Locality of temperature
M. Kliesch,1 C. Gogolin,1 M. J. Kastoryano,1 A. Riera,1, 2 and J. Eisert1
arXiv:1309.0816v4 [quant-ph] 1 Aug 2014
2
1
Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany
Max Planck Institute for Gravitational Physics (Albert Einstein Institute), Am Mühlenberg 1, 14476 Potsdam-Golm, Germany
This work is concerned with thermal quantum states of Hamiltonians on spin and fermionic lattice systems
with short range interactions. We provide results leading to a local definition of temperature, thereby extending
the notion of “intensivity of temperature” to interacting quantum models. More precisely, we derive a perturbation formula for thermal states. The influence of the perturbation is exactly given in terms of a generalized
covariance. For this covariance, we prove exponential clustering of correlations above a universal critical temperature that upper bounds physical critical temperatures such as the Curie temperature. As a corollary, we
obtain that above the critical temperature, thermal states are stable against distant Hamiltonian perturbations.
Moreover, our results imply that above the critical temperature, local expectation values can be approximated
efficiently in the error and the system size.
I.
INTRODUCTION
The ongoing miniaturization of devices, with structures
reaching the nanoscale, has lead to the development of extremely small thermometers [1, 2], some of which are so small
that they can only be read out with powerful electron microscopes [3]. Even small thermal machines working in the quantum regime have been suggested [4, 5]. In order to understand
the working of such devices, it is necessary to formulate a
theory of statistical mechanics and thermodynamics at the microscopic and mesoscopic scales. A prerequisite for this is a
good understanding of the limitations of the concept of temperature at small scales.
The problem with assigning locally a temperature to a small
subsystem of a globally thermal system is the following: Interactions between the subsystem and its environment that generate correlations can lead to noticeable deviations of the state
of the subsystem from a thermal state (see Figure 1). Hence,
given only a subsystem state, there is no canonical way to assign a temperature to the subsystem. We call this the locality
of temperature problem.
The first steps toward a solution of the locality of temperature problem have been taken in Refs. [6–8], and more recently, within the mindset of quantum information theory, in
Ref. [9]. The general locality of temperature problem is, however, still open. In this work, we conclusively solve it for spin
and fermionic lattice systems.
More precisely, we first show that the locality of temperature problem is equivalent to a decay of correlations measured by an averaged generalized covariance that precisely
captures the response of expectation values to perturbations
of the Hamiltonian. We expect the corresponding equality to
be useful for applications beyond the scope of this article.
We then provide conditions under which the generalized covariance decays exponentially with the distance, including a
detailed analysis of the preasymptotic, and of the finite-size
regime. In particular, this exponential decay holds above a
universal critical temperature that only depends on the “connectivity” of the underlying graph of the model and is an upper
bound on physically relevant critical temperatures such as the
Curie temperature.
While, in the low-temperature regime, quantum lattice
?
?? ?
?
T
Figure 1. The locality of temperature problem: Subsystems of thermal states are themselves, in general, not in a state with a locally
well-defined temperature. Down to which length scale can temperature be an intensive quantity?
models exhibit a great diversity of phases, many of which involve the emergence of long-range or topological order [10],
in the high-temperature regime, exponential clustering of correlations is expected. Our rigorous results help to delineate the
boundary between these two regimes. They build upon and go
significantly beyond previous results on the clustering of correlations in classical systems [11], for quantum gases [12],
i.e., translation-invariant Hamiltonians in the continuum, and
cubic lattices [13–15].
Mathematically, we significantly contribute to the problem of whether and under which precise conditions thermal
quantum states are stable against distant Hamiltonian perturbations. This is particularly relevant in the broader scheme
of phase transitions in classical and quantum lattice models
[14, 16] as well as for the foundations of statistical mechanics and the equilibration and thermalization behavior of closed
quantum systems [17–25]. In the light of the recent surge of
interest in these topics, developing a better understanding of
the properties of thermal states has become a timely issue.
A major obstacle to progress on some of the most interesting open questions in this context, such as equilibration time
scales in closed quantum systems, is the limited set of mathematical tools available for exploiting the structure of locally
interacting Hamiltonians [25]. Our results are among the first
that explicitly exploit properties of local Hamiltonians, without being limited to very specific models.
For quantum Monte Carlo simulations [26], our results provide a guideline as to how large the finite system size has to
2
be taken in order to be able to sample from the right partition
function and, conversely, to identify observables that are best
suited to detect long-range correlations.
In fact, our results are reminiscent of known statements
about ground states. If a Hamiltonian has a unique ground
state and is gapped, correlations in its ground state cluster
exponentially and faraway regions become essentially uncorrelated. This is rigorously proven using information theory
inspired methods such as Lieb-Robinson bounds and quasiadiabatic continuation [27–29]. These rigorous results allow
for certified algorithms that efficiently approximate ground
states of gapped Hamiltonians on classical computers [30].
In the same spirit, we are able to show that an exponential
decay of correlations renders thermal states locally efficiently
simulatable.
The rest of this paper is structured as follows: In Section II,
we formulate the precise setting and explain the main results
and their implications. In Section III, we discuss connections to known results on phase transitions, thermalization in
closed quantum systems, and matrix product operator approximations. Then, in Section IV, we discuss basic properties
of the generalized covariance, explain how our results can be
made applicable to finite-range k-body interactions, and state
the results for fermionic lattices. We proceed with proving
all theorems in Section V and conclude in Section VI. In the
Appendix, we provide a detailed proof of two bounds on truncated cluster expansions, one of which is an important ingredient to the proof of clustering of correlations.
II.
tum state ρ, and parameter τ ∈ [0, 1] as
covτρ (A, A′ ) := Tr ρτ A ρ1−τ A′ − Tr(ρ A) Tr(ρ A′ ) . (2)
We discuss various properties of this covariance and generalizations to arbitrary-rank quantum states in Section IV A.
