normal phase. The result matters because it gives rare fully analytic control over a symmetry-breaking phase transition in strongly coupled systems with non-relativistic scaling and hyperscaling violation.

What carries the argument

The load-bearing object is the Yang-Mills ansatz $B^a\tau^a = b(u)\tau^3 dt + w(u)\tau^1 dx^1$, whose nonzero components combine the chemical potential $b(u)$ with a spatially directed vector order $w(u)$. Under the three conditions $d(\alpha+1)=3$, $z=1$, and $\sqrt{3}\mu=4$, the Yang-Mills equations reduce to a Sturm-Liouville problem whose first nontrivial solution is exactly $w_1(u)=u^2/(1+u^2)^2$; the paper then solves the Einstein and scalar equations order by order in the small parameters $\varepsilon$ (order parameter amplitude) and $\delta=\kappa_D/g_{YM}$ (backreaction strength), keeping the leading $\delta^2\varepsilon^2$ corrections as closed-form functions $N_2$, $\sigma_2$, $H_2$, $J_2$ and $\varphi_2$. Holographic entanglement entropy is evaluated from the minimal-surface prescription, expanded for small subsystems, and matched to the energy change to extract the entanglement temperature.

What would settle it

Numerically solve the coupled Yang-Mills equations for $D=3$ ($\alpha=2$) and $D=4$ ($\alpha=1/2$) at chemical potentials slightly above and below $4/\sqrt{3}$ and look for a nontrivial $w_1(u)$ solution: if it exists at any other $\mu$, the claim that the transition must sit at the critical point fails. Alternatively, compute the backreacted entanglement entropy to the next order in $\varepsilon^2\delta^2$ and check whether the positive leading correction that makes the anisotropic phase preferred changes sign.

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Extended reading notes

Core claim

The paper claims that in $D=3$ with $(\alpha,z)=(2,1)$ and in $D=4$ with $(\alpha,z)=(1/2,1)$, the Einstein-scalar–U(1)×SU(2) Yang-Mills system admits analytic Yang-Mills solutions describing a vector order parameter along one spatial direction, provided $d(\alpha+1)=3$ and the rescaled chemical potential takes the special value $\mu=4/\sqrt{3}$. At this value, the leading vector profile is $w_1(u)=u^2/(1+u^2)^2$, and the backreacted metric functions $N$, $\sigma$, $H$, $J$ and scalar $\varphi$ can be written in closed rational forms plus simple logarithms. Computing holographic entanglement entropy for line segments and narrow straps, the paper finds that the entropy change relative to the pure hyperscaling-violating background is positive at leading order, concluding that the superfluid/normal-fluid phase transition occurs at the critical point and that the anisotropic phase is preferred. The first law of entanglement entropy holds with an entanglement temperature inversely proportional to subsystem size, with the same coefficient in directions parallel and perpendicular to the order.

Load-bearing premise

The argument assumes that the specially tuned point ($d(\alpha+1)=3$, $z=1$, $\mu=4/\sqrt{3}$) is where the transition actually happens, and that stopping backreaction at second order in the small parameters does not change the phase preference.

