# Excitation spectra of quantum matter
without quasiparticles II: random $t$-$J$ models

Maria Tikhanovskaya, Haoyu Guo, Subir Sachdev, Grigory Tarnopolsky  Department
of Physics, Harvard University, Cambridge MA, 02138, USA

###### Abstract

We present numerical solutions of the spectral functions of $t$-$J$ models
with random and all-to-all exchange and global SU($M$) spin rotation symmetry.
The solutions are obtained from the saddle-point equations of the large volume
limit, followed by the large $M$ limit. These saddle point equations involve
Green’s functions for fractionalized spinons and holons carrying emergent U(1)
gauge charges, obeying relations similar to those of the Sachdev-Ye-Kitaev
(SYK) models. The low frequency spectral functions are compared with an
analytic analysis of the operator scaling dimensions, with good agreement. We
also compute the low frequency and temperature behavior of gauge-invariant
observables: the electron Green’s function, the local spin susceptibility and
the optical conductivity; along with the temperature dependence of the d.c.
resistivity. The time reparameterization soft mode (equivalent to the boundary
graviton in holographically dual models of two-dimensional quantum gravity)
makes important contributions to all observables, and provides a linear-in-
temperature contribution to the d.c. resistivity.

###### Contents

  1. I Introduction
  2. II $t$-$J$ model and fractionalization
    1. II.1 Model Hamiltonian
    2. II.2 Fermionic spinon + Bosonic holon
    3. II.3 Bosonic Spinon + Fermionic holon
  3. III Operator Spectrum and Corrections to Conformality
    1. III.1 Corrections to Conformality
    2. III.2 Operator Spectrum
  4. IV Gauge Invariant Observables
    1. IV.1 Electron Spectral Density
      1. IV.1.1 Zero temperature
      2. IV.1.2 Finite Temperature
    2. IV.2 Conductivity
      1. IV.2.1 Zero temperature Optical conductivity
      2. IV.2.2 Finite temperature DC conductivity
      3. IV.2.3 Finite temperature optical conductivity at linear order
    3. IV.3 Spin Spectral Density
      1. IV.3.1 Zero temperature
      2. IV.3.2 Finite temperature
  5. V Numerical results for the $t$-$J$ model
  6. VI Conclusions
  7. A Zero temperature numerics for $t-J$ model
    1. A.1 Luttinger constraints and fermionic charge vs chemical potential
  8. B Corrections to Conformality
    1. B.0.1 Kernel of the $t$-$J$ Model in the bosonic spinon $+$ fermionic holon case
    2. B.0.2 Linear order correction at zero temperature
    3. B.0.3 Nonlinear order correction at zero temperature
  9. C Qualitative analysis of potentially relevant operators in the $t$-$J$ model
  10. D Calculation of Optical Conductivity

##  I Introduction

This paper extends our recent analysis in Ref. Tikhanovskaya  _et al._ (2020)
from insulating spin models to metallic states of a random $t$-$J$ model. This
yields a rare solvable model of a metallic system without quasiparticle
excitations, and so it is worthwhile to explore its properties in detail.

In the condensed matter literature, extensions of the Sachdev-Ye-Kitaev (SYK)
models Sachdev and Ye (1993); Kitaev (2015); Sachdev (2015) provide solvable
theories of ‘strange’ and ‘bad’ metal behavior. Extensions which include
single particle hopping Song  _et al._ (2017); Zhang (2017); Chowdhury  _et
al._ (2018); Patel  _et al._ (2018); Patel and Sachdev (2019) have regimes of
intermediate temperature ($T$) with the paradigmatic linear-in-$T$
resistivity, but with a bad metal resistivity i.e. larger than the quantum of
resistance $h/e^{2}$ (in two dimensions). Upon lowering $T$, these models
enter a disordered Fermi liquid regime with resistivity $\sim T^{2}$ and well-
defined quasiparticles, once the resistivity becomes smaller than $h/e^{2}$. A
strange metal regime, i.e. a linear-in-$T$ resistivity smaller than $h/e^{2}$,
only appears possible in special models with ‘resonant’ interactions Patel and
Sachdev (2019).

In recent work, Joshi et al. Joshi  _et al._ (2020) argued that a shortcoming
of the above SYK models Song  _et al._ (2017); Zhang (2017); Chowdhury  _et
al._ (2018); Patel  _et al._ (2018); Patel and Sachdev (2019) is the absence
of ‘Mottness’ i.e. a strong local repulsion between electrons that allows an
interaction-driven insulator at commensurate densities. Mottness was present
in the original Sachdev-Ye model Sachdev and Ye (1993) of an insulator, and
this can be extended to metals by considering a SYK-like version of the
$t$-$J$ model with random exchange interactions Parcollet and Georges (1999);
Joshi  _et al._ (2020); Guo  _et al._ (2020); Shackleton  _et al._ (2020).
Ref. Joshi  _et al._ (2020) argued that upon tuning the electron density, such
$t$-$J$ models have critical points or phases in which the SYK criticality
extends down to $T=0$.

Ref. Joshi  _et al._ (2020) also proposed a large $M$ limit of a $t$-$J$ model
with SU($M$) symmetry, when the solution reduces to a set of SYK-like saddle-
point equations for the Green’s functions of fractionalized particles carrying
emergent U(1) gauge charges: these are the spinons and holons carrying the
spin and charge of the underlying electrons. The leading low frequency
behavior of the Green’s functions were determined in Ref. Joshi  _et al._
(2020). Here, we shall provide a full numerical analysis of the saddle-point
equations, and show that solutions with the proposed low frequency
singularities do exist over some range of parameters. We shall also
analytically compute the subleading corrections to the low frequency behavior,
along the lines of Refs. Guo  _et al._ (2020); Tikhanovskaya  _et al._ (2020),
and show that it is consistent with our numerical results.

With the knowledge of the Green’s functions of the fractionalized particles,
we can compute various physical gauge-invariant observables. We list the
properties of the electron spectral weight, $\rho_{e}(\omega)$, the spin
spectral density $\rho_{Q}(\omega)$, and the optical conductivity
$\sigma(\omega)$ at frequency $\omega$ and at temperature $T=0$ ($J$ is the
root-mean-square exchange interaction):

$\displaystyle\rho_{e}(\omega)$ $\displaystyle=$ $\displaystyle
A^{\pm}\left(1-\sum_{h}A_{h}^{\pm}|\omega/J|^{h-1}-\sum_{hh^{\prime}}A_{hh^{\prime}}^{\pm}|\omega/J|^{h+h^{\prime}-2}\right)\,,$
$\displaystyle\rho_{Q}(\omega)$ $\displaystyle=$ $\displaystyle
B^{\pm}\left(1-\sum_{h}B_{h}|\omega/J|^{h-1}-\sum_{hh^{\prime}}B_{hh^{\prime}}|\omega/J|^{h+h^{\prime}-2}\right)\,,$
$\displaystyle\mbox{Re}\,\sigma(\omega)$ $\displaystyle=$
$\displaystyle\sigma_{0}\left(1-\sum_{h}C_{h}|\omega/J|^{h-1}-\sum_{hh^{\prime}}C_{hh^{\prime}}|\omega/J|^{h+h^{\prime}-2}\right)\,,$
(1)

where $\pm$ superscripts refer to $\omega\gtrless 0$. Here the summation is
over the spectrum of irrelevant operators, and $A,B,C,\sigma_{0}$’s are
coefficients that will be given in the main text in (72), (IV.3.1) and (81).
In the above expressions, we include the leading term arising from the
conformal solution, and the subleading corrections. Example of the latter
includes a leading irrelevant operator with scaling dimension $h_{1}<2$ (see
Fig. 3), and the time reparameterization mode (i.e. the boundary graviton in
dual models of two-dimensional quantum gravity Maldacena and Stanford (2016);
Maldacena  _et al._ (2016); Kitaev and Suh (2018)) with scaling dimension
$h_{0}=2$. We note that the presence of the mode with $h_{1}<2$ is a new
feature of the doped $t$-$J$ model, and was not present in the undoped
antiferromagnet examined earlier Tikhanovskaya  _et al._ (2020); the leading
irrelevant operator in the latter case was the $h_{0}=2$ boundary graviton.

We can also extend the results in (1) to non-zero $T$: each term will be
multiplied by a scaling function of $\omega/T$. The expressions for these
scaling functions are rather complicated at general scaling dimension $h$ and
are given in the body of the paper. But the scaling functions do simplify for
the $h_{0}=2$ case. Including only these $T$-dependent corrections, the non-
zero $T$ form of (1) is, for $\omega,T\ll J$ ($\beta\equiv 1/T$)

$\displaystyle\rho_{e}(\omega)=\frac{C_{e}\cosh(\frac{\beta\omega}{2})}{J\cosh(\frac{\beta\omega}{2}-\pi\mathscr{E}_{e})}\left(1-\frac{\pi\alpha_{0}}{\beta
J}\Big{(}\frac{\beta\omega}{2\pi}-\mathscr{E}_{e}\Big{)}\big{(}4\tanh(\frac{\beta\omega}{2}-\pi\mathscr{E}_{e})-3(\sin(2\theta_{f})-\sin(2\theta_{b}))\big{)}\right)\,,$
$\displaystyle\rho_{Q}(\omega)=\frac{C_{Q}}{J}\tanh\big{(}\frac{\beta\omega}{2}\big{)}\left(1-\frac{2\alpha_{0}\omega}{J}\tanh\big{(}\frac{\beta\omega}{2}\big{)}\right)\,,$
$\displaystyle\mbox{Re}\,\sigma(\omega)=\sigma_{0}\left(1-\frac{2\alpha_{0}\omega}{J}\coth(\frac{\beta\omega}{2})\right)\,.$
(2)

In these expressions, the terms outside the square brackets arise from the
leading conformal theory, while the terms inside the square brackets are the
contributions of the boundary graviton mode (with $h_{0}=2$). Here $C_{e}$,
$C_{Q}$ and $\sigma_{0}$ are normalization constants that depend on the UV
details. $\alpha_{0}$ is a dimensionless number parameterizing the strength of
$h_{0}=2$ correction: this is also a UV parameter and is extracted from the
numerics. $\theta_{f}$ and $\theta_{b}$ parameterize the conformal solution
(22) and are related to the U(1) charges. $\mathscr{E}_{e}$ is the spectral
asymmetry of the electron Green’s function (71), which can be calculated from
$\theta_{f}$ and $\theta_{b}$. We also include schematic plots of the
functions in (2) in Fig. 1.

Finally, we present the results for the d.c. resistivity, $\rho(T)$. Note from
(1) that the zero frequency conductivity from the scaling limit solution is a
constant i.e. the conformal solution yields a residual resistivity. So all the
$T$ dependence arises from corrections to conformality, and these are (from
(88))

$\rho(T)=\frac{1}{\sigma_{0}}\left[1+\sum_{h}\alpha_{h}(v_{hf+}+v_{hf-}+v_{hb+}+v_{hb-})R(h)\left(\frac{T}{J}\right)^{h-1}\right]\,.$
(3)

Note that the time reparameterization mode, with $h_{0}=2$, yields a linear-
in-temperature correction. Further results on the numerical values of the
coefficients in (1) and (3) appear in Section V.

Figure 1: Plots of the functions in Eq. (2). We used $\beta J=20$,
$\theta_{f}=0.1\pi$, $\theta_{b}=0.3\pi$, $\alpha_{0}=0.05$.

The outline of our paper is as follows. We define the $t$-$J$ model in Section
II, along with its large $M$ limit using fractionalized spinons and holons. We
also present the SYK-like saddle-point equations and their conformal
solutions. Section III turns to our new results on the operator spectrum and
the associated corrections to conformality for the spinon and holon Green’s
functions. We also find regimes where the conformal solutions are unstable:
this can happen by the appearance of operators with scaling dimension $h<1$,
and by complex values of $h$. Section IV combines the fractionalized results
of Section III to compute the spectral functions of various gauge invariant
observables and their corrections to conformality. Finally, Section V computes
the full numerical solution of the saddle-point equations and compares them
with our conformal expansions.

##  II $t$-$J$ model and fractionalization

###  II.1 Model Hamiltonian

We consider the following Hamiltonian of the $t$-$J$ model

$H=\sum_{\left\langle
ij\right\rangle,l,\alpha}t_{ij}c_{il\alpha}^{\dagger}c_{jl\alpha}+h.c.+\sum_{\left\langle
ij\right\rangle,\alpha\beta}J_{ij}\left(S_{i\alpha\beta}S_{j\beta\alpha}-\frac{1}{M}S_{i\alpha\alpha}S_{j\beta\beta}\right).$
(4)

Here $c_{il\alpha}$ is electron operator, with $i$ denoting site, $l$ denotes
an auxiliary SU$(M^{\prime})$ orbital label, and $\alpha$ denotes SU($M$)
spin. The $S_{i\alpha\beta}=S_{i\beta\alpha}^{\dagger}$ is the spin operator
on site $i$, and the $1/M$ term (which will be dropped in large $M$ limit) is
added to ensure it transform in the adjoint of SU$(M)$.

We assume the system lives on a Bethe lattice of coordination number $z$ with

$t_{ij}=\frac{t}{\sqrt{Mz}}\,,$ (5)

and $J_{ij}$’s are Gaussian random variables with

$\overline{J_{ij}}=0,\quad\overline{J_{ij}^{2}}=\frac{J^{2}}{Mz}\,.$ (6)

Implementing dynamical mean-field approximation ($z\to\infty$), and assuming a
diagonal mean-field ansatz, the Hamiltonian induces the following interaction
terms in the single-site action

$\begin{split}S\supset&\frac{t^{2}}{M}\sum_{l\alpha}\int_{0}^{1/T}{\rm
d}\tau{\rm
d}\tau^{\prime}\sum_{l\alpha}c_{l\alpha}^{\dagger}G_{c}(\tau,\tau^{\prime})c_{l\alpha}(\tau^{\prime})-\frac{J^{2}}{2M}\sum_{\alpha\beta}\int_{0}^{1/T}{\rm
d}\tau{\rm
d}\tau^{\prime}S_{\alpha\beta}(\tau)Q(\tau,\tau^{\prime})S_{\beta\alpha}(\tau^{\prime})\,.\end{split}$
(7)

Here the mean-field ansatz $G_{e}$ and $Q$ should satisfy the following self-
consistency equations

$G_{e}(\tau,\tau^{\prime})=\frac{-1}{MM^{\prime}}\sum_{l\alpha}\langle
c_{l\alpha}(\tau)c_{l\alpha}^{\dagger}(\tau^{\prime})\rangle\,,\quad
Q(\tau,\tau^{\prime})=\frac{1}{M^{2}}\sum_{\alpha\beta}\left\langle
S_{\beta\alpha}(\tau)S_{\alpha\beta}(\tau^{\prime})\right\rangle\,.$ (8)

We also assume large $M,M^{\prime}$ limit with fixed

$k=\frac{M^{\prime}}{M}\,.$ (9)

The full action also contains WZW terms for slave particles and Langrangian
multipliers, which will be detailed shortly.

