Deriving density-matrix functionals for excited states

2025-04-22 0 0 544.92KB 13 页 10玖币

侵权投诉

Julia Liebert1, 2 and Christian Schilling1, 2, ∗

1Department of Physics, Arnold Sommerfeld Center for Theoretical Physics,

Ludwig-Maximilians-Universit¨at M¨unchen, Theresienstrasse 37, 80333 M¨unchen, Germany

2Munich Center for Quantum Science and Technology (MCQST), Schellingstrasse 4, 80799 M¨unchen, Germany

(Dated: May 22, 2023)

We initiate the recently proposed w-ensemble one-particle reduced density matrix functional

theory (w-RDMFT) by deriving the ﬁrst functional approximations and illustrate how excitation

energies can be calculated in practice. For this endeavour, we ﬁrst study the symmetric Hubbard

dimer, constituting the building block of the Hubbard model, for which we execute the Levy-Lieb

constrained search. Second, due to the particular suitability of w-RDMFT for describing Bose-

Einstein condensates, we demonstrate three conceptually diﬀerent approaches for deriving the uni-

versal functional in a homogeneous Bose gas for arbitrary pair interaction in the Bogoliubov regime.

Remarkably, in both systems the gradient of the functional is found to diverge repulsively at the

boundary of the functional’s domain, extending the recently discovered Bose-Einstein condensation

force to excited states. Our ﬁndings highlight the physical relevance of the generalized exclusion

principle for fermionic and bosonic mixed states and the curse of universality in functional theories.

I. INTRODUCTION

The Penrose and Onsager criterion [1] identiﬁes

one-body reduced density matrix functional theory

(RDMFT) as a potentially ideal approach to describe

Bose-Einstein condensation (BEC). Indeed, BEC is

present whenever one eigenvalue of the one-particle re-

duced density matrix (1RDM) is proportional to the to-

tal particle number N. The 1RDM in turn is the natural

variable in RDMFT, which is, at least in-principle, an

exact approach to describe interacting N-particle quan-

tum systems. The absence of complete condensation for

interacting bosons is strongly tied to the concept of quan-

tum depletion which characterizes the fraction of bosons

outside the BEC ground state [2]. It has been one of the

recent achievements of RDMFT [3, 4] to provide a uni-

versal explanation for quantum depletion, independently

of the microscopic details of the system: The distinctive

shape of the universal functional reveals the existence of a

BEC force which explains from a purely geometric point

of view why not all bosons can condense.

This and other recent progress in the ﬁeld of ground

state RDMFT for bosons [3–7] and w-ensemble RDMFT

for excited states [8–10] urges us to systematically de-

rive in this paper the ﬁrst functionals for w-ensemble

RDMFT. To initiate the development of w-ensemble

functionals in diﬀerent ﬁelds of physics we derive ana-

lytically the universal functional for both the symmetric

Hubbard dimer with on-site interaction and the homo-

geneous Bose gas in the Bogoliubov regime. The lat-

ter functional constitutes the bosonic analogue of the

Hartree-Fock functional for fermions [11]. Both systems

do not only allow us to obtain an analytic expression

for the universal functional but are also well-suited to

illustrate conceptually diﬀerent routes for their deriva-

tion. Besides illustrating the application of w-ensemble

∗c.schilling@physik.uni-muenchen.de

RDMFT for the ﬁrst time, we extend the concept of a

BEC force based on the diverging gradient of the func-

tional close to the boundary of its domain to excited state

RDMFT.

The paper is structured as follows. To keep our work

self-contained, we recall in Sec. II the basic formalism of

w-ensemble RDMFT. We illustrate w-ensemble RDMFT

and derive the exact universal functionals for the sym-

metric Hubbard dimer in Sec. III and the homogeneous

BECs in Sec. IV.

