This is the accepted manuscript made available via CHORUS. The article has been
published as:

Spin, Charge, and mathxmlns="http://www.w3.org/1998/Math/MathML"
display="inline">mi>η/mi>/math>-Spin Separation in
One-Dimensional Photodoped Mott Insulators
Yuta Murakami, Shintaro Takayoshi, Tatsuya Kaneko, Andreas M. Läuchli, and Philipp
WernerPhys. Rev. Lett. 130, 106501 — Published 10 March 2023
DOI: 10.1103/PhysRevLett.130.106501

Spin, charge and η-spin separation in one-dimensional photo-doped Mott insulators
Yuta Murakami,1 Shintaro Takayoshi,2 Tatsuya Kaneko,3 Andreas M. Läuchli,4, 5 and Philipp Werner61

Center for Emergent Matter Science, RIKEN, Wako, Saitama 351-0198, Japan2
Department of Physics, Konan University, Kobe 658-8501, Japan3
Department of Physics, Osaka University, Toyonaka, Osaka 560-0043, Japan
4
Laboratory for Theoretical and Computational Physics,
Paul Scherrer Institute, 5232 Villigen, Switzerland
5
Institute of Physics, Ecole Polytechnique Fedérale de Lausanne (EPFL), 1015 Lausanne, Switzerland
6
Department of Physics, University of Fribourg, 1700 Fribourg, Switzerland(Dated: February 6, 2023)
We show that effectively cold metastable states in one-dimensional photo-doped Mott insulators
described by the extended Hubbard model exhibit spin, charge and η-spin separation. Namely,
their wave functions in the large on-site Coulomb interaction limit can be expressed as |Ψi =|Ψcharge i|Ψspin i|Ψη−spin i, which is analogous to the Ogata-Shiba states of the doped Hubbard modelin equilibrium. Here, the η-spin represents the type of photo-generated pseudoparticle (doublon
or holon). |Ψcharge i is determined by spinless free fermions, |Ψspin i by the isotropic Heisenberg
model in the squeezed spin space, and |Ψη−spin i by the XXZ model in the squeezed η-spin space.
In particular, the metastable η-pairing and charge-density-wave (CDW) states correspond to thegapless and gapful states of the XXZ model. The specific form of the wave function allows us toaccurately determine the exponents of correlation functions. The form also suggests that the central
charge of the η-pairing state is 3 and that of the CDW phase is 2, which we numerically confirm.
Our study provides analytic and intuitive insights into the correlations between active degrees of
freedom in photo-doped strongly correlated systems.