###  II.2 Fermionic spinon + Bosonic holon

We fractionalize the model by introducing fermionic spinon and bosonic holon:

$c_{il\alpha}=f_{i\alpha}b_{il}^{\dagger},\quad
S_{i\alpha\beta}=f_{i\alpha}^{\dagger}f_{i\beta}\,.$ (10)

Here the boson $b_{il}$ transforms in anti-fundamental of SU($M^{\prime}$)
such that the electron $c_{il\alpha}$ transforms in fundamental of
SU($M^{\prime}$). This induces a U(1) gauge symmetry

$f_{i\alpha}(\tau)\to f_{i\alpha}(\tau)e^{i\phi_{i}(\tau)},\quad
b_{il}(\tau)\to b_{il}(\tau)e^{i\phi_{i}(\tau)}.$ (11)

For the theory to be consistent, the physical Hilbert space must be U(1)
gauge-symmetric, which implies that the gauge charge is conserved, and we
consider the representation

$\sum_{\alpha}f_{i\alpha}^{\dagger}f_{i\alpha}+\sum_{l}b_{il}^{\dagger}b_{il}=\kappa
M.$ (12)

We also implement the doping density $p$ by

$\frac{1}{M^{\prime}}\sum_{l}\langle b_{il}^{\dagger}b_{il}\rangle=p.$ (13)

The full action is thus

$\displaystyle S[f,b,\lambda]$ $\displaystyle=$
$\displaystyle\int_{0}^{1/T}d\tau\left[\sum_{\ell}b_{\ell}^{\dagger}\left(\frac{\partial}{\partial\tau}+i\lambda\right)b_{\ell}+\sum_{\alpha}f_{\alpha}^{\dagger}\left(\frac{\partial}{\partial\tau}+\epsilon_{0}+i\lambda\right)f_{\alpha}-i\lambda\frac{M}{2}\right]$
(14) $\displaystyle~{}+\frac{t^{2}}{M}\sum_{\ell,\alpha}\int_{0}^{1/T}d\tau
d\tau^{\prime}f^{\dagger}_{\alpha}(\tau)b_{\ell}(\tau)G_{c}(\tau-\tau^{\prime})b^{\dagger}_{\ell}(\tau^{\prime})f_{\alpha}(\tau^{\prime})$
$\displaystyle~{}-\frac{J^{2}}{2M}\sum_{\alpha,\beta}\int_{0}^{1/T}d\tau
d\tau^{\prime}Q(\tau-\tau^{\prime})f_{\alpha}^{\dagger}(\tau)f_{\beta}(\tau)f_{\beta}^{\dagger}(\tau^{\prime})f_{\alpha}(\tau^{\prime})\,.$

Introducing bilocal fields

$G_{f}(\tau,\tau^{\prime})=\frac{-1}{M}\sum_{\alpha}f_{\alpha}(\tau)f^{\dagger}_{\alpha}(\tau^{\prime}),\quad
G_{b}(\tau,\tau^{\prime})=\frac{-1}{M^{\prime}}\sum_{l}b_{l}(\tau)b^{\dagger}_{l}(\tau^{\prime})$
(15)

the saddle-point equations for the slave particles $b_{\ell}$ and $f_{\alpha}$
are given as follows,

$\displaystyle G_{b}(i\omega_{n})$ $\displaystyle=$
$\displaystyle\frac{1}{i\omega_{n}+\mu_{b}-\Sigma_{b}(i\omega_{n})},$ (16)
$\displaystyle\Sigma_{b}(\tau)$ $\displaystyle=$
$\displaystyle-t^{2}G_{f}(\tau)G_{f}(-\tau)G_{b}(\tau),$ (17) $\displaystyle
G_{f}(i\omega_{n})$ $\displaystyle=$
$\displaystyle\frac{1}{i\omega_{n}+\mu_{f}-\Sigma_{f}(i\omega_{n})},$ (18)
$\displaystyle\Sigma_{f}(\tau)$ $\displaystyle=$
$\displaystyle-J^{2}G_{f}(\tau)^{2}G_{f}(-\tau)+kt^{2}G_{f}(\tau)G_{b}(\tau)G_{b}(-\tau).$
(19)

Here $\mu_{f}$ and $\mu_{b}$ are chemical potentials, determined by
$\epsilon_{0}$ and the saddle point value of $\lambda$, and chosen to satisfy

$\left\langle f^{\dagger}f\right\rangle=\kappa-kp\quad,\quad\left\langle
b^{\dagger}b\right\rangle=p\,.$ (20)

The electron Green’s functions is a product

$G_{e}(\tau)=-G_{f}(\tau)G_{b}(-\tau)\,.$ (21)

Eqs.(16)-(19) admit the following infra-red (IR) conformal saddle point
solution at zero temperature Joshi  _et al._ (2020)

$\quad G^{c}_{a}(\tau)=-\begin{pmatrix}e^{\pi\mathscr{E}_{a}}\\\ \zeta
e^{-\pi\mathscr{E}_{a}}\end{pmatrix}\frac{b_{a}^{\Delta_{a}}}{|J\tau|^{2\Delta_{a}}},\quad
G_{a}^{c}(i\omega)=-\frac{iC_{a}}{J}\begin{pmatrix}e^{-i\theta_{a}}\\\
-e^{i\theta_{a}}\end{pmatrix}|\omega/J|^{2\Delta_{a}-1}\,,$ (22)

where $c$ in the superscript means conformal and subscript $a=b,f$ indexes
boson and fermion respectively. We are following the convention of writing
functions as two component vectors, where the first component is for $\tau>0$
or $\omega>0$, and the second one is for $\tau<0$ or $\omega<0$. Also
$\mathscr{E}_{b}$, $\mathscr{E}_{f}$ and $\theta_{b},\theta_{f}$ are asymmetry
parameters and angles and $\zeta$ factor is $\zeta_{b}=1$ and $\zeta_{f}=-1$.
To pass from the coordinate space to the frequency one we use the Fourier
transform in our plus/minus basis

$\bigintssss\begin{pmatrix}a_{+}\\\
a_{-}\end{pmatrix}|\tau|^{-\alpha}e^{i\omega\tau}d\tau=M(\alpha)\begin{pmatrix}a_{+}\\\
a_{-}\end{pmatrix}|\omega|^{\alpha-1}\,,\quad
M(\alpha)=\Gamma(1-\alpha)\begin{pmatrix}i^{1-\alpha}&~{}i^{\alpha-1}\\\
i^{\alpha-1}&~{}i^{1-\alpha}\end{pmatrix}\,.$ (23)

We will restrict our attention here to the the case where the boson and
fermion scaling dimensions are $\Delta_{f}=\Delta_{b}=1/4$, which is the case
proposed for the deconfined critical point in Ref. Joshi  _et al._ , 2020. The
large $M$ equations also admit a solution in a critical phase Joshi  _et al._
(2020) with $1/4<\Delta_{f}<1/2$, $\Delta_{f}+\Delta_{b}=1/2$ which we will
not explicitly consider here: we expect similar results to also apply to this
critical phase. For the case $\Delta_{f}=\Delta_{b}=1/4$, the pre-factors
satisfy

$\begin{split}&(t^{2}/J^{2})C_{f}^{2}C_{b}^{2}\cos(2\theta_{f})=\pi,\\\
&C_{f}^{4}\cos(2\theta_{f})-k(t^{2}/J^{2})C_{f}^{2}C_{b}^{2}\cos(2\theta_{b})=\pi\end{split}$
(24)

and we can find

$C_{f}=\Big{(}\frac{\pi(1-\eta)}{\cos(2\theta_{f})}\Big{)}^{1/4},\quad
C_{b}=\frac{J}{t}\Big{(}\frac{\pi}{\cos(2\theta_{f})(1-\eta)}\Big{)}^{1/4},\quad\eta\equiv-k\frac{\cos(2\theta_{b})}{\cos(2\theta_{f})}\,.$
(25)

Positivity constraints on the spectral densities impose restrictions on
asymmetry angles: $-\pi\Delta_{f}<\theta_{f}<\pi\Delta_{f}$,
$\pi\Delta_{b}<\theta_{b}<\pi/2$. Therefore we see that $\eta>0$ and since
$C_{b}$ and $C_{f}$ are defined to be real positive numbers, we find another
constraint for $\theta_{b}$ and $\theta_{f}$:

$\eta<1\,.$ (26)

For completeness we also list expressions for $b_{f}$ and $b_{b}$:

$b_{f}=\frac{\cos(2\theta_{f})}{4\pi}(1-\eta),\quad
b_{b}=\frac{\cos(2\theta_{f})}{4\pi}\frac{\eta^{2}J^{4}/t^{4}}{k^{2}(1-\eta)}\,$
(27)

and the asymmetry parameters $\mathscr{E}_{a}$ are related to asymmetry angles
$\theta_{a}$ as

$e^{2\pi\mathscr{E}_{a}}=\frac{\sin(\theta_{a}+\pi\Delta_{a})}{\sin(\theta_{a}-\pi\Delta_{a})}\zeta,\quad
e^{-2i\theta_{a}}=\frac{\sin\pi(i\tilde{\mathscr{E}}_{a}+\Delta_{a})}{\sin\pi(i\tilde{\mathscr{E}}_{a}-\Delta_{a})},$
(28)

where
$\tilde{\mathscr{E}}_{b}=\mathscr{E}_{b},\tilde{\mathscr{E}}_{f}=\mathscr{E}_{f}+1/2$.

Thus all parameters of the IR solution are fully determined by the two
asymmetry angles $\theta_{f}$ and $\theta_{b}$. We can relate these angles to
the boson and fermion densities by Luttinger constraints. According to Georges
_et al._ (2001); Gu  _et al._ (2020), for a SYK-type model with UV source
$\sigma$ we can define a U(1) charge at zero temperature

$\mathcal{Q}=-\int_{-\infty}^{\infty}{\rm d}\tau\tau\sigma(\tau)G(-\tau)\,.$
(29)

For the case of our model,
$\sigma(\tau)=\delta^{\prime}(\tau)-\mu\delta(\tau)$, which yields

$\mathcal{Q}=\frac{G(0^{+})+G(0^{-})}{2}.$ (30)

Matching this with the definition of Green’s functions, we get

$\mathcal{Q}_{f}=\left\langle
f^{\dagger}f\right\rangle-\frac{1}{2},\quad\mathcal{Q}_{b}=-\left\langle
b^{\dagger}b\right\rangle-\frac{1}{2}\,.$ (31)

Also the charge can be calculated entirely from the IR asymptotics, related to
$\theta$-parameter by

$\mathcal{Q}(\theta)=-\frac{\theta}{\pi}-\left(\frac{1}{2}-\Delta\right)\frac{\sin(2\theta)}{\sin(2\pi\Delta)}\,.$
(32)

Therefore, from (20), $\theta_{f},\theta_{b}$ satisfy the following Luttinger
constraints Georges  _et al._ (2001); Gu  _et al._ (2020)

$\displaystyle\frac{\theta_{f}}{\pi}+\left(\frac{1}{2}-\Delta_{f}\right)\frac{\sin(2\theta_{f})}{\sin(2\pi\Delta_{f})}$
$\displaystyle=$ $\displaystyle\frac{1}{2}-\kappa+kp,$ (33)
$\displaystyle\frac{\theta_{b}}{\pi}+\left(\frac{1}{2}-\Delta_{b}\right)\frac{\sin(2\theta_{b})}{\sin(2\pi\Delta_{b})}$
$\displaystyle=$ $\displaystyle\frac{1}{2}+p\,.$ (34)

###  II.3 Bosonic Spinon + Fermionic holon

We also discuss the other fractionalization scheme, consisting of bosonic
spinons and fermionic holons,

$c_{il\alpha}=\mathfrak{b}_{i\alpha}\mathfrak{f}_{il}^{\dagger},\quad
S_{i\alpha\beta}=\mathfrak{b}_{i\alpha}^{\dagger}\mathfrak{b}_{i\beta}\,,$
(35)

with the gauge charge constraint

$\sum_{\alpha}\mathfrak{b}^{\dagger}_{i\alpha}\mathfrak{b}_{i\alpha}+\sum_{l}\mathfrak{f}_{il}^{\dagger}\mathfrak{f}_{il}=\kappa
M,$ (36)

and filling density

$\frac{1}{M^{\prime}}\sum_{l}\langle\mathfrak{f}_{il}^{\dagger}\mathfrak{f}_{il}\rangle=p\,.$
(37)

But most of our analysis will be carried out in the fractionalization scheme
in Section II.2. Following the similar discussion of the other scheme, we can
now derive the action to be

$\displaystyle S[\mathfrak{f},\mathfrak{b},\lambda]$
$\displaystyle=\int_{0}^{1/T}{\rm
d}\tau\left[\sum_{\alpha}\mathfrak{b}^{\dagger}_{\alpha}\left(\frac{\partial}{\partial\tau}+i\lambda\right)\mathfrak{b}_{\alpha}+\sum_{\alpha}\mathfrak{f}_{l}\left(\frac{\partial}{\partial\tau}+\epsilon_{0}+i\lambda\right)\mathfrak{f}_{l}-i\lambda\frac{M}{2}\right]$
$\displaystyle+\frac{t^{2}}{M}\sum_{l\alpha}\int_{0}^{1/T}{\rm d}\tau{\rm
d}\tau^{\prime}\mathfrak{f}(\tau)\mathfrak{b}_{\alpha}^{\dagger}(\tau)G_{c}(\tau,\tau^{\prime})\mathfrak{b}_{\alpha}(\tau^{\prime})\mathfrak{f}^{\dagger}_{l}(\tau^{\prime})$
$\displaystyle-\frac{J^{2}}{2M}\sum_{\alpha\beta}\int_{0}^{1/T}{\rm d}\tau{\rm
d}\tau^{\prime}\mathfrak{b}_{\alpha}^{\dagger}(\tau)\mathfrak{b}_{\beta}(\tau)Q(\tau,\tau^{\prime})\mathfrak{b}^{\dagger}_{\beta}(\tau^{\prime})\mathfrak{b}_{\alpha}(\tau^{\prime})\,.$
(38)

The self consistency relations are

$\displaystyle G_{e}(\tau,\tau^{\prime})$
$\displaystyle=\frac{-1}{MM^{\prime}}\sum_{l\alpha}\langle
c_{l\alpha}(\tau)c^{\dagger}_{l\alpha}(\tau^{\prime})\rangle$
$\displaystyle=\frac{-1}{MM^{\prime}}\sum_{l\alpha}\left\langle\mathfrak{b}_{\alpha}(\tau)\mathfrak{f}^{\dagger}_{l}(\tau)\mathfrak{f}_{l}(\tau^{\prime})\mathfrak{b}^{\dagger}_{\alpha}(\tau^{\prime})\right\rangle\,,$
(39) $\displaystyle Q(\tau,\tau^{\prime})$
$\displaystyle=\frac{1}{M^{2}}\sum_{\alpha\beta}\left\langle
S_{\beta\alpha}(\tau)S_{\alpha\beta}(\tau^{\prime})\right\rangle$
$\displaystyle=\frac{1}{M^{2}}\sum_{\alpha\beta}\left\langle\mathfrak{b}^{\dagger}_{\beta}(\tau)\mathfrak{b}_{\alpha}(\tau)\mathfrak{b}^{\dagger}_{\alpha}(\tau^{\prime})\mathfrak{b}_{\beta}(\tau^{\prime})\right\rangle\,.$
(40)