The generalized covariance appears naturally in our first
theorem about the response of expectation values to perturbations. More precisely, when we are given an unperturbed
Hamiltonian H0 and perturbed Hamiltonian H, then the difference of expectation values in the corresponding thermal
states is captured by that covariance:
Theorem 1 (Perturbation formula). Let H0 and H be Hamiltonians acting on the same Hilbert space. For s ∈ [0, 1], define the interpolating Hamiltonian by H(s) := H0 + s (H −
H0 ) and denote its thermal state by gs := g[H(s)]. Then,
Tr A g0 (β) − Tr A g(β)
Z 1 Z 1
(3)
=β
dτ
ds covτgs (β) (H − H0 , A)
0
0
for any operator A.
The proof of the theorem, which is presented in Section V A, relies on the fundamental theorem of calculus and
Duhamel’s formula. We refer to the double integral over the
covariance in Eq. (3) as the averaged (generalized) covariance.
SETTING AND MAIN RESULTS
B.
In this section, we introduce the setting, state the locality of
temperature problem more formally, and state our results.
A. Perturbation formula for thermal states
As the first result, we state a perturbation formula, which
is a general statement about the response of the expectation
value of an observable in the thermal state, upon changes in
the system Hamiltonian. It does not make any reference to
the locality structure of the Hamiltonian but turns out to be
especially useful when correlations between local observables
decay rapidly with distance.
Throughout the paper, we assume the Hilbert space to be
finite dimensional [31] and denote the thermal state, or Gibbs
state, of a Hamiltonian H at inverse temperature β by
g(β) :=
e−β H
,
Z(β)
(1)
with Z(β) := Tr(e−β H ) being the partition function. If
we mean the thermal state or partition function of a different
Hamiltonian H ′ , we write g[H ′ ](β) or Z[H ′ ](β).
We measure correlations by the (generalized) covariance
that we define for any two operators A and A′ , full-rank quan-
Spin lattice systems
In the remainder of this work, we will be concerned with
spin and fermionic lattice systems. We will only write out
everything for spin systems and then later, in Sections IV C
and V C, explain the necessary modifications for fermionic
systems. In the case
N of spin lattice systems, the Hilbert space
is given by H = x∈V Hx , where V is called the vertex set
and is assumed to be finite. To make the presentation more accessible, many of the following definitions are highlighted in
Figure 2. A local Hamiltonian with interaction (hyper)graph
(V, E) is a sum
H=
X
hλ
(4)
λ∈E
of local Hamiltonian terms hλ acting on H. The (hyper)edge
set E is the set of supports λ = supp(hλ ) ⊂ V of the
local terms hλ . For
P any subset of edges F ⊂ E, we denote by HF :=
λ∈F hλ the Hamiltonian only containing the interactions in F , and for any subsystem B ⊂ V ,
we define the truncated Hamiltonian to be H↾B := HE(B) ,
where E(B) ⊂ {λ ∈ E : λ ⊂ B} is the restricted edge
set and N
we take H↾B to be an operator on the Hilbert space
HB := x∈B Hx .
Given some subsystem S ⊂ V there are two natural thermal
states associated with it:
3
∂B
supported on B,
Tr AB g↾B (β) − Tr A g(β)
Z 1 Z 1
(5)
=β
dτ
ds covτgs (β) (H∂B , A).
0
B
S E(S)
F
0
Now we choose S ⊂ B ⊂ V (see Figure 2). If, for a given
inverse temperature β, correlations over the distance between
S and ∂B are negligible, then the corollary clearly implies
that
Tr[A g(β)] ≈ Tr[AB g↾B (β)]
Figure 2. A 2D square lattice: The boxes indicate subsystems S ⊂
B ⊂ V . The edges in S are E(S), boundary edges of B are ∂B,
and F is a shortest path connecting S and ∂B; hence, d(S, ∂B) =
|F | = 2. The set of edges E(S) is an example for an animal of size
|E(S)| = 7, while ∂B is not connected and hence not an animal.
(i) g↾S (β) := g[H↾S ](β) denotes the thermal state of S
alone, i.e., the thermal state of the truncated Hamiltonian
H↾S .
(ii) g S (β) := TrS c (g(β)) denotes the full thermal state reduced to S.
For a non-interacting Hamiltonian, these two states coincide,
but, in general, this is not the case due to correlations between
S and its environment. This discrepancy raises the question
of how to define temperature locally as an intensive quantity,
i.e., the locality of temperature problem.
C. Locality of temperature
In order to locally assign a temperature to the subsystem
S ⊂ V it was suggested, e.g., in Ref. [9], to extend S by a
buffer region and define the temperature of S via the thermal
state of the Hamiltonian truncated outside the extended region
B, see Figure 2. The role of the buffer region B is to remove
the boundary effects and the correlations with the rest of the
system that are intuitively the reason for the locality of temperature problem. Nevertheless, it is not obvious how these
correlations should be quantified and how large this buffer region needs to be. We will see shortly that Theorem 1 answers
these questions.
By ∂B ⊂ E, we denote the set of boundary edges of B,
i.e., the edges having overlap with both B and its complement
B c := V \ B. Then, by choosing H0 = H − H∂B in Theorem 1, using that g0 = g↾B ⊗ g↾B c , and tracing over B c , we
obtain the following:
Corollary 1 (Truncation formula). Let H be a local Hamiltonian, let B ⊂ V be a subsystem, and denote the corresponding boundary Hamiltonian by H∂B , and the interpolating Hamiltonian by H(s) := H −(1−s) H∂B with its thermal
state gs := g[H(s)]. Then, for any operator A = AB ⊗ 1B c
(6)
for any observable AB = AS ⊗ 1B\S on S. Also note that
such an approximate equality does not hold whenever average
correlations over lengths exceeding the distance between S
and ∂B are non-negligible.
Hence, we have the following equivalence for the temperature defined via thermal states:
Implication 1 (Locality of temperature). Temperature is intensive on a given length scale if and only if correlations
(measured by the averaged generalized covariance) are negligible compared to 1/β on that length scale.
In order to fully exploit Corollary 1, it is necessary to bound
the generalized covariance, which we will do for high temperatures in the next section.
