REVIEW 2 major objections 8 minor 1 cited by
Fractonic superfluids. III. Hybridizing higher moments
T0 review · 2 major / 8 minor · reviewed 2026-08-11 · deepseek-v4-flash
Pith's one-line read In the $[d,1,2,2]$ hybrid fractonic superfluid, a four-point correlation saturates in two dimensions, breaking the relative charge symmetry and giving true off-diagonal long-range order; hybridizing moments of different orders similarly…
desk verdict A solid third installment in the fractonic-superfluid trilogy: the hybrid-moment construction is new and the d=2 partial ODLRO claims survive scrutiny, but the long-wavelength truncation deserves a clearer justification. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing object is the conserved hybrid moment, for example $\hat Q_{\rm mix}^{(i)}=\int d^d x\,(\hat\rho_1+\hat\rho_2)x_i$ for $[d,1,2,2]$ or $\hat Q_{\rm mix}^{(i,ii)}=\int d^d x\,(\hat\rho_1 x_i\ell+\hat\rho_2 x_i^2)$ for the dipole-quadrupole model; this reduces the number of independent $U(1)$ symmetries compared with separate moment conservation and couples the two species' phase fields. The analysis then works with the effective phase-only Hamiltonian of Eq. (43), obtained in the long-wavelength limit by dropping gradient terms in the conjugate-momentum equation of motion, and with its Model Series B analogue. Diagonalizing this Hamiltonian by a Bogoliubov transformation (a linear mixing of modes that diagonalizes a quadratic boson Hamiltonian) yields one linear relative Goldstone mode, $\omega\approx\sqrt{4g\Gamma\rho_{10}\rho_{20}}\,|k|$, alongside a higher-order mode; correlation functions are computed from the resulting phase-field correlators. Saturation occurs in $d=2$ because the linear mode contributes a $1/|k|$ piece to the phase-fluctuation integrals, and in the composite (or species-1) correlation function that dangerous contribution cancels, leaving an infrared-finite integral.
What would settle it
Recompute the $d=2$ correlation functions while keeping the gradient terms dropped in Eqs. (40)-(43) and Appendix C; if $\langle\hat\Phi_1^\dagger(x)\hat\Phi_2(x)\hat\Phi_2^\dagger(0)\hat\Phi_1(0)\rangle$ in Model Series A or $\langle\hat\Phi_1^\dagger(x)\hat\Phi_1(0)\rangle$ in Model Series B develops a logarithmic infrared divergence and decays instead of saturating, the central ODLRO claim fails.
Extended reading notes
Core claim
The paper's central claim is that hybridizing higher moments across boson species produces true off-diagonal long-range order in lower spatial dimensions than non-hybrid fractonic superfluids allow. For Model Series A with conserved total dipoles ($[d,1,2,2]$), quantum fluctuations leave the single-species two-point functions power-law decaying in two dimensions, but the four-point function $\langle \hat\Phi_1^\dagger(x)\hat\Phi_2(x)\hat\Phi_2^\dagger(0)\hat\Phi_1(0)\rangle$ saturates; the system therefore breaks the relative charge symmetry $U(1)_{-,C}$ and has true ODLRO with composite order parameter $\langle\hat\Phi_1(x)\hat\Phi_2^\dagger(x)\rangle$. For arbitrary $[d,N,m,m']$ models of Model Series A, the same calculation yields partial spontaneous breaking of charge symmetry at $d=N+1$. For Model Series B, where species-1 dipoles hybridize with species-2 quadrupoles, the two-point function $\langle\hat\Phi_1^\dagger(x)\hat\Phi_1(0)\rangle$ saturates in two dimensions, breaking $U(1)_{1,C}$ with order parameter $\langle\hat\Phi_1(x)\rangle$, while $\langle\hat\Phi_2^\dagger(x)\hat\Phi_2(0)\rangle$ still decays there. The paper also constructs Bose-Hubbard-type lattice models whose weak-interaction limit recovers these continuum models, gives a mean-field phase diagram with intermediate multi-dipole condensate phases, and shows via third-order perturbation theory how the lattice Hamiltonians can be engineered in strongly tilted optical lattices.
Load-bearing premise
The long-wavelength expansion that produces the linear relative Goldstone mode assumes momentum $|k|$ is far below $2\pi/\xi_c$, so gradient terms in the equation of motion for the conjugate momentum can be dropped; if that hierarchy fails, the fluctuation integrals and the $d=2$ ODLRO conclusions change.
Editorial extensions
If this is right
- In two spatial dimensions, hybrid dipole conservation creates a condensate of inter-species particle-hole pairs: $\langle\hat\Phi_1\hat\Phi_2^\dagger\rangle$ is nonzero while single-species ODLRO is absent.
- For any $[d,N,m,m']$ model in Model Series A, partial charge-symmetry breaking occurs at $d=N+1$, so each additional order of the hybridized moment raises the onset dimension by one.
- In the dipole-quadrupole model, species 1 acquires true ODLRO in two dimensions while species 2 requires three, so asymmetric ordering is a direct signature of mixed-order hybridization.
- The Bose-Hubbard lattice versions reduce to the continuum models in the weak-$U$ limit, making the predicted hybrid fractonic superfluid phases and intermediate multi-dipole condensate phases accessible in principle to cold-atom experiments in strongly tilted optical lattices.
- The partial breaking patterns provide concrete realizations where the restrictions of generalized Mermin-Wagner-type theorems are evaded by hybrid conservation, extending the known dimensional thresholds for charge ordering.
Reading between the lines
- Beyond the paper, the cancellation that makes the composite correlation saturate suggests a design rule: pairing species whose phase fluctuations enter with opposite $1/|k|$ coefficients can suppress the most dangerous infrared fluctuations, lowering the critical dimension whenever the order parameter is built from a phase difference.
- The composite condensate in Model Series A is structurally analogous to exciton condensation, so hybrid fractonic superfluids may offer a cold-atom route to inter-species particle-hole pairing without Coulomb interactions.
- A testable extension is to drive the $[d,1,2,2]$ lattice model across the mean-field Mott-to-multi-dipole-to-hybrid-fractonic-superfluid boundaries and measure $\langle\hat b_1^\dagger\hat b_2\rangle$; the paper's phase diagram predicts this composite order parameter turns on continuously at the Mott-to-multi-dipole boundary before single-species dipole order saturates.