Introducing bilocal fields

$G_{\mathfrak{b}}(\tau,\tau^{\prime})=\frac{-1}{M}\sum_{\alpha}\mathfrak{b}_{\alpha}(\tau)\mathfrak{b}^{\dagger}_{\alpha}(\tau^{\prime}),\quad
G_{\mathfrak{f}}(\tau,\tau^{\prime})=\frac{-1}{M^{\prime}}\sum_{l}\mathfrak{f}_{l}(\tau)\mathfrak{f}^{\dagger}_{l}(\tau^{\prime})\,,$
(41)

and the corresponding self-energies
$\Sigma_{\mathfrak{b}}(\tau,\tau^{\prime})$,
$\Sigma_{\mathfrak{f}}(\tau,\tau^{\prime})$, we can obtain the saddle point
equations

$\displaystyle G_{\mathfrak{b}}(i\omega_{n})$ $\displaystyle=$
$\displaystyle\frac{1}{i\omega_{n}+\mu_{\mathfrak{b}}-\Sigma_{\mathfrak{b}}(i\omega_{n})}\,,$
(42) $\displaystyle G_{\mathfrak{f}}(i\omega_{n})$ $\displaystyle=$
$\displaystyle\frac{1}{i\omega_{n}+\mu_{\mathfrak{f}}-\Sigma_{\mathfrak{f}}(i\omega_{n})}\,,$
(43) $\displaystyle\Sigma_{\mathfrak{b}}(\tau)$ $\displaystyle=$
$\displaystyle-
kt^{2}G_{\mathfrak{f}}(\tau)G_{\mathfrak{f}}(-\tau)G_{\mathfrak{b}}(\tau)+J^{2}G_{\mathfrak{b}}(\tau)^{2}G_{\mathfrak{b}}(-\tau)\,,$
(44) $\displaystyle\Sigma_{\mathfrak{f}}(\tau)$ $\displaystyle=$
$\displaystyle
t^{2}G_{\mathfrak{f}}(\tau)G_{\mathfrak{b}}(-\tau)G_{\mathfrak{b}}(\tau)\,,$
(45)

where we have substituted the self consistent relations

$G_{e}(\tau)=G_{\mathfrak{f}}(-\tau)G_{\mathfrak{b}}(\tau),\quad
Q(\tau)=G_{\mathfrak{b}}(\tau)G_{\mathfrak{b}}(-\tau)\,.$ (46)

The chemical potentials should be adjusted to satisfy

$\left\langle\mathfrak{f}^{\dagger}\mathfrak{f}\right\rangle=p,\quad\left\langle\mathfrak{b}^{\dagger}\mathfrak{b}\right\rangle=\kappa-
kp\,.$ (47)

Again we assume a conformal ansatz for the solution

$\quad G^{c}_{a}(\tau)=-\begin{pmatrix}e^{\pi\mathscr{E}_{a}}\\\ \zeta
e^{-\pi\mathscr{E}_{a}}\end{pmatrix}\frac{b_{a}^{\Delta_{a}}}{|J\tau|^{2\Delta_{a}}},\quad
G_{a}^{c}(i\omega)=-\frac{iC_{a}}{J}\begin{pmatrix}e^{-i\theta_{a}}\\\
-e^{i\theta_{a}}\end{pmatrix}|\omega/J|^{2\Delta_{a}-1}\,,$ (48)

where $c$ in the superscript means conformal and subscript
$a=\mathfrak{b},\mathfrak{f}$ indexes boson and fermion respectively. Assuming
$\Delta_{\mathfrak{f}}=\Delta_{\mathfrak{b}}=1/4$, the coefficients
$C_{\mathfrak{b}}$ and $C_{\mathfrak{f}}$ satisfy

$\begin{split}&(t^{2}/J^{2})C_{\mathfrak{f}}^{2}C_{\mathfrak{b}}^{2}\cos(2\theta_{\mathfrak{b}})=-\pi\,,\\\
&C_{\mathfrak{b}}^{4}\cos(2\theta_{\mathfrak{b}})-k(t^{2}/J^{2})C_{\mathfrak{b}}^{2}C_{\mathfrak{f}}^{2}\cos(2\theta_{\mathfrak{f}})=-\pi\end{split}$
(49)

and we find

$C_{\mathfrak{b}}=\Big{(}\frac{\pi(1-\eta)}{-\cos(2\theta_{\mathfrak{b}})}\Big{)}^{1/4},\quad
C_{\mathfrak{f}}=\frac{J}{t}\Big{(}\frac{\pi}{-\cos(2\theta_{\mathfrak{b}})(1-\eta)}\Big{)}^{1/4},\quad\eta\equiv-k\frac{\cos(2\theta_{\mathfrak{f}})}{\cos(2\theta_{\mathfrak{b}})}\,.$
(50)

The unitarity constraints are
$-\pi\Delta_{\mathfrak{f}}<\theta_{\mathfrak{f}}<\pi\Delta_{\mathfrak{f}}$,
$\pi\Delta_{\mathfrak{b}}<\theta_{\mathfrak{b}}<\pi/2$ and the condition for
real $C_{\mathfrak{b}}$ and $C_{\mathfrak{f}}$ is $\eta<1$ as before.

The Luttinger constraints for $\theta_{\mathfrak{f}},\theta_{\mathfrak{b}}$
are

$\displaystyle\frac{\theta_{\mathfrak{f}}}{\pi}+\left(\frac{1}{2}-\Delta_{\mathfrak{f}}\right)\frac{\sin(2\theta_{\mathfrak{f}})}{\sin(2\pi\Delta_{\mathfrak{f}})}$
$\displaystyle=$ $\displaystyle\frac{1}{2}-p\,,$ (51)
$\displaystyle\frac{\theta_{\mathfrak{b}}}{\pi}+\left(\frac{1}{2}-\Delta_{\mathfrak{b}}\right)\frac{\sin(2\theta_{\mathfrak{b}})}{\sin(2\pi\Delta_{\mathfrak{b}})}$
$\displaystyle=$ $\displaystyle\frac{1}{2}+\kappa-kp\,.$ (52)

If we ignore the Luttinger constraints, the saddle point equations above are
identical to that of the other fractionalization scheme by substitution

$G_{f}(\tau)\leftrightarrow G_{\mathfrak{b}}(\tau),\quad
G_{b}(\tau)\leftrightarrow G_{\mathfrak{f}}(\tau),\quad 0<\tau<1/T\,,$ (53)

and we transform $\tau<0$ into $0<\tau<\beta$ using KMS relation. In
particular, when
$\Delta_{\mathfrak{f}}=\Delta_{b}=\Delta_{\mathfrak{b}}=\Delta_{f}=1/4$, we
could identify the Green’s functions in the two schemes by

$\theta_{\mathfrak{f}}=\frac{\pi}{2}-\theta_{b},\quad\theta_{\mathfrak{b}}=\frac{\pi}{2}-\theta_{f},\quad
C_{\mathfrak{f}}=C_{b},\quad C_{\mathfrak{b}}=C_{f}.$ (54)

However, we point out that (54) is incompatible with Luttinger constraints.
For example, we can add (34) and (51) together and using (54) to see
$\sin(2\theta_{b})=1$, which leads to $p=0$, but (52) has no solution for
$p=0$. This reflects fact that at large $M$ limit the two fractionalization
schemes are two different theories, and they are identical only at
$M=2,M^{\prime}=1$.

##  III Operator Spectrum and Corrections to Conformality

###  III.1 Corrections to Conformality

The saddle point solutions $G_{f}$ and $G_{b}$ of the $t$-$J$ model can be
viewed as a correlation functions in conformal field theory, which is deformed
by infinite series of irrelevant operators Gross and Rosenhaus (2017);
Klebanov and Tarnopolsky (2017); Klebanov  _et al._ (2018). Therefore the two-
point functions receive corrections from these operators. We shall focus here
on the fermionic spinon $+$ bosonic holon model and the results can be easily
generalized to the other case. The general expression for the two-point
function at zero temperature reads

$\displaystyle G(\tau)$
$\displaystyle=G^{c}(\tau)\left(1-\sum_{h}\frac{\alpha_{h}v_{h}}{|J\tau|^{h-1}}-\sum_{h,h^{\prime}}\frac{a_{hh^{\prime}}\alpha_{h}\alpha_{h^{\prime}}}{|J\tau|^{h+h^{\prime}-2}}-\sum_{h,h^{\prime},h^{\prime\prime}}\frac{a_{hh^{\prime}h^{\prime\prime}}\alpha_{h}\alpha_{h^{\prime}}\alpha_{h^{\prime\prime}}}{|J\tau|^{h+h^{\prime}+h^{\prime\prime}-3}}-\dots\right)\,,$
(55)

where $v_{h}$, $a_{hh^{\prime}}$, $a_{hh^{\prime}h^{\prime\prime}}$, etc are
four-component vectors in $b/f$ and $+/-$ basis, so
$v_{h}=(v_{hb+},v_{hb-},v_{hf+},v_{hf-})^{\textrm{T}}$. For instance the
fermionic Green’s function for $\tau>0$ reads

$G_{f}(\tau)=G^{c}_{f}(\tau)\left(1-\sum_{h}\frac{\alpha_{h}v_{hf+}}{|J\tau|^{h-1}}-\sum_{h,h^{\prime}}\frac{a_{hh^{\prime}f+}\alpha_{h}\alpha_{h^{\prime}}}{|J\tau|^{h+h^{\prime}-2}}-\sum_{h,h^{\prime},h^{\prime\prime}}\frac{a_{hh^{\prime}h^{\prime\prime}f+}\alpha_{h}\alpha_{h^{\prime}}\alpha_{h^{\prime\prime}}}{|J\tau|^{h+h^{\prime}+h^{\prime\prime}-3}}-\dots\right)\,.$
(56)

The real numbers $\alpha_{h}$ are the same for all components $b/f$ and $+/-$.
We are able to find these numbers only numerically. On the other hand the
vectors $v_{h}$ and anomalous dimensions $h$ can be found analytically from
solving the equation

$K_{G}(h)v_{h}=v_{h}\,,$ (57)

where the $4\times 4$ matix $K_{G}(h)=W_{\Sigma}(h)W_{G}(h)$ is a product of
two matrices $W_{\Sigma}(h)$ and $W_{G}$ which in the $(b/f)\times(+/-)$ basis
are given by formulas

$W_{\Sigma}(h)=\frac{\sec(\pi h)}{(1-2h)}\begin{pmatrix}\sin(\pi
h+2\theta_{b})&\sin(2\theta_{b})-1\\\ -\sin(2\theta_{b})-1&\sin(\pi
h-2\theta_{b})\end{pmatrix}\oplus\begin{pmatrix}\sin(\pi
h+2\theta_{f})&\sin(2\theta_{f})-1\\\ -\sin(2\theta_{f})-1&\sin(\pi
h-2\theta_{f})\end{pmatrix},$ (58)
$W_{G}=\left(\begin{array}[]{cccc}1&0&1&1\\\ 0&1&1&1\\\
\eta&\eta&2-\eta&1-\eta\\\ \eta&\eta&1-\eta&2-\eta\\\ \end{array}\right),$
(59)

and we remind that $\eta=-k\cos(2\theta_{b})/\cos(2\theta_{f})$. We notice
that the matrix $W_{\Sigma}(h)$ is block-diagonal in $b/f$ space and $W_{G}$
does not depend on $h$. For $\eta=0$ the fermionic sector is independent on
the bosonic one and its Green’s function is exactly the same as in the complex
SYK model.

The results described above are based on the Kitaev-Suh theory Kitaev and Suh
(2018); Guo  _et al._ (2020); Tikhanovskaya  _et al._ (2020). The derivation
of the kernel $K_{G}$ is done in Guo  _et al._ (2020). The kernel $K_{G}$ for
bosonic spinon + fermionic holon case is obtained by substitution
$\theta_{f}\to\theta_{\mathfrak{b}}$, $\theta_{b}\to\theta_{\mathfrak{f}}$,
for details see Appendix B. The coefficients $a_{hh^{\prime}}$,
$a_{hh^{\prime}h^{\prime\prime}}$ can be calculated using the recursion
procedure described in Tikhanovskaya  _et al._ (2020). We present explicit
derivation of $a_{hh^{\prime}}$ for the $t$-$J$ model in Appendix B.

The linear order zero-temperature correction (55) was generalized to finite
temperature in Guo  _et al._ (2020); Tikhanovskaya  _et al._ (2020). The
result takes the form on the interval $\tau\in(0,\beta)$:

$G_{a}(\tau)=G^{c}_{a}(\tau)\left(1-\frac{v_{ha+}+v_{ha-}}{2}\frac{\alpha_{h}}{(\beta
J)^{h-1}}f_{h}^{\mathrm{A}}(\tau)-\frac{v_{ha+}-v_{ha-}}{2}\frac{\alpha_{h}}{(\beta
J)^{h-1}}f_{h}^{\mathrm{S}}(\tau)\right)\,,$ (60)

where $G^{c}_{a}(\tau)$ is the finite temperature conformal Green’s function

$G^{c}_{a}(\tau)=-b_{a}^{\frac{1}{4}}\left(\frac{\beta
J}{\pi}\sin\frac{\pi\tau}{\beta}\right)^{-\frac{1}{2}}e^{2\pi\mathscr{E}_{a}(\frac{1}{2}-\frac{\tau}{\beta})}\,,$
(61)

and the functions $f_{h}^{\textrm{A}/\textrm{S}}(\tau)$ are given by the
formula

$\displaystyle f_{h}^{\textrm{A}}(\tau)$
$\displaystyle=\frac{(2\pi)^{h-1}\Gamma(h)^{2}}{\Gamma(2h-1)\cos(\frac{\pi
h}{2})}\left(A_{h}(e^{i\frac{2\pi\tau}{\beta}})+A_{h}(e^{-i\frac{2\pi\tau}{\beta}})\right)\,,$
$\displaystyle f_{h}^{\textrm{S}}(\tau)$
$\displaystyle=\frac{(2\pi)^{h-1}\Gamma(h)^{2}}{\Gamma(2h-1)\sin(\frac{\pi
h}{2})}\left(iA_{h}(e^{i\frac{2\pi\tau}{\beta}})-iA_{h}(e^{-i\frac{2\pi\tau}{\beta}})\right)\,,$
(62)

where $A_{h}(u)=(1-u)^{h}\mathbf{F}(h,h,1;u)$ and $\mathbf{F}$ is the
regularized hypergeometric function. We stress that there are no vectors in
the formula (60) and it is correct for both bosons and fermions $a=b,f$ with
their corresponding conformal two-point functions $G^{c}(\tau)$ and components
$v_{h\pm}$.

