Area law of noncritical ground states in 1D long-range interacting systems


Derivation of (pmb{bar{alpha },ge,alpha,-,2})

We here show the proof of (bar{alpha },ge,alpha,-,2) for Hamiltonian (3). More general cases including fermionic systems are given in Supplementary Note 1. For the proof, we estimate the upper bound of

$$parallel {V}_{X,Y}parallel le sum _{i,in,X}sum _{j,in,Y}parallel {h}_{i,j}parallel le Jsum _{i,in,X}sum _{j,in,Y}{r}_{i,j}^{-alpha },$$

where we use the power-law decay of the interaction as (parallel {h}_{i,j}parallel le J/{r}_{i,j}^{alpha }). Let us define dist(XY) = r. Then, we obtain

$$Jsum _{i,in,X}sum _{j,in,Y}{r}_{i,j}^{-alpha }le Jsum_{x,=,0}^{infty }sum_{y,=,0}^{infty }{(r,+,x,+,y)}^{-alpha },$$

where we use the fact that X and Y are concatenated subsets. For arbitrary integer ({r}_{0}in {mathbb{N}}), we have

$$mathop{sum }limits_{x,=,0}^{infty }{(x,+,{r}_{0})}^{-alpha }le {r}_{0}^{-alpha },+,int_{{r}_{0}}^{infty }{x}^{-alpha }dxle frac{alpha}{alpha -1}{r}_{0}^{-alpha +1},$$

and hence

$$Jsum_{x,=,0}^{infty }sum_{y,=,0}^{infty }{(r,+,x,+,y)}^{-alpha }le frac{alpha J}{alpha -2}{r}^{-alpha +2}.$$

We thus prove that VX,Y decays at least faster than rα + 2.

Proof sketch of the main result

We here show the sketch of the proof for area-law inequality (6). The full proof is quite intricate, and we show the details in Supplementary Notes 24. In Fig. 3, we have summarized a flow of the discussions in this section.

Fig. 3: Flowchart of the area-law proof.

The proof consists of several key claims. The details of the proof for these claims are given in Supplementary Notes 2–4.

For the proof, we take the approximate-ground-state-projection (AGSP) approach5,6. The AGSP operator K is roughly given by the operator that satisfies (Kleft|0rightrangle,simeq,left|0rightrangle) and (parallel K(1-left|0rightrangle leftlangle 0right|)parallel,simeq,0). The ground state (left|0rightrangle) does not change by the AGSP K, while any excited state approximately vanishes by K. In more formal definitions, the AGSP is defined by three parameters δK, ϵK, and DK. Let (left|{0}_{K}rightrangle) be a quantum state that does not change by K, namely (Kleft|{0}_{K}rightrangle =left|{0}_{K}rightrangle). Then, the three parameters are defined by the following three inequalities:

$$parallel left|0rightrangle -left|{0}_{K}rightrangle parallel ,le, {delta }_{K},quad parallel K(1-left|{0}_{K}rightrangle leftlangle {0}_{K}right|)parallel,le,{epsilon }_{K},quad {rm{SR}}(K),le,{D}_{K},$$

where SR(K) is the Schmidt rank of K with respect to the given partition Λ = LR. The essential point of this approach is that a good AGSP ensures the existence of a quantum state that has a small Schmidt rank and large overlap with the ground state. It is mathematically formulated by the following statement:

Claim 1

(Proposition 2 in Supplementary Note 2) Let K be an AGSP operator for (left|0rightrangle) with the parameters (δKϵKDK). If we have ({epsilon }_{K}^{2}{D}_{K}le (1/2)), there exists a quantum state (left|psi rightrangle) with ({rm{SR}}(left|psi rightrangle ),le,{D}_{{K}}) such that

$$leftVert left|psi rightrangle -left|0rightrangle rightVert le {epsilon }_{K}sqrt{2{D}_{K}}+{delta }_{K}.$$

(9)

where ({rm{SR}}(left|psi rightrangle )) is the Schmidt rank of (left|psi rightrangle) with respect to the given partition.

From this statement, the primary problem reduces to one of finding a good AGSP to satisfy the condition ({epsilon }_{K}^{2}{D}_{K}le (1/2)).

