Complete Phase Diagrams for a Holographic Superconductor/Insulator System

Benson Way; Gary T. Horowitz

REVIEW

Complete Phase Diagrams for a Holographic Superconductor/Insulator System

Gary T. Horowitz, Benson Way

High Energy Physics - Theory · Submitted 21 Jul 2010 · 21 references

Pith's verdict: A solid backreacted completion of the soliton/black-hole phase diagram, with a genuinely new low-temperature window; the word 'complete' in the abstract is doing more work than the body supports.

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arXiv ID1007.3714 (version 2)

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CitationBenson Way, Gary T. Horowitz. "Complete Phase Diagrams for a Holographic Superconductor/Insulator System." arXiv:1007.3714 (2010). https://pith.science/paper/F4RAHIGO

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The thing to know: this is the backreacted version of Nishioka-Ryu-Takayanagi's probe-limit soliton superconductor, and the payoff is real. With backreaction included, the four phases (soliton insulator, soliton superconductor, charged black hole, hairy black hole) typically meet in two triple points, and as you lower the scalar charge q the triple points merge into a quadruple point and then pass through each other. That opens a narrow window of chemical potential where cooling takes you from black hole conductor to black hole superconductor to soliton insulator. That first-order superconductor-to-insulator transition on cooling is not in the probe-limit paper and is the reason to read this one. The q = 0 boson-star byproduct is also handled with unusual care, including an honest account of why the double analytic continuation gives a theory with negative kinetic energy rather than a free lunch. Nothing is fitted to the phase structure; the free energies come from independently solved bulk saddles, so the circularity worry is basically nil. The equations, ansatz, tip and horizon boundary conditions, and shooting parameters are all written out, which is more than a lot of this literature bothers to do. Soft spots, in order of how much they matter. First, 'complete' in the abstract and title outruns the body: the diagrams stop at q = 1, and Section 6 explicitly defers q < 1 to higher-precision work while arguing the low-temperature phase there is probably a hairy black hole. The authors flag this themselves, so it is an overclaim in the framing rather than a hidden gap. Second, there is no grid-refinement or shooting-convergence study, and the new window is narrow (about 0.07 in mu at q = 1). The free-energy gaps in Figure 7 are 0.03 to 0.1, plausibly above noise, but the exact edges of the window are not demonstrated to be robust. The qualitative existence of the window is probably safe; the quoted numbers are not. Third, the claim that the soliton superconductor exists at all chemical potentials rests on the plotted range plus a curvature heuristic, not a derivation. Minor. On the psi_(1) = 0 choice: that is a defining choice of which dual operator condenses, not a hidden fragility, and I would not weight it as the reader does. For holographic-superconductor people and anyone thinking about confined-phase phase structure. Worth a serious referee.

Referee report

Significance

If the phase structure holds, this is a useful and well-motivated extension of a known result: it promotes the probe-limit superconductor/insulator transition to a fully backreacted phase diagram and identifies a genuinely new first-order superconductor-to-insulator transition at low temperature, which is a falsifiable prediction for the dual field theory. The q = 0 boson-star construction is a nice byproduct with a clear gauge-theory interpretation. The equations of motion, boundary conditions, and shooting parameters are written explicitly (Eqs. 3.1–3.12, 4.1–4.7), and the free-energy comparison follows the standard Gibbs construction, so the calculation is reproducible in principle. The main strengths are the completeness of the phase-diagram survey for q ≥ 1 and the clean identification of the quadruple-point mechanism; the main weaknesses are the absence of a numerical convergence study for the narrow-window claim and the restriction to a single scalar quantization.

Major objections (3)