###  III.2 Operator Spectrum

In this subsection we describe solutions $h$ of the equation (57). Such values
of $h$ correspond to scaling dimensions of the bilinear operators
$\mathcal{O}_{h}$ in the $t$-$J$ model Tikhanovskaya  _et al._ (2020). We can
rewrite the equation for $h$ in the form

$\det(1-K_{G}(h))=0\,.$ (63)

We label positive solutions of the above equation by $h_{0},h_{1},h_{2},\dots$
and notice that the sums in (55) are taken over this series. Several remarks
are in order. First we notice that there are always two $h=1$ solutions which
are related to $U(1)$ global symmetry and $U(1)$ gauge symmetry. The $h=1$
modes are not included in the series $h_{0},h_{1},h_{2},\dots$. The effect of
these modes is already taken into account in the asymmetry parameters
$\mathscr{E}_{f}$ and $\mathscr{E}_{b}$ in $G_{f}^{c}$ and $G_{b}^{c}$ Gu  _et
al._ (2020); Guo  _et al._ (2020). Second there is another parameter-invariant
solution $h=2$ which corresponds to emergent time-reparametrization symmetry
of the Eqs.(16)-(19). We label this solution by $h_{0}$, so we always have
$h_{0}=2$. The rest of the modes are ordered and their values depend on
$k,\theta_{f},\theta_{b}$, so we have

$h_{0}=2,\quad h_{1}\leq h_{2}\leq h_{3}\leq h_{4}\leq\dots\,.$ (64)

We will use notation $\alpha_{i}$ for $\alpha_{h_{i}}$ and $v_{i}$ for
$v_{h_{i}}$. The eigenvector $v_{0}$ has a simple form

$v_{0}=\begin{pmatrix}1-\frac{3}{2}\sin(2\theta_{b})\\\
1+\frac{3}{2}\sin(2\theta_{b})\\\ 1-\frac{3}{2}\sin(2\theta_{f})\\\
1+\frac{3}{2}\sin(2\theta_{f})\end{pmatrix}$ (65)

and is not normalized to unity. We assume that all other vectors are
normalized to unity, so $v_{i}^{\textrm{T}}v_{i}=1$ for $i=1,2,3,\dots$.

Our interest is in modes which can be potentially smaller than the time-
reparameterization mode $h_{0}=2$. In the $t$-$J$ model the mode $h_{1}$ is
less than $3/2$ for arbitrary parameters and thus it is always more relevant
than $h_{0}=2$ mode. In addition, the mode $h_{2}$ can become smaller than
$h_{0}$ for some regions of parameters. A qualitative plot of these
potentially relevant operators for the fermionic spinon case of Section II.2
is shown in Fig. 2 (the derivation is in Appendix. C).

Figure 2: Scaling dimensions $h$ of potentially dominant operators in the
$(\theta_{f},\theta_{b})$ plane. Here $k=0.2$. Regions corresponding to
different ranges of $h$ are coded with different colors, with boundaries given
by Eqs.(146),(147),(149),(150). There is a region dubbed “No conformality”
meaning (26) is violated. On the right side, we also label different values of
$p$ obtained from the Luttinger constraint (34). In particular, the maximal
$p_{\textrm{max}}\simeq 0.053$ is reached when
$\theta_{b}=\frac{1}{2}\arccos\left(\frac{-2}{\pi}\right)$. The colored lines
are constant $\kappa$ contours calculated from the Luttinger constraints
(33),(34).

We remind the readers that $\theta_{f}$ and $\theta_{b}$ are constrained by
unitarity $-\pi/4<\theta_{f}<\pi/4$, $\pi/4<\theta_{b}<\pi/2$. The mode
$h_{1}$ (see Fig. 3) has $h_{1}=3/2$ at zero doping $p=0$, and its scaling
dimensions decreases as $p$ grows, and it can become relevant $h_{1}<1$ or
complex $h_{1}=1/2+is,~{}s\in\mathbf{R}$.

Figure 3: The scaling dimension $h_{1}$ plotted versus doping density $p$. The
plot is obtained by taking three vertical cuts from Fig. 2 at
$\theta_{f}=0,0.1\pi,0.2\pi$. The scaling dimension $h_{1}=3/2$ when $p=0$ and
reaches $h_{1}=1$ when $p=p_{\textrm{max}}\simeq 0.053$. Then $h_{1}$ starts
to decrease with $p$ until it reaches $h_{1}=1/2$ and becomes complex. Here we
used $k=0.2$.

At first glance, it seems that the low-energy properties would be dominated by
this operator. However, as we will see in Section V, the prefactor
$\alpha_{1}$ accompanying this mode is small compared to $\alpha_{0}$ of the
$h_{0}=2$ mode when doping $p$ is small. A partial explanation is the
coefficient $\alpha_{1}$ is proportional to $k_{G}^{\prime}(h_{1})^{-1}$ (see
(134)), and when $p$ is small, $h_{1}$ is close to the pole at $k_{G}(3/2)$,
so $k_{G}^{\prime}(h_{1})$ is large. Therefore, we expect time
reparameterization mode $h_{0}=2$ to be important above some threshold
temperature $T>T_{1}(p)$, where $T_{1}(p)\to 0$ as $p\to 0$. There is another
potentially relevant operator $h_{2}$ which can move below 2, but it only
appears in a restricted range of parameters.

Figure 4: As in Fig. 2, but now for the bosonic spinon model.

The analysis of conformal corrections and operator spectrum in the bosonic
spinon $+$ fermionic holon case of Section II.3 is very similar to the
fermionic spinon + bosonic holon model. The kernel $K_{G}$ can be obtained
from the previous model by changing $\theta_{f}\to\theta_{\mathfrak{b}}$,
$\theta_{b}\to\theta_{\mathfrak{f}}$ (see Appendix. B). The discussions of
operators at $h=1$ and $h=2$ in the previous subsection still applie here. The
analysis of other potentially relevant operators is shown in Fig. 4. In this
case the second potentially relevant operator $h_{2}$ appears in a larger
region of parameter space.

##  IV Gauge Invariant Observables

In this section we present results for some gauge invariant observables
including electron spectral density $\rho_{e}(\omega)$, spin spectral density
$\rho_{Q}(\omega)$ and optical conductivity $\text{Re}~{}\sigma(\omega)$. We
shall be focusing here on the fermionic spinon + bosonic holon
fractionalization of Section II.2.

###  IV.1 Electron Spectral Density

The electron Green’s function is a product of the spinon and holon Green’s
functions

$G_{e}(\tau)=-G_{f}(\tau)G_{b}(-\tau)\,.$ (66)

To obtain the electron spectral density, we shall first analytically continue
to retarded-Green’s function from imaginary time to the real one

$G_{e,R}(t)=i\theta(t)(G_{e}(it+0)-G_{e}(it-0))\,,$ (67)

where $\theta(t)$ is the Heaviside step function. Then we can find
$G_{e,R}(\omega)$ in the frequency space and extract the spectral density as

$\rho_{e}(\omega)=-\frac{1}{\pi}\text{Im}G_{e,R}(\omega)\,.$ (68)

The electron spectral density can also be written in terms of the spectral
densities of spinon $\rho_{f}=-\frac{1}{\pi}\text{Im}G_{f,R}$ and holon
$\rho_{b}=-\frac{1}{\pi}\text{Im}G_{b,R}$ as

$\rho_{e}(\Omega)=\int_{-\infty}^{+\infty}{\rm
d}\nu(n_{F}(\Omega+\nu)+n_{B}(\nu))\rho_{f}(\Omega+\nu)\rho_{b}(\nu)\,,$ (69)

where $n_{F/B}(\omega)=1/(e^{\beta\omega}-\zeta)$ is the Fermi or Bose
distribution. We will perform the calculation at both zero temperature at
finite temperature. At zero temperature we include nonlinear corrections, and
at finite temperature we will restrict expressions to linear order
corrections.

####  IV.1.1 Zero temperature

Using (55) and (66), we can write the electron Green’s function as

$G_{e}(\tau)=G_{e}^{c}(\tau)\bigg{(}1-\sum_{h}\frac{\alpha_{h}}{|J\tau|^{h-1}}(v_{hf}+\bar{v}_{hb})-\sum_{hh^{\prime}}\frac{\alpha_{h}\alpha_{h^{\prime}}}{|J\tau|^{h+h^{\prime}-2}}(a_{hh^{\prime}f}+\bar{a}_{hh^{\prime}b}-v_{hf}\bar{v}_{h^{\prime}b})-\dots\bigg{)}.$
(70)

Here $v_{hf(b)}$ means taking the fermionic (bosonic) component of the
4-component vector $v_{h}$, and similar meanings apply to
$a_{hh^{\prime}f(b)}$. $\bar{v}_{hb}$ means switching the $\pm$ components of
$v_{hb}$ and similar for $\bar{a}_{hh^{\prime}b}$. $G_{e}^{c}(\tau)$ is the
conformal electron Green’s function

$\quad G^{c}_{e}(\tau)=-\begin{pmatrix}e^{\pi\mathscr{E}_{e}}\\\
-e^{-\pi\mathscr{E}_{e}}\end{pmatrix}\frac{C_{e}}{|J\tau|}\,,\quad
C_{e}=\frac{J}{t}\Big{(}\frac{-\cos(2\theta_{b})}{4\pi}\Big{)}^{1/2}\,,$ (71)

where $\mathscr{E}_{e}=\mathscr{E}_{f}-\mathscr{E}_{b}$ and we have used that
$\Delta_{f}=\Delta_{b}=1/4$. Analytically continuing to real-time and extract
the spectral density, we obtain

$\rho_{e}(\omega)=\rho_{e}^{c}(\omega)\bigg{(}1-\sum_{h}\frac{\alpha_{h}(v_{hf}+\bar{v}_{hb})}{\Gamma(h)}\left|\frac{\omega}{J}\right|^{h-1}-\sum_{hh^{\prime}}\frac{\alpha_{h}\alpha_{h^{\prime}}(a_{hh^{\prime}f}+\bar{a}_{hh^{\prime}b}-v_{hf}\cdot\bar{v}_{h^{\prime}b})}{\Gamma(h+h^{\prime}-1)}\left|\frac{\omega}{J}\right|^{h+h^{\prime}-2}-\dots\bigg{)}.$
(72)

Here $\rho_{e}^{c}(\omega)$ is the electron-spectral weight of the conformal
solution

$\rho_{e}^{c}(\omega)=\frac{C_{e}}{J}\begin{pmatrix}e^{\pi\mathscr{E}_{e}}\\\
e^{-\pi\mathscr{E}_{e}}\end{pmatrix}\,,$ (73)

where $C_{e}$ is defined in (71).

####  IV.1.2 Finite Temperature

We will work on the interval $0<\tau<\beta$. The electron Green’s function to
linear order in $\alpha_{h}$ is

$\displaystyle G_{e}(\tau)=$ $\displaystyle
G_{e}^{c}(\tau)\bigg{(}1-\sum_{h}\frac{\alpha_{h}}{2(\beta
J)^{h-1}}\big{(}(v_{hf+}+v_{hf-}+v_{hb+}+v_{hb-})f_{h}^{\textrm{A}}(\tau)$
$\displaystyle+(v_{hf+}-v_{hf-}-v_{hb+}+v_{hb-})f_{h}^{\textrm{S}}(\tau)\big{)}-\dots\bigg{)}\,,$
(74)

where the conformal electron Green’s function at finite temperature is

$G_{e}^{c}(\tau)=-C_{e}\Big{(}\frac{\beta
J}{\pi}\sin\frac{\pi\tau}{\beta}\Big{)}^{-1}e^{2\pi\mathscr{E}_{e}(\frac{1}{2}-\frac{\tau}{\beta})}\,,$
(75)

and the functions $f_{h}^{\textrm{A}}(\tau)$ and $f_{h}^{\textrm{S}}(\tau)$
are given in (62).

To find electron spectral density at finite temperature we analytically
continue $G_{e}(\tau)$ to real time using (67) and then take the Fourier
transform to find $G_{e,R}(\omega)$ in the frequency space. Finally using (68)
we find for the electron spectral density

$\displaystyle\rho_{e}(\omega)=$
$\displaystyle\rho_{e}^{c}(\omega)\bigg{(}1-\sum_{h}\frac{\alpha_{h}}{2(\beta
J)^{h-1}}\Big{(}(v_{hf+}+v_{hf-}+v_{hb+}+v_{hb-})\mathcal{R}_{h}^{\textrm{A}}\big{(}\frac{\beta\omega}{2\pi}-\mathscr{E}_{e}\big{)}$
$\displaystyle+(v_{hf+}-v_{hf-}-v_{hb+}+v_{hb-})\mathcal{R}_{h}^{\textrm{S}}\big{(}\frac{\beta\omega}{2\pi}-\mathscr{E}_{e}\big{)}\Big{)}-\dots\bigg{)}\,,$
(76)

where the conformal electron spectral density is

$\rho_{e}^{c}(\omega)=\frac{C_{e}\cosh(\frac{\beta\omega}{2})}{J\cosh(\frac{\beta\omega}{2}-\pi\mathscr{E}_{e})}\,,$
(77)

and the functions $\mathcal{R}_{h}^{\textrm{A}}(\omega)$ and
$\mathcal{R}_{h}^{\textrm{S}}(\omega)$ are

$\begin{split}\mathcal{R}_{h}^{\textrm{A}}(\omega)&=\frac{2\left(\frac{\pi}{2}\right)^{h}\Gamma(h)}{\sqrt{\pi}\sin\left(\frac{\pi
h}{2}\right)\Gamma\left(h-\frac{1}{2}\right)}\textrm{Re}\,{}_{3}\textbf{F}_{2}\biggl{(}\genfrac{.}{.}{0.0pt}{}{h\mskip
8.0mu1-h\mskip 8.0mu\frac{1}{2}+i\omega}{1\mskip 8.0mu1};1\biggr{)}\,,\\\
\mathcal{R}_{h}^{\textrm{S}}(\omega)&=\frac{2\left(\frac{\pi}{2}\right)^{h}\Gamma(h)}{\sqrt{\pi}\cos\left(\frac{\pi
h}{2}\right)\Gamma\left(h-\frac{1}{2}\right)}\textrm{Im}\,{}_{3}\textbf{F}_{2}\biggl{(}\genfrac{.}{.}{0.0pt}{}{h\mskip
8.0mu1-h\mskip 8.0mu\frac{1}{2}+i\omega}{1\mskip
8.0mu1};1\biggr{)}\,,\end{split}$ (78)

where $\,{}_{3}\textbf{F}_{2}$ is the regularized hypergeometric function. For
derivation of similar to (76) results see Tikhanovskaya  _et al._ (2020).
Retaining only $h_{0}=2$ mode in (76) we find

$\displaystyle\rho_{e}(\omega)=\frac{C_{e}\cosh(\frac{\beta\omega}{2})}{J\cosh(\frac{\beta\omega}{2}-\pi\mathscr{E}_{e})}\left(1-\frac{\pi\alpha_{0}}{\beta
J}\Big{(}\frac{\beta\omega}{2\pi}-\mathscr{E}_{e}\Big{)}\Big{(}4\tanh(\frac{\beta\omega}{2}-\pi\mathscr{E}_{e})-3(\sin(2\theta_{f})-\sin(2\theta_{b}))\Big{)}\right)\,,$
(79)

where we have used the explicit expression (65) for the eigenvector $v_{0}$.

