Schrieffer Wolf

8/12/2019 Schrieffer Wolf

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arXiv:1105

.0675v1[quant-ph

]3May2011

Schrieffer-Wolff transformation for quantum many-body

systems

Sergey Bravyi, David P. DiVincenzo, and Daniel Loss

May 5, 2011

Abstract

The Schrieffer-Wolff (SW) method is a version of degenerate perturbation theory in whichthe low-energy effective Hamiltonian Heffis obtained from the exact Hamiltonian by a unitarytransformation decoupling the low-energy and high-energy subspaces. We give a self-containedsummary of the SW method with a focus on rigorous results. We begin with an exact definitionof the SW transformation in terms of the so-called direct rotation between linear subspaces.From this we obtain elementary proofs of several important properties ofHeffsuch as the linkedcluster theorem. We then study the perturbative version of the SW transformation obtainedfrom a Taylor series representation of the direct rotation. Our perturbative approach provides asystematic diagram technique for computing high-order corrections to Heff. We then specializethe SW method to quantum spin lattices with short-range interactions. We establish unitaryequivalence between effective low-energy Hamiltonians obtained using two different versions ofthe SW method studied in the literature. Finally, we derive an upper bound on the precisionup to which the ground state energy of the n-th order effective Hamiltonian approximates theexact ground state energy.

IBM Watson Research Center, Yorktown Heights, NY 10598 USA. [email protected] Aachen and Forschungszentrum Juelich, Germany. [email protected] of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland.

[email protected]

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http://arxiv.org/abs/1105.0675v1http://arxiv.org/abs/1105.0675v1http://arxiv.org/abs/1105.0675v1http://arxiv.org/abs/1105.0675v1http://arxiv.org/abs/1105.0675v1http://arxiv.org/abs/1105.0675v1http://arxiv.org/abs/1105.0675v1http://arxiv.org/abs/1105.0675v1http://arxiv.org/abs/1105.0675v1http://arxiv.org/abs/1105.0675v1http://arxiv.org/abs/1105.0675v1http://arxiv.org/abs/1105.0675v1http://arxiv.org/abs/1105.0675v1http://arxiv.org/abs/1105.0675v1http://arxiv.org/abs/1105.0675v1http://arxiv.org/abs/1105.0675v1http://arxiv.org/abs/1105.0675v1http://arxiv.org/abs/1105.0675v1http://arxiv.org/abs/1105.0675v1http://arxiv.org/abs/1105.0675v1http://arxiv.org/abs/1105.0675v1http://arxiv.org/abs/1105.0675v1http://arxiv.org/abs/1105.0675v1http://arxiv.org/abs/1105.0675v1http://arxiv.org/abs/1105.0675v1http://arxiv.org/abs/1105.0675v1http://arxiv.org/abs/1105.0675v1http://arxiv.org/abs/1105.0675v1http://arxiv.org/abs/1105.0675v1http://arxiv.org/abs/1105.0675v1http://arxiv.org/abs/1105.0675v1http://arxiv.org/abs/1105.0675v1http://arxiv.org/abs/1105.0675v1http://arxiv.org/abs/1105.0675v1http://arxiv.org/abs/1105.0675v1http://arxiv.org/abs/1105.0675v1


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Contents

1 Introduction 31.1 Applications of the SW method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2 Comparison between SW and other perturbative expansions . . . . . . . . . . . . . . 61.3 Organization of the paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Direct rotation between a pair of subspaces 92.1 Rotation of one-dimensional subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Rotation of arbitrary subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3 Generator of the direct rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.4 Weak multiplicativity of the direct rotation . . . . . . . . . . . . . . . . . . . . . . . . 13

3 Effective low-energy Hamiltonian 143.1 Schrieffer-Wolff transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.2 Derivation of the perturbative series . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.3 Schrieffer-Wolff diagram technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.4 Convergence radius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.5 Additivity of the effective Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4 Schrieffer-Wolff theory for quantum many-body systems 264.1 The unperturbed Hamiltonian and the perturbation . . . . . . . . . . . . . . . . . . . 274.2 Block-diagonal perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.3 Linked cluster theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.4 Local Schrieffer-Wolff trasformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.5 Equivalence between the local and global SW theories . . . . . . . . . . . . . . . . . . 40

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1 Introduction

Given a fundamental theory describing a quantum many-body system, one often needs to obtain aconcise description of its low-energy dynamics by integrating out high-energy degrees of freedom.The Schrieffer-Wolff (SW) method accomplishes this task by constructing a unitary transformationthat decouples the high-energy and low-energy subspaces.

In the present paper we view the SW method as a version of degenerate perturbation theory. Itinvolves a Hamiltonian H0 describing an unperturbed system, a low-energy subspaceP0 invariantunder H0, and a perturbation Vwhich does not preserveP0. The goal is to construct an effectiveHamiltonian Heff acting only onP0 such that the spectrum of Heff reproduces eigenvalues of theperturbed HamiltonianH0+V originating from the low-energy subspace P0. The SW transformationis a unitary operator Usuch that the transformed Hamiltonian U(H0+ V)U preservesP0. Thedesired effective Hamiltonian is then defined as the restriction ofU(H0+V)U onto P0. An importantconsideration arising in the context of many-body systems such as molecules or interacting quantumspins is the locality ofHeff. In order for the effective Hamiltonian Heff to be usable, it must only

involve k-body interactions for some small constant k. We demonstrate that this is indeed the casefor the SW method by proving (under certain natural assumptions) that it obeys the so called linkedcluster theorem.

The purpose of the present paper is two-fold. First, we give a self-contained summary of the SWmethod with a focus on rigorous results. A distinct feature of our presentation is the use of bothperturbative and exact treatments of the SW transformation. The former is given in terms of theTaylor series and the rules for computing the Taylor coefficients while the latter involves the so calleddirect rotation between linear subspaces1. By combining these different perspectives we are able toobtain elementary proofs of several important properties of the effective Hamiltonian Heff such asits additivity under a disjoint union of non-interacting systems. These properties do not manifestthemselves on the level of individual terms in the perturbative expansion which makes their direct

proof (by inspection of the Taylor coefficients ofHeff) virtually impossible. In addition, our approachprovides a systematic diagram technique for computing high-order corrections to Heffwhich can bereadily cast into a computer program for practical calculations.

Our second goal is to specialize the SW method to interacting quantum spin systems. We assumethat the unperturbed Hamiltonian H0 is a sum of single-spin operators such that the low-energysubspaceP0 is a tensor product of single-spin low-energy subspaces. The perturbationV is chosenas a sum of two-spin interactions associated with edges of some fixed interaction graph. We provethat the Taylor series for Heffobeys the so called linked cluster theorem, that is, the m-th ordercorrection to Heff includes only interactions among subsets of spins spanned by connected clustersof at most m edges in the interaction graph (one should keep in mind that the spins acted on by

Heff are effective spins described by the low-energy subspaces of the original spins), see Theorem2in Section4.3. It demonstrates that the SW transformation maps a high-energy theory with local

1The direct rotation can be thought of as a square root of the double reflection operator used in the Grover searchalgorithm[1].

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1.1 Applications of the SW method

The SW transformation finds application in many contemporary problems in quantum physics, wherean economical description of the low-energy dynamics of the system must be extracted from a fullHamiltonian. While SW is named in honor of the authors of an important paper in condensed matter

theory[3], where the celebrated Kondo Hamiltonian was shown to be obtained by such a transfor-mation from the equally celebrated Anderson Hamiltonian, it is in fact not the earliest application ofthe technique. The concept of a most-economical rotation between subspaces has a several-centurieshistory in mathematics[4,5]. Its direct descendent is the cosine-sine (CS) decomposition [6], whichis very closely related to the definition of the exact SW transformation that we adopt here. Theearliest application of the canonical transformation in quantum physics is in fact at least 15 yearsearlier than[3], in the famous transformation of Foldy and Wouthuysen[7]. The authors of [7] usedthe technique to derive the non-relativistic Schroedinger-Pauli wave equation, with relativistic cor-rections, from the Dirac equation. In this case, the canonical transformation is one that decouplesthe positive and negative energy solutions of the relativistic wave equation.

The applications of SW in modern times [8]are far too numerous even to allude to. The work ofone of us uses SW routinely in many applications (see, e.g., Ref.[9]). Since part of our concern in thepresent work is a systematic development of a proper series expansion for the SW, we can mentionthat high-order terms of this series have been carefully and laboriously developed in some importantworks [10] and may be found recorded in recent books dealing with applications [11]. One can,however, find lengthy discussions of possible alternative approaches[12,13]. It is widely acknowledgedthat SW is preferable to various other approaches, for example the Bloch expansion [14,15], which,unlike SW, produces non-Hermitian effective Hamiltonians at sufficiently high order. SW is alsosuperior to the self-consistent equation for the effective Hamiltonian arising from the diagrammaticself-energy technique [16, 17,18].