D. Clustering of correlations at high temperatures
For small temperatures, correlations can be arbitrarily longranged, as is, e.g., the case for the ferromagnetic Ising model
in two or higher dimensions below the Curie temperature. On
the other hand, above a universal critical temperature, depending only on a local property of the interaction graph, correlations cluster exponentially, as we will see next. Given the
combinatorial nature of parts of the arguments leading to this
result, we need additional notation related to edges and vertices of the lattice. Most of the following definitions can be
understood intuitively, as is shown in Figure 2.
We say that two subsystems X, Y ⊂ V overlap if X ∩ Y 6=
∅, a set X ⊂ V and a set F ⊂ E overlap if F contains an edge
that overlaps with X, and two sets F, F ′ ⊂ E overlap if F
overlaps with any of the edges in F ′ . A subset of edges F ⊂
E connects X and Y if F contains a sequence of pairwise
overlapping edges such that the first overlaps with X and the
last overlaps with Y and similarly for the case where X and/or
Y are just vertices.
The graph distance on V , and also the induced distance on
subsets of V , are denoted by d. The distance d(X, F ) of a
subset X ⊂ V and a subset F ⊂ E is 0 if X and F overlap
and otherwise equal to the size of the smallest subset of E
that connects X and F . Sometimes, we denote the support
of an operator by the operator itself, e.g., for two operators A
and A′ , their distance is d(A, A′ ) := d(supp A, supp A′ ) and
∂A ⊂ E are the edges across the boundary of supp(A).
A subset of edges F ⊂ E that connects all pairs of its elements λ, λ′ ∈ F is called connected. Such a connected set
4
F is also called an (edge) animal. The size |F | of an animal F is given by the number of edges contained in F . The
results presented here apply to Hamiltonians with interaction
graphs (V, E) whose number am of lattice animals of size m
containing some fixed edge is exponentially bounded. With
S
S
B
am := sup |{F ⊂ E connected : λ ∈ F, |F | = m}| , (7)
λ∈E
the growth constant α is the smallest constant satisfying
am ≤ α m .
For example, the growth constant of a D-dimensional cubic
lattice can be bounded as α ≤ 2 D e (Lemma 2 in Ref. [32]),
where e is Euler’s number. Moreover, α is finite for any regular lattice [33]. Upper bounds to growth constants for socalled spread-out graphs [32] render our results applicable for
the case of bounded-range two-body interactions. By a simple
embedding argument, one can also bound the growth constant
for the case of local k-body interactions on a regular lattice,
which we explain in Section IV B in detail.
For any operator A and p ∈ [1, ∞], we denote by kAkp its
Schatten p-norm; e.g., kAk∞ is the operator norm and kAk1
is trace norm of A. We call J := maxλ∈E khλ k∞ the local
interaction strength of a local Hamiltonian, as given in Eq. (4).
We are able to provide a universal inverse critical temperature β ∗ , which is, in particular, independent of the system size,
below which correlations decay exponentially with a thermal
correlation length ξ(β):
Theorem 2 (Clustering of correlations at high temperatures).
Let g(β) be the thermal state at inverse temperature β of a
local Hamiltonian with finite interaction (hyper)graph (V, E)
having growth constant α and local interaction strength J.
Define the quantities
p
β ∗ := ln 1 + 1 + 4/α /2 /(2 J)
(9)
and
ξ(β) := ln α e
2 |β| J
e
2 |β| J
−1
−1
.
(10)
Then, for every |β| < β ∗ , parameter τ ∈ [0, 1], every two operators A and B with d(A, B) ≥ L0 (β, a) [given in Eq. (50)],
and a := min{| ∂A|, | ∂B|},
| covτg(β) (A, B)| ≤
4 a kAk∞ kBk∞ − d(A,B)/ξ(β)
e
.
ln(3) (1 − e−1/ξ(β) )
(11)
The proof is given in Section V B.
In the following sections, we outline some of the applications of Theorem 2.
E.
d(S, ∂B)
(8)
Universal locality and stability at high temperatures
If one is interested in the state g S (β) of some subsystem S,
then one can truncate the Hamiltonian to S extended by some
buffer region and obtain the approximation via the thermal
Figure 3. The truncation from Corollary 2 and 3: For β < β ∗ and
d(S, ∂B) ≪ ξ(β) Corollary 2 implies that g S (β), depicted on the
S
left, and g↾B
(β), depicted on the right, are approximately equal.
state of the truncated Hamiltonian. The following theorem
implies that the approximation error is exponentially small in
the width of the buffer region.
For any operator ρ, we denote its reduction to a subsystem
S ⊂ V by ρS := TrS c [ρ] and note that
ρS
= sup {| Tr[A ρ]| : supp(A) = S, kAk∞ = 1} .
(12)
Then, as a consequence of Corollary 1 and Theorem 2, we
obtain the following:
1
Corollary 2 (Universal locality at high temperatures). Let H
be a Hamiltonian satisfying the conditions of Theorem 2, let
|β| < β ∗ , and let S ⊂ B ⊂ V be subsystems with d(S, ∂B) ≥
L0 (β, | ∂S|). Then,
S
g S (β) − g↾B
(β)
1
≤
v |β| J
e− d(S,∂B)/ξ(β) , (13)
1 − e−1/ξ(β)
S
denotes the thermal state of B reduced to S and
where g↾B
v := 4 | ∂S| | ∂B|/ ln(3).
Similarly, as a corollary of Theorems 1 and 2 we obtain the
following:
Implication 2 (Stability). Below the critical inverse temperature β ∗ [from Eq. (9)], thermal states of local Hamiltonians
are exponentially stable against distant locally bounded perturbations.
F. Efficient approximation
Corollary 2 on the universal locality of thermal states also
has the following complexity theoretic consequence:
Implication 3 (Efficient approximation). For |β| < β ∗ , local
expectation values can be approximated with a computational
cost independent of the system size and bounded polynomially
in the reciprocal error.
