Composite order-higher-charge-condensation-charge-4e-vestigial-order-early-and-recent-works-overview

BRIEF LITERATURE OVERVIEW UNDER CONSTRUCTION  

 

States with order parameters made of four- and higher- electron composites.

 

 

These composite orders can emerge in systems

with a superconducting ground state, resulting in an

order parameter formed by four– (or more) rather than two–

electrons. While standard Bardeen–Cooper–Schrieffer

(BCS) theory forbids their formation. Early work

on superconducting systems [6,7]

 have  shown that higher-order condensates form under certain conditions

 when two key assumptions of BCS theory are

violated: (i) the superconducting order parameter

breaks multiple symmetries, and (ii) strong fluctuations

invalidate the BCS mean-field approximation,

producing a regime of incoherent Cooper pairs, i.e.

no order in bilinear fields <c_ic_i>, which still preserve higher-than-

bilinear broken composite symmetries such as

<c_ic_ic_jc_j> or <c_ic_ic_j^\dagger c_j^\dagger> . Thus, in

this regime, the only symmetries spontaneously broken

are composite symmetries constructed from quartic

or higher-order electronic operators.

This type  of order have been discussed under

different names: composite orders, charge-4e condensates, vestigial

order, quadrupling condensates.   

In connection with high-Tc

superconductors  and nematic superconductors

such phases are often termed "vestigial order" [35,36,37]

In related but distinct microscopic settings, analogous

phases have been labeled symmetric mass generation

[3] and paired phases [4, 5].

 

 

Different microscopic mechanisms have been proposed for   stabilization of this type of order

including gauge-field–mediated intercomponent

coupling [6, 7] and partial melting of pair-density wave

order [8–11]. Recent works [1, 12–14] have demonstrated that simplified analytical approaches

may lead to false positives, predicting composite

order in systems where it does not actually occur.

However in two- and three–dimensional London and Ginzburg–

Landau models has been demonstrated numerically in

several classes of superconductors with broken symmetries in large-scale Monte-Carlo simulations.

These include (i) U(1)xU(1) → U(1)

[4, 7, 15, 16], (ii) U(1)xZ2 → Z2 as in s + is,

s+id, d+id, and p+ip superconductors [14, 17–21],

(iii)SU(2) → O(3) [5, 22, 23] (iv) SU(N) → SUn(N) [24–28].

Here, the remaining broken symmetries

U(1),Z2,O(3) are associated with a four-electron

composite order. The trend observed is that it

is typically harder to stabilize these phases when

associated to higher broken symmetries.

 

Recently a microscopic

demonstration of   electron quadrupling states was published [49]

 

 

 

[6] E. Babaev, Phase diagram of planar U (1)xU (1)

superconductors: condensation of vortices with fractional

flux and a superfluid state, arXiv preprint condmat/

0201547 (2002).

[7] E. Babaev, A. Sudb., and N. W. Ashcroft, A superconductor

to superfluid phase transition in liquid

metallic hydrogen, Nature 431, 666 (2004), number:

7009 Publisher: Nature Publishing Group.

[3] N. Butt, S. Catterall, and G. C. Toga, Symmetric

mass generation in lattice gauge theory, Symmetry

13, 2276 (2021).

[4] A. B. Kuklov, N. V. Prokof’ev, B. V. Svistunov, and

M. Troyer, Deconfined criticality, runaway flow in the

two-component scalar electrodynamics and weak firstorder

superfluid-solid transitions, Annals of Physics

July 2006 Special Issue, 321, 1602 (2006).

[5] A. B. Kuklov, M. Matsumoto, N. V. Prokof’ev, B. V.

Svistunov, and M. Troyer, Deconfined Criticality:

Generic First-Order Transition in the SU(2) Symmetry

Case, Physical Review Letters 101, 050405 (2008),

publisher: American Physical Society.

[8] D. F. Agterberg and H. Tsunetsugu, Dislocations and

vortices in pair-density-wave superconductors, Nature

Physics 4, 639 (2008).