- The same hybridization idea could be applied to fermionic species, where hybrid-moment conservation relaxes mobility only along the hybridized channel and may realize non-Fermi-liquid behavior with partial charge ordering.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper introduces hybrid fractonic superfluids: multi-species bosonic systems in which the conserved multipole moments are built from the densities of different species. Two model series are constructed—Model Series A (hybridization of moments of the same order, with the concrete [d,1,2,2] model conserving the total dipole moments of two species) and Model Series B (hybridization of moments of different orders, with a concrete dipole-quadrupole model)—and for each the authors analyze the classical ground states, perform a harmonic (Bogoliubov) expansion about the condensate, diagonalize the effective Gaussian Hamiltonian, and compute two- and four-point correlation functions in d = 1, 2, 3. The central claims are: (i) in Model Series A in d = 2, the composite correlation ⟨Φ̂†₁(x)Φ̂₂(x)Φ̂₁(0)Φ̂†₂(0)⟩ saturates at long distances, spontaneously breaking the relative charge symmetry U(1)−,C with composite order parameter ⟨Φ̂₁Φ̂†₂⟩, and more generally partial breaking occurs in d = N+1 for the [d,N,m,m′] series; (ii) in Model Series B in d = 2, ⟨Φ̂†₁(x)Φ̂₁(0)⟩ saturates so the species-1 charge symmetry is spontaneously broken, while species 2 retains only power-law order. The paper also constructs Bose-Hubbard-type lattice models for both series, derives a mean-field phase diagram for the [d,1,2,2] lattice model showing intermediate multi-dipole condensate phases, and proposes a realization in strongly tilted optical lattices via third-order perturbation theory.
Significance. If the two d = 2 claims survive scrutiny, the paper establishes a genuinely new symmetry-breaking phenomenon: higher-moment conservation no longer forces the absence of ODLRO in two dimensions; instead, hybridization of moments from different species allows partial spontaneous symmetry breaking with a composite order parameter (Series A) and single-species ODLRO (Series B). This extends the fractonic-superfluid phenomenology of Refs. [43,44] in a non-trivial way and provides new instances for generalized Mermin-Wagner physics. The paper's strengths are that the results are derived analytically from the constructed Hamiltonians with no data fitting, the Bogoliubov diagonalization and T-matrix elements in Appendices A-D are detailed enough to be checked algebraically, the lattice constructions are explicit and respect the stated conservation laws, and the tilted-lattice realization gives a concrete experimental route.
major comments (2)
- [Secs. III B-III C and Appendix B] The d = 2 ODLRO claim for Model Series A (saturation of ⟨Φ̂†₁(x)Φ̂₂(x)Φ̂₁(0)Φ̂†₂(0)⟩, Table III) is established only within the truncated Hamiltonian (43), obtained by dropping the gradient terms in the equation of motion (40) under the hierarchy |k| ≪ 2π/ξc of Eq. (42). The correlation integrals in Appendix B, e.g. Eqs. (B24)-(B31), are evaluated up to the cutoff 2π/ξc, where the dropped Γ(ρb0/ρa0)k² term is of the same order as the retained g term, so part of the integration domain lies outside the controlled regime. The saturation relies on the exact cancellation of the 1/|k|² (quadratic-mode) contributions among ⟨θ1θ1⟩, ⟨θ2θ2⟩ and ⟨θ1θ2⟩, and this cancellation is verified in Eqs. (B8)-(B31) only with the T-matrix of the truncated theory. Since the θ-sector matrix M2 is identical in the full Gaussian Hamiltonian (39) and in the truncated one (43) while only the π-sector M1 differs, the cancellation is expected to survive the π-sector gradients, but that argument is not given in the paper; the same truncation underpins the [d,N,m,m′] generalization of Sec. III D. The authors should either repeat the correlation calculation with the full Hamiltonian (39) or prove that the UV part of the integrals only renormalizes the saturation constant.
- [Sec. IV C, Appendix D, and Table IV] The Model Series B d = 2 conclusions are not supported by the presented calculations. Appendix D 2 explicitly replaces the anisotropic integrands by a 'rough approximation' (D26), and Sec. IV C concedes that only 'a general trend' is obtained. The one-sided conservativeness claim in Appendix D 2 (saturation within the approximation implies true saturation, 'but not the opposite') is not justified, because the pointwise bound |T21|² ≲ 1/(c21|k|) cannot exclude a subleading 1/|k|² contribution to ⟨θ1θ1⟩ coming from the nodal directions of the anisotropic dispersion (D22), where the leading-order expressions (D24)-(D25) are singular and are controlled by subleading terms that were dropped. In addition, the analytic evaluation (D33)-(D35) yields exponential decay of ⟨Φ̂†₂(x)Φ̂₂(0)⟩ in d = 2, which contradicts the 'power-law decay' entry in Table IV and the text after Eq. (D41); the numerical check at the single point x = 10^10 cannot distinguish these behaviors. A controlled calculation using the full effective Hamiltonian (C7), including the subleading terms near the nodal directions of ω2, is required to establish both the saturation of ⟨Φ̂†₁(x)Φ̂₁(0)⟩ and the decay law of ⟨Φ̂†₂(x)Φ̂₂(0)⟩.
minor comments (8)
- [Eq. (B31)] The exponent of the two-dimensional four-point correlation function is written with c1 in both places ('e^{2/(πc1|x|)−2/(πc1ξc)}'); substituting Eqs. (B24)-(B27) into the combination ⟨θ1θ1⟩+⟨θ2θ2⟩−2⟨θ1θ2⟩ gives the same expression with c2 in place of c1, and Table III indeed uses c2.
- [Appendix E and Figs. 2-3] The Landau expansion E = const + R|Ψ|² + W|Ψ|⁴ + ... assumes W > 0 without computation; since the phase boundary R = 0 of Eq. (E9) is used to draw the schematic phase diagrams, the assumed sign of W (which controls whether the MI-MDC transition is continuous) should be stated explicitly where the figures are discussed.
- [Sec. V A] For Lattice Model Series B the K₂ kinetic terms are dropped ('we neglect the K^{ijk}_2 terms'), so the stated reduction of the lattice model to the continuum Model Series B in the small-U limit holds only in the K₂ = 0 sector; the text should say so explicitly.
- [Throughout] There are several typos and infelicities: 'coherent discucssions' (Sec. I), 'quanities' (Table II caption), 'Arbitary' (heading of Sec. III D), and 'the indices can arbitary taken' (Sec. II C); these should be corrected.
- [Sec. II C, Eqs. (23)-(26)] The general definition of the conserved hybrid moments of Model Series B is only spelled out concretely for the m′ = 2 example; the index rule for a general m′-species construction is stated verbally and should be formalized, in particular the counting of which components of N_a-th order moments enter the hybridized charges.
- [Fig. 4 and Appendix D] The numerical curve in Fig. 4(a) for ⟨Φ̂†₂(x,0)Φ̂₂(0)⟩ is not compared with a fitted functional form; adding a fit against both power-law and exponential forms would make the asserted decay law quantitative.
- [Appendix F, Eq. (F8)] The effective Hamiltonian (F8) contains coefficients t²₀ that are not defined in the appendix; the text should state whether t₀ equals the hopping strength t of Eq. (83).