In this sense, the error bound (see Figure 4) of Corollary 2
is reminiscent of the quasi-locality of dynamics, as, e.g., presented in Ref. [34], which is a consequence of Lieb-Robinson
5
A. Critical temperatures and phase transitions
Figure 4. One can obtain slightly tighter error bounds in Corollaries 2 and 5 by using Eq. (47) directly. The plot shows this bound on
S
the approximation error g S (β) − g↾B
(β) 1 for the case of S being a single site on a 2D square lattice as a function of the inverse
temperature β in units of the critical temperature and the width of
the buffer region L. This can be seen as an imaginary time LiebRobinson “cone” with diverging width as β → β ∗ .
bounds [35, 36]. The quasi-locality theorem [34] allows for an
approximation of time evolved local observables by truncating
the Hamiltonian in the time evolution operator at a distance
L > 0 far away from the space time cone of the observable’s
support and has an approximation error that is exponentially
small in L.
G.
Fermions
In Ref. [37], it was shown for fermionic systems that twopoint functions of observables that are odd polynomials in
the fermionic operators decay exponentially with a correlation
length proportional to the inverse temperature. Here, we obtain an exponential decay of the covariance above the critical
temperature for all operators.
Observation 1 (Fermions). All results also hold for locally
interacting fermions on a lattice. See Theorem 4 and Corollaries 4 and 5 in Section IV C for the precise statements.
III.
RELATIONS TO KNOWN RESULTS
In this section, we discuss the critical temperature from the
clustering theorem, the connection of this work to concepts related to thermalization, and approximations of thermal states
with so-called matrix product operators. As a last point, we
briefly mention similarities with local topological quantum order.
Our results show that the quantity β ∗ , as defined in Eq. (9),
provides a potentially coarse but universal and completely
general upper bound on physical critical temperatures like the
Curie temperature. For the ferromagnetic two-dimensional
isotropic Ising model without externalpfield, our bound yields,
for example. 1/(β ∗ J) = 2/ ln((1 + 1 + 1/e)/2) ≈ 24.58,
whereas the phase transition between the disordered paramagnetic and the ordered ferromagnetic
√ phases is known to really
happen at 1/(βc J) = 2/ ln(1+ 2) ≈ 2.27 [16]. Our universal bound is about an order of magnitude higher than the actual
value for this example. To put this discrepancy into perspective, it is worth pointing out that it is generally a very difficult
task to estimate physical critical temperatures — numerically
or analytically. In fact, analytic expressions for critical temperatures or even just bounds on their values are known only
for very few models.
One of the few known general statements is the MerminWagner-Hohenberg theorem [38]. It states that in certain lowdimensional systems with short-range interactions there cannot be any phase transition involving the spontaneous breaking of a continuous symmetry at any non-zero temperature.
However, such systems can still have a low-temperature phase
with quasi-long-range order characterized by power-law-like
decaying correlations. Consequently, even for systems covered by the Mermin-Wagner-Hohenberg theorem, our Theorem 2 is nontrivial. For example, it implies an upper bound on
the critical temperature of the Kosterlitz-Thouless transition
in the two-dimensional XY -model [39].
In this work, we have concentrated on the general picture,
but it seems likely that refinements of the methods employed
and developed here can yield much tighter bounds on critical
temperatures if more specific properties of a model are taken
into account. At the same time, it remains an open problem to
actually find a model with a phase transition with long-range
order at the universal highest possible temperature.
B.
Foundations of statistical mechanics
The recent years have seen a large number of numerical and
experimental (see Ref. [21] for a review) as well as analytical investigations (see, for example, Refs. [17–19, 22–24]) of
equilibration and thermalization in closed quantum systems.
In the focus of these works are the approach to equilibrium
or properties of energy eigenstates. The current work complements this body of literature in that it shows fundamental properties of systems in thermal equilibrium. A feature
that makes the current work unique is that, contrary to essentially all other works, the results derived here explicitly
use the structure of locality interacting systems (noteworthy
exceptions are Ref. [24] and, albeit in a very special setting,
Ref. [18]).
The locality of thermal states is also of interest for recent
results [24] on the dynamical thermalization of translationinvariant lattice models: Our Corollary 2 guarantees the existence of a “unique phase” [24] for all temperatures above our
6
critical temperature. Hence, it implies that at sufficiently high
temperatures, Theorems 1, 2, and 3 of Ref. [24] are applicable
for any translation-invariant Hamiltonian.
There is also an interesting connection of our locality and
stability results to the so-called eigenstate thermalization hypothesis (ETH) [20, 21]. The ETH essentially conjectures that
the expectation values of certain physically relevant observables (for example local ones) in energy eigenstates of sufficiently complex Hamiltonians should be very similar to the
expectation values in thermal states with the same average energy. Corollary 2 and Implication 2 thus imply that the eigenstates of a Hamiltonian in the center of the spectrum (which
correspond to high-temperature thermal states) must, if the
Hamiltonian fulfills the ETH, also be locally stable against
perturbations of the Hamiltonian. This insight could put constraints on the class of Hamiltonians that fulfills the ETH, provide new insights into the properties of their eigenstates, and
open up new ways to test the ETH.
C. MPO approximation of thermal states
Matrix Product Operators (MPOs) are a certain class of
operators that are tractable on classical computers for onedimensional systems. Therefore, they play an important role
in numerical simulations based on so-called tensor networks.
An important ingredient to our proof of Theorem 2 on clustering of correlations will be a bound on a truncated cluster
expansion (Lemma 1). The original result on the cluster expansion (Lemma 2 in the Appendix) is due to Hastings and
was first used to approximate thermal states with inverse temperature below 2 β ∗ by MPOs [40]. This approximation is
summarized in the next theorem.
In one spatial dimension, this MPO approximation yields a
tensor size bounded polynomially in the system size and the
approximation error (see the subsequent corollary). In higher
dimensions, however, the MPO approximation yields a tensor size bounded only subexponentially in the system size and
is hence not computationally efficient, albeit exponentially
cheaper than storing the full density matrix g(β). In order
to explain this in more detail, we start the discussion with a
slightly non-standard definition of MPOs:
Definition 1 (Matrix product operator (MPO)).