Editorial extensions

If this is right

  • The phase transition from normal to superfluid occurs exactly at the critical chemical potential $\mu=4/\sqrt{3}$, and the anisotropic phase is thermodynamically preferred at leading order.
  • The same analytic vector profile $u^2/(1+u^2)^2$ controls p-wave order in three different bulk dimensions ($D=5,4,3$), with hyperscaling-violating asymptotics for $D=4$ and $D=3$.
  • The first law of entanglement entropy holds in the anisotropic phase, with entanglement temperature $T\sim 1/L$ for a line segment and $T\sim 1/W$ for a strap, with universal coefficients independent of direction.
  • The explicit backreacted metrics provide a ready-made arena for computing transport and thermodynamic observables in anisotropic strongly coupled systems with hyperscaling violation.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Beyond the paper, I would expect the same Sturm-Liouville mechanism to yield analytic vector profiles at other allowed values of $d$ if $z$ is allowed to vary; the paper fixes $z=1$, so testing $z\neq 1$ is a natural next step.
  • Beyond the paper, because the entanglement-temperature coefficient comes out identical for straps parallel and perpendicular to the order, one testable prediction is that directional transport coefficients, such as shear viscosity or conductivity, show a characteristic anisotropy pattern derivable from these metrics.
  • Beyond the paper, the leading-order positivity of $\Delta S$ could be checked to next order in $\varepsilon^2\delta^2$; if the sign persists, the phase preference is robust, and if it flips, the conclusion may be an artifact of truncation.
  • Beyond the paper, the analytic backgrounds could be used to compute entanglement entropy for larger subsystems numerically, where the small-size expansion used here is not required.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper constructs perturbative anisotropic black brane solutions in D=3 and D=4 Einstein-Scalar-U(1)×SU(2) Yang-Mills theory with hyperscaling-violating asymptotics. At z=1 and with α fixed by d(α+1)=3 (α=2 for d=1, α=1/2 for d=2), and at rescaled chemical potential μ=4/√3, the SU(2) vector profile w1(u)=u²/(1+u²)² solves the same Sturm-Liouville problem as in [21]. The authors compute the leading δ²ε² backreaction on the metric, then holographic entanglement entropies for a line segment (D=3) and for straps parallel and perpendicular to the order (D=4), verify an entanglement first law, and conclude that the positive sign of the leading correction means the anisotropic phase is preferred and that the transition must occur at μ=4/√3.

Significance. If corrected and appropriately scoped, the paper would provide rare fully analytic leading-order backreacted anisotropic solutions in non-AdS holography, together with explicit minimal-surface computations and a check of the entanglement first law. The authors are transparent about integration constants and fix them by horizon regularity and asymptotic conditions, which is a strength. However, the thermodynamic interpretation is the weakest part: the phase-preference and 'must occur at the critical point' claims are not consequences of the computations shown.

major comments (3)
  1. [§4.1, Eq. (60)] The final D=3 metric function N2(u) as printed does not satisfy the stated horizon condition N2(1)=0. At u=1 the numerator of Eq. (60) is -279+838+1680-282-281=1676, so N2(1)=1676/20160≠0. Substituting C3=-281/840 into Eq. (56) instead gives a numerator -279-838u²+1680u⁴-282u⁶-281u⁸, so the sign in front of the u² term in Eq. (60) is wrong. Since N(u) enters the minimal-surface integrals (88)-(92), the D=3 entanglement entropies in §5.1 and the first-law check in §5.2 are based on a metric whose horizon is no longer at u=1. This must be corrected and the numerics rechecked.
  2. [§5.1–§5.5] The phase-preference conclusion is inferred from the sign of the δ²ε² coefficient in ΔS, e.g. Eq. (103) for D=3 and Eqs. (116), (124), (129) for D=4, with the text stating that a positive leading term means the systems prefer the anisotropic phase. This is not a valid thermodynamic selection rule. At fixed temperature and chemical potential the preferred phase minimizes the grand potential, i.e. the renormalized on-shell Euclidean action; ΔS is the entanglement entropy of a small spatial subsystem, not the thermal entropy. The first law verified in §5.2 and §5.5 uses the entanglement temperature ΔE/ΔS∼1/L, which is a geometric consistency relation, not a stability condition. The existence of a normalizable profile at μ=4/√3 locates a candidate branch; it does not show the isotropic branch is unstable or that the anisotropic phase is selected. The abstract's claim to 'confirm that the superfluid/normalfluid phase transition must occur at the critical point' is therefore not established by the computations presented. I ask the authors either to compute the free-energy difference between the two branches or to remove/reword the thermodynamic claim.
  3. [§3.1, conditions 1–3; Appendix B] The Yang-Mills profiles b0(u)=4(1-u^{-2}), w1(u)=u²/(1+u²)² and b2(u) are the same as those obtained in [21] for AdS5; the new content is the backreacted geometry. This is acceptable, but the paper should state more explicitly that conditions 1–3 are sufficient choices that reduce the D=3 and D=4 Yang-Mills equations to the known Sturm-Liouville problem, and that they are not shown to be necessary for a phase transition. In particular, the conclusion that the transition 'must occur at the critical point' requires showing that no nontrivial branch exists for μ≠4/√3, or a free-energy comparison. A concrete test is a small-μ expansion of Eqs. (37)–(38) around the normal phase to identify the onset of the zero mode; without it, the 'must' in the claim is unjustified.
minor comments (4)
  1. [Table 1] The last row should be D=3, not D=2, since d=1 implies D=d+2=3.
  2. [§5.3] The title contains a duplicated word: 'lying along along x-axis'; there are also several typographical errors elsewhere ('alytic', 'soltuions', 'precisley', 'st rep').
  3. [§5.1–§5.4] The subtraction term is called 'pure AdS3/4', but the reference geometry is the uncharged hyperscaling-violating background with the same α and z, which is not AdS; please clarify the terminology.
  4. [§4.2] The choice H2(u)=1−2u²/(24(1+u²)⁴)−... in Eq. (76) and the resulting J2(u)=−H2(u) in Eq. (79) should be explained as a gauge choice, since H2 is not fixed by the equations alone; the physical meaning of this choice is currently implicit.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: backreaction and entanglement entropy are solved from the equations of motion, and the imported Yang-Mills solution rests on independent prior work.