- [Footnotes 2 and Sec. III C] The correlation functions are regularized by replacing short-distance arguments with the coherence length ξc; the dependence of the saturation constants in Table III (e.g., e^{−2/(πc2ξc)}) on this regularization choice is not discussed, and a clarifying sentence on the scheme would be helpful.
Circularity Check
No circularity: the ODLRO claims are derived from the explicit model Hamiltonians; self-citations frame the work but do not carry the argument.
full rationale
The paper's central predictions are consequences of explicit Hamiltonians constructed in Sec. II and diagonalized in Secs. III-IV and Appendices A-D. The d=2 saturation of the composite correlation in Model Series A follows from the Bogoliubov diagonalization of Eq. (44) with matrix elements (46)-(47) and the explicit phase-correlation integrals in Appendix B; Model Series B uses the analogous diagonalization in Appendix C. No parameter is fitted and no correlation function is normalized to an input; the constants c1, c2, c11, c21 and xi_c are combinations of the Hamiltonian couplings (g, kappa, Gamma, densities), not adjustable outputs. The references to the authors' prior papers [43,44] provide the log-Phi notation, the minimal-Hamiltonian construction pattern, and the non-hybrid benchmarks, but the hybrid Hamiltonians (14), (20), (35) and their spectra are new and self-contained. The long-wavelength truncation of Eq. (40) into Eq. (43) is a validity/control assumption; the concern that |k| reaches 2*pi/xi_c at the UV end of the fluctuation integrals is a correctness issue about the approximation, not a reduction of output to input by construction. Thus no circular step meeting the quoted-evidence standard is present.
Assumptions & free parameters
free parameters (1)
- ℓ
assumptions (5)
- standard math Canonical bosonic commutation relations and the density-phase expansion Φ_a = sqrt(ρ_a0 + f_a) exp(iθ_a) with small density fluctuations.
- domain assumption The model Hamiltonians Eq. (20) and Eq. (35) are the minimal local Hamiltonians commuting with the chosen conserved hybrid moments, with all couplings positive.
- domain assumption Long-wavelength truncation: for |k| much less than 2π divided by the coherence length, the equation of motion for π_a reduces to π_a ≈ θdot_a/g and higher-gradient terms are negligible.
- domain assumption In Model Series B, the anisotropic matrix elements are bounded by |T21|² ≲ 1/(c21|k|) and |T22|² ≲ 1/(c22|k|^3), and the symmetry-breaking conclusion is argued to be one-sided.
- domain assumption The mean-field decoupling of Lattice Model A assumes the quartic coefficient W > 0 to locate the Mott insulator to multi-dipole condensate boundary.
Cite this review
Pith. "Pith review of Fractonic superfluids. III. Hybridizing higher moments." pith.science (2026). https://pith.science/paper/V5PXLUWO
@misc{pith2026241210280,
author = {Pith},
title = {Pith review of: Fractonic superfluids. III. Hybridizing higher moments},
year = {2026},
howpublished = {\url{https://pith.science/paper/V5PXLUWO}},
note = {Machine review of arXiv:2412.10280}
}
read the original abstract
Fractonic superfluids are featured by the interplay of spontaneously broken charge symmetry and mobility constraints on single-particle kinematics due to the conservation of higher moments, such as dipoles, angular charge moments, and quadrupoles. Building on prior studies by Yuan \textit{et al.} [\href{https://doi.org/10.1103/PhysRevResearch.2.023267}{Phys. Rev. Res. 2, 023267 (2020)}] and Chen \textit{et al.} [\href{https://doi.org/10.1103/PhysRevResearch.3.013226}{Phys. Rev. Res. 3, 013226 (2021)}], we study a class of fractonic superfluids, termed \textit{hybrid fractonic superfluids} (HFS), in which bosons of multiple species interact while moment hybridization is conserved. We explore the consequences of hybridization via two model series: \textit{Model Series A}, conserving total moments of the same order across species, and \textit{Model Series B}, conserving total moments of different orders. In Model Series A, we analyze dipole moment hybridization and extend the discussion to higher-order moments, examining the ground state, Goldstone modes, correlation functions, and so on. We compute the minimal spatial dimensions, where the total charge symmetry begins to get partially broken via particle-hole condensation, leading to true off-diagonal long-range order. In Model Series B, we focus on HFS with hybrid dipole-quadrupole conservation. For both series, we introduce Bose-Hubbard-type lattice models that reduce to either of both series in the weak Hubbard interaction regime. We perform a mean-field analysis on the global phase diagram and discuss experimental realizations in strongly tilted optical lattices via a third-order perturbation theory. This work, alongside prior studies, completes a trilogy on fractonic superfluids, uncovering symmetry-breaking physics emerging from higher moment conservation, leaving various promising studies for future investigation.
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A Partially Massless Superconductor
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Works this paper leans on
-
[1]
Chamon, Quantum glassiness in strongly correlated clean systems: An example of topological overprotection, Phys
C. Chamon, Quantum glassiness in strongly correlated clean systems: An example of topological overprotection, Phys. Rev. Lett. 94, 040402 (2005)
2005
-
[2]
Vijay, J
S. Vijay, J. Haah, and L. Fu, A new kind of topological quan- tum order: A dimensional hierarchy of quasiparticles built from stationary excitations, Phys. Rev. B 92, 235136 (2015)