###  IV.2 Conductivity

The Kubo formula for conductivity on Bethe lattice is derived in Guo  _et al._
(2020):

$\text{Re}~{}\sigma(\omega)=\frac{2\pi
M^{\prime}e^{2}t^{2}a^{2-d}}{z}\int_{-\infty}^{+\infty}{\rm
d}\Omega\rho_{e}(\Omega)\rho_{e}(\Omega+\omega)\frac{n_{F}(\Omega)-n_{F}(\Omega+\omega)}{\omega}\,,$
(80)

where $a$ is lattice constant and $d$ is spatial dimension. We can evaluate
the expression in three scenarios as below:

####  IV.2.1 Zero temperature Optical conductivity

At zero temperature, we can evaluate the conductivity to nonlinear order in
$\alpha_{h}$ using (72), the result is

$\begin{split}&\text{Re}~{}\sigma(\omega)=\sigma_{0}\Bigg{[}1-\sum_{h}\frac{2\alpha_{h}}{\Gamma(h+1)}(v_{hf+}+v_{hf-}+v_{hb+}+v_{hb-})|\omega/J|^{h-1}-\sum_{hh^{\prime}}\frac{2\alpha_{h}\alpha_{h^{\prime}}|\omega/J|^{h+h^{\prime}-2}}{\Gamma(h+h^{\prime})}\\\
&\times[a_{hh^{\prime}f+}+a_{hh^{\prime}f-}+a_{hh^{\prime}b+}+a_{hh^{\prime}b-}-v_{hf+}v_{h^{\prime}b-}-v_{hf-}v_{h^{\prime}b+}-(v_{hf+}+v_{hb-})(v_{h^{\prime}f-}+v_{h^{\prime}b+})/2]\Bigg{]}.\end{split}$
(81)

Here $\sigma_{0}$ is the $T=0$ DC conductivity given by

$\sigma_{0}=-\frac{M^{\prime}e^{2}a^{2-d}\cos(2\theta_{b})}{2z}\,,$ (82)

where we used (73) for $\rho_{e}^{c}(\Omega)$.

####  IV.2.2 Finite temperature DC conductivity

The finite temperature DC conductivity can be evaluated by

$\sigma_{DC}=\frac{2\pi
M^{\prime}e^{2}t^{2}a^{2-d}}{z}\int_{-\infty}^{+\infty}{\rm
d}\Omega\rho_{e}(\Omega)^{2}\left(-\frac{\partial
n_{F}}{\partial\Omega}\right)\,,$ (83)

The conformal contribution is given by

$\begin{split}\sigma_{DC}^{c}&=\frac{2\pi
M^{\prime}e^{2}t^{2}a^{2-d}}{z}\int_{-\infty}^{+\infty}{\rm
d}\Omega\rho_{e}^{c}(\Omega)^{2}\left(-\frac{\partial
n_{F}}{\partial\Omega}\right)=\sigma_{0}\,,\end{split}$ (84)

where for $\rho_{e}^{c}(\Omega)$ we used (77). We can also include conformal
corrections up to linear order

$\delta_{h}\sigma_{DC}=\sigma_{0}\frac{\int{\rm
d}\Omega\,2\rho_{e}^{c}(\Omega)\delta_{h}\rho_{e}(\Omega)\left(-\frac{\partial
n_{F}}{\partial\Omega}\right)}{\int{\rm
d}\Omega\rho_{e}^{c}(\Omega)^{2}\left(-\frac{\partial
n_{F}}{\partial\Omega}\right)}\,.$ (85)

Here $\delta_{h}\rho_{e}(\Omega)$ contains two terms given in (76), so we have

$\frac{\delta_{h}\sigma_{DC}}{\sigma_{0}}=-\frac{\alpha_{h}}{(\beta
J)^{h-1}}(v_{hf+}+v_{hf-}+v_{hb+}+v_{hb-})R(h)\,,$ (86)

where

$R(h)=\frac{\pi}{2}\int_{-\infty}^{+\infty}{\rm
d}\omega\frac{\mathcal{R}_{h}^{\textrm{A}}(\omega)}{\cosh^{2}(\pi\omega)}=-\frac{2\left(\frac{\pi}{2}\right)^{h-1}\cos\left(\frac{\pi
h}{2}\right)\Gamma(h-1)}{\sqrt{\pi}h\Gamma\left(h-\frac{1}{2}\right)}\,.$ (87)

It is clear from (78) that $\mathcal{R}_{h}^{\textrm{A/S}}(\omega)$ is an
even/odd function of $\omega$, and therefore only
$\mathcal{R}_{h}^{\textrm{A}}$ contributes. Thus, after replacing variable
$\omega=\frac{\beta\Omega}{2\pi}-\mathscr{E}_{e}$ and taking the integral we
arrive to (87). A numerical plot of $R(h)$ is shown in Fig. 5.

Figure 5: The function $R(h)$ defined in (87). Some special values are
$R(1)=R(2)=1$.

Therefore the DC conductivity is

$\sigma_{DC}=\sigma_{0}\left[1-\sum_{h}\alpha_{h}(v_{hf+}+v_{hf-}+v_{hb+}+v_{hb-})R(h)\left(\frac{T}{J}\right)^{h-1}-\dots\right].$
(88)

In particular, from the $h_{0}=2$ mode we can recover the linear resistivity
obtained in Guo  _et al._ (2020)

$\rho_{DC}=\frac{1}{\sigma_{0}}\left(1+\frac{4\alpha_{0}T}{J}-\dots\right),$
(89)

where we have used the eigenvector (65). There is a factor of two difference
from Guo  _et al._ (2020) because we have normalized the eigenvector
differently.

####  IV.2.3 Finite temperature optical conductivity at linear order

For the optical conductivity at finite temperature, we can analytically
determine the contribution from the $\alpha_{h}$ correction. The result is

$\text{Re}\sigma(\omega)=\sigma_{0}\left(1-\sum_{h}\frac{\alpha_{h}}{(\beta
J)^{h}}(v_{hf+}+v_{hf-}+v_{hb+}+v_{hb-})\Sigma_{h}\left(\frac{\beta\omega}{2\pi}\right)-\dots\right),$
(90)

where

$\Sigma_{h}(x)=\frac{2\left(\frac{\pi}{2}\right)^{h}\Gamma(h)}{\sqrt{\pi}\sin\left(\frac{\pi
h}{2}\right)\Gamma\left(h-\frac{1}{2}\right)}\text{Re}\,{}_{3}\textbf{F}_{2}\biggl{(}\genfrac{.}{.}{0.0pt}{}{h\mskip
8.0mu1-h\mskip 8.0mu1-ix}{1\mskip 8.0mu2};1\biggr{)}\,.$ (91)

In particular, for the $h_{0}=2$ mode we have

$\text{Re}\sigma(\omega)=\sigma_{0}\left(1-\frac{2\alpha_{0}\omega}{J}\coth(\frac{\beta\omega}{2})-\dots\right)\,.$
(92)

Here we have used the eigenvectors (65). The detail of the calculation is in
Appendix D.

###  IV.3 Spin Spectral Density

In the large $M$ limit, the spin-spin correlator factorizes as
$Q(\tau)=G_{f}(\tau)G_{f}(-\tau)$, and the spin spectral density is defined as
$\rho_{Q}=-\frac{1}{\pi}\text{Im}Q_{R}(\omega)$. It can be expressed in terms
of spinon spectral weight $\rho_{f}$ as

$\rho_{Q}(\omega)=\int_{-\infty}^{\infty}d\nu\rho_{f}(\nu)\rho_{f}(\nu-\omega)(n_{F}(\nu-\omega)-n_{F}(\nu))\,.$
(93)

The computation of this quantity is the same as the previous paper
Tikhanovskaya  _et al._ (2020), and we report the results here.

####  IV.3.1 Zero temperature

At zero temperature, the conformal spin spectral density is

$\rho_{Q}^{c}(\omega)=\frac{C_{Q}}{J}{\rm sgn\,}\omega,\quad
C_{Q}=\Big{(}\frac{\cos(2\theta_{f})}{4\pi}(1-\eta)\Big{)}^{1/2}\,.$ (94)

Including the conformal corrections, we have

$\displaystyle\rho_{Q}(\omega)=$
$\displaystyle\rho_{Q}^{c}(\omega)\bigg{(}1-\sum_{h}\frac{\alpha_{h}(v_{hf+}+v_{hf-})}{\Gamma(h)}\left|\frac{\omega}{J}\right|^{h-1}$
$\displaystyle-\sum_{h,h^{\prime}}\frac{\alpha_{h}\alpha_{h^{\prime}}(a_{hh^{\prime}f+}+a_{hh^{\prime}f-}-v_{hf+}v_{h^{\prime}f-})}{\Gamma(h+h^{\prime}-1)}\left|\frac{\omega}{J}\right|^{h+h^{\prime}-2}-\dots\bigg{)}\,,$
(95)

####  IV.3.2 Finite temperature

At finite temperature, we write the spin spectral weight to linear order in
$\alpha_{h}$ as

$\rho_{Q}(\Omega)=\rho_{Q}^{c}(\omega)\bigg{(}1-\sum_{h}(v_{fh+}+v_{fh-})\frac{\alpha_{h}}{(\beta
J)^{h-1}}\mathcal{R}_{h}^{\textrm{A}}(\omega)-\dots\bigg{)}\,,$ (96)

where the conformal contribution to spin spectral weight is

$\displaystyle\rho_{Q}^{c}(\omega)$
$\displaystyle=\frac{C_{Q}}{J}\tanh(\frac{\beta\omega}{2})\,.$ (97)

and the function $\mathcal{R}_{h}^{\textrm{A}}(\omega)$ defined in (78).
Considering only the correction from $h_{0}=2$ mode, we have

$\displaystyle\rho_{Q}(\omega)=\frac{C_{Q}}{J}\tanh\big{(}\frac{\beta\omega}{2}\big{)}\left(1-\frac{2\alpha_{0}\omega}{J}\tanh\big{(}\frac{\beta\omega}{2}\big{)}-\dots\right)\,.$
(98)

##  V Numerical results for the $t$-$J$ model

In this section we numerically study the saddle point equations at zero
temperature for the $t-J$ model, and use the solutions to constrain the
coefficients of the conformal perturbations. We find that the contribution of
the linear term from the $h_{0}=2$ mode dominates in the IR limit for the spin
spectral density, but that the $h_{1}$ and $h_{0}$ modes are comparable in
other observables. We focus on the case of fermionic spinon and bosonic holon
model discussed in the Section II.2. The solution in the case of bosonic
spinon and fermonic holon can be derived and solved in a straightforward way
using the approach discussed here by changing asymmetry angles
$\theta_{f}\to\theta_{\mathfrak{b}}$ and $\theta_{b}\to\theta_{\mathfrak{f}}$.

The saddle point equations of interest at zero temperature are those in
(16)-(19). To solve these equations numerically, we use an approach similar to
the one used in the previous paper Tikhanovskaya  _et al._ (2020). The
detailed treatment of the equations is discussed in the Appendix A. Our focus
is to solve the saddle point equations for bosonic and fermionic spectral
densities first, and then present results for gauge invariant observables such
as electron and spin spectral densities, and optical conductivity.

We focus on the asymmetry parameters of the model $\theta_{f}$, $\theta_{b}$
that satisfy the Luttinger constraints. In this section we are interested in a
particular case of $\Delta_{b}=\Delta_{f}=1/4$, and thus we assume it
everywhere. Therefore, we can write the Luttinger constraints (34) in the
following form

$\displaystyle\frac{\theta_{f}}{\pi}+\frac{1}{4}{\sin(2\theta_{f})}$
$\displaystyle=$ $\displaystyle\frac{1}{2}-\kappa+kp,$ (99)
$\displaystyle\frac{\theta_{b}}{\pi}+\frac{1}{4}{\sin(2\theta_{b})}$
$\displaystyle=$ $\displaystyle\frac{1}{2}+p\,.$ (100)

The solutions with the constrains will correspond to the region crossed by one
of the lines at chosen $\kappa$ in the Fig. 2. Although there is a large
region of asymmetry parameters where an analytical solution approximates a
numerical result well, in this section we analyze only a portion of the full
parameter space. We choose the asymmetry angles in the region of
$\theta_{b}=0.26\pi,0.32\pi$ and $0<\theta_{f}\leq 0.15\pi$.

(a) (b)

Figure 6: Fermioic (a) and bosonic (b) spectral densities as solutions of the
equations (16)-(19). In both plots the chosen parameters:
$\theta_{b}=0.26\pi$, $\theta_{f}=0.03\pi$. Solid magenta lines are the
numerical solution for these parameters. Black dashed lines are given by
analytical solution with nonlinear corrections. The values of the fitting
parameters are given in the Fig. 8.

(a) (b)

Figure 7: Fermioic (a) and bosonic (b) spectral densities as solutions of the
equations (16)-(19). As in the Fig. 6 except the chosen parameters are
$\theta_{b}=0.32\pi$, $\theta_{f}=0.15\pi$.

We now show that the numerical solution with the constraints implied in small
enough $\omega$-region has behavior given by the results of the theoretical
approach in the previous section. We discuss the behavior of fermionic and
bosonic spectral densities as solutions of the saddle point equations
(16)-(19) as well as gauge invariant observables such as electron, spin
spectral densities and optical conductivity at zero temperature.

Figure 8: Approximate values of exponents $h$ and coefficients $\alpha_{h}$
for different asymmetry angles $\theta_{f}$, $\theta_{b}$ obtain by fits to
the numerical solution.

We first find the solution for bosonic $\rho_{b}(\omega)$ and fermionic
$\rho_{f}(\omega)$ spectral densities of the equations (100)-(100) at zero
temperature at various sets of parameters $\theta_{f}$ and $\theta_{b}$ and
fixed $k=0.2$, $t=J=1$. The method of solving the equations at zero
temperature is explained in detail in Appendix A as well as in Tikhanovskaya
_et al._ (2020). Amongst the range of parameters for which we solve the
equations, we show two solutions that represent the behavior of the spectral
densities at large ($\theta_{b}=0.32\pi$, $\theta_{f}=0.15\pi$) and small
($\theta_{b}=0.26\pi$, $\theta_{f}=0.03\pi$) asymmetry angles. The solutions
are presented in Figs. 6,7, respectively.

The solutions found in this section compared to an analytical fit at small
frequencies described in the previous sections and in Ref. Tikhanovskaya  _et
al._ , 2020. Using the numerical solutions, we find the threshold of asymmetry
angles where it is still possible to get the solution as discussed in Appendix
A. Below we describe a behavior of the gauge invariant observables and show
that an analytical solution follows a numerical curve well enough at small
frequencies.