full rationale

The derivation is essentially self-contained for the new content. Sections 4 and 5 start from the stated action (1), impose the ansatz (6)-(9), reduce the equations of motion, and solve the backreaction order by order in epsilon and delta, fixing integration constants by horizon regularity and asymptotic hyperscaling-violation boundary conditions (e.g., C3S=0, C3=-281/840, C12=D3S=0 in D=3; CN=281/1680 with the other constants zero in D=4). No parameter is fitted to a data set and then renamed a prediction; the sign of the delta^2 epsilon^2 correction in Delta S follows from those boundary conditions, not from a prior fit to the conclusion. The one imported ingredient is the Yang-Mills solution w1(u)=u^2/(1+u^2)^2 at mu=4/sqrt(3), which is reduced by conditions 1-3 to the same Sturm-Liouville problem as in [21]. This is a genuine reduction, not a circular definition: the cited result is prior published work with stated assumptions (AdS5, z=1, alpha=0, d=3) that do not include the D=3/4 hyperscaling-violating target, so by the review rules it is real evidence and does not raise the circularity score. The empty bracket in Appendix B ('The answer is in precedent researches[].') is a missing-reference and omitted-proof flaw that should be fixed, but it is not a circular step because the Sturm-Liouville equation is displayed in the manuscript and the intended source is external prior work. The abstract's claim to 'confirm that the superfluid/normalfluid phase transition must occur at the critical point' overstates what the entanglement-entropy calculation shows (positive Delta S is not a free-energy comparison), but that is a thermodynamic-criterion and correctness concern, not a circularity of the derivation. Overall, no significant circularity is present.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The construction rests on the standard holographic dictionary, the known hyperscaling-violating black brane background borrowed from [12,23], and a sequence of choices (z=1, alpha from d(alpha+1)=3, mu=4/sqrt(3), H2 gauge) that make the equations integrable. No new entities are introduced. The perturbative expansion in epsilon and delta is truncated at leading order without error control.