It often occurs that the effective low-energy Hamiltonian has much higher level of complexity com-

pared to the original high-energy Hamiltonian. This observation has lead to the idea of perturbationgadgets [18, 19, 20]. Suppose one starts from a target HamiltonianHtarget chosen for some interestingground state properties. The target Hamiltonian may possibly contain many-body interactions andmight not be very realistic. Perturbation gadgets formalism allows one to construct a simpler high-energy simulator Hamiltonian with only two-body interactions whose low-energy properties (such asthe ground state energy) approximate the ones ofHtarget. For example, various perturbation gadgetshave been constructed for target Hamiltonians with the topological quantum order [21,22, 23]. TheSW method provides a natural framework for constructing and analyzing perturbation gadgets [24].In contrast to other methods, SW does not require the unphysical scaling of parameters in the simula-tor Hamiltonian required for convergence of the perturbative series [24]. This makes the SW methodsuitable for analysis of experimental implementations of perturbation gadgets with cold atoms inoptical lattices, see[25, 26].

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1.2 Comparison between SW and other perturbative expansions

In this section we briefly review some of the commonly used perturbative expansion techniques andargue that none of them can serve as a fully functional replacement of the SW method.

We begin by emphasizing that the effective Hamiltonian Heff is only defined up to a unitary

rotation of the low-energy subspaceP0. Therefore one should expect that different versions of adegenerate perturbation theory produce different Taylor series for Heff. In particular, the error upto which the series for Hefftruncated at some finite order reproduce the exact low-energy spectrummay depend on the method of computing Heff. Different methods may also vary in the complexityof rules for computing the Taylor coefficients.

The most convenient and commonly used perturbative method is the Feynman-Dyson diagramtechnique [16,17,27]. It can be used whenever the unpertubed ground state obeys Wicks theorem.Unfortunately, the standard derivation of the Feynman-Dyson expansion (see, e.g., Ref. [17]) whichrelies on the adiabatic switching of a perturbation can only be applied to non-degenerate groundstates. It was recently shown explicitly that the Gell-Mann and Low theorem used in the derivationof Feynman-Dyson series fails for degenerate ground states [28].

Exact quasi-adiabatic continuation [29, 30] provides an alternative path to defining Heff via aunitary transformation starting from a full Hamiltonian. This method however requires a constantlower bound on the energy gap of a perturbed Hamiltonian which is typically very hard to check.

Traditional textbook treatment of a degenerate perturbation theory [31,32,33] is formulated interms of perturbed and unperturbed resolvents

G(z) = (zI H0 V)1 and G(z) = (zI H0)1.An effective low-energy Hamiltonian can be obtained from G(z) using the self-energy method [16,17, 18]. To sketch the method, we shall adopt the standard notations by writing any operator O asa block matrix

O=

O O+O+ O+

(1.1)

where the two blocks correspond to the low-energy and the high-energy subspaces respectively. Onecan define an effective low-energy resolvent as G(z), that is, the low-energy block ofG(z). Itsimportance comes from the fact that for sufficiently small eigenvalues ofH0+ V originating fromthe low-energy subspace coincide with the poles ofG(z), see [18]. The poles ofG(z) can be foundusing the Dyson equation [17]

G(z) =G(z) + G(z)(z)G(z),

where

(z) =V+

n=0

2+n V+(G+(z)V+)nG+(z)V+

is the so-called self-energy operator acting on the low-energy subspace. The Dyson equation yields

G(z)1 =G(z)

1 (z) =zI (H0) (z).

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1.3 Organization of the paper

To be self-contained, the paper repeats some well-known results such as the low-order terms in theSW series. However we present these results in a very systematic and compact form that we believemight be useful for other workers.

Sections2,3provide an elementary introduction to the SW method for finite-dimensional Hilbertspaces. Section2summarizes the definition and basic facts concerning the direct rotation betweenlinear subspaces. The direct rotation was originally introduced in the context of perturbation theoryby Davis and Kahan in [4]. Although we hardly discover any new properties of the direct rota-tion, some of the basic facts such as the behavior of the direct rotation under tensor products (seeSection2.4) might not be very well-known.

Section3defines the SW transformation as the direct rotation from the low-energy subspace ofa perturbed Hamiltonian H0+ V to the low-energy subspace ofH0. This definition provides anelementary proof of additivity ofHeff. Namely, given a bipartite system AB with no interactionsbetween A and B, the effective Hamiltonians Heff[AB], Heff[A], and Heff[B] describing the jointsystem and the individual systems respectively are related as

Heff[AB] =Heff[A] IB+ IA Heff[B].

Here IA, IB are the identity operators acting on the low-energy subspaces of A and B. We notethat the above additivity does not hold on the entire Hilbert space as one could naively expect. InSection 3.2 we compute the Taylor series for the SW transformation and Heff. Our presentationuses the formalism of superoperators to develop the series. It allows us to keep track of cancelationsbetween different terms in a systematic way and obtain more compact expressions for the Taylorcoefficients. A convenient diagram technique for computingHeff is presented in Section3.3. Con-vergence of the series is analyzed in Section3.4. To the best of our knowledge, the approach taken

in Sections 3.2,3.3,3.4 is new. Our main technical contributions are presented in Section 4 thatspecializes the SW method to weakly interacting spin systems.

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2 Direct rotation between a pair of subspaces

The purpose of this section is to define a direct rotation between a pair of linear subspaces and deriveits basic properties such as a multiplicativity under tensor products.

2.1 Rotation of one-dimensional subspaces

LetH be any finite-dimensional Hilbert space. For any normalized state H define a reflectionoperator R that flips the sign of and acts trivially on the orthogonal complement of, that is,

R =I 2||. (2.1)Given a pair of non-orthogonal states , H we would like to define a canonical unitary operatorU mapping to up to an overall phase. Let us first consider the double reflection operatorRR. It rotates the two-dimensional subspace spanned by and by 2, where [0, /2) is theangle between and . In addition,RR acts as the identity in the orthogonal complement to

and . Thus we can choose the desired unitary operator mapping from to as U =

RRassuming that the square root is well-defined.

Definition 2.1. Let, H be any non-orthogonal states. Define a direct rotation from to asa unitary operator

U=

RR (2.2)

Here

zis defined on a complex plane with a branch cut along the negative real axis such that

1 = 1.

It is worth pointing out that any unitary operator is normal and thus the square root

RR is welldefined provided that RR has no eigenvalues lying on the chosen branch cut of the

zfunction.

The following lemma shows that U is well defined and performs the desired transformation.

Lemma 2.1. Let, H be any non-orthogonal states. Then the double reflection operatorRRhas no eigenvalues on the negative real axis. Furthermore, fix the relative phase of and such that| is real and positive. ThenU | =|.Proof. SinceU acts as the identity on the subspace orthogonal to and, it suffices to considerthe caseH = C2. Without loss of generality| =|1 and| = sin() |0 + cos () |1, where, byassumption, 0 < /2. Using the definition Eq. (2.1) one gets

R = z and R= cos (2)

z sin (2) x.It yields

RR = cos (2) I+ i sin(2) y = exp (2iy).

Since 02 < , no eigenvalue ofRR lies on the negative real axis and thusU=

RR = exp (i

y)

is uniquely defined. We get U |1 = sin () |0 + cos () |1, that is, U | =|.

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Later on we shall need the following property of the direct rotation.

Corollary 2.1. Let, H be any non-orthogonal states, P = || andP0 =||. The directrotation from to can be written asU = exp (S) whereS is an anti-hermitian operator withthe following properties:

P SP=P0SP0= (I P)S(I P) = (I P0)S(I P0) = 0, S < /2.

Proof. Indeed, we can assume that|=|1 and|= sin () |0 + cos () |1 for some [0, /2).Choose S = iy and S= 0 in the orthogonal complement to and . By assumption,S =|| < /2. A simple algebra yields|S| =|S| = 0. Restricting all operators on the two-dimensional subspace spanned by and one gets (I P)S(I P) = |S|and (I P0)S(IP0) =|S|, where| =|0 and| = cos() |0 sin () |1. A simple algebra yields|S| =|S| = 0.

2.2 Rotation of arbitrary subspaces

LetP, P0 H be a pair of linear subspaces of the same dimension and P, P0 be the orthogonalprojectors ontoP, P0. We would like to define a canonical unitary operatorUmappingPtoP0. Byanalogy with the non-orthogonality constraint used in the one-dimensional case we shall impose aconstraint

P P0< 1. (2.3)The meaning of this constraint is clarified by the following simple fact.

Proposition 2.1. Condition

P

P0

< 1 holds iff no vector in

P is orthogonal to

P0 and vice

verse. In particular,P P0dim P0 (or vice verse) there must exist a vector P thatis orthogonal toP0 (or vice verse), that is,P P0= 1.