Introduction– Doping charge carriers into strongly correlated insulators provides a pathway to produce intriguing emergent phenomena such as high-Tc superconductivity [1, 2]. In equilibrium, the doping concentration
can be chemically controlled. An alternative nonequilibrium way of introducing charge carriers is photo-doping,where electrons are excited across the gap [3–7]. The
photo-doping of Mott insulators creates novel pseudoparticle excitations such as doublons and holons (in the
single-band case), while the equilibrium system can host
only one type of charge carrier. Such additional degrees of freedom can lead to intriguing properties andnonthermal phases. Important examples include photoinduced insulator-metal transitions [8–13] and charge
density waves [14–16], and the control of magnetic [17, 18]
and superconducting orders [19–26].
In systems with a large Mott gap, the life-time of thephoto-doped pseudoparticles becomes exponentially enhanced [27–33]. In such a situation, an intraband cooling of the photo-doped pseudoparticles may occur, while
their density remains approximately constant. This results in a metastable steady state (a pseudoequilibrium
state) [19, 34–41], analogous to the case of photo-doped
semiconductors [42–44], see Fig. 1(a). It has been shownthat such metastable states can host unique phases suchas η-pairing [38, 40], chiral superconducting phases [45],and exotic spin/orbital orders [18, 39, 46]. Since different
types of charge carriers are present in photo-doped systems, it is crucial to understand the correlations between
the active degrees of freedoms. However, the metastablestates of photo-doped strongly correlated systems havebeen mainly studied numerically so far [34–41, 47], andanalytical or intuitive insights are limited.
Here we reveal the nature of the metastable states andthe correlations between the active degrees of freedom inphoto-doped one-dimensional Mott insulators. We showthat the wave functions of the metastable states in thelimit of large on-site Coulomb interaction exhibit spin,
charge and η-spin separation, see Fig. 1(b). The η-spin
represents the type of pseudoparticle: doubly occupied
site (doublon) or empty site (holon). Our results provide
a comprehensive understanding of the character of thephoto-induced metastable phases in one-dimensional systems and reveal the similarities and differences between
photo-doped and chemically-doped systems.
Results– We focus on the one-dimensional extended
Hubbard model,
X †Ĥ = −thop(ĉi,σ ĉi+1,σ + h.c.) + ĤU + ĤV ,(1)i,σand assume that electrons are excited across the Mott
gap via photo-excitation. Similar setups
can be considP
ered with cold atoms [19]. ĤU = U i (n̂i↑ − 21 )(n̂i↓ − 21 ) isP
the on-site interaction and ĤV = V i (n̂i − 1)(n̂i+1 − 1)is the nearest-neighbor interaction. ĉ†iσ is the creation
operator of a fermion with spin σ at site i, n̂iσ = ĉ†iσ ĉiσ ,n̂i = n̂i↑ + n̂i↓ , and thop is the hopping parameter. Whenthe Mott gap is large enough, the recombination time of
the created doublons and holons becomes exponentiallylong [27–32]. Thus, intraband relaxation due to scattering events and coupling to the environment is expected tobring the system into a (intraband thermalized) steadystate with a fixed number of doublons and holons, see2
Photo-doping(a)doublons), the model corresponds to the t-J model whenĤ3−site is neglected [53]. In the following, we denote themodel without Ĥ3−site by Ĥeff2 . When V = 0, Ĥ, Ĥeffand Ĥeff2 host an SU(2) symmetry of the doublon-holon
sector [50, 54] that corresponds to the spin SU(2) symmetry via a particle-hole (Shiba) transformation [55].

Metastable state

UHB

Intrabandcooling
LHB

(b)

|

h

d

h

i

d

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FIG. 1. (a) Schematic picture of the photo-doing and intraband cooling processes that result in a metastable stateof the large-gap Mott insulator. UHB (LHB) stands for upper (lower) Hubbard band. (b) The wave function of themetastable state in the limit of U → ∞ can be expressed asa direct product of the charge wave function, the spin wave
function in the squeezed space and the η-spin wave functionin the squeezed space. The green shaded circles in the charge
wave function represent spinless fermions, while “h” and “d”
stand for a holon and a doublon, respectively.

Fig. 1(a). As previously discussed, such a quasi-steadystate can be described with the effective Hamiltonian obtained by a Schrieffer-Wolff transformation [48] from theoriginal Hamiltonian (1) [19, 34–40, 47], see also the Supplemental Material (SM) [49]. This effective Hamiltonian
explicitly conserves the number of doublons and holons.
Up to O(t2hop /U ), it takes the formr∈ro
: #r = #σ = Ns and #η = Nη .

(3)

Here r, σ and η are sets of space, spin and η-spin indices,ĉ†r is a creation operator of a spin-less fermion (SF), and# indicates the number of elements. η takes the values↑ or ↓ and r = {rNs , · · · , r1 } with L ≥ rNs > rNs −1 >· · · > r1 ≥ 1. We identify this Hilbert space with theoriginal Hilbert space using the unitary transformationÛ : H → H0 defined by
Û