free parameters (4)
  • Chemical potential mu = 4/sqrt(3)
    Fixed by condition 3 in Sec. 3.1: mu r0^alpha sqrt(3) e^{-sqrt(alpha/6) phi0}=4, with phi0=sqrt(6 alpha) log r0. This is the Sturm-Liouville eigenvalue, not fitted to data, but it is a special value chosen to make the ansatz integrable.
  • Scalar boundary value phi0 = sqrt(6 alpha) log r0
    Chosen in eq (39) to simplify condition 3 and to set the hyperscaling-violating asymptotics.
  • Lifshitz exponent z and hyperscaling exponent alpha = z=1; alpha=2 (D=3), alpha=1/2 (D=4)
    Conditions 1-2 (Sec. 3.1) set z=1 and d(alpha+1)=3. These values satisfy the null energy condition but are chosen by hand to reduce the Yang-Mills equations to the known Sturm-Liouville form.
  • Gauge function H2 = H2=0 (D=3); H2 as in eq (76) (D=4)
    The equations of motion leave H2 undetermined. The authors exercise radial gauge freedom to set it, and this choice enters the entropy computation.
assumptions (5)
  • domain assumption Gauge/gravity duality, including the Ryu-Takayanagi minimal-surface formula for entanglement entropy
    The paper interprets bulk solutions as dual to boundary superfluids and computes entanglement entropy from extremal surface areas (Sec. 5). This dictionary is assumed, not derived.
  • domain assumption The Einstein-Scalar-U(1)xSU(2) action and the isotropic hyperscaling-violating black brane background are valid starting points
    The action (1) and the background solution (10)-(17) are taken from prior work [12,23]; no UV completion is provided.
  • domain assumption Null energy condition restricts the allowed values of alpha and z
    Used in Sec. 3.1 and Appendix A to select d=1,2,3 and to justify condition 1.
  • ad hoc to paper Perturbative expansions in epsilon and delta are uniformly valid at leading order delta^2 epsilon^2
    The backreaction and entanglement results are computed to this order with no convergence or error estimates supplied (Secs. 3-5).
  • domain assumption The order parameter along x1 only captures the relevant phase transition
    The ansatz (7)-(9) is the standard p-wave superfluid ansatz; no proof is given that other symmetry-breaking channels are subleading.

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Cite this review

Pith. "Pith review of Analytic approaches to anisotropic holographic superfluids in asymptotically hyperscaling violation geometry." pith.science (2026). https://pith.science/paper/TBEHTIBO

@misc{pith2026250413635,
  author       = {Pith},
  title        = {Pith review of: Analytic approaches to anisotropic holographic superfluids in asymptotically hyperscaling violation geometry},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TBEHTIBO}},
  note         = {Machine review of arXiv:2504.13635}
}
abstract

We explore anisotropic holographic superfluidity and find analytic solutions near critical point where superfluid/normalfluid phase transition appears in a holographic dual fluid system. In arXiv:hep-th/1109.4592, the authors obtained such an analytic solution in 5-dimensional Einstein-SU(2)Yang-Mills system, and it is asymptotically AdS$_5$. What is more in this note is that we get analytic solutions near the critical point in 3- and 4- dimensional Einstein-Scalar-U(1)$\times$SU(2)Yang-Mills systems, which become asymptotically hyperscaling violation geometry. We also get leading order back reactions to the background geometry, which clearly show spatial anisotropy of the bulk geometry. To explore the properties of the spacetime, we compute holographic entanglement entropy and confirm that the superfluid/normalfluid phase transition must occur at the critical point.

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Reference graph

Works this paper leans on

47 extracted references · 6 canonical work pages · cited by 1 Pith paper

  1. [21]

    Basu and J

    P. Basu and J. H. Oh, JHEP 07, 106 (2012) doi:10.1007/JHEP07(2012)106 [arXiv:1109.4592 [hep-th]]

    arXiv 2012

  2. [1]

    Aharony, S

    O. Aharony, S. S. Gubser, J. M. Maldacena, H. Ooguri and Y. Oz , Phys. Rept. 323, 183-386 (2000) doi:10.1016/S0370-1573(99)00083-6 [arXiv:hep-th/9905 111 [hep-th]]

    doi:10.1016/s0370-1573(99)00083-6 2000

  3. [2]