[Section 5, Figures 6 and 7] The central new claim [abstract:s4] rests on a narrow window of chemical potential: 1.69 ≲ μ ≲ 1.71 for q = 1.1 (width 0.02) and 1.74 ≲ μ ≲ 1.81 for q = 1 (width 0.07), as stated in the Figure 6 caption. Figure 7 shows free-energy differences of order 0.03–0.1 at μ ≈ 1.79. The manuscript reports no grid-refinement study, no tolerance for the shooting parameters, and no estimate of the numerical error in F_bhsc − F_sl at the crossing temperature. Without such a check, the sign of the free-energy difference that places the new first-order superconductor/insulator boundary is an assumption rather than a demonstrated result. Please report the radial grid size and tolerance used, and show that halving the step (or otherwise refining) leaves the crossing temperature and the width of the window stable at the quoted precision.
[Section 3.1] The manuscript states 'For numerical simplicity, we only consider the case ψ_(1) = 0.' At m² = −15/4 the Δ = 3/2 branch is equally normalizable, and the entire phase diagram is a comparison of free energies among solution branches. The paper does not discuss whether the Δ = 3/2 quantization shifts the critical chemical potentials, the triple/quadruple point locations, or the width of the new low-temperature window enough to change the qualitative phase ordering. At minimum, add a discussion of why the qualitative structure is expected to survive; ideally, repeat one representative case (e.g. q = 1.1) in the other quantization and report whether the quadruple point and the new boundary persist.
[Abstract [abstract:s2] and Section 6] The abstract claims to 'construct the complete phase diagram for this system,' but Section 6 explicitly defers q < 1 to future work and argues that for 1.86 < μ < μ_c the hairy black hole should dominate at T = 0, i.e. that the low-temperature phase ordering changes qualitatively in that regime. The authors acknowledge this limitation in Section 6, so the remaining issue is the unqualified 'complete' in the abstract. Please qualify the abstract (e.g. 'for q ≥ 1') or state the q < 1 expectation there, so that the headline claim matches the content of the paper.

Minor objections (6)

[Sections 3.1 and 4] The numerical method is not specified: it is unclear whether the boundary-value problem is solved by shooting, relaxation, or collocation, what radial grid is used, and what convergence criterion is applied. A short methods paragraph would make the results reproducible.
[Figure 6] The figure does not distinguish first-order from second-order phase boundaries by line style, and the caption does not say which boundaries are first order. Since the paper's main point is that the soliton superconductor transition and the new superconductor/insulator transition become first order at small q, marking this in the figure would make the phase diagrams much easier to read.
[Section 5] The statement that 'the superconducting soliton phase seems to exist at all chemical potentials' is ambiguous: it is unclear whether this refers to the existence of the solution branch or to the phase being thermodynamically favored. Clarifying this, and showing explicitly how the probe limit is recovered as q → ∞, would connect the results more directly to [11].
[Section 3.2] The authors note that 'It is not clear what physically distinguishes these two phases of the superconductor' across the discontinuity at μ_dis. This is an honest admission, but a brief discussion of whether the optical conductivity or another gauge-invariant observable distinguishes the two branches would substantially increase the physical value of the phase diagram.
[Section 5, Eq. (5.4)] The soliton superconductor free energy is written as F_slsc = ε_slsc − μ ρ_slsc, but the asymptotic expansion of the soliton metric from which ε_slsc is extracted is not given (only the ψ and φ expansions in Eqs. 3.7–3.8 are). Adding the relevant metric asymptotics would make the free-energy computation checkable.
[Section 6] The expectation that the hairy black hole must dominate for 1.86 < μ < μ_c at q < 1 is phrased as 'it seems clear'; since this is an extrapolation beyond the computed range, it should be flagged explicitly as a conjecture rather than a result.

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The paper

The paper claims

Including backreaction yields the complete temperature-chemical potential phase diagram of a holographic superconductor/insulator system, revealing two triple points that merge into a quadruple point and, at low scalar charge, a first-order superconductor-to-insulator transition on cooling.

The reading

In the holographic approach to strongly interacting matter, a gravitational system in five dimensions is used as a stand-in for a quantum field theory in four. Earlier work showed that a particular five-dimensional geometry, the AdS soliton, behaves like an insulator, and that adding a chemical potential makes it unstable to growing a charged scalar field, which turns it into a superconductor. That earlier analysis ignored the gravitational pull of the scalar and the electric field, which is only valid when the scalar's charge is taken very large. This paper puts the backreaction back in and solves the coupled Einstein-Maxwell-scalar equations numerically for a range of scalar charges q from 0 up to 10, with the scalar mass fixed at the value -15/4 and one spatial direction compactified to a circle of length pi.

Extended reading notes

Core claim

The full backreacted phase diagram of the five-dimensional Einstein-Maxwell-scalar theory contains four phases (soliton insulator, soliton superconductor, black hole conductor, black hole superconductor) whose boundaries typically meet in two triple points; as the scalar charge q is lowered the triple points merge into a quadruple point and then separate, opening a window of chemical potential where cooling takes the system from black hole conductor to black hole superconductor to soliton insulator. See Figure 6 and the summary in Section 6: 'there is a new first order superconductor/insulator phase transition that opens up at low temperature.'

