On the Equivalence between Spin and Charge Dynamics of the Fermi Hubbard Model

Utilizing the Fermi gas microscope, recently the MIT group has measured the spin transport of the Fermi Hubbard model starting from a spin-density-wave state, and the Princeton group has measured the charge transport of the Fermi Hubbard model starting from a charge-density-wave state. Motivated by these two experiments, we prove a theorem that shows under certain conditions, the spin and charge transports can be equivalent to each other. The proof makes use of the particle-hole transformation of the Fermi Hubbard model and a recently discovered symmetry protected dynamical symmetry. Our results can be directly verified in future cold atom experiment with the Fermi gas microscope.

Quantum gas microscope is one of the most significant developments in the cold atom physics during the past decade. It opens up a new avenue for studying strongly correlated physics, because it allows one not only to detect the system in situ with single-site resolution, but also to prepare an eigenstate of real space density operators, with which the nonequilibrium dynamics of strongly correlated systems can be studied. Recently, the MIT group and the Princeton group have prepared the Fermi Hubbard model (FHM) initially in a spin-density-wave state and a charge-density-wave state, respectively, and the subsequent spin or charge dynamics has been measured [1,2]. From these two measurements, they extracted the spin diffusion constant and the charge diffusion constant, respectively [1,2].
This article is to prove that, under certain conditions, the spin and the charge transport measurements can be equivalent to each other for the Fermi Hubbard model. To be specific, we first write down the FHM that these two groups have simulated by loading ultracold fermionic atoms in square optical lattices, that iŝ where J is the hopping amplitude between two nearest neighbouring sites of the square lattice, and U is the on-site interaction strength. Here the interaction term is written in a particlehole symmetric form. Taking J as the energy unit, the model is characterized by one single parameter U, together with two conserved quantities: N ↑ +N ↓ −N s (N s denotes the total number of sites), known as the doping from half filling; and N ↑ − N ↓ , known as the spin imbalance. First, let us start with a real space spin-density-wave state written as which is shown schematically in the upper panel of Fig. 1.
Here, there is no constraint on the choices of region A and B. Neither of them has to be single-connected or has equal size to the other. For instance, if one considers a (π, π) antiferromagnetic state alongẑ direction on a square lattice, then (Dated: August 18, 2018) Fermi gas microscope, recently the MIT group has measured the spin transport of the Fermi l starting from a spin-density-wave state, and the Princeton group has measured the charge transi Hubbard model starting from a charge-density-wave state. This paper is to prove that these easurements are in fact equivalent to each other. The proof makes use of the particle-hole symermi Hubbard model and a recently discovered symmetry protected dynamical symmetry. We general condition for the equivalence of spin and charge transport in the Fermi Hubbard model. tum gas microscope is one of the nts in cold atom physics during renew avenue for studying strongly rth emphasizing that the quantum llows one to detect the system in on, but also allows one to prepare density operators, with which one m dynamics of strongly correlated t the MIT group and the Princeton pared a Fermi Hubbard model inity-wave state or a charge-densityre the subsequent spin and charge ct the spin di↵usion constant and rom these two measurements, rethat these two transport measurelent to each other. To be specific, i Hubbard model that two groups ionic atom loaded in square opti- litude between nearest neighboure, and U is the on-site interaction interaction term in a particle-hole as energy unit, the model is charameter U, together with two con-N s (N s denotes the total number of half-filling and N " N # known rts with a real space spin-density an ideal version of such state as Y on the choice of region A and B. to be single-connected and they do . For instance, if one considers an ate on a square lattice, A denotes es the other sublattices. For MIT polarized up and B denotes the rest regions where spins are polarized down. Then this state will evolve under the Fermi Hubbard Hamiltonian and then at certain time t, one measures the local spin density alongẑ-direction as The Princeton experiment starts with a real space chargedensity wave state [2]. Similarly, we can write a simple version of such state that region A is double occupied and region B is empty, that is, Then this state is also evolved under the Fermi Hubbard Hamiltonian and one can measure the local total density, or its deviation from half-filling at certain time t, i.e. n i (t) = CDW h |e iĤt (n i" +n i# 1)e iĤt | i CDW .
Theorem. For the Fermi Hubbard Model, measurement of local spin density S i (t) defined by Eq. 15 with parameter U 0 and conserved quantities N " + N # N s = x and N " N # = y always equals to measurement of charge density n i (t) defined by Eq. 16 for the same parameter U 0 and conserved quantities N " + N # N s = y and N " N # = x.
That is to say, we state that the charge and spin dynamics are equivalent for the Fermi Hubbard model with same hopping and interaction, with the doping and the spin imbalance exchanging their parameters. For instance, when one measures the spin dynamics of a half-filling Fermi Hubbard model with spin imbalance, it is equivalent to measure charge dynamics of the same Fermi Hubbard model doped away from half-filling with equal spin population. More specifically, for half-filling without spin population imbalance, the spin and charge dynamics defined above are always identical.
The proof of this theorem follows from two steps.
Step 1: We consider a well-known particle-hole transfor-mationP defined aŝ Utilizing the Fermi gas microscope, recently the MIT group has measured the spin transport of the Fermi Hubbard model starting from a spin-density-wave state, and the Princeton group has measured the charge transport of the Fermi Hubbard model starting from a charge-density-wave state. This paper is to prove that these two transport measurements are in fact equivalent to each other. The proof makes use of the particle-hole symmetry of the Fermi Hubbard model and a recently discovered symmetry protected dynamical symmetry. We also present the general condition for the equivalence of spin and charge transport in the Fermi Hubbard model. development of quantum gas microscope is one of the gnificant developments in cold atom physics during rears. It opens up a new avenue for studying strongly ted physics. It is worth emphasizing that the quantum croscope not only allows one to detect the system in h single site resolution, but also allows one to prepare nstate of real space density operators, with which one dy the non-equilibrium dynamics of strongly correlated . For instance, recent the MIT group and the Princeton ave respectively prepared a Fermi Hubbard model inieither a spin-density-wave state or a charge-densityate, and them measure the subsequent spin and charge ics [1,2]. They extract the spin di↵usion constant and di↵usion constant from these two measurements, reely [1,2].
article is to prove that these two transport measurere essentially equivalent to each other. To be specific, write down the Fermi Hubbard model that two groups e with ultracold fermionic atom loaded in square optices, that iŝ J is the hopping amplitude between nearest neighbours of the square lattice, and U is the on-site interaction h. Here we write the interaction term in a particle-hole tric form. Taking J as energy unit, the model is chard by one single parameter U, together with two conquantities N " + N # N s (N s denotes the total number of nown as doping from half-filling and N " N # known imbalance.