Ns
Y
i=1

ĉ†ri ,σi

Nη
 Y
j=1

â†r̄j ,ηj



!
|vaci

=

Y
r∈r


ĉ†r |vaci|σi|ηi.
(4)Ĥeff =ĤU + Ĥkin + ĤV+ Ĥspin,ex + Ĥdh,ex + ĤU,shift + Ĥ3−site ,We consider an effectively cold system with arbitrary
filling, whose state is described by the ground state of
the effective Hamiltonian for a given number of doublons and holons, i.e., we assume that the system is thermalized into the lowest energy state for the given constraint. We show that the corresponding wave function
can be expressed as the direct product of the charge,
spin and η-spin wave functions in the limit of Jex → 0
with V /Jex = const, similar to the Ogata-Shiba state ofthe doped Hubbard model in equilibrium [56, 57]. To be
more specific, we set the system size to L and the numberof singly occupied sites to Ns , so that the number of doublons and holons (the number of η spins) is Nη = L − Ns .
Now we introduce the Hilbert spacenY 
H0 = |ri|σi|ηi ≡ĉ†r |vaci|σi|ηi(2)Pˆ i,σ̄ (ĉ†i,σ ĉj,σ + h.c.)n̄ˆ j,σ̄ −where Ĥkin = −thop hi,ji,σ n̄P†thop hi,ji,σ n̂i,σ̄ (ĉi,σ ĉj,σ + h.c.)n̂j,σ̄ represents the hopping of a doublon or a holon, σ̄ is the opposite spin ofˆ i,σ = 1 − n̂i,σ . The other terms are proporσ, and n̄
P
4t2
tional to Jex ≡ Uhop . Ĥspin,ex = Jex hi,ji ŝi · ŝj is thePspin exchange term, and Ĥdh,ex = −Jex hi,ji [η̂ix η̂jx +η̂iy η̂jy + η̂iz η̂jz ] is the exchange term for doublons andholons on P
neighboring sites. Here the spin operators
are ŝ = 12 α,β=↑,↓ ĉ†α σ αβ ĉβ with σ denoting the Paulimatrices, and we introduced the η-spin operators asη̂i+ = (−)i ĉ†i↓ ĉ†i↑ , η̂i− = (−)i ĉi↑ ĉi↓ and η̂iz = 12 (n̂i − 1) [50–52]. ĤU,shift describes the shift of the local interactionand Ĥ3−site represents three-site terms such as correlated
doublon hoppings, see SM [49]. In equilibrium (without

Here, r̄ = {r̄Nη , · · · r̄1 } with L ≥ r̄Nη > r̄Nη −1 > · · · >r̄1 ≥ 1, r ∪ r̄ = {L, L − 1, · · · , 1}, â†r̄,↑ = (−)r̄ ĉ†r̄↓ ĉ†r̄↑

and â†r̄,↓ = 1. With this identification, |ri is the basis of SF and |σi (|ηi) is the basis of the squeezed spin
(η-spin) space. Note that the η-spin configuration represents the sequence of doublons and holons. As shown
below, Hamiltonians ruling the σ and η spaces are notfully symmetric due to the staggering of the doublons.
The wave function in the limit of Jex → 0 can beconstructed by degenerate perturbation theory [57]. For
Jex = V = 0, the eigenstates of Ĥeff are degenerate withrespect to the configurations of spins and η-spins. Thisis because Ĥkin never exchanges the positions of spins or
those of doulons and holons.
P Specifically, one can showthat Û Ĥkin Û † = −thop hi,ji (ĉ†i ĉj + h.c.) (≡ Ĥ0,SF ).
This means that in the representation of H0 the groundstate for Jex = V = 0 can be described as |ΨGS
SF i|Ψσ,η i,

3where |ΨGS
SF i is the ground state of Ĥ0,SF and |Ψσ,η i is anarbitrary spin and η-spin wave function. The remaining
spin/η-spin degeneracy of 2Ns · 2Nη is lifted by the termsof O(Jex ). Within lowest-order degenerate perturbation
theory, the wave function of the spin and η-spin is obtained by the O(Jex ) terms projected to |ΨGS
SF i|σi|ηi. In
the resultant projected Hamiltonian, the squeezed spinand η-spin spaces are decoupled, and the corresponding
Hamiltonians become (SQ stands for squeezed space)X(SQ)sĤspin = Jexŝi+1 · ŝi ,
i