II. RECAP OF ENSEMBLE RDMFT FOR

NEUTRAL EXCITATIONS

Before deriving the ﬁrst w-ensemble functionals in

Secs. III and IV we introduce in this section the required

foundational concepts of w-ensemble RDMFT which has

recently been proposed by us for bosons in Ref. [10] and

for fermions in Ref. [8, 9]. From a general perspective,

RDMFT is based on the observation that in each ﬁeld of

physics the interaction Wbetween the particles is usually

kept ﬁxed. As a consequence, one considers all Hamil-

tonians ˆ

Hon the D-dimensional N-boson Hilbert space

HNthat are parameterized by the one-particle Hamilto-

nian ˆ

H(ˆ

h)≡ˆ

h+ˆ

W . (1)

To arrive at a corresponding functional theory, the en-

semble RDMFT for excited states combines a variational

principle proposed by Gross, Oliveira and Kohn (GOK)

[12–14] with the Levy-Lieb constrained search [15, 16].

In the GOK variational principle, the weighted sum

Ew≡PjwjEjof the increasingly ordered eigenener-

gies Ei,E1≤E2≤. . . ≤ED, of the Hamiltonian ˆ

and decreasingly ordered weights w1≥w2≥... ≥wD

with PD

i=1 wi= 1 follows from minimizing the energy

TrN[ˆ

Γˆ

H] over all N-boson/fermion density operators

arXiv:2210.00964v3 [cond-mat.quant-gas] 19 May 2023

with spectrum given by the weight vector, spec↓(ˆ

Γ) = w.

This spectral condition deﬁnes the set EN(w) of N-

particle density operators

EN(w)≡ {ˆ

Γ|ˆ

Γ = ˆ

Γ†,ˆ

Γ≥0,TrN[ˆ

Γ] = 1,spec↓(ˆ

Γ) = w}.

(2)

Then, the GOK variational principle reads [12]

Ew≡

j=1

wjEj= min

Γ∈EN(w)

TrNhˆ

Γˆ

Hi.(3)

Applying the Levy-Lieb constrained search [15, 16] to

this variational principle for excited states yields

Ew(ˆ

h) = min

Γ∈EN(w)

TrN[(ˆ

h+ˆ

W)ˆ

Γ]

= min

ˆγ∈E1

N(w)hmin

EN(w)3ˆ

Γ7→ˆγ

TrN[(ˆ

h+ˆ

W)ˆ

Γ]i

= min

ˆγ∈E1

N(w)hTr1[ˆ

hˆγ] + Fw(ˆγ)i,

(4)

where we deﬁned the universal functional Fwwhose do-

main is given by E1

N(w) = NTrN−1(EN(w)). For sim-

plicity we used in Eq. (3) the same symbol for the one-

particle Hamiltonian ˆ

hon the N-particle and the one-

particle Hilbert space. It is worth stressing here that the

set E1

N(w) is typically not convex [17, 18]. Moreover, Fw

is usually not (locally) convex, i.e., there exist convex re-

gions on which Fwis not convex, even for those special

cases with convex domain E1

N(w). A well-known example

for the latter scenario is the ground state Hubbard dimer

functional for the singlet spin sector which is recovered

for the weight vector w= (1,0, . . .) [6, 19, 20].

One of the main achievements of Refs. [8–10, 18]

was to overcome the too intricate w-ensemble N-

representability constraints that deﬁne the domain

N(w) of the universal functional Fw. In analogy to Val-

one’s ground state RDMFT [21] this was achieved by per-

forming an exact convex relaxation. Indeed the energy

Ew(ˆ

h) remains unaﬀected by replacing the non-convex

sets EN(w) and E1

N(w) by their respective convex hulls,

EN(w)≡conv EN(w),

N(w)≡NTrN−1EN(w)= conv E1

N(w).(5)

In particular, inserting Eq. (5) in the Levy-Lieb con-

strained search replaces Fwby its lower convex envelope

Fw(ˆγ)≡min

(w)3ˆ

Γ7→ˆγ

TrN[ˆ

Γˆ

= conv (Fw(ˆγ)) .(6)

It was exactly this convex relaxation which allowed us in

Refs. [8–10] to obtain a feasible functional theory thanks

to a comprehensive characterization of the set E1

N(w) for

bosons and fermions. To be more speciﬁc, we derived a

compact description of the corresponding spectral set

Σ(w)≡spec( E1

N(w)) ,(7)

in terms of ﬁnitely many linear constraints [8–10, 18].