    J. M. Maldacena, Adv. Theor. Math. Phys. 2, 231-252 (1998) doi:10.1023/A:1026654312961 [arXiv:hep-th/9711200 [hep-th]]

    arXiv 1998

  4. [3]

    Gauge-gravity Duality and its Applications to Cosmology a nd Fluid Dynamics

    J. H. Oh,“Gauge-gravity Duality and its Applications to Cosmology a nd Fluid Dynamics”

    work page

  5. [4]

    Kovtun, D

    P. Kovtun, D. T. Son and A. O. Starinets, Phys. Rev. Lett. 94, 111601 (2005) doi:10.1103/PhysRevLett.94.111601 [arXiv:hep-th/0405231 [hep-t h]]

    arXiv 2005

  6. [5]

    Bhattacharyya, V

    S. Bhattacharyya, V. E. Hubeny, S. Minwalla and M. Rangamani, J HEP 02, 045 (2008) doi:10.1088/1126-6708/2008/02/045 [arXiv:0712.2456 [hep-th]]

    arXiv 2008

  7. [6]

    Bhattacharyya, R

    S. Bhattacharyya, R. Loganayagam, S. Minwalla, S. Nampuri, S. P. Trivedi and S. R. Wa- dia, JHEP 02, 018 (2009) doi:10.1088/1126-6708/2009/02/018 [arXiv:0806.000 6 [hep-th]]

    work page doi:10.1088/1126-6708/2009/02/018 2009

  8. [7]

    Benincasa, A

    P. Benincasa, A. Buchel and R. Naryshkin, Phys. Lett. B 645, 309-313 (2007) doi:10.1016/j.physletb.2006.12.030 [arXiv:hep-th/0610145 [hep-th]]

    arXiv 2007

Show all 47 references
  1. [8]

    Iqbal and H

    N. Iqbal and H. Liu, Phys. Rev. D 79, 025023 (2009) doi:10.1103/PhysRevD.79.025023 [arXiv:0809.3808 [hep-th]]

    2009 arXiv

  2. [9]

    S. A. Hartnoll, Class. Quant. Grav. 26, 224002 (2009) doi:10.1088/0264-9381/26/22/224002 [arXiv:0903.3246 [hep-th]]

    2009 arXiv

  3. [10]

    C. P. Herzog, J. Phys. A 42, 343001 (2009) doi:10.1088/1751-8113/42/34/343001 [arXiv:0904.1975 [hep-th]]

    2009 arXiv

  4. [11]

    G. T. Horowitz, Lect. Notes Phys. 828, 313-347 (2011) doi:10.1007/978-3-642-04864-7 10 [arXiv:1002.1722 [hep-th]]

    2011 arXiv

  5. [12]

    Alishahiha, E

    M. Alishahiha, E. O Colgain and H. Yavartanoo, JHEP 11, 137 (2012) doi:10.1007/JHEP11(2012)137 [arXiv:1209.3946 [hep-th]]

    2012 arXiv

  6. [13]

    S. S. Gubser, Phys. Rev. D 78, 065034 (2008) doi:10.1103/PhysRevD.78.065034 [arXiv:0801.2977 [hep-th]]

    2008 arXiv

  7. [14]

    S. S. Gubser, Phys. Rev. Lett. 101, 191601 (2008) doi:10.1103/PhysRevLett.101.191601 [arXiv:0803.3483 [hep-th]]

    2008 arXiv

  8. [15]

    S. A. Hartnoll, C. P. Herzog and G. T. Horowitz, JHEP 12, 015 (2008) doi:10.1088/1126- 6708/2008/12/015 [arXiv:0810.1563 [hep-th]]

    2008 arXiv

  9. [16]

