# Spin incoherent liquid and interaction-driven criticality in 1D Hubbard
model

Jia-Jia Luo  State Key Laboratory of Magnetic Resonance and Atomic and
Molecular Physics, Wuhan Institute of Physics and Mathematics, APM, Chinese
Academy of Sciences, Wuhan 430071, China  University of Chinese Academy of
Sciences, Beijing 100049, China.  Han Pu<EMAIL_ADDRESS>Department of Physics
and Astronomy, and Rice Center for Quantum Materials, Rice University,
Houston, Texas 77251-1892, USA  Xi-Wen Guan<EMAIL_ADDRESS>State Key
Laboratory of Magnetic Resonance and Atomic and Molecular Physics, Wuhan
Institute of Physics and Mathematics, APM, Chinese Academy of Sciences, Wuhan
430071, China  NSFC-SPTP Peng Huanwu Center for Fundamental Theory, Xi’an
710127, China  Department of Theoretical Physics, Research School of Physics
and Engineering, Australian National University, Canberra ACT 0200, Australia

###### Abstract

Although the one dimensional (1D) repulsive Fermi-Hubbard model has been
intensively studied over many decades, a rigorous understanding of many
aspects of the model is still lacking. In this work, based on the solutions to
the thermodynamic Bethe ansatz equations, we provide a rigorous study on the
following: (1) We calculate the fractional excitations of the system in
various phases, from which we identify the parameter regime featuring the spin
incoherent Luttinger liquid (SILL). We investigate the universal properties
and the asymprotic of correlation functions of the SILL. (2) We study the
interaction-driven phase transition and the associated criticality, and build
up an essential connection between the Contact susceptibilities and the
variations of density, magnetization and entropy with respect to the
interaction strength. As an application of these concepts, which hold true for
higher dimensional systems, we propose a quantum cooling scheme based on the
interaction-driven refrigeration cycle.

One-dimensional (1D) Fermi-Hubbard model, describing strongly correlated
electrons in a 1D lattice, has become increasingly important in ultracold
atoms, condensed matter and quantum metrology. Owing to the Bethe ansatz exact
solution of the model Lieb:1968 ; Lieb:2003 , a variety of strongly correlated
many-body phenomena have been extensively studied for over forty years,
including Tomonaga-Luttinger liquid (TLL) Giamarchi:book ; Imambekov:2012 ;
Ess05 , spin-charge separation Boll:2016 ; Hilker:2017 ; Vijayan:2020 ;
Spar:2022 , thermal and magnetic properties Krivnov:1975 ; Bogoliubov:1988 ;
Lee:1988 ; Penc ; Woynarovich:1983 ; Sacramento:1994 ; Essler:1994 , the
Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) Fulde:1964 ; Larkin:1965  pairing
correlation Yang:2001 ; Tezuka:2008 ; Feiguin:2007 ; Kajala:2011 ; Cheng:2017A
; Cheng:2017B , etc. On the experimental side, 1D exactly solvable models have
been successfully realized in the lab, allowing us to compare elegant and
sophisticated exact solutions directly with experimental measurements
Cazalilla:2011 ; Guan:2013 ; Batchelor:2016 ; Mistakidis:2022 , Significant
new experimental developments cover a broad range of physics such as the
generalized hydrodynamics Kinoshita:2006 ; Langen:2015 ; Schemmer:2019 ,
dynamical fermionization Wilson:2020 , TLL Yang:2017 ; YangTL:2018 ;
Meinert:2015 , fractional exclusion statistics Zhang:2022 , quantum holonomy
Haller:2009 ; Kao:2021 , $p$-wave interacting fermions Chang:2020 ; Ahmed-
Braun:2021 ; Jackson:2022 , high spin symmetry magnetism Pagano:2014 ;
Song:2020 , etc.. For a recent review, see Ref. Guan:2022 .

Despite of these tremendous efforts, many aspects of the model still lack
rigorous understanding. In particular, phase transitions have been extensively
studied in the context of varying external potentials such as chemical
potential and magnetic field. However, interaction-driven phase transitions
have not received much attention even though interaction plays an essential
role in many-body systems. This can be attributed to the fact that interaction
strength is hardly tunable in traditional solid materials. The advent of cold
atoms completely changed this situation as interaction strengths in atomic
systems have been routinely controlled via Feshbach resonance. A notable
recent example is the demonstration of the spin-charge separation Haldane:1981
; Recati:PhysRevLett.90.020401 ; Guan:2012 ; Mestyan:2019spin ;
Patu:PhysRevB.101.035149  in a 1D continuum Fermi gas where the spin and the
charge velocities are shown to exhibit distinct dependence on the interaction
strength Senaratne:2022 .

Motivated by this, in this Letter, we show that the tunability of interaction
strength allows further exploration of the spin incoherent Luttinger liquid
(SILL) Fiete:2004 ; Fiete:2007 ; Cheianov:2004 ; Cavazos-Cavazos:2022  and
interaction-driven quantum phase transitions in the Hubbard model.
Specifically, we present rigorous results of fractional charge and spin
excitations, analytical results on the asymptotic of single-particle Green’s
function and pair correlation functions of the SILL, and interaction-driven
criticality. Furthermore, inspired by the notion of the partial wave Contact
in ultracold Fermi gas Tan:2008 ; ZhangSZ:2009 , we build up general relations
between Contact susceptibilities and the variation of density, magnetization
and entropy with respect to the interaction strength, using which we propose a
quantum cooling scheme based on the interaction-driven refrigeration cycle.

Figure 1: Phase diagram represented by the contour plot of the Wilson ratio
$R^{\chi_{s}}_{w}$ at temperature $T=0.005$ and $u=1$. The corresponding
phases are: empty lattice I, partially filled and fully-polarized phase II,
fully-filled and -polarized phase III, partially-filled and -polarized phase
IV, and fully-filled and partially-polarized phase V (Mott insulator). The
dotted lines represent analytic solution of BA equations obtained at zero
temperature. The orange symbols indicates the locations of excitations plotted
in Fig. 2.

1D Hubbard model – The 1D single-band Hubbard model is described by the
Hamiltonian Lieb:1968 ; Ess05

$\displaystyle H$ $\displaystyle=$
$\displaystyle-t\sum^{L}_{j=1,a=\uparrow,\downarrow}\left(c_{j,a}^{\dagger}c_{j+1,a}+{\rm
h.c.}\right)$ (1)
$\displaystyle+u\sum_{j=1}^{L}\left(2n_{j,\uparrow}-1\right)\left(2n_{j,\downarrow}-1\right)-\mu\hat{n}-2B\hat{S}^{z},$

where $c_{j,a}^{\dagger}$ ($c_{j,a}$) is the creation (annihilation) operator
of an electron with spin $a$ ($a=\uparrow$ or $\downarrow$) at site $j$ on a
1D lattice of length $L$, satisfying the standard anticommutation relations.
$t$, $\mu$ and $B$ are hopping amplitude, chemical potential and magnetic
field, respectively. In this work, we will only consider repulsive interaction
with $u>0$ and take $t=1$ as the unit of the system. $\mu$ and $B$ are
renormalized accordingly and become dimensionless. Meanwhile
$n_{j,a}=c_{j,a}^{\dagger}c_{j,a}$ and
$\hat{n}=\frac{\hat{N}}{L}=\frac{1}{L}\sum_{j,a}n_{j,a}$ are the density
operator and the average fermion number per lattice site, respectively. We
denote the magnetization
$\hat{S}^{z}=\sum_{j,a}\left(n_{j,\uparrow}-n_{j,\downarrow}\right)$. For
vanishing external potentials ($\mu=0,\,B=0$) and even $L$, Hamiltonian (S1)
possesses $SO(4)\cong SU(2)\times SU(2)/\mathbb{Z}_{2}$ symmetry, preserving
spin rotational and $\eta$-pairing symmetries Ess05 ; YangCN:1989 ;
YangCN:1990 , i.e. ${[H,S^{\alpha}]}=0=[H,\eta^{\alpha}]$ with $\alpha=x,y,z$.
The spin and $\eta$-pair operators can be transformed to each other via Shiba
transformation, showing the connection between spin and charge Ess05 , also
see Supplementary Materials SM . We will use spin and $\eta$-pair
magnetizations $S^{z}$, $\eta^{z}=\frac{1}{2}\left(N-L\right)$ to characterize
the fractional spin and charge excitations.

In 1968 Lieb and Wu Lieb:1968  derived the BA equations for the 1D Hubbard
model by means of Bethe’s hypothesis Bethe:1931 . Takahashi Takahashi:1972
later found the root patterns of the BA equations, i.e. real $k$, length-$n$
$\Lambda$ string (known as spinon bound state) composed of $n$ spin-down
electrons, length-$m$ $k$-$\Lambda$ string containing $m$ down-spin and $m$
up-spin particles, which determine both the ground and the excited states of
the model. Building on Takahashi’s string hypothesis, and using the Yang-Yang
method YangCN:1969 , one can obtain the thermodynamic Bethe ansatz (TBA)
equations of the model Takahashi:1972  (for convenience, see SM ). In
principle, all thermodynamic properties of the model can be obtained from the
TBA. However, solving the infinite number of nonlinear integral TBA equations
poses a tremendous theoretical challenge. Therefore many important questions
remain to be answered.

Figure 2: Elementary fractional spin and charge excitations. The orange areas
in (a) and (b) represent particle-hole excitations of charge, whereas the
green parts show the two-spinon excitations with quantum numbers
$\left(\Delta\eta^{z},\Delta S^{z}\right)=(0,1)$ induced from spin flipping.
(c) Fractional antiholon-spinon excitations $(\frac{1}{2},-\frac{1}{2})$ ,
i.e. adding an extra spin-down electron to create an antiholon and a spinon.
(d) Gapped excitation spectra for length-1 $k$-$\Lambda$ string and a length-2
$\Lambda$ string. All graphs are drawn in the first Brillouin zone with
interaction $u=1$ and the parameters (a): $B=0,\mu=-0.6619$, density
$n=0.9801$ (near half-filled band); (b): $B=0,\mu=-3.8508$, density $n=0.1389$
(dilute limit); (c): $B=0.555,\mu=-1$; (d): $B=0.555,\mu=-1.32$.

Low-temperature phase diagram and fractional excitations – A rich phase
diagram of the 1D Hubbard model (S1) in magnetic field-chemical potential
plane can be obtained from either the BA or the TBA equations at zero
temperature. We find that the dimensionless Wilson ratio (WR)
$R^{\chi_{s}}_{w}=\frac{4}{3}\left(\frac{\pi
k_{B}}{\mu_{B}g}\right)^{2}\frac{\chi_{s}}{C_{v}/T}$, where $\chi_{s}$ is the
spin susceptibility and $C_{v}$ the specific heat, conveniently characterizes
the TLLs. Here $k_{B}$, $\mu_{B}$ and $g$ are the Boltzmann constant, the Bohr
magneton and the Landé factor, respectively, which we set to be unity in our
calculation. The value of the WR is temperature independent at low energy and
exhibits a sudden change in the vicinities of the phase boundaries. This
feature naturally maps out the full phase diagram of the 1D repulsive Hubbard
model, as we show in Fig. 1. Specifically, we observe that the values of the
WR in Fig. 1 confirm the bosonization result Fiete:2007 , i.e.
$R^{\chi_{s}}_{w}=2v_{c}K_{s}/(v_{s}+v_{c}),\,2$ and $4K_{s}$ for the TLLs in
Phase IV, II and V, respectively, where $K_{s}$ is the Luttinger parameter for
spin, $v_{c,s}$ are sound velocities for charge and spin, respectively. The WR
is zero for Phases I and III, see SM SM  for more details.

Figure 2 (a) and (b) demonstrate excitations at zero magnetic field in charge
and spin degrees of freedom near the half-filled lattice and the dilute limit,
respectively. The particle-hole excitation of charge (orange) forms continuum
spectra within the first Brillouin zone. Flipping one spin leads to excitation
spectrum (green) of two deconfined spinons with a fractional
spin-$\frac{1}{2}$. In the long wavelength limit, i.e. $\Delta K\to 0$, both
charge and spin excitations exhibit linear dispersion: $\Delta
E_{c,s}=v_{c,s}\hbar|\Delta K|$, where $v_{c}=0.5995,v_{s}=1.2403$ in (a) and
$v_{c}=0.7206,v_{s}=0.1454$ in (b), showing spin and charge separated
excitations. However, subtle differences between these two limits are
observed, i.e. for (a), the charge excitation displays a single-particle
nature due to the vanishing of the charge Fermi sea; for (b), the spin and
charge excitations are significantly separated, making this preferred region
to observe spin-charge separation SM . Later, we will further demonstrate the
existence of the SILL in region (b) for temperature $E_{s}\ll k_{B}T\ll
E_{F}$, where $E_{s,c}\sim k_{F}v_{s,c}$ with $k_{F}=\pi n_{c}$ are the spin
and charge energies, respectively. Fig. 2(c) shows the fractional antiholon-
spinon excitation spectra with the $\eta$-pair and spin magnization
$(\Delta\eta^{z},\Delta S^{z})=(\frac{1}{2},-\frac{1}{2})$ by adding an
antiholon particle $N_{e}=N+1$ superposed with one spinon particle in $M_{1}$
sector, which are outside of the spin-charge separated TLL regime SM . Fig.
2(d) shows the two gapped excitations, i.e., length-$1$ $k$-$\Lambda$ string
and length-$2$ $\Lambda$ spinon bound states, forming a gapped continuum band.

Universal scaling laws, SILL, correlation functions – Rigorous results on
quantum criticality of the repulsive Hubbard model remain largely unknown. At
zero temperature, the phase transition occurs at a quantum critical point
(QCP) where a degree of freedom appears, disappears or reaches saturation. At
finite temperature, the QCP fans out into the V-shaped quantum critical
regime, in which the free energy takes universal form. By considering the
relevant degrees of freedom, we can simplify the TBA equations and find such
universal forms. In the SM SM , we have derived analytically the free energies
of all quantum critical regions associated with various phase transitions of
the 1D Hubbard model. Here, we only write down the free energy at quantum
criticality for the II-IV transition:

$\displaystyle f=f_{0}-\frac{\pi
T^{2}}{6v_{c}}+T^{\frac{3}{2}}\pi^{\frac{1}{2}}\sigma_{1}(0)\left(\frac{\varepsilon^{{}^{\prime\prime}}_{1}(0)}{2}\right)^{-\frac{1}{2}}\mathrm{Li}_{\frac{3}{2}}\left(-\mathrm{e}^{\frac{-\varepsilon_{1}(0)}{T}}\right),$
(2)

where $\mathrm{Li}_{n}$ denotes the polylog functions, $f_{0}$ is the ground-
state energy, the $T^{2}$ terms represents the contributions from the
collective excitations of the background degree of freedom near the QCP,
$\sigma_{1}(0)$ denotes the spin density at $\Lambda=0$,
$\varepsilon_{1}(\Lambda)$ is the dressed energy of length-$1$ string, and
$\varepsilon^{{}^{\prime\prime}}_{1}(0)\equiv\left.\frac{d^{2}\varepsilon_{1}}{d\Lambda^{2}}\right|_{\Lambda=0}$,
with

$\displaystyle\varepsilon_{1}(0)$ $\displaystyle=-$
$\displaystyle\alpha_{B}\Delta B-\alpha_{\mu}\Delta\mu-\alpha_{u}\Delta u,$
(3)

denoting the spin dressed energy gap away from the QCP, where $\Delta
B=B-B_{c},\,\Delta\mu=\mu-\mu_{c},\,\Delta u=u-u_{c}$ are distances away from
the QCP ($B_{c},\,\mu_{c},\,u_{c}$). The analytic expressions of the factors
$\alpha_{B,\mu,u}$ are rather cumbersome and can be found in the SM SM . As we
will show below, the free energy in Eq. (2) elegantly leads to and provides a
rigorous understanding of the universal thermodynamic properties of the TLL,
the SILL and the quantum scaling laws at criticality.

