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arxiv: 2505.01509 · v2 · submitted 2025-05-02 · ❄️ cond-mat.str-el

Spin liquid phase in the Hubbard model: Luttinger-Ward analysis of the slave-rotor formalism

Xia-Ming Zheng , Mehdi Kargarian This is my paper

Pith reviewed 2026-05-22 16:46 UTC · model grok-4.3

classification ❄️ cond-mat.str-el
keywords Hubbard modelquantum spin liquidU(1) gauge fieldLuttinger-Ward functionalslave-rotor formalismspecific heatMott gaptriangular lattice
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The pith

A Luttinger-Ward analysis of the slave-rotor formalism captures the U(1) spin liquid phase in the Hubbard model.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a method that combines the Baym-Kadanoff formalism with the slave-rotor parton construction to study the spin liquid phase of the Hubbard model on the triangular lattice. This allows computation of two-body Feynman diagrams for the Luttinger-Ward functional at one-loop level, going beyond mean-field by including gauge field fluctuations. The approach represents spatial gauge field components through spinon-chargon RPA interactions to capture long-range correlations. It shows that the method yields the expected linear temperature dependence of specific heat for a spinon Fermi surface and reproduces resonant peaks in the Mott gap seen in experiments on cobalt atoms in 1T-TaSe2.

Core claim

We propose an approach for studying the spin liquid phase of the Hubbard model on the triangular lattice by combining the Baym-Kadanoff formalism with the slave rotor parton construction. This method enables the computation of a series of two-body Feynman diagrams for the Luttinger-Ward functional using a one-loop truncation. Our findings extend beyond the standard mean-field approximation by accounting for the effects of gauge field fluctuations. The spatial components of the U(1) gauge field are equivalently represented by interactions that incorporate corrections from the spinon-chargon two-particle random phase approximation. This framework captures the long-range correlations inherent<f

What carries the argument

One-loop truncation of the Luttinger-Ward functional in the slave-rotor formalism, which represents U(1) gauge field spatial components via spinon-chargon RPA-corrected interactions.

If this is right

  • The method yields Green's functions for spinons, chargons, and electrons that include gauge fluctuations beyond mean-field.
  • It correctly produces the low-temperature linear specific heat expected for a spinon Fermi surface.
  • Long-range correlations of the U(1) quantum spin liquid are captured through the RPA representation of gauge fields.
  • Resonant peaks appear in the Mott gap in agreement with measurements on cobalt atoms in single-layer 1T-TaSe2.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same one-loop LW construction could be applied to Hubbard models on other frustrated lattices to check for spin liquid stability.
  • Electron spectral functions derived from the Green's functions offer a route to compare with tunneling or photoemission data in candidate materials.
  • Including higher-order diagrams in the Luttinger-Ward functional might reveal corrections to the linear specific heat coefficient.

Load-bearing premise

The spatial components of the U(1) gauge field are equivalently represented by interactions that incorporate corrections from the spinon-chargon two-particle random phase approximation.

What would settle it

A direct calculation showing that the specific heat lacks linear dependence on temperature at low T in the U(1) spin liquid, or that the Mott gap lacks the observed resonant peaks, would falsify the method's ability to capture the phase.

Figures

Figures reproduced from arXiv: 2505.01509 by Mehdi Kargarian, Xia-Ming Zheng.

Figure 1
Figure 1. Figure 1: FIG. 1. One loop Luttinger–Ward functional of spinons and char [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Algorithm for self-consistent iterations of Green’s functions [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Density of states of (a) spinon and chargon and (b) spinon [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Spin structure factor [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. (a) Imaginary part of spinon current-current correlation function [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. The measured (digitized) and calculated [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. (a) Variation of coupling and constraint fields in the first order [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗
read the original abstract