MIT experiment starts with a real space spin-density ate [1]. Let us write an ideal version of such state as The Princeton experiment starts with a real space char density wave state [2]. Similarly, we can write a simple v sion of such state that region A is double occupied and reg B is empty, that is, Then this state is also evolved under the Fermi Hubb Hamiltonian and one can measure the local total density, its deviation from half-filling at certain time t, i.e. n i (t) = CDW h |e iĤt (n i" +n i# 1)e iĤt | i CDW .
Theorem. For the Fermi Hubbard Model, measurement local spin density S i (t) defined by Eq. 15 with parameter and conserved quantities N " + N # N s = x and N " N # = always equals to measurement of charge density n i (t) defin by Eq. 16 for the same parameter U 0 and conserved quantit N " + N # N s = y and N " N # = x.
That is to say, we state that the charge and spin dyna ics are equivalent for the Fermi Hubbard model with sa hopping and interaction, with the doping and the spin imb ance exchanging their parameters. For instance, when o measures the spin dynamics of a half-filling Fermi Hubb model with spin imbalance, it is equivalent to measure cha dynamics of the same Fermi Hubbard model doped away fr half-filling with equal spin population. More specifically, half-filling without spin population imbalance, the spin a charge dynamics defined above are always identical.
The proof of this theorem follows from two steps.
Step 1: We consider a well-known particle-hole transf mationP defined aŝ Utilizing the Fermi gas microscope, recently the MIT group has measured the spin transport of Hubbard model starting from a spin-density-wave state, and the Princeton group has measured the ch port of the Fermi Hubbard model starting from a charge-density-wave state. This paper is to prove two transport measurements are in fact equivalent to each other. The proof makes use of the particlemetry of the Fermi Hubbard model and a recently discovered symmetry protected dynamical symm also present the general condition for the equivalence of spin and charge transport in the Fermi Hubb The development of quantum gas microscope is one of the most significant developments in cold atom physics during recent years. It opens up a new avenue for studying strongly correlated physics. It is worth emphasizing that the quantum gas microscope not only allows one to detect the system in situ with single site resolution, but also allows one to prepare an eigenstate of real space density operators, with which one can study the non-equilibrium dynamics of strongly correlated system. For instance, recent the MIT group and the Princeton group have respectively prepared a Fermi Hubbard model initially in either a spin-density-wave state or a charge-densitywave state, and them measure the subsequent spin and charge dynamics [1,2]. They extract the spin di↵usion constant and charge di↵usion constant from these two measurements, respectively [1,2].
This article is to prove that these two transport measurements are essentially equivalent to each other. To be specific, we first write down the Fermi Hubbard model that two groups simulate with ultracold fermionic atom loaded in square optical lattices, that iŝ where J is the hopping amplitude between nearest neighbouring sites of the square lattice, and U is the on-site interaction strength. Here we write the interaction term in a particle-hole symmetric form. Taking J as energy unit, the model is characterized by one single parameter U, together with two conserved quantities N " + N # N s (N s denotes the total number of sites) known as doping from half-filling and N " N # known as spin imbalance. The MIT experiment starts with a real space spin-density wave state [1]. Let us write an ideal version of such state as Here there is no constraint on the choice of region A and B.
The Princeton experiment starts wi density wave state [2]. Similarly, we sion of such state that region A is doub B is empty, that is, Then this state is also evolved unde Hamiltonian and one can measure the its deviation from half-filling at certain n i (t) = CDW h |e iĤt (n i" +n i# Theorem. For the Fermi Hubbard M local spin density S i (t) defined by Eq. and conserved quantities N " + N # N always equals to measurement of char by Eq. 16 for the same parameter U 0 an N " + N # N s = y and N " N # = x.
That is to say, we state that the ch ics are equivalent for the Fermi Hub hopping and interaction, with the dopi ance exchanging their parameters. F measures the spin dynamics of a half model with spin imbalance, it is equiva dynamics of the same Fermi Hubbard m half-filling with equal spin population half-filling without spin population im charge dynamics defined above are alw The proof of this theorem follows fr Step 1: We consider a well-known mationP defined aŝ Utilizing the Fermi gas microscope, recently the MIT group has measured Hubbard model starting from a spin-density-wave state, and the Princeton grou port of the Fermi Hubbard model starting from a charge-density-wave state. two transport measurements are in fact equivalent to each other. The proof ma metry of the Fermi Hubbard model and a recently discovered symmetry pro also present the general condition for the equivalence of spin and charge transp The development of quantum gas microscope is one of the most significant developments in cold atom physics during recent years. It opens up a new avenue for studying strongly correlated physics. It is worth emphasizing that the quantum gas microscope not only allows one to detect the system in situ with single site resolution, but also allows one to prepare an eigenstate of real space density operators, with which one can study the non-equilibrium dynamics of strongly correlated system. For instance, recent the MIT group and the Princeton group have respectively prepared a Fermi Hubbard model initially in either a spin-density-wave state or a charge-densitywave state, and them measure the subsequent spin and charge dynamics [1,2]. They extract the spin di↵usion constant and charge di↵usion constant from these two measurements, respectively [1,2].
This article is to prove that these two transport measurements are essentially equivalent to each other. To be specific, we first write down the Fermi Hubbard model that two groups simulate with ultracold fermionic atom loaded in square optical lattices, that iŝ where J is the hopping amplitude between nearest neighbouring sites of the square lattice, and U is the on-site interaction strength. Here we write the interaction term in a particle-hole symmetric form. Taking J as energy unit, the model is characterized by one single parameter U, together with two conserved quantities N " + N # N s (N s denotes the total number of sites) known as doping from half-filling and N " N # known as spin imbalance. The MIT experiment starts with a real space spin-density wave state [1]. Let us write an ideal version of such state as Here there is no constraint on the choice of region A and B.
The Princeton e density wave state sion of such state B is empty, that is Then this state i Hamiltonian and its deviation from Theorem. For t local spin density and conserved qua always equals to m by Eq. 16 for the s N " + N # N s = y That is to say, ics are equivalent hopping and inter ance exchanging measures the spin model with spin im dynamics of the sa half-filling with eq half-filling withou charge dynamics d The proof of thi Step 1: We con mationP defined Schematics of a real space spin-density-wave state (upper panel) and the real space charge-density-wave state (lower panel) as the initial state for measuring spin and charge dynamics, respectively.