Figure 3: Contour plot of specific heat in the $T$-$u$ plane at
$\mu=-2,B=0.55$ for II-IV transition. The blue dashed lines present the
critical temperatures determined by the maximum values of the specific heat
(Eq.(4)). The yellow lines with square symbols mark the TLL phase boundary,
below which the specific heat shows a linear temperature dependence. Crossover
regimes between the blue dashed and the yellow lines denote the spin-gapped
phase (on the left) and the SILL phase (on the right).

In the quantum critical regime, $T\gg\Delta u$, and from Eq. (2), the specific
heat can be readily derived as

$\displaystyle C_{v}/T$ $\displaystyle=$ $\displaystyle
c_{0}+c_{1}T^{-1/2}\left[\frac{3}{4}\mathrm{Li}_{\frac{3}{2}}\left(-\mathrm{e}^{x}\right)-x\mathrm{Li}_{\frac{1}{2}}\left(-\mathrm{e}^{x}\right)\right.$
(4)
$\displaystyle\left.+x^{2}\mathrm{Li}_{-\frac{1}{2}}\left(-\mathrm{e}^{x}\right)\right]+O((\Delta
u/T)^{5/2}),$

where $x\equiv\alpha_{u}\Delta u/T$, and $c_{0,1}$ denote the regular part and
a constant depending on the critical point $u_{c}$, respectively. In Fig. 3,
we display the contour plot of the specific heat in the plane spanned by $T$
and $u$ at $\mu=-2,\,B=0.55$. The specific heat shows a bimodal structure,
whose local maxima mark the critical temperatures. The local maxima can be
determined by $\partial C_{v}/\partial u=0$, leading to $x_{1}=-1.5629$ and
$x_{2}=3.6205$, corresponding to the two blue dashed lines in Fig. 3. These
two lines join at $u_{c}=1$ at $T=0$ and the quantum critical regime resides
between them, displaying a universal free fermion criticality, i.e. dynamical
and correlation critical exponents $Z=2,\,\nu=1/2$, respectively.

The yellow line with square symbols in Fig. 3 represents the boundary of the
TLL region which lies below the line. In the TLL, $C_{v}$ is linear in $T$. To
the left of $u_{c}$, the TLL (labelled as TLLC) contains only the charge
component and $C_{v}=\frac{\pi}{3}\frac{T}{v_{c}}$. To the right of $u_{c}$,
the TLL (label as TLLSC) contains both the spin and the charge component with
$C_{v}=\frac{\pi}{3}\left(\frac{1}{v_{c}}+\frac{1}{v_{s}}\right)T$. This is
the regime where spin-charge separation Haldane:1981 ;
Recati:PhysRevLett.90.020401 ; Guan:2012 ; Mestyan:2019spin ;
Patu:PhysRevB.101.035149  can be observed. Above TLLC, the spin sector is
gapped and the specific heat is given by
$C_{v}\approx\alpha_{1}T+\alpha_{2}T^{1/2}\mathrm{e}^{\alpha_{3}\Delta u/T}$,
where $\alpha_{1,\,2,\,3}$ are constants. By contrast, above TLLSC, we find a
region of SILL in the temperature range given by $k_{F}v_{s}\ll k_{B}T\ll
k_{F}v_{c}$, exhibiting a propagating charge mode but not a spin mode with the
corresponding specific heat $C_{v}\approx\beta_{1}T+\beta_{2}T^{3}$ with
$\beta_{1,2}$ being constants SM , showing a gas-liquid co-existence in the
SILL.

In the SILL regime, the spin excitation is suppressed and hence the spin
sector is non-dynamic, while charge maintains relevant at low energy. Taking
the reasonable limits $|x\pm iv_{c}t|\ll v_{c}/T$ and $|x\pm iv_{s}t|\gg
v_{s}/T$, we can calculate the finite-temperature single-particle Green’s
function and pair correlation function

$\displaystyle G^{\uparrow}$ $\displaystyle\\!\approx\\!$
$\displaystyle\mathrm{e}^{-\mathrm{i}k_{F,\uparrow}x}{\cal
C}_{\uparrow}^{-}\left(x-\mathrm{i}v_{c}t\right)\langle
S_{R}^{+}\left(x,t\right)S_{R}(0,0)\rangle+h.c.,$ $\displaystyle G^{p}$
$\displaystyle\\!\approx\\!$
$\displaystyle\mathrm{e}^{-\mathrm{i}\left(k_{F,\uparrow}+k_{F,\downarrow}\right)x}{\cal
C}_{p}^{-}\left(x-\mathrm{i}v_{c}t\right){\cal
C}_{p}^{+}\left(x+\mathrm{i}v_{c}t\right)$ (5) $\displaystyle\times\langle
S_{R}^{+}\left(x,t\right)S_{R}(0,0)\rangle+h.c.,$

where the charge correlations ${\cal C}_{\uparrow}^{-}\left(Z\right)\sim
1/Z^{2\Delta_{c}^{+}}$, ${\cal C}_{p}^{\pm}\left(Z\right)\sim
1/Z^{2\Delta_{c}^{\mp}}$ decay as a power-law of distance, whereas the spin
mode correlation $\langle
S_{R}^{+}\left(x,t\right)S_{R}(0,0)\rangle\sim\left(2\pi\alpha
k_{F}\right)^{2\Delta^{+}_{s}+2\Delta^{-}_{s}}\mathrm{e}^{-\pi\alpha\left(2\Delta^{+}_{s}+2\Delta^{-}_{s}\right)k_{F}x}$
decays exponentially. Here $\Delta_{c}^{\pm}$ are the conformal dimensions
which can be calculated analytically and numerically SM , $\alpha$ is a
constant. For the particular case $B=0$, our results agree with those given in
Ref. Cheianov:2004 . We comment that the SILL has been theoretically studied
under the framework of bosonization Fiete:2004 ; Fiete:2007 ; Cheianov:2004 .
Our work here provides a rigorous underpinning of the SILL based on the TBA.

Figure 4: (a) Contour plot of the entropy in $T-u$ plane for
$B=0.15,\,\mu=-2.5$. Black dotted curve is the isentropic line for $s=0.01$.
When the interaction increases, the system enters sequentially into Phases II,
IV and V. (b1) Contour plot of density $n$ near the IV-V phase transition with
$u=1,\,\mu=-0.82724$ and $B_{c}=0.82714$. (b2) Scaling behaviour of charge
susceptibilities near phase transition from IV to V driven by interaction.

Contact susceptibilities and quantum cooling – In analogy to the Contact for
quantum gases Tan:2008 ; ZhangSZ:2009 , here we define the lattice version of
the Contact $C=\partial f/\partial u=4d-2n+1$, where
$n=\langle\hat{n}\rangle$, and $d=\frac{1}{N}\sum_{i}\left\langle
n_{i,\uparrow}n_{i,\downarrow}\right\rangle$ is the average double occupancy,
a quantity which can also depict the phase diagram SM . It is, however, more
essential to define Contact susceptibilities with respect to the external
potentials. Using the Maxwell relations, we may build up general relations
between Contact susceptibilities and interaction-driven variations of entropy,
density and magnetization SM :

$\frac{\partial s}{\partial u}=-\frac{\partial C}{\partial
T},\;\;\;\frac{\partial n}{\partial u}=-\frac{\partial
C}{\partial\mu},\;\;\;\frac{\partial m}{\partial u}=-\frac{\partial
C}{\partial(2B)}.$ (6)

These relations provide deep insights into the interaction effects and
universal behaviour of phase transitions.

As a specific example, we now use the first relation in Eqs. (S94) to
investigate interaction-driven quantum cooling. Fig. 4 (a) shows a contour
plot of entropy in the $T$-$u$ plane for fixed $B$ and $\mu$. The interaction-
driven phase transitions from I to II, II to IV, and IV to V occur
sequentially with increasing interaction strength. We observe a single-
component charge TLLC in II, a spin and charge separated TLLSC in IV, and a
spin TLLS in the Mott phase V. Conducting the total derivative of entropy with
respect to the interaction $u$, the phase points on the isentropic line in the
$T$-$u$ plane admit the relation $\frac{C_{v}}{T}\frac{\partial T}{\partial
u}=\frac{\partial C}{\partial T}.$ Thus the interaction-driven Grüneisen
parameter Yu:2020  defined by $\Gamma_{int}=\frac{u}{C_{v}}\frac{\partial
C}{\partial T}$ quantifies the efficiency of interaction-driven refrigeration.
Near a critical point, local maximum of the entropy leads to a local
temperature minimum in an isentropic process, and using the condition
$\frac{\partial C}{\partial T}=0$, we have
$\frac{1}{2}\mathrm{Li}_{\frac{1}{2}}\left(-\mathrm{e}^{x}\right)-x\mathrm{Li}_{-\frac{1}{2}}\left(-\mathrm{e}^{x}\right)=0$
that gives a general solution $x\equiv:\alpha_{u}\Delta u/T\approx 1.3117$.
Using the free energy Eq. (2), we can obtain the explicit expression of the
maximum entropy near the transition point from II to IV
$s_{c1}\approx\lambda_{1}\pi^{1/2}\sigma_{1}(0)(\varepsilon^{{}^{\prime\prime}}_{1}(0)/2)^{-1/2}T_{c1}^{1/2}$,
where
$\lambda_{1}=x\mathrm{Li}_{1/2}\left(-\mathrm{e}^{x}\right)-3/2\mathrm{Li}_{3/2}\left(-\mathrm{e}^{x}\right)\approx
1.3467$. Similarly, for phase transition from the Mott phase V to the phase
IV, the maximum entropy is given by
$s_{c2}\approx\lambda_{1}\pi^{1/2}\rho(\pi)(-\kappa^{{}^{\prime\prime}}(\pi)/2)^{-1/2}T_{c2}^{1/2}$,
where $\rho(\pi)$ denotes the charge density at $k=\pi$ and
$\kappa^{{}^{\prime\prime}}(\pi)\equiv\left.\frac{d^{2}\kappa}{dk^{2}}\right|_{k=\pi}$,
with $\kappa(k)$ the charge dressed energy SM . On the other hand, the entropy
in the Luttinger liquid phases TTLC and TTLS are given by $s_{L1}=\pi
T_{L1}/{3v_{c}}$ and $s_{L2}=\pi T_{L2}/{3v_{s}}$, respectively. Therefore
through an interaction-driven refrigeration cycle near phase transitions from
II to IV and from V to IV in the $T$-$u$ plane, we can show that the reachable
minimum temperatures are given by

$\displaystyle\frac{T_{c1}^{1/2}}{T_{L1}}$ $\displaystyle=$
$\displaystyle\frac{\pi^{1/2}(\varepsilon^{{}^{\prime\prime}}_{1}(0)/2)^{1/2}}{3\lambda_{1}v_{c}\sigma_{1}(0)},$
(7) $\displaystyle\frac{T_{c2}^{1/2}}{T_{L2}}$ $\displaystyle=$
$\displaystyle\frac{\pi^{1/2}(-\kappa^{{}^{\prime\prime}}(\pi)/2)^{1/2}}{3\lambda_{1}v_{s}\rho(\pi)},$
(8)

respectively. The minimum temperature in the $T$-$u$ plane is governed by the
relation $\alpha_{u}\Delta u/T\approx 1.3117$ SM . We remark that efficient
cooling in lattice is a significant experimental challenge in ultracold atomic
gases, the lack of which poses as a roadblock for realizing some exotic
quantum phases.

On the other hand, the other two relations in Eqs. (S94) provide essential
insights for charge (IV-V) and spin (II-IV) phase transitions, respectively.
Using these, we find two useful relations among the parameters
$\alpha_{u,\mu,B}$ in Eq. (3)

$\frac{\alpha_{u}}{\alpha_{\mu}}=-\frac{\partial\mu}{\partial
u},\quad\frac{\alpha_{u}}{\alpha_{B}}=-\frac{\partial B}{\partial u},$ (9)

that provide us deep insights into the quantum criticality driven by dynamical
interaction and external poentials. For example, Fig. 4(b1) shows the phase
transition from Phase IV to the Mott phase V in the $\mu-u$ plane, where
$\frac{\partial\mu}{\partial u}$ is the slope along the transition line $n=1$.
From Eq. (9) with fixed $B_{c}=0.82714$ around $u_{c}=1,\,\mu_{c}=-0.8272$ in
Fig. 4(b1), we may numerically get $\alpha_{u}\approx-1.9627$. With this and
using the scaling form of free energy given in SM , we can obtain the scaling
behaviour of the compressibility in terms of $u$, which is in excellent
agreement with numerical calculation from TBA equations, see Fig. 4(b2).

Summary – We have presented new rigorous results of the 1D repulsive Hubbard
model. We focus on the interaction-driven quantum criticality which has been
largely ignored in previous studies. We studied the fractional excitations
from which the SILL regime is identified and carefully investigated. We
introduced several Contact susceptibilities and show how they provide crucial
new insights into the system. Finally, we proposed a quantum cooling scheme
based on the interaction-driven refrigeration cycle, which can potentially
open up new avenues of research in reaching unprecedented low temperatures in
lattice quantum gases. We note that some of the key concepts developed here
will hold true in higher dimensions.

## Acknowledgement

J.J.L and X.W.G. is supported by the NSFC key grant No. 12134015, the NSFC
grant No. 11874393 and No. 12121004. H.P. acknowledges support from the US NSF
(PHY-2207283) and the Welch Foundation (Grant No. C-1669).