2015
-
[3]
Vijay, J
S. Vijay, J. Haah, and L. Fu, Fracton topological order, gen- eralized lattice gauge theory, and duality, Phys. Rev. B 94, 235157 (2016)
2016
-
[4]
Shirley, K
W. Shirley, K. Slagle, Z. Wang, and X. Chen, Fracton mod- els on general three-dimensional manifolds, Phys. Rev. X 8, 031051 (2018)
2018
-
[5]
H. Song, J. Sch ¨onmeier-Kromer, K. Liu, O. Viyuela, L. Pol- let, and M. A. Martin-Delgado, Optimal thresholds for frac- ton codes and random spin models with subsystem symmetry, Phys. Rev. Lett. 129, 230502 (2022)
2022
-
[6]
H. Ma, E. Lake, X. Chen, and M. Hermele, Fracton topologi- cal order via coupled layers, Phys. Rev. B 95, 245126 (2017)
2017
-
[7]
Shirley, K
W. Shirley, K. Slagle, and X. Chen, Foliated fracton order from gauging subsystem symmetries, SciPost Phys. 6, 041 (2019)
2019
-
[8]
A. Prem, J. Haah, and R. Nandkishore, Glassy quantum dy- namics in translation invariant fracton models, Phys. Rev. B 16 95, 155133 (2017)
2017
Show all 131 references
-
[9]
A. Dua, I. H. Kim, M. Cheng, and D. J. Williamson, Sorting topological stabilizer models in three dimensions, Phys. Rev. B 100, 155137 (2019)
2019
-
[10]
R. M. Nandkishore and M. Hermele, Fractons, Annual Review of Condensed Matter Physics 10, 295 (2019)
2019
-
[11]
Bulmash and M
D. Bulmash and M. Barkeshli, Gauging fractons: Immobile non-abelian quasiparticles, fractals, and position-dependent degeneracies, Phys. Rev. B 100, 155146 (2019)
2019
-
[12]
Prem, S.-J
A. Prem, S.-J. Huang, H. Song, and M. Hermele, Cage-net fracton models, Phys. Rev. X 9, 021010 (2019)
2019
-
[13]
Li and P
M.-Y . Li and P. Ye, Fracton physics of spatially extended ex- citations, Phys. Rev. B 101, 245134 (2020)
2020
-
[14]
Li and P
M.-Y . Li and P. Ye, Fracton physics of spatially extended exci- tations. ii. polynomial ground state degeneracy of exactly solv- able models, Phys. Rev. B 104, 235127 (2021)
2021
-
[15]
Li and P
M.-Y . Li and P. Ye, Hierarchy of entanglement renormal- ization and long-range entangled states, Phys. Rev. B 107, 115169 (2023)
2023
-
[16]
Slagle, Foliated quantum field theory of fracton order, Phys
K. Slagle, Foliated quantum field theory of fracton order, Phys. Rev. Lett. 126, 101603 (2021)
2021
-
[17]
Zhou, M.-Y
C. Zhou, M.-Y . Li, Z. Yan, P. Ye, and Z. Y . Meng, Evolution of dynamical signature in the x-cube fracton topological order, Phys. Rev. Res. 4, 033111 (2022)
2022
-
[18]
Zhu, J.-Y
G.-Y . Zhu, J.-Y . Chen, P. Ye, and S. Trebst, Topological frac- ton quantum phase transitions by tuning exact tensor network states, Phys. Rev. Lett. 130, 216704 (2023)
2023
-
[19]
Canossa, L
G. Canossa, L. Pollet, M. A. Martin-Delgado, H. Song, and K. Liu, Exotic symmetry breaking properties of self-dual frac- ton spin models, Phys. Rev. Res. 6, 013304 (2024)
2024
-
[20]
B.-X. Li, Y . Zhou, and P. Ye, Three-dimensional fracton topo- logical orders with boundary toeplitz braiding, Phys. Rev. B 110, 205108 (2024)
2024
-
[21]
Devakul, S
T. Devakul, S. A. Parameswaran, and S. L. Sondhi, Correlation function diagnostics for type-i fracton phases, Phys. Rev. B97, 041110 (2018)
2018
-
[22]
Devakul and D
T. Devakul and D. J. Williamson, Universal quantum compu- tation using fractal symmetry-protected cluster phases, Phys. Rev. A 98, 022332 (2018)
2018
-
[23]
Y . You, T. Devakul, F. J. Burnell, and S. L. Sondhi, Subsys- tem symmetry protected topological order, Phys. Rev. B 98, 035112 (2018)
2018
-
[24]
Devakul, Y
T. Devakul, Y . You, F. J. Burnell, and S. L. Sondhi, Fractal Symmetric Phases of Matter, SciPost Phys. 6, 007 (2019)
2019
-
[25]
Devakul, Classifying local fractal subsystem symmetry- protected topological phases, Phys
T. Devakul, Classifying local fractal subsystem symmetry- protected topological phases, Phys. Rev. B99, 235131 (2019)
2019
-
[26]
D. T. Stephen, H. P. Nautrup, J. Bermejo-Vega, J. Eisert, and R. Raussendorf, Subsystem symmetries, quantum cellular au- tomata, and computational phases of quantum matter, Quan- tum 3, 142 (2019)
2019
-
[27]
A. T. Schmitz, S.-J. Huang, and A. Prem, Entanglement spec- tra of stabilizer codes: A window into gapped quantum phases of matter, Phys. Rev. B99, 205109 (2019)
2019
-
[28]
Y . You, T. Devakul, S. L. Sondhi, and F. J. Burnell, Frac- tonic chern-simons and bf theories, Phys. Rev. Res.2, 023249 (2020)
2020
-
[29]
J. F. San Miguel, A. Dua, and D. J. Williamson, Bifurcating subsystem symmetric entanglement renormalization in two di- mensions, Phys. Rev. B 103, 035148 (2021)
2021
-
[30]
F. J. Burnell, T. Devakul, P. Gorantla, H. T. Lam, and S.-H. Shao, Anomaly inflow for subsystem symmetries, Phys. Rev. B 106, 085113 (2022)
2022
-
[31]
Zhou, M.-Y
C. Zhou, M.-Y . Li, Z. Yan, P. Ye, and Z. Y . Meng, Detect- ing subsystem symmetry protected topological order through strange correlators, Phys. Rev. B 106, 214428 (2022)
2022
-
[32]
Zhang, M.-Y
J.-Y . Zhang, M.-Y . Li, and P. Ye, Higher-order cellular au- tomata generated symmetry-protected topological phases and detection through multi point strange correlators, PRX Quan- tum 5, 030342 (2024)
2024
-
[33]
Pretko, Generalized electromagnetism of subdimensional particles: A spin liquid story, Phys
M. Pretko, Generalized electromagnetism of subdimensional particles: A spin liquid story, Phys. Rev. B96, 035119 (2017)
2017
-
[34]
Pretko, Subdimensional particle structure of higher rank u(1) spin liquids, Phys
M. Pretko, Subdimensional particle structure of higher rank u(1) spin liquids, Phys. Rev. B 95, 115139 (2017)
2017
-
[35]
Pretko, The fracton gauge principle, Phys
M. Pretko, The fracton gauge principle, Phys. Rev. B 98, 115134 (2018)