A denotes one sublattice and B denotes the other. Or A denotes a group of domains where spins are polarized up, and B denotes the rest regions where spins are polarized down. This SDW state will then evolve under the FHM Hamiltonian and, at certain time t, one measures the local spin density alonĝ z-direction as Similarly, we can write down an ideal version of chargedensity-wave state that region A is doubly occupied while region B is empty, that is, This state is shown schematically in the lower panel of Fig.  1. The evolution of this state is also governed by the FHM Hamiltonian, and at certain time t, one can measure the local total density, or its deviation from half filling, i.e., Theorem. For the FHM on a square lattice, the measurement of the local spin density S z i (t) defined by Eq. 3 with parameter U 0 and conserved quantities N ↑ + N ↓ − N s = x and N ↑ − N ↓ = y always equals to the measurement of the local charge density n i (t) defined by Eq. 5 with the same U 0 and conserved quantities N ↑ + N ↓ − N s = y and N ↑ − N ↓ = x. arXiv:1808.07225v2 [cond-mat.quant-gas] 23 Sep 2018 2 doping and spin imbalance as (iii) It also changes the spin-density-wave state | i SDW defined in Eq. 2 to the charge-density-wave state defined in Eq. 4. Thus, by using (i-iii) particle-hole symmetryP, the conclusion of Step 1 is that the spin dynamics for starting from spindensity-wave state with interaction parameter U 0 and conserved quantities N " + N # N s = x and N " N # = y is equivalent to the charge dynamics starting from charge-density-wave state with interaction parameter U 0 and conserved quantities N " + N # N s = y and N " N # = x.
Step 2. The step 2 follows from another theorem we proved in Ref. [3] which we termed as "symmetry protected dynamical symmetry". It states as follows: Considering the HamiltonianĤ =Ĥ 0 +V, hereV denotes the interaction term andĤ 0 denotes the rest terms, if we can find an antiunitary operatorŜ =RŴ, whereR is the (antiunitary) time-reversal operator andŴ is a unitary operator that satisfies the following conditions: (i)Ŝ anticommutes withĤ 0 and commutes withV, i.e.
(ii) The initial state | 0 i only acquires a global phase factor underŜ, i.e.Ŝ (iii) The measurement operator is a Hermitian operatorÔ that is even or odd under symmetry operation byŜ, i.e.
density-wave state with interaction parameter U 0 , with the same conserved quantities N " + N # N s = y and N " N # = x. By combining the conclusion from Step 1 and Step 2, the theorem is proved. From this prove, we can also see that the results can be more general in the sense that it does not depend on the specific choice of initial state | i SDW and | i CDW introduced in Eq. 2 and Eq. 4. We can start to measure spin dynamics from | i 1 as ad to measure the charge dynamics from | i 2 as The theorem will still be hold as long as | i 1 and | i 2 satisfy following two conditions: (i) | i 1 and | i 2 are related by the particle-hole symmetrŷ P.
(ii) | i 2 is invariant underŜ =RŴ up to a phase, witĥ R being time-reversal symmetry andŴ being the bipartite lattice symmetry.
So far, as reported in Ref. [1] and Ref. [2], MIT group only reported spin transport coe cient measured for halffilled Hubbard model with zero spin population imbalance, and the Princeton group only reported data doped away from half-filling. Thus these two set of data can not be directly compared. But it will be straightforward to extend their measurement to regions with both finite doping and finite spin population imbalance. Then they can compare measurements and our theorem can be confirmed experimentally. This newly established relation between spin and charge dynamics may also shed light on experiments on cuprates.
Thus, by using (i-iii) particle-hole symmetryP, the conclusion of Step 1 is that the spin dynamics for starting from spindensity-wave state with interaction parameter U 0 and conserved quantities N " + N # N s = x and N " N # = y is equivalent to the charge dynamics starting from charge-density-wave state with interaction parameter U 0 and conserved quantities N " + N # N s = y and N " N # = x.
Step 2. The step 2 follows from another theorem we proved in Ref. [3] which we termed as "symmetry protected dynamical symmetry". It states as follows: Considering the HamiltonianĤ =Ĥ 0 +V, hereV denotes the interaction term andĤ 0 denotes the rest terms, if we can find an antiunitary operatorŜ =RŴ, whereR is the (antiunitary) time-reversal operator andŴ is a unitary operator that satisfies the following conditions: (i)Ŝ anticommutes withĤ 0 and commutes withV, i.e.
(ii) The initial state | 0 i only acquires a global phase factor underŜ, i.e.Ŝ (iii) The measurement operator is a Hermitian operatorÔ that is even or odd under symmetry operation byŜ, i.e.
ad to measure the charge dynamics from | i 2 as The theorem will still be hold as long as | i 1 and | i 2 satisfy following two conditions: (i) | i 1 and | i 2 are related by the particle-hole symmetrŷ P.
(ii) | i 2 is invariant underŜ =RŴ up to a phase, witĥ R being time-reversal symmetry andŴ being the bipartite lattice symmetry.
So far, as reported in Ref. [1] and Ref. [2], MIT group only reported spin transport coe cient measured for halffilled Hubbard model with zero spin population imbalance, and the Princeton group only reported data doped away from half-filling. Thus these two set of data can not be directly compared. But it will be straightforward to extend their measurement to regions with both finite doping and finite spin population imbalance. Then they can compare measurements and our theorem can be confirmed experimentally. This newly established relation between spin and charge dynamics may also shed light on experiments on cuprates.
⇤ hzhai@tsinghua.edu.cn nges the spin-density-wave state | i SDW deo the charge-density-wave state defined in Eq.
g (i-iii) particle-hole symmetryP, the conclus that the spin dynamics for starting from spintate with interaction parameter U 0 and cons N " + N # N s = x and N " N # = y is equivae dynamics starting from charge-density-wave ction parameter U 0 and conserved quantities y and N " N # = x. tep 2 follows from another theorem we proved ch we termed as "symmetry protected dynami-It states as follows: the HamiltonianĤ =Ĥ 0 +V, hereV denotes term andĤ 0 denotes the rest terms, if we can ary operatorŜ =RŴ, whereR is the (antiunirsal operator andŴ is a unitary operator that lowing conditions: mutes withĤ 0 and commutes withV, i.e.
l state | 0 i only acquires a global phase factor asurement operator is a Hermitian operatorÔ dd under symmetry operation byŜ, i.e.
results can be more general in the sense that it does not de-pend on the specific choice of initial state | i SDW and | i CDW introduced in Eq. 2 and Eq. 4. We can start to measure spin dynamics from | i 1 as ad to measure the charge dynamics from | i 2 as The theorem will still be hold as long as | i 1 and | i 2 satisfy following two conditions: (i) | i 1 and | i 2 are related by the particle-hole symmetrŷ P.
(ii) | i 2 is invariant underŜ =RŴ up to a phase, witĥ R being time-reversal symmetry andŴ being the bipartite lattice symmetry.