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## Supplementary material: Spin incoherent liquid and interaction-driven
criticality in 1D Hubbard model

Jia-Jia Luo, Han Pu, Xi-Wen Guan

##  I The Hubbard model

The Hamiltonian of 1D Hubbard model is

$H=-t\sum_{j=1}^{L}\sum_{a=\uparrow\downarrow}\left(c_{j,a}^{\dagger}c_{j+1,a}+c_{j+1,a}^{\dagger}c_{j,a}\right)+u\sum_{j=1}^{L}(1-2n_{j\uparrow})(1-2n_{j\downarrow})-\mu\hat{N}-2B\hat{S}_{z},$
(S1)

where $c_{j,a}^{\dagger}$ and $c_{j,a}$ are creation and annihilation
operators of electrons with spin $a(\uparrow,\downarrow)$ situated in site $j$
on a 1D lattice of length $L$. $t$, $\mu$ and $B$ are hopping amplitude,
chemical potential and magnetic field, respectively.
$n_{ja}=c_{j,a}^{\dagger}c_{j,a}$ is particle density operator,
$\hat{N}=\sum_{j=1}^{L}(n_{j\uparrow}+n_{j\downarrow})$ is the total number
particle operator, and
$\hat{S}_{z}=\frac{1}{2}\sum_{j=1}^{L}(n_{j\uparrow}-n_{j\downarrow})$ is the
magnetization operator. $u$ denotes interaction strength and here, we focus on
repulsive case $u>0$. In later analysis the hopping strength $t$ and
$k_{B},\,h$ are set to unity. Thus, all thermodynamic quantities are
dimensionless in all calculations.

For vanishing external potentials ($\mu=0,\,B=0$) and even lattice sites,
Hamiltonian (S1) possesses $SO(4)\cong SU(2)\times SU(2)/\mathbb{Z}_{2}$
symmetry, preserving spin rotational and $\eta$-pairing symmetries, i.e.
${[H,S^{\alpha}]}=0$, $[H,\eta^{\alpha}]=0$ with $\alpha=x,y,z$, where the
spin and $\eta$-pair operators are given in Ref. Ess05 ,

$\displaystyle S^{\alpha}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{j=1}^{L}\sum_{a,b=\uparrow,\downarrow}c_{j,a}^{\dagger}(\sigma^{\alpha})|_{b}^{a}c_{j,b},$
(S2) $\displaystyle\eta^{x}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\sum_{j=1}^{L}(-1)^{j}\left(c_{j,\uparrow}^{\dagger}c_{j,\downarrow}^{\dagger}+c_{j,\uparrow}c_{j,\downarrow}\right),$
(S3) $\displaystyle\eta^{y}$ $\displaystyle=$
$\displaystyle\frac{i}{2}\sum_{j=1}^{L}(-1)^{j}\left(c_{j,\uparrow}^{\dagger}c_{j,\downarrow}^{\dagger}-c_{j,\uparrow}c_{j,\downarrow}\right),$
(S4) $\displaystyle\eta^{z}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(N-L\right).$ (S5)

In terms of these two discrete symmetries, the spin and the charge degrees of
freedom can be transformed into each other in some ways, rendering fractional
spin and charge excitations characterized by $S^{z}$ and $\eta^{z}$.

The Bethe ansatz equations for the 1D Hubbard model was derived by Lieb and Wu
Lieb:1968 . Here we write down explicitly the Bethe ansastz equations in terms
of Takahashi Takahashi:1972  the root patterns, i.e. real $k$, length-$n$
$\Lambda$ string composed of $n$ spin-down electrons, length-$m$ $k$-$\Lambda$
string containing $m$ down-spin and $m$ up-spin particles. As being introduced
in the main text, let us denote $M_{e},\,M_{n},\,M^{\prime}_{n}$ respectively
the numbers of real $k$, length-$n$ $\Lambda$ and length-$n$ $k-\Lambda$
strings, thus the total particle number $N$ and spin down electron number $M$
are given by Ess05  $N=\mathcal{M}_{e}+\sum_{n=1}^{\infty}2nM_{n}^{\prime}$
and $M=\sum_{n=1}^{\infty}n\left(M_{n}+M_{n}^{\prime}\right)$, respectively.
We further denote $\rho^{p},\sigma_{n}^{p},\sigma_{n}^{\prime p}$
($\rho^{h},\sigma_{n}^{h},\sigma_{n}^{\prime h}$) as the densities for
particles (holes) of $k,\,\Lambda,k-\Lambda$ sector, the root distributions of
these types of strings are given by Takahashi:1972

$\displaystyle\rho^{p}(k)+\rho^{h}(k)$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi}+\cos
k\sum_{n=1}^{\infty}\int_{-\infty}^{\infty}\mathrm{d}\Lambda
a_{n}(\Lambda-\sin k)\left(\sigma_{n}^{\prime
p}(\Lambda)+\sigma_{n}^{p}(\Lambda)\right),$ (S6)
$\displaystyle\sigma_{n}^{h}(\Lambda)$ $\displaystyle=$
$\displaystyle-\sum_{m=1}^{\infty}A_{nm}*\sigma_{m}^{p}(\Lambda)+\int_{-\pi}^{\pi}\mathrm{d}ka_{n}(\sin
k-\Lambda)\rho^{p}(k),$ (S7) $\displaystyle\sigma_{n}^{\prime h}(\Lambda)$
$\displaystyle=$
$\displaystyle\frac{1}{\pi}\operatorname{Re}\frac{1}{\sqrt{1-(\Lambda-
inu)^{2}}}-\sum_{m=1}^{\infty}A_{nm}*\sigma_{m}^{\prime
p}(\Lambda)-\int_{-\pi}^{\pi}\mathrm{d}ka_{n}(\sin k-\Lambda)\rho^{p}(k),$
(S8)

where $a_{n}(x)=\frac{1}{2\pi}\frac{2nu}{(nu)^{2}+x^{2}}$, and * stands for
the convolution

$A_{nm}*f|_{x}=\int_{-\infty}^{\infty}\frac{\mathrm{d}y}{2\pi}\frac{\mathrm{d}}{\mathrm{d}x}\Theta_{nm}\left(\frac{x-y}{u}\right)f(y).$
(S9)

The derivation of the function $\Theta_{nm}$ reads

$\frac{1}{2\pi}\frac{\mathrm{d}}{\mathrm{d}x}\Theta_{nm}\left(\frac{x-y}{u}\right)=\left\\{\begin{array}[]{l}a_{|n-m|}\left(x-y\right)+2a_{|n-m|+2}\left(x-y\right)+\cdots+2a_{n+m-2}\left(x-y\right)+a_{n+m}\left(x-y\right),\text{
if }n\neq m\\\
2a_{2}\left(x-y\right)+2a_{4}\left(x-y\right)+\cdots+2a_{2n-2}\left(x-y\right)+a_{2n}\left(x-y\right),\text{
if }n=m\end{array}\right..$ (S10)

The above equations can be used to study ground state properties and
excitations in spin and charge.

Building on the above Takahashi string solutions Takahashi:1972 , finite
temperature thermodynamics of the model can be obtained from the TBA equations
in terms of the dressed energies
$\kappa(k),\varepsilon_{n}(\Lambda),\varepsilon_{n}^{\prime}(\Lambda)$,
associated with the product of the logarithm of the ratio of hole density to
particle density and temperature
$\kappa=T\ln\left(\frac{\rho^{h}}{\rho^{p}}\right),\varepsilon_{n}=T\ln\left(\frac{\sigma_{n}^{h}}{\sigma_{n}^{p}}\right),\varepsilon_{n}^{\prime}=T\ln\left(\frac{\sigma_{n}^{\prime
h}}{\sigma_{n}^{\prime p}}\right)$, namely, explicit TBA equations are given
by

$\displaystyle\kappa(k)$ $\displaystyle=$ $\displaystyle-2\cos
k-\mu-2u-B+\sum_{n=1}^{\infty}\int_{-\infty}^{\infty}\mathrm{d}\Lambda
a_{n}(\sin
k-\Lambda)T\ln\left(1+\mathrm{e}^{-\frac{\varepsilon_{n}^{\prime}(\Lambda)}{T}}\right)$
$\displaystyle-\sum_{n=1}^{\infty}\int_{-\infty}^{\infty}\mathrm{d}\Lambda
a_{n}(\sin
k-\Lambda)\ln\left(1+\mathrm{e}^{-\frac{\varepsilon_{n}(\Lambda)}{T}}\right),$
$\displaystyle\varepsilon_{n}(\Lambda)$ $\displaystyle=$ $\displaystyle
2nB-\int_{-\pi}^{\pi}\mathrm{d}k\cos ka_{n}(\sin
k-\Lambda)T\ln\left(1+\mathrm{e}^{-\frac{\kappa(k)}{T}}\right)+\sum_{m=1}^{\infty}A_{nm}*T\ln\left(1+\mathrm{e}^{-\frac{\varepsilon_{m}(\Lambda)}{T}}\right),$
(S12) $\displaystyle\varepsilon_{n}^{\prime}(\Lambda)$ $\displaystyle=$
$\displaystyle 4\textmd{Re}\sqrt{1-(\Lambda-
inu)^{2}}-2n\mu-4nu-\int_{-\pi}^{\pi}\mathrm{d}k\cos ka_{n}(\sin
k-\Lambda)T\ln\left(1+\mathrm{e}^{-\frac{\kappa(k)}{T}}\right)$
$\displaystyle+\sum_{m=1}^{\infty}A_{nm}*T\ln\left(1+\mathrm{e}^{-\frac{\varepsilon_{m}^{\prime}(\Lambda)}{T}}\right).$

The Gibbs free energy per site is given in terms of the dressed energies

$\displaystyle f$ $\displaystyle=$
$\displaystyle-T\int_{-\pi}^{\pi}\frac{\mathrm{d}k}{2\pi}\ln\left(1+\mathrm{e}^{-\frac{\kappa(k)}{T}}\right)+u$
(S14)
$\displaystyle-T\sum_{n=1}^{\infty}\int_{-\infty}^{\infty}\frac{\mathrm{d}\Lambda}{\pi}\operatorname{Re}\frac{1}{\sqrt{1-(\Lambda-
inu)^{2}}}\ln\left(1+\mathrm{e}^{-\frac{\varepsilon_{n}^{\prime}(\Lambda)}{T}}\right).$

From Eq. (S14), thermal and magnetic properties of the model can be obtained
by using the standard statistical relations.

In the limit of zero temperature, the dressed energy of bulk kinds and lengths
of strings is always positive in the parameter space. Consequently, they offer
no contributions to the integrals due to the logarithm function in the
integral kernel. The only survivals are $k$ and length-1 $\Lambda$ string, and
the values of arguments $k,\Lambda$ should be cut off at points where dressed
energies switch the sign. Thereby, coupled equations for densities and dressed
energiesLieb:1968  are greatly simplified

$\displaystyle\kappa(k)$ $\displaystyle=$
$\displaystyle-2\cos{k}-\mu-2u-B+\int_{-A}^{A}\mathrm{d}\Lambda
a_{1}(\sin{k}-\Lambda)\varepsilon_{1}(\Lambda),$ (S15)
$\displaystyle\varepsilon_{1}(\Lambda)$ $\displaystyle=$ $\displaystyle
2B+\int_{-Q}^{Q}\mathrm{d}k\cos
ka_{1}(\sin{k}-\Lambda)\kappa(k)-\int_{-A}^{A}\mathrm{d}\Lambda^{\prime}a_{2}\left(\Lambda-\Lambda^{\prime}\right)\varepsilon_{1}(\Lambda^{\prime}).$
(S16)

And total quantities
$\rho=\rho^{p}+\rho^{k},\sigma_{1}=\sigma_{1}^{p}+\sigma_{1}^{h}$ can be
recast in a more concise configuration

$\displaystyle\rho(k)$ $\displaystyle=$ $\displaystyle\frac{1}{2\pi}+\cos
k\int_{-A}^{A}\mathrm{d}\Lambda a_{1}(\Lambda-\sin k)\sigma_{1}(\Lambda),$
(S17) $\displaystyle\sigma_{1}(\Lambda)$ $\displaystyle=$
$\displaystyle-\int_{-A}^{A}\mathrm{d}\Lambda^{\prime}a_{2}(\Lambda-\Lambda^{\prime})\sigma_{1}(\Lambda^{\prime})+\int_{-Q}^{Q}\mathrm{d}ka_{1}(\sin
k-\Lambda)\rho(k).$ (S18)

In general, The momentum of paticles for $|k|\leq Q,|\Lambda|\leq A$ or holons
if $|k|>Q,|\Lambda|>A$ with relative parameter
$k_{j},\Lambda_{\alpha}^{n},\Lambda_{\alpha}^{\prime n}$ can be calculated
through

$\displaystyle p(k)=\frac{2\pi
I_{k}}{L}=2\pi\int_{0}^{k}\mathrm{d}k^{\prime}\rho\left(k^{\prime}\right),$
(S19) $\displaystyle p_{1}(\Lambda)=\frac{2\pi
J_{1}}{L}=2\pi\int_{0}^{\Lambda}\mathrm{d}\Lambda^{\prime}\sigma_{1}\left(\Lambda^{\prime}\right),$
$\displaystyle p_{n}(\Lambda)=\frac{2\pi
J_{n}}{L}=2\pi\int_{0}^{\Lambda}\mathrm{d}\Lambda^{\prime}\sigma_{n}^{h}\left(\Lambda^{\prime}\right),\
n\geq 2,$ $\displaystyle p_{n}^{\prime}(\Lambda)=\frac{2\pi
J_{n}^{\prime}}{L}=-2\pi\int_{0}^{\Lambda}\mathrm{d}\Lambda^{\prime}\sigma_{n}^{\prime
h}\left(\Lambda^{\prime}\right)+\pi(n+1),\ n\geq 1,$

which will be used to study elementary excitations.

The aim of this Supplementary Materials is to provide derivations of the key
results which are for the first time to be reported in our main text, i.e.
analytical results of spin incoherent Luttinger liquid, asymptotic of
correlation functions, interaction-driven criticality, as well as the lattice
Contact susceptibilities and quantum cooling.

##  II Phase diagram

By the fact that thermodynamic properties feature dramatic fluctuations around
quantum critical point (QCP), the ground state phase diagram can be
characterized by dimensionless parameters grouped with thermodynamic
quantities at low temperature. One dimensionless quantity is Wilson ratio

$R^{\chi_{s}}_{w}=\frac{4}{3}\left(\frac{\pi
k_{B}}{\mu_{B}g}\right)^{2}\frac{\chi_{s}}{C_{v}/T},$ (S20)

which characterizes the nature of the TLLs, remarkably map out the full phase
diagram of the 1D repulsive Hubbard model, consisting of five phases, see the
Fig.1 in main text. A significant aspect of these ratio is that it is
approximate constant away from the critical point. In Ref. LPG:preparation ,
we analytically calculate relevant thermodynamic quantities in all TLL and
quantum criticality regions. The numerical simulation in Fig.1 of main text
and analytical results in Ref. LPG:preparation  both show that the Wilson
ratio in these five phases have the values of

$\displaystyle\text{II:}\quad R^{\chi_{s}}_{w}$ $\displaystyle\approx$
$\displaystyle 2,$ (S21) $\displaystyle\text{IV:}\quad R^{\chi_{s}}_{w}$
$\displaystyle>$ $\displaystyle 2,$ (S22) $\displaystyle\text{V:}\quad
R^{\chi_{s}}_{w}$ $\displaystyle>$ $\displaystyle 4,$ (S23)
$\displaystyle\text{I and III:}\quad R^{\chi_{s}}_{w}$ $\displaystyle=$
$\displaystyle 0.$ (S24)

Moreover, in terms of bosonization results of the magnetic susceptibility,
specific heat, Luttinger parameter and velocity, the Wilson ratios are given
by

$\displaystyle\text{II:}\quad R^{\chi_{s}}_{w}$ $\displaystyle\approx$
$\displaystyle 2,$ (S25) $\displaystyle\text{IV:}\quad R^{\chi_{s}}_{w}$
$\displaystyle\approx$ $\displaystyle 2v_{c}K_{s}/(v_{s}+v_{c}),$ (S26)
$\displaystyle\text{V:}\quad R^{\chi_{s}}_{w}$ $\displaystyle\approx$
$\displaystyle 4K_{s},$ (S27) $\displaystyle\text{I and III:}\quad
R^{\chi_{s}}_{w}$ $\displaystyle=$ $\displaystyle 0,$ (S28)

where $K_{s}$ is the Luttinger parameter for spin, and $v_{c,s}$ are sound
velocities for charge and spin, respectively. However, the $K_{s}$ cannot be
completely determined at finite magnetic field using bosonination. From
bosonination theory, we only know that spin Luttinger parameter $K_{s}>1$ for
repulsive Hubbard model at finite magnetic field. Therefore, the numerical
results are in agreement with the bosonination theory.