(SQ)
η
Ĥη−spin = −JX

X
X
y
x
z
(η̂j+1η̂jx + η̂j+1
η̂jy ) + JZηη̂j+1η̂jz ,jjηswith Jex= (x̃ − x̃0 )Jex , JX
= (ỹ − ỹ 0 )Jex and JZη = −(ỹ −00ỹ )Jex + 4ỹV . Here x̃, x̃ , ỹ and ỹ 0 are the renormalization
factors determined by |ΨGS
SF i. With ns = Ns /L and nη =Nη /L and in the limit L → ∞ they can be expressed assin(2πns ) sin2 (πns )sin2 (πns ) 0
,
x̃
=
−
,
π 2 ns2π
π 2 nssin(2πnη ) sin2 (πnη )sin2 (πnη ) 0
,
ỹ
=
−
.
ỹ = nη −π 2 nη
2π
π 2 nη

x̃ = ns −Here x̃ and ỹ are the contributions from the 2-site termsof O(Jex ), while x̃0 and ỹ 0 are those from the 3-site terms.
(SQ)
Note that Ĥη−spin becomes the ferromagnetic Heisenbergηmodel (JX
= −JZη > 0) for V = 0. Thus, the wave
function (in H0 ) takes the formGSGS
|Ψi = |ΨGS
SF i|Ψσ i|Ψη i,

(5)

(SQ)

GS
where |ΨGS
σ i is the ground state of Ĥspin and |Ψη i is
(SQ)

that of Ĥη−spin . For more details, see SM [49]. The formGSof |ΨGS
SF i and |Ψσ i is independent of the ratio of doublons and holons, and, in particular, these states are thesame as those in the equilibrium doped Hubbard modelat the doping level nholes = nη [56, 57]. This implies thatthe effects of photo-doping and chemical doping on thespins are essentially the same, which is consistent withprevious numerical analyses [40, 58, 59].
Now we focus on half filling and discuss the implications of the exact form of the wave function for the originof the different phases. The η-spin sector hosts the phasetransition between the gapless and gapful phases of the
XXZ model, which is controlled by the ratio between
Jex and V . As seen below, these states are characterized by the behavior of the correlation functions of theη-spins, i.e. χη,a (r) ≡ hη̂ a (r)η̂ a (0)i. Namely, the gaplessphase corresponds to the η-pairing phase, where the paircorrelation χη−pair ≡ χη,x is dominant. On the other
hand, the gapful phase corresponds to the CDW phase,where the charge correlation χcharge ≡ χη,z is dominant.
True long-range order (LRO) is realized at V = 0 for

CDW(string)

CDW
(LRO)

SDW
η pairing(power law)η pairing(LRO)

FIG. 2. Phase diagram of the photo-doped one-dimensional
Mott insulator described by Ĥeff in the limit Jex → 0. The
phase boundary (black solid line) corresponds to an SU(2)0(SQ). The horizonsymmetric point of Ĥη−spin , i.e. V /Jex = ỹ−ỹ2ỹtal dashed line indicates the phase boundary for the systemdescribed by Ĥeff2 .

the η-pairing phase [60], while a LRO CDW is realized
at nη = 1 and V > J2ex . Apart from these limits, wehave quasi-long-range orders (power law decay of correlations). Note that the appearance of η-pairing in nonthermal states has been recently discussed [22, 26, 38, 61–63]
in relation with the photo-induced superconducting-like
phases [20, 64–67]. Furthermore, we emphasize that LRO
is realized in the squeezed η-spin space for the CDW
phase, which is reminiscent of the string order in the
Haldane phase [68]. The phase transition occurs at the(SQ)ηSU(2) point of Ĥη−spin (JX= JZη > 0), see Fig. 2. ForηĤeff2 (without Ĥ3−site ), ∆(≡ JZη /JX) and thus the phaseboundary are independent of the filling, which consistently explains a previous numerical result [40]. On theother hand, for Ĥeff , the ratio ∆ depends on the filling
due to the effects of the 3-site term ỹ 0 . In particular, the3-site term is found to favor the η-pairing phase.
The exact form of the wave function allows us toevaluate the asymptotic behavior of the correlation
functions analytically or numerically.
Here we extend the analyses for spin correlations of the equilibrium Hubbard model [69, 70]. Since the spin correlations of the metastable state are the same as thosefor the equilibrium Hubbard model, i.e. hŝa (r)ŝa (0)i ∝31cos(πns r)r− 2 (ln r) 2 , we focus on the η-spin correlation
functions χη,a (r). Note that despite the apparent similarity between the squeezed spin and η space there are
crucial differences in the pairing correlations. Using expression (5), the correlation functions are expressed asχη,a (r) =

r+1
X
m=2

Q̄rSF (m)χ(SQ)
η,a (m − 1).