So far, as reported in Ref. [1] and Ref. [2], MIT group only reported spin transport coe cient measured for halffilled Hubbard model with zero spin population imbalance, and the Princeton group only reported data doped away from half-filling. Thus these two set of data can not be directly compared. But it will be straightforward to extend their measurement to regions with both finite doping and finite spin population imbalance. Then they can compare measurements and our theorem can be confirmed experimentally. This newly established relation between spin and charge dynamics may also shed light on experiments on cuprates.
⇤ hzhai@tsinghua.edu.cn and charge onstant and rements, rert measurebe specific, two groups square opti- . (1) t neighbourinteraction article-hole del is charth two conl number of N # known spin-density ch state as (2) el of Fig. 1. n A and B. and they do onsiders an , A denotes s. For MIT iltonian and one can measure the local total density, or eviation from half-filling at certain time t, i.e.
heorem. For the Fermi Hubbard Model, measurement of l spin density S i (t) defined by Eq. 15 with parameter U 0 conserved quantities N " + N # N s = x and N " N # = y ys equals to measurement of charge density n i (t) defined q. 16 for the same parameter U 0 and conserved quantities N # N s = y and N " N # = x. hat is to say, we state that the charge and spin dynamare equivalent for the Fermi Hubbard model with same ping and interaction, with the doping and the spin imbalexchanging their parameters. For instance, when one sures the spin dynamics of a half-filling Fermi Hubbard el with spin imbalance, it is equivalent to measure charge amics of the same Fermi Hubbard model doped away from -filling with equal spin population. More specifically, for -filling without spin population imbalance, the spin and ge dynamics defined above are always identical. he proof of this theorem follows from two steps. tep 1: We consider a well-known particle-hole transfor-ionP defined aŝ rt measure-be specific, two groups square opti- neighbourinteraction article-hole del is charth two conl number of N # known pin-density h state as (2) n A and B. and they do onsiders an , A denotes s. For MIT re spins are Hamiltonian and one can measure the local total density, or its deviation from half-filling at certain time t, i.e. n i (t) = CDW h |e iĤt (n i" +n i# 1)e iĤt | i CDW .
Theorem. For the Fermi Hubbard Model, measurement of local spin density S i (t) defined by Eq. 15 with parameter U 0 and conserved quantities N " + N # N s = x and N " N # = y always equals to measurement of charge density n i (t) defined by Eq. 16 for the same parameter U 0 and conserved quantities N " + N # N s = y and N " N # = x.
That is to say, we state that the charge and spin dynamics are equivalent for the Fermi Hubbard model with same hopping and interaction, with the doping and the spin imbalance exchanging their parameters. For instance, when one measures the spin dynamics of a half-filling Fermi Hubbard model with spin imbalance, it is equivalent to measure charge dynamics of the same Fermi Hubbard model doped away from half-filling with equal spin population. More specifically, for half-filling without spin population imbalance, the spin and charge dynamics defined above are always identical.
The proof of this theorem follows from two steps.
Step 1: We consider a well-known particle-hole transfor-mationP defined aŝ nt to each other. To be specific, Hubbard model that two groups nic atom loaded in square opti- ude between nearest neighbourand U is the on-site interaction teraction term in a particle-hole energy unit, the model is chareter U, together with two con-(N s denotes the total number of half-filling and N " N # known with a real space spin-density ideal version of such state as the choice of region A and B. be single-connected and they do or instance, if one considers an on a square lattice, A denotes the other sublattices. For MIT up of domains where spins are its deviation from half-filling at certain time t, i.e. n i (t) = CDW h |e iĤt (n i" +n i# 1)e iĤt | i CDW .
Theorem. For the Fermi Hubbard Model, measurement of local spin density S i (t) defined by Eq. 15 with parameter U 0 and conserved quantities N " + N # N s = x and N " N # = y always equals to measurement of charge density n i (t) defined by Eq. 16 for the same parameter U 0 and conserved quantities N " + N # N s = y and N " N # = x.
That is to say, we state that the charge and spin dynamics are equivalent for the Fermi Hubbard model with same hopping and interaction, with the doping and the spin imbalance exchanging their parameters. For instance, when one measures the spin dynamics of a half-filling Fermi Hubbard model with spin imbalance, it is equivalent to measure charge dynamics of the same Fermi Hubbard model doped away from half-filling with equal spin population. More specifically, for half-filling without spin population imbalance, the spin and charge dynamics defined above are always identical.
The proof of this theorem follows from two steps.
Step 1: We consider a well-known particle-hole transfor-mationP defined aŝ essentially equivalent to each other. To be specific, rite down the Fermi Hubbard model that two groups with ultracold fermionic atom loaded in square optis, that is s the hopping amplitude between nearest neighbourof the square lattice, and U is the on-site interaction Here we write the interaction term in a particle-hole c form. Taking J as energy unit, the model is charby one single parameter U, together with two conantities N " + N # N s (N s denotes the total number of wn as doping from half-filling and N " N # known balance. IT experiment starts with a real space spin-density e [1]. Let us write an ideal version of such state as n i (t) = CDW h |e iĤt (n i" +n i# 1)e iĤt | i CDW .
Theorem. For the Fermi Hubbard Model, measurement of local spin density S i (t) defined by Eq. 15 with parameter U 0 and conserved quantities N " + N # N s = x and N " N # = y always equals to measurement of charge density n i (t) defined by Eq. 16 for the same parameter U 0 and conserved quantities N " + N # N s = y and N " N # = x.
That is to say, we state that the charge and spin dynamics are equivalent for the Fermi Hubbard model with same hopping and interaction, with the doping and the spin imbalance exchanging their parameters. For instance, when one measures the spin dynamics of a half-filling Fermi Hubbard model with spin imbalance, it is equivalent to measure charge dynamics of the same Fermi Hubbard model doped away from half-filling with equal spin population. More specifically, for half-filling without spin population imbalance, the spin and charge dynamics defined above are always identical.
The proof of this theorem follows from two steps.
Step 1: We consider a well-known particle-hole transfor-mationP defined aŝ measuring spin and charge dynamics, respectively experiment, A denotes a group of domains where spins are polarized up and B denotes the rest regions where spins are polarized down. Then this state will evolve under the Fermi Hubbard Hamiltonian and then at certain time t, one measures the local spin density alongẑ-direction as The Princeton experiment starts with a real space chargedensity wave state [2]. Similarly, we can write a simple version of such state that region A is double occupied and region B is empty, that is, This state is schematically shown in the lower panel of Fig.  1. Then this state is also evolved under the Fermi Hubbard Hamiltonian and one can measure the local total density, or its deviation from half-filling at certain time t, i.e.