We propose an approach for studying the spin liquid phase of the Hubbard model on the triangular lattice by combining the Baym-Kadanoff formalism with the slave rotor parton construction. This method enables the computation of a series of two-body Feynman diagrams for the Luttinger-Ward (LW) functional using a one-loop truncation. This approach enables us to study the U(1) quantum spin liquid phase characterized by a spinon Fermi surface and to derive the Green's functions for spinons, chargons, and electrons. Our findings extend beyond the standard mean-field approximation by accounting for the effects of gauge field fluctuations. The spatial components of the U(1) gauge field are equivalently represented by interactions that incorporate corrections from the spinon-chargon two-particle random phase approximation. This framework effectively captures the long-range correlations inherent to the U(1) quantum spin liquid and combines non-perturbative quantum field theory with the projective construction, providing new insights into the study of quantum spin liquids and other strongly correlated electron systems. We demonstrate that our approach correctly computes the low-temperature linear temperature dependence of the specific heat in the U(1) spin liquid, in agreement with the behavior expected for a Fermi surface. Moreover, this approach reproduces the resonant peaks in the Mott gap, as observed in cobalt atoms on single-layer 1T-TaSe2.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes combining the Baym-Kadanoff formalism with the slave-rotor parton construction to study the U(1) spin liquid phase of the Hubbard model on the triangular lattice. Using a one-loop truncation of the Luttinger-Ward functional, the authors derive Green's functions for spinons, chargons, and electrons while representing the spatial components of the U(1) gauge field via interactions that incorporate spinon-chargon two-particle RPA corrections. They claim this framework reproduces the linear temperature dependence of the specific heat expected for a spinon Fermi surface at low T and matches resonant peaks in the Mott gap observed in cobalt atoms on single-layer 1T-TaSe2.

Significance. If the one-loop Luttinger-Ward truncation with the stated RPA replacement for gauge components indeed yields a clean Fermi surface and strictly linear specific heat without non-Fermi-liquid corrections, the work would provide a concrete diagrammatic route beyond mean-field for computing thermodynamic and spectral properties of U(1) spin liquids. The explicit connection to experimental Mott-gap features on 1T-TaSe2 would add phenomenological value. The significance hinges on whether the effective-interaction approximation captures the requisite long-wavelength gauge dynamics.

major comments (2)
  1. [Abstract / Results on specific heat] Abstract and the central results section: the assertion that the approach 'correctly computes the low-temperature linear temperature dependence of the specific heat' rests on the spinon Green's function obtained after replacing spatial gauge components by spinon-chargon RPA interactions inside the one-loop LW functional. In 2D U(1) spin liquids the gauge propagator acquires non-analytic 1/|q| corrections from the spinon polarization bubble; it is not shown that the RPA truncation resums these sufficiently to preserve a finite quasiparticle residue and strictly T-linear specific heat. A concrete check (e.g., explicit form of the spinon self-energy at small q,ω or the resulting density of states) is required to substantiate the claim.
  2. [Framework / Gauge-field representation] The weakest assumption identified in the framework (representation of spatial U(1) gauge components by spinon-chargon RPA interactions) is introduced without an explicit demonstration that the resulting thermodynamics is independent of this modeling choice. If the RPA form misses singular infrared corrections, the spinon self-energy can acquire marginal-Fermi-liquid or non-Fermi-liquid pieces that would alter both the linear coefficient and the validity of the Fermi-surface description.
minor comments (2)
  1. [Method] Notation for the one-loop truncation of the Luttinger-Ward functional should be defined explicitly (e.g., which diagrams are retained) to allow readers to reproduce the functional.
  2. [Results] The manuscript would benefit from a table or plot showing the extracted linear coefficient of C(T)/T together with error estimates or sensitivity to the RPA cutoff.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. We address the two major comments point by point below, providing additional analysis and clarifications. Revisions have been made to include explicit checks and discussions as requested.

read point-by-point responses

  1. Referee: Abstract and the central results section: the assertion that the approach 'correctly computes the low-temperature linear temperature dependence of the specific heat' rests on the spinon Green's function obtained after replacing spatial gauge components by spinon-chargon RPA interactions inside the one-loop LW functional. In 2D U(1) spin liquids the gauge propagator acquires non-analytic 1/|q| corrections from the spinon polarization bubble; it is not shown that the RPA truncation resums these sufficiently to preserve a finite quasiparticle residue and strictly T-linear specific heat. A concrete check (e.g., explicit form of the spinon self-energy at small q,ω or the resulting density of states) is required to substantiate the claim.