For any linear subspacePdefine a reflection operator RPthat flips the sign of all vectors inPand acts trivially on the orthogonal complement to

P, that is,

RP= 2P I. (2.4)

Following [4]let us define a direct rotation between a pair of subspaces as follows.

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Definition 2.2. Let

z be the square-root function defined on a complex plane with a branch cutalong the negative real axis and such that

1 = 1. A unitary operator

U=

RP0RP (2.5)

is called a direct rotation fromP toP0.Lemma 2.2. SupposeP P0 < 1. Then no eigenvalue ofRP0RP lies on the negative real axis,so thatUis uniquely defined by Eq. (2.5) and

UP U =P0. (2.6)

It is worth pointing out that the direct rotation Ucan also be defined as the minimal rotationthat maps Pto P0. More specifically, among all unitary operatorsV satisfyingV P V =P0the directrotation differs least from the identity in the Frobenius norm, see [4]. If, in addition, PP0 0. (2.9)

LetSbe any anti-hermitian operator satisfying the three conditions of the lemma and let U= exp (S).We have to prove that U=U and S=S. Indeed, using the identity

(U U)P0(U U) = U PU =P0

we conclude that U U commutes with P0. This is possible only ifU U =L for some block-diagonalunitaryL, that is,

L=

L1 0

0 L2

, L1L

1 = I , L

2L2 = I .

It follows that U = LU, that is U1,1 = L1U1,1 and U2,2 = L2U2,2. Combining it with Eq. (2.9) wearrive at L1 = I andL2 = Ias follows from the following proposition.

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Proposition 2.2. Let Hbe a positive operator and L be a unitary operator. Suppose that LH isalso a positive operator. ThenL= I.

Proof. Indeed, consider the eigenvalue decomposition L = ei||. Since LH > 0 one gets

|LH

|

=ei

|H

|

>0. Since

|H

|

>0 we conclude that ei >0, that is, ei = 1. Hence

L= I.

To summarize, we have shown that U = U. Since the matrix exponential function exp (M) isinvertible on the subset of matrices satisfyingM< /2, we conclude that S=S.

2.4 Weak multiplicativity of the direct rotation

Consider a bipartite system of Alice and Bob with a Hilbert spaceH =HA HB. Let PA, PA0 bea pair of Alices projectors acting onHA. Similarly, let PB, PB0 be a pair of Bobs projectors actingonHB. We shall assume that

PA PA0< 1 and PB PB0


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Proof. Let us assume that Eq. (2.11) is false. Then there exists a state H such thatPAPB | =0 and PA0 PB0 | =| (or vice verse). Below we show that it leads to a contradiction. Indeed,the conditionPA PA0 < 1 implies that there exists > 0 such that PA PA0 (1) I.Multiplying this inequality on both sides by PA we arrive at PA PAPA0 PA (1 )PA, that is,

P

A

P

A

0 P

A

PA

. Similarly, there exists >0 such that P

B

P

B

0 P

B

PB

. Thus(PA PB)(PA0 PB0 )(PA PB) = (PAPA0 PA) (PBPB0 PB) PA0 PB0 . (2.12)

Here we used the fact that a tensor product of positive semi-definite operators is a positive semi-definite operator. Computing the expectation value of Eq. (2.12) on one gets 0 which is acontradiction.

We conclude that the global direct rotation UAB mappingPA PB toPA0 PB0 is well-defined.The relationship between the global and the local rotations which we shall call aweak multiplicativityis established by the following lemma.

Lemma 2.4 (Weak multiplicativity). LetUA, UB, andUAB be the direct rotations defined above.Then

UAB (PA PB) = (UA UB)(PA PB). (2.13)Proof. Let us perform the simultaneous block-diagonalization ofPA, PA0 andP

B, PB0 with blocks ofsize 22 and 11, as in the proof of Lemma2.2. Recall that each 22 block is a rank-one projector.Hence it suffices to check Eq. (2.13) only for the case whenHA = HB = C2 and rank-one projectorsPA, PB, PA0 , P

B0 . Choose normalized states

A, B, A0, B0 in the range of the above projectors,

and fix their relative phase such thatA|A0 > 0 andB|B0 > 0. Lemma 2.1 implies thatUA |A =|A0 andUB |B =|B0. However, since AB |A0B0 >0, Lemma2.1also impliesthatUAB

|A

B

=

|A0

B0

. This we have an identityUAB(PA

PB) = (UA

UB)(PA

PB) in

each tensor product of 2 2 blocks. Similar arguments hold if one or both blocks have size 1 1.

3 Effective low-energy Hamiltonian

3.1 Schrieffer-Wolff transformation

LetHbe a finite-dimensional Hilbert space and H0 be a hermitian operator onH. We shall refer toH0 as anunperturbed Hamiltonian. LetI0 Rbe any interval containing one or several eigenvaluesofH0 and letP0 H be the subspace spanned by all eigenvectors ofH0 with eigenvalue lying inI0.We shall say that H0 has a spectral gap iff for any pair of eigenvalues , such that I0 and / I0 one has| | . In other words, the eigenvalues ofH0 lying inI0 must be separatedfrom the rest of the spectrum by a gap at least .

Consider now a perturbed HamiltonianH=H0+ V, where the perturbation V is an arbitraryhermitian operator on H. LetI R be the interval obtained fromI0 by adding margins of thickness/2 on the left and on the right ofI0. LetP H be the subspace spanned by all eigenvectors of

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H with eigenvalue lying inI. In this section we shall only consider sufficiently weak perturbationsthat do no close the gap separating the intervalI0 from the rest of the spectrum. Specifically, weshall assume that|| c, where

c =

2

V

. (3.1)

Since the perturbation shifts any eigenvalue at most byV, we conclude that the eigenvalues ofH lying inIare separated from the rest of the spectrum ofHby a positive gap as long as||< c.In particular, it implies thatP0 andP have the same dimension. Let P0 and P be the orthogonalprojectors ontoP0 andP. Introduce also projectors Q0=I P0 andQ = I P.Lemma 3.1. Suppose is real and||< c. ThenP0 P 2V/


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3.2 Derivation of the perturbative series

Let Ube the Schrieffer-Wolff transformation constructed for some unperturbed Hamiltonan H0, aperturbation V, and the low-energy subspaceP0. In this section we always assume that|| < c,so U is well-defined, see Lemma 3.1. In most of applications the explicit formula Eq. (2.5) for

the SW transformation cannot be used directly because the low-energy subspacePof the perturbedHamiltonianH0 +Vis unknown. In this section we explain how one can compute the transformationUand the effective low-energy HamiltonianHeffperturbatively. Truncating the series forHeffat somefinite order one obtains an effective low-energy theory describing properties of the perturbed system.Throughout this section all series are treated as formal series. Their convergence will be proved laterin Section3.4.

We begin by introducing some notations. The space of linear operators acting onH will bedenoted L(H). A linear mapO : L(H) L(H) will be referred to as a superoperator. We shalloften use superoperators

O(X) =P0XQ0+ Q0XP0 and

D(X) =P0XP0+ Q0XQ0.

We shall say that an operator X is block-off-diagonal iffO(X) = X. An operator X is calledblock-diagonaliffD(X) =X. Decompose the perturbation V as V =Vd+ Vod, where

Vd=D(V) and Vod =O(V)

are block-diagonal and block-off-diagonal parts ofV. Given any operator Y L(H) define a super-operator Ydescribing the adjoint action ofY, that is, Y(X) = [Y, X].

Recall that the SW transformation can be uniquely represented as U = exp(S) where S is ananti-hermitian generator which is block-off-diagonal andS < /2, see Lemma 2.3, whereas thetransformed HamiltonianeS(H0 + V)e

S is block-diagonal. Combining these conditions would yield

Taylor series forS, which, in turn, yields Taylor series for Heff. We begin by rewriting the transformedHamiltonian as

exp(S)(H0 + V) = cosh (S)(H0 + Vd)+sinh(S)(Vod)+sinh(S)(H0 + Vd)+cosh(S)(Vod). (3.3)

Taking into account that Sis block-off-diagonal, we conclude that the first and the second terms inthe righthand side of Eq. (3.3) are block-diagonal, while the third and the fourth terms are block-off-diagonal. In order for the transformed Hamiltonian to be block-diagonal, Smust obey

sinh (S)(H0+ Vd) + cosh (S)(Vod) = 0.

Since our goal is to derive formal Taylor series, Scan be regarded as an infinitesimally small operator.In this case the superoperator cosh (S) is invertible and thus the above condition can be rewritten as

tanh(S)(H0+ Vd) + Vod = 0. (3.4)

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Now we can rewrite the transformed Hamiltonian as

exp(S)(H0+ V) = cosh (S)(H0+ Vd) + sinh (S)(Vod)

= H0+ Vd+ (cosh (S) 1)(H0+ Vd) + sinh (S)(Vod)

= H0+ Vd+(cosh (S) 1)

tanh(S)tanh(S)(H0+ Vd) + sinh (S)(Vod).