Let us first rewrite the analytical expressions of the polynomials from
section IV that, as we show below, approximate the exact numerical solution at
small frequencies. Following the notations in the previous sections, we use
$h_{1}$, $h_{2}$ to denote the exponents in the fitting functions. We consider
the polynomials for each of the gauge invariant observables truncated at the
order of $h_{2}-1$ in frequency with all non-linear corrections included. A
truncated expression for electron and spin spectral densities (72), (IV.3.1)
can be written as

$\begin{split}\rho_{e,Q}(\omega)=\rho_{e,Q}^{c}(\omega)\left(1-\sum_{h}^{h_{2}}\alpha_{h}A^{e,Q}_{h}|\omega/{J}|^{h-1}-\sum_{hh^{\prime}}^{h_{2}}\alpha_{h}\alpha_{h^{\prime}}A^{e,Q}_{hh^{\prime}}|\omega/{J}|^{h+h^{\prime}-2}\right).\end{split}$
(101)

We write the similar expression for optical conductivity $\sigma(\omega)$
truncating the full series (81) in a similar way

$\sigma(\omega)=\sigma_{0}(\omega)\left(1-\sum_{h}^{h_{2}}\alpha_{h}A^{\sigma}_{h}|\omega/{J}|^{h-1}-\sum_{hh^{\prime}}^{h_{2}}\alpha_{h}\alpha_{h^{\prime}}A^{\sigma}_{hh^{\prime}}|\omega/{J}|^{h+h^{\prime}-2}\right)$
(102)

with different coefficients $A^{\sigma}_{h}$. In the above,
$\rho_{e,Q}^{c}(\omega)$ are the electron and spin spectral weights of the
conformal solutions (73), (94), and $\sigma_{0}$ is given by (82). The
coefficients $A^{e,Q,\sigma}_{h}$ are known exactly and have analytical forms
for electron spectral density (72), for spin spectral density (IV.3.1) and for
conductivity (81).

Figure 9: Table of computed analytical expressions of spin and electron
spectral densities (101) with all parameters fixed for a range of asymmetry
angles $(\theta_{f},\theta_{b})$. The polynomials approximate a corresponding
numerical solution at small frequencies. The exponents are given by $h-1$
where $h$ are presented in the Fig. 8.

(a) (b)

Figure 10: Spin (a) and electron (b) spectral densities. Solid magenta lines
are numerical solutions. Black dashed lines are given by analytical solution
(101). Here $\theta_{b}=0.26\pi$ and $\theta_{f}=0.03\pi$. The exponents and
fitting functions are given in Figs. 8, 9 respectively.

(a) (b)

Figure 11: Spin (a) and electron (b) spectral densities. Same as in the Fig.10
except the parameters $\theta_{b}=0.32\pi$ and $\theta_{f}=0.15\pi$.

We note that the analytical solution has unfixed parameters $\alpha_{h}$. In
the series we consider, there are three parameters the values of which can be
obtained by fitting an analytical expression into to a numerical curve at
corresponding parameters. For a range of parameters we discuss in this
section, the values of coefficients $\alpha_{h}$ as well as exponents $h$ are
presented in Fig. 8.

Using the values of the coefficients $\alpha_{h}$ we obtain analytical
solutions with all parameters fixed. The analytical formulas for spin and
electron spectral densities are presented in the Figs. 9, 12. From the
polynomials in the tables we notice that increasing the asymmetry angles leads
to an increasing a linear coefficient compared to the coefficient of the
$h_{1}-1$ exponent and coefficients of its non-linear corrections. By making
it big enough, the linear coefficient becomes significantly larger than other
lower order coefficients. We show two examples of fitting the corresponding
analytical expressions of the spin and electron spectral densities in the
Figs. 10,11 for different asymmetry angles.

We conclude this section by discussing the optical and d.c. conductivity. In
the Table 12 we present the analytical expressions for optical conductivity at
zero temperature and finite temperature d.c. conductivity with all parameters
fixed. We note that even though we do not discuss the finite temperature exact
numerical solution in this paper, the coefficients $\alpha_{h}$ are fixed to
their zero temperature values, and therefore, we can write down the analytical
formulas at finite temperature as well. We again notice increasing the
asymmetry angles leads to increasing the linear coefficient. Within the
allowed range of asymmetry angles ($\theta_{f}\leq 0.15\pi$, as discussed in
the Appendix A) we see that the linear term becomes dominant. We explicitly
show a behavior of optical conductivity at zero temperature in Fig. 13 for
different asymmetry angles.

Figure 12: Table of computed analytical expressions of optical conductivity
(102) at zero temperature (second column) and DC conductivity (88) at finite
temperature with all parameters fixed for a range of asymmetry angles
$(\theta_{f},\theta_{b})$. The polynomials approximate a corresponding
numerical solution at small frequencies. The exponents are given by $h-1$
where the values of $h$ are presented in the Fig. 8.

(a) (b)

Figure 13: Optical conductivity for (a) $\theta_{b}=0.26\pi$ and
$\theta_{f}=0.03\pi$ and (b) $\theta_{b}=0.32\pi$ and $\theta_{f}=0.15\pi$.
Numerical solutions in both plots are given by the solid magenta lines and the
black dashed lines are the analytical solutions presented in Fig. 12.

##  VI Conclusions

We have presented numerical and analytic solutions of the non-Fermi liquid
phase random $t$-$J$ model without quasiparticle excitations, with good
agreement between the two approaches. The solutions were obtained the large
$M$ limit of a model with SU($M$) spin symmetry, and follow our previous
analyses of the undoped insulator in Ref. Tikhanovskaya  _et al._ , 2020. The
theory has the structure of the SYK model for fractionalized spinons and
holons carrying gauge charges, which have random $q=4$-body interaction terms
between them. Note, however, that expressed in terms of the underlying gauge-
invariant electrons, the interactions are not of the $q=4$ SYK form.

Our attention focused on the leading and sub-leading singularities of several
gauge-invariant physical observables at low frequencies and temperatures.
These are controlled by operators with scaling dimension $h$ which perturb the
leading conformally invariant solution. As the theory is in $0+1$ spacetime
dimensions, we require $h>1$ for the operator to be irrelevant in the
infrared, and contribute to the sub-leading singularities.

An important operator that appears in all the subleading corrections is the
time-reparameterization mode, or the boundary graviton of the holographic
dual. This operator is universal in SYK criticality, and has $h_{0}=2$
exactly; no corrections are expected to this scaling dimension at all orders
in $1/M$. It was also present in the previous analysis of the insulating model
Tikhanovskaya  _et al._ (2020). This operator leads to relative corrections in
observables which scale $\omega^{h_{0}-1}$ or $T^{h_{0}-1}$ in the IR. In
particular, as the leading contribution to the resistivity is $T$-independent
in the $t$-$J$, the first contribution from the graviton mode is a linear-
in-$T$ resistivity Guo  _et al._ (2020).

We also found that the $t$-$J$ model has an irrelevant operator with scaling
dimension $1<h_{1}<2$ over the regime where scaling solutions were found
numerically (see Fig. 8). This will therefore lead to corrections which scale
as $\omega^{h_{1}-1}$ or $T^{h_{1}-1}$ in the IR, which are more singular than
those from the graviton mode. This operator is special to the doped case, and
was not found in insulator studied previously Tikhanovskaya  _et al._ (2020).
However, it should be noted that this operator is not protected, and we do
expect $1/M$ corrections to the value of $h_{1}$. We have not studied these
corrections, and the possibility remains that such corrections will increase
the value of $h_{1}$ above 2, and return the graviton mode as the dominant
correction. Moreover, we found that the numerical co-efficient of the $h_{1}$
correction was quite small in the spin spectral density, and significantly
smaller than that of the $h_{0}$ correction (see Fig. 9).

A recent exact-diagonalization study of the random $t$-$J$ model with SU(2)
spin symmetry Shackleton  _et al._ (2020) computed the spin spectral density
and found that it matched the spectral density similar to Fig. 10a over a wide
range of frequencies near the critical doping where the spin-glass order
vanished (spin-glass order is present at $M=2$ for low doping). The numerics
show
$\rho_{Q}(\omega)\sim\mbox{sgn}(\omega)\left[c_{0}-c_{1}|\omega|+\ldots\right]$
at small $\omega$, consistent with the numerical results in Table 9 where the
$h_{1}$ contribution is small.

We note from Figs. 9 and 12 that the contributions of $h_{1}$ mode are more
significant for the other observables, including the d.c. resistivity, and can
be comparable to the $h_{0}=2$ contribution. It would be interesting to study
the relative contributions of these modes at higher order in $1/M$ to
understand the situation for SU(2) better.

### Acknowledgements

We thank Yingfei Gu, Igor R. Klebanov, and Henry Shackleton for valuable
discussions. This research was supported by the U.S. Department of Energy
under Grant DE-SC0019030.

##  Appendix A Zero temperature numerics for $t-J$ model

We consider Dyson-Schwinger equations for the retarded Green’s function for
$t-J$ model, which is obtained by analytic continuation from the Matsubara
frequency $i\omega_{n}\to\omega+i0$. The first pair of the Dyson-Schwinger
equations reads

$\displaystyle G_{f,R}(\omega)^{-1}=\omega+i0+\mu_{f}-\Sigma_{f,R}(\omega)\,;$
(103) $\displaystyle
G_{b,R}(\omega)^{-1}=\omega+i0+\mu_{b}-\Sigma_{b,R}(\omega)\,.$ (104)

As before, we define analytic in the upper half plane Green’s functions $G(z)$
and self energy $\Sigma(z)$, which are expressed through the spectral
densities $\rho(\omega)$ and $\sigma(\omega)$ as

$\displaystyle
G(z)=\int_{-\infty}^{+\infty}d\omega\frac{\rho(\omega)}{z-\omega}\,,\quad\Sigma(z)=\int_{-\infty}^{+\infty}d\omega\frac{\sigma(\omega)}{z-\omega}\,.$
(105)

The Matsubara and retarded Green’s functions can be obtained from these
functions by taking $z=i\omega_{n}$ and $z=\omega+i0$. We can find the
spectral density as $\rho(\omega)=-\frac{1}{\pi}\textrm{Im}G_{R}(\omega)$.
Also using the representation (105) we can obtain Green’s function in
imaginary time expressed through integral over the spectral density

$\displaystyle
G_{a}(\tau)=\frac{1}{\beta}\sum_{n}G_{a}(i\omega_{n})e^{-i\omega_{n}\tau}=-\int_{-\infty}^{+\infty}d\omega\frac{\rho(\omega)e^{-\omega\tau}}{1-\zeta
e^{-\beta\omega}},\quad\tau\in(0,\beta)\,,$ (106)

where $a=f,b$ and $\zeta_{f}=-1$, $\zeta_{b}=1$. We notice that
$G(0^{+})-\zeta G(\beta^{-})=-1=-\int_{-\infty}^{+\infty}d\omega\rho(\omega)$
for arbitrary temperature.

Here we consider an original model with fermionic spinon and bosonic holon,
that gives the following second pair of the Dyson-Schwinger equations

$\displaystyle\Sigma_{f}(\tau)=J^{2}G_{f}^{2}(\tau)G_{f}(\beta-\tau)+kt^{2}G_{f}(\tau)G_{b}(\tau)G_{b}(\beta-\tau)\,;$
(107)
$\displaystyle\Sigma_{b}(\tau)=t^{2}G_{f}(\tau)G_{f}(\beta-\tau)G_{b}(\tau)\,.$
(108)

To obtain the second Dyson-Schwinger equation for the retarded self-energy
$\Sigma_{R}(\omega)$ we consider the equations in the Matsubara space above
and use (106) to write it in terms of the spectral densities for fermions

$\displaystyle\Sigma_{f}(i\omega_{n})$
$\displaystyle=-J^{2}\int_{-\infty}^{+\infty}\prod_{i=1}^{3}(d\omega_{i}\rho_{f}(\omega_{i}))\frac{n_{f}(\omega_{1})n_{f}(\omega_{2})n_{f}(-\omega_{3})+n_{f}(-\omega_{1})n_{f}(-\omega_{2})n_{f}(\omega_{3})}{\omega_{1}+\omega_{2}-\omega_{3}-i\omega_{n}}$
(109)
$\displaystyle+kt^{2}\int_{-\infty}^{+\infty}\prod_{i=1}^{3}d\omega_{i}\rho_{f}(\omega_{1})\rho_{b}(\omega_{2})\rho_{b}(\omega_{3})\frac{n_{f}(\omega_{1})n_{b}(\omega_{2})n_{b}(-\omega_{3})+n_{f}(-\omega_{1})n_{b}(-\omega_{2})n_{b}(\omega_{3})}{\omega_{1}+\omega_{2}-\omega_{3}-i\omega_{n}}$
(110)

and for bosons

$\displaystyle\Sigma_{b}(i\omega_{n})$
$\displaystyle=t^{2}\int_{-\infty}^{+\infty}\prod_{i=1}^{3}d\omega_{i}\rho_{f}(\omega_{1})\rho_{b}(\omega_{2})\rho_{f}(\omega_{3})\frac{n_{f}(\omega_{1})n_{b}(\omega_{2})n_{f}(-\omega_{3})+n_{f}(-\omega_{1})n_{b}(-\omega_{2})n_{f}(\omega_{3})}{\omega_{1}-\omega_{2}+\omega_{3}-i\omega_{n}}$
(111)

where $n_{a}(\omega)=1/(e^{\beta\omega}-\zeta)$ is the Bose or Fermi
distribution and we can get
$\Sigma_{R}(\omega)=\Sigma(i\omega_{n}=\omega+i0)$. At zero temperature
$\beta=\infty$ we can replace $n_{b}(\omega)$ by $-\theta(-\omega)$ and
$n_{f}(\omega)$ by $\theta(-\omega)$. Though $n_{b}(\omega)$ is divergent for
$\omega\to 0$, we assume that this divergence does not play any role.
Functions $G_{R}(\omega)$ and $\Sigma_{R}(\omega)$ are complex valued and
further we will adopt notations for their real and imaginary parts
$G_{R}(\omega)=G^{\prime}(\omega)+iG^{\prime\prime}(\omega)$ and
$\Sigma_{R}(\omega)=\Sigma^{\prime}(\omega)+i\Sigma^{\prime\prime}(\omega)$.
So for $\beta=\infty$ for fermions we find