Theorem. For the Fermi Hubbard Model, measurement of local spin density S i (t) defined by Eq. 15 with parameter U 0 and conserved quantities N " + N # N s = x and N " N # = y always equals to measurement of charge density n i (t) defined by Eq. 16 for the same parameter U 0 and conserved quantities N " + N # N s = y and N " N # = x.
That is to say, we state that the charge and spin dynamics are equivalent for the Fermi Hubbard model with same Thus, by using (i-iii) particle-hole symmetryP, the conclusion of Step 1 is that the spin dynamics for starting from spindensity-wave state with interaction parameter U 0 and conserved quantities N " + N # N s = x and N " N # = y is equivalent to the charge dynamics starting from charge-density-wave state with interaction parameter U 0 and conserved quantities N " + N # N s = y and N " N # = x.
Step 2. The step 2 follows from another theorem we proved in Ref. [3] which we termed as "symmetry protected dynamical symmetry". It states as follows: Considering the HamiltonianĤ =Ĥ 0 +V, hereV denotes the interaction term andĤ 0 denotes the rest terms, if we can find an antiunitary operatorŜ =RŴ, whereR is the (antiunitary) time-reversal operator andŴ is a unitary operator that satisfies the following conditions: (i)Ŝ anticommutes withĤ 0 and commutes withV, i.e.

{Ŝ,Ĥ
(ii) The initial state | 0 i only acquires a global phase factor underŜ, i.e.Ŝ (iii) The measurement operator is a Hermitian operatorÔ that is even or odd under symmetry operation byŜ, i.e.
ad to measure the charge dynamics from | i 2 as n i (t) = 2 h |e iĤt (n i" +n i# 1)e iĤt | i 2 .
The theorem will still be hold as long as | i 1 and | i 2 satisfy following two conditions: (i) | i 1 and | i 2 are related by the particle-hole symmetrŷ P.
(ii) | i 2 is invariant underŜ =RŴ up to a phase, witĥ R being time-reversal symmetry andŴ being the bipartite lattice symmetry.
So far, as reported in Ref. [1] and Ref. [2], MIT group only reported spin transport coe cient measured for halffilled Hubbard model with zero spin population imbalance, and the Princeton group only reported data doped away from half-filling. Thus these two set of data can not be directly compared. But it will be straightforward to extend their measurement to regions with both finite doping and finite spin population imbalance. Then they can compare measurements and our theorem can be confirmed experimentally. This newly established relation between spin and charge dynamics may also shed light on experiments on cuprates.
⇤ hzhai@tsinghua.edu.cn doping and spin imbalance as (iii) It also changes the spin-density-wave state | i SDW defined in Eq. 2 to the charge-density-wave state defined in Eq. 4. Thus, by using (i-iii) particle-hole symmetryP, the conclusion of Step 1 is that the spin dynamics for starting from spindensity-wave state with interaction parameter U 0 and conserved quantities N " + N # N s = x and N " N # = y is equivalent to the charge dynamics starting from charge-density-wave state with interaction parameter U 0 and conserved quantities N " + N # N s = y and N " N # = x.
Step 2. The step 2 follows from another theorem we proved in Ref. [3] which we termed as "symmetry protected dynamical symmetry". It states as follows: Considering the HamiltonianĤ =Ĥ 0 +V, hereV denotes the interaction term andĤ 0 denotes the rest terms, if we can find an antiunitary operatorŜ =RŴ, whereR is the (antiunitary) time-reversal operator andŴ is a unitary operator that satisfies the following conditions: (i)Ŝ anticommutes withĤ 0 and commutes withV, i.e.

{Ŝ,Ĥ
(ii) The initial state | 0 i only acquires a global phase factor underŜ, i.e.Ŝ (iii) The measurement operator is a Hermitian operatorÔ that is even or odd under symmetry operation byŜ, i.e.
0 same conserved quantities N " + N # N s = y and N " N # = x. By combining the conclusion from Step 1 and Step 2, the theorem is proved. From this prove, we can also see that the results can be more general in the sense that it does not depend on the specific choice of initial state | i SDW and | i CDW introduced in Eq. 2 and Eq. 4. We can start to measure spin dynamics from | i 1 as S i (t) = 1 h |e iĤt (n i" n i# )e iĤt | i 1 .
The theorem will still be hold as long as | i 1 and | i 2 satisfy following two conditions: (i) | i 1 and | i 2 are related by the particle-hole symmetrŷ P.
(ii) | i 2 is invariant underŜ =RŴ up to a phase, witĥ R being time-reversal symmetry andŴ being the bipartite lattice symmetry.
So far, as reported in Ref. [1] and Ref. [2], MIT group only reported spin transport coe cient measured for halffilled Hubbard model with zero spin population imbalance, and the Princeton group only reported data doped away from half-filling. Thus these two set of data can not be directly compared. But it will be straightforward to extend their measurement to regions with both finite doping and finite spin population imbalance. Then they can compare measurements and our theorem can be confirmed experimentally. This newly established relation between spin and charge dynamics may also shed light on experiments on cuprates.
(iii) It also changes the spin-density-wave state | i SDW defined in Eq. 2 to the charge-density-wave state defined in Eq. 4. Thus, by using (i-iii) particle-hole symmetryP, the conclusion of Step 1 is that the spin dynamics for starting from spindensity-wave state with interaction parameter U 0 and conserved quantities N " + N # N s = x and N " N # = y is equivalent to the charge dynamics starting from charge-density-wave state with interaction parameter U 0 and conserved quantities N " + N # N s = y and N " N # = x.
Step 2. The step 2 follows from another theorem we proved in Ref. [3] which we termed as "symmetry protected dynamical symmetry". It states as follows: Considering the HamiltonianĤ =Ĥ 0 +V, hereV denotes the interaction term andĤ 0 denotes the rest terms, if we can find an antiunitary operatorŜ =RŴ, whereR is the (antiunitary) time-reversal operator andŴ is a unitary operator that satisfies the following conditions: (i)Ŝ anticommutes withĤ 0 and commutes withV, i.e.
(ii) The initial state | 0 i only acquires a global phase factor underŜ, i.e.Ŝ (iii) The measurement operator is a Hermitian operatorÔ that is even or odd under symmetry operation byŜ, i.e.
results can be more general in the sense that it does not de-pend on the specific choice of initial state | i SDW and | i CDW introduced in Eq. 2 and Eq. 4. We can start to measure spin dynamics from | i 1 as ad to measure the charge dynamics from | i 2 as The theorem will still be hold as long as | i 1 and | i 2 satisfy following two conditions: (i) | i 1 and | i 2 are related by the particle-hole symmetrŷ P.
