Stability of sliding Luttinger liquid phases against the complete set of multiwire, marginal, and disorder perturbations

Sliding Luttinger liquid (SLL) phases — the smectic metal of parallel quantum-wire arrays, the crossed sliding Luttinger liquid (CSLL), and their three-dimensional stacks — are perturbatively constructed non-Fermi-liquid metallic states in two and three dimensions. Their original demonstration (Mukhopadhyay, Kane, and Lubensky, Phys. Rev. B 64, 045120 (2001)) established stability only against a restricted set of perturbations: all two-wire operators and a subset of multiwire superconducting terms, while explicitly deferring the treatment of arbitrary multiwire (higher-order) interchain operators. The open problem is to determine the true low-temperature fate of these fixed points under the complete set of symmetry-allowed perturbations — arbitrary multiwire charge-density-wave and superconducting operators, marginal spin-dependent interchain couplings in the gapless-spin case, and impurity forward scattering in the transverse transport — together with the robustness of the fine parameter tuning required in three dimensions and of the commensurability and long-range-interaction assumptions used to build the effective theory. A later line of work (2017–2022) already contests the stability conclusion at the two-wire level for physically natural interactions, so the completeness question is not merely technical: it decides whether these sliding phases are genuine stable phases of matter or artifacts of a truncated operator analysis.

Background

A one-dimensional interacting electron gas is not a Fermi liquid: it forms a Luttinger liquid, a gapless state with spin-charge separation, collective bosonic modes, and power-law correlations with continuously varying exponents. In the bosonization description, each wire $j$ is characterized by conjugate fields $\theta_j$ (superconducting phase) and $\phi_j$ (charge-density wave (CDW) phase), with the Gaussian fixed-point action controlled by a Luttinger parameter $\kappa$ and a velocity $v$ . Whether a perturbation, e.g. a Josephson coupling $\cos(\sqrt{2\pi}(\theta_j-\theta_{j+n}))$ or a CDW coupling $\cos(\sqrt{2\pi}(\phi_j-\phi_{j+n}))$ , grows or shrinks under renormalization is decided by its scaling dimension $\Delta$ : in the $1+1$ -dimensional action the coupling is relevant (destabilizes the fixed point) if $\Delta<2$ and irrelevant if $\Delta>2$ .

Since no fully controlled theory of non-Fermi-liquid metals exists in dimensions above one, arrays of weakly coupled quantum wires have been studied as anisotropic (quasi-one-dimensional) candidates. Mukhopadhyay, Kane, and Lubensky established, by explicit computation of scaling dimensions, that a window of interwire density-density and current-current forward-scattering interactions renders all single-particle, two-wire Josephson, and two-wire CDW perturbations irrelevant: each wire remains a Luttinger liquid while the array as a whole slides, the quantum analog of the sliding phases of stacked classical XY models. For an array of parallel wires this is the smectic metal; for two perpendicular interpenetrating arrays it is the crossed sliding Luttinger liquid (CSLL), which at finite temperature has a long-wavelength in-plane conductivity that is isotropic and diverges as a power of $T$ as $T\to 0$ , mimicking a two-dimensional Luttinger liquid; a three-dimensional stack of crossed arrays is similarly stable within the analysis. The mechanism is a transverse-momentum-dependent stiffness $\kappa(q_\perp)$ : by tuning $\kappa$ to be small near some incommensurate wavevector $q_0$ , both the Josephson and CDW two-wire dimensions can simultaneously be made large, frustrating interwire locking.