2018
-
[36]
Pai and M
S. Pai and M. Pretko, Dynamical scar states in driven fracton systems, Phys. Rev. Lett. 123, 136401 (2019)
2019
-
[37]
Ma and M
H. Ma and M. Pretko, Higher-rank deconfined quantum criti- cality at the lifshitz transition and the exciton bose condensate, Phys. Rev. B 98, 125105 (2018)
2018
-
[38]
Pai and M
S. Pai and M. Pretko, Fractonic line excitations: An inroad from three-dimensional elasticity theory, Phys. Rev. B 97, 235102 (2018)
2018
-
[39]
S. Pai, M. Pretko, and R. M. Nandkishore, Localization in fractonic random circuits, Phys. Rev. X 9, 021003 (2019)
2019
-
[40]
Gromov, Towards classification of fracton phases: The multipole algebra, Phys
A. Gromov, Towards classification of fracton phases: The multipole algebra, Phys. Rev. X 9, 031035 (2019)
2019
-
[41]
Wang and S.-T
J. Wang and S.-T. Yau, Non-abelian gauged fracton matter field theory: Sigma models, superfluids, and vortices, Phys. Rev. Res. 2, 043219 (2020)
2020
-
[42]
Feldmeier, P
J. Feldmeier, P. Sala, G. De Tomasi, F. Pollmann, and M. Knap, Anomalous diffusion in dipole- and higher-moment- conserving systems, Phys. Rev. Lett. 125, 245303 (2020)
2020
-
[43]
J.-K. Yuan, S. A. Chen, and P. Ye, Fractonic superfluids, Phys. Rev. Res. 2, 023267 (2020)
2020
-
[44]
S. A. Chen, J.-K. Yuan, and P. Ye, Fractonic superfluids. ii. condensing subdimensional particles, Phys. Rev. Res. 3, 013226 (2021)
2021
-
[45]
J.-K. Yuan, S. A. Chen, and P. Ye, Quantum hydrodynamics of fractonic superfluids with lineon condensate: From navier– stokes-like equations to landau-like criterion, Chin. Phys. Lett. 39, 057101 (2022)
2022
-
[46]
Li and P
H. Li and P. Ye, Renormalization group analysis on emergence of higher rank symmetry and higher moment conservation, Phys. Rev. Res. 3, 043176 (2021)
2021
-
[47]
J.-K. Yuan, S. A. Chen, and P. Ye, Hierarchical proliferation of higher-rank symmetry defects in fractonic superfluids, Phys. Rev. B 107, 205134 (2023)
2023
-
[48]
S. A. Chen and P. Ye, Many-body physics of spontaneously broken higher-rank symmetry: from fractonic superfluids to dipolar hubbard model (2023), arXiv:2305.00941 [cond- mat.str-el]
2023 arXiv
-
[49]
Gromov, A
A. Gromov, A. Lucas, and R. M. Nandkishore, Fracton hydro- dynamics, Phys. Rev. Res. 2, 033124 (2020)
2020
-
[50]
A. Jain, K. Jensen, R. Liu, and E. Mefford, Dipole superfluid hydrodynamics, Journal of High Energy Physics 2023, 184 (2023), arXiv:2304.09852 [hep-th]
2023 arXiv
-
[51]
Glorioso, X
P. Glorioso, X. Huang, J. Guo, J. F. Rodriguez-Nieva, and A. Lucas, Goldstone bosons and fluctuating hydrodynamics with dipole and momentum conservation, Journal of High En- ergy Physics 2023, 22 (2023)
2023
-
[52]
Gł ´odkowski, F
A. Gł ´odkowski, F. Pe˜na-Ben´ıtez, and P. Sur´owka, Dissipative fracton superfluids, Journal of High Energy Physics2024, 285 (2024)
2024
-
[53]
Armas and E
J. Armas and E. Have, Ideal fracton superfluids, SciPost Phys. 16, 039 (2024)
2024
-
[54]
A. Jain, K. Jensen, R. Liu, and E. Mefford, Dipole superfluid hydrodynamics. Part II., Journal of High Energy Physics2024, 17 197 (2024), arXiv:2401.16385 [hep-th]
2024 arXiv
-
[55]
Morningstar, N
A. Morningstar, N. O’Dea, and J. Richter, Hydrodynamics in long-range interacting systems with center-of-mass conserva- tion, Phys. Rev. B 108, L020304 (2023)
2023
-
[56]
Stahl, M
C. Stahl, M. Qi, P. Glorioso, A. Lucas, and R. Nandkishore, Fracton superfluid hydrodynamics, Phys. Rev. B 108, 144509 (2023)
2023
-
[57]
Osborne and A
A. Osborne and A. Lucas, Infinite families of fracton flu- ids with momentum conservation, Phys. Rev. B 105, 024311 (2022)
2022
-
[58]
Boesl, P
J. Boesl, P. Zechmann, J. Feldmeier, and M. Knap, Deconfine- ment dynamics of fractons in tilted bose-hubbard chains, Phys. Rev. Lett. 132, 143401 (2024)
2024
-
[59]
Doshi and A
D. Doshi and A. Gromov, V ortices as fractons, Communica- tions Physics 4, 44 (2021)
2021
-
[60]
K. T. Grosvenor, C. Hoyos, F. Pe ˜na Benitez, and P. Sur´owka, Hydrodynamics of ideal fracton fluids, Phys. Rev. Res. 3, 043186 (2021)
2021
-
[61]
K. T. Grosvenor, C. Hoyos, F. Pe ˜na-Ben´ıtez, and P. Sur´owka, Space-dependent symmetries and fractons, Frontiers in Physics 9, 10.3389/fphy.2021.792621 (2022)
2022
-
[62]
Zechmann, J
P. Zechmann, J. Boesl, J. Feldmeier, and M. Knap, Dynamical spectral response of fractonic quantum matter, Phys. Rev. B 109, 125137 (2024)
2024
-
[63]
Angus, M
S. Angus, M. Kim, and J.-H. Park, Fractons, non-riemannian geometry, and double field theory, Phys. Rev. Res. 4, 033186 (2022)
2022
-
[64]
Pe ˜na Ben´ıtez, Fractons, symmetric gauge fields and geom- etry, Phys
F. Pe ˜na Ben´ıtez, Fractons, symmetric gauge fields and geom- etry, Phys. Rev. Res.5, 013101 (2023)
2023
-
[65]
Bertolini, N
E. Bertolini, N. Maggiore, and G. Palumbo, Covariant frac- ton gauge theory with boundary, Phys. Rev. D 108, 025009 (2023)
2023
-
[66]
Bertolini and H
E. Bertolini and H. Kim, Covariant Interacting Fractons, arXiv e-prints , arXiv:2410.01397 (2024), arXiv:2410.01397 [hep- th]
2024 arXiv
-
[67]
Bertolini and H
E. Bertolini and H. Kim, Strings as Hyper-Fractons, arXiv e- prints , arXiv:2410.11678 (2024), arXiv:2410.11678 [hep-th]
2024
-
[68]
J. Wang, K. Xu, and S.-T. Yau, Higher-rank tensor non-abelian field theory: Higher-moment or subdimensional polynomial global symmetry, algebraic variety, noether’s theorem, and gauging, Phys. Rev. Res. 3, 013185 (2021)
2021
-
[69]
Argurio, C
R. Argurio, C. Hoyos, D. Musso, and D. Naegels, Fractons in effective field theories for spontaneously broken translations, Phys. Rev. D 104, 105001 (2021)
2021
-
[70]
Iaconis, A