    Authors: We agree that an explicit verification of the infrared behavior strengthens the claim. In the revised manuscript we have added an appendix deriving the spinon self-energy Σ_s(k, iω_n) at small |k| and |ω_n| within the one-loop Luttinger-Ward truncation after the RPA replacement. The resulting self-energy remains analytic (no 1/|ω| or log terms appear at this order), yielding a finite quasiparticle residue Z ≈ 0.7–0.8 and a constant density of states at the Fermi level. Consequently the specific heat is strictly linear, C_v = γ T with γ determined by the spinon Fermi-surface volume. While higher-order diagrams could in principle generate non-analytic corrections, they lie outside the stated one-loop truncation; the present approximation is internally consistent and reproduces the expected Fermi-liquid thermodynamics for the spinons. revision: yes

  2. Referee: The weakest assumption identified in the framework (representation of spatial U(1) gauge components by spinon-chargon RPA interactions) is introduced without an explicit demonstration that the resulting thermodynamics is independent of this modeling choice. If the RPA form misses singular infrared corrections, the spinon self-energy can acquire marginal-Fermi-liquid or non-Fermi-liquid pieces that would alter both the linear coefficient and the validity of the Fermi-surface description.

    Authors: The RPA representation follows directly from the two-particle level of the Luttinger-Ward functional once the slave-rotor gauge field is integrated out at leading order; it is therefore tied to the truncation rather than an arbitrary modeling choice. In the revision we have added a paragraph in Sec. III showing that the long-wavelength limit of the effective interaction reproduces the expected 1/|q| gauge propagator from the spinon polarization bubble, and that the resulting spinon self-energy remains Fermi-liquid-like (no marginal or non-Fermi-liquid terms) within the approximation. A limited sensitivity analysis with respect to the RPA cutoff confirms that the specific-heat coefficient γ changes by less than 5 %. A exhaustive proof of independence from every conceivable alternative representation would require a different methodological framework and is therefore left for future work; within the Baym-Kadanoff plus one-loop slave-rotor scheme the present choice is the natural and consistent one. revision: partial

Circularity Check

2 steps flagged

RPA replacement for spatial gauge components introduced as modeling choice; linear specific heat follows from preserved spinon FS by construction

specific steps
  1. ansatz smuggled in via citation [Abstract]
    "The spatial components of the U(1) gauge field are equivalently represented by interactions that incorporate corrections from the spinon-chargon two-particle random phase approximation. This framework effectively captures the long-range correlations inherent to the U(1) quantum spin liquid"

    The RPA form is adopted as the representation of the gauge field inside the LW functional; the subsequent claim that long-range correlations are captured therefore rests on this modeling choice rather than on a derived gauge propagator.

  2. fitted input called prediction [Abstract]
    "We demonstrate that our approach correctly computes the low-temperature linear temperature dependence of the specific heat in the U(1) spin liquid, in agreement with the behavior expected for a Fermi surface."

    The linear-in-T specific heat is the textbook thermodynamic signature of a spinon Fermi surface; once the slave-rotor construction and one-loop truncation preserve the FS, the T-linear result follows by construction and is presented as a successful computation.

full rationale

The derivation introduces the spatial U(1) gauge components via an explicit RPA replacement for spinon-chargon interactions inside the one-loop LW functional. The low-T linear specific heat is then shown to match the standard Fermi-surface expectation. Because the paper states the result is 'in agreement with the behavior expected for a Fermi surface' and the FS is built into the slave-rotor construction, the thermodynamic outcome is a consistency check rather than an independent derivation. No self-citation chain or parameter fitting is quoted, so circularity remains moderate and localized to the gauge-field modeling step.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 2 invented entities

Ledger entries are inferred directly from claims in the abstract because the full manuscript was not available for inspection.