    C. P. Herzog and S. S. Pufu, JHEP 04, 126 (2009) doi:10.1088/1126-6708/2009/04/126 [arXiv:0902.0409 [hep-th]]. 29

    2009 arXiv

  10. [17]

    P. Basu, J. He, A. Mukherjee and H. H. Shieh, JHEP 11, 070 (2009) doi:10.1088/1126- 6708/2009/11/070 [arXiv:0810.3970 [hep-th]]

    2009 arXiv

  11. [18]

    Ryu and T

    S. Ryu and T. Takayanagi, Phys. Rev. Lett. 96, 181602 (2006) doi:10.1103/PhysRevLett.96.181602 [arXiv:hep-th/0603001 [hep-t h]]

    2006 arXiv

  12. [19]

    Ryu and T

    S. Ryu and T. Takayanagi, JHEP 08, 045 (2006) doi:10.1088/1126-6708/2006/08/045 [arXiv:hep-th/0605073 [hep-th]]

    2006 arXiv

  13. [20]

    Erdmenger, P

    J. Erdmenger, P. Kerner and H. Zeller, Phys. Lett. B 699, 301-304 (2011) doi:10.1016/j.physletb.2011.04.009 [arXiv:1011.5912 [hep-th]]

    2011 arXiv

  14. [22]

    J. H. Oh, JHEP 06, 103 (2012) doi:10.1007/JHEP06(2012)103 [arXiv:1201.5605 [hep-t h]]

    2012 arXiv

  15. [23]

    M. Park, J. Park and J. H. Oh, Eur. Phys. J. C 77, no.11, 810 (2017) doi:10.1140/epjc/s10052-017-5382-8 [arXiv:1609.08241 [hep-th]]

    2017 arXiv

  16. [24]

    C. Park, G. Kim, J. s. Chae and J. H. Oh, JHEP 02, 182 (2023) doi:10.1007/JHEP02(2023)182 [arXiv:2210.08919 [hep-th]]

    2023 arXiv

  17. [25]

    S. N. Solodukhin, Phys. Lett. B 665, 305-309 (2008) doi:10.1016/j.physletb.2008.05.071 [arXiv:0802.3117 [hep-th]]

    2008 arXiv

  18. [26]

    L. Y. Hung, R. C. Myers and M. Smolkin, JHEP 04, 025 (2011) doi:10.1007/JHEP04(2011)025 [arXiv:1101.5813 [hep-th]]

    2011 arXiv

  19. [27]

    Casini, M

    H. Casini, M. Huerta and R. C. Myers, JHEP 05, 036 (2011) doi:10.1007/JHEP05(2011)036 [arXiv:1102.0440 [hep-th]]

    2011 arXiv

  20. [28]

    Bhattacharya, M

    J. Bhattacharya, M. Nozaki, T. Takayanagi and T. Ugajin, Ph ys. Rev. Lett. 110, no.9, 091602 (2013) doi:10.1103/PhysRevLett.110.091602 [arXiv:1212.11 64 [hep-th]]

    2013 doi

  21. [29]

    Bianchi and R

    E. Bianchi and R. C. Myers, Class. Quant. Grav. 31, 214002 (2014) doi:10.1088/0264- 9381/31/21/214002 [arXiv:1212.5183 [hep-th]]

    2014 arXiv

  22. [30]

    Nozaki, T

    M. Nozaki, T. Numasawa, A. Prudenziati and T. Takayanagi, Ph ys. Rev. D 88, no.2, 026012 (2013) doi:10.1103/PhysRevD.88.026012 [arXiv:1304.7100 [he p-th]]

    2013 arXiv

  23. [31]

    Allahbakhshi, M

    D. Allahbakhshi, M. Alishahiha and A. Naseh, JHEP 08, 102 (2013) doi:10.1007/JHEP08(2013)102 [arXiv:1305.2728 [hep-th]]

    2013 arXiv

  24. [32]