On the other hand, Wilson ratio gives rise to a dimensionless scaling function
in the quantum critical region, see our later calculation with the
eqs.(S42)-(S46):

$R^{\chi_{s}}_{w}\approx\frac{2\pi^{2}\alpha^{2}_{B}}{3}\frac{\mathrm{Li}_{-\frac{1}{2}}\left(-\mathrm{e}^{x}\right)}{\frac{3}{4}\mathrm{Li}_{\frac{3}{2}}\left(-\mathrm{e}^{x}\right)-x\mathrm{Li}_{\frac{1}{2}}\left(-\mathrm{e}^{x}\right)+x^{2}\mathrm{Li}_{-\frac{1}{2}}\left(-\mathrm{e}^{x}\right)},$
(S29)

where the meaning of $\alpha_{B},x$ can be seen from eq.(S63). The spin-
incoherent Luttinger liquid will be shown to occur near a transition of from
phase IV to II in dilute limit, i.e. a crossover region between TLL phase
boundary and the critical temperature in the $T-u$ or $T-\mu$ plane. The value
of Wilson ratio is in a range between the ones determined by the eq.(S26) and
eq.(S29).

###  II.1 Two-holon excitations and antiholon-spinon excitations

Flipping one spin-down particle upon the ground state will result in two-
spinon fractional excitation. Together with particle-hole excitation of
charge, these excitation modes exhibit spin-charge separation with quantum
numbers $\left(\Delta\eta^{z},\Delta S^{z}\right)=(0,1)$. Therefore, each of
the two spinons carry no charge but spin-$\frac{1}{2}$. Meanwhile, in the
charge sector, one particle can be removed from inside to outside the Fermi
surface, corresponding to particle-hole excitation with quantum numbers
$\left(\Delta\eta^{z},\Delta S^{z}\right)=(0,0)$. In the long wavelength
limit, i.e. $\Delta K\to 0$, both charge and spin excitations exhibit linear
dispersion: $\Delta E_{c,s}=v_{c,s}\hbar|\Delta K|$. Hence, the slopes of such
spin and charge low-energy excitations determine the velocities of the spin
and charge density waves, respectively. The energies and momenta for these two
two-parametric excitations are

$\displaystyle E_{ph}$ $\displaystyle=$
$\displaystyle\kappa(k_{p})-\kappa(k_{h}),\quad P_{ph}=p(k_{p})-p(k_{h}),$
(S30) $\displaystyle E_{ss}$ $\displaystyle=$
$\displaystyle-\varepsilon_{1}(\Lambda_{h1})-\varepsilon_{1}(\Lambda_{h2}),\quad
P_{ss}=-p_{1}(\Lambda_{h1})-p_{1}(\Lambda_{h2})+\pi n_{c},$ (S31)

where $k_{p},\,k_{h}$ are the positions of particle and hole in charge degrees
of freedom, $\Lambda_{h1},\,\Lambda_{h2}$ denote two hole positions in the
spin sector, $\pi n_{c}$ comes from the parity change. To obtain these two-
parametric spectra, firstly fix one of the components at a certain value, and
plot the single particle excitation corresponding to the other component. Then
let the fixed value change, and repeat the previous step to draw a single
particle excitation for another quantity. Repeating this procedure again and
agin until finally the values of every components are combined to obtain a
complete excitation spectra in the first Brillouin zone, see Fig.2 in the main
text. In such elementary excitaions, spin and charge modes are fully separated
and characterised by different $v_{c},v_{s}$. Especially, in the low-density
limit, spin sector is highly suppressed. Thus at certain finite temperatures,
one situation can occur: the temperature is larger than the characteristic
energy scale of spin $E_{s}\sim k_{F}v_{s}$, but still lower than typical
energy scale of charge $E_{c}\sim k_{F}v_{s}$, such that the collective
excitation still remains in the charge sector, but no such collective motion
in the spin sector, as reflected in Fig.2 in the main text. Here the
temperature scale is beyond the typical energy scale in spin, rendering the
spin as highly excited state. This case is termed as the spin incoherent
Luttinger liquid (SILL). A comprehensive description of the SILL has been
theoretically studied in terms of bosonization and field theory, see
Fiete:2007 ; Cheianov:2004 .

Let us consider the antiholon-spinon excitation by adding a particle with spin
up or down over the ground state. These two cases are classified by the charge
and spin magnetization values $(\Delta\eta^{z},\Delta
S^{z})=(\frac{1}{2},-\frac{1}{2})$ and $(\frac{1}{2},\frac{1}{2})$, indicating
the emergence of fractional excitations of charge and spin originated from the
discrete symmetries characterised by the spin flip and Shiba transformation
Ess05 . Note that existence of charge fractional excitation is unique in
lattice models and absent in continuous systems. With the help of eqs.(S19),
the excitation spectra are given by

$\displaystyle E_{\overline{h}\overline{s}}$ $\displaystyle=$
$\displaystyle\kappa(k_{p})+\varepsilon_{1}(\Lambda_{p}),\quad
P_{\overline{h}\overline{s}}=p(k_{p})+p_{1}(\Lambda_{p})+\pi n_{c},$ (S32)
$\displaystyle E_{\overline{h}s}$ $\displaystyle=$
$\displaystyle\kappa(k_{p})-\varepsilon_{1}(\Lambda_{h}),\quad
P_{\overline{h}s}=p(k_{p})-p_{1}(\Lambda_{h}),$ (S33)

where $k_{p}$ is the quasimomentum of the added particle in charge degree,
$\Lambda_{p},\Lambda_{h}$ denote particle and hole quasimomenta in spin
sector, $\pi n_{c}$ comes from the parity change. We draw the excitation
spectra following the same as the method used for obtaining the two-spinon
excitation. We observe that these excitations correspond to spin-charge
scattering state, which strongly coupled together and outside of the TLL
regime. It is of importance to experimentally measure such excitations via
photoemission experiment like Angle Resolved Photoemission Spectroscopy.

##  III Spin-charge separation in low temperature

Under the condition of $B/T\gg 1$ and $|\mu|/T\gg 1$, the TBA equations can be
simplified to the following coupled equations for real $k$ and length-$1$
$\Lambda$ string

$\displaystyle\kappa(k)$ $\displaystyle=$
$\displaystyle-2\cos{k}-\mu-2u-B-\int_{-\infty}^{\infty}\mathrm{d}\Lambda
a_{1}(\sin{k}-\Lambda)T\ln(1+\mathrm{e}^{-\frac{\varepsilon_{1}(\Lambda)}{T}}),$
(S34) $\displaystyle\varepsilon_{1}(\Lambda)$ $\displaystyle=$ $\displaystyle
2B-\int_{-\pi}^{\pi}\mathrm{d}k\cos
ka_{1}(\sin{k}-\Lambda)T\ln(1+\mathrm{e}^{-\frac{\kappa(k)}{T}})$ (S35)
$\displaystyle+\int_{-\infty}^{\infty}\mathrm{d}\Lambda^{\prime}a_{2}\left(\Lambda-\Lambda^{\prime}\right)T\ln(1+\mathrm{e}^{-\frac{\varepsilon_{1}(\Lambda^{\prime})}{T}}),$

and the free energy is given by

$f=-\int_{-\pi}^{\pi}\frac{\mathrm{d}k}{2\pi}T\ln(1+\mathrm{e}^{-\frac{\kappa(k)}{T}})+u.$
(S36)

By observing the structure of the expressions, the main challenge is the
simplification of the integral terms, and one way to deal with these integrals
is using Sommerfeld expansion. Denote the fermi point of
$\kappa(k),\varepsilon(\Lambda)$ as $Q,A$ which satisfy
$\kappa(Q)=0,\varepsilon(A)=0$. Define $\frac{\partial{\bar{a}_{1}(\sin
k,\Lambda)}}{\partial{\Lambda}}=a_{1}(\sin k-\Lambda)$,
$\frac{\partial{\bar{a}(\sin k,\Lambda)}}{\partial{\Lambda}}=a_{1}(\sin
k-\Lambda)+a_{1}(\sin k+\Lambda)$ and the second term in eq.(S34) is given by

$\displaystyle-\int_{-\infty}^{\infty}\mathrm{d}\Lambda
a_{1}(\sin{k}-\Lambda)T\ln(1+\mathrm{e}^{-\frac{\varepsilon_{1}(\Lambda)}{T}})$
(S37) $\displaystyle=$ $\displaystyle-\bar{a}_{1}(\sin
k,\Lambda)T\ln(1+\mathrm{e}^{-\frac{\varepsilon_{1}(\Lambda)}{T}})|_{-\infty}^{\infty}-\int_{-\infty}^{\infty}\mathrm{d}\Lambda\bar{a}_{1}(\sin{k},\Lambda)\frac{1}{1+\mathrm{e}^{\frac{\varepsilon_{1}(\Lambda)}{T}}}\frac{\varepsilon_{1}(\Lambda)}{\partial{\Lambda}}$
$\displaystyle=$
$\displaystyle-\int_{0}^{\infty}\mathrm{d}\Lambda\bar{a}(\sin{k},\Lambda)\frac{1}{1+\mathrm{e}^{\frac{\varepsilon_{1}(\Lambda)}{T}}}\frac{\varepsilon_{1}(\Lambda)}{\partial{\Lambda}}$
$\displaystyle=$
$\displaystyle-\int_{\varepsilon_{1}(0)}^{\varepsilon_{1}(\infty)}\mathrm{d}\varepsilon_{1}\bar{a}(\sin{k},\Lambda(\varepsilon_{1}))\frac{1}{1+\mathrm{e}^{\frac{\varepsilon_{1}}{T}}}$
$\displaystyle=$
$\displaystyle-\int_{\frac{\varepsilon_{1}(0)}{T}}^{\frac{\varepsilon_{1}(A)}{T}}\mathrm{d}xT\bar{a}(\sin{k},\Lambda(Tx))\frac{1}{1+\mathrm{e}^{x}}-\int_{\frac{\varepsilon_{1}(A)}{T}}^{\frac{\varepsilon_{1}(\infty)}{T}}\mathrm{d}xT\bar{a}(\sin{k},\Lambda(Tx))\frac{1}{1+\mathrm{e}^{x}}$
$\displaystyle=$
$\displaystyle-\int_{\frac{\varepsilon_{1}(0)}{T}}^{\frac{\varepsilon_{1}(A)}{T}}\mathrm{d}xT\bar{a}(\sin{k},\Lambda(Tx))\left(1-\frac{1}{1+\mathrm{e}^{-x}}\right)-\int_{\frac{\varepsilon_{1}(A)}{T}}^{\frac{\varepsilon_{1}(\infty)}{T}}\mathrm{d}xT\bar{a}(\sin{k},\Lambda(Tx))\frac{1}{1+\mathrm{e}^{x}}$
$\displaystyle=$
$\displaystyle-\int_{\frac{\varepsilon_{1}(0)}{T}}^{0}\mathrm{d}xT\bar{a}(\sin{k},\Lambda(Tx))+\int_{0}^{-\frac{\varepsilon_{1}(0)}{T}}\mathrm{d}xT\bar{a}(\sin{k},\Lambda(-Tx))\frac{1}{1+\mathrm{e}^{x}}-\int_{0}^{\frac{\varepsilon_{1}(\infty)}{T}}\mathrm{d}xT\bar{a}(\sin{k},\Lambda(Tx))\frac{1}{1+\mathrm{e}^{x}}$
$\displaystyle=$
$\displaystyle-\int_{\varepsilon_{1}(0)}^{0}\mathrm{d}\varepsilon_{1}\bar{a}(\sin{k},\Lambda(\varepsilon_{1}))+\int_{0}^{\infty}\mathrm{d}xT\frac{\bar{a}(\sin{k},\Lambda(-Tx))-\bar{a}(\sin{k},\Lambda(Tx))}{1+\mathrm{e}^{x}}$
$\displaystyle=$
$\displaystyle\int_{\varepsilon_{1}(0)}^{0}\mathrm{d}\varepsilon_{1}\frac{\partial{\Lambda}}{\partial{\varepsilon_{1}}}\varepsilon_{1}(a_{1}(\sin{k}-\Lambda)+a_{1}(\sin{k}+\Lambda))-\int_{0}^{\infty}\mathrm{d}xT\frac{(a_{1}(\sin{k}-\Lambda(0))+a_{1}(\sin{k}+\Lambda(0)))\Lambda^{{}^{\prime}}(0)2Tx}{1+\mathrm{e}^{x}}$
$\displaystyle=$
$\displaystyle\int_{0}^{A}\mathrm{d}\Lambda\varepsilon_{1}(\Lambda)(a_{1}(\sin{k}-\Lambda)+a_{1}(\sin{k}+\Lambda))-2T^{2}(a_{1}(\sin{k}-A)+a_{1}(\sin{k}+A))\Lambda^{{}^{\prime}}(0)\frac{\pi^{2}}{12}$
$\displaystyle=$
$\displaystyle\int_{0}^{A}\mathrm{d}\Lambda\varepsilon_{1}(\Lambda)(a_{1}(\sin{k}-\Lambda)+a_{1}(\sin{k}+\Lambda))-\frac{\pi^{2}T^{2}}{6}\frac{1}{\varepsilon_{1}^{{}^{\prime}}(A)}(a_{1}(\sin{k}-A)+a_{1}(\sin{k}+A)).$

In the above calculations, the following equations are used:

$\displaystyle\frac{\bar{a}(\sin{k},\Lambda(Tx))-\bar{a}(\sin{k},\Lambda(-Tx))}{2Tx}=\frac{\partial{\bar{a}}}{\partial{\Lambda}}|_{\Lambda=\Lambda(0)}\Lambda^{{}^{\prime}}(0),$
$\displaystyle\Lambda^{{}^{\prime}}(0)=\frac{1}{\frac{d\varepsilon_{1}}{d\Lambda}|_{\varepsilon_{1}=0}}=\frac{1}{\varepsilon^{{}^{\prime}}_{1}(A)},\qquad\int_{0}^{\infty}\mathrm{d}x\frac{x}{1+\mathrm{e}^{x}}=\frac{\pi^{2}}{12}.$
(S38)