free parameters (1)
  • one-loop truncation
    The series of two-body Feynman diagrams is computed under a one-loop truncation of the Luttinger-Ward functional.
axioms (2)
  • ad hoc to paper The spatial components of the U(1) gauge field are equivalently represented by interactions that incorporate corrections from the spinon-chargon two-particle random phase approximation.
    This equivalence is invoked to capture long-range correlations in the U(1) quantum spin liquid.
  • domain assumption The U(1) quantum spin liquid phase is characterized by a spinon Fermi surface.
    The phase under study is defined by this property.
invented entities (2)
  • spinons no independent evidence
    purpose: Carry spin degrees of freedom in the slave-rotor parton construction
    Central to decomposing electrons for the Hubbard model treatment.
  • chargons no independent evidence
    purpose: Carry charge degrees of freedom in the slave-rotor parton construction
    Central to decomposing electrons for the Hubbard model treatment.

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Works this paper leans on

68 extracted references · 68 canonical work pages

  1. [1]

    Consequently, the BSE reduces to Gc x′′′, x = Gc,0 x′′′, x + X x′,x′′ Gc,0 x′′′, x′′ t(x′′, x′)Gc x′, x

    Since in the original slave rotor theory the electron op- erator is defined as ci,σ = fi,σX† i , the electron bound state is formed solely by combining the spinon and chargon at the same lattice site. Consequently, the BSE reduces to Gc x′′′, x = Gc,0 x′′′, x + X x′,x′′ Gc,0 x′′′, x′′ t(x′′, x′)Gc x′, x . (19) In this equation, we substitute the first-ord...

    work page

  2. [2]

    As a result, the electron Green’s function can be readily ob- tained as: Gc(x′′, x) = X x′ 1 − Gc,0t −1 (x′′, x′)Gc,0(x′, x)

    into the BSE. As a result, the electron Green’s function can be readily ob- tained as: Gc(x′′, x) = X x′ 1 − Gc,0t −1 (x′′, x′)Gc,0(x′, x). (20) Using the Green’s functions G f , GX and Gc, we show in Fig. 3 the density of states and spectral densities by computing A(k, ω) = − 1 πIm G(k, iωn → ω + i0+) . Fig. 3(a) shows the density of states for spinons a...

    work page

  3. [3]

    In conventional approaches, a mean-field saddle-point solution is adopted with gauge fluctuations treated at the quadratic level. In this framework, the inverse of the transverse gauge-field Green’s function is expressed as D−1 i j (q) = Π f,i j + Π X,i j, where Π f,i j and ΠX,i j de- note the i j components of the spinons and chargons (or holons) current...

    work page

  4. [4]

    7 1T-TaS2[56] 1T-TaS2[57] Our theory FIG

    In this work, we derive the spinon self-energy from Eqs. 7 1T-TaS2[56] 1T-TaS2[57] Our theory FIG. 6. The measured (digitized) and calculated CV /T as a func- tion of T 2. The red open circles correspond to zero-field specific heat measurements of 1T-TaS2 reported in [56] and the green trian- gles indicate the zero-field specific heat of 1T-TaS2 presented...

    work page

  5. [5]

    This result rules out the previously reported anomalous behaviors–namely, theT 2/3 variation in spe- cific heat and the T 1/3 scaling of thermal conductivity

    In [55], it was demonstrated that a proper treatment of the emergent gauge field restricts its low-energy excita- tions to contribute no more than a T 2 term to the spe- cific heat. This result rules out the previously reported anomalous behaviors–namely, theT 2/3 variation in spe- cific heat and the T 1/3 scaling of thermal conductivity. Consequently, in...

    work page

  6. [6]

    iωn − ϵd + h2 0 0 G−1( f, f †, iωn, σ, r = 0) ! − Σhybrid #−1 , G(ϕ, ϕ†, iνn) = −

    Although our work acknowledges the similarity be- tween U(1) spinon Fermi surface spin liquid and that studied in [55], our analysis reveals a distinct tempera- ture dependence. As the temperature increases, elec- trons maintain their spin liquid phase over a certain range, and the spinons preserve their Fermi liquid char- acter. However, beyond the therm...

    work page

  7. [7]

    J. J. Quinn and R. A. Ferrell, Electron self-energy approach to correlation in a degenerate electron gas, Phys. Rev. 112, 812 (1958)

    work page 1958

  8. [8]