    G. Wong, I. Klich, L. A. Pando Zayas and D. Vaman, JHEP 12, 020 (2013) doi:10.1007/JHEP12(2013)020 [arXiv:1305.3291 [hep-th]]

    2013 arXiv

  25. [33]

    Momeni, M

    D. Momeni, M. Raza, H. Gholizade and R. Myrzakulov, Int. J. The or. Phys. 55, no.11, 4751-4758 (2016) doi:10.1007/s10773-016-3098-4 [arXiv:1505.00 215 [hep-th]]. 30

    2016 doi

  26. [34]

    Park, Adv

    C. Park, Adv. High Energy Phys. 2013, 389541 (2013) doi:10.1155/2013/389541 [arXiv:1209.0842 [hep-th]]

    2013 arXiv

  27. [35]

    Park, Phys

    C. Park, Phys. Rev. D 93, no.8, 086003 (2016) doi:10.1103/PhysRevD.93.086003 [arXiv:1511.02288 [hep-th]]

    2016 arXiv

  28. [36]

    K. S. Kim and C. Park, Phys. Rev. D 95, no.10, 106007 (2017) doi:10.1103/PhysRevD.95.106007 [arXiv:1610.07266 [hep-th]]

    2017 arXiv

  29. [37]

    H. S. Jeong, K. Y. Kim and Y. W. Sun, JHEP 06, 078 (2022) doi:10.1007/JHEP06(2022)078 [arXiv:2203.07612 [hep-th]]

    2022 arXiv

  30. [38]

    L. Y. Hung, R. C. Myers, M. Smolkin and A. Yale, JHEP 12, 047 (2011) doi:10.1007/JHEP12(2011)047 [arXiv:1110.1084 [hep-th]]

    2011 arXiv

  31. [39]

    Huerta, Phys

    M. Huerta, Phys. Lett. B 710, 691-696 (2012) doi:10.1016/j.physletb.2012.03.044 [arXiv:1112.1277 [hep-th]]

    2012 arXiv

  32. [40]

    Banerjee, Y

    S. Banerjee, Y. Nakaguchi and T. Nishioka, JHEP 03, 048 (2016) doi:10.1007/JHEP03(2016)048 [arXiv:1508.00979 [hep-th]]

    2016 arXiv

  33. [41]

    M. P. Hertzberg and F. Wilczek, Phys. Rev. Lett. 106, 050404 (2011) doi:10.1103/PhysRevLett.106.050404 [arXiv:1007.0993 [hep-th]]

    2011 arXiv

  34. [42]

    Lewkowycz, R

    A. Lewkowycz, R. C. Myers and M. Smolkin, JHEP 04, 017 (2013) doi:10.1007/JHEP04(2013)017 [arXiv:1210.6858 [hep-th]]

    2013 arXiv

  35. [43]

    Rosenhaus and M

    V. Rosenhaus and M. Smolkin, JHEP 12, 179 (2014) doi:10.1007/JHEP12(2014)179 [arXiv:1403.3733 [hep-th]]

    2014 arXiv

  36. [44]

    Rosenhaus and M

    V. Rosenhaus and M. Smolkin, JHEP 02, 015 (2015) doi:10.1007/JHEP02(2015)015 [arXiv:1410.6530 [hep-th]]

    2015 arXiv

  37. [45]

    Park, Phys

    C. Park, Phys. Rev. D 92, no.12, 126013 (2015) doi:10.1103/PhysRevD.92.126013 [arXiv:1505.03951 [hep-th]]

    2015 arXiv

  38. [46]

    R. E. Arias and I. S. Landea, JHEP 01, 157 (2013) doi:10.1007/JHEP01(2013)157 [arXiv:1210.6823 [hep-th]]

    2013 arXiv

  39. [47]

    R. G. Cai, S. He, L. Li and Y. L. Zhang, JHEP 07, 027 (2012) doi:10.1007/JHEP07(2012)027 [arXiv:1204.5962 [hep-th]]. 31

    2012 arXiv

Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.