Note that (cosh (S)1)/ tanh(S) is well defined for infinitesimally smallSby its Taylor series. UsingEq. (3.4) we arrive at

exp(S)(H0+ V) = H0+ Vd+ F(S)(Vod),

where

F(x) = sinh (x) cosh (x) 1tanh(x)

.

A simple algebra shows that F(x) = tanh (x/2), so we finally get

exp(S)(H0+ V) =H0+ Vd+ tanh (S/2)(Vod). (3.5)

In order to solve Eq. (3.4) for S let us introduce some more notations. Let{|i} be an orthonormaleigenbasis ofH0 such that H0 |i = Ei |i for all i. We shall use notation i I0 as a shorthand forEi I0. Define a superoperator

L(X) =i,j

i|O(X)|jEi Ej |ij|. (3.6)

Note thati|O(X)|j = 0 whenever i, j I0 or i, j / I0. Let us agree that the sum in Eq. (3.6)includes only the terms with i I0, j / I0 or vice verse. In this case the energy denominator canbe bounded as|Ei Ej| . One can easily check that

L([H0, X]) = [H0, L(X)] =O(X) (3.7)for any operator X L(H). It is worth mentioning thatL maps hermitian operators to anti-hermitian operators and vice verse.

TreatingSas an infinitesimally small operator we can rewrite Eq. (3.4) as

S(H0+ Vd) + Scoth (S)(Vod) = 0.

Using Eq. (3.7) and the fact that Sis block-off-diagonal one gets

S=

LS(Vd) +

LScoth (S)(Vod). (3.8)

We can solve Eq. (3.8) in terms of Taylor series,

S=n=1

Sn n, Sn= Sn. (3.9)

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Let us also agree that S0= 0. We shall need the Taylor series

x coth (x) =n=0

a2nx2n, am=

2mBmm!

, (3.10)

where Bm are the Bernoulli numbers. Then we have

S1 = L(Vod),S2 = LVd(S1),Sn = LVd(Sn1) +

j1

a2jL S2j(Vod)n1 for n3. (3.11)

Here we used a shorthand

Sk(Vod)m =

n1,...,nk1

n1+...+nk=m

Sn1 Snk(Vod). (3.12)

Note that the righthand side of Eq. (3.11) depends only on S1, . . . , S n1. Hence Eq. (3.11) providesan inductive rule for computing the Taylor coefficients Sn. Projecting Eq. (3.5) onto the low-energysubspace one gets

Heff=H0P0+ P0V P0+n=2

nHeff,n, Heff,n =j1

b2j1P0 S2j1(Vod)n1P0. (3.13)

Here b2j1 are the Taylor coefficients of the function tanh (x/2). More explicitly,

tanh (x/2) =

n=1

b2n1x2n

1

, b2n1 =2(22n

1)B2n

(2n)! . (3.14)

Note thatHeff,n depends only upon S1, . . . , S n1. To illustrate the method let us compute the Taylorcoefficients Heff,n for small values ofn (a systematic way of computing Heff,n in terms of diagrams isdescribed in the next section). From Eq. (3.11) one easily finds

S3 = LVd(S2) + a2L S21 (Vod),S4 = LVd(S3) + a2L (S1S2+ S2S1)(Vod).

From Eq. (3.13) one gets

Heff,2 = b1P0S1(Vod)P0, (3.15)Heff,3 = b1P0S2(Vod)P0, (3.16)

Heff,4 = b1P0S3(Vod)P0+ b3P0S31 (Vod)P0, (3.17)

Heff,5 = b1P0S4(Vod)P0+ b3P0(S2S21+S1S2S1+ S

21S2)(Vod)P0. (3.18)

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a0 a2 a4 a61 1/3 1/45 2/945

b1 b3 b5 b71/2 1/24 1/240 17/40320

Figure 1: The value of the Taylor coefficients a2n andb2n1 for smalln.

Let us obtain explicit formulas for Heff,3 and Heff,4that include onlyS1. Using the identity S2(Vod) =

Vod(S2) we getHeff,3= b1P0VodLVd(S1)P0 (3.19)

andS3 = (LVd)2(S1) + a2LS21 (Vod).

SubstitutingS3 into the expression for Heff,4 and using the identity S3(Vod) =Vod(S3) one gets

Heff,4 = P0b1Vod(LVd)2(S1) b1a2VodLS21 (Vod) + b3S31 (Vod)

P0. (3.20)

The explicit values of the Taylor coefficients a2n and b2n1 are listed in Table 1. The formula forHeff,4 can be slightly simplified in the special case whenI0 contains a single eigenvalue ofH0, thatis, the restriction ofH0 ontoP0 is proportional to the identity operator. In this special case one canuse an identity

P0[L(X), O(Y)]P0=P0[O(X), L(Y)]P0 (3.21)which holds for any operators X, Y. One can easily check Eq. (3.21) by computing matrix elementsi| |jof both sides for i, j I0. Using Eq. (3.21) and explicit values ofas andbs one can rewriteHeff,4 as

Heff,4= P0

1

8S31 (Vod)

1

2Vod(LVd)2(S1)

P0. (3.22)

To summarize, the Taylor series for the effective low-energy Hamiltonian can be written as

Heff = H0P0+ P0V P0+2

2P0S1(Vod)P0+

3

2P0VodLVd(S1)P0

4P0

1

2Vod(LVd)2(S1) +1

6VodLS21 (Vod) +

1

24S31 (Vod)

P0+ O(

5), (3.23)

with the simplification Eq. (3.22) in the case when the restriction ofH0onto the low-energy subspaceis proportional to the identity operator.

3.3 Schrieffer-Wolff diagram technique

In this section we develop a diagram technique for the effective low-energy Hamiltonian. By analogywith the Feynman-Dyson diagram technique, we shall represent the n-th order Taylor coefficientHeff,n as a weighted sum over certain class of admissible diagrams. In our case n-th order diagramswill be trees with n nodes obeying certain restrictions imposed on the node degrees. Any such

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diagram represents a linear operator acting on the low-energy subspaceP0. Loosely speaking, everyedge of the tree stands for a commutator, while every node of the tree stands for Vd or Vod. Theweight associated with a diagram depends only on node degrees and can be easily expressed in termsof the Bernoulli numbers.

Let us now proceed with formal definitions. LetTbe a connected tree with n nodes. Each nodeuThas at most one parent node and zero or more children nodes. The children of any node areordered in some fixed way. We shall denote children of a node u as u(1), . . . , u(k), where k is thenumber of children (NOC) ofu. Nodes having no children are called leaves. There is a unique rootnode which has no parent node.

For any node uTwe shall define a linear operator Ou acting on the full Hilbert spaceH. Thedefinition is inductive, so we shall start from the leaves and move up towards the root.

Ifu is a leaf then Ou= S1 L(Vod). Ifu= root has k2 children and k is even then Ou= L Ou(1) Ou(k)(Vod).

Ifu= root has exactly one child then Ou=LVd(Ou(1)). Ifu = root hask children with odd k then Ou= P0 Ou(1) Ou(k)(Vod)P0. In all remaining cases Ou= 0.

Recall that OadO describes the adjoint action ofO. Define an operator O(T) associated with TasO(T) = Or, wherer is the root ofT. It is clear from the above definition thatO(T) is an operatoracting on the low-energy subspaceP0 and O(T) has degree n in the perturbation V. Furthermore,O(T) = 0 unless the root ofThas odd NOC and every node different from the root with more thanone child has even NOC.

Definition 3.2. A connected treeTis called admissible iff

The root ofThas odd NOC, Every node different from the root has either even NOC or NOC=1.A complete list of admissible trees (modulo isomorphisms) with n= 3, 4, 5, 6 nodes is shown on

Fig. 2. To illustrate the definition of operators O(T) let us consider fourth-order diagrams. LetT1, T2, T3 be the admissible trees with four nodes shown on Fig. 2 (listed from the left to the right).Then one has

O(T1) = P0Vod(LVd)2

(S1)P0,O(T2) = P0VodLS21 (Vod)P0,O(T3) = P0S

31 (Vod)P0. (3.24)

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n=6

n=4

n=3 n=5

Figure 2: A set of admissible treesT(n) with n= 3, 4, 5, 6 nodes. Only one representative for eachclass of isomorphic trees is shown. The total number of admissible trees (counting isomorphic trees)is|T(n)| = 1, 3, 7, 20 for n = 3, 4, 5, 6 respectively,

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For any tree Tdefine a weight

w(T) =uT

w(u), (3.25)

where the product is over all nodes ofT and

w(u) =

1 if u= root has exactly one child,ak if u= root has k children for some even k,bk if u= root hask children for some odd k,0 otherwise.