$\displaystyle\Sigma_{f}^{\prime\prime}(\omega)=\begin{cases}\pi\int_{0}^{\omega_{1}+\omega_{2}\leq\omega}d\omega_{1}d\omega_{2}\rho_{f}(\omega_{1})\left(kt^{2}\rho_{b}(\omega_{2})\rho_{b}(\omega_{1}+\omega_{2}-\omega)-J^{2}\rho_{f}(\omega_{2})\rho_{f}(\omega_{1}+\omega_{2}-\omega)\right),\quad\omega>0\\\
\pi\int^{0}_{\omega_{1}+\omega_{2}\geq\omega}d\omega_{1}d\omega_{2}\rho_{f}(\omega_{1})\left(kt^{2}\rho_{b}(\omega_{2})\rho_{b}(\omega_{1}+\omega_{2}-\omega)-J^{2}\rho_{f}(\omega_{2})\rho_{f}(\omega_{1}+\omega_{2}-\omega)\right),\quad\omega<0\,\end{cases}$
(112)

and for bosons

$\displaystyle\Sigma_{b}^{\prime\prime}(\omega)=\begin{cases}-\pi
t^{2}\int_{0}^{\omega_{1}+\omega_{2}\leq\omega}d\omega_{1}d\omega_{2}\rho_{f}(\omega_{1})\rho_{b}(\omega_{2})\rho_{f}(\omega_{2}+\omega_{1}-\omega),\quad\omega>0\\\
-\pi
t^{2}\int_{\omega_{1}+\omega_{2}\geq\omega}^{0}d\omega_{1}d\omega_{2}\rho_{f}(\omega_{1})\rho_{b}(\omega_{2})\rho_{f}(\omega_{2}+\omega_{1}-\omega),\quad\omega<0\,.\end{cases}$
(113)

We anticipate that at zero temperature the functions $\rho(\omega)$ and
$\Sigma^{\prime\prime}(\omega)$ will have discontinuity. So it will be
convenient to use a new set of functions defined separately for $\omega>0$ and
$\omega<0$

$\displaystyle\rho_{a}(\omega)=\begin{cases}\frac{g_{a}^{+}(\omega)}{\sqrt{\omega}},\quad\omega>0\\\
\frac{g_{a}^{-}(-\omega)}{\sqrt{-\omega}},\quad\omega<0\end{cases}\,,\quad\Sigma_{a}^{\prime\prime}(\omega)=\begin{cases}4\pi\sqrt{\omega}s_{a}^{+}(\omega),\quad\omega>0\\\
4\pi\sqrt{-\omega}s_{a}^{-}(-\omega),\quad\omega<0\end{cases}\,.$ (114)

We make change of variables $\omega_{1}=\omega\sin^{2}u\cos^{2}\phi$ and
$\omega_{2}=\omega\sin^{2}u\sin^{2}\phi$ in (112) and obtain

$\displaystyle s^{\pm}_{f}(\omega)=$
$\displaystyle\int_{0}^{\frac{\pi}{2}}du\sin
u\int_{0}^{\frac{\pi}{2}}d\phi\;g^{\pm}_{f}(\omega\sin^{2}u\cos^{2}\phi)(kt^{2}g^{\pm}_{b}(\omega\sin^{2}u\sin^{2}\phi)g^{\mp}_{b}(\omega\cos^{2}u)$
$\displaystyle-J^{2}g^{\pm}_{f}(\omega\sin^{2}u\sin^{2}\phi)g^{\mp}_{f}(\omega\cos^{2}u))\,,$
(115)

and for bosons

$\displaystyle s^{\pm}_{b}(\omega)=-t^{2}\int_{0}^{\frac{\pi}{2}}du\sin
u\int_{0}^{\frac{\pi}{2}}d\phi\;g^{\pm}_{f}(\omega\sin^{2}u\cos^{2}\phi)g^{\pm}_{b}(\omega\sin^{2}u\sin^{2}\phi)g^{\mp}_{f}(\omega\cos^{2}u)\,,$
(116)

and we notice that $s_{\pm}(x)$ and $g_{\pm}(x)$ are defined only for a
positive argument. Now it is left to find a real part
$\Sigma^{\prime}(\omega)$ of the self-energy. For this we use Kramers-Kronig
transform

$\displaystyle\Sigma^{\prime}(\omega)$
$\displaystyle=\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{-\infty}^{+\infty}\frac{d\nu}{\pi}\frac{\Sigma^{\prime\prime}(\nu)-\Sigma^{\prime\prime}(\omega)}{\nu-\omega}\,.$
(117)

Defining $\Sigma^{\prime(\pm)}_{a}(\omega)$ as
$\Sigma_{a}^{\prime}(\omega)=\Sigma^{\prime(+)}_{a}(\omega)\theta(\omega)+\Sigma^{\prime(-)}_{a}(-\omega)\theta(-\omega)$
we find

$\displaystyle\Sigma^{\prime(\pm)}_{a}(\omega)$
$\displaystyle=\pm\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{0}^{+\infty}\frac{d\nu}{\pi}\left(\frac{\Sigma^{\prime\prime(\pm)}_{a}(\nu)-\Sigma^{\prime\prime(\pm)}_{a}(\omega)}{\nu-\omega}-\frac{\Sigma^{\prime\prime(\mp)}_{a}(\nu)-\Sigma^{\prime\prime(\pm)}_{a}(\omega)}{\nu+\omega}\right)\,.$
(118)

At zero temperature we set chemical potential
$\mu_{a}=\Sigma_{a}^{\prime}(\omega=0)$, so introducing $h^{\pm}_{a}(\omega)$
as

$\displaystyle\Sigma_{a}^{\prime}(\omega)-\Sigma_{a}^{\prime}(0)=\begin{cases}4\sqrt{\omega}h^{+}_{a}(\omega),\quad\omega>0\\\
4\sqrt{-\omega}h^{-}_{a}(-\omega),\quad\omega<0\end{cases}\,.$ (119)

and simplifying expressions we finally obtain

$\displaystyle
h^{\pm}_{a}(\omega)=\pm\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{0}^{+\infty}d\nu\left(\frac{\sqrt{\omega}s^{\pm}_{a}(\nu)-\sqrt{\nu}s^{\pm}_{a}(\omega)}{\sqrt{\nu}(\nu-\omega)}+\frac{\sqrt{\omega}s^{\mp}_{a}(\nu)+\sqrt{\nu}s^{\pm}_{a}(\omega)}{\sqrt{\nu}(\nu+\omega)}\right)\,.$
(120)

Now using the first pair of the Dyson-Schwinger equations (103),(104) we can
get $g^{\pm}_{a}$ from $s^{\pm}_{a}$ and $h^{\pm}_{a}$

$\displaystyle
g^{\pm}_{a}(\omega)=-\frac{4s^{\pm}_{a}(\omega)}{(4h^{\pm}_{a}(\omega)\mp\sqrt{\omega})^{2}+16\pi^{2}(s^{\pm}_{a}(\omega))^{2}}\,.$
(121)

We solve Dyson-Schwinger equations iteratively using (115), (116), (120) and
(121) and also imposing the initial conditions

$\displaystyle g^{\pm}_{a}(0)=\frac{C_{a}}{\pi
J}\sin(\frac{\pi}{4}\pm\theta_{a}),\quad s_{a}^{\pm}(0)=-\frac{1}{4\pi
C_{a}}\sin(\frac{\pi}{4}\pm\theta_{a}),\quad
h_{a}^{\pm}(0)=\mp\frac{1}{4C_{a}}\cos(\frac{\pi}{4}\pm\theta_{a})\,.$ (122)

where the constants $C_{a}$ are given in (25).

###  A.1 Luttinger constraints and fermionic charge vs chemical potential

We can numerically investigate the solution similarly to the previous paper
Tikhanovskaya  _et al._ (2020) computing the Luttinger constraints (99),(100)
both numerically and analytically that we rewrite as follows

$\displaystyle\mathcal{Q}_{f}(\theta_{f})$ $\displaystyle=$
$\displaystyle-\frac{\theta_{f}}{\pi}-\frac{1}{4}{\sin(2\theta_{f})}=\left\langle
f^{\dagger}f\right\rangle-\frac{1}{2}\,,$ (123)
$\displaystyle\mathcal{Q}_{b}(\theta_{b})$ $\displaystyle=$
$\displaystyle-\frac{\theta_{b}}{\pi}-\frac{1}{4}{\sin(2\theta_{b})}=-\frac{1}{2}-\left\langle
b^{\dagger}b\right\rangle\,.$ (124)

Numerically we compute the areas under fermionic and bosonic spectral density
curves respectively at fixed parameters. We show the comparison of the
analytical and numerical results in the Fig.14.

Figure 14: The Luttinger constraints for fermions (left) and bosons (right).
Blue dots are numerically computed Luttinger constraints and the dashed black
lines are the analytical functions (123),(124). Here we fix the parameters
$k=0.2$, and $\theta_{b}=0.26\pi$ (left), $\theta_{f}=0.1\pi$ (right). We note
that the precision for each point is in the range of $10^{-5}-10^{-7}$.

In addition to the Luttinger constraints, we numerically compute the fermionic
charge as a function of the chemical potential that is given by the real part
of the self energy at zero frequency $\mu=\Sigma^{\prime}(\omega=0)$. The
result is presented in the Fig.15.

Figure 15: Left: Charge as a function of fermionic chemical potential at
$T=0$. The function $\mathcal{Q}_{f}$ depends on $\mu_{f}$ through
$\theta_{f}$. Right: Compressibility as a function of charge at $T=0$.
Parameters for both plots: $\theta_{b}=0.26\pi$, $k=0.2$, $J=1$ and $t=1$.
Dashed line on the left plot is numerical line connecting the blue points. The
divergence of compressibility happens at $\theta_{f}\sim 0.15\pi$.

The behavior of the fermionic charge as a function of chemical potential is
similar to the SYK case Tikhanovskaya  _et al._ (2020). We do not see special
features in the bosonic charge $\mathcal{Q}_{b}$.

##  Appendix B Corrections to Conformality

The notation we adopt here is the same as Gu  _et al._ (2020); Guo  _et al._
(2020); Tikhanovskaya  _et al._ (2020), where we represent the functions as a
four-component vector indexed by boson/fermion and $\pm\tau$ branches. We use
the following basis for the UV-source and the corrections of Green’s function
and self-energy:

$\sigma_{h}(\tau)=\begin{pmatrix}\delta\sigma_{b+}\Sigma_{b}^{c}(\tau)\\\
\delta\sigma_{b-}\Sigma_{b}^{c}(\tau)\\\
\delta\sigma_{f+}\Sigma_{f}^{c}(\tau)\\\
\delta\sigma_{f-}\Sigma_{f}^{c}(\tau)\end{pmatrix}|J\tau|^{1-h},$ (125)
$\delta G(\tau)=\begin{pmatrix}\delta G_{b+}G_{b}^{c}(\tau)\\\ \delta
G_{b-}G_{b}^{c}(\tau)\\\ \delta G_{f+}G_{f}^{c}(\tau)\\\ \delta
G_{f-}G_{f}^{c}(\tau)\end{pmatrix}|J\tau|^{1-h},\qquad\delta\Sigma(\tau)=\begin{pmatrix}\delta\Sigma_{b+}\Sigma_{b}^{c}(\tau)\\\
\delta\Sigma_{b-}\Sigma_{b}^{c}(\tau)\\\
\delta\Sigma_{f+}\Sigma_{f}^{c}(\tau)\\\
\delta\Sigma_{f-}\Sigma_{f}^{c}(\tau)\end{pmatrix}|J\tau|^{1-h}.$ (126)

Here $\pm$ means positive/negative $\tau$ branches, and $b,f$ denote boson and
fermion respectively.

####  B.0.1 Kernel of the $t$-$J$ Model in the bosonic spinon $+$ fermionic
holon case

In this part we compute the kernel $K_{G}$ of the $t$-$J$ model in bosonic
spinon $+$ fermionic holon case. The basic ingredient is $W_{\Sigma}$ and
$W_{G}$ as defined in Gu  _et al._ (2020); Guo  _et al._ (2020). We elaborate
the definition by explicitly writing down on indices

$\displaystyle W_{\Sigma}(\tau_{1},\tau_{2};\tau_{3},\tau_{4})_{a\tilde{a}}$
$\displaystyle=\left.\frac{\delta
G_{*a}(\tau_{1},\tau_{2})}{\delta\Sigma_{\tilde{a}}(\tau_{3},\tau_{4})}\right|_{\Sigma^{c}},$
(127) $\displaystyle W_{G}(\tau_{1},\tau_{2};\tau_{3},\tau_{4})_{a\tilde{a}}$
$\displaystyle=\left.\frac{\delta\Sigma_{*a}(\tau_{1},\tau_{2})}{\delta
G_{\tilde{a}}(\tau_{3},\tau_{4})}\right|_{G^{c}},$

where $a,\tilde{a}=\mathfrak{f},\mathfrak{b}$ denote fermion and boson. Using
Eq.(127) and the conformal saddle point equations, we obtain

$W_{\Sigma}(1,2;3,4)=\begin{pmatrix}G_{\mathfrak{f}}^{13}G_{\mathfrak{f}}^{42}&0\\\
0&G_{\mathfrak{b}}^{13}G_{\mathfrak{b}}^{42}\end{pmatrix},$ (128)

and diagramatically

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}\pgfsys<EMAIL_ADDRESS>(129)

Here we use dashed line to denote fermion and solid line to denote boson.

$W_{G}$ differs from the other scheme by a switching
$\mathfrak{b}\leftrightarrow f,~{}\mathfrak{f}\leftrightarrow b$ and an
overall minus sign.

$W_{G}(1,2;3,4)=\begin{pmatrix}t^{2}G_{\mathfrak{b}}^{12}G_{\mathfrak{b}}^{21}\delta^{24;13}&t^{2}G_{\mathfrak{b}}^{43}G_{\mathfrak{f}}^{12}(\delta^{24;13}+\delta^{23;14})\\\
-kt^{2}G_{\mathfrak{f}}^{43}G_{\mathfrak{b}}^{12}(\delta^{24;13}+\delta^{23;14})&-kt^{2}G_{\mathfrak{f}}^{12}G_{\mathfrak{f}}^{21}\delta^{24;13}+J^{2}\left(2G_{\mathfrak{b}}^{12}G_{\mathfrak{b}}^{21}\delta^{24;13}+(G_{\mathfrak{b}}^{12})^{2}\delta^{23;14}\right)\end{pmatrix}$
(130)

where notation
$\delta^{ij;kl}=\delta(\tau_{i}-\tau_{j})\delta(\tau_{k}-\tau_{l})$.
Diagrammatically, $W_{G}$ has the following representation

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}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
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{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
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(131)

where a dotted line without arrow represents a $\delta$-function. In the basis
given by (125) and (126) $W_{\Sigma}$ and $W_{\Sigma}$ take the familiar form

$\begin{split}W_{\Sigma}(h)&=w(\Delta_{\mathfrak{f}},\theta_{\mathfrak{f}};h)\oplus
w(\Delta_{\mathfrak{b}},\theta_{\mathfrak{b}};h),\\\
w(\Delta,\theta;h)&=\frac{\Gamma(2\Delta-1+h)\Gamma(2\Delta-h)}{\Gamma(2\Delta)\Gamma(2\Delta-1)\sin(2\pi\Delta)}\begin{pmatrix}\sin(\pi
h+2\theta)&-\sin(2\pi\Delta)+\sin(2\theta)\\\
-\sin(2\pi\Delta)-\sin(2\theta)&\sin(\pi h-2\theta)\end{pmatrix},\end{split}$
(132)

and

$W_{G}(h)=\left(\begin{array}[]{cccc}1&0&1&1\\\ 0&1&1&1\\\ -k\frac{\cos
2\theta_{\mathfrak{f}}}{\cos 2\theta_{\mathfrak{b}}}&-k\frac{\cos
2\theta_{\mathfrak{f}}}{\cos 2\theta_{\mathfrak{b}}}&k\frac{\cos
2\theta_{\mathfrak{f}}}{\cos 2\theta_{\mathfrak{b}}}+2&k\frac{\cos
2\theta_{\mathfrak{f}}}{\cos 2\theta_{\mathfrak{b}}}+1\\\ -k\frac{\cos
2\theta_{\mathfrak{f}}}{\cos 2\theta_{\mathfrak{b}}}&-k\frac{\cos
2\theta_{\mathfrak{f}}}{\cos 2\theta_{\mathfrak{b}}}&k\frac{\cos
2\theta_{\mathfrak{f}}}{\cos 2\theta_{\mathfrak{b}}}+1&k\frac{\cos
2\theta_{\mathfrak{f}}}{\cos 2\theta_{\mathfrak{b}}}+2\\\ \end{array}\right).$
(133)

Finally $K_{G}(h)=W_{\Sigma}(h)W_{G}(h)$ and
$K_{\Sigma}(h)=W_{G}(h)W_{\Sigma}(h)$. It turns out that in this scheme the
kernel is identical to that of the fermionic spinon + bosonic holon theory if
we substitute $\theta_{f}\to\theta_{\mathfrak{b}}$,
$\theta_{b}\to\theta_{\mathfrak{f}}$.