Load-bearing premise

All solutions are restricted to the psi_(1) = 0 quantization of the scalar, i.e. only the Delta = 5/2 operator is turned on. Section 3.1 states: 'For numerical simplicity, we only consider the case psi_(1) = 0.' The paper never checks whether the Delta = 3/2 branch, which is equally normalizable at m^2 = -15/4, produces different critical chemical potentials, different triple point locations, or a different low-temperature phase ordering. Since the entire phase diagram is a comparison of free energies among these branches, a shift in any one of them moves the boundaries that constitute the central claim.

Original abstract

The gravitational dual of an insulator/superconductor transition driven by increasing the chemical potential has recently been constructed. However, the system was studied in a probe limit and only a part of the phase diagram was obtained. We include the backreaction and construct the complete phase diagram for this system. For fixed chemical potential there are typically two phase transitions as the temperature is lowered. Surprisingly, for a certain range of parameters, the system first becomes a superconductor and then becomes an insulator as the temperature approaches zero. As a byproduct of our analysis, we also construct the gravitational dual of a Bose-Einstein condensate of glueballs in a confining gauge theory.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

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What the abstract itself states, as probabilities: claims first 0.11 · states limitation 0.13 · human participants 0.01 · controlled comparison 0.05 · randomized assignment 0.01 · states quantitative result 0.05 · compares with prior results 0.87 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; show every role.

The gravitational dual of an insulator/superconductor transition driven by increasing the chemical potential has recently been constructed. However, the system was studied in a probe limit and only a part of the phase diagram was obtained. We include the backreaction and construct the complete phase diagram for this system. For fixed chemical potential there are typically two phase transitions as the temperature is lowered. Surprisingly, for a certain range of parameters, the system first becomes a superconductor and then becomes an insulator as the temperature approaches zero. As a byproduct of our analysis, we also construct the gravitational dual of a Bose-Einstein condensate of glueballs in a confining gauge theory.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	The gravitational dual of an insulator/superconductor transition driven by increasing the chemical potential has recently been constructed.	background 1.00	0.04	0.04	asserted 1.00
s1	However, the system was studied in a probe limit and only a part of the phase diagram was obtained.	gap 0.91	0.07	0.03	asserted 1.00
s2	We include the backreaction and construct the complete phase diagram for this system.	contribution 1.00	0.97	0.33	asserted 1.00
s3	For fixed chemical potential there are typically two phase transitions as the temperature is lowered.	result 1.00	0.85	0.18	asserted 0.87
s4	Surprisingly, for a certain range of parameters, the system first becomes a superconductor and then becomes an insulator as the temperature approaches zero.	result 1.00	0.89	0.36	asserted 0.99
s5	As a byproduct of our analysis, we also construct the gravitational dual of a Bose-Einstein condensate of glueballs in a confining gauge theory.	contribution 0.51	0.94	0.05	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.92 · evidence from simulation or computation 0.55.

What was examined, and what was not

sources read

arXiv listing abstract v2

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** The value of the condensate (left), and the charge d

**Figure 2.** The free energy densities of the soliton with scala

**Figure 3.** The value of the condensate (left plots), and the ch

**Figure 4.** The energy density (left), and the condensate (rig

**Figure 5.** The value of the condensate (left), and the charge d

**Figure 6.** Starting from the top left, the phase diagrams for

**Figure 7.** The free energy density vs temperature for the

Figures reproduced from arXiv:1007.3714 by the authors.