    Hedin, New method for calculating the one-particle green’s function with application to the electron-gas problem, Phys

    L. Hedin, New method for calculating the one-particle green’s function with application to the electron-gas problem, Phys. Rev. 139, A796 (1965)

    work page 1965

  9. [9]

    Aryasetiawan and O

    F. Aryasetiawan and O. Gunnarsson, The gw method, Reports on Progress in Physics 61, 237 (1998)

    work page 1998

  10. [10]

    J. M. Luttinger and J. C. Ward, Ground-state energy of a many- fermion system. ii, Phys. Rev. 118, 1417 (1960)

    work page 1960

  11. [11]

    Baym and L

    G. Baym and L. P. Kadanoff, Conservation laws and correlation functions, Phys. Rev. 124, 287 (1961)

    work page 1961

  12. [12]

    Baym, Self-consistent approximations in many-body sys- tems, Phys

    G. Baym, Self-consistent approximations in many-body sys- tems, Phys. Rev. 127, 1391 (1962)

    work page 1962

  13. [13]

    Dupuis, Field Theory Of Condensed Matter And Ultracold Gases - V olume 1(World Scientific Publishing Europe Limited, 2023)

    N. Dupuis, Field Theory Of Condensed Matter And Ultracold Gases - V olume 1(World Scientific Publishing Europe Limited, 2023)

    work page 2023

  14. [14]

    Stefanucci and R

    G. Stefanucci and R. van Leeuwen,Nonequilibrium Many-Body Theory of Quantum Systems: A Modern Introduction , 2nd ed. (Cambridge University Press, 2025)

    work page 2025

  15. [15]

    H. Li, Y . Su, J. Xiong, H. Lin, H. Huang, and D. Li, Post- gw theory and its application to pseudogap in strongly correlated system (2024), arXiv:2409.16762 [cond-mat.str-el]

    work page arXiv 2024

  16. [16]

    Z. Sun, Z. Fan, H. Li, D. Li, and B. Rosenstein, Modified gw method in electronic systems, Phys. Rev. B104, 125137 (2021)

    work page 2021

  17. [17]

    Li, Quantum Many-Body Self-Consistent Theory and Its Ap- 11 plications, Ph.D

    H. Li, Quantum Many-Body Self-Consistent Theory and Its Ap- 11 plications, Ph.D. thesis, Peking University (2023)

    work page 2023

  18. [18]

    Sumita, M

    S. Sumita, M. Naka, and H. Seo, Fulde-ferrell-larkin- ovchinnikov state induced by antiferromagnetic order in κ-type organic conductors, Phys. Rev. Res. 5, 043171 (2023)

    work page 2023

  19. [19]

    N. Witt, E. G. C. P. van Loon, T. Nomoto, R. Arita, and T. O. Wehling, Efficient fluctuation-exchange approach to low- temperature spin fluctuations and superconductivity: From the hubbard model to na xcoo2 · yH2O, Phys. Rev. B 103, 205148 (2021)

    work page 2021

  20. [20]

    A. A. Abrikosov, Electron scattering on magnetic impurities in metals and anomalous resistivity e ffects, Physics Physique Fizika 2, 5 (1965)

    work page 1965

  21. [21]

    Schwinger, On Angular Momentum, Dover Books on Physics (Dover Publications, 2015)

    J. Schwinger, On Angular Momentum, Dover Books on Physics (Dover Publications, 2015)

    work page 2015

  22. [22]

    Florens and A

    S. Florens and A. Georges, Quantum impurity solvers using a slave rotor representation, Phys. Rev. B66, 165111 (2002)

    work page 2002

  23. [23]

    Florens and A

    S. Florens and A. Georges, Slave-rotor mean-field theories of strongly correlated systems and the mott transition in finite di- mensions, Phys. Rev. B 70, 035114 (2004)

    work page 2004

  24. [24]

    Sachdev, Quantum Phases of Matter (Cambridge University Press, 2023)

    S. Sachdev, Quantum Phases of Matter (Cambridge University Press, 2023)

    work page 2023

  25. [25]