2
Let (b[x](j) )dj=1 be a basis for the operators on Hx and write
an arbitrary operator A on H in the product basis as
X
O
b[x](kx ) ,
(14)
A=
Ak
k∈[d2 ]V
x∈V
with expansion coefficients Ak ∈ C and where [d2 ] :=
{1, 2, . . . , d2 }. If the Ak are of the form
Ak =
Y
a[x](k) ,
(15)
x∈V
where every a[x](k) only depends on at most r of the |V | indices kx , then A is called an MPO with tensor size d2r .
Thermal states can be approximated by such MPOs. The
following theorem is a consequence of Lemma 2, which we
will prove in the Appendix along with Lemma 1.
Theorem
P 3 (MPO approximation of thermal states [40]). Let
H = λ∈E hλ be a local Hamiltonian with finite interaction
graph (V, E) having a growth constant α, and local interaction strength J = maxλ∈E khλ k∞ , and define b(βJ) :=
α e|βJ| e|βJ| − 1 . Moreover, let β be small enough such
that b(βJ) < 1. Then, for each L ∈ Z+ , there exists a
self-adjoint MPO ρ(β, L) [given in Eq. (A.3)] with tensor size
d2 N (L) , where
N (L) := sup |{x ∈ V : d(x, x0 ) < L}|
(16)
x0 ∈V
is the number of vertices within a distance less than L. The
approximation error is bounded as
b(βJ)L
kg(β) − ρ(β, L)k1 ≤ exp |E|
− 1 ; (17)
1 − b(βJ)
i.e., for fixed |βJ| < b−1 (1), the trace norm difference scales
as O(|E| e−| ln[b(βJ)]| L ) for large enough L.
In particular, the theorem implies the following:
Corollary 3 (Bound on the tensor size). Let D be the spatial
dimension of the Hamiltonian’s interaction graph (V, E), let
n := |E| be the system size, and β < 2 β ∗ with β ∗ from
Eq. (9). Then, the MPO approximation in Theorem 3 gives
rise to a tensor size of the MPO ρ(β, L) scaling as
logd (tensor size) ≤ O(ln(C n/ǫ)D ) ,
(18)
with some β-dependent constant C. In particular, for D = 1,
the bound on the tensor size scales polynomially with n/ǫ.
Let us consider a one-dimensional system and suppose we
are explicitly given the MPO tensors a′ [x] [see Eq. (15)] of an
approximation to a state ρ and, similarly, an observable A of
MPO form with MPO tensors a[x]. If the tensor sizes of both
MPOs scale at most polynomially in the system size, then one
can compute the corresponding approximation to the expectation value Tr(ρA) with a computational cost scaling polynomially in the system size. This means that, for instance, global
product observables can be approximated efficiently, which is
not guaranteed by our Implication 3. The problem with the
MPO approximation, however, is that Theorem 3 only guarantees the existence of the MPO tensors but it is not obvious
how they can be computed (efficiently).
Proof of Corollary 3. The condition β < 2 β ∗ is equivalent to
b(βJ) < 1. Let us denote the bound to the approximation
error in Eq. (17) by ǫ. Note that the upper bound in Eq. (17)
satisfies
b(βJ)L
ǫ := exp |E|
− 1 ≤ C n b(βJ)L
(19)
1 − b(βJ)
for distances L being at least logarithmically large in n =
|E| and some β-dependent constant C. Then, the distance L
7
necessary to reach ǫ must asymptotically be at least as large
as
L≥
ln(C n/ǫ)
.
| ln[b(βJ)]|
(20)
Bounding N (L) in terms of the spatial dimension D as
N (L) ≤ M LD with some constant M yields a tensor size
bounded as
D
ln (C n/ǫ)
logd (tensor size) ≤ 2 M
.
(21)
ln (1/ b(βJ))
D. Local topological quantum order
It is worth mentioning that Corollary 2 and Implication 2
are very reminiscent of the local topological quantum order
condition for open quantum systems introduced in Ref. [41]
and the results on the local stability of stationary states of local Liouvillians in Ref. [42]. A slightly different family of local topological quantum order conditions for closed quantum
systems [41–44] has played a very important role in the theory
of locally stable (topological) lattice systems and for rigorous
proofs of entropic area laws. Corollary 2 similarly characterizes the regime where local perturbations cannot drive any
thermal phase transition.
IV. DETAILS
In this section, we first discuss the generalized covariance
and then provide details concerning the applicability of our
results to Hamiltonians with k-body interactions. Finally, we
justify Observation 1 by stating the fermionic versions of our
results.
A. The generalized covariance
The generalized covariance defined in Eq. (2), which depends on a parameter τ ∈ [0, 1], provides more information
about the correlations between two observables than the standard covariance in a similar way as the class of Rényi entropies characterizes more completely the entanglement properties of a state than simply the von Neumann entropy [45].
While it occurs quite naturally in the perturbation formula
Theorem 1, other possible applications are to be explored.
Here, we discuss possible generalizations of the generalized
covariance to operators of arbitrary rank, show that for operators A and A′ they are always bounded by kAk∞ kA′ k∞ ,
and comment on convexity and a symmetrized version of the
generalized covariance.
A definition of the generalized covariance for states of arbitrary rank is not relevant for this work because for non-zero
temperatures thermal states are full-rank operators. However,
the discussion of possible generalizations also hints at the behaviour of covτ at the end points of the unit interval. On the
open interval τ ∈ ]0, 1[, it is natural to simply keep the definition from Eq. (2). There are two natural ways to define ρ0 :
Either as, ρ0 := 1 or as ρ0+ := limτ →0 ρτ , where ρ0+ turns
out to be the projector onto the image of the operator ρ. For
each end point τ = 0 and τ = 1, there are hence two natural
ways to define covτ , either such that the generalized covariance is continuous or such that cov0ρ (A, A′ ) = covρ (A′ , A)
and cov1ρ (A, A′ ) = covρ (A, A′ ), where
covρ (A, A′ ) := Tr(ρ A A′ ) − Tr(ρ A) Tr(ρ A′ )
(22)
defines the standard covariance.