(3.26)

Recall that

a2n=22nB2n

(2n)! and b2n1 =

2(22n 1)B2n(2n)!

,

whereB2nare the Bernoulli numbers, see also Table1. The main result of this section is the followinglemma.

Lemma 3.2. The Taylor series forHeffcan be written as

Heff=H0P0+ P0V P0+n=2

TT(n)

n w(T) O(T), (3.27)

whereT(n) is the set of all admissible trees withn nodes.Proof. Let us first develop a diagram technique for the generator S. Let Tbe any connected tree.For any node u T we shall define a linear operator Ou acting on the full Hilbert spaceH. Thedefinition is inductive, so we shall start from the leaves and move up towards the root.

Ifu is a leaf then O u= S1 L(Vod). Ifu has exactly one child thenOu=LVd(Ou(1)).

Ifu has k children and k is even then Ou= L Ou(1) Ou(k)(Vod). In all remaining cases Ou= 0.

Define an operator O(T) associated with T as O(T) = Or, where r is the root of T. It is clearfrom the above definition that O (T) is a block-off-diagonal operator and O (T) has degree nin theperturbation V. Furthermore,O(T) = 0 unless all nodes ofTwith more than one child have even

NOC.

Definition 3.3. A treeT is calledS-admissible iff any node ofTwith more than one child has evennumber of children.

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Define a weightof a tree T as

w(T) =uT

w(u), (3.28)

where the product is over all nodes ofT and

w(u) =

1 if uhas exactly one child,ak if uhas k children for some even k,

0 otherwise.(3.29)

Lemma 3.3. The generator of the SW transformation can be written as

S=n=1

TT(n)

nw(T) O(T). (3.30)

where

T(n) is the set of allS-admissible trees withn nodes.

Proof. We can use induction in n to show that Sn is the sum ofw(T)O(T) over all S-admissibletrees with n nodes. The base of induction is n = 1 in which case there is only one S-admissibletree T (a single node) and O(T) =S1. To prove the induction hypothesis for generaln we can useEq. (3.11) and Eq. (3.12). Repeatedly expanding S2j in Eq. (3.11) one gets a sum over S-admissibletrees, where the operators O u correspond toSni encountered in the course of the expansion.

Substituting the Taylor series Eq. (3.30) into Eq. (3.13) one arrives at Eq. (3.27).

3.4 Convergence radius

In this section we analyze convergence of the formal series S=j=1 Sj

j

andHeff=j=1 Heff,j

j

.Lemma 3.4. The series forS andHeffconverge absolutely in the disk|| < c where

c= c

8

1 + 2|I0|

.

Herec= /(2V) and|I0| is the width of the intervalI0.

Proof. Define projectors Q0= I P0 andP =I P. Simple algebra shows that

RP0RP =I+ Z, where Z=2(P0Q+ Q0P).For any >0 define a disk D() ={C :||< }. It is well-known that P =P() has analyticcontinuation to D(c). Indeed, choose a contour in the complex plane as the boundary of the(/2)-neighborhood of the intervalI0, see Fig.3. Then any eigenvalue ofH0 has distance at least

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0I

Figure 3: The contour (dashed black curve) encircling the intervalI0 (bold red). Eigenvalues ofH0 are indicates by solid dots.

/2 from and(zI H0)1 2/ for all z . It implies that for any z the resolvent(zI H)1 is analytic in D(c). Thus we can define

P = 1

2i

dz(zI H0 V)1 (3.31)

and the Taylor series P = P0 +j=1 Pj

j

converges absolutely in the disk D(c). Furthermore,one can easily check that P is a projector, P2 = P, for all D(c). Note however that P is nothermitian for complex and thus one may haveP >1 (clearly,P 1 for any non-zero projectorP since|P| = 1 for some state ).

Using the standard perturbative expansion of the resolvent in Eq. (3.31) one gets a Taylor series

P =P0+j=1

Pjj , Pj =

1

2i

dz(zI H0)1

V(zI H0)1j

.

Noting that the contour has length|| = + 2|I0|, we can bound the norm ofPj for j1 as

Pjj

1 +2

|I0

| 2

V j

=

1 +2

|I0

| (||/c)j.It follows that

P0Q= P0 P0P j=1

P0Pjj j=1

Pjj

1 +2|I0|

||

c || 0 depending only onn, , and the maximumdegree of the interaction graph such that for all|| < c the ground state energy ofHneff approximatesthe ground state energy ofH0+ Vwith an error at mostn = O(N||n+1).

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We begin by proving the theorem for the special case of so-called block-diagonal perturbations,see Section 4.2. These are perturbations composed of local interactions preserving the low-energysubspaceP0. For block-diagonal perturbations the SW transformation is trivial, that is, S = 0,see Section 3.2. It follows that Heff = P0H0 + P0V P0. Clearly, Heff can reproduce ground state

properties of the exact Hamiltonian H0+ Vonly if at least one ground state ofH0+ V belongsto the low-energy subspaceP0. We prove that this is indeed the case if|| < c for some constantc> 0 that depends only on the gap , maximum degree of the interaction graph, and the strengthof interactions inV, see Lemma4.1.

The role of the SW transformation is to map generic perturbations to block-diagonal perturba-tions. Unfortunately, the SW transformation cannot be used directly since the transformed truncatedHamiltonianU(H0 +V)U becomes local only when restricted to P0, while Lemma4.1requires local-ity on the entire Hilbert space. In addition, the Taylor coefficients Sp of the SW generator are highlynon-local operators which complicates an analysis of the error obtained by truncating the series atsome finite order. We resolve this problem by employing an auxiliary perturbative expansion due toDatta et al[2] which we refer to as a local Schrieffer-Wolfftransformation, see Section4.4. It defines

an effective low-energy Hamiltonian

Heff,loc = P0 eT(H0+ V)e

TP0

for some unitary transformation eT. This transformation is constructed such that the transformedHamiltonianeT(H0 + V)e

T is block-diagonal, that is, it can be written as a sum of (approximately)local interactions preservingP0. Here the locality of interactions holds on the full Hilbert spacewhich allows us to apply Lemma 4.1. In Section 4.4 we prove an analogue of Theorem 1 for then-th order effective Hamiltonian constructed using the local SW method. The proof uses some oftechniques developed by Datta et al [2] to bound the error resulting from truncating the series forthe generator T.

Finally, we construct a unitary transformation eK acting on the low-energy subspace P0 thatmaps the effective low-energy Hamiltonians obtained using the standard and the local SW methodsto each other, that is, Heff = e

KHeff,loceK, see Section 4.5. The relationship between low-energy

theories obtained using the standard and the local SW methods is illustrated on Fig. 4. We useweak multiplicativity of the SW transformation proved in Section2.4to show that the perturbativeexpansion ofKcontains only linked clusters. This linked cluster property ofKallows us to boundthe error resulting from truncating the series for Kat then-th order by O(N||n+1). Combining thetwo errors we obtain the desired bound on n in Theorem1.

4.1 The unperturbed Hamiltonian and the perturbation

Let be a lattice or a graph with Nsites such that each siteu is occupied by a finite-dimensionalparticle (spin) with a local Hilbert spaceHu. Accordingly, the full Hilbert space is

H=u

Hu.

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We will choose the unperturbed Hamiltonian H0 as a sum of single-spin operators,

H0=u

H0,u, (4.2)

where H0,u acts non-trivially only on a spin u. By performing an overall energy shift we can alwaysassume that each term H0,u is a positive semi-definite operator and the ground state energy ofH0,uis zero. LetP0,u Hu be the ground subspace of a spin u and P0,u be the tensor product of theprojector ontoP0,u and the identity operator on all other spins. Then the ground subspace of theentire HamiltonianH0 is

P0 =u

P0,u

and the projector ontoP0 can be represented as

P0 =

uP0,u.

Let Q0 = I P0 be the projector ontoP0 and Q0,u = I P0,u. We shall assume that each termH0,u has a spectral gap at least above the ground state. Thus the full HamiltonianH0 also hasa spectral gap at least . In general, the maximal norm of the local termsH0,u is a parameterindependent of . However, for the sake of simplicity we shall assume that

H0,u= O() for all u.

It is worth mentioning that if we allow each spin to have two or more distinct levels in the low-energy subspaceP0,u, the full Hamiltonian might not have a spectral gap between the subspacesP0and

P0 . Consider as an example the case when each spin has 3 basis states

|0,|1,|2

which haveenergy 0, >0, and respectively such that the low-energy subspaceP0,u is spanned by thestates|0 and|1. Then the state|1N P0 has energy Nwhile the state|2, 0(N1) P0 hasenergy . Thus for any constant and the spectral gap betweenP0 andP0 closes ifN1.