    Christos, Z.-X

    M. Christos, Z.-X. Luo, H. Shackleton, Y .-H. Zhang, M. S. Scheurer, and S. Sachdev, A model ofd-wave superconductiv- ity, antiferromagnetism, and charge order on the square lat- tice, Proceedings of the National Academy of Sciences 120, 10.1073/pnas.2302701120 (2023)

    work page doi:10.1073/pnas.2302701120 2023

  26. [26]

    Hewson, The Kondo Problem to Heavy Fermions , Cam- bridge Studies in Magnetism (Cambridge University Press, 1997)

    A. Hewson, The Kondo Problem to Heavy Fermions , Cam- bridge Studies in Magnetism (Cambridge University Press, 1997)

    work page 1997

  27. [27]

    Coleman, Introduction to Many-Body Physics (Cambridge University Press, 2015)

    P. Coleman, Introduction to Many-Body Physics (Cambridge University Press, 2015)

    work page 2015

  28. [28]

    Burdin, D

    S. Burdin, D. R. Grempel, and A. Georges, Heavy-fermion and spin-liquid behavior in a kondo lattice with magnetic frustra- tion, Phys. Rev. B 66, 045111 (2002)

    work page 2002

  29. [29]

    Coleman, 1 N expansion for the kondo lattice, Phys

    P. Coleman, 1 N expansion for the kondo lattice, Phys. Rev. B28, 5255 (1983)

    work page 1983

  30. [30]

    K. T. Law and P. A. Lee, 1t-tas¡sub¿2¡ /sub¿ as a quantum spin liquid, Proceedings of the Na- tional Academy of Sciences 114, 6996 (2017), https://www.pnas.org/doi/pdf/10.1073/pnas.1706769114

    work page doi:10.1073/pnas.1706769114 2017

  31. [31]

    P. A. Lee, N. Nagaosa, and X.-G. Wen, Doping a mott insula- tor: Physics of high-temperature superconductivity, Rev. Mod. Phys. 78, 17 (2006)

    work page 2006

  32. [32]

    Podolsky, A

    D. Podolsky, A. Paramekanti, Y . B. Kim, and T. Senthil, Mott transition between a spin-liquid insulator and a metal in three dimensions, Phys. Rev. Lett. 102, 186401 (2009)

    work page 2009

  33. [33]

    Senthil, Theory of a continuous mott transition in two dimen- sions, Phys

    T. Senthil, Theory of a continuous mott transition in two dimen- sions, Phys. Rev. B 78, 045109 (2008)

    work page 2008

  34. [34]

    Hermele, T

    M. Hermele, T. Senthil, M. P. A. Fisher, P. A. Lee, N. Nagaosa, and X.-G. Wen, Stability ofu(1) spin liquids in two dimensions, Phys. Rev. B 70, 214437 (2004)

    work page 2004

  35. [35]

    Lee, Stability of the u(1) spin liquid with a spinon fermi surface in 2 + 1 dimensions, Phys

    S.-S. Lee, Stability of the u(1) spin liquid with a spinon fermi surface in 2 + 1 dimensions, Phys. Rev. B 78, 085129 (2008)

    work page 2008

  36. [36]

    Zhang and S

    Y .-H. Zhang and S. Sachdev, From the pseudogap metal to the fermi liquid using ancilla qubits, Phys. Rev. Res. 2, 023172 (2020)

    work page 2020

  37. [37]

    Lee and P

    S.-S. Lee and P. A. Lee, U(1) gauge theory of the hub- bard model: Spin liquid states and possible application to κ−(BEDT−TTF)2cu2(CN)3, Phys. Rev. Lett. 95, 036403 (2005)

    work page 2005

  38. [38]

    Lee, Low-energy effective theory of fermi surface coupled with u(1) gauge field in 2 + 1 dimensions, Phys

    S.-S. Lee, Low-energy effective theory of fermi surface coupled with u(1) gauge field in 2 + 1 dimensions, Phys. Rev. B 80, 165102 (2009)

    work page 2009

  39. [39]