(ii) | i 2 is invariant underŜ =RŴ up to a phase, witĥ R being time-reversal symmetry andŴ being the bipartite lattice symmetry.
So far, as reported in Ref. [1] and Ref. [2], MIT group only reported spin transport coe cient measured for halffilled Hubbard model with zero spin population imbalance, and the Princeton group only reported data doped away from half-filling. Thus these two set of data can not be directly compared. But it will be straightforward to extend their measurement to regions with both finite doping and finite spin population imbalance. Then they can compare measurements and our theorem can be confirmed experimentally. This newly established relation between spin and charge dynamics may also shed light on experiments on cuprates.
⇤ hzhai@tsinghua.edu.cn e total number of N " N # known ace spin-density of such state as (2) r panel of Fig. 1. region A and B. ected and they do one considers an attice, A denotes attices. For MIT model with spin imbalance, it is equivalent to measure charge dynamics of the same Fermi Hubbard model doped away from half-filling with equal spin population. More specifically, for half-filling without spin population imbalance, the spin and charge dynamics defined above are always identical.
The proof of this theorem follows from two steps.
Step 1: We consider a well-known particle-hole transfor-mationP defined aŝ ace spin-density f such state as (2) region A and B. cted and they do one considers an ttice, A denotes ttices. For MIT where spins are model with spin imbalance, it is equivalent to measure charge dynamics of the same Fermi Hubbard model doped away from half-filling with equal spin population. More specifically, for half-filling without spin population imbalance, the spin and charge dynamics defined above are always identical. The proof of this theorem follows from two steps.
Step 1: We consider a well-known particle-hole transfor-mationP defined aŝ starts with a real space spin-density ite an ideal version of such state as nt on the choice of region A and B. ve to be single-connected and they do ze. For instance, if one considers an state on a square lattice, A denotes otes the other sublattices. For MIT group of domains where spins are dynamics of the same Fermi Hubbard model doped away from half-filling with equal spin population. More specifically, for half-filling without spin population imbalance, the spin and charge dynamics defined above are always identical. The proof of this theorem follows from two steps.
Step 1: We consider a well-known particle-hole transfor-mationP defined aŝ e MIT experiment starts with a real space spin-density state [1]. Let us write an ideal version of such state as there is no constraint on the choice of region A and B. of them does not have to be single-connected and they do ave to have equal size. For instance, if one considers an ) anti-ferromagnetic state on a square lattice, A denotes sublattice and B denotes the other sublattices. For MIT riment, A denotes a group of domains where spins are dynamics of the same Fermi Hubbard model doped away from half-filling with equal spin population. More specifically, for half-filling without spin population imbalance, the spin and charge dynamics defined above are always identical. The proof of this theorem follows from two steps.
Step 1: We consider a well-known particle-hole transfor-mationP defined aŝ This state is schematically shown in the lower panel of Fig.  1. Then this state is also evolved under the Fermi Hubbard Hamiltonian and one can measure the local total density, or its deviation from half-filling at certain time t, i.e. n i (t) = CDW h |e iĤt (n i" +n i# 1)e iĤt | i CDW .
Theorem. For the Fermi Hubbard Model, measurement of local spin density S i (t) defined by Eq. 15 with parameter U 0 and conserved quantities N " + N # N s = x and N " N # = y always equals to measurement of charge density n i (t) defined by Eq. 16 for the same parameter U 0 and conserved quantities N " + N # N s = y and N " N # = x.
That is to say, we state that the charge and spin dynamics are equivalent for the Fermi Hubbard model with same = sublattices. This transformation does following things: (i) It leaves the hopping term invariant, and inverts the sign of interaction term by change U to U. (ii) Moreover, it interchanges doping and spin imbalance as (iii) It also changes the spin-density-wave state | i SDW defined in Eq. 2 to the charge-density-wave state defined in Eq. 4. Thus, by using (i-iii) particle-hole symmetryP, the conclusion of Step 1 is that the spin dynamics for starting from spindensity-wave state with interaction parameter U 0 and conserved quantities N " + N # N s = x and N " N # = y is equivalent to the charge dynamics starting from charge-density-wave state with interaction parameter U 0 and conserved quantities N " + N # N s = y and N " N # = x.
Step 2. The step 2 follows from another theorem we proved in Ref. [3] which we termed as "symmetry protected dynamical symmetry". It states as follows: Considering the HamiltonianĤ =Ĥ 0 +V, hereV denotes the interaction term andĤ 0 denotes the rest terms, if we can find an antiunitary operatorŜ =RŴ, whereR is the (antiunitary) time-reversal operator andŴ is a unitary operator that satisfies the following conditions: (i)Ŝ anticommutes withĤ 0 and commutes withV, i.e.
(ii) The initial state | 0 i only acquires a global phase factor underŜ, i.e.Ŝ (iii) The measurement operator is a Hermitian operatorÔ that is even or odd under symmetry operation byŜ, i.e.
starting from charge-density-wave with interaction parameter U 0 equals to the charge dynamics from the same chargedensity-wave state with interaction parameter U 0 , with the same conserved quantities N " + N # N s = y and N " N # = x.
By combining the conclusion from Step 1 and Step 2, the theorem is proved. From this prove, we can also see that the results can be more general in the sense that it does not depend on the specific choice of initial state | i SDW and | i CDW introduced in Eq. 2 and Eq. 4. We can start to measure spin dynamics from | i 1 as S i (t) = 1 h |e iĤt (n i" n i# )e iĤt | i 1 .
The theorem will still be hold as long as | i 1 and | i 2 satisfy following two conditions: (i) | i 1 and | i 2 are related by the particle-hole symmetrŷ P.
(ii) | i 2 is invariant underŜ =RŴ up to a phase, witĥ R being time-reversal symmetry andŴ being the bipartite lattice symmetry.
So far, as reported in Ref. [1] and Ref. [2], MIT group only reported spin transport coe cient measured for halffilled Hubbard model with zero spin population imbalance, and the Princeton group only reported data doped away from half-filling. Thus these two set of data can not be directly compared. But it will be straightforward to extend their measurement to regions with both finite doping and finite spin population imbalance. Then they can compare measurements and our theorem can be confirmed experimentally. This newly established relation between spin and charge dynamics may also shed light on experiments on cuprates.
where i = (i x , i y ) labels each site. This transformation leaves spin-up unchanged but makes a particle-hole transformation for spin-down particle, accompanied by a sign change on one sublattices. This transformation does following things: (i) It leaves the hopping term invariant, and inverts the sign of interaction term by change U to U. (ii) Moreover, it interchanges doping and spin imbalance as (iii) It also changes the spin-density-wave state | i SDW defined in Eq. 2 to the charge-density-wave state defined in Eq. 4. Thus, by using (i-iii) particle-hole symmetryP, the conclusion of Step 1 is that the spin dynamics for starting from spindensity-wave state with interaction parameter U 0 and conserved quantities N " + N # N s = x and N " N # = y is equivalent to the charge dynamics starting from charge-density-wave state with interaction parameter U 0 and conserved quantities N " + N # N s = y and N " N # = x.