Note that for product states and operators with disjoint support, all versions of the generalized covariance vanish. Moreover, for pure states, the continuous version of the generalized
covariance vanishes also, meaning that classical correlations
are needed to yield a non-zero value.
Next, we show that the generalized covariance is always
bounded as
covτρ (A, A′ ) ≤ kAk∞ kA′ k∞ ,
(23)
irrespective of which definitions are chosen for cov0 and cov1 .
We consider a state ρ and define Ā := A − Tr(ρA). Then,
covτρ (A, A′ ) = Tr ρτ Ā ρ1−τ A′ .
(24)
Hölder’s inequality generalized to several operators and the
1/p
fact that kXkp = k|X|p k1 then imply that
covτρ (A, A′ ) ≤ kρτ k1/τ kĀk∞ kρτ k1/(1−τ ) kA′ k∞ (25)
= kĀk∞ kA′ k∞
(26)
and, by noting that kĀk∞ = kAk∞ , the bound (23) is proven
for the continuous version of the generalized covariance. For
the non-continuous versions, the bound follows similarly.
The variance covτρ (A, A) induced by the continuous version of the covariance is convex in τ . This can be seen by
writing out ρ in its eigenbasis. As one can change the sign
of covτρ (A, A′ ) by just changing the sign of A′ , the generalized covariance is not convex in τ . But, it might be that its
magnitude | covτρ (A, A′ )| is convex, which is unclear. If this
were the case, it would be enough to prove the clustering Theorem 2 only for the end points τ ∈ {0, 1}, and hence the proof
could be significantly simplified.
Similarly, as there is a symmetrized version of the standard
covariance, one can also symmetrize the generalized covariance with respect to the two operators. Because of the cyclicity of the trace, the generalized covariance satisfies the symmetry property
covτρ (A, A′ ) = cov1−τ
(A′ , A)
ρ
(27)
Hence, one can define the symmetrized version of the generalized covariance as follows:
covτρ (A, A′ ) :=
1
covτρ (A, A′ ) + covτρ (A′ , A) .
2
(28)
8
Our results can also be phrased in terms of this symmetrized
version, since the averaged generalized covariance in the perturbation formula Theorem 1 can easily be rewritten in terms
of cov, and a bound analogous to the clustering Theorem 2
holds also for the symmetrized quantity.
B.
Bound on the growth constant for local k-body interactions
In this section, we show that regular hyperlattices also have
a finite growth constant, which renders our results applicable
to Hamiltonians with local k-body interactions.
In the case of k-body interactions, the Hamiltonian is again
a sum of local terms hλ whose supports are hyperedges λ =
supp(hλ ) ⊂ V with |λ| ≤ k. As before, V denotes the vertex
set and E the set of hyperedges.
We assume that the interaction hypergraph (V, E) is a regular hyperlattice, i.e., that it can be embedded into a regular
hypercubic lattice of a certain dimension D with hyperedges
of hypercubic form. Let us denote by R the edge length of the
resulting hypercubes. Note that such an embedding is, in general, not unique and changes both the number of terms in the
Hamiltonian and the local interaction strength of H. Moreover, the grouping changes the values of the metric d in our
results.
In order to find an exponential upper bound to the number
am of hyperanimals composed of m hypercubes, let us define a spread-out graph of range R as the graph with the edge
set consisting of all pairs {x, y} with 0 < kx − yk∞ ≤ R
and x, y ∈ ZD (see Ref. [32]). Notice that as any hypercube is uniquely specified by the coordinates of its “lower left
corner”, any hyperanimal of size m corresponds to a lattice
animal of size m − 1 and range R in the spread-out graph.
It follows from Lemma 2 in Ref. [32] that am ≤ (Ke)m
with K = (2 R + 1)D − 1 being the coordination number.
Hence, the hyperlattice has a growth constant bounded by
α ≤ ((2 R + 1)D − 1) e.
The bound obtained is for most models, far from optimal, in
particular, in situations where the supports of the local Hamiltonian terms are very different from hypercubes. For such
cases, one can derive tighter but more specific bounds from
known results about lattice animals in a similar way.
C. Fermionic versions of the main results
To make Observation 1 about fermions precise, we introduce the setting of interacting fermions on lattices. For each
site x ∈ V , the corresponding fermionic operators, i.e., the
creation and annihilation operators fx† and fx , act on the
fermionic Fock space and satisfy
{fx , fy† } = δx,y 1 ,
{fx , fy } = 0 ,
(29)
(30)
where {A, B} := A B+B A is the anti-commutator. For such
systems, all operators can be given in terms of polynomials in
the fermionic operators. A monomial of fermionic operators
is called even (odd) if it can be written as a product of an even
(odd) number of fermionic operators fx and fy† . A polynomial
of fermionic operators is called even (odd) if it can be written
as a linear combination of only even (odd) monomials, and an
operator is called even (odd) if it can be written as an even
(odd) polynomial of fermionic operators. According to the
fermion number parity superselection rule, only operators that
are even polynomials in the fermionic operators are physical
observables and Hamiltonians.
As with spin lattice systems, we have again a finite interaction graph (V, E); however, the support of an operator is now
to be understood in the picture of second quantization as follows: The support of any operator A being a polynomial in
the fermionic operators is the set of vertices of the fermionic
operators that occur in the polynomial. Correspondingly, we
denote the algebra of the even operators supported on a region
X ⊂ V by G X and denote G := G V for short. The Hamiltonian of a fermionic lattice system is of the form
H=
X
hλ
(31)
λ∈E
with hλ ∈ G λ . For B ⊂ V , the truncated Hamiltonian H↾B is
similarly the sum only over the edges contained in B. As for
spin systems, H∂B is the sum over the boundary edges of B.
Theorem 1 also holds for such fermionic lattice systems,
and we can prove statements analogous to Corollary 1, Theorem 2, and Corollary 2. Hence, all implications stated in
Section II, also hold. All proofs are presented in Section V C.