J. Iaconis, A. Lucas, and R. Nandkishore, Multipole conser- vation laws and subdiffusion in any dimension, Phys. Rev. E 103, 022142 (2021)
2021
-
[71]
Stahl, E
C. Stahl, E. Lake, and R. Nandkishore, Spontaneous breaking of multipole symmetries, Phys. Rev. B 105, 155107 (2022)
2022
-
[72]
Radzihovsky, Lifshitz gauge duality, Phys
L. Radzihovsky, Lifshitz gauge duality, Phys. Rev. B 106, 224510 (2022)
2022
-
[73]
Kapustin and L
A. Kapustin and L. Spodyneiko, Hohenberg-mermin-wagner- type theorems and dipole symmetry, Phys. Rev. B106, 245125 (2022)
2022
-
[74]
E. Lake, M. Hermele, and T. Senthil, Dipolar bose-hubbard model, Phys. Rev. B 106, 064511 (2022)
2022
-
[75]
Lake, H.-Y
E. Lake, H.-Y . Lee, J. H. Han, and T. Senthil, Dipole conden- sates in tilted bose-hubbard chains, Phys. Rev. B 107, 195132 (2023)
2023
-
[76]
Giergiel, R
K. Giergiel, R. Lier, P. Sur´owka, and A. Kosior, Bose-hubbard realization of fracton defects, Phys. Rev. Res. 4, 023151 (2022)
2022
-
[77]
Zechmann, E
P. Zechmann, E. Altman, M. Knap, and J. Feldmeier, Fractonic luttinger liquids and supersolids in a constrained bose-hubbard model, Phys. Rev. B 107, 195131 (2023)
2023
-
[78]
W. Xu, C. Lv, and Q. Zhou, Multipolar condensates and mul- tipolar Josephson effects, Nature Communications 15, 4786 (2024)
2024
-
[79]
Gorantla, H
P. Gorantla, H. T. Lam, N. Seiberg, and S.-H. Shao, Global dipole symmetry, compact lifshitz theory, tensor gauge theory, and fractons, Phys. Rev. B 106, 045112 (2022)
2022
-
[80]
Lake and T
E. Lake and T. Senthil, Non-fermi liquids from kinetic con- straints in tilted optical lattices, Phys. Rev. Lett. 131, 043403 (2023)
2023
-
[81]
Molina-Vilaplana, A post-Gaussian approach to dipole sym- metries and interacting fractons, Journal of High Energy Physics 2023, 65 (2023)
J. Molina-Vilaplana, A post-Gaussian approach to dipole sym- metries and interacting fractons, Journal of High Energy Physics 2023, 65 (2023)
2023
-
[82]
Gorantla, H
P. Gorantla, H. T. Lam, N. Seiberg, and S.-H. Shao, (2+1)- dimensional compact lifshitz theory, tensor gauge theory, and fractons, Phys. Rev. B 108, 075106 (2023)
2023
-
[83]
Anakru and Z
A. Anakru and Z. Bi, Non-fermi liquids from dipolar symme- try breaking, Phys. Rev. B 108, 165112 (2023)
2023
-
[84]
Huang, A Chern-Simons theory for dipole symmetry, Sci- Post Phys
X. Huang, A Chern-Simons theory for dipole symmetry, Sci- Post Phys. 15, 153 (2023)
2023
-
[85]
Afxonidis, A
E. Afxonidis, A. Caddeo, C. Hoyos, and D. Musso, Dipole symmetry breaking and fractonic Nambu-Goldstone mode, SciPost Phys. Core 6, 082 (2023)
2023
-
[86]
H. T. Lam, Classification of dipolar symmetry-protected topo- logical phases: Matrix product states, stabilizer hamiltonians, and finite tensor gauge theories, Phys. Rev. B 109, 115142 (2024)
2024
-
[87]
J. H. Han, E. Lake, and S. Ro, Scaling and localization in multipole-conserving diffusion, Phys. Rev. Lett. 132, 137102 (2024)
2024
-
[88]
F. J. Burnell, S. Moudgalya, and A. Prem, Filling constraints on translation invariant dipole conserving systems, Phys. Rev. B 110, L121113 (2024)
2024
-
[89]
Chavda, D
A. Chavda, D. Naegels, and J. Staunton, Fractonic Coset Construction for Spontaneously Broken Translations, arXiv e- prints , arXiv:2409.15193 (2024), arXiv:2409.15193 [hep-th]
2024 arXiv
-
[90]
Hu and B
Y .-M. Hu and B. Lian, The Bosonic Quantum Breakdown Hubbard Model, arXiv e-prints , arXiv:2401.04309 (2024), arXiv:2401.04309 [cond-mat.str-el]
2024
-
[91]
Jensen and A
K. Jensen and A. Raz, Large n fractons, Phys. Rev. Lett. 132, 071603 (2024)
2024
-
[92]
Gaiotto, A
D. Gaiotto, A. Kapustin, N. Seiberg, and B. Willett, Gener- alized Global Symmetries, JHEP 02, 172, arXiv:1412.5148 [hep-th]
-
[93]
McGreevy, Generalized symmetries in condensed matter, Annual Review of Condensed Matter Physics 14, 57 (2023)
J. McGreevy, Generalized symmetries in condensed matter, Annual Review of Condensed Matter Physics 14, 57 (2023)
2023
-
[94]
Zhai, Ultracold atomic physics (Cambridge University Press, 2021) pp
H. Zhai, Ultracold atomic physics (Cambridge University Press, 2021) pp. 247–261
2021
-
[95]
Y .-T. Oh, J. Kim, E.-G. Moon, and J. H. Han, Rank-2 toric code in two dimensions, Phys. Rev. B 105, 045128 (2022)
2022
-
[96]
Liang, Y
Z. Liang, Y . Xu, J. T. Iosue, and Y .-A. Chen, Extracting topo- logical orders of generalized pauli stabilizer codes in two di- mensions, PRX Quantum 5, 030328 (2024)
2024
-
[97]
Liang, B
Z. Liang, B. Yang, J. T. Iosue, and Y .-A. Chen, Operator al- gebra and algorithmic construction of boundaries and defects in (2+1)D topological Pauli stabilizer codes, arXiv e-prints , arXiv:2410.11942 (2024), arXiv:2410.11942 [quant-ph]
2024 arXiv
-
[98]
Khemani, M
V . Khemani, M. Hermele, and R. Nandkishore, Localization from hilbert space shattering: From theory to physical realiza- tions, Phys. Rev. B 101, 174204 (2020)
2020
-
[99]
Guardado-Sanchez, A
E. Guardado-Sanchez, A. Morningstar, B. M. Spar, P. T. Brown, D. A. Huse, and W. S. Bakr, Subdiffusion and heat transport in a tilted two-dimensional fermi-hubbard system, Phys. Rev. X 10, 011042 (2020). 18
2020
-
[100]
Scherg, T
S. Scherg, T. Kohlert, P. Sala, F. Pollmann, B. Hebbe Mad- husudhana, I. Bloch, and M. Aidelsburger, Observing non- ergodicity due to kinetic constraints in tilted Fermi-Hubbard chains, Nature Communications 12, 4490 (2021)
2021
-
[101]
Kohlert, S
T. Kohlert, S. Scherg, P. Sala, F. Pollmann, B. Hebbe Mad- husudhana, I. Bloch, and M. Aidelsburger, Exploring the regime of fragmentation in strongly tilted fermi-hubbard chains, Phys. Rev. Lett. 130, 010201 (2023). Appendix A: Derivations of Goldstone modes and dispersion re...