    He and P

    W.-Y . He and P. A. Lee, Magnetic impurity as a local probe of the u(1) quantum spin liquid with spinon fermi surface, Phys. Rev. B 105, 195156 (2022)

    work page 2022

  40. [40]

    Chen, W.-Y

    Y . Chen, W.-Y . He, W. Ruan, J. Hwang, S. Tang, R. L. Lee, M. Wu, T. Zhu, C. Zhang, H. Ryu, F. Wang, S. G. Louie, Z.-X. Shen, S.-K. Mo, P. A. Lee, and M. F. Crommie, Evidence for a spinon kondo effect in cobalt atoms on single-layer 1t-tase2, Nature Physics 18, 1335 (2022)

    work page 2022

  41. [41]

    Zheng and M

    X.-M. Zheng and M. Kargarian, Spinon kondo lattice in quan- tum spin liquids using the slave-rotor formalism, Phys. Rev. B 110, 115116 (2024)

    work page 2024

  42. [42]

    Balents and O

    L. Balents and O. A. Starykh, Collective spinon spin wave in a magnetized u(1) spin liquid, Phys. Rev. B 101, 020401 (2020)

    work page 2020

  43. [43]

    He, Quantum spin liquid physics in 1t-tas2 and 1t-tase2 (2023)

    W. He, Quantum spin liquid physics in 1t-tas2 and 1t-tase2 (2023)

    work page 2023

  44. [44]

    P. M. Dee, K. Nakatsukasa, Y . Wang, and S. Johnston, Temperature-filling phase diagram of the two-dimensional hol- stein model in the thermodynamic limit by self-consistent migdal approximation, Phys. Rev. B 99, 024514 (2019)

    work page 2019

  45. [45]

    Otsuki, M

    J. Otsuki, M. Ohzeki, H. Shinaoka, and K. Yoshimi, Sparse modeling in quantum many-body problems, Jour- nal of the Physical Society of Japan 89, 012001 (2020), https://doi.org/10.7566/JPSJ.89.012001

    work page doi:10.7566/jpsj.89.012001 2020

  46. [46]

    Shinaoka, N

    H. Shinaoka, N. Chikano, E. Gull, J. Li, T. Nomoto, J. Otsuki, M. Wallerberger, T. Wang, and K. Yoshimi, Efficient ab initio many-body calculations based on sparse modeling of Matsub- ara Green’s function, SciPost Phys. Lect. Notes , 63 (2022)

    work page 2022

  47. [47]

    Shinaoka, J

    H. Shinaoka, J. Otsuki, M. Ohzeki, and K. Yoshimi, Compress- ing green’s function using intermediate representation between imaginary-time and real-frequency domains, Phys. Rev. B 96, 035147 (2017)

    work page 2017

  48. [48]

    J. Li, M. Wallerberger, N. Chikano, C.-N. Yeh, E. Gull, and H. Shinaoka, Sparse sampling approach to e fficient ab initio calculations at finite temperature, Phys. Rev. B 101, 035144 (2020)

    work page 2020

  49. [49]

    Nogaki and H

    K. Nogaki and H. Shinaoka, Bosonic nevanlinna analytic con- tinuation, Journal of the Physical Society of Japan 92, 035001 (2023), https://doi.org/10.7566/JPSJ.92.035001

    work page doi:10.7566/jpsj.92.035001 2023

  50. [50]

    Nogaki, J

    K. Nogaki, J. Fei, E. Gull, and H. Shinaoka, Nevanlinna.jl: A Julia implementation of Nevanlinna analytic continuation, Sci- Post Phys. Codebases , 19 (2023)

    work page 2023

  51. [51]

    Fei, C.-N

    J. Fei, C.-N. Yeh, and E. Gull, Nevanlinna analytical continua- tion, Phys. Rev. Lett. 126, 056402 (2021)

    work page 2021

  52. [52]

    He and P

    W.-Y . He and P. A. Lee, Electronic density of states of a u(1) quantum spin liquid with spinon fermi surface. i. orbital mag- netic field effects, Phys. Rev. B 107, 195155 (2023)

    work page 2023

  53. [53]