Corollary 4 (Fermionic truncation formula). Let H =
P
λ∈E hλ be a fermionic local Hamiltonian with local terms
hλ ∈ G, let B ⊂ V be a subsystem, and let the interpolating
Hamiltonian by H(s) := H − (1 − s) H∂B with its thermal
state gs := g[H(s)]. Then, for any operator A with support
supp(A) ⊂ B,
Tr A g[H↾B ](β) − Tr A g(β)
Z 1 Z 1
=β
dτ
ds covτgs (β) (H∂B , A).
0
0
(32)
Theorem 4 (Clustering of correlations in fermionic systems).
Let g(β) be the thermal state at inverse
P temperature β of a
local fermionic Hamiltonian H =
λ∈E hλ with finite interaction graph (V, E) having growth constant α, local terms
hλ ∈ G, and local interaction strength J. Define the functions
β ∗ , ξ, and L0 as in Eqs. (10), (9), and (50). Then, for every
|β| < β ∗ , τ ∈ [0, 1], and every two operators A and B with
d(A, B) ≥ L0 (β, a), where a := min{| ∂A|, | ∂B|},
|covτg(β) (A, B)| ≤
4a kAk∞ kBk∞ − d(A,B)/ξ(β)
e
.
ln(3) (1 − e−1/ξ(β) )
(33)
Corollary 5 (Locality of fermionic thermal states). Let H
be a Hamiltonian satisfying the conditions of Theorem 4,
let |β| < β ∗ , and let S ⊂ B ⊂ V be subsystems with
9
d(S, ∂B) ≥ L0 (β, | ∂S|). Then,
B.
v |β| J
e− d(S, ∂B)/ξ(β) ,
1 − e−1/ξ(β)
(34)
where v = 4 | ∂S| | ∂B|/ ln(3).
g S (β) − g S [H↾B ](β)
1
≤
Proof of Theorem 2 on clustering of correlations
The proof of Theorem 2 builds on and develops further a
cluster expansion of the power series of e−β H in terms of
summands of the form
h(w) := hw1 hw2 . . . hw|w| ,
V. PROOFS
We start this section with the proofs of Theorems 1 and 2.
One important stepping stone for the proof of the latter is
a tailored version of a bound on a truncated cluster expansion (Lemma 1) from Ref. [40]. Both versions are proven
in the Appendix. In the last part of the section, we prove the
fermionic versions of our main results, Therorem 4 and Corollaries 4 and 5.
A. Proof of the perturbation formula (Theorem 1)
The two main ingredients in the proof of Theorem 1 are
the fundamental theorem of calculus, and Duhamel’s formula.
The generalized covariance appears as a natural measure of
correlations.
Proof of Theorem 1. Using the fundamental theorem of calculus we obtain
Z 1
d e−β H(s)
Tr[A g0 (β)] − Tr[A g1 (β)] = − Tr A
ds
0 ds Zs (β)
with Zs := Z[H(s)]. The derivative can be written as
d e−β H(s)
d −βH(s)
1 d −βH(s) gs (β)
=
e
−
Tr
e
.
ds Zs (β)
Zs (β) ds
Zs (β)
ds
After applying Duhamel’s formula to both derivatives, i.e., using that
d −βH(s)
e
ds
Z 1
τ d
1−τ
−βH(s)
e
=−β
e−βH(s)
dτ ,
H(s)
ds
0
we obtain
Tr[A g0 ] − Tr[A g]
Z 1 Z 1
= −β Tr A
−gsτ (H − H0 ) gs1−τ
0
0
+ gs Tr[gsτ (H − H0 ) gs1−τ ] dτ ds .
(35)
Together with the the cyclicity of the trace and the definition
of the generalized covariance in Eq. (2), this finishes the proof.
(36)
where wj ∈ E. For the sake of a compact presentation, we
refer to edges as letters, to the edge set E as an alphabet, and
call sequences of edgesSwords. For any sub-alphabet F ⊂ E,
∞
we denote by F ∗ := l=0 F l the set of words with letters
in F and arbitrary length l, where the length |w| of a word
w ∈ E ∗ is the total number of letters it contains. For two
words w, v ∈ E ∗ , their concatenation is denoted by w ◦ v :=
(w1 , w2 , . . . , w|w| , v1 , v2 , . . . , v|v| ). We call a word c ∈ E ∗
connected or a cluster if the set of letters in c is an animal,
i.e., connected. So, clusters are connected sequences of edges
where the edges can also occur multiple times, while animals
are sets of edges without any order or repetition. A word v is
called a sub-sequence of w ∈ E ∗ if v can be obtained from
w by omitting letters, i.e., if there is an increasing sequence
j1 < j2 < . . . < j|v| such that vi = wji . This will be
denoted by v ⊂ w. A connected sub-sequence c ⊂ w is
called a maximal cluster of w if c is not a sub-sequence of any
other connected sub-sequence of w. Importantly, for any word
w ∈ E ∗ , one can permute its letters to a new word w′ such
that h(w′ ) = h(w) irrespective of the choice of the local terms
hλ and such that w′ = c1 ◦ c2 ◦ · · · ◦ ck is a concatenation of
maximal clusters cj ⊂ w of w. Note that this decomposition
is unique up to the order of the cj .
In the following, we will consider systems that are either
n = 2 or n = 4 copies of the original system with Hilbert
space H. For any operator A on H, we denote by A(j)
the operator on H⊗n that acts as A on the jth copy, e.g.,
A(2) := 1 ⊗ A for n = 2. By S (i,j) , we denote the swap
operator on H⊗n that swaps the ith and jth tensor factors,
e.g., S 1,2 |k1 , k2 , k3 , k4 i = |k2 , k1 , k3 , k4 i for n = 4. For
n = 2, we write S instead of S 1,2 .
We can now state the subsequent lemma, which is a bound
on a truncated cluster expansion that is based on a more general, but for our purposes not tight enough bound, used previously in Ref. [40] (see Lemma 2 in the Appendix). The
lemma will play an important role in the subsequent proof of
Theorem 2.