(6)Pˆ 0 n̄
ˆ r δ( rl=0 n̄ˆ l − m)iSF , which is deHere Q̄rSF (m) = hn̄GStermined by |ΨSF i, is the probability that the sys(SQ)
tem has m doublons or holons in [0, r]. χη,a (m) =aahη̂ (m)η̂ (0)iη−spin,squeezed is the correlation function in4(a)<latexit sha1_base64="JspsCJDmXtZ/ruVXo2ZT9qhVtx4=">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</latexit>

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FIG. 4. Central charge of the ground state of Ĥeff2 for Jex =0.4 and the indicated values of nη . To evaluate the central
charge, we apply Eq. (8) to the iTEBD results with D ∈[200, 1600]. The red shaded area indicates the stability regionof the η-pairing phase for Jex → 0, while the blue shaded areashows that of the CDW phase. The inset plots the relation
between SE and ξD and the corresponding linear fits.
FIG. 3. (a)(b) Asymptotic behavior analytically obtainedfor (a) χcharge and (b) χη−pair . The dashed line at ∆ = 21
in (a) is the boundary of different expressions. The thickgreen line corresponds to the SDW, the thick blue line to the
CDW with LRO and the thick red line to the η-pairing phasewith LRO. (c)(d) Numerically evaluated correlation functionsfor (c) the η-pairing phase and (d) the CDW phase using
Eq. (6) and the iTEBD results for the XXZ model (blue circles). The corresponding points are indicated with crosses inpanels (a)(b). We also show the correlations estimated by(SQ)
the conjecture χη−pair ' n2η χη (rnη ) as well as the fit with
1− 1
−2
−2
α cos(πnη r).
C1 r + C2 r

the squeezed η-spin space. Numerically, Q̄rSF (m) and(SQ)χη,a (m) can be efficiently evaluated in the thermodynamic limit. We use the expression for the Fourier components and perform an inverse Fourier transformation to
obtain Q̄rSF (m) [49, 70], while the infinite time-evolving
block decimation (iTEBD) [71] for the XXZ model can(SQ)
be used to calculate χη,a (m). Moreover, we can also
gain analytic insights using the knowledge of the asymptotic behavior of the correlation functions of the XXZ
model [72–74] and the moments of Q̄rSF (m) up to thesecond one [69]. The former can be expressed withα ≡ 1 − π1 arccos(∆), which is a control parameter of
the Tomonaga-Luttinger liquid, and the latter indicatesthat most of the weight of Q̄rSF (m) is at rnη . From these(SQ)
facts, if the asymptotic form of χη (m) is (−)m f (m)with f (m) being a smooth function, one can prove that
( r+1
)
r+1
X
X
Q̄rSF (m)(−)m f (m) 'Q̄rSF (m)(−)m f (hmi).
m=2

m=2

(7)
(SQ)

Here hmi = nη r + 1. If χη(m) ' f (m), the equa-tion without (−)m is satisfied. See SM for the detailed
meaning of the equality ' and the derivation. Since
Pr+1cos(πnη r)[69, 75] andwe have { m=2 Q̄rSF (m)(−)m } ∝1
r2
Pr+1 r2
m=2 Q̄SF (m) = nη (in leading order in r), one can obtain the asymptotic form of the correlation functions.
Equation (7) shows that the decay of η-spin correlations in real space originates from that in the squeezed
space and the contribution from the intercalated singlyoccupied sites. The latter is determined by |ΨGS
SF i, andhas a different impact depending on whether the correlation functions in the squeezed space are staggered ornot. In particular, the pairing correlation is not affectedby the SF background, while the charge correlations canbe affected like the spin correlations.
The asymptotic forms obtained analytically for χchargeand χη−pair are summarized in Figs. 3(a)(b). The magnitude relation of the exponents of these correlation func(SQ)
tions changes at the SU(2) point of Ĥη−spin . Note that
this SU(2) symmetry is an emergent symmetry in thesqueezed space, which is absent in the original Hamiltonian. For 0 < nη < 1, χcharge shows an exponent of 1/2
in the CDW phase due to the contribution from the SF
part, although it shows LRO in the squeezed space. On
the other hand, the analytic argument based on Eq. (7)does not allow to make exact statements for the components decaying faster than O( lnr2r ). To analyze thispoint, we numerically evaluate the correlation functions,
see Figs. 3(c)(d). Firstly, our results verify the con(SQ)
jecture χη−pair (r) ' n2η χη,x (rnη ) and its applicability
even in the CDW regime, where χη−pair (r) decays exponentially, see Fig. 3(c). Secondly, Fig. 3(d) shows thatEq. (7) is practically applicable for the leading and the
sub-leading terms of χcharge decaying faster than O( lnr2r ),1
1
i.e. χcharge (r) ' C1 r−2 + C2 r− 2 − α cos(πnη r).