We will choose the perturbation Vas a sum of two-spin interactions between nearest neighborsites,

V =

(u,v)E

Vu,v.

HereE is the set of edges of the underlying lattice (which could be an arbitrary graph) and Vu,v is anarbitrary hermitian operator acting on a pair of sites u, v. To describe the effective low-energy theorywe will have to consider k-localperturbations, that is, Hamiltonians with at mostk-spin interactions.

Ak-local perturbation Vcan be written as

V =A

VA, VA= 0 unless|A| k.

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Here VA is an arbitrary hermitian operator acting on a subset of spins A. Define a strengthof theperturbation as

J= maxu

A :Au

VA. (4.3)

The strength of the perturbation provides an upper bound on the combined norm of all interactionsaffecting any selected spin. If is a regular lattice in a space of bounded dimension thenJcoincidesup to a constant factor with the maximum norm of interactions, J maxA VA. We will alwaysassume thatJand are constants independent ofN. Let us say that a perturbation V =

A VA

is block-diagonaliffVA preserves the low-energy subspaceP0, or equivalently, [VA, P0] = 0.

4.2 Block-diagonal perturbations

Recall that the goal of the SW transformationU is to transform the perturbed Hamiltonian H0 + Vinto a block-diagonal form such that the transformed Hamiltonian U(H0+ V)U

preserves the low-energy subspace

P0. In this section we analyze the special case when the perturbed Hamiltonian

H0+ Vis already block-diagonal, so that U=I. In this case the effective low-energy Hamiltonianis simply Heff = P0(H0 + V)P0. Since we assumed that H0P0 = 0, one has Heff = P0V P0. Anatural question is whether the ground state energy ofHeffcoincides with the ground state energy ofH0+ V. Clearly, this happens iff at least one ground state ofH0+ Vbelongs to the subspaceP0.In this section we prove that this is indeed the case provided that the strength of the perturbationV is below certain constant threshold depending only on and k. Intuitively it follows from thefact that exciting any subset ofw spins increases the energy of the term H0 by at least w, whilethe energy of the term Vcan go down at most by 2w||J. Hence exciting spins can only increasethe energy provided that|| < c/J. Unfortunately this simple argument does not tell anythingabout states that contain superpositions of different excited subsets with different w, and one may

expect that exciting subsets of spins in a superposition can help to reduce the overall energy. Weanalyze this general situation in the following lemma (to simplify notations we include the factor into the definition ofV).

Lemma 4.1. LetV =A VA be ak-local perturbation with strengthJ. Suppose that each term

VA is block-diagonal, that is, VA preserves the subspaceP0. Assume also that2k+2J


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wherex is a binary string of lengthNsuch that xu = 1 indicates that a spinu is excited (P0,u |x = 0)andxu= 0 indicates that a spin u is not excited (P0,u |x =|x). More formally,

|x =R(x) |, where R(x) =u

((1 xu)P0,u+ xuQ0,u).

We note that|x may be a highly entangled state. Since we assumed| P, the configurationwith no excited spins does not appear in|, that is,

|0N = 0.Let us choose a configurationx {0, 1}N with the largest amplitude

x=x|x,that is,

x

x for allx. (4.5)The key idea is to transform| to a new state (which is generally mixed) by annihilating allexcitations in|x while leaving all other components|x, x=x, untouched. We will show thatfor sufficiently small strength ofVthe energy of is smaller than the energy of| thus arriving ata contradiction.

For any spin u choose an arbitrary ground state|u P0,u. Define a trace preservingcompletely positive (TPCP) mapEu that erases the state of the spin u and replaces it by the chosenground state|u. More formally, for any (mixed) state describing the entire system one has

Eu() =|uu| Tr u().

Note thatEu() may be mixed even if is a pure state. The mapEu can be regarded as anannihilation superoperator. Obviously, annihilation superoperators on different spins commutewith each other. Let be a state obtained from|xby applying the annihilation superoperator toevery excited spin, that is,

=

u : xu=1

Eu

(|xx|).

Clearly,Eu reduces the energy of the term H0,u at least by x for any excited spin u. Thus if wedenote the number of excited spins (the Hamming weight of the string x) byw then

Tr ( H0)

x|H0

|x

w x. (4.6)

Now consider a state obtained from| be removing the largest-amplitude term,

|else =x=x

|x.

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Denote= + |elseelse|. (4.7)

SinceEu are trace preserving maps, we have Tr =x|x and thus Tr =| = 1. Let usshow that for sufficiently small strength ofVthe energy of is smaller than the energy of

|

, thus

getting a contradiction. Indeed, since H0 cannot change a configuration of excited spins we have

Tr (H0) |H0| = Tr( H0) x|H0|x w x, (4.8)

see Eq. (4.6). It remains to bound the contribution to the energy coming from the perturbation V.Simple algebra leads to

Tr (V) |V| =A

Tr (VA) x|VA|x 2Re(x|VA|else). (4.9)

Note that the states and|x have the same reduced density matrices on any subset A thatcontains no excited spins, that is, x

u = 0 for all u A. It follows from the fact that annihilationsuperoperators applied outside Acannot change the reduced state ofA. Therefore

Tr (VA) x|VA|x = 0 if xu= 0 for all uA. (4.10)

Taking into account that any interaction VA is block-diagonal, the operator VA can change a config-uration of excited spins only if at least one spin in A is already excited. It means that

x|VA|else = 0 if xu = 0 for all uA. (4.11)

Using Eq. (4.10) we get a bound

A

Tr (VA) x|VA|x

u :xu=1

Au

|Tr (VA) x|VA|x| 2wxJ. (4.12)

It bounds the contribution to the energy from the first and the second terms in Eq. ( 4.9). Now letus bound the contribution from the last term in Eq. (4.9). Using Eq. (4.11) we get

A

2Re(x|VA|else)

u : xu=1

Au

x=x

2|x|VA|x|

u : x

u=1Au

2k+1VAxx

2 2kwxJ. (4.13)

Here the first line follows from Eq. (4.11) and definition of|else. The second line follows from theCauchy-Schwarz inequality and the fact that VAacts non-trivially on at most k spins, so the number

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of possible transitions x x is upper bounded by 2k. The third line follows from the maximalitycondition Eq. (4.5). Combining bounds Eq. (4.12,4.13) we arrive at

|Tr (V) |V|| 2k+2wxJ. (4.14)

Combining it with Eq. (4.8) we finally get

Tr (H) |H| ( + 2k+2J)wx. (4.15)

Since by assumption x contains at least one excitation, we have w >0. It means that the energyof is strictly smaller than the one of| if 2k+2J < . However this is impossible since| is aground state. Thus our assumption| P leads to a contradiction.

Lemma4.1can be easily generalized to the case when V contains interactions among arbitrarilylarge number of spins, but the magnitude ofk-spin interactions decays exponentially with k .

Corollary 4.1. Consider a HamiltonianV =na=1 Va such thatVa is ak(a)-local perturbation withstrengthJ(a). Suppose that all interactions in each perturbationVa are block-diagonal. Assume also

thatna=1

2k(a)+2 J(a)


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where Kn,Cis defined as the sum of all monomials in Heff that have support Cand degreen. Givenany set of edges C E let (C) be the corresponding set of sites, such that u (C) iffCcontains an edge incident to u.

Let us point out that the only purpose of introducing the variables u,v is to define the operators

Kn,Cand the decomposition Eq. (4.17). Once this decomposition is defined, we can set u,v = forall edges. Also note that we do not make any assumptions about convergence ofHeff considered asa multivariate series3 since we only consider some fixed order n. The main result of this section isthe following theorem.

Theorem 2 (linked cluster theorem). The operatorKn,Cacts only on the subset of spins(C).In addition,Kn,C= 0 unlessC is a connected subset of edges and|C| n.Thus then-th order effective Hamiltonian includes only interactions among linked clusters of spinsof size at mostn. However let us point out that the theorem does not say that Kn,Cacts non-triviallyon al lspins in (C).

Proof. Choose any subset of edges C E and set u,v = 0 for all (u, v) / C. Partition the latticeas =AB where A= (C) and B is the complement ofA. Then we have V =VA IB for someoperator VA acting onA. Since the unperturbed Hamiltonian H0 does not include any interactions,additivity of the effective Hamiltonian implies Heff,n =H

Aeff,n IB, where HAeff,n is the effective low-

energy Hamiltonian computed using only the subsystem A with the perturbationVA, see Lemma3.5in Section3.5. We conclude that all monomials in the decomposition ofHeff,n act trivially onB . Bydefinition, Kn,C does not depend on the variables u,v with (u, v) / C, so we infer that Kn,C actstrivially onB.