####  B.0.2 Linear order correction at zero temperature

Following derivations in Tikhanovskaya  _et al._ (2020), the first order
conformal corrections at zero temperature take the following form :

$\begin{split}\delta G(\tau)&=\sum_{h}\delta_{h}G(\tau)\,,\\\
\delta_{h}G(\tau)&=-\alpha_{h}v_{h}\frac{G^{c}(\tau)}{|J\tau|^{h-1}}\,,\quad\alpha_{h}=-\frac{w_{h}W_{\Sigma}(h)\vec{\sigma}_{h}}{k_{G}^{\prime}(h)}\,.\end{split}$
(134)

Here the sum is over the operator spectrum of the $t$-$J$ model determined by
(63), and $v_{h}$ and $w_{h}$ are the corresponding right and left
eigenvectors of $K_{G}(h)$ respectively which have eigenvalue $k_{G}(h)=1$ and
normalized as $w_{h}v_{h}=1$, and
$\vec{\sigma}_{h}=(\delta\sigma_{b+},\delta\sigma_{b-},\delta\sigma_{f+},\delta\sigma_{f-})^{T}$.

It can be verified that the eigenspace corresponding to the two $h=1$
operators are spanned by $\delta G^{c}_{a}/\delta\theta_{b}$ and $\delta
G^{c}_{a}/\delta\theta_{b}$ (i.e. changes of $\theta_{b}$ and $\theta_{f}$)
which by the Luttinger relations (33),(34) are related to
$\delta\mathcal{Q}_{f}$ and $\delta\mathcal{Q}_{b}$ respectively. Therefore,
there is no need to include these corrections if we use the renormalized value
of $\theta_{f},\theta_{b}$.

####  B.0.3 Nonlinear order correction at zero temperature

It is also possible to obtain corrections of higher order in $\alpha$
following derivations in Tikhanovskaya  _et al._ (2020). The result takes the
form

$\displaystyle G(\tau)$
$\displaystyle=G^{c}(\tau)\bigg{(}1-\sum_{h}\frac{\alpha_{h}v_{h}}{|J\tau|^{h-1}}-\sum_{h,h^{\prime}}\frac{a_{hh^{\prime}}\alpha_{h}\alpha_{h^{\prime}}}{|J\tau|^{h+h^{\prime}-2}}-\sum_{h,h^{\prime},h^{\prime\prime}}\frac{a_{hh^{\prime}h^{\prime\prime}}\alpha_{h}\alpha_{h^{\prime}}\alpha_{h^{\prime\prime}}}{|J\tau|^{h+h^{\prime}+h^{\prime\prime}-3}}-\dots\bigg{)}\,,$
(135)

where $v_{h}$, $a_{hh^{\prime}}$, $a_{hh^{\prime}h^{\prime\prime}}$, etc are
four-component vectors. The coefficients $a_{hh^{\prime}}$,
$a_{hh^{\prime}h^{\prime\prime}}$ can be calculated using the recursion
procedure described in Tikhanovskaya  _et al._ (2020). As an example, we
perform a calculation for $a_{hh^{\prime}}$, using Eq. (4.32) of Tikhanovskaya
_et al._ (2020):

$\delta^{2}G=\frac{1}{1-W_{\Sigma}W_{G}}\left[W_{\Sigma}\bar{\delta}^{2}\Sigma_{*}[G]+\bar{\delta}^{2}G_{*}[\Sigma]\right],$
(136)

where $\Sigma_{*}[G]$ and $G_{*}[\Sigma]$ denote the RHS of Schwinger-Dyson
equations (16)-(19) with $(i\omega_{n}+\mu)$ dropped. The meaning of
$\bar{\delta}^{2}\Sigma_{*}[G]$ is that we compute the second-order variation
of $\Sigma_{*}$ using only first-order variation of $G$.

The second term in the bracket of (136) can be computed using the same
procedure as Tikhanovskaya  _et al._ (2020), with the result

$\displaystyle\frac{\bar{\delta}^{2}G_{*}(\tau)}{G^{c}(\tau)}=\sum_{h,h^{\prime}}F(h+h^{\prime}-1)^{-1}\big{(}F(h)v_{h}\cdot
F(h^{\prime})v_{h^{\prime}}\big{)}\frac{\alpha_{h}\alpha_{h^{\prime}}}{|J\tau|^{h+h^{\prime}-2}}\,,$
(137)

where “$\cdot$” means multiply two vectors entry-wise and $F(h)$ is a 4 by 4
matrix that acts on $v_{h}$:

$\begin{split}F(h)&=f(h,\theta_{b})\oplus f(h,\theta_{f}),\\\
f(h,\theta)&=-\frac{\Gamma(3/2-h)}{2\sqrt{\pi}}\left(\begin{array}[]{cc}-e^{-\frac{1}{2}i\pi
h}\left(e^{2i\theta}+i\right)&-e^{\frac{i\pi
h}{2}}\left(e^{2i\theta}-i\right)\\\
i\left(e^{2i\theta}+i\right)e^{\frac{1}{2}i(\pi
h-4\theta)}&\left(-1-ie^{2i\theta}\right)e^{-\frac{1}{2}i(\pi h+4\theta)}\\\
\end{array}\right).\end{split}$ (138)

The first term in the bracket of (136) can be computed by expanding the
Schwinger-Dyson equation $\Sigma=\Sigma_{*}[G]$ to second order, with the
result

$\frac{\bar{\delta}^{2}\Sigma_{*}(\tau)}{\Sigma^{c}(\tau)}=\sum_{h,h^{\prime}}\frac{\alpha_{h}\alpha_{h^{\prime}}}{|J\tau|^{h+h^{\prime}-2}}d_{hh^{\prime}}\,,$
(139)

and $d_{hh^{\prime}}$ is a four-component vector whose entries are

$\displaystyle d_{hh^{\prime};b+}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(v_{hb+}(v_{h^{\prime}f-}+v_{h^{\prime}f+})+v_{hf-}(v_{h^{\prime}b+}+v_{h^{\prime}f+})+v_{hf+}(v_{h^{\prime}b+}+v_{h^{\prime}f-}))\,,$
(140) $\displaystyle d_{hh^{\prime};b-}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(v_{hb-}(v_{h^{\prime}f-}+v_{h^{\prime}f+})+v_{hf-}(v_{h^{\prime}b-}+v_{h^{\prime}f+})+v_{hf+}(v_{h^{\prime}b-}+v_{h^{\prime}f-}))\,,$
(141) $\displaystyle d_{hh^{\prime};f+}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\eta\Big{(}v_{hb-}(v_{h^{\prime}b+}+v_{h^{\prime}f+})+v_{hb+}(v_{h^{\prime}b-}+v_{h^{\prime}f+})-2v_{hf-}v_{h^{\prime}f+}$
$\displaystyle+v_{hf+}(v_{h^{\prime}b-}+v_{h^{\prime}b+}-2(v_{h^{\prime}f-}+v_{h^{\prime}f+}))\Big{)}+v_{hf-}v_{h^{\prime}f+}+v_{hf+}(v_{h^{\prime}f-}+v_{h^{\prime}f+})$
$\displaystyle d_{hh^{\prime};f-}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\eta\Big{(}v_{hb-}(v_{h^{\prime}b+}+v_{h^{\prime}f-})+v_{hb+}(v_{h^{\prime}b-}+v_{h^{\prime}f-})-2v_{hf+}v_{h^{\prime}f-}\,,$
$\displaystyle+v_{hf-}(v_{h^{\prime}b-}+v_{h^{\prime}b+}-2(v_{h^{\prime}f-}+v_{h^{\prime}f+}))\Big{)}+v_{hf-}(v_{h^{\prime}f-}+v_{h^{\prime}f+})+v_{hf+}v_{h^{\prime}f-}\,.$

We remind the reader that $\eta=-k\cos(2\theta_{b})/\cos(2\theta_{f})$.
Combining (137) and (139), we obtain

$\displaystyle a_{hh^{\prime}}=$
$\displaystyle-\big{(}1-W_{\Sigma}(h+h^{\prime}-1)W_{G}\big{)}^{-1}\Big{(}F(h+h^{\prime}-1)^{-1}\big{(}F(h)v_{h}\cdot
F(h^{\prime})v_{h^{\prime}}\big{)}+W_{\Sigma}(h+h^{\prime}-1)d_{hh^{\prime}}\Big{)}\,.$
(144)

##  Appendix C Qualitative analysis of potentially relevant operators in the
$t$-$J$ model

The goal of this analysis is to determine the range of parameters regarding
the existence and stability of the saddle conformal solution, and to study the
relevance of time-reparameterization mode $h=2$. The outcome of the analysis
is in Fig. 2 and Fig. 4. We will mainly discuss the fermionic spinon + bosonic
holon model. The analysis of the fermionic holon + bosonic spinon model is
completely parallel as the kernel is related to by
$\theta_{f}\to\theta_{\mathfrak{b}}$, $\theta_{b}\to\theta_{\mathfrak{f}}$ .

We will be analyzing the function

$F(h)\equiv\det(K_{G}(h)-1)=0,$ (145)

whose zeroes determine the scaling dimension of irrelevant operators appearing
in the corrections. The conformal saddle exists if the Eq.(24) has real
solutions for $C_{f}$ and $C_{b}$, which implies

$\cos(2\theta_{f})+k\cos(2\theta_{b})>0.$ (146)

Here $\theta_{f}\in[-\pi/4,\pi/4]$ and $\theta_{b}\in[\pi/4,\pi/4]$ are
assumed to respect unitarity constraint.

By numerically inspecting the function $F(h)$, we found that there is almost
always a solution with scaling dimension $\text{Re}~{}h_{1}<\frac{3}{2}$.
Depending on parameters, this solution may be on the real axis within the
range $\frac{1}{2}<h_{1}<\frac{3}{2}$, or it may move onto the imaginary axis
such that $h_{1}=1/2\pm is,~{}s>0$.

Qualitative behaviors of the solution can be described by simple inequalities,
which we describe below. First, whether the solution $h_{1}$ is real or
complex depends on the sign of $F(h)$ near the pole $h=1/2$. When $h_{1}$
moves from real to complex, the pole switches sign. Therefore the existence of
a complex solution corresponds to

$\begin{split}0>&16\pi^{2}\lim_{h\to
1/2}(h-\frac{1}{2})^{4}F(h)=8\left(4\cos\left(2\theta_{b}\right)+\pi\right)\cos\left(2\theta_{f}\right)+16k\cos\left(4\theta_{b}\right)+16k+\pi(3\pi-4)\\\
&+2(\pi-2)k\cos\left(2\theta_{b}\right)\left(8\cos\left(2\theta_{b}\right)+3\pi-4\right)\sec\left(2\theta_{f}\right)+4(3\pi-4)(k+1)\cos\left(2\theta_{b}\right).\end{split}$
(147)

Second, generically $F(h)$ has a double zero at $h=1$, corresponding two gauge
charges $\mathcal{Q}_{f},~{}\mathcal{Q}_{b}$ which have scaling dimension 1.
When $h_{*}$ crosses 1, $F(1)$ will be a triple zero such that

$0=F^{\prime\prime}(1)=16\left(\pi\cos\left(2\theta_{b}\right)+2\right)\left(\pi\cos\left(2\theta_{f}\right)+2\right)\sec\left(2\theta_{f}\right)\left(k\cos\left(2\theta_{b}\right)+\cos\left(2\theta_{f}\right)\right).$
(148)

Assuming, (146) holds, the only factor that could change sign is
$2+\pi\cos(2\theta_{b})$, so the condition for $1/2<h_{1}<1$ is

$\theta_{b}>\frac{1}{2}\arccos\left(\frac{-2}{\pi}\right)\simeq 0.3598\pi.$
(149)

We also remark that this is the maximum of the LHS of Luttinger constraint
(34). When $h_{1}=\frac{3}{2}$, it will be absorbed by the pole of $F(h)$ at
$h=\frac{3}{2}$, so the mode disappears (it has zero OPE with the $f$ and
$b$). Physically, this corresponds to $\theta_{b}=\pi/4$, i.e. the holon
density is zero.

Additionally, there can be a second solution near $h_{2}$ near $h=2$. When
this mode crosses $h=2$, $F^{\prime}(2)$ changes sign. The condition that
$h_{2}<2$ is

$0>F^{\prime}(2)=\frac{16}{81}\sec\left(2\theta_{f}\right)\left(-4k\cos\left(2\theta_{b}\right)-3\pi
k\cos\left(4\theta_{b}\right)+4\cos\left(2\theta_{f}\right)+3\pi\cos\left(4\theta_{f}\right)-3\pi
k+3\pi\right).$ (150)

The above constraints define all the boundaries in Fig. 2. Same inequalties
with $\theta_{f}\to\theta_{\mathfrak{b}}$,
$\theta_{b}\to\theta_{\mathfrak{f}}$ define the boundaries in Fig. 4.

##  Appendix D Calculation of Optical Conductivity

In this appendix we give details on the evaluation of (90). For convenience,
we will work in units where $\beta=2\pi$.

Our starting point is the Kubo formula (80), recast into the following form

$\text{Re}\sigma(\omega)=\frac{\sigma_{0}}{C_{e}^{2}/J^{2}}\int{\rm
d}\Omega\frac{1}{\omega}\left(\frac{1}{e^{2\pi\Omega}+1}-\frac{1}{e^{2\pi(\Omega+\omega)}+1}\right)\rho_{e}(\Omega)\rho_{e}(\Omega+\omega)\,,$
(151)