The authors were explicit that this stability is only with respect to a restricted operator set. First, the Hamiltonian also allows multiwire (higher-order) operators acting on three or more wires at once, of the form $\cos(\sqrt{2\pi}\sum_p s_p\,\theta_{i+p})$ and $\cos(\sqrt{2\pi}\sum_p s_p\,\phi_{i+p})$ with integer weights $s_p$ satisfying $\sum_p s_p=0$ ; only a subset of superconducting ones was analyzed, pairwise CDW couplings were retained, and 'the relevance of higher order terms of this form is a subtle issue' deferred to a future publication that never appeared. Second, in three dimensions stability with respect to the complete operator set requires an even finer tuning of the generalized current-current coupling $\kappa(q_y,q_z)$ , namely a shallow minimum of value $K\delta$ (here $K$ is the value of $\kappa$ away from the minimum, i.e. the mean intra-wire Luttinger stiffness, and $\delta$ is the relative depth of the minimum) with both $\delta$ and the curvature $\partial^2\kappa/K$ much smaller than unity, so that the stable manifold may be of finite codimension and the practical realizability of such tuning is unclear. Third, in the case where spin excitations are gapless (two Luttinger parameters $\kappa_\rho,\kappa_\sigma$ , with $\kappa_\sigma=1$ fixed by SU(2) spin symmetry), marginal spin-spin interchain couplings compatible with SU(2) were simply omitted from the fixed-point Hamiltonian, and their effect has not been analyzed. Fourth, the perpendicular (interwire) conductivity calculation ignores disorder: at the lowest temperatures, forward scattering from impurities — the process that degrades transport in a single Luttinger liquid — is expected to modify both the exponents and prefactors. Fifth, the crossed-array effective theory assumes the periodic potential seen by each wire from the crossing array is commensurate with the wire band structure, and long-range bare interactions are truncated, leaving the influence of incommensurability, longer-range couplings, and microscopic band details on the fixed points undetermined.

The stakes of the completeness question were raised by later work. Vishwanath and Carpentier independently found a stable two-dimensional anisotropic non-Fermi-liquid phase of coupled spinless Luttinger liquids by a related mechanism (transverse interactions at an incommensurate wavevector frustrating both CDW and superconducting lock-in). Later, Fleurov, Kagalovsky, Lerner, and Yurkevich re-examined the phase diagram for physically natural interwire interactions (screened Coulomb and Coulomb-blockade type) and concluded that the combination of inter-wire single-particle hybridization, CDW, and superconducting couplings always destroys the SLL for repulsive intra-wire interactions, a stability region surviving only when single-particle tunneling is suppressed; Sur and Yang found Coulomb-coupled wire arrays unstable to weak interwire backscattering toward CDW and Wigner-crystal states; Jose and Uchoa, in a microscopic model of wires as domain walls of a two-dimensional Dirac system, found only smectic-metal and Fermi-liquid phases, with the smectic metal stable in the ideal-wire limit. On the constructive side, sliding Luttinger liquids have been revived as candidate states of moiré systems (twisted bilayer WTe $_2$ , anisotropically flattened twisted bilayers, symmetry-mismatched moiré interfaces), where the same restricted-operator stability analyses are repeated in new settings, and millikelvin-temperature experiments on coupled-wire platforms report two-dimensional anisotropic Luttinger-liquid phenomenology. In parallel, a different line (perfect-metal phases of many-channel wires) has shown that fixed points stable against all perturbations up to any fixed finite order in fermion operators provably exist, demonstrating that complete-operator stability analyses are feasible in principle — but for a different mechanism and not for the sliding phases.