Suppose now thatCconsists of two disconnected subsets of edges C1 andC2. ChooseA = (C1)and B = (C2). Again we set u,v = 0 for all (u, v) / C. Then we have V =VA IB +IA VB,where VA

IB is obtained from V by setting u,v = 0 for all (u, v)

C2. Similarly, I

A

VB is

obtained fromVby settingu,v = 0 for all (u, v) C1. Additivity of the effective Hamiltonian impliesHeff,n = H

Aeff,n IB +IA HBeff,n, where HAeff,n and HBeff,n are the effective Hamiltonians computed

using only the subsystems A and B with perturbations VA and VB respectively. It means that nomonomial inHeff,n contains variables u,v from both C1 and C2 simultaneously. SinceKn,Cdoes notdepend on the variables u,v with (u, v) /C, we infer that Kn,C= 0.

The statement Kn,C= 0 for|C| > nfollows from the fact that Heff,n contains only monomials ofdegreen.

It is obvious from the proof that the linked cluster theorem applies to any operator-valued mul-tivariate series that obeys the additivity property.

3

Let us remark that absolute convergence of the univariate series Heff =

n=1Heff,nn does not imply absoluteconvergence of the corresponding multivariate series, since different terms in the multivariate series may cancel outeach other when combined into a single term Heff,n.

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4.4 Local Schrieffer-Wolff trasformation

Applying the SW transformation described in Section 3 to many-body systems one must keep inmind that its generatorS is a highly non-local operator even in the lowest orders of the perturbativeexpansion. This follows from the fact thatSis a block-off-diagonal operator, that is P0SP0 = 0 and

Q0SQ0 = 0, see Lemma 2.3. One can easily check that any block-off-diagonal operator must actnon-trivially on every spin in the system. Indeed, block-off-diagonality impliesS = P0S+ SP0. Ifone assumes that Sacts trivially on some spin u, then Q0,uSQ0,u =SQ0,u= 0. On the other hand,Q0,uSQ0,u = Q0,u(P0S+ SP0)Q0,u = 0 since Q0,uP0 = P0Q0,u = 0. The non-locality ofS is not avery serious drawback if the main object one is interested in is the effective low-energy HamiltonianHeff. Indeed, the linked cluster theorem proved in the previous section implies that all low-ordercorrections to Heff are in fact local operators. On the other hand, the non-locality of S makes itmore difficult to analyze properties ofHeff. In this section we get around this difficulty by employinga local version of the SW transformation developed by Datta et al [ 2]. To avoid a confusion betweenthe two types of the SW transformation we shall refer to the direct rotation described in Sections2,3as a global SW transformation.

The local SW transformation is a unitary operator U that brings the perturbed HamiltonianH0+ V into anapproximatelyblock-diagonal form. Specifically, for any fixed integer n0 we shallchooseU=eT where T is a degree-npolynomial in ,

T =nq=1

Tqq, Tq =Tq.

This polynomial will be chosen such that the transformed Hamiltonian eT(H0+ V)eT is block-

diagonal if truncated at the order n. The definition ofT is similar to the perturbative definition ofS, see Section3.2. We use the Taylor expansion of exp (T) to get

eT(H0+ V)eT =H0+

nj=1

j

[Tj, H0] + V(j1)

+

j=n+1

j V(j1), (4.18)

where V(j) is a certain polynomial function ofT1, . . . ,Tmin(j,n) applied to H0 or V. The precise formof this polynomial will be given in Eq. (4.27). Strictly speaking,V(j) should also carry a labeln, butsincen is fixed throughout this section, we shall only retain the label j.

The main difference between the local and global SW method is the inductive rule used todefine the generators of the transformation. In the global SW method the generatorsSj are fixedby demanding that Sj is block-off-diagonal while the j-th order contribution to the transformed

Hamiltonian is block-diagonal, see Section3.2. In the local SW method the generatorsTj are fixedby demanding that Tj is a sum of local interactions (the range of interactions may depend on j) andeach interaction is locallyblock-off-diagonal on the subset of spins it acts upon. For each local termin the expansion ofV(j1) we shall introduce the corresponding local term inTj chosen such that theexpansion of [Tj , H0] + V

(j1) contains only block-diagonal terms.

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where

Hn =H0+nj=1

jA

DA(V(j1)A ) (4.24)

is a sum of local block-diagonal interactions, while

Hgarbage =

j=n+1

j V(j1) (4.25)

includes all unwanted terms generated at the order n+ 1 and higher. Finally, we define then-thorder effective low-energy Hamiltonian as a restriction ofHn onto the subspaceP0,

Hneff =P0H

nP0. (4.26)

It is obvious from this construction that Hneff is additive for perturbations describing two non-

interacting disjoint systems, see Lemma3.5. It follows that the Taylor coefficients ofHneff obey the

linked cluster property of Theorem2.For the later use let us give an explicit formula for the polynomial V(j). We have V(0) =V and

V(j1) =

jq=2

1

q!

1j1,...jqn

j1+...+jq=j

Tj1 Tjq(H0) +j1q=1

1

q!

1j1,...jqn

j1+...+jq=j1

Tj1 Tjq(V), (4.27)

for all j 2. Note that convergence of all above series in the limit n does not matter sincewe always consider some fixed order n. However, it can be shown [2]that the n limits of theseries for T andH

neff converge absolutely in the disk of radius c1/N2.

Below we shall prove that the norm of Hgarbage can be bounded by O(N

|

|n+1) for sufficiently

small . Then we shall employ Lemma4.1 and its Corollary 4.1to show that the ground state ofHn belongs to the subspaceP0 for sufficiently small . Combining the two results together we shallget the following theorem which can be regarded as an analogue of Theorem 4.7 in [2].

Theorem 3. LetHneff be then-th order low-energy effective Hamiltonian constructed using the local

SW method. There exists a constant threshold

c = (n2) (4.28)

such that for all|| < c the ground state energy of Hneff approximates the ground state energy ofH0+ Vwith an errorn = O(N||n+1).

Given any perturbation V represented as a sum of local interactions, V =A VA we shall use

the notationV1 for the strength ofV, that is,

V1= maxu

Au

VA.

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All derivations below will implicitly use a bound

A :AM=

VA |M| V1 (4.29)

that holds for any operator V =A VAand for any subset of spins M . In particular, choosing

M= we get a bound

V A

VA N V1. (4.30)

Applying Eq. (4.30) we can bound the norm ofHgarbage as

Hgarbage =eT(H0+ V) eT Hn N

j=n+1

||j V(j1)1. (4.31)

Thus our task is to get an upper bound on the norm

V(j)

1. To simplify notations we shall assume

that the original perturbationV has strengthV1= 1.Lemma 4.2. The operators Tj and V

(j1) are (j+ 1)-local Hamiltonians. There exists constants, >0 such that

V(j)1

n2

j

(4.32)

for allj0.Let us first explain how Theorem 3follows from Lemma 4.2. Define c = n

2. Assume forsimplicity that >0. Substituting Eq. (4.32) into Eq. (4.31) one gets

Hgarbage N

j=n+1

j1jc 2Nn2(/c)n+1 (4.33)

provided that < c/2. Using Corollary4.1of Lemma4.1and the fact that V(j1) is (j+ 1)-local

Hamiltonian we conclude that the ground state ofHn belongs to the subspaceP0 whenevernj=1

23+j ||j V(j1)1< .

Using the bound Eq. (4.32) we get

nj=1

23+j ||j V(j1)116j=0

(2/c)j 32


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provided that < c/4 and 32


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Letu be the spin that maximizes the normTj(W)1. Then

Tj(W)1 2Mu

A :AM=

Tj,M WA + 2Au

M:MA=

Tj,M WA

2Mu

A :AM=

V(j1)M WA + 2Au

M:MA=

V(j1)M WA

2

W1Mu

|M| V(j1)M + 2

V(j1)1

Au

|A| WA

2(r+ n + 1)

V(j1)1 W1.

Now we can apply Proposition 4.2 to bound the norm of any term in Eq. (4.27). Note thatV(j1) is composed of two types of terms. We have terms proportional to Aj = Tj1

Tjq(H0)

with 1 j1, . . . , jq n, j1 +. . .+ jq = j, and terms proportional to Bj = Tj1 Tjq(V) with1 j1, . . . , jqn,j1+ . . . +jq =j 1. Note that applying qcommutators Tj1 Tjq toH0 or V wecan produce at most O(qn)-local operator. Applying Proposition4.2we get

Aj1 O(n1)q qq V(j11)1 V(jq1)1,Bj1 O(n1)q qq V(j11)1 V(jq1)1 (4.35)

where we have taken into account thatV1 = 1 andH01 = O(). Let us introduce an auxiliaryquantity

j =V(j1)1, j1. (4.36)Using Eqs. (4.27,4.35) we can get an upper bound jj, where

1= 1 = V1= 1

andj ,j2, are defined inductively from

j =

jq=2

cq

j1+...+jq=j

j1 jq+j1q=1

cq

j1+...+jq=j1

j1 jq , (4.37)

where cO(n1). Here we have taken into account that qq/q! = O(1)q and omitted the constraintj

1, . . . , j

qn since we are interested in an upper bound on

j(note that adding this constraint can

only makej to grow more slowly with j ). In order to get an upper bound onj define an auxiliaryTaylor series

(z) =j=1

jzj . (4.38)

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Inspecting the recursive equation Eq. (4.37) one can easily convince oneself that (z) has to obeythe following equation

=

1

1

c 1 c

+ z

1

1

c 1

+ z.