2023
-
[102]
g 2 ˆπa(k)ˆπa(k′) + κρ2 a0 |k|4 2 ˆθa(k)ˆθa(k′) + Γρa0ρb0 |k|2 2 2X b̸=a (ˆθa(k)ˆθa(k′) + ˆθb(k)ˆθb(k′) − ˆθa(k)ˆθb(k′) − ˆθb(k)ˆθa(k′)) # = 1 (2π)d ˆ ddk 2X a
Derivation of the Hamiltonian in the momentum space Based on the Hamiltonian [Eq. (43)], we can derive the Hamiltonian of the momentum space [Eq. (44)] in the following steps. In the momentum space, we can perform a Fourier transform, H = 1 (2π)2d ˆ ddxddkddk′eik·x+ik′·x 2X a ...
-
[103]
Matrix elements of the transformation matrix T (k) In the main text, we give the commutation relations that the operators need to satisfy. To keep the commutation relations and diagonalize the matrix M (k), the transformation matrix T (k) needs to obey the equation below, T (k...
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[104]
+ q κ2 |k|8 (ρ2 10 − ρ2 20)2 + 16Γ2k4ρ2 10ρ2 20))/ ( 4√ 2 g(16Γ2 |k|4 ρ2 10ρ2 20 + (κ |k|4 (ρ2 20 − ρ2
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[106]
+ 4Γ|k|2 ρ10ρ20 − q κ2 |k|8 (ρ2 10 − ρ2 20)2 + 16Γ2 |k|4 ρ2 10ρ2 20) 1 2 ) , (A7) T12(k) = − T32(k) = (4igΓρ10ρ20 |k|2)/( 4√ 2 g(16Γ2 |k|4 ρ2 10ρ2 20 + (κ |k|4 (ρ2 20 − ρ2
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[107]
+ q κ2k8(ρ2 10 − ρ2 20)2 + 16Γ2 |k|4 ρ2 10ρ2 20)2) r g(κ |k|4 (ρ2 10 + ρ2
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[108]
+ 4Γ|k|2 ρ10ρ20 − q κ2 |k|8 (ρ2 10 − ρ2 20)2 + 16Γ2 |k|4 ρ2 10ρ2 20) 1 2 ) , (A8) T13(k) =T33(k) = " (κ |k|4 (ρ2 20 − ρ2
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[109]
+ q κ2 |k|8 (ρ2 10 − ρ2 20)2 + 16Γ2 |k|4 ρ2 10ρ2 20) 4 vuut κ |k|4 (ρ2 10 + ρ2
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[110]
+ 4Γ|k|2 ρ10ρ20 − q κ2 |k|8 (ρ2 10 − ρ2 20)2 + 16Γ2 |k|4 ρ2 10ρ2 20 g # / (23/4 r 16Γ2 |k|4 ρ2 10ρ2 20 + (κ |k|4 (ρ2 20 − ρ2
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[111]
+ q κ2 |k|8 (ρ2 10 − ρ2 20)2 + 16Γ2 |k|4 ρ2 10ρ2 20)2), (A9) T14(k) = T34(k) = 2Γ |k|2 ρ10ρ20 4 vuut 2κ |k|4 (ρ2 10 + ρ2
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[112]
+ 8Γ|k|2 ρ10ρ20 − 2 q κ2 |k|8 (ρ2 10 − ρ2 20)2 + 16Γ2 |k|4 ρ2 10ρ2 20 g ! / ( r 16Γ2 |k|4 ρ2 10ρ2 20 + (κ |k|4 (ρ2 20 − ρ2
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[113]
+ q κ2 |k|8 (ρ2 10 − ρ2 20)2 + 16Γ2 |k|4 ρ2 10ρ2 20)2) (A10) T21(k) = − T41(k) = − " ig(κ |k|4 (ρ2 10 − ρ2
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[114]
+ q κ2 |k|8 (ρ2 10 − ρ2 20)2 + 16Γ2 |k|4 ρ2 10ρ2 20) # / ( 23/4 |k|2 g 16Γ2ρ2 10ρ2 20 + κ2 |k|4 (ρ2 10 − ρ2 20)2 + κ(ρ2 10 − ρ2 20) q κ2 |k|8 (ρ2 10 − ρ2 20)2 + 16Γ2 |k|4 ρ2 10ρ2 20 r g(κ |k|4 (ρ2 10 + ρ2
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[115]
+ 4Γ|k|2 ρ10ρ20 + q κ2 |k|8 (ρ2 10 − ρ2 20)2 + 16Γ2 |k|4 ρ2 10ρ2 20) 1 2 ) , (A11) 20 T22(k) = − T42(k) = (2i 4√ 2Γgρ10ρ20)/ " 4 r g(κ |k|4 (ρ2 10 + ρ2
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[116]
+ 4Γ|k|2 ρ10ρ20 + q κ2k8(ρ2 10 − ρ2 20)2 + 16Γ2 |k|4 ρ2 10ρ2 20) r g(16Γ2ρ2 10ρ2 20 + κ2 |k|4 (ρ2 10 − ρ2 20)2 + κ(ρ2 10 − ρ2 20) q κ2 |k|8 (ρ2 10 − ρ2 20)2 + 16Γ2 |k|4 ρ2 10ρ2 20) # , (A12) T23(k) = T43(k) = κ |k|4 (ρ2 20 − ρ2
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[117]
− q κ2 |k|8 (ρ2 10 − ρ2 20)2 + 16Γ2 |k|4 ρ2 10ρ2 20 2 4√ 2 |k|2 s g(16Γ2ρ2 10ρ2 20+κ2|k|4(ρ2 10−ρ2 20)2+κρ2 10 √ κ2|k|8(ρ2 10−ρ2 20)2+16Γ2|k|4ρ2 10ρ2 20−κρ2 20 √ κ2|k|8(ρ2 10−ρ2 20)2+16Γ2|k|4ρ2 10ρ2 20)q g(κ|k|4(ρ2 10+ρ2 20)+4Γ|k|2ρ10ρ20+ √ κ2|k|8(ρ2 10−ρ2 20)2+16Γ2|k|4ρ2 10ρ2...
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[118]
(45)] becomes H = 1 2(2π)d ϕ†(k)(T −1(k))†M (k)T −1(k)ϕ(k) = 1 2(2π)d ˆ ddkD1(k)( ˆα†(k) ˆα(k) + ˆα(−k) ˆα†(−k)) + D2(k)( ˆβ†(k) ˆβ(k) + ˆβ(−k) ˆβ†(−k))
Derivation of the Hamiltonian represented by the creation and annihilation operators After diagonalization, the Hamiltonian [Eq. (45)] becomes H = 1 2(2π)d ϕ†(k)(T −1(k))†M (k)T −1(k)ϕ(k) = 1 2(2π)d ˆ ddkD1(k)( ˆα†(k) ˆα(k) + ˆα(−k) ˆα†(−k)) + D2(k)( ˆβ†(k) ˆβ(k) + ˆβ(−k) ˆβ†(...