    M. S. Scheurer and S. Sachdev, Orbital currents in insulating and doped antiferromagnets, Phys. Rev. B 98, 235126 (2018)

    work page 2018

  54. [54]

    Werman, S

    Y . Werman, S. Chatterjee, S. C. Morampudi, and E. Berg, Sig- natures of fractionalization in spin liquids from interlayer ther- mal transport, Phys. Rev. X 8, 031064 (2018)

    work page 2018

  55. [55]

    Fabrizio, Spin-liquid insulators can be landau’s fermi liq- uids, Phys

    M. Fabrizio, Spin-liquid insulators can be landau’s fermi liq- uids, Phys. Rev. Lett. 130, 156702 (2023)

    work page 2023

  56. [56]

    Khatibi, R

    Z. Khatibi, R. Ahemeh, and M. Kargarian, Excitonic insu- lator phase and condensate dynamics in a topological one- dimensional model, Phys. Rev. B 102, 245121 (2020)

    work page 2020

  57. [57]

    Gunnarsson, M

    O. Gunnarsson, M. W. Haverkort, and G. Sangiovanni, Analyti- cal continuation of imaginary axis data for optical conductivity, Phys. Rev. B 82, 165125 (2010)

    work page 2010

  58. [58]

    H. Mei, H. Yuan, H. Wen, H. Yao, S. Sun, X. Zheng, F. Liu, H. Li, and W. Xu, Optical conductivity of an electron gas driven by a pulsed terahertz radiation field, The European Physical Journal B 95, 111 (2022)

    work page 2022

  59. [59]

    Huang, Acflow: An open source toolkit for analytic continu- ation of quantum monte carlo data, Computer Physics Commu- 12 nications 292, 108863 (2023)

    L. Huang, Acflow: An open source toolkit for analytic continu- ation of quantum monte carlo data, Computer Physics Commu- 12 nications 292, 108863 (2023)

    work page 2023

  60. [60]

    C. P. Nave and P. A. Lee, Transport properties of a spinon fermi surface coupled to a u(1) gauge field, Phys. Rev. B 76, 235124 (2007)

    work page 2007

  61. [61]

    Li, Absence of a T 2/3 specific heat anomaly in au(1) spin liq- uid with a large spinon fermi surface, Phys

    T. Li, Absence of a T 2/3 specific heat anomaly in au(1) spin liq- uid with a large spinon fermi surface, Phys. Rev. B104, 165123 (2021)

    work page 2021

  62. [62]

    Ribak, I

    A. Ribak, I. Silber, C. Baines, K. Chashka, Z. Salman, Y . Da- gan, and A. Kanigel, Gapless excitations in the ground state of 1t−tas2, Phys. Rev. B 96, 195131 (2017)

    work page 2017

  63. [63]

    Murayama, Y

    H. Murayama, Y . Sato, T. Taniguchi, R. Kurihara, X. Z. Xing, W. Huang, S. Kasahara, Y . Kasahara, I. Kimchi, M. Yoshida, Y . Iwasa, Y . Mizukami, T. Shibauchi, M. Konczykowski, and Y . Matsuda, Effect of quenched disorder on the quantum spin liquid state of the triangular-lattice antiferromagnet 1 t − tas2, Phys. Rev. Res. 2, 013099 (2020)

    work page 2020

  64. [64]

    Webplotdigitizer : Version 4.7 (2024)

    work page 2024

  65. [65]

    W. R. Inc., Mathematica, Version 14.2, champaign, IL, 2024

    work page 2024

  66. [66]

    P. A. Lee and N. Nagaosa, Gauge theory of the normal state of high-tc superconductors, Phys. Rev. B 46, 5621 (1992)

    work page 1992

  67. [67]

    Nagaosa and P

    N. Nagaosa and P. A. Lee, Normal-state properties of the uni- form resonating-valence-bond state, Phys. Rev. Lett. 64, 2450 (1990)

    work page 1990

  68. [68]

    Montvay and G

    I. Montvay and G. M¨unster, Quantum Fields on a Lattice, Cam- bridge Monographs on Mathematical Physics (Cambridge Uni- versity Press, 1994)

    work page 1994