Theoretical attempts to explain the origin of high temperature superconductivity are challenged by the complexity of the normal state, which exhibits three regimes with increasing hole doping: a pseudo-gap regime when underdoped, strange power laws near optimal doping and more conventional metallic behavior when heavily overdoped. We suggest that the origin of this behavior is linked to a zero temperature quantum phase transition separating the overdoped Fermi liquid from a spin-charge separated underdoped phase. Central to our analysis is a new $Z_2$ gauge theory formulation, which supports topological vortex excitations - dubbed visons. The visons are gapped in the underdoped phase, splitting the electron's charge and Fermi statistics into two separate excitations. Superconductivity occurs when the resulting charge $e$ boson condenses. The visons are condensed in the overdoped phase, thereby confining the charge and statistics of the electron leading to a Fermi liquid phase. Right at the quantum confinement transition the visons are in a critical state, leading to power law behavior for both charge and spin.
Theoretical attempts to explain the origin of high temperature superconductivity are challenged by the complexity of the normal state, which exhibits three regimes with increasing hole doping: a pseudo-gap regime when underdoped, strange power laws near optimal doping and more conventional metallic behavior when heavily overdoped. We suggest that the origin of this behavior is linked to a zero temperature quantum phase transition separating the overdoped Fermi liquid from a spin-charge separated underdoped phase. Central to our analysis is a new Z2 gauge theory formulation, which supports topological vortex excitations - dubbed visons. The visons are gapped in the underdoped phase, splitting the electron's charge and Fermi statistics into two separate excitations. Superconductivity occurs when the resulting charge e boson condenses. The visons are condensed in the overdoped phase, thereby confining the charge and statistics of the electron leading to a Fermi liquid phase. Right at the quantum confinement transition the visons are in a critical state, leading to power law behavior for both charge and spin.
The Gaussian fluctuations in slave-boson mean-field theory of t−J model is analyzed in this paper in the low-doping regime where the superconducting to normal (pseudo-gap phase) transition is driven by vanishing of bose-condensation amplitude. By eliminating the boson and constraint fields exactly in the linear response regime we show that the Gaussian theory describes a Fermi liquid superconductor where the superconducting to normal transition is actually a spin-charge separation transition characterized by a change of Landau parameter from F1>−1 in the superconducting phase to F1=−1 in the pseudo-gap phase. Consequences of this proposal are discussed.
Abstract We review a theoretical framework for the cuprate superconductors, rooted in a fractionalized Fermi liquid (FL*) description of the intermediate-temperature pseudogap phase at low doping. The FL* theory predicted hole pockets each of fractional area p / 8 at hole doping p , in contrast to the area p / 4 in spin density wave theory. Magnetotransport measurements, including observation of the Yamaji angle, show clear evidence of hole pocket quasiparticles which can tunnel coherently between square lattice layers, and are consistent with the FL* description. The FL* phase of a single-band model is described using a layer construction with a pair of ancilla qubits on each site: the Ancilla layer model (ALM). Its mean field theory yields hole pockets of area p / 8 , and matches the gapped photoemission spectrum in the anti-nodal region of the Brillouin zone. Fluctuations are described by the SU(2) gauge theory of a background spin liquid with critical Dirac spinons. A Monte Carlo study of the thermal SU(2) gauge theory transforms the hole pockets into Fermi arcs in photoemission. One route to confinement of FL* upon lowering temperature yields a d -wave superconductor via a Kosterlitz–Thouless transition of h / ( 2 e ) vortices, with nodal Bogoliubov quasiparticles featuring anisotropic velocities and vortices surrounded by charge order halos. An alternative route yields a charge-ordered metallic state that has quantum oscillations consistent with observations. These confinement transitions are driven by the condensation of a SU(2) fundamental Higgs field, which also provides a fractionalized description of intertwined orders. Increasing doping from the FL* phase in the ALM drives a transition to a conventional FL at large doping, passing through an intermediate strange metal regime. We formulate a theory of the FL*-FL metal-metal transition without a symmetry-breaking order parameter, using a critical quantum ‘charge’ liquid of mobile electrons in the presence of disorder, developed via an extension of the Sachdev–Ye–Kitaev model to two spatial dimensions. At low temperatures, and across optimal and over doping, we address the regimes of extended non-FL behavior by Griffiths effects near quantum phase transitions in disordered metals. Partly based on lectures by S S at Boulder School 2025, Dynamics of Strongly Correlated Electrons , 14–18 July. Lecture videos . Joint ICTP-WE Heraeus School and Workshop on Advances in Quantum Matter: Pushing the Boundaries , ICTP, Trieste, 4, 6 August 2025. Lecture videos . School on Quantum Dynamics of Matter, Light and Information , ICTP, Trieste, 18, 19 August 2025. Lecture videos . Croucher Advanced Study Institute for Fractional Chern Insulators , University of Hong Kong, 4, 5 September 2025. Lecture slides . Advanced School and Conference on Quantum Matter , ICTP Trieste, 1–12 December 2025. Lecture Notes . Lecture videos .