In the construction of the AGSP operator with the desired properties, we usually utilize a polynomial of the Hamiltonian. The obstacle here is that the long-range interactions induce an infinitely large Schmidt rank in the thermodynamic limit; that is, the Hamiltonian H has the Schmidt rank of poly(n). In order to avoid this, we truncate the long-range interactions of the Hamiltonian. If we truncate all the long-range interactions, the norm difference between the original Hamiltonian and the truncated one is on the order of ({mathcal{O}}(n)), and hence the spectral gap condition cannot be preserved. The first central idea in the proof is to truncate the long-range interaction only around the boundary (see Fig. 4). In more detail, we first decompose the total system into (q + 2) blocks with q an even integer. The edge blocks B0 and Bq + 1 have arbitrary sizes, but the bulk blocks B1B2, …, Bq have the size l (i.e., Bs = l). Then, we truncate all the interactions between nonadjacent blocks, which yields the Hamiltonian Ht as

$${H}_{{rm{t}}},=,mathop{sum }limits_{s,=,0}^{q,+,1}{h}_{s}+mathop{sum }limits_{s,=,0}^{q}{h}_{s,s,+,1},$$

(10)

where hs is the internal interaction in the block Bs, and hs,s + 1 is the interaction between two blocks Bs and Bs + 1. By using notation (4), we have ({h}_{s,s,+,1},=,{V}_{{B}_{s},{B}_{s,+,1}}). In the Hamiltonian Ht, long-range interactions only around the boundary are truncated, and hence the norm difference between the original and the truncated Hamiltonian can be sufficiently small for large l.

Fig. 4: Interaction-truncated Hamiltonian Ht.
figure4

We truncate the long-range interactions only around the boundary. In the figure above, the interactions between nonadjacent blocks (i.e., ({{{B}_{s}}}_{s,=,0}^{7})) are truncated. By this truncation, the properties of the Hamiltonian Ht are proved to be almost the same as the original one H, as long as (q{l}^{-bar{alpha }},lesssim, 1) (see Claim 2).

Claim 2

(Lemmas 3 and 4 in Supplementary Note 2) The norm distance between H and Ht is bounded from above by

$$parallel H-{H}_{{rm{t}}}parallel,le,{g}_{0}q{l}^{-bar{alpha }}.$$

Also, the spectral gap Δt of Ht and the norm difference between (left|0rightrangle) and (left|{0}_{{rm{t}}}rightrangle) are upper-bounded by

$${Delta }_{{rm{t}}}ge Delta -2{g}_{0}q{l}^{-bar{alpha }},quad parallel left|0rightrangle -left|{0}_{{rm{t}}}rightrangle parallel,le, frac{parallel H-{H}_{{rm{t}}}parallel }{Delta -4parallel H-{H}_{{rm{t}}}parallel },$$

where (left|{0}_{{rm{t}}}rightrangle) is the ground state of Ht.

From this statement, if (q{l}^{-bar{alpha }},lesssim ,1), the truncated Hamiltonian Ht possesses almost the same properties as the original one.

The second technical obstacle is the norm of the Hamiltonian. The gap condition provides us an efficient construction of the AGSP operator, which is expressed by the following statement:

Claim 3

(Lemma 11 in Supplementary Note 2) By using the Chebyshev polynomial, we can find a m-degree polynomial K(m, Ht) such that

$$parallel K(m,{H}_{{rm{t}}})(1-left|{0}_{{rm{t}}}rightrangle leftlangle {0}_{{rm{t}}}right|)parallel le 2exp left(-frac{2m}{sqrt{parallel {H}_{{rm{t}}}parallel /{Delta }_{{rm{t}}}}}right),$$

(11)

where the explicit form of the polynomial K(m,x) is given in Supplemental Lemma 11.

We notice that the gap condition plays a crucial role in this claim. Here, the norm of Ht is as large as ({mathcal{O}}(n)), which necessitates the polynomial degree of (m={mathcal{O}}(sqrt{n})). Polynomials with such a large degree cannot be utilized to prove the condition for the AGSP in Claim 1. To overcome this difficulty, we aim to construct an effective Hamiltonian with a small norm that retains the similar low-energy properties to the original Hamiltonian. For this purpose, in each of the blocks, we cut off the energy spectrum up to some truncation energy (see Fig. 5). Then, the block–block interactions (i.e., hs,s + 1) do not change, and the internal Hamiltonian hs is transformed to ({tilde{h}}_{s}). By this energy cutoff, the total norm of the effective Hamiltonian ({tilde{H}}_{{rm{t}}}) is roughly given by qτ. The question is whether this effective Hamiltonian possesses the ground-state property similar to H. By extending the original result in ref. 48, which considers a cutoff in a Hamiltonian of one region, we prove the statement as follows:

Fig. 5: Effective Hamiltonian ({tilde{boldsymbol{H}}}_{{mathbf{t}}}) by multienergy cutoff.
figure5

In each of the internal Hamiltonians ({h_{s}}_{s,=,0}^{q,+,1}), we perform the energy cutoff up to the energy τs = Es,0 + τ. Here, ({{E}_{s,j},|{E}_{s,j}rangle }) are the energy eigenvalues and the corresponding eigenstates of hs, respectively. The internal Hamiltonians hs and ({tilde{h}}_{s}) have the same eigenstates ({|{E}_{s,j}rangle }) and the same eigenvalues (i.e., ({E}_{s,j},=,{tilde{E}}_{s,j})), as long as Es,j ≤ τs, above which the eigenvalues differ from each other.

Claim 4

(Theorem 5 in Supplementary Note 2) Let us choose τ such that

$$tau,gtrsim,mathrm{log},(q).$$

Then, the spectral gap ({tilde{Delta}}_{{rm{t}}}) of the effective Hamiltonian is preserved as

$${tilde{Delta }}_{{rm{t}}},ge, {mathcal{O}}({Delta }_{{rm{t}}}).$$

Moreover, the norm distance between the original ground state (left|{0}_{{rm{t}}}rightrangle) and the effective one (left|{tilde{0}}_{{rm{t}}}rightrangle) is exponentially small with respect to the cut-off energy τ:

$$parallel left|{tilde{0}}_{{rm{t}}}rightrangle -left|{0}_{{rm{t}}}rightrangle parallel le , {e}^{-{mathcal{O}}(tau )}.$$

As long as τ is larger than ({mathcal{O}}({rm{log}}(q))), the spectral gap is preserved, and the norm of the effective Hamiltonian is as large as (q,mathrm{log},(q)), namely (parallel {tilde{H}}_{{rm{t}}}parallel lesssim ;q,mathrm{log},(q)). In the standard construction of the effective Hamiltonian6,48, we perform the energy cutoff only in the edge blocks (i.e., B0 and Bq+1). However, this simple procedure allows us to prove the long-range area law only in the short-range power-exponent regimes (i.e., α > 3). The multienergy cutoff is crucial to prove the area law even in the long-range power-exponent regimes (i.e., 2 < α ≤ 3).

By using the polynomial K(m,x) in (11) with (x={tilde{H}}_{{rm{t}}}), we can obtain the AGSP operator Kt for the ground state (left|{0}_{{rm{t}}}rightrangle) of Ht. Before showing the AGSP parameter for Kt, we discuss the Schmidt rank of the polynomial of the Hamiltonian. Now, the effective Hamiltonian ({tilde{H}}_{{rm{t}}}) is given by the form of (mathop{sum }nolimits_{s,=,0}^{q,+,1}{tilde{h}}_{s}+mathop{sum }nolimits_{s,=,0}^{q}{h}_{s,s,+,1}). By extending the Schmidt rank estimation in refs. 5,6, we can derive the following statement:

Claim 5

(Proposition 4 in Supplementary Note 2) The Schmidt rank of the power of the effective Hamiltonian ({rm{SR}}({H}_{{rm{t}}}^{m})) is bounded from above by

$${rm{SR}}({tilde{H}}_{{rm{t}}}^{m}),le,{e}^{{mathcal{O}}(ql)+{mathcal{O}}(m/q){mathrm{log}},(ql)}.$$

This inequality gives the upper bound of the Schmidt rank for ({K}(m,{tilde{H}}_{{rm{t}}})).

We have obtained all the ingredients to estimate the parameters ({delta }_{{K}_{{rm{t}}}}), ({epsilon }_{{K}_{{rm{t}}}}), and ({D}_{{K}_{{rm{t}}}}) for the AGSP ({K}_{{rm{t}}}={K}(m,{tilde{H}}_{{rm{t}}})). They are given by Claim 4, inequality (11), and Claim 5 as follows:

$$begin{array}{rcl}&&{delta }_{{K}_{{rm{t}}}}={e}^{-{mathcal{O}}(tau )},quad {epsilon }_{{K}_{{rm{t}}}}={e}^{-{mathcal{O}}(m)/sqrt{q{,}{mathrm{log}},(q)}},\ &&{rm{and}}quad {D}_{{K}_{{rm{t}}}}={e}^{{mathcal{O}}(ql)+{mathcal{O}}(m/q){mathrm{log}},(ql)},end{array}$$