Problem Statement

Determine the true stability of the sliding Luttinger liquid phases — the smectic metal of parallel arrays of coupled quantum wires, the crossed sliding Luttinger liquid (CSLL) of two interpenetrating perpendicular arrays, and their three-dimensional stacks, in both the spin-gapped case (the Luther-Emery state: a one-dimensional metal with gapped spin modes and gapless charge modes) and the gapless-spin case — with respect to the complete set of perturbations allowed by symmetry in the microscopic Hamiltonian, and thereby decide whether these phases are genuine stable low-temperature phases of coupled-wire systems or artifacts of the restricted operator analysis in the original construction. Concretely, the problem comprises, as one linked stability program: (i) computing or rigorously bounding the scaling dimensions of arbitrary multiwire (higher-order) interchain operators $\cos(\sqrt{2\pi}\sum_p s_p\,\theta_{i+p})$ and $\cos(\sqrt{2\pi}\sum_p s_p\,\phi_{i+p})$ with integer weights $s_p$ satisfying $\sum_p s_p=0$ (superconducting, CDW, and single-particle multiwire generalizations), for general transverse stiffness profiles $\kappa(q_\perp)$ , and establishing whether all of them are irrelevant throughout the claimed stability windows, or exhibiting a relevant one; (ii) the same question for the three-dimensional stacks, where stability was claimed only under a fine tuning of $\kappa(q_y,q_z)$ to a shallow minimum $K\delta$ (with $K$ the value of $\kappa$ away from the minimum, $\delta\ll 1$ the relative depth of the minimum, and the curvature $\partial^2\kappa/K$ also small), including determining the codimension of the stable manifold and whether any generic mechanism avoids the tuning; (iii) the gapless-spin case with the marginal spin-spin interchain couplings compatible with SU(2) symmetry ( $\kappa_\sigma=1$ ) included in the fixed-point Hamiltonian and their effect on the stability criteria; (iv) the low-temperature correction of the perpendicular (interwire) conductivity of the crossed and stacked phases by impurity forward scattering and other disorder processes, i.e. the corrected exponents and prefactors; and (v) the effect of relaxing the construction's simplifications — incommensurate (rather than commensurate) band structure for crossed wires, longer-range bare interwire interactions, and realistic microscopic band structure — on the fixed points. The sub-parts (i)–(v) form one linked program. Stability requires all of (i)–(v) jointly: irrelevance of the complete symmetry-allowed operator set, control of the codimension and genericity of the three-dimensional stable manifold, inclusion of the marginal spin-spin couplings and of disorder in the perpendicular transport, and control of the relaxed microscopic assumptions, together with the resulting low-temperature phase diagram and transport exponents. The stability question is resolved negatively by a single explicit witness: at least one symmetry-allowed perturbation belonging to one of (i)–(v) whose scaling dimension, computed for the corresponding transverse stiffness profile and wire geometry, is relevant at some stiffness profile within the claimed stability window.