Conducting similar manipulations in the integrals of
$\varepsilon_{1}(\Lambda)$, the dressed energies can be converted to

$\displaystyle\kappa(k)$ $\displaystyle=$ $\displaystyle-2\cos{k}-\mu-2u-B$
(S39)
$\displaystyle+\int_{-A}^{A}\mathrm{d}\Lambda\varepsilon_{1}(\Lambda)a_{1}(\sin{k}-\Lambda)-\frac{\pi^{2}T^{2}}{6}\frac{1}{\varepsilon_{1}^{{}^{\prime}}(A)}(a_{1}(\sin{k}-A)+a_{1}(\sin{k}+A)),$
$\displaystyle\varepsilon_{1}(\Lambda)$ $\displaystyle=$ $\displaystyle
2B-\frac{\pi^{2}T^{2}}{6\kappa^{{}^{\prime}}(Q)}\cos
Q(a_{1}(\sin{Q}-\Lambda)+a_{1}(\sin{Q}+\Lambda))+\frac{\pi^{2}T^{2}}{6\varepsilon_{1}^{{}^{\prime}}(A)}(a_{2}(\Lambda-A)+a_{2}(\Lambda+A))$
(S40) $\displaystyle+\int_{-Q}^{Q}\mathrm{d}k\kappa(k)\cos
ka_{1}(\sin{k}-\Lambda)-\int_{0}^{A}\mathrm{d}\Lambda^{{}^{\prime}}\varepsilon_{1}(\Lambda^{{}^{\prime}})(a_{2}(\Lambda-\Lambda^{{}^{\prime}})+a_{2}(\Lambda+\Lambda^{{}^{\prime}})).$

Comparing above equations with root densities (S17), (S18), we gain insight
that there is an implicit connection between the two sets of equations which
are highly symmetric. In order to derive a compact and closed form, combine
eq.(S17), (S18) with (S39), (S40), the free energy can be associated with
velocities:

$\displaystyle f$ $\displaystyle=$
$\displaystyle-\int_{-\pi}^{\pi}\frac{\mathrm{d}k}{2\pi}T\ln(1+\mathrm{e}^{-\frac{\kappa(k)}{T}})+u-\frac{\pi^{2}T^{2}}{6\kappa^{{}^{\prime}}(Q)}+\int_{0}^{Q}\frac{\mathrm{d}k}{\pi}\kappa(k)+u$
(S41) $\displaystyle=$
$\displaystyle-\frac{\pi^{2}T^{2}}{6\kappa^{{}^{\prime}}(Q)}+\int_{-Q}^{Q}\mathrm{d}k\left[-2\cos{k}-\mu-2u-B-\frac{\pi^{2}T^{2}}{6}\frac{1}{\varepsilon_{1}^{{}^{\prime}}(A)}(a_{1}(\sin{k}-A)+a_{1}(\sin{k}+A))\right]\rho(k)+u$
$\displaystyle+\int_{-A}^{A}\mathrm{d}\Lambda\left[2B-\frac{\pi^{2}T^{2}}{6\kappa^{{}^{\prime}}(Q)}\cos
Q(a_{1}(\sin{Q}-\Lambda)+a_{1}(\sin{Q}+\Lambda))+\frac{\pi^{2}T^{2}}{6\varepsilon_{1}^{{}^{\prime}}(A)}(a_{2}(\Lambda-A)+a_{2}(\Lambda+A))\right]\sigma_{1}(\Lambda)$
$\displaystyle=$
$\displaystyle-\frac{\pi^{2}T^{2}}{6\kappa^{{}^{\prime}}(Q)}+f_{0}-\frac{\pi^{2}T^{2}}{6\varepsilon_{1}^{{}^{\prime}}(A)}\left[\sigma_{1}(A)+\sigma_{1}(-A)+\int_{-A}^{A}\mathrm{d}\Lambda(a_{2}(\Lambda-A)+a_{2}(\Lambda+A))\sigma_{1}(\Lambda)\right]$
$\displaystyle-\frac{\pi^{2}T^{2}}{6\kappa^{{}^{\prime}}(Q)}\left[\rho(Q)+\rho(-Q)-\frac{1}{\pi}\right]+\frac{\pi^{2}T^{2}}{6\varepsilon_{1}^{{}^{\prime}}(A)}\int_{-A}^{A}\mathrm{d}\Lambda(a_{2}(\Lambda-A)+a_{2}(\Lambda+A))\sigma_{1}(\Lambda)$
$\displaystyle=$ $\displaystyle
f_{0}-\frac{\pi^{2}T^{2}\rho(Q)}{3\kappa^{{}^{\prime}}(Q)}-\frac{\pi^{2}T^{2}\sigma_{1}(A)}{3\varepsilon_{1}^{{}^{\prime}}(A)}$
$\displaystyle=$ $\displaystyle f_{0}-\frac{\pi
T^{2}}{6}\left(\frac{1}{v_{c}}+\frac{1}{v_{s}}\right),$

where
$f_{0}=\int_{-Q}^{Q}\mathrm{d}k\left(-2\cos{k}-\mu-2u-B\right)\rho(k)+2B\int_{-A}^{A}\mathrm{d}\lambda\sigma_{1}(\lambda)+u$
is the background contribution from ground state. $v_{c},v_{s}$ denote the
velocities of charge and spin by the definition
$v_{c}=\frac{\kappa^{{}^{\prime}}(Q)}{2\pi\rho(Q)},v_{s}=\frac{\varepsilon_{1}^{{}^{\prime}}(A)}{2\pi\sigma_{1}(A)}$.
Specific heat is directly derived $C_{v}=\frac{\pi
T}{3}\left(\frac{1}{v_{c}}+\frac{1}{v_{s}}\right)$. In spin polarized band II,
spin degree vanishes. The free energy at low temperature is simplified as
$f=f_{0}-\frac{\pi T^{2}}{6}\frac{1}{v_{c}}$ with specific heat
$C_{v}=\frac{\pi T}{3}\frac{1}{v_{c}}$ accordingly. While in phase V with
charge half filled, the free energy is denoted as $f=f_{0}-\frac{\pi
T^{2}}{6}\frac{1}{v_{s}}$, specific heat $C_{v}=\frac{\pi
T}{3}\frac{1}{v_{s}}$.

##  IV Universal scaling functions near quantum criticality point

At zero temperature, the phase transition occurs when certain degrees of
freedom appears, disappears or reaches saturation. At finite temperature,
these quantum criticality span into the V-shaped critical region. In this
section we show that the free energies at quantum criticality share the
following universal forms:

$\displaystyle\text{I-II: }f$ $\displaystyle=$ $\displaystyle
u+T^{\frac{3}{2}}\pi^{\frac{1}{2}}\rho(0)\left(\frac{\kappa^{{}^{\prime\prime}}(0)}{2}\right)^{-\frac{1}{2}}\mathrm{Li}_{\frac{3}{2}}\left(-\mathrm{e}^{-\frac{\kappa(0)}{T}}\right),$
(S42) $\displaystyle\text{II-III: }f$ $\displaystyle=$ $\displaystyle
f_{0}+T^{\frac{3}{2}}\pi^{\frac{1}{2}}\rho(\pi)\left(\frac{-\kappa^{{}^{\prime\prime}}(\pi)}{2}\right)^{-\frac{1}{2}}\mathrm{Li}_{\frac{3}{2}}\left(-\mathrm{e}^{\frac{\kappa(\pi)}{T}}\right),$
(S43) $\displaystyle\text{V-III: }f$ $\displaystyle=$ $\displaystyle
f_{0}+T^{\frac{3}{2}}\pi^{\frac{1}{2}}\sigma_{1}(0)\left(\frac{\varepsilon^{{}^{\prime\prime}}_{1}(0)}{2}\right)^{-\frac{1}{2}}\mathrm{Li}_{\frac{3}{2}}\left(-\mathrm{e}^{-\frac{\varepsilon_{1}(0)}{T}}\right),$
(S44) $\displaystyle\text{II-IV: }f$ $\displaystyle=$ $\displaystyle
f_{0}-\frac{\pi
T^{2}}{6v_{c}}+T^{\frac{3}{2}}\pi^{\frac{1}{2}}\sigma_{1}(0)\left(\frac{\varepsilon^{{}^{\prime\prime}}_{1}(0)}{2}\right)^{-\frac{1}{2}}\mathrm{Li}_{\frac{3}{2}}\left(-\mathrm{e}^{-\frac{\varepsilon_{1}(0)}{T}}\right),$
(S45) $\displaystyle\text{V-IV: }f$ $\displaystyle=$ $\displaystyle
f_{0}-\frac{\pi
T^{2}}{6v_{s}}+T^{\frac{3}{2}}\pi^{\frac{1}{2}}\rho(\pi)\left(\frac{-\kappa^{{}^{\prime\prime}}(\pi)}{2}\right)^{-\frac{1}{2}}\mathrm{Li}_{\frac{3}{2}}\left(-\mathrm{e}^{\frac{\kappa(\pi)}{T}}\right),$
(S46)

where $f_{0}$ comes from the ground state, the terms with $T^{2}$ reflect the
properties of TLL for noncritical degrees of freedom, $\sigma_{1}(0)$ denotes
the spin density at $\Lambda=0$,
$\varepsilon^{{}^{\prime\prime}}_{1}(0)\equiv\left.\frac{d^{2}\varepsilon_{1}}{d\Lambda^{2}}\right|_{\Lambda=0}$,
$\rho(0),\rho(\pi)$ denotes the charge density at $k=0,\pi$, and
$\kappa^{{}^{\prime\prime}}(0)\equiv\left.\frac{d^{2}\kappa}{dk^{2}}\right|_{k=0},\kappa^{{}^{\prime\prime}}(\pi)\equiv\left.\frac{d^{2}\kappa}{dk^{2}}\right|_{k=\pi}$.
Special function $\mathrm{Li}_{n}$ denotes the polylog function. In this note,
we mainly concentrate on eqs.(S45), (S46). The functions
$-\varepsilon_{1}(0),-\kappa(0),\kappa(\pi)$ are independent of temperature in
the polylog function $\mathrm{Li}_{\frac{3}{2}}$. They depend on the energy
gaps away from the QCPs, i.e.

$\displaystyle-\varepsilon_{1}(0),-\kappa(0),\kappa(\pi)$
$\displaystyle\approx$ $\displaystyle\alpha_{B}\Delta
B+\alpha_{\mu}\Delta\mu+\alpha_{u}\Delta u,$ (S47)

where $\Delta B=B-B_{c},\Delta\mu=\mu-\mu_{c},\Delta u=u-u_{c}$ denote the
distances away from QCPs $(B_{c},\mu_{c},u_{c})$; The factors
$\alpha_{(B,\mu,u)}$ in front of $\Delta(B,\mu,u)$ represent the paths across
the QCPs. These expressions eqs.(S42)-(S46) are universal scaling forms for
the second order phase transitions associated with the critical dynamical
experiment $z=2$ and correlation length exponent $\nu=1/2$. The coefficients
$\alpha_{B,\mu,u}$ can be derived from the free energy and the low temperature
TBA equations. Here we just give the proof for eqs.(S45), (S46).

###  IV.1 The transition of IV-II:

In the vicinity of the phase transition from phase IV to phase II, the spin
degrees of freedom vanish gradually, and play a leading role at criticality.
Thus we can safely expand the kernels $a_{n}(\sin
k-\Lambda),\,a_{n}(\Lambda^{{}^{\prime}}-\Lambda)$ around the point of
$\Lambda=0$ in TBA eqs.(S15), (S16) to get the terms in the powers of
$\Lambda^{n}$. This procedure decouples the convolution terms. Firstly,
$a_{1}(\sin k-\Lambda)$ expands as

$a_{1}(\sin
k-\Lambda)\approx-\frac{A_{1}(k)}{2}-\frac{A_{2}(k)}{2}\Lambda^{2}+O(\Lambda^{4}),$
(S48)

where $A_{1}(k)=-2a_{1}(\sin k),\,A_{2}(k)=2\pi a_{1}^{2}(\sin
k)/u-8\pi^{2}\sin^{2}ka_{1}^{3}(\sin k)/u^{2}$. We keep the orders up to
$O(\Lambda^{2})$ in whole calculations. Applying this expansion form into
eqs.(S34), (S35), we get the leading terms

$\displaystyle\kappa(k)$ $\displaystyle=$ $\displaystyle-2\cos
k-\mu-2u-B+A_{1}(k)I_{1}+A_{2}(k)I_{2},$ (S49)
$\displaystyle\varepsilon_{1}(\Lambda)$ $\displaystyle=$ $\displaystyle
D_{1}\Lambda^{2}+D_{2},$ (S50)

where $I_{1},I_{2}$ are integrals term with respect to
$\varepsilon_{1}(\Lambda)$, which denoted by

$\displaystyle I_{1}$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}\mathrm{d}\Lambda
T\ln(1+\mathrm{e}^{-\frac{\varepsilon_{1}(\Lambda)}{T}})=-\frac{T^{\frac{3}{2}}\pi^{\frac{1}{2}}}{2D^{\frac{1}{2}}_{1}}\mathrm{Li}_{\frac{3}{2}}\left(-\mathrm{e}^{\frac{-D_{2}}{T}}\right),$
(S51) $\displaystyle I_{2}$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}\mathrm{d}\Lambda\Lambda^{2}T\ln(1+\mathrm{e}^{-\frac{\varepsilon_{1}(\Lambda)}{T}})=-\frac{T^{\frac{5}{2}}\pi^{\frac{1}{2}}}{4D^{\frac{3}{2}}_{1}}\mathrm{Li}_{\frac{5}{2}}\left(-\mathrm{e}^{\frac{-D_{2}}{T}}\right),$
(S52)

and $D_{1},D_{2}$ are functions of $\kappa(k),I_{1},I_{2}$ (explicit
expressions in Ref. LPG:preparation ). In the limit of $\Lambda\rightarrow 0$,
the spin density at $\Lambda=0$ has simple form
$\sigma(0)=\int_{-Q}^{Q}\mathrm{d}k/(2\pi)a_{1}(\sin k)$. Conducting
Sommerfeld expansion in free energy eq.(S36) firstly, and then substituting
eq.(S49) into free energy eq.(S36), we have