Those conditions represent nothing else than a complete

generalization of Pauli’s exclusion principle to mixed

states of bosons and fermions, respectively. Therefore,

the challenging task addressed in this paper is to derive

the ﬁrst w-ensemble functionals. This should initiate

the development of more elaborated functional approxi-

mations in analogy to the developments of ground state

RDMFT functionals for fermions [22–34] which were in-

spired by or even based on the Hartree-Fock functional

introduced in the seminal work by Lieb [11].

III. DERIVATION OF THE UNIVERSAL

FUNCTIONAL FOR THE SYMMETRIC

BOSE-HUBBARD DIMER

As our ﬁrst proof-of-principle for w-ensemble RDMFT,

we derive in this section the exact w-ensemble func-

tional for the symmetric Bose-Hubbard dimer. Due to

the equivalence of this system to the Fermi-Hubbard

dimer for two electrons in their singlet sector, the cor-

responding results can be translated to fermionic w-

ensemble RDMFT in a straightforward manner. Un-

derstanding the w-ensemble functional and its domain

for this model is particularly interesting since the Bose-

Hubbard dimer constitutes the building block of the Hub-

bard model widely used in the ﬁeld of ultracold quan-

tum gases. Besides this, the Hubbard dimer model is

widely used throughout RDMFT and density functional

theory to illustrate conceptual aspects of functional the-

ory [19, 20, 35–50]. The Hamiltonian for spinless bosons

on two lattice sites reads

H=−tˆa†

LˆaR+ ˆa†

RˆaL+UX

j=L,R

ˆnj(ˆnj−1) ,(8)

where the ﬁrst term describes hopping at a rate tbe-

tween the left (L) and right (R) lattice site. The second

term describes the Hubbard on-site interaction with cou-

pling strength Uand ˆnj= ˆa†

jˆajis the occupation number

operator.

In the case of periodic boundary conditions, the Hamil-

tonian in Eq. (8) is translationally invariant. This implies

that the total momentum Pis conserved, i.e., Pis a good

quantum number. As a result, the minimization in the

constrained search formalism in Eq. (4) can be restricted

to all ˆ

Γ∈ EN(w, P ) in the chosen symmetry sector with

ﬁxed P. Then, it is possible to establish a separate func-

tional in each symmetry sector instead of a single more

involved functional referring to all P. Moreover, every

1RDM ˆγis diagonal in momentum representation. In the

following, we consider the case of N= 2 spinless bosons

and restrict to repulsive interactions, i.e., U > 0. Then,

the natural occupation numbers are given by the momen-

tum occupation numbers np≥0 restricted through the

normalization Ppnp= 2.

It is also worth noticing that for the symmetric Bose-

Hubbard dimer the translational invariance is equivalent

to the inversion symmetry. To explain this, we now

skip the periodic boundary conditions and instead re-

strict to the even symmetry sector. The corresponding

symmetry-adapted one-boson basis consists of the two

states |ei= (|Li+|Ri)/√2 and |oi= (|Li − |Ri)/√2

which actually coincide with the two one-particle mo-

mentum states. The two-dimensional subspace with even

inversion-symmetry is then spanned by the two basis

states |e, eiand |o, oi. This implies that the 1RDM ˆγ

is diagonal and thus depends on only one free parame-

ter ne, the occupation number of |ei. Thus, the resulting

w-ensemble functional Fwis equivalent to Fwin the sym-

metric Bose-Hubbard dimer with periodic boundary con-

ditions and restricted to P= 0 in (10) with n0replaced

by ne.

In the following, we consider the P= 0 momentum

sector. For two lattice sites, the single particle momen-

tum can take the discrete values pν=πν with ν= 0,1.

We denote the creation (annihilation) operator referring

to the momentum νby ˆa†

ν(ˆaν) and the occupation num-

ber operators by ˆnν= ˆa†

νˆaν. The only two conﬁgurations

satisfying Pν=1,2ν(mod2) = 0 are (0,0) and (1,1) cor-

responding to the two basis states |1i=1

√2(ˆa†

0)2|0iand

|2i=1

√2(ˆa†

1)2|0i, where |0idenotes the vacuum state.