References

R. Mukhopadhyay, C. L. Kane, and T. C. Lubensky, "Sliding Luttinger liquid phases", Phys. Rev. B 64, 045120 (2001). DOI: 10.1103/PhysRevB.64.045120, https://doi.org/10.1103/PhysRevB.64.045120 (arXiv:cond-mat/0102163, https://arxiv.org/abs/cond-mat/0102163)
R. Mukhopadhyay, C. L. Kane, and T. C. Lubensky, "Crossed sliding Luttinger liquid phase", Phys. Rev. B 63, 081103(R) (2001). DOI: 10.1103/PhysRevB.63.081103, https://doi.org/10.1103/PhysRevB.63.081103 (arXiv:cond-mat/0007039, https://arxiv.org/abs/cond-mat/0007039)
A. Vishwanath and D. Carpentier, "Two-Dimensional Anisotropic Non-Fermi-Liquid Phase of Coupled Luttinger Liquids", Phys. Rev. Lett. 86, 676 (2001). DOI: 10.1103/PhysRevLett.86.676, https://doi.org/10.1103/PhysRevLett.86.676 (arXiv:cond-mat/0003036, https://arxiv.org/abs/cond-mat/0003036)
V. Fleurov, V. Kagalovsky, I. V. Lerner, and I. V. Yurkevich, "Instability of the sliding Luttinger liquid", J. Phys.: Condens. Matter 30, 055601 (2018). DOI: 10.1088/1361-648x/aab980, https://doi.org/10.1088/1361-648x/aab980 (arXiv:1711.01639, https://arxiv.org/abs/1711.01639)
S. Sur and K. Yang, "Coulomb interaction driven instabilities of sliding Luttinger liquids", Phys. Rev. B 96, 075131 (2017). DOI: 10.1103/PhysRevB.96.075131, https://doi.org/10.1103/PhysRevB.96.075131 (arXiv:1705.00417, https://arxiv.org/abs/1705.00417)
S. Begum, V. Fleurov, V. Kagalovsky, and I. V. Yurkevich, "Sliding Luttinger liquid with alternating interwire couplings", J. Phys.: Condens. Matter 31, 425601 (2019). DOI: 10.1088/1361-648x/ab2a56, https://doi.org/10.1088/1361-648x/ab2a56 (arXiv:1903.02920, https://arxiv.org/abs/1903.02920)
G. Jose and B. Uchoa, "Stability of the smectic phase in arrays of parallel quantum wires", Phys. Rev. B 106, 125140 (2022). DOI: 10.1103/PhysRevB.106.125140, https://doi.org/10.1103/PhysRevB.106.125140 (arXiv:2202.11744, https://arxiv.org/abs/2202.11744)
A. L. Chudnovskiy, V. Kagalovsky, and I. V. Yurkevich, "Metal-insulator transition in a sliding Luttinger liquid with line defects", Phys. Rev. B 96, 165111 (2017). DOI: 10.1103/PhysRevB.96.165111, https://doi.org/10.1103/PhysRevB.96.165111 (arXiv:1706.03550, https://arxiv.org/abs/1706.03550)
C. Chen, A. H. Castro Neto, and V. M. Pereira, "Correlated states of a triangular net of coupled quantum wires: Implications for the phase diagram of marginally twisted bilayer graphene", Phys. Rev. B 101, 165431 (2020). DOI: 10.1103/PhysRevB.101.165431, https://doi.org/10.1103/PhysRevB.101.165431 (arXiv:1907.11450, https://arxiv.org/abs/1907.11450)
E. Plamadeala, M. Mulligan, and C. Nayak, "Perfect metal phases of one-dimensional and anisotropic higher-dimensional systems", Phys. Rev. B 90, 241101(R) (2014). DOI: 10.1103/PhysRevB.90.241101, https://doi.org/10.1103/PhysRevB.90.241101 (arXiv:1404.4367, https://arxiv.org/abs/1404.4367)
C. Murthy and C. Nayak, "Almost Perfect Metals in One Dimension", Phys. Rev. Lett. 124, 136801 (2020). DOI: 10.1103/PhysRevLett.124.136801, https://doi.org/10.1103/PhysRevLett.124.136801 (arXiv:1909.04371, https://arxiv.org/abs/1909.04371)
Y. Hu, Y. Xu, and B. Lian, "Twisted coupled wire model for a moiré sliding Luttinger liquid", Phys. Rev. B 110, L201106 (2024). DOI: 10.1103/PhysRevB.110.L201106, https://doi.org/10.1103/PhysRevB.110.L201106 (arXiv:2310.04070, https://arxiv.org/abs/2310.04070)
A. Sarma, "Sliding Luttinger liquid and topological flat bands in symmetry-mismatched moiré interfaces", Phys. Rev. B 111, 205147 (2025). DOI: 10.1103/PhysRevB.111.205147, https://doi.org/10.1103/PhysRevB.111.205147 (arXiv:2412.17973, https://arxiv.org/abs/2412.17973)
Y.-M. Wu, C. Murthy, and S. A. Kivelson, "Possible Sliding Regimes in Twisted Bilayer WTe2", Phys. Rev. Lett. 133, 246501 (2024). DOI: 10.1103/PhysRevLett.133.246501, https://doi.org/10.1103/PhysRevLett.133.246501 (arXiv:2405.09616, https://arxiv.org/abs/2405.09616)
H. Wang, Z. Liu, Y. Jiang, and J. Wang, "Giant anisotropic band flattening in twisted $\Gamma$ valley semiconductor bilayers", Phys. Rev. B 108, L201120 (2023). DOI: 10.1103/PhysRevB.108.L201120, https://doi.org/10.1103/PhysRevB.108.L201120 (arXiv:2307.09300, https://arxiv.org/abs/2307.09300)
G. Yu, P. Wang, A. J. Uzan-Narovlansky, Y. Jia, M. Onyszczak, R. Singha, X. Gui, T. Song, Y. Tang, K. Watanabe, T. Taniguchi, R. J. Cava, L. M. Schoop, and S. Wu, "Evidence for two dimensional anisotropic Luttinger liquids at millikelvin temperatures", Nat. Commun. 14, 7025 (2023). DOI: 10.1038/s41467-023-42821-2, https://doi.org/10.1038/s41467-023-42821-2 (arXiv:2307.15881, https://arxiv.org/abs/2307.15881)