$\displaystyle f$ $\displaystyle=$ $\displaystyle-\frac{\pi
T^{2}}{6\kappa^{{}^{\prime}}(Q)}+\int_{0}^{Q}\frac{\mathrm{d}k}{\pi}\kappa(k)+u$
(S56) $\displaystyle=-\frac{\pi
T^{2}}{6\kappa^{{}^{\prime}}(Q)}+f_{0}+I_{1}\int_{0}^{Q}\frac{\mathrm{d}k}{\pi}A_{1}(k)$
$\displaystyle=-\frac{\pi
T^{2}}{6\kappa^{{}^{\prime}}(Q)}+f_{0}-2\sigma_{1}(0)I_{1}$
$\displaystyle=f_{0}-\frac{\pi
T^{2}}{6v_{c}}+T^{\frac{3}{2}}\pi^{\frac{1}{2}}\sigma_{1}(0)\left(\frac{\varepsilon^{{}^{\prime\prime}}_{1}(0)}{2}\right)^{-\frac{1}{2}}\mathrm{Li}_{\frac{3}{2}}\left(-\mathrm{e}^{-\frac{\varepsilon_{1}(0)}{T}}\right),$

where $f_{0}=\int_{0}^{Q}\mathrm{d}k/\pi\left(-2\cos{k}-\mu-2u-B\right)+u$ and
$D_{1}=\varepsilon^{{}^{\prime\prime}}_{1}(0)/2,D_{2}=\varepsilon_{1}(0)$ were
used. By the analysis of critical behavior, the argument function
$\varepsilon_{1}(0)$ in the polylog functions can be deduced from the TBA
equations,

$\varepsilon_{1}(0)=2B-\int_{-\pi}^{\pi}\mathrm{d}k\cos ka_{1}(\sin
k-\Lambda)T\ln(1+\mathrm{e}^{-\frac{\kappa(k)}{T}})|_{\Lambda=0}.$ (S57)

The phase transition occurs at $\varepsilon_{1}(0)=0$ with charge expressed as
$\kappa(k)=-2\cos k-\mu-2u-B$. Let $Q$ denote the Fermi point of $\kappa(k)$,
and the boundary condition satisfies $2\cos Q=-\mu-2u-B$. Utilizing this
boundary condition at zero temperature, $\varepsilon_{1}(0)$ has the following
form

$\varepsilon_{1}(0)=2B-\frac{4u}{\pi}\int_{0}^{Q}\mathrm{d}k\frac{\cos
k}{u^{2}+\sin^{2}k}(\cos k-\cos Q)+O(T^{2}).$ (S58)

By further performing expansion around the critical points
$(B_{c},\mu_{c},u_{c})$, one can give

$\cos Q\approx\cos Q_{c}-\sin Q_{c}\left[(\partial Q/\partial
B)|_{Q=Q_{c}}\Delta B+(\partial Q/\partial\mu)|_{Q=Q_{c}}\Delta\mu+(\partial
Q/\partial u)|_{Q=Q_{c}}\Delta u\right],$ (S59)

with $Q_{c}=\arccos\left(-\frac{1}{2}(\mu_{c}+2u_{c}+B_{c})\right)$. Through
some algebraic calculations, $\alpha_{B},\alpha_{\mu},\alpha_{u}$ can thus be
obtained for the transition from II to IV

$\displaystyle\alpha_{B}$ $\displaystyle=$
$\displaystyle-2\left[1-\frac{1}{\pi}\arctan\left(\frac{\sin
Q_{c}}{u}\right)\right],$ (S60) $\displaystyle\alpha_{\mu}$ $\displaystyle=$
$\displaystyle\frac{2}{\pi}\arctan\left(\frac{\sin Q_{c}}{u}\right),$ (S61)
$\displaystyle\alpha_{u}$ $\displaystyle=$
$\displaystyle\frac{4}{\pi}\left[\int_{0}^{Q_{c}}\mathrm{d}k\frac{\cos^{2}k}{u_{c}^{2}+\sin^{2}k}-2u_{c}^{2}\int_{0}^{Q_{c}}\mathrm{d}k\frac{\cos^{2}k}{(u_{c}^{2}+\sin^{2}k)^{2}}\right.$
(S62) $\displaystyle+\left.\arctan\left(\frac{\sin
Q_{c}}{u_{c}}\right)+\frac{\sin(2Q_{c})}{2(u_{c}^{2}+\sin Q_{c}^{2})}\right].$

We observe from above equations that $\alpha_{B},\alpha_{\mu}$ can be directly
calculated, although a closed form of $\alpha_{u}$ cannot be found. We note
that the interaction strength $u$ appears in the kernel of the integral,
leading to a significant challenge in the calculation of quantum scaling
functions. Nevertheless at quantum criticality, the heat capacity can be
obtained through the second partial of the free energy with respect to
temperature

$C_{v}/T=c_{0}+c_{1}T^{-1/2}\left[\frac{3}{4}\mathrm{Li}_{\frac{3}{2}}\left(-\mathrm{e}^{x}\right)-x\mathrm{Li}_{\frac{1}{2}}\left(-\mathrm{e}^{x}\right)+x^{2}\mathrm{Li}_{-\frac{1}{2}}\left(-\mathrm{e}^{x}\right)\right],$
(S63)

where in the transition of phase IV to II, $c_{0}=\pi/(3v_{c})$ presents the
contribution from the background,
$c_{1}=-\pi^{\frac{1}{2}}\sigma_{1}(0)(\varepsilon^{{}^{\prime\prime}}_{1}(0)/2)^{-1/2}$,
and $x=-\varepsilon_{1}(0)/T=(\alpha_{B}\Delta
B+\alpha_{\mu}\Delta\mu+\alpha_{u}\Delta u)/T$. By numerical simulations, we
observe that specific heat displays a bimodal structure, whose local maxima
mark the critical crossover temperatures fanning out from the critical point.
The local maxima can be determined by the condition $\partial C_{v}/\partial
B(\mu,u)=0$, leading the condition

$\frac{1}{4}\mathrm{Li}_{\frac{1}{2}}\left(-\mathrm{e}^{x}\right)-x\mathrm{Li}_{-\frac{1}{2}}\left(-\mathrm{e}^{x}\right)-x^{2}\mathrm{Li}_{-\frac{3}{2}}\left(-\mathrm{e}^{x}\right)=0,$
(S64)

that gives two solutions $x_{1}=-1.5629,\,x_{2}=3.6205$. We also observe that
the analytic result (S63) agrees well with the numerical results, showing that
our estimations are rather efficient. Fig.3 in the main text shows the contour
plot of specific heat in the temperature-interaction strength plane. In this
critical region, $T\gg\Delta u=u-u_{c}$, thermodynamic properties of the
system can be cast into universal scaling forms, for example, see eq.(S63).

The yellow line with square symbols in Fig.3 in the main text represents the
boundary of the TLL region, which resides below the line, characterized by a
specific heat $C_{v}$ linearly dependent on $T$. To the left of $u_{c}=1$, the
TLL contains only one component. To the right of $u_{c}$, the TLL contains
both the spin and the charge degrees of freedom. In this regime, spin-charge
separation Haldane:1981 ; Recati:PhysRevLett.90.020401 ; Guan:2012 ;
Mestyan:2019spin ; Patu:PhysRevB.101.035149  can be observed. Above the TLLC
phase the spin sector is gapped, utilizing the asymptotic behaviour of polylog
functionmohankumar2007two  and expanding the results from QC part eq.(S63),
the specific heat is given by

$C_{v}\approx\frac{\pi
T}{3v_{c}}+\frac{3\pi^{\frac{1}{2}}}{4}\sigma_{1}(0)\left(\frac{\varepsilon^{{}^{\prime\prime}}_{1}(0)}{2}\right)^{-\frac{1}{2}}T^{\frac{1}{2}}\mathrm{e}^{\alpha_{u}\Delta
u/T}+O\left(\Delta u\mathrm{e}^{\alpha_{u}\Delta u/T}\right).$ (S65)

In contrast, for the temperature $E_{s}\sim k_{F}v_{s}\ll k_{B}T\ll E_{c}\sim
k_{F}v_{c}$ above the TLLSC phase boundary, the specific heat is given by

$C_{v}\approx\frac{\pi
T}{3v_{c}}+\frac{\pi^{2}\sigma_{1}(0)\left(\varepsilon^{{}^{\prime\prime}}_{1}(0)/2\right)^{-\frac{1}{2}}(-\varepsilon_{1}(0))^{-\frac{1}{2}}T}{3}\left[1+\frac{21\pi^{2}}{40}(-\varepsilon_{1}(0))^{-2}T^{2}\right]+O(T^{4}),$
(S66)

which showing a gas-liquid co-existence.

###  IV.2 Correlation function in SILL

In the TLL regime $T\ll E_{c},E_{s}$, the finite size corrections to the
energy of low-lying excited states in terms of the changes of particle
numbers, backward scattering, particle and hole are expressed by Ess05 ;
Giamarchi:book ; Guan:2013

$\Delta E=\frac{2\pi}{L}\sum_{\alpha={\rm
c,s}}v_{\alpha}(N_{\alpha}^{-}+N_{\alpha}^{+})+\frac{\pi}{2L}\sum_{\alpha={\rm
c,s}}\left\\{v_{\alpha}(\hat{Z}^{-1}\Delta\vec{N})_{\alpha}^{2}+v_{\alpha}[\hat{Z}^{\rm
t}(2\vec{D})]_{\alpha}^{2}\right\\},$ (S67)

where $v_{\alpha}$ are Fermi velocities,
$N_{\alpha}^{\pm},\Delta\vec{N},\vec{D}$ are related to the three types of
excitations: adding particle at left and right Fermi points, change total
particle number, backward scattering, and $\hat{Z}$ denotes dressed charge
matrix Ess05 . In contrast, the bosonization Hamiltonian is given by
Giamarchi:book ; Guan:2013

$\displaystyle H$ $\displaystyle=$ $\displaystyle\sum_{\alpha={\rm
c,s}}\sum_{q\neq
0}v_{\alpha}|q|\hat{b}^{\dagger}_{\alpha,q}\hat{b}_{\alpha,q}~{}+\frac{\pi}{2L}\sum_{\alpha={\rm
c,s}}\left(~{}v^{\alpha}_{\rm N}\Delta N_{\alpha}^{2}~{}+~{}v^{\alpha}_{\rm
J}J_{\alpha}^{2}~{}\right)$
$\displaystyle+\frac{2c}{L}\sum_{k_{1},k_{2},p}\sum_{r=\pm
1}\hat{\psi}^{{\dagger}}_{\uparrow,r,k_{1}}\hat{\psi}^{\dagger}_{\downarrow,-r,k_{2}}\hat{\psi}_{\downarrow,r,k_{2}+2rk_{\rm
F}+p}\hat{\psi}_{\uparrow,-r,k_{1}-2rk_{\rm F}-p},$

where $\hat{b}^{\dagger}\,(\hat{b})$ denotes creation (annihilation) operator
of an boson with quasimomentuem $q$, $v^{\alpha}_{\rm N},v^{\alpha}_{\rm J}$
denotes density stiffness and phase stiffness, $c$ is relevant to coupling
strength, $J_{\alpha}$ is the current operator and $\hat{\psi}$ the Fermi
field operator. Based on finite-size correction, the two-point correlation
functions of the primary field at finite temperature are expressed in terms of
conformal dimensions

$\displaystyle\left\langle\phi(x,t)\phi(0,0)\right\rangle_{T}$
$\displaystyle=$ $\displaystyle\sum
A(D_{c},D_{s},N^{\pm}_{c},N^{\pm}_{s})\text{exp}(-2\mathrm{i}D_{c}k_{F,\uparrow}x)\text{exp}(-2\mathrm{i}(D_{c}+D_{s})k_{F,\downarrow}x)$
(S69) $\displaystyle\times\left(\frac{\pi T}{v_{c}\text{sinh}(\pi
T(x-\mathrm{i}v_{c}t)/{v_{c}})}\right)^{2\Delta^{+}_{c}}\left(\frac{\pi
T}{v_{c}\text{sinh}(\pi
T(x+\mathrm{i}v_{c}t)/{v_{c}})}\right)^{2\Delta^{-}_{c}}$
$\displaystyle\times\left(\frac{\pi T}{v_{s}\text{sinh}(\pi
T(x-\mathrm{i}v_{s}t)/{v_{s}})}\right)^{2\Delta^{+}_{s}}\left(\frac{\pi
T}{v_{s}\text{sinh}(\pi
T(x+\mathrm{i}v_{s}t)/{v_{s}})}\right)^{2\Delta^{-}_{s}},$

where $\Delta_{c}^{\pm},\Delta_{s}^{\pm}$ are the conformal dimensions
associated with elementary excitations and dressed charge matrix $\hat{Z}$
given in Ref. Ess05 .

However, in the SILL regime, the spin excitation is suppressed severely due to
the temperature scale $T>E_{s}$, and hence the spin sector is non-dynamic,
while charge maintains relevant low-energy behaviour. Although the TLL theory
holds under the condition $T\ll E_{c},E_{s}$, we can separately treat the
energy scales of charge and spin degrees of freedom in the SILL regime, also
see Fig. 3 in the main text. For the temperature $E_{s}\ll T\ll E_{c}$
Cheianov:2004 , the finite-temperature correlation functions of the conformal
field theory eq.(S69) still remain valid for the charge and spin degrees of
freedom under different conditions

$\displaystyle|x\pm\mathrm{i}v_{c}t|\ll
v_{c}/T,\qquad|x\pm\mathrm{i}v_{s}t|\gg v_{s}/T,$ (S70)

which is essential to capture the asymptotic behaviour of the SILL. Here we
remark that the typical energy scale of spin can be given by $E_{s}\sim
J\sim\left(k_{F\uparrow}+k_{F\downarrow}\right)/2\cdot v_{s}\equiv
k_{F}v_{s}$, where $J$ stands for the effective exchange coupling of spin
chain induced by interaction Guan:2013 . Thus under the condition (S70), the
single-particle Green’s function and pair correlation function are formally
given by

$\displaystyle G^{\uparrow}$ $\displaystyle\\!\approx\\!$
$\displaystyle\mathrm{e}^{-\mathrm{i}k_{F,\uparrow}x}{\cal
C}_{\uparrow}^{-}\left(x-\mathrm{i}v_{c}t\right)\langle
S_{R}^{+}\left(x,t\right)S_{R}(0,0)\rangle+h.c.,$ (S71) $\displaystyle G^{p}$
$\displaystyle\\!\approx\\!$
$\displaystyle\mathrm{e}^{-\mathrm{i}\left(k_{F,\uparrow}+k_{F,\downarrow}\right)x}{\cal
C}_{p}^{-}\left(x-\mathrm{i}v_{c}t\right){\cal
C}_{p}^{+}\left(x+\mathrm{i}v_{c}t\right)\langle
S_{R}^{+}\left(x,t\right)S_{R}(0,0)\rangle+h.c.,$ (S72)

where the charge correlations ${\cal C}_{\uparrow}^{-}\left(Z\right)\sim
1/Z^{2\Delta_{c}^{+}}$, ${\cal C}_{p}^{\pm}\left(Z\right)\sim
1/Z^{2\Delta_{c}^{\mp}}$ decay as a power-law of distance, whereas the spin
mode correlation $\langle
S_{R}^{+}\left(x,t\right)S_{R}(0,0)\rangle\sim\left(2\pi\alpha
k_{F}\right)^{2\Delta^{+}_{s}+2\Delta^{-}_{s}}\mathrm{e}^{-\pi\alpha\left(2\Delta^{+}_{s}+2\Delta^{-}_{s}\right)k_{F}x}$
decays exponentially. Here $\alpha$ is a constant to be determined. In the
limit of high density and low spin-down density, the conformal dimensions can
be given in the order no more than $O(\hat{n}_{c},n_{\downarrow})$
LPG:preparation