Since (0,0) and (1,1) are the only allowed conﬁgurations

in the P= 0 sector, it follows from Ref. [10] that the

larger value of n0and n1= 2 −n0is bounded from

above by 2w1and the lower one from below by 2w2. In

particular, this means that the domain E1

N(w) of the w-

ensemble functional Fwis already convex,

Σ(w, P = 0) = {n0|0≤2w2≤n0≤2w1≤2}.(9)

Then, minimizing the expectation value Tr2[ˆ

Wˆ

Γ] accord-

ing to the constrained search formalism, where ˆ

Wis the

second term in the Hamiltonian (8), leads to (see Ap-

pendix A)

Fw(n0) = U1−pn0(2 −n0)−4w1w2

=U1−p(n0−2w2) (2w1−n0).(10)

Since Fw(n0) is already convex, it is equal to the relaxed

functional Fw(n0).

The two equivalent expressions of Fw(n0) in Eq. (10)

illustrate two diﬀerent properties of the universal func-

tional. From the ﬁrst line together with (9) it follows

immediately that Fw(n0) is symmetric around n0= 1.

The second expression in (10) emphasizes the diverging

behaviour of the gradient of Fw(n0) at the boundary of

its domain for each weight w1= 1 −w2. Indeed, the

derivative of Fw(n0) with respect to n0diverges at the

boundary ∂Σ of the domain of Fwas



∂Fw(n0)

∂n0∼1

pdist(n0, ∂Σ) .(11)

The sign of the gradient reveals that the corresponding

force is repulsive, i.e., it prevents n0from ever reaching

the boundary ∂Σ.

The universal functional Fw(n0) is illustrated for sev-

eral values of w1in Fig. 1. This also demonstrates the

inclusion relation [9, 10]

w0≺w⇔ E1

N(w0)⊂ E1

N(w).(12)

Indeed, for w0≺w(corresponding here to w0

1≤w1) we

have Σ(w0, P = 0) ⊂Σ(w, P = 0).

FIG. 1. Illustration of the n0dependence of the w-ensemble

functional F

wfor the symmetric Bose-Hubbard dimer with

total momentum P= 0 for U= 1/2 and diﬀerent values of

the weight w1(recall that w1+w2= 1).

To further illustrate w-ensemble RDMFT we calculate

the energy Ewby minimizing the total energy functional

Tr1[ˆ

tˆγ] + Fw(n0), where ˆ

tis given by the ﬁrst term in (8)

and Tr1[ˆ

tˆγ] = −2t(n0−1). Solving

∂

∂n0

(−2t(n0−1) + Fw(n0))n0=˜n0

= 0 (13)

for the minimizer ˜n0and substituting the result into the

energy functional yields for the weighted sum Ewof the

two eigenenergies E1, E2according to Eqs. (3) and (4),

Ew=w1U−p4t2+U2+w2U+p4t2+U2.

(14)

Note that this result is in agreement with the eigenener-

gies E1and E2obtained from an exact diagonalization

of the Hamiltonian ˆ

Hin Eq. (8) (see Appendix A for

further details).

IV. w-RDMFT FOR BOSE-EINSTEIN

CONDENSATES

The application of w-ensemble RDMFT to Bose-

Einstein condensates (BECs) is appealing due to a num-

ber of reasons. First, as already stressed in the introduc-

tion, the Penrose and Onsager criterion [1, 3] for BEC

identiﬁes RDMFT as a particularly suitable approach to

BECs. Second, recent analyses of their ground states

have revealed an intriguing new concept, namely the ex-

istence of a BEC-force [3, 4, 6]. The question arises

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Derivingdensity-matrixfunctionalsforexcitedstatesJuliaLiebert1,2andChristianSchilling1,2,1DepartmentofPhysics,ArnoldSommerfeldCenterforTheoreticalPhysics,Ludwig-Maximilians-UniversitatMunchen,Theresienstrasse37,80333Munchen,Germany2MunichCenterforQuantumScienceandTechnology(MCQST),Schellingstras...

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