Abstract The mysterious metallic phase showing T-linear resistivity and a universal scattering rate 1 / τ = α P k B T / ℏ with a universal prefactor α P ∼ 1 and logarithmic-in-temperature singular specific heat coefficient, the so-called ‘Planckian metal phase’ was observed in various overdoped high- T c cuprate superconductors over a finite range in doping. Revealing the mystery of the Planckian metal state is believed to be the key to understanding the mechanism for high- T c superconductivity. Here, we propose a generic microscopic mechanism for this state based on quantum-critical local bosonic charge Kondo fluctuations coupled to both spinon and a heavy conduction-electron Fermi surface within the heavy-fermion formulation of the slave-boson t–J model. By a controlled perturbative renormalization group analysis, we examine the competition between the pseudogap phase, characterized by Anderson’s Resonating-Valence-Bond spin-liquid, and the Fermi-liquid state, modeled by the electron hopping (effective charge Kondo effect). We find a quantum-critical metallic phase with a universal Planckian ℏ ω / k B T scaling in scattering rate near an extended localized-delocalized (pseudogap-to-Fermi liquid) charge-Kondo breakdown transition. The d-wave superconducting ground state emerges near the transition. Unprecedented qualitative and quantitative agreements are reached between our theoretical predictions and various experiments, including optical conductivity, universal doping-independent field-to-temperature scaling in magnetoresistance, specific heat coefficient, marginal Fermi-liquid spectral function observed in ARPES, and Fermi surface reconstruction observed in Hall coefficients in various overdoped cuprates. Our mechanism offers a microscopic understanding of the quantum-critical Planckian metal phase observed in cuprates and its link to the pseudogap, d-wave superconducting, and Fermi liquid phases. It offers a promising route for understanding how d-wave superconductivity emerges from such a strange metal phase in cuprates–one of the long-standing open problems in condensed matter physics since 1990s–as well as shows a broader implication for the Planckian strange metal states observed in other correlated unconventional superconductors.
The mysterious metallic phase showing perfect T-linear resistivity and a universal scattering rate 1/τ=αPkBT/ℏ with a universal prefactor αP∼1 and logarithmic-in-temperature singular specific heat coefficient, so-called Planckian metal phase was observed in various overdoped high-Tc cuprate superconductors over a finite range in doping. Here, we propose a microscopic mechanism for this exotic state based on quantum-critical bosonic charge Kondo fluctuations coupled to both spinon and a heavy conduction-electron Fermi surfaces within the heavy-fermion formulation of the slave-boson t-J model. Using a controlled perturbative renormalization group (RG) analysis, we examine the competition between the pseudogap phase, characterized by Anderson's Resonating-Valence-Bond spin-liquid, and the Fermi-liquid state, characterized by the electron hoping (effective charge Kondo effect). We find a quantum-critical metallic phase with a universal Planckian ℏω/kBT scaling in scattering rate near a localized-delocalized (pseudogap-to-Fermi liquid) charge Kondo breakdown transition. Our results are in excellent agreement with the recent experimental observations on optical conductivity (without fine-tuning) in Nat. Commun. 14, 3033 (2023), universal doping-independent field-to-temperature scaling in magnetoresistance in Nature 595, 661 (2021), and the marginal Fermi-liquid spectral function observed in ARPES (Science 366, 1099 (2019)) as well as Hall coefficient in various overdoped cuprates in Nature 595, 661 (2021) and Annu. Rev. Condens. Matter Phys. 10, 409 (2019). Our mechanism offers a microscopic understanding of the quantum-critical Planckian metal phase observed in cuprates d-wave superconducting, and Fermi liquid phases.
Electrical and magnetic properties of underdoped cuprates in the spin gap phase, a precursor to high Tc superconducting state, is riddled with puzzles. We propose a novel reference state to study this phase, where each hole dopant adds one real holon and one real spinon, with Haldane exclusion statistics gh=1 and gs=1/2, to a fairly inert (pseudo gaped) spin liquid vacuum, resulting in a low density spin-charge liquid. Spins and charges interact and form a novel collective state, a \textit{3/2-Fermi liquid}. A holon pairs with a spinon; this hole like fermion composite carries charge +e and a spin-\hlf moment and novel exclusion statistics ghole=3/2. We explain an anomalous expansion of Fermi sea area by 3/2, seen in recent quantum oscillation experiments at two different dopings.
Insulating quantum spin liquids can undergo a confinement transition to a valence bond solid via the condensation of topological excitations of the associated gauge theory. We extend the theory of such transitions to fractionalized Fermi liquids (FL*): These are metallic doped spin liquids in which the Fermi surfaces only have gauge neutral quasiparticles. Using insights from a duality transform on a doped quantum dimer model for the U(1)-FL* state, we show that projective symmetry group of the theory of the topological excitations remains unmodified, but the Fermi surfaces can lead to additional frustrating interactions. We propose a theory for the confinement transition of Z2-FL* states via the condensation of visons. A variety of confining, incommensurate density wave states are possible, including some that are similar to the incommensurate d-form factor density wave order observed in several recent experiments on the cuprate superconductors.