[9] L. Radzihovsky and A. Vishwanath, Quantum Liquid

Crystals in an Imbalanced Fermi Gas: Fluctuations

and Fractional Vortices in Larkin-Ovchinnikov

States, Physical Review Letters 103, 010404 (2009),

publisher: American Physical Society.

Charge-4e superconductivity from pair-density-wave order in certain high-temperature superconductors E Berg, E Fradkin, SA Kivelson - Nature Physics, 2009

[10]E Berg, E Fradkin, SA Kivelson Charge-4e superconductivity from pair-density-wave order in certain high-temperature superconductors Nature Physics, 2009 .

[11] D. F. Agterberg, M. Geracie, and H. Tsunetsugu,

Conventional and charge-six superfluids from melting

hexagonal Fulde-Ferrell-Larkin-Ovchinnikov phases

in two dimensions, Physical Review B 84, 014513

(2011), publisher: American Physical Society.

[12] A. C. Yuan, Absence of floating phase in superconductors

with time-reversal symmetry breaking on any

lattice, Physical Review B 109, 094509 (2024), publisher:

American Physical Society.

[13] P. T. How and S. Yip, Superfluid transition of a ferromagnetic

bose gas, Phys. Rev. Res. 6, L022030 (2024).

[14] P. T. How and S. K. Yip, Broken time reversal symmetry

vestigial state for a two-component superconductor

in two spatial dimensions, Physical Review B

110, 054519 (2024), publisher: American Physical Society.

[15] E. Sm.rgrav, E. Babaev, J. Smiseth, and A. Sudb.,

Observation of a Metallic Superfluid in a Numerical

Experiment, Physical Review Letters 95, 135301

(2005).

[16] E. V. Herland, E. Babaev, and A. Sudb., Phase transitions

in a three dimensional U ( 1 ) °ø U ( 1 ) lattice

London superconductor: Metallic superfluid and

charge- 4 e superconducting states, Physical Review

B 82, 134511 (2010).

[17] T. A. Bojesen, E. Babaev, and A. Sudb., Time reversal

symmetry breakdown in normal and superconducting

states in frustrated three-band systems, Physical

Review B 88, 220511 (2013), publisher: American

Physical Society.

[18] I. Maccari, J. Carlstr.m, and E. Babaev, Prediction of

time-reversal-symmetry breaking fermionic quadrupling

condensate in twisted bilayer graphene, Physical

Review B 107, 064501 (2023), publisher: American

Physical Society.

[19] T. A. Bojesen, E. Babaev, and A. Sudb., Phase transitions

and anomalous normal state in superconductors

with broken time-reversal symmetry, Physical

Review B 89, 104509 (2014), publisher: American

Physical Society.

[20] V. Grinenko, D. Weston, F. Caglieris, C. Wuttke,

C. Hess, T. Gottschall, I. Maccari, D. Gorbunov,

S. Zherlitsyn, J. Wosnitza, A. Rydh, K. Kihou, C.-H.

Lee, R. Sarkar, S. Dengre, J. Garaud, A. Charnukha,

R. Hühne, K. Nielsch, B. Büchner, H.-H. Klauss, and

E. Babaev, State with spontaneously broken timereversal

symmetry above the superconducting phase

transition, Nature Physics , 1 (2021).

[21] I. Maccari and E. Babaev, Effects of intercomponent

couplings on the appearance of time-reversal symmetry

breaking fermion-quadrupling states in twocomponent

London models, Physical Review B 105,

214520 (2022), publisher: American Physical Society.

[22] O. I. Motrunich and A. Vishwanath, Comparative

study of Higgs transition in one-component

and two-component lattice superconductor models,

arXiv:0805.1494 [cond-mat] (2008), arXiv:

0805.1494.

[23] E. V. Herland, T. A. Bojesen, E. Babaev, and

A. Sudb., Phase structure and phase transitions in

a three-dimensional $SU(2)$ superconductor, Physical

Review B 87, 134503 (2013), publisher: American

Physical Society.

[24] D. Weston and E. Babaev, Composite order in

$Ämathrm{SU}(N)$ theories coupled to an Abelian

gauge field, Physical Review B 104, 075116 (2021),

publisher: American Physical Society.