Step 2. The step 2 follows from another theorem we proved in Ref. [3] which we termed as "symmetry protected dynamical symmetry". It states as follows: Considering the HamiltonianĤ =Ĥ 0 +V, hereV denotes the interaction term andĤ 0 denotes the rest terms, if we can find an antiunitary operatorŜ =RŴ, whereR is the (antiunitary) time-reversal operator andŴ is a unitary operator that satisfies the following conditions: (i)Ŝ anticommutes withĤ 0 and commutes withV, i.e.

{Ŝ,Ĥ
(ii) The initial state | 0 i only acquires a global phase factor underŜ, i.e.Ŝ (iii) The measurement operator is a Hermitian operatorÔ that is even or odd under symmetry operation byŜ, i.e.
holes and only adds the extra minus sign on one sublattice to both two spin components. It is straightforward to check that with this choice ofŴ and with the initial state chosen as the charge-density-wave state, the condition (i)-(iii) are satisfied. Moreover, the two conserved quantities N " + N # N s and N " N # are both invariant underŴ. Thus, the conclusion of Step 2 is that the charge dynamics starting from charge-density-wave with interaction parameter U 0 equals to the charge dynamics from the same chargedensity-wave state with interaction parameter U 0 , with the same conserved quantities N " + N # N s = y and N " N # = x.
By combining the conclusion from Step 1 and Step 2, the theorem is proved. From this prove, we can also see that the results can be more general in the sense that it does not depend on the specific choice of initial state | i SDW and | i CDW introduced in Eq. 2 and Eq. 4. We can start to measure spin dynamics from | i 1 as ad to measure the charge dynamics from | i 2 as The theorem will still be hold as long as | i 1 and | i 2 satisfy following two conditions: (i) | i 1 and | i 2 are related by the particle-hole symmetrŷ P.
(ii) | i 2 is invariant underŜ =RŴ up to a phase, witĥ R being time-reversal symmetry andŴ being the bipartite lattice symmetry.
So far, as reported in Ref. [1] and Ref. [2], MIT group only reported spin transport coe cient measured for halffilled Hubbard model with zero spin population imbalance, and the Princeton group only reported data doped away from half-filling. Thus these two set of data can not be directly compared. But it will be straightforward to extend their measurement to regions with both finite doping and finite spin population imbalance. Then they can compare measurements and our theorem can be confirmed experimentally. This newly established relation between spin and charge dynamics may also shed light on experiments on cuprates.
⇤ hzhai@tsinghua.edu.cn antities N " + N # N s = x and N " N # = y measurement of charge density n i (t) defined same parameter U 0 and conserved quantities and N " N # = x. we state that the charge and spin dynamt for the Fermi Hubbard model with same raction, with the doping and the spin imbaltheir parameters. For instance, when one dynamics of a half-filling Fermi Hubbard mbalance, it is equivalent to measure charge ame Fermi Hubbard model doped away from qual spin population. More specifically, for ut spin population imbalance, the spin and defined above are always identical. is theorem follows from two steps. nsider a well-known particle-hole transforas and conserved quantities N " + N # N s = x and N " N # = y always equals to measurement of charge density n i (t) defined by Eq. 16 for the same parameter U 0 and conserved quantities N " + N # N s = y and N " N # = x. That is to say, we state that the charge and spin dynamics are equivalent for the Fermi Hubbard model with same hopping and interaction, with the doping and the spin imbalance exchanging their parameters. For instance, when one measures the spin dynamics of a half-filling Fermi Hubbard model with spin imbalance, it is equivalent to measure charge dynamics of the same Fermi Hubbard model doped away from half-filling with equal spin population. More specifically, for half-filling without spin population imbalance, the spin and charge dynamics defined above are always identical.
The proof of this theorem follows from two steps.
Step 1: We consider a well-known particle-hole transfor-mationP defined aŝ nearest neighbouron-site interaction in a particle-hole the model is charther with two conthe total number of d N " N # known space spin-density of such state as i. (2) f region A and B. nected and they do f one considers an lattice, A denotes lattices. For MIT s where spins are always equals to measurement of charge density n i (t) defined by Eq. 16 for the same parameter U 0 and conserved quantities N " + N # N s = y and N " N # = x. That is to say, we state that the charge and spin dynamics are equivalent for the Fermi Hubbard model with same hopping and interaction, with the doping and the spin imbalance exchanging their parameters. For instance, when one measures the spin dynamics of a half-filling Fermi Hubbard model with spin imbalance, it is equivalent to measure charge dynamics of the same Fermi Hubbard model doped away from half-filling with equal spin population. More specifically, for half-filling without spin population imbalance, the spin and charge dynamics defined above are always identical.
The proof of this theorem follows from two steps.
Step 1: We consider a well-known particle-hole transfor-mationP defined aŝ amplitude between nearest neighbourlattice, and U is the on-site interaction e the interaction term in a particle-hole ng J as energy unit, the model is chare parameter U, together with two con-N # N s (N s denotes the total number of from half-filling and N " N # known t starts with a real space spin-density rite an ideal version of such state as raint on the choice of region A and B. have to be single-connected and they do size. For instance, if one considers an tic state on a square lattice, A denotes enotes the other sublattices. For MIT s a group of domains where spins are always equals to measurement of charge density n i (t) defined by Eq. 16 for the same parameter U 0 and conserved quantities N " + N # N s = y and N " N # = x.
That is to say, we state that the charge and spin dynamics are equivalent for the Fermi Hubbard model with same hopping and interaction, with the doping and the spin imbalance exchanging their parameters. For instance, when one measures the spin dynamics of a half-filling Fermi Hubbard model with spin imbalance, it is equivalent to measure charge dynamics of the same Fermi Hubbard model doped away from half-filling with equal spin population. More specifically, for half-filling without spin population imbalance, the spin and charge dynamics defined above are always identical.
The proof of this theorem follows from two steps.
The Princeton experiment starts with a real space chargedensity wave state [2]. Similarly, we can write a simple version of such state that region A is double occupied and region B is empty, that is, This state is schematically shown in the lower panel of Fig.  1. Then this state is also evolved under the Fermi Hubbard Hamiltonian and one can measure the local total density, or its deviation from half-filling at certain time t, i.e. n i (t) = CDW h |e iĤt (n i" +n i# 1)e iĤt | i CDW .