$\displaystyle G^{\uparrow}$ $\displaystyle:$
$\displaystyle\,2\Delta^{+}_{c}\approx 0,\,\,2\Delta^{-}_{c}\approx
1-\zeta_{1}n_{\downarrow},\,\,2\Delta^{+}_{s}\approx\frac{1}{4}-\frac{1}{u}\hat{n}_{c}-\frac{3}{4}\zeta_{1}n_{\downarrow},2\Delta^{-}_{s}\approx\frac{1}{4}-\frac{1}{u}\hat{n}_{c}+\frac{1}{4}\zeta_{1}n_{\downarrow};$
(S73) $\displaystyle G^{p}$ $\displaystyle:$
$\displaystyle\,2\Delta^{+}_{c}\approx\frac{9}{4}-\frac{3}{u}\hat{n}_{c},\,\,2\Delta^{-}_{c}\approx\frac{1}{4}-\frac{1}{u}\hat{n}_{c},$
(S74) $\displaystyle
2\Delta^{+}_{s}\approx\frac{1}{4}+\frac{1}{u}\hat{n}_{c}-\frac{3}{4}\zeta_{1}n_{\downarrow},\,\,2\Delta^{-}_{s}\approx\frac{1}{4}-\frac{1}{u}\hat{n}_{c}-\frac{3}{4}\zeta_{1}n_{\downarrow},$

where $\hat{n}_{c}=1-n,\zeta_{1}=\sqrt{1+u^{2}}/u$ with $n,n_{\downarrow}$
particle number and down-spin number. For the special case $B=0$, we can
compare the Luttinger parameter determined from the Wilson ratio and the
conformal dimensions are given in Ref. Ess05 ,

$G^{\uparrow}:\,2\Delta^{+}_{c}\approx\frac{1}{16},\,\,2\Delta^{-}_{c}\approx\frac{9}{16},\,\,2\Delta^{+}_{s}\approx
0,2\Delta^{-}_{s}\approx\frac{1}{2},$ (S75)

and observe that these results eq.(S75) agree with those given in Ref.
Cheianov:2004 , corresponding to the case of spinless fermions with Luttinger
parameter $K_{c}=1/2$. Although the bosonization expressions of the conformal
dimensions were already given in Cheianov:2004 ; Fiete:2007 , here our results
give new insights into such novel SILL in the 1D Hubbard model from the Bethe
ansatz perspective.

###  IV.3 The transition of IV-V:

In the phase transition of IV-V, the charge is gradually saturated as
$\kappa(\pi)$ approaching zero. Thus we expand the kernels $a_{n}(\sin
k-\Lambda)$ around the point of $\sin k=0$ in TBA eqs.(S15) and (S16) in terms
of the power of $\sin^{n}k$. Then charge dispersion can be written as the
functions of $\cos k$ and $\sin k$. We first expand the kernal function
$a_{1}(\sin k-\Lambda)$ in the form

$a_{1}(\sin k-\Lambda)\approx a_{1}(\Lambda)-(\pi
a^{2}_{1}(\Lambda)/u-4\pi^{2}\Lambda^{2}a^{3}_{1}(\Lambda)/u^{2})\sin^{2}k+O(\sin^{4}k).$
(S76)

Applying this expansion form into eqs.(S34), (S35), we have the result

$\displaystyle\kappa(k)$ $\displaystyle=$ $\displaystyle-2\cos
k+2C_{1}\sin^{2}k+C_{2},$ (S77) $\displaystyle\varepsilon_{1}(\Lambda)$
$\displaystyle=$ $\displaystyle
2B-\int_{-\pi}^{\pi}\mathrm{d}k2\cos^{2}ka_{1}(\sin
k-\Lambda)+\int_{-\infty}^{\infty}\mathrm{d}\Lambda^{\prime}a_{2}\left(\Lambda-\Lambda^{\prime}\right)T\ln(1+\mathrm{e}^{-\frac{\varepsilon_{1}(\Lambda^{\prime})}{T}})$
(S78)
$\displaystyle-2J_{1}a_{1}(\Lambda)+2J_{2}\left[\frac{\pi}{u}a^{2}_{1}(\Lambda)-\frac{4\pi^{2}}{u^{2}}\Lambda^{2}a^{3}_{1}(\Lambda)\right],$

where $C_{1},C_{2}$ are integral terms with respect to
$\varepsilon_{1}(\Lambda)$

$\displaystyle C_{1}$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}\mathrm{d}\Lambda\left[\frac{\pi}{u}a^{2}_{1}(\Lambda)-\frac{4\pi^{2}}{u^{2}}\Lambda^{2}a^{3}_{1}(\Lambda)\right]T\ln(1+\mathrm{e}^{-\frac{\varepsilon_{1}(\Lambda)}{T}}),$
(S79) $\displaystyle C_{2}$ $\displaystyle=$
$\displaystyle-\mu-2u-B-\int_{0}^{\infty}\mathrm{d}\Lambda
2a_{1}(\Lambda)T\ln(1+\mathrm{e}^{-\frac{\varepsilon_{1}(\Lambda)}{T}}).$
(S80)

And $J_{1},J_{2}$ are integral terms with respect to $\kappa(k)$ and can be
calculated by integration by parts:

$\displaystyle J_{1}$ $\displaystyle=$ $\displaystyle
T\int_{0}^{\pi}\mathrm{d}k\cos
k\ln(1+\mathrm{e}^{\frac{\kappa(k)}{T}})=\frac{T^{\frac{3}{2}}}{\sqrt{1-2C_{1}}}\Gamma(\frac{3}{2})\mathrm{Li}_{\frac{3}{2}}\left(-\mathrm{e}^{\frac{2+C_{2}}{T}}\right)-\frac{T^{\frac{5}{2}}}{8(1-2C_{1})^{\frac{5}{2}}}\Gamma(\frac{5}{2})\mathrm{Li}_{\frac{5}{2}}\left(-\mathrm{e}^{\frac{2+C_{2}}{T}}\right),$
(S81) $\displaystyle J_{2}$ $\displaystyle=$ $\displaystyle
T\int_{0}^{\pi}\mathrm{d}k\cos
k\sin^{2}k\ln(1+\mathrm{e}^{\frac{\kappa(k)}{T}})=\frac{T^{\frac{5}{2}}}{3(1-2C_{1})^{\frac{3}{2}}}\Gamma(\frac{5}{2})\mathrm{Li}_{\frac{5}{2}}\left(-\mathrm{e}^{\frac{2+C_{2}}{T}}\right).$
(S82)

The second integral in eq.(S78) can be obtained by the Sommerfeld expansion,
i.e.

$\int_{-\infty}^{\infty}\mathrm{d}\Lambda^{\prime}a_{2}\left(\Lambda-\Lambda^{\prime}\right)T\ln(1+\mathrm{e}^{-\frac{\varepsilon_{1}(\Lambda^{\prime})}{T}})=-\int_{-A}^{A}\mathrm{d}\Lambda^{\prime}a_{2}\left(\Lambda-\Lambda^{\prime}\right)\varepsilon_{1}(\Lambda^{\prime})+\frac{\pi^{2}T^{2}}{6\varepsilon_{1}^{{}^{\prime}}(A)}(a_{2}(\Lambda-A)+a_{2}(\Lambda+A)).$
(S83)

On the other hand, the density functions in the limit of $k\rightarrow\pi$
have the form

$\displaystyle\rho(\pi)$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi}-\int_{-A}^{A}\mathrm{d}\Lambda
a_{1}(\Lambda)\sigma_{1}(\Lambda),$ (S84) $\displaystyle\sigma_{1}(\Lambda)$
$\displaystyle=$
$\displaystyle-\int_{-A}^{A}\mathrm{d}\Lambda^{\prime}a_{2}(\Lambda-\Lambda^{\prime})\sigma_{1}(\Lambda^{\prime})+\frac{1}{2\pi}\int_{-\pi}^{\pi}\mathrm{d}ka_{1}(\sin
k-\Lambda).$ (S85)

Substituting the expressions of densities eq.(S84), (S85) into free energy,
then we calculate the contributions from ground state near Mott phase

$\displaystyle f_{0}$ $\displaystyle=$
$\displaystyle\int_{-\pi}^{\pi}\mathrm{d}k\left(-2\cos{k}-\mu-2u-B\right)\rho(k)+2B\int_{-A}^{A}\mathrm{d}\lambda\sigma_{1}(\lambda)+u$
(S86) $\displaystyle=$ $\displaystyle-\mu-
u-B+\int_{-\pi}^{\pi}\mathrm{d}k\int_{-A}^{A}\mathrm{d}\lambda\sigma_{1}(\lambda)\cos
k(-2\cos k-\mu-2u-B)a_{1}(\sin
k-\Lambda)+2B\int_{-A}^{A}\mathrm{d}\lambda\sigma_{1}(\lambda)$
$\displaystyle=$ $\displaystyle-\mu-
u-B-\int_{-A}^{A}\mathrm{d}\lambda\sigma_{1}(\lambda)\int_{-\pi}^{\pi}\mathrm{d}k2\cos^{2}ka_{1}(\sin
k-\Lambda)+2B\int_{-A}^{A}\mathrm{d}\lambda\sigma_{1}(\lambda)$
$\displaystyle=$ $\displaystyle-\mu-
u-B+\int_{-A}^{A}\mathrm{d}\lambda\sigma_{1}(\lambda)\varepsilon^{0}(\Lambda),$

where
$\varepsilon^{0}(\Lambda)=2B-\int_{-\pi}^{\pi}\mathrm{d}k2\cos^{2}ka_{1}(\sin
k-\Lambda)$. Therefore the free energy at low temperature via integration by
part is given by

$\displaystyle f$ $\displaystyle=$
$\displaystyle-T\int_{-\pi}^{\pi}\frac{\mathrm{d}k}{2\pi}\ln\left(1+\mathrm{e}^{\frac{-\kappa(k)}{T}}\right)+u$
(S87) $\displaystyle=$
$\displaystyle\int_{-\pi}^{\pi}\frac{\mathrm{d}k}{2\pi}\kappa(k)+u-T\int_{-\pi}^{\pi}\frac{\mathrm{d}k}{2\pi}\ln\left(1+\mathrm{e}^{\frac{\kappa(k)}{T}}\right)$
$\displaystyle=$
$\displaystyle\int_{-\pi}^{\pi}\frac{\mathrm{d}k}{2\pi}\kappa(k)+u+\frac{1}{\pi}\int_{0}^{\pi}\mathrm{d}k\frac{(k-\pi)}{1+\mathrm{e}^{-\frac{\kappa(k)}{T}}}\frac{\partial\kappa(k)}{\partial
k}-\frac{T}{\pi}(k-\pi)\ln\left(1+\mathrm{e}^{\frac{\kappa(k)}{T}}\right)|_{0}^{\pi}$
$\displaystyle=$
$\displaystyle\int_{-\pi}^{\pi}\frac{\mathrm{d}k}{2\pi}\kappa(k)+u-T\ln(1+\mathrm{e}^{\frac{\kappa(0)}{T}})+\frac{J_{1}}{\pi}+\frac{J_{2}}{2\pi}.$

Using eq.(S34), the first integral in eq.(S87) can be written as

$\displaystyle\int_{-\pi}^{\pi}\frac{\mathrm{d}k}{2\pi}\kappa(k)=-\mu-2u-B-\int_{-\pi}^{\pi}\frac{\mathrm{d}k}{2\pi}\int_{-\infty}^{\infty}\mathrm{d}\Lambda
a_{1}(\sin{k}-\Lambda)T\ln(1+\mathrm{e}^{-\frac{\varepsilon_{1}(\Lambda)}{T}})$
(S88) $\displaystyle=$
$\displaystyle-\mu-2u-B+\int_{-\pi}^{\pi}\frac{\mathrm{d}k}{2\pi}\left\\{\int_{-A}^{A}\mathrm{d}\Lambda
a_{1}(\sin{k}-\Lambda)\varepsilon_{1}(\Lambda)-\frac{\pi^{2}T^{2}}{6\varepsilon_{1}^{{}^{\prime}}(A)}(a_{1}(\sin
k-A)+a_{2}(\sin k+A))\right\\},$

where the second term can be dealt with through the density eq.(S85)

$\displaystyle\int_{-\pi}^{\pi}\frac{\mathrm{d}k}{2\pi}\int_{-A}^{A}\mathrm{d}\Lambda
a_{1}(\sin{k}-\Lambda)\varepsilon_{1}(\Lambda)$ (S89) $\displaystyle=$
$\displaystyle\int_{-A}^{A}\mathrm{d}\Lambda\varepsilon_{1}(\Lambda)\left[\sigma_{1}(\Lambda)+\int_{-A}^{A}\mathrm{d}\Lambda^{\prime}a_{2}(\Lambda-\Lambda^{\prime})\sigma_{1}(\Lambda^{\prime})\right]$
$\displaystyle=$
$\displaystyle\int_{-A}^{A}\mathrm{d}\Lambda\sigma_{1}(\Lambda)\left[\varepsilon_{1}(\Lambda)+\int_{-A}^{A}\mathrm{d}\Lambda^{\prime}a_{2}(\Lambda-\Lambda^{\prime})\varepsilon_{1}(\Lambda^{\prime})\right]$
$\displaystyle=$
$\displaystyle\int_{-A}^{A}\mathrm{d}\Lambda\sigma_{1}(\Lambda)\left[\varepsilon^{0}(\Lambda)-2J_{1}a_{1}(\Lambda)+\frac{\pi^{2}T^{2}}{6\varepsilon_{1}^{{}^{\prime}}(A)}(a_{2}(\Lambda-A)+a_{2}(\Lambda+A))\right]$
$\displaystyle=$ $\displaystyle
f_{0}+(\mu+u+B)+2J_{1}\left(\rho(\pi)-\frac{1}{2\pi}\right)+\frac{\pi^{2}T^{2}}{6\varepsilon_{1}^{{}^{\prime}}(A)}\int_{-A}^{A}\mathrm{d}\Lambda\sigma_{1}(\Lambda)(a_{2}(\Lambda-A)+a_{2}(\Lambda+A)),$

where we ignore the terms related to $J_{2}$ in the subleading order to
$J_{1}$. Thus the free energy finally can be given by

$\displaystyle f$ $\displaystyle=$ $\displaystyle
f_{0}+2J_{1}\left(\rho(\pi)-\frac{1}{2\pi}\right)+\frac{J_{1}}{\pi}$ (S90)
$\displaystyle+\frac{\pi^{2}T^{2}}{6\varepsilon_{1}^{{}^{\prime}}(A)}\left\\{\int_{-A}^{A}\mathrm{d}\Lambda\sigma_{1}(\Lambda)(a_{2}(\Lambda-A)+a_{2}(\Lambda+A))-\int_{-\pi}^{\pi}\frac{\mathrm{d}k}{2\pi}(a_{1}(\sin
k-A)+a_{1}(\sin k+A))\right\\}$ $\displaystyle=$ $\displaystyle
f_{0}+2\rho(\pi)J_{1}-\frac{\pi^{2}T^{2}\sigma(A)}{3\varepsilon_{1}^{{}^{\prime}}(A)}$
$\displaystyle=$ $\displaystyle f_{0}-\frac{\pi
T^{2}}{6v_{s}}+T^{\frac{3}{2}}\pi^{\frac{1}{2}}\rho(\pi)\left(\frac{-\kappa^{{}^{\prime\prime}}(\pi)}{2}\right)^{-\frac{1}{2}}\mathrm{Li}_{\frac{3}{2}}\left(-\mathrm{e}^{\frac{\kappa(\pi)}{T}}\right),$

where the term $\kappa(\pi)$ is given by

$\kappa(\pi)=2-\mu-2u-B+\int_{-A}^{A}\mathrm{d}\Lambda
a_{1}(\Lambda)\varepsilon_{1}(\Lambda).$ (S91)

Here, $\varepsilon_{1}(\Lambda)$ is given by

$\varepsilon_{1}(\Lambda)=2B-\int_{-\pi}^{\pi}\mathrm{d}k2\cos^{2}ka_{1}(\sin
k-\Lambda)-\int_{-A}^{A}\mathrm{d}\Lambda^{\prime}a_{2}(\Lambda-\Lambda^{\prime})\varepsilon_{1}(\Lambda^{\prime}).$
(S92)

It is indicated from above two equations (S91), (S92) that

$\alpha_{\mu}=-1,$ (S93)

which only appears in the leading term of charge. However, it’s hard to derive
explicit expressions of $\alpha_{B}$ and $\alpha_{u}$ due to the coupling of
the two degrees of freedom. We will give another method to solve this problem
in next section.