References

Works cited (21)

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Forward citations

Cited by 11 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Landau-Ginzburg Theory of a Holographic Superconductor: A Horizon Perspective on the Zero-Temperature Ground State
hep-th 2026-11 accept novelty 7.0 of 10

The quartic term removes the Lifshitz region in three distinct stages, and the horizon effective potential of Landau-Ginzburg form determines the critical point.
Competition and coexistence of baryonic condensates in AdS hardwall model
hep-th 2026-10 conditional novelty 6.0 of 10

In a hardwall AdS/QCD model with two charged scalars, diquark and dibaryon condensates coexist only within narrow windows of operator dimension, widening with charge and closing when the operators coincide.
Holographic subregion complexity in insulator/superconductor transition
hep-th 2026-08 accept novelty 6.0 of 10

Holographic subregion complexity signals the insulator-superconductor transition but its qualitative features are sensitive to subsystem scale and entanglement-wedge topology.
A Holographic P-wave Superconductor Model
hep-th 2013-09 conditional novelty 6.0 of 10

A charged complex vector field in four-dimensional anti-de Sitter gravity, solved with full backreaction, yields second-, first-, and zeroth-order phase transitions plus a retrograde condensed phase whose free energy ...
Gregory-Laflamme as the confinement/deconfinement transition in holographic QCD
hep-th 2011-07 conditional novelty 6.0 of 10

The deconfined phase of holographic QCD is dual to a localized solitonic D3 brane produced by a Gregory-Laflamme transition, not to the black D4 brane, because only the former carries the correct centre-symmetry realization.

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Numerical Methods for Finding Stationary Gravitational Solutions
hep-th 2015-10 conditional novelty 5.5 of 10

A practical guide to the Einstein-DeTurck method for stationary gravitational boundary value problems, with new numerical constructions of rotating Gregory-Laflamme onsets and lumpy AdS black holes.
Holographic Entanglement Entropy in Insulator/Superconductor Transition
hep-th 2012-03 conditional novelty 5.0 of 10

In a fully back-reacted holographic insulator/superconductor model, the entanglement entropy is non-monotonic in chemical potential: it jumps in slope at the critical point, rises to a maximum, then decreases.
A model of a Fermi liquid using gauge-gravity duality
hep-th 2011-07 conditional novelty 5.0 of 10

A hard-wall anti-de Sitter model with bulk fermions treated exactly yields a confining Fermi liquid whose Fermi surface area matches the charge density through Gauss's law.
Electron stars for holographic metallic criticality
hep-th 2010-08 accept novelty 5.0 of 10

Charged fermion fluid backreaction in AdS4 produces 'electron star' geometries with an emergent infrared Lifshitz scaling whose dynamical exponent is set by the fermion mass and density, and whose conductivity scales ...
What can gauge-gravity duality teach us about condensed matter physics?
cond-mat.str-el 2011-08 accept novelty 3.0 of 10

Holography can compute low-frequency transport at strongly coupled quantum critical points and can generate candidate compressible quantum states, but the mapping to real condensed matter systems is not yet settled.
Introduction to Holographic Superconductor Models
hep-th 2015-02 accept novelty 2.0 of 10

Bottom-up holographic models with scalar, vector, and tensor bulk fields reproduce s-wave, p-wave, and d-wave superconducting phenomenology, and these orders can compete and coexist in the dual field theory.

How later papers used it

64 later papers cite it

Evidence: substantial. 60 of the citing papers share no author with it, and 17 of those carry a confident label from Pith's citation classifier.

Pith read 19 of the 126 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

5 later papers by other authors are listed below: 3 used its method or tool and 2 compared their own results with it.

Uses its method or tool 3 of 4 papers with this label

Used this paper's metric ansatz for the AdS soliton, with B(r) vanishing at some radius r0 that is the tip of the soliton.

Phase transition in the holographic model of superfluidity with backreactions (2012) · 0.84 · paraphrased: the licence does not allow quoting
Following this paper, took linear combinations of its equations.

Kibble-Zurek Scaling in Holographic Quantum Quench : Backreaction (2014) · 0.99 · paraphrased: the licence does not allow quoting
Following this paper, found that the Einstein equations give its equations.

Towards A Holographic Model of Color Superconductivity (2011) · 0.85 · paraphrased: the licence does not allow quoting

Compares with it 2 of 14 papers with this label

Concluded that small charge q<4.5 could trigger first order flat space/boson star transitions similar to cases in this paper's holographic insulator/superconductor system.

Studies of a general flat space/boson star transition model in a box through a language similar to holographic superconductors (2017) · 0.99 · paraphrased: the licence does not allow quoting
Another somewhat surprising observation that can be made is that the phase diagrams presented in Section [ref] bear a remarkable similarity to the phase diagrams obtained in [ref] which are those of a Holographic superconductor by relating the quark mass of our work and chemical potential in those studies.