Theorem. For the Fermi Hubbard Model, measurement of local spin density S i (t) defined by Eq. 15 with parameter U 0 and conserved quantities N " + N # N s = x and N " N # = y always equals to measurement of charge density n i (t) defined by Eq. 16 for the same parameter U 0 and conserved quantities N " + N # N s = y and N " N # = x.
That is to say, we state that the charge and spin dynamics are equivalent for the Fermi Hubbard model with same spin-up unchanged but makes a particle-hole transformation for spin-down particle, accompanied by a sign change on one sublattices. This transformation does following things: (i) It leaves the hopping term invariant, and inverts the sign of interaction term by change U to U. (ii) Moreover, it interchanges doping and spin imbalance as (iii) It also changes the spin-density-wave state | i SDW defined in Eq. 2 to the charge-density-wave state defined in Eq. 4. Thus, by using (i-iii) particle-hole symmetryP, the conclusion of Step 1 is that the spin dynamics for starting from spindensity-wave state with interaction parameter U 0 and conserved quantities N " + N # N s = x and N " N # = y is equivalent to the charge dynamics starting from charge-density-wave state with interaction parameter U 0 and conserved quantities N " + N # N s = y and N " N # = x.
Step 2. The step 2 follows from another theorem we proved in Ref. [3] which we termed as "symmetry protected dynamical symmetry". It states as follows: Considering the HamiltonianĤ =Ĥ 0 +V, hereV denotes the interaction term andĤ 0 denotes the rest terms, if we can find an antiunitary operatorŜ =RŴ, whereR is the (antiunitary) time-reversal operator andŴ is a unitary operator that satisfies the following conditions: (i)Ŝ anticommutes withĤ 0 and commutes withV, i.e.
(ii) The initial state | 0 i only acquires a global phase factor underŜ, i.e.Ŝ (iii) The measurement operator is a Hermitian operatorÔ that is even or odd under symmetry operation byŜ, i.e.
N # are both invariant under W. Thus, the conclusion of Step 2 is that the charge dynamics starting from charge-density-wave with interaction parameter U 0 equals to the charge dynamics from the same chargedensity-wave state with interaction parameter U 0 , with the same conserved quantities N " + N # N s = y and N " N # = x.
By combining the conclusion from Step 1 and Step 2, the theorem is proved. From this prove, we can also see that the results can be more general in the sense that it does not depend on the specific choice of initial state | i SDW and | i CDW introduced in Eq. 2 and Eq. 4. We can start to measure spin dynamics from | i 1 as S i (t) = 1 h |e iĤt (n i" n i# )e iĤt | i 1 .
The theorem will still be hold as long as | i 1 and | i 2 satisfy following two conditions: (i) | i 1 and | i 2 are related by the particle-hole symmetrŷ P.
(ii) | i 2 is invariant underŜ =RŴ up to a phase, witĥ R being time-reversal symmetry andŴ being the bipartite lattice symmetry.
So far, as reported in Ref. [1] and Ref. [2], MIT group only reported spin transport coe cient measured for halffilled Hubbard model with zero spin population imbalance, and the Princeton group only reported data doped away from half-filling. Thus these two set of data can not be directly compared. But it will be straightforward to extend their measurement to regions with both finite doping and finite spin population imbalance. Then they can compare measurements and our theorem can be confirmed experimentally. This newly established relation between spin and charge dynamics may also shed light on experiments on cuprates.

Particle-Hole Transformation
Step 2: That is to say, the kind of charge and spin dynamics defined above are equivalent for the FHM of the same hopping and interaction parameters, with the doping and the spin imbalance quantities interchanging with each other. For instance, if one measures the spin dynamics of Eq. 3 for a half-filled FHM with spin imbalance, it is equivalent to measuring the charge dynamics of Eq. 5 for the same FHM with balanced spin population yet doped away from half filling. In particular, for a half-filled and spin-balanced FHM, the spin and charge dynamics defined above are always identical.
The proof of this theorem follows from two steps.
Step 1: We consider a well-known particle-hole transfor-mationP defined as [3] c i↑ →ĉ i↑ ,ĉ † i↑ →ĉ † i↑ (6) where i = (i x , i y ) labels each site. This transformation leaves the spin-up field operators unchanged while makes a particlehole transformation for the spin-down ones, accompanied by a sign change on one sublattice. This transformation does the following things. (i) It leaves the hopping term invariant, and inverts the sign of interaction term, i.e. U → −U. (ii) Moreover, it interchanges the local spin density with the local particle density deviation from unity, (n i↑ −n i↓ ) −→ (n i↑ +n i↓ − 1), and it also interchanges the doping and spin imbalance of the system (iii) It also transforms the spin-density-wave state |Ψ SDW defined in Eq. 2 to the charge-density-wave state |Ψ CDW defined in Eq. 4. As a result, the conclusion of the Step 1 is that the spin dynamics starting from the spin-density-wave state of Eq. 2 with interaction parameter U 0 and conserved quantities N ↑ + N ↓ − N s = x and N ↑ − N ↓ = y is equivalent to the charge dynamics starting from the charge-density-wave state of Eq. 4 with interaction parameter −U 0 and conserved quantities N ↑ + N ↓ − N s = y and N ↑ − N ↓ = x.
Step 2. This step follows from another theorem we proved in Ref. [4], which we termed as "symmetry protected dynamical symmetry". It states as follows.
then we can conclude Here O(t) ±U denotes the expectation value ofÔ under the time-dependent wave function |Ψ(t) = e iĤt |Ψ with interaction strength ±U inĤ, respectively.
UnlikeP, this transformation does not exchange particles and holes. Instead, it only introduces an extra minus sign on one sublattice for both two spin components. It is straightforward to check that with this choice ofŴ and with the initial state chosen as the charge-density-wave state defined in Eq. 4, conditions (i)-(iii) are satisfied. Moreover, the two conserved quantities N ↑ + N ↓ − N s and N ↑ − N ↓ are both invariant under W.
Thus, the conclusion of the Step 2 is that the charge dynamics starting from the charge-density-wave state of Eq. 4 with interaction parameter −U 0 equals the charge dynamics from the same charge-density-wave state with interaction parameter U 0 , with the same conserved quantities N ↑ + N ↓ − N s = y and N ↑ − N ↓ = x.
Combining the conclusions from the Step 1 and the Step 2, the theorem is now proved. The theorem, as well as two steps of proof, is schematically shown in Fig. 2. From the proof, we can also see that the results can be more general in the sense that it does not depend on the specific choices of the initial