5
The expression (5) also provides valuable insights into
the physical nature of the metastable phases. One important quantity that characterizes one-dimensional systems is the central charge (c), which counts to the number of gapless degrees of freedoms [72]. In equilibrium,
the doped Hubbard model exhibits c = 2, because of
the massless modes both in the spin and charge sectors [76, 77]. On the other hand, the exact form of thewave-function (5) suggests that the metastable state possesses three degrees of freedoms. The wave functions of
the charge and spin sectors are those of gapless states(i.e. doped free fermions and the isotropic Heisenberg
model), while that of the η-spin sector corresponds to
the gapless state or the gapful state of the η-XXZ modelfor the η-pairing state and the CDW state, respectively.
Thus, one naturally expects that c = 3 in the η-pairing
state and c = 2 in the CDW state. To confirm this, we
perform iTEBD simulations on the effective model Ĥeff2
for various cut-off dimension (D) and extract c from therelation [78]SE =cln(ξD ) + s0 .
6(8)Here SE is the entanglement entropy, s0 is a constantand ξD is the correlation length evaluated from thesecond-largest eigenvalue of the transfer matrix, see SM.
In Fig. 4, we show the central charge for Ĥeff2 with
Jex = 0.4, which is extracted using Eq. (8) and the linear fit of the iTEBD results [see the inset of Fig. 4]. The
results indeed confirm the above expectation. We emphasize that the emergence of a c = 3 state in the Hubbard
model is hardly expected in equilibrium and reflects themetastable nature of the state.
Conclusion– We showed that the additional degrees of
freedom activated by photo-doping lead to peculiar typesof quantum liquids absent in equilibrium. In particular,
we revealed the intriguing structure of the correlations
between active degrees of freedom in photo-doped onedimensional strongly correlated systems, i.e. the spincharge-η-spin separation. Our results open a new avenue
for studying metastable states in one-dimensional systems and raise interesting questions. Firstly, in contrast
to the equilibrium Hubbard model, the weak coupling
regime is not well-defined, and the relation between thelattice model and the corresponding conformal field theory is not clear. Construction of the field theory for themetastable states is an important future task. Secondly,
we provide a rigorous basis for the future developmentof a bosonization approach [79, 80]. With such an approach, one can better understand the spectral featuresof the photo-doped systems and the implications of thespin-charge-η-spin separation for dynamical properties.
Thirdly, various concepts developed for one-dimensional
systems in equilibrium can be extended to understandthe physics of metastable states. For example, extending
the spin incoherent Luttinger liquids [81] may be help-

ful for understanding effectively cold, but not ultracold
systems.
Finally but not least, our analytical and intuitive insights provide a useful reference for the study of photodoped Mott insulators in higher dimensions, where theseparation of spin, charge and η-spin is not expected, but
a crossover from high-dimensional to one-dimensional behavior can occur in anisotropic systems.
We thank R. Arita for inspiring comments and A. J.
Millis, Z. Sun, D. Golež and D. Baeriswyl for fruitful discussions. This work was supported by Grantin-Aid for Scientific Research from JSPS, KAKENHI
Grant Nos. JP20K14412 (Y.M.), JP21H05017 (Y.M.),
21K03412 (S. T.), JP18K13509 (T.K.) JP20H01849(T.K.), JST CREST Grant No. JPMJCR1901 (Y.M.),
JPMJCR19T3 (Y. M. and S. T.), and ERC Consolidator
Grant No. 724103 (P.W.).

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