(12)

where we omit the (bar{alpha })-dependence of the parameters. Let us apply Claim 1 to the AGSP Kt and the ground state (left|{0}_{{rm{t}}}rightrangle). Under the condition of (q{l}^{-bar{alpha }},lesssim, 1), we can find q, m, and l such that ({epsilon }_{{K}_{{rm{t}}}}^{2}{D}_{{K}_{{rm{t}}}}le (1/2)), where {qml} have quantities of ({mathcal{O}}(1)). This leads to the following statement:

Claim 6

(Proposition 6 in Supplementary Note 3) There exists a quantum state (left|phi rightrangle) such that (parallel left|0rightrangle -left|phi rightrangle parallel le 1/2) with

$${mathrm{log}},left[{rm{SR}}(left|phi rightrangle )right]le {c}^{* }{{mathrm{log}},}^{2}(d){left(frac{{mathrm{log}},(d)}{Delta }right)}^{1+2/bar{alpha }}{mathrm{log},}^{3+3/bar{alpha }}left(frac{{mathrm{log}},(d)}{Delta }right),$$

(13)

where c* is a constant that depends only on g0 and (bar{alpha }), which is finite in the limit of (bar{alpha }to infty).

Finally, we construct a set of the AGSP operators ({{{K}_{p}}}_{p,=,1}^{infty }) for the ground state (left|0rightrangle), where the AGSP parameters are denoted by δp, ϵp, and Dp. The errors ϵp and δp decrease with the index p, namely ϵ1 ≥ ϵ2 ≥   and δ1 ≥ δ2 ≥  . In the limit of p → , the AGSP Kp approaches the exact ground-state projection as ({K}_{infty }=left|0rightrangle leftlangle 0right|), namely ({mathrm{lim},}_{pto infty }{delta }_{p}=0) and ({mathrm{lim},}_{pto infty }{epsilon }_{p}=0). These AGSP operators allow the derivation of an upper bound of the entanglement entropy, as well as the approximation of the ground state by quantum states with small Schmidt ranks.

Claim 7

(Proposition 3 in Supplementary Note 2) Let (left|phi rightrangle) be an arbitrary quantum state with (parallel left|0rightrangle -left|phi rightrangle parallel le 1/2). Also, let ({{{K}_{p}}}_{p,=,1}^{infty }) be AGSP operators defined as above. Then, we prove for each of ({{{K}_{p}}}_{p,=,1}^{infty })

$$leftVert frac{{K}_{p}{e}^{-i{theta }_{p}}left|phi rightrangle }{parallel {K}_{p}left|phi rightrangle parallel }-left|0rightrangle rightVert le {gamma }_{p}:=frac{{epsilon }_{p}}{1/2-{delta }_{p}}+{delta }_{p},$$

where the phase ({theta }_{p}in {mathbb{R}}) is appropriately chosen. Moreover, under the condition γp ≤ 1 for all p, the entanglement entropy is bounded from above by

$$S(left|0rightrangle ),le,{mathrm{log}},left[{rm{SR}}(left|phi rightrangle )right],-,mathop{sum }limits_{p,=,0}^{infty }{gamma }_{p}^{2}{mathrm{log}},frac{{gamma }_{p}^{2}}{3{D}_{p,+,1}},$$

where we set γ0 := 1.

In Proposition 7 of Supplementary Note 3, we show a construction of the AGSP set ({{{K}_{p}}}_{p,=,1}^{infty }) such that ({gamma }_{p}^{2}=1/{p}^{2}) and

$${mathrm{log}},(3{D}_{p})le {c}_{1}{bar{alpha }}^{-1}frac{{mathrm{log},}^{5/2}(3p/Delta )}{sqrt{Delta }}+{c}_{2}frac{{mathrm{log},}^{3/2}(3p/Delta ){mathrm{log}},(d)}{sqrt{Delta }},$$

(14)

where c1 and c2 are constants that depend on g0. We have obtained the quantum state (left|phi rightrangle) with the Schmidt rank as in (13), and hence from Claim 7, the above AGSP operators give the upper bound of the entanglement entropy in (6). This completes the proof of the area law in long-range interacting systems. □

MPS approximation of the ground state

We here prove inequality (8). For simplicity, let us consider X to be the total system (i.e., X = Λ). Generalization to X Λ is straightforward. Our proof relies on the following statement:

Claim 8

(Lemma 1 in ref. 49) Let (left|psi rightrangle) be an arbitrary quantum state. We define the Schmidt decomposition between the subsets {1, 2, …, i} and {i + 1, i + 2, …, n}, as follows:

$$left|psi rightrangle,=,mathop{sum }limits_{m,=,1}^{infty }{mu }_{m}^{(i)}left|{psi }_{le i,m}rightrangle otimes left|{psi }_{ ,{> },i,m}rightrangle ,$$

(15)

where ({{{mu }_{m}^{(i)}}}_{m,=,1}^{infty }) are the Schmidt coefficients in the descending order. Then, there exists an MPS approximation (left|{psi }_{D}rightrangle) with the bond dimension D that approximates the quantum state (left|psi rightrangle) as

$$parallel left|psi rightrangle -left|{psi }_{D}rightrangle parallel^2,le,2mathop{sum }limits_{i,=,1}^{n,-,1}{delta }_{i},quad {delta }_{i},:=,sum_{m ,{> },D}| {mu }_{m}^{(i)}{| }^{2}.$$

From this claim, if we can obtain the truncation error of the Schmidt rank, we can also derive the approximation error by the MPS.

In the following, we give the truncation error by using  Claim 7. Let us consider a fixed decomposition as Λ = LR. Then,  Claim 7  ensures the existence of the approximation of the ground state (left|0rightrangle) with the approximation error γp, which is achieved by the quantum state (|{psi }_{p}rangle :={K}_{p}{e}^{i{theta }_{p}}|phi rangle /parallel {K}_{p}|phi rangle parallel) with its Schmidt rank of

$$mathrm{log},[{rm{SR}}(|{psi }_{p}rangle )]le mathrm{log},({D}_{p})+mathrm{log},[{rm{SR}}(left|phi rightrangle )],$$

where (left|phi rightrangle) has the Schmidt rank of (13) at most. We have already proved that for γp = 1/p2, the quantity Dp is upper-bounded by (14). Thus, for p ≥ (1/δ)1/4 or (γp ≤ δ1/2), the Schmidt rank (mathrm{log},[{rm{SR}}(left|{psi }_{p}rightrangle )]) satisfies the following inequality:

$${mathrm{log}},[{rm{SR}}(|{psi }_{p}rangle )] , lesssim , ({bar{alpha }}^{-1}mathrm{log},(1/delta )+1){mathrm{log},}^{3/2}(1/delta )$$

(16)

for (1/Delta ={mathcal{O}}(1)) and sufficiently small δ 1, where we use the fact that (mathrm{log},[{rm{SR}}(left|phi rightrangle )]) is a constant of ({mathcal{O}}(1)).

In order to connect inequality (16) to the truncation error of the Schmidt decomposition, we use the following statement:

Claim 9

(Eckart–Young theorem50) Let us consider a normalized state (left|psi rightrangle) as in Eq. (15). Then, for an arbitrary quantum state (left|psi ^{prime} rightrangle), we have the inequality of ({sum }_{m ,{> },{rm{SR}}(left|psi ^{prime} rightrangle )}| {mu }_{m}^{(i)}{| }^{2},le,parallel left|psi rightrangle -left|psi ^{prime} rightrangle {parallel }^{2}), where the Schmidt rank ({rm{SR}}(left|psi ^{prime} rightrangle )) is defined for the decomposition of {1, 2, …, i} and {i + 1, i + 2, …, n}.

In the above claim, we choose (left|{0}rightrangle, |{psi }_{p}rangle) as (left|psi rightrangle, left|psi ^{prime} rightrangle), respectively, and obtain the inequality of

$$sum_{m ,{> },{rm{SR}}(left|{psi }_{p}rightrangle )}| {mu }_{m}^{(i)}{|}^{2},le,{gamma }_{p}^{2},$$

(17)

where we use (parallel |{psi }_{p}rangle,-,left|0rightrangle parallel,le,{gamma }_{p}). By applying inequalities (16) and (17) to Claim 8, we can achieve

$$parallel left|0 rightrangle,-,|{psi }_{D}rangle parallel^2,le,2ndelta,$$

if ({mathrm{log}},(D)) is as large as ({bar{alpha }}^{-1}{mathrm{log},}^{5/2}(1/delta )) [(bar{alpha }={mathcal{O}}(1))]. This completes the proof. □



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