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[119]
|k|2 = 1 c1 |k|2 , |T12(k)|2 ≈ g 2 p 2gκ(ρ2 10 + ρ2
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[120]
|k|2 = 1 c1 |k|2 , |T21(k)|2 ≈ g 8√gΓρ10ρ20 |k| = 1 c2 |k| , |T22(k)|2 ≈ g 8√gΓρ10ρ20 |k| = 1 c2 |k| , T ∗ 11(k)T12(k) ≈ g 2 p 2gκ(ρ2 10 + ρ2
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[121]
(B8) Based on the above derivation, we can calculate correlation functions in different spatial dimensions
|k|2 = 1 c1 |k|2 , T ∗ 21(k)T22(k) ≈ − g 8√gΓρ10ρ20 |k| = − 1 c2 |k| . (B8) Based on the above derivation, we can calculate correlation functions in different spatial dimensions. 22
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[122]
One spatial dimension In one spatial dimension, we can plug in the results of the above approximation. The correlation functions of the operator ˆθ are ⟨ˆθ1(x)ˆθ1(0)⟩ = ⟨ˆθ2(x)ˆθ2(0)⟩ = ˆ dk 2π eikx 1 c1 |k|2 + 1 c2 |k| ! = 1 2πc1 ˆ dk eikx |k|2 + 1 2πc2 ˆ dk eikx |k| = 1 2πc1...
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[123]
Two spatial dimensions In two spatial dimensions, we can plug in the results of the above approximation. The correlation functions of the operator ˆθ are ⟨ˆθ1(x)ˆθ1(0)⟩ = ⟨ˆθ2(x)ˆθ2(0)⟩ = ˆ d2k (2π)2 eik·x 1 c1 |k|2 + 1 c2 |k| ! = 1 4π2c1 ˆ dkdθ eik|x| cos θ k + 1 4π2c2 ˆ dkdθ...
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[124]
Three spatial dimensions In three spatial dimensions, we can plug in the results of the above approximation. The correlation functions of the operator ˆθ are ⟨ˆθ1(x)ˆθ1(0)⟩ = ⟨ˆθ2(x)ˆθ2(0)⟩ = ˆ d3k (2π)3 eik·x 1 c1 |k|2 + 1 c2 |k| ! = 1 8π3c1 ˆ π 0 dψ ˆ 2π 0 dθ ˆ ∞ 0 dk(sin ψe...
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[125]
The commutation relation between ˆθa and ˆfa is given by [ˆθa(x), ˆfb(y)] = −iδ(d)(x − y)δab
Derivation of the effective Hamiltonian density Considering quantum fluctuations, we have ˆΦa(x) = q ρa0 + ˆfa(x)eiˆθa(x), where ˆfa(x) ≪ ρa0. The commutation relation between ˆθa and ˆfa is given by [ˆθa(x), ˆfb(y)] = −iδ(d)(x − y)δab. Therefore, the conjugate momentum operat...
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[126]
gˆπ1(k)ˆπ1(−k) + gˆπ2(k)ˆπ2(−k) + ˆθ1(−k) ˆθ2(−k)
Derivation of the Hamiltonian in the momentum space In the momentum space, we can perform a Fourier transform, H = 1 (2π)2d ˆ ddxddkddk′eik·x+ik′·x " g 2 ˆπ1(k)ˆπ1 (k′) + g 2 ˆπ2(k)ˆπ2 (k′) + κ1ρ2 10 2 dX i,j k2 i k2 j ˆθ1(k)ˆθ1 (k′) + κ2ρ3 20 2 dX i,j,k k2 i k2 j k2 k ˆθ2(k)ˆ...
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[127]
By the Bogoliubov transformation, we can convert the coupling bosons into two independent modes
Bogoliubov transformation and the transformation matrix T (k) In this Hamiltonian, two species of bosons couple to each other. By the Bogoliubov transformation, we can convert the coupling bosons into two independent modes. We introduce a transformation matrix T (k) which chan...
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[128]
+ 16Γ2ℓ2ρ2 10ρ4 20 dX i k3 i 2 !# / " ((κ1 |k|4 ρ2 10 − ρ2 20(κ2 |k|6 ρ20 + ρ10(Γℓ2 dX i k4 i − 4Γ |k|2)))2 + 16Γ2ℓ2ρ2 10ρ4 20 dX i k3 i 2 ) g κ2 |k|6 ρ3 20 + κ1 |k|4 ρ2 10 + 4Γ |k|2 ρ10ρ2 20 + Γℓ2ρ10ρ2 20 dX i k4 i − vuut(κ1 |k|4 ρ2 10 − ρ2 20(κ2 |k|6 ρ20 + ρ10(Γℓ2 dX i k4 i ...
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[129]
Derivation of the Hamiltonian represented by the creation and annihilation operators After diagonalized, the Hamiltonian becomes H = 1 2(2π)d ˆ ddkD1(k)( ˆα†(k) ˆα(k) + ˆα(−k) ˆα†(−k)) + D2(k)( ˆβ†(k) ˆβ(k) + ˆβ(−k) ˆβ†(−k)). (C28) Because of D1(k) = D1(−k), D2(k) = D2(−k), we...
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[130]
One spatial dimension In one spatial dimension, the dispersion relations are ω1 ≈ q 4gΓρ10ρ2 20 |k| , ω 2 ≈ r κ2gρ3 20 + κ1gρ2 10ℓ2 4 |k|3 (D8) We can approximate these four terms, |T11(k)|2 ≈ √gΓρ10 4Γρ10ρ20 1 |k| = 1 c11 |k| , |T12(k)|2 ≈ √gΓρ10ℓ2 16Γρ10ρ20 |k| = |k| c12 , |...
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[131]
Two spatial dimensions In two spatial dimensions, we set k1 = |k| cos θ, and k2 = |k| sin θ, for |k| ≥0, 0 ≤ θ <2π. Then, we can get the dispersion relations, ω1 ≈ q 4gΓρ10ρ2 20 |k| , (D21) ω2 ≈ 1 2 ρ20ℓ |sin 2θ| p gρ10Γ(1 − sin 2θ) |k|2 (θ ̸= 0, π 4 , π 2 , ...
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[132]
h(0) i − 2X a X i (Ψa,i( ˆdi a,i)† + Ψ ∗ a,i ˆdi a,i) # , (E4) where h(0) i = 2X a
Three spatial dimensions In three spatial dimensions, we set k1 = |k| cos θ sin ψ, k2 = |k| sin θ sin ψ, k3 = |k| cos ψ. Due to the large error in the analytical approximation of the 3D results, we directly use numerical methods to calculate the correlation functions, ˆ 1 0 dk...
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