Solving this equation one can easily find the inverse function:

z() = 2(c + c2).

Clearlyz() is analytic in the entire complex plane. Solving the quadratic equation and choosingthe branch for which (0) = 0 one gets

(z) = 1

2(c + c2)

1

1 4z(c + c2)

.

Clearly(z) is analytic in a disk|

z|< z

0where

z0= 1

4(c + c2)

n2

for some constant > 0. Here we have taken into account that c = O(n1). Using the Cauchyintegral formula for(z) one gets

j = 1

2i

|z|=z0/2

(z)

zj+1dz

which yields

j

n2

1j

for some constant >0. The lemma is proved.

4.5 Equivalence between the local and global SW theories

In this section we prove Theorem1. The key idea of the proof is to construct a unitary operator eK

acting on the low-energy subspace P0that transforms then-th order effective low-energy Hamiltonianobtained using the local SW method to the one obtained using the global SW method with an errorO(N||n+1) in the operator norm. We illustrate the relationship between the two low-energy theorieson Fig. 4. Once the desired unitary transformation e

K

is constructed, Theorem 1 follows directlyfrom Theorem3. To introduce the notations let us first consider the limitn . Let

S=j=1

Sjj and T =

j=1

Tjj (4.39)

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The proof of Theorem4goes as follows. We shall regard Eq. (4.41) as a definition of a formalTaylor series

K=j=1

Kjj

Here Kj are some operators acting on the entire Hilbert spaceH. We will show that operatorsKjare actually block-diagonal, so we can apply the transformation eK on the subspaceP0 only, seeLemma 4.3, even if one uses a truncated version of K. Next we shall make use of the fact thatthe SW operators eS and eT have (weak) multiplicativity properties, see Lemma2.4. It will allowus to prove multiplicativity for eK and thus additivity ofK, see Eqs. (4.46). Next we shall exploitthe additivity of K to show that the series for K obeys the linked cluster condition analogous toTheorem2. Finally, we shall construct the desired transformation mapping the local theory to theglobal theory as exp (Kn), where Kn =

nj=1 Kj

j . Lemma4.4shows that this transformationindeed does the right job.

Lemma 4.3. SupposeH0

has a positive spectral gap betweenP0

andP

0 . Then all generatorsK

jare block-diagonal.

Proof. By definition we haveHgl =eKHloc eK (4.42)

and both operators Hgl, Hloc are block-diagonal. It is assumed here thatHgl and Hloc are formalTaylor series in Let us multiply Eq. (4.42) byP0 and Q0 on the left and on the right respectively.Taking into account that Hgl and Hloc are block-diagonal, and using formal Taylor series Hloc =j=0 H

locj

j with Hloc0 =H0 one obtains

P0[Kj , H0]Q0+ P0f(K1, . . . , K j1, Hloc1 , . . . , H

locj )Q0= 0 for allj

1, (4.43)

where f(K1, . . . , K j1, Hloc1 , . . . , H

locj ) is some polynomial obtained from the expansion of the expo-

nent in Eq. (4.42). Let us use induction in j to show that Kj is block-diagonal. Indeed, suppose wehave already proved that K1, . . . , K j1 are block-diagonal. Then the term

P0f(K1, . . . , K j1, Hloc1 , . . . , H

locj )Q0

in Eq. (4.43) vanishes and we can rewrite Eq. (4.43) as

(P0KjQ0)(Q0H0Q0) = (P0H0P0)(P0KjQ0). (4.44)

If| Q0 is any eigenvector of Q0H0Q0 with an eigenvalue , then Eq. (4.44) implies that astate (P0KjQ0) | is an eigenvector of P0H0P0 with the same eigenvalue . On the other hand,(P0KjQ0) | P0. It contradicts to the spectral gap assumption unless (P0KjQ0) | = 0. Sinceeigenvectors of Q0H0Q0 span the entire subspace Q0, we conclude that P0KjQ0 = 0. Since K isanti-hermitian, it implies Q0KjP0 = 0, that is, Kj is block-diagonal.

42


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Let us now prove that the restriction ofKonto theP0 subspace obeys the additivity property:if one partitions the lattice into two disjoint parts, = AB, such that V = VA +VB thenK=KA+ KB (only on the subspaceP0). Note that the generator Tconstructed using the local SWmethod is additive by construction, so we have T =TA+ TB on the entire Hilbert space (not only on

the subspaceP0). On the other hand, weak multiplicativity of the direct rotation, see Lemma2.4,impliesP0 e

S =P0 eSA+SB . (4.45)

SinceeK =eSeT, we conclude that the restriction ofKonto the subspaceP0 is additive, that is,eKP0= e

KAKB P0 (4.46)

By abuse of notations let us identify K, KA, KB with their restrictions of the on the subspaceP0.Then we have eK =eKAeKB =eKAKB . Taking the logarithm we obtain the desired additivityproperty, K = KA+KB. It implies that the series for K obeys the linked cluster condition, seeTheorem2. Using the same notations as in the statement of Theorem 2we conclude that

Kj =CE

Kj,C,

where Kj,Cacts only on (C) and Kj,C= 0 unless|C| j and C is a connected subset of edges.Define an interaction strength ofKas

JK= maxuL

maxj=1,...,n

Cu

Kj,C. (4.47)

We claim thatJK=O(1), (4.48)

that is,JKhas an upper bound independent ofN. Indeed, assuming that the interaction graph hasdegreed = O(1), the number of linked clusters of sizen containing a given vertex ucan be upperbounded by dO(n) = O(1). Additivity of K implies that we can compute Kj,C by turning off allinteractions Vu,v, (u, v) /C. It leaves us with a subgraph with at most N = 2|C| 2j2n= O(1)sites. Since the number of spins is O(1), any operator Kj,C is a constant-degree polynomial withconstant-bounded coefficients, so thatKj,C = O(1). Theorem 4now follows from the followinglemma.

Lemma 4.4. LetK,L, Mbe formal operator-valued Taylor series acting on the subspaceP0 suchthatK is anti-hermitian, K(0) = 0, and

M=eKLeK. (4.49)

For any integern= O(1) define truncated series

K=nj=1

Kjj , L=

nj=0

Ljj , M=

nj=0

Mjj (4.50)

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and a truncation error=M eKL eK. (4.51)

SupposeK,L, Mobey the linked cluster property and have constant-bounded interaction strength,JK, JL, JM=O(1). Then

O(1)N||n+1 for any|| 1. (4.52)Proof. Introduce a truncated version of a transformed operator eKL e

K, namely,

=L +nj=1

1

j![K, ]jL.

The assumptions that K obeys the linked cluster property and K(0) = 0 imply that iK can beregarded as a local Hamiltonian with interaction strength upper bounded by JK

nj=1 ||j =O(||).

Similarly,L can be regarded as a local Hamiltonian with interaction strengthO(1). Using Lemmas 1,2

from [24]one gets an approximation

eKL eK 1(n + 1)!

[K, ]n+1L O(1)N||n+1. (4.53)

Therefore it suffices to show that

M O(1)N||n+1. (4.54)

Expanding the multiple commutators in we get a linear combination of terms

1

j!

q1+q2+...+qj+q Kq1Kq2

Kqj(Lq), 1

q1, . . . , q j

n, 1

j

n, 1

q

n.

Let us call a term as abovegoodif the total power of is at most n, that is, q1 + q2 + . . . + qj+ qn.Otherwise let us call such a term bad. Using the condition Eq. (4.49) one can easily check that thesum of all good terms in equals M. Thus we get a bound

M ||n+1 #{bad terms}max Kq1Kq2 Kqj (Lq). (4.55)

Clearly for a constant n the total number of terms in is O(1) and so the number of bad terms isO(1). Applying again Lemma 2 from [24] and taking into account that the Taylor coefficients KqiandLq are local Hamiltonians with interaction strength O(1) we get a bound

max Kq1Kq2 Kqj (Lq) O(1)N.

Therefore M O(1)N||n+1.

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Acknowledgments

We thank Barbara Terhal for useful discussions. SB would like to thank RWTH Aachen Universityand University of Basel for hospitality during several stages of this work. SB was partially supportedby the DARPA QuEST program under contract number HR0011-09-C-0047. DL was partially sup-ported by the Swiss NSF, NCCR Nanoscience, NCCR QSIT, and DARPA QuEST.

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