Phases of a 10-D Holographic hard wall model (2022) · Discussion · 1.00

How the literature uses it

64 arXiv papers cite this in 126 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 75 sentences without a confident label

Uses its method or tool 4 papers

The symmetries of the system allow gauge choices leading to the following forms of the fields [ref] [equation] The complete equations of motion are given in the Appendix.

Kibble-Zurek Scaling in Holographic Quantum Quench : Backreaction · Gravity Backreaction · 0.91
We use the metric ansatz for the AdS soliton [ref] [equation] with $B(r)$ vanishes at some radius $r_0$, which is the tip of the soliton.

Phase transition in the holographic model of superfluidity with backreactions · The condensation in AdS Soliton background · 0.84
Starting from the action ([ref]) with scalar derivatives replaced by covariant derivatives, the scalar and Maxwell equations are: [equation] [equation] Following [ref] , we find that the Einstein equations give: [equation] [equation] [equation]

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In addition, there are flat space/boson star transitions corresponding to holographic insulator/superconductor and global AdS/boson star systems in AdS gravity [ref] .

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This is also the situation which has been observed in [ref] ( and also in [ref] for s-wave) where it was shown that the phase transition becomes first order at the critical point.

P-Wave Holographic Insulator/Superconductor Phase Transition · Phase structure · 0.83
However, unlike the s-wave case, there is no additional first order transition occurring in the superconducting phase [ref] .

Entanglement Entropy in Holographic P-Wave Superconductor/Insulator Model · Thermodynamics and phase transition · 0.91

Cites it as support 1 paper

However, the soliton becomes unstable to the formation of scalar hair if the chemical potential is large enough [ref] .

Stability of black holes and solitons in Anti-de Sitter space-time · Planar case, $k=0$ · 0.96

Cites it as background 18 papers

The properties of this model, investigated in [ref] in the probe limit and in [ref] including the backreaction of the scalar on the metric, depend on the value of the chemical potential $μ$, which corresponds to the time component of the gauge field at the boundary (see below).

Normal modes and time evolution of a holographic superconductor after a quantum quench · The model · 0.87
And then the holographic insulator/superconductor transiton was also established in the AdS soliton spacetime [ref] .

A general holographic insulator/superconductor model away from the probe limit · Introduction · 0.85
Therefore, many authors have tried to study the holographic superconductors away from the probe limit [ref] .

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What it cites · 11 sentences without a confident label

Uses the method or tool of 1 paper

We can calculate the free energy through [ref] [equation] We can solve [ref] for $r_+$ and substitute the result into $[ref]$ to obtain $F_bh$ as a function of temperature and chemical potential.

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Compares with 4 papers

This is very different from the superconductor/insulator transition studied in [ref] .

Holographic Superconductor/Insulator Transition at Zero Temperature · Introduction · 0.98
The black holes are always unstable to forming scalar hair along a line of constant $T/μ$ exactly analogous to previous studies of the four dimensional case [ref] .

Holographic Superconductors · Introduction · 0.93
These results are similar to results for the four-dimensional superconductors studied in [ref] .

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Cites as background 14 papers

The insulator is described by the AdS soliton [ref] , which has a mass gap and has previously been used to describe a confining vacuum of a dual gauge theory.

The AdS/CFT Correspondence and a New Positive Energy Conjecture for General Relativity · Introduction · 0.89
The insulator is described by the AdS soliton [ref] , which has a mass gap and has previously been used to describe a confining vacuum of a dual gauge theory.

Anti-de Sitter Space, Thermal Phase Transition, And Confinement In Gauge Theories · Introduction · 0.89
Recently, this correspondence has led to gravitational descriptions of superfluids and superconductors [ref] .

Building an AdS/CFT superconductor · Introduction · 0.99

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

J. M. Maldacena, “The large N limit of superconformal ﬁeld theorie s and supergrav- ity,” Adv. Theor. Math. Phys. 2, 231 (1998) [Int. J. Theor. Phys. 38, 1113 (1999)] [arXiv:hep-th/9711200]

View on Pith Pith review arXiv 1998

[2] [2]

S. S. Gubser, I. R. Klebanov and A. M. Polyakov, “Gauge theory correlators from non-critical string theory,” Phys. Lett. B 428, 105 (1998) [arXiv:hep-th/9802109]

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