[25] C. Bonati, A. Pelissetto, and E. Vicari, Threedimensional

abelian and non-abelian gauge higgs theories,

Physics Reports 1133, 1 (2025).

[26] C. Bonati, A. Pelissetto, and E. Vicari, Lattice

abelian-higgs model with noncompact gauge fields,

Physical Review B 103, 085104 (2021).

[27] C. Bonati, A. Pelissetto, and E. Vicari, Abelian higgs

gauge theories with multicomponent scalar fields and

multiparameter scalar potentials, Physical Review B

108, 245154 (2023).

[28] C. Bonati, A. Pelissetto, and E. Vicari, Coulombhiggs

phase transition of three-dimensional lattice

abelian higgs gauge models with noncompact gauge

variables and gauge fixing, Physical Review E 108,

044125 (2023).

[29] A. B. Kuklov and B. V. Svistunov, Counterflow Superfluidity

of Two-Species Ultracold Atoms in a Commensurate

Optical Lattice, Physical Review Letters

90, 100401 (2003), publisher: American Physical Society.

[30] E. Altman, W. Hofstetter, E. Demler, and M. D.

Lukin, Phase diagram of two-component bosons on an

optical lattice, New Journal of Physics 5, 113 (2003).

[31] A. Kuklov, N. Prokof’ev, and B. Svistunov,

Superfluid-Superfluid Phase Transitions in a Two-

Component Bose-Einstein Condensate, Physical Review

Letters 92, 030403 (2004).

[32] S. G. S.yler, B. Capogrosso-Sansone, N. V. Prokof’ev,

and B. V. Svistunov, Sign-alternating interaction mediated

by strongly correlated lattice bosons, New

Journal of Physics 11, 073036 (2009).

[33] K. Sellin and E. Babaev, Superfluid drag in the twocomponent

Bose-Hubbard model, Physical Review B

97, 094517 (2018), publisher: American Physical Society.

[34] E. Blomquist, A. Syrwid, and E. Babaev, Borromean

Supercounterfluidity, Physical Review Letters 127,

255303 (2021), publisher: American Physical Society.

[35]  R. M. Fernandes, P. P. Orth, and J. Schmalian, Intertwined

vestigial order in quantum materials: Nematicity

and beyond, Annual Review of Condensed

Matter Physics 10, 133 (2019).

[36] E. Fradkin, S. A. Kivelson, and J. M. Tranquada, Colloquium:

Theory of intertwined orders in high temperature

superconductors, Reviews of Modern Physics

[37] J. Shen, W.-Y. He, N. F. Q. Yuan, Z. Huang, C.-w.

Cho, S. H. Lee, Y. S. Hor, K. T. Law, and R. Lortz,

Nematic topological superconducting phase in Nbdoped

Bi2Se3, npj Quantum

87, 457 (2015), publisher: American Physical Society.

[44] A. Kuklov, L. Radzihovsky, and B. Svistunov,

Field theory of borromean super-counterfluids, arXiv

preprint arXiv:2507.21766 (2025).

[45] Y.-G. Zheng, A. Luo, Y.-C. Shen, M.-G. He, Z.-H.

Zhu, Y. Liu, W.-Y. Zhang, H. Sun, Y. Deng, Z.-S.

Yuan, and J.-W. Pan, Counterflow superfluidity in

a two-component Mott insulator, Nature Physics 21,

208 (2025), publisher: Nature Publishing Group.

[46] C. Bonati, A. Pelissetto, and E. Vicari, Diverse

universality classes of the topological deconfinement

transitions of three-dimensional noncompact lattice

abelian higgs models, Physical Review D 109, 034517

(2024).

[47] C. Bonati, A. Pelissetto, and E. Vicari, Deconfinement

transitions in three-dimensional compact lattice

abelian higgs models with multiple-charge scalar

fields, Physical Review E 109, 044146 (2024).

[48] N. Butt, S. Catterall, and A. Hasenfratz, Symmetric

Mass Generation with Four SU(2) Doublet Fermions,

Physical Review Letters 134, 031602 (2025), publisher:

American Physical Society.

 

[49] Albert Samoilenka, Egor Babaev     Microscopic theory of electron quadrupling condensates arXiv:2505.12542