##  V Contact susceptibility

In quantum many-body systems, interaction plays the central role. In analogy
to the Contact for quantum atomic gases, we define the lattice version of the
Contact $C=\partial f/\partial u$, i.e. the derivative of the free energy with
respect to the interaction. We observe that the Contact remarkably marks the
phase boundaries. This can be seen in Fig. S1, where we use the contour plot
of the Contact to re-draw the phase diagram as in Fig.1 in the main text. The
two dark orange dashed lines represent the Contact $C=0$ and $C=-0.65$,
showing a sudden change at phase boundaries (black dashed lines). Here, we
define the Contact susceptibilities with respect to the external potentials.
Using the Maxwell relations, i.e., the derivative orders are commutative
$\frac{\partial}{\partial T}\left(\frac{\partial f}{\partial
u}\right)=\frac{\partial}{\partial u}\left(\frac{\partial f}{\partial
T}\right)$,$\frac{\partial}{\partial B}\left(\frac{\partial f}{\partial
u}\right)=\frac{\partial}{\partial u}\left(\frac{\partial f}{\partial
B}\right)$,$\frac{\partial}{\partial\mu}\left(\frac{\partial f}{\partial
u}\right)=\frac{\partial}{\partial u}\left(\frac{\partial
f}{\partial\mu}\right)$, we may build up general relations between Contact
susceptibilities and interaction-driven variations of density, magnetization
and entropy:

$\displaystyle\frac{\partial s}{\partial u}$ $\displaystyle=$
$\displaystyle-\frac{\partial C}{\partial T},$ (S94)
$\displaystyle\frac{\partial n}{\partial u}$ $\displaystyle=$
$\displaystyle-\frac{\partial C}{\partial\mu},$ (S95)
$\displaystyle\frac{\partial m}{\partial u}$ $\displaystyle=$
$\displaystyle-\frac{\partial C}{\partial(2B)}.$ (S96)

These three relations relate interaction-induced phase transitions with
magnetic-/chemical-induced phase transitions, and can be used to determine
$\alpha_{u}$ appeared in scaling functions.

Figure S1: Contour plot of the Contact in $B-\mu$ plane for $T=0.005$ and
$u=1$. The Contact displays a sudden changes at phase transitions, consistent
with the analytical phase boundaries (black dashed lines at zero temperature,
see Fig.1 in the main text.). The dark orange dashed lines mark the contours
of $C=0$ and $C=-0.65$.

###  V.1 Quantum cooling

It is precisely the first equality eq.(S94) of Maxwell relations that allows
us to drive interaction to refrigeration isentropically. Consider the entropy
as functions of $s=s(u,T)$ for fixed magnetic field, and thus the total
derivative is

$\mathrm{d}s=\frac{\partial s}{\partial u}\mathrm{d}u+\frac{\partial
s}{\partial T}\mathrm{d}T=0.$ (S97)

Substitute eq.(S94) into above expression and using $\frac{\partial
s}{\partial T}=\frac{C_{v}}{T}$ with $C_{v}$ is specific heat, the points on
the isentrope line in the $(u,T)$ coordinates admit

$\frac{C_{v}}{T}\frac{\partial T}{\partial u}=\frac{\partial C}{\partial T},$
(S98)

where $C_{v}$ is given by eq.(S63), $\partial C/\partial T$ can be derived
with the help of eq.(S45), (S46), for example for II-V:

$\frac{\partial C}{\partial
T}=T^{-\frac{1}{2}}\alpha_{u}\pi^{\frac{1}{2}}\sigma_{1}(0)\left(\frac{\varepsilon^{{}^{\prime\prime}}_{1}(0)}{2}\right)^{-\frac{1}{2}}\left[\frac{1}{2}\mathrm{Li}_{\frac{1}{2}}\left(-\mathrm{e}^{x}\right)-x\mathrm{Li}_{-\frac{1}{2}}\left(-\mathrm{e}^{x}\right)\right],$
(S99)

where $x=\alpha_{u}\Delta u/T$. Near a critical point, by the fact that local
maximum of the entropy leads to a local temperature minimum in an isentropic
process, and using the condition $\frac{\partial C}{\partial T}=0$, we have

$\frac{1}{2}\mathrm{Li}_{\frac{1}{2}}\left(-\mathrm{e}^{x}\right)-\frac{\alpha_{u}\Delta
u}{T}\mathrm{Li}_{-\frac{1}{2}}\left(-\mathrm{e}^{x}\right)=0.$ (S100)

The approximate solution of the quantity $x\approx 1.3117$. The entropy for
criticality regions can be obtained from the derivatives of eqs.(S45), (S46).
And the minimum temperature can be obtained, see the main text.

###  V.2 Calculation of $\alpha_{u}$ in the transition IV-V

In the discussion of the phase transition of IV-V, the coefficient
$\alpha_{u}$ associated with $\Delta u$ in the argument of the scaling
function is important to determine the scaling functions of physical
properties with respect to the interaction strength. Meanwhile, $\alpha_{u}$
also determines the lowest temperature which can be reached in an interaction-
driven quantum cooling process, i.e. $x_{0}=\alpha_{u}\Delta u/T\approx
1.3117$. Here we present a proposal of numerical estimation with the help of
the eq.(S95). Note that the demarcation line between phases IV and V is
described by constant density $n=1$. Considering the chemical potential $\mu$
and interaction $u$ driven phase transitions, we conduct a total derivative of
$n$

$\mathrm{d}n=\frac{\partial n}{\partial u}\mathrm{d}u+\frac{\partial
n}{\partial\mu}\mathrm{d}\mu=0.$ (S101)

Substitute eq.(S95) into above expression and using $\frac{\partial
n}{\partial\mu}=\chi_{c}$ with $\chi_{c}$ compressibility, the phase points on
the constant density line $n=1$ in the $(u,\mu)$ coordinates have

$\chi_{c}\frac{\partial\mu}{\partial u}=\frac{\partial C}{\partial\mu},$
(S102)

where $\chi_{c},\,\partial C/\partial\mu$ are near the critical points and
therefore can be obtained through the derivatives of free energy eqs.(S46)

$\displaystyle\chi_{c}$ $\displaystyle=$
$\displaystyle-T^{-\frac{1}{2}}\alpha^{2}_{\mu}\pi^{\frac{1}{2}}\rho(\pi)\left(\frac{-\kappa^{{}^{\prime\prime}}(\pi)}{2}\right)^{-\frac{1}{2}}\mathrm{Li}_{-\frac{1}{2}}\left(-\mathrm{e}^{\frac{\kappa(\pi)}{T}}\right),$
(S103) $\displaystyle\frac{\partial C}{\partial\mu}$ $\displaystyle=$
$\displaystyle
T^{-\frac{1}{2}}\alpha_{\mu}\alpha_{u}\pi^{\frac{1}{2}}\rho(\pi)\left(\frac{-\kappa^{{}^{\prime\prime}}(\pi)}{2}\right)^{-\frac{1}{2}}\mathrm{Li}_{-\frac{1}{2}}\left(-\mathrm{e}^{\frac{\kappa(\pi)}{T}}\right).$
(S104)

Substituting these expression of $\chi_{c}$ and $\partial C/\partial\mu$ into
eq.(S102), we get

$\alpha_{u}=-\alpha_{\mu}\frac{\partial\mu}{\partial
u}=\frac{\partial\mu}{\partial u},$ (S105)

in which we have used the identity $\alpha_{\mu}=-1$, see eq.(S93). It is
expected from eq.(S105) that the slope of the crossover line in the $(u,\mu)$
coordinates for the Mott transition can evaluate the coefficient
$\alpha_{u}(u_{c},\mu_{c})$. In view of the relation (S105), we plot the
contour diagram of density at fixed magnetic field $B_{c}=0.82714$ around
$u_{c}=1,\mu_{c}=-0.8272$, seen Fig.4 (b1) in the main text. Thus a series of
transition points $(u_{c},\mu_{c})$ in the $(u,\mu)$ coordinate are identified
at which the density firstly equals unity $n=1$. By artificially choosing two
adjacent points $(u_{1},\mu_{1}),(u_{2},\mu_{2})$ near $u_{c}=1$, then we
approximately get differential
$(\mu_{1}-\mu_{2})/(u_{1}-u_{2})\approx-1.9627$. Therefore we have
$\alpha_{u}(u_{c}=1)\approx-1.9627$. Using this value, we plot the
compressibility near the phase transition driven by interaction, see Fig.4(b2)
in the main text. A good agreement between the scaling function of the
compressibility with this analytical result of the $\alpha_{u}$ and numerical
result from the TBA result is seen in Fig.4(b2) in the main text. The relation
(S105) reveals deep insights into the transition from IV to V, i.e. the
emergence of Mott insulator with constant density and compressibility.

###  V.3 Calculation of $\alpha_{u}$ in the transition IV-II

In analogy with the application of eq.(S95), the relation eq.(S96) is also
possible to derive the coefficient $\alpha_{u}$ associated with $\Delta u$ in
the transition from II to V. This relation eq.(S96) is useful to study the
phase transition with a sudden change of the density of state in spin-down
electrons. The total derivatives on the spin-down particle density
$n_{\downarrow}$ is zero at the transition line at a constant chemical
poential, i.e.

$\mathrm{d}n_{\downarrow}=\frac{\partial n_{\downarrow}}{\partial
u}\mathrm{d}u+\frac{\partial n_{\downarrow}}{\partial(2B)}\mathrm{d}(2B)=0.$
(S106)

Due to the Maxwell relation eqs.(S96) and (S95) relate density $n$ and
magnetization $m$ with interaction strength $u$, we had better convert
$n_{\downarrow}$ into $n,\,m$. Using $n_{\downarrow}=n/2-m$, thus total
derivative eq.(S106) is transformed into the form below

$\mathrm{d}n_{\downarrow}=\left(\frac{1}{2}\frac{\partial n}{\partial
u}-\frac{\partial m}{\partial
u}\right)\mathrm{d}u+\left(\frac{1}{4}\frac{\partial n}{\partial
B}-\frac{1}{2}\frac{\partial m}{\partial B}\right)\mathrm{d}(2B)=0.$ (S107)

Substituting eq.(S96) into the above expression and using $\frac{\partial
m}{\partial B}=\chi_{s}$ with $\chi_{s}$ being the spin susceptibility, the
points on the constant density line of $n_{\downarrow}=0$ in the $(u,B)$
coordinates meet

$\left(\frac{1}{2}\frac{\partial n}{\partial u}+\frac{\partial
C}{\partial(2B)}\right)+\left(\frac{1}{2}\frac{\partial n}{\partial
B}-\chi_{s}\right)\frac{\partial B}{\partial u}=0.$ (S108)

Using eq.(S45), the analytic expressions for above derivatives are given by

$\displaystyle\frac{\partial n}{\partial u}$ $\displaystyle=$
$\displaystyle-T^{-\frac{1}{2}}\alpha_{u}\alpha_{\mu}\pi^{\frac{1}{2}}\sigma_{1}(0)\left(\frac{\varepsilon^{{}^{\prime\prime}}_{1}(0)}{2}\right)^{-\frac{1}{2}}\mathrm{Li}_{\frac{1}{2}}\left(-\mathrm{e}^{-\frac{\varepsilon_{1}(0)}{T}}\right),$
(S109) $\displaystyle\frac{\partial C}{\partial(2B)}$ $\displaystyle=$
$\displaystyle\frac{1}{2}T^{-\frac{1}{2}}\alpha_{u}\alpha_{B}\pi^{\frac{1}{2}}\sigma_{1}(0)\left(\frac{\varepsilon^{{}^{\prime\prime}}_{1}(0)}{2}\right)^{-\frac{1}{2}}\mathrm{Li}_{\frac{1}{2}}\left(-\mathrm{e}^{-\frac{\varepsilon_{1}(0)}{T}}\right),$
(S110) $\displaystyle\frac{\partial n}{\partial B}$ $\displaystyle=$
$\displaystyle-T^{-\frac{1}{2}}\alpha_{B}\alpha_{\mu}\pi^{\frac{1}{2}}\sigma_{1}(0)\left(\frac{\varepsilon^{{}^{\prime\prime}}_{1}(0)}{2}\right)^{-\frac{1}{2}}\mathrm{Li}_{\frac{1}{2}}\left(-\mathrm{e}^{-\frac{\varepsilon_{1}(0)}{T}}\right),$
(S111) $\displaystyle\chi_{s}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}T^{-\frac{1}{2}}\alpha^{2}_{B}\pi^{\frac{1}{2}}\sigma_{1}(0)\left(\frac{\varepsilon^{{}^{\prime\prime}}_{1}(0)}{2}\right)^{-\frac{1}{2}}\mathrm{Li}_{\frac{1}{2}}\left(-\mathrm{e}^{-\frac{\varepsilon_{1}(0)}{T}}\right).$
(S112)

Then substituting these expressions into eq.(S108), we derive $\alpha_{u}$

$\alpha_{u}=-\alpha_{B}\frac{\partial B}{\partial u},$ (S113)

which is dependent of $u_{c},\,B_{c}$. Our analysis provides an alternative
way to measure themodynamic quantities based on interaction-induced quantum
transition. Apart from the quantum transition in terms of chemical potential
and magnetic field, here we present the critical properties of the 1D Hubbard
model from perspective of the interaction.

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