Lemma
P1 (Truncated cluster expansion). Let τ ∈ [0, 1] and
H = λ∈E hλ be a local Hamiltonian on H with finite interaction graph (V, E) having growth constant α and local
interaction strength J = maxλ∈E khλ k∞ . We denote by
H̃ the Hamiltonian of two weighted copies with local terms
(1)
(2)
h̃λ := τ hλ + (1 − τ ) hλ . Considertwo operators A and
B on H, define b(x) := α e|x| e|x| − 1 , and let |β| be small
enough such that b(βJ) < 1. For some set of edges F ⊂ E,
let C ≥L(F ) ⊂ E ∗ be the set of words containing at least one
cluster c that contains at least one letter of F and has size
|c| ≥ L and let us denote the corresponding truncated cluster
10
2 covτρ (A, B) =
F
A⊗1−1⊗A B⊗1−1⊗B
gτ
gτ
g 1−τ
g 1−τ
Figure 6. The “multiple swap-trick”: Eq. (40) as a tensor network.
where ρ4 := ρτ ⊗ ρτ ⊗ ρ1−τ ⊗ ρ1−τ . For the case ρ = g(β),
the operator ρ4 turns out to be
(+)
→ ∈ C ≥5(F ), C 2≥4(F )
ρ4 =
→ ∈ C ≥4(F )
Figure 5. A 2D square lattice. Three different sub-alphabets are indicated: Words that contain all letters in those alphabets are members
of different sets C ≥L(F ).
covτg(β) (A, B) =
X (−β)|w|
1
t(w)
2 Z(β)2
|w|!
∗
expansion of e−β H̃ by
w∈C ≥L(F )
(−β)|w|
h̃(w) ,
|w|!
with
(+)
(37)
with h̃(w) := h̃w1 h̃w2 . . . h̃w|w| . Then, for all τ ∈ [0, 1],
Tr S A(1) B (2) Ω[H̃](β)
b(βJ)L
≤ exp |F |
− 1.
kAk∞ kBk∞ Z(β)
1 − b(βJ)
(38)
In the Appendix, we provide a detailed proof of this lemma.
The terms resulting from the expansion of the exponential series of e−βH are classified according to whether they contain a cluster of size at least L that contains a letter from
F . One can then show that there is a percolation transition
at β ∗ = b−1 (1)/(2J) such that for |β| < β ∗ , the contribution
of long clusters is exponentially suppressed.
In the following proof of the exponential clustering, we will
use the so-called swap-trick: For any two operators A and B,
it holds that
Tr(A B) = Tr(S(A ⊗ B)) ,
(39)
which can be checked by a straightforward calculation.
Proof of Theorem 2. Fix some τ ∈ [0, 1]. For any operator
A : H → H, we define A(±) := A ⊗ 1 ± 1 ⊗ A and Ã(+) :=
τ A(1) + A(2) + (1 − τ ) A(3) + A(4) .
As the first step, we write the covariance as
1
covτρ (A, B) = Tr A(−) (ρτ ⊗ ρτ ) B (−) ρ1−τ ⊗ ρ1−τ .
2
Using the swap-trick (39) yields (see Figure 6)
1
covτρ (A, B) = Tr S 1,3 S 2,4 (A(−) ⊗ B (−) ) ρ4 , (40)
2
(42)
w∈E
h
i
t(w) := Tr S 1,3 S 2,4 (A(−) ⊗ B (−) ) h̃(+) (w)
X
(41)
Writing out ρ4 as a power series yields
→ 6∈ C ≥L(F ) for L ≥ 1
Ω[H̃](β) :=
e−β H̃
.
Z(β)2
(+)
(43)
(+)
and h̃(+) (w) := h̃w1 h̃w2 . . . h̃w|w| . Next, we argue that t(w)
vanishes whenever w does not contain a cluster connecting the
supports of A and B. Without loss of generality, we assume
that | ∂A| ≤ | ∂B| and consider C ≥L(∂A)c = E ∗ \ C ≥L(∂A),
the set of words that do not contain a cluster containing
an edge in ∂A of size L := d(A, B) or larger. The set
C ≥L(∂A)c hence contains no words with clusters that connect supp(A) and supp(B). Any word w ∈ C ≥L(∂A)c can
be replaced by a concatenation of two words wA and wB
such that h̃(+) (w) = h̃(+) (wA ) h̃(+) (wB ), where wA contains all maximal clusters of w that overlap with supp(A)
and wB all other maximal clusters of w. The operators
h̃(+) (wA ) and 1 ⊗ 1 ⊗ B (−) =: B̂, and h̃(+) (wB ) and
A(−) ⊗ 1 ⊗ 1 =: Â then have disjoint supports, respectively,
and the trace in Eq. (43) factorizes into a product of two traces,
one over the subsystem X := supp(Â) ∪ supp(h̃(+) (wA ))
and the other over the rest of the system. It turns out that
both vanish: By using the symmetries  = −S 1,2  S 1,2 ,
h̃(+) (wA ) = S 1,2 S 3,4 h̃(+) (wA ) S 3,4 S 1,2 , Â S 3,4 = S 3,4 Â,
2
and that S i,j = 1, one can show, e.g., that
Tr S 1,3 S 2,4 Â h̃(+) (wA ) = − Tr S 1,3 S 2,4 Â h̃(+) (wA ) .
(44)
This implies that for every w ∈ C ≥L(∂A)c ,
t(w) ∝ Tr S 1,3 S 2,4 Â h̃(+) (wA ) = 0 .
(45)
Together with Eq. (42), realizing that Z(β)2 = Z[H (+) ](β),
and using the notation from Eq. (37) with F = ∂A and L =
d(A, B), it follows that
1,3 2,4
S S Â B̂ (+)
τ
covg(β) (A, B) = Tr
Ω H̃
(β) . (46)
2 Z(β)2
11
After applying Lemma 1 and using that kÂk∞ ≤ 2 kAk∞ ,
and similarly for B, we obtain
| covτg(β) (A, B)|
b(2 βJ)L
| ∂A| 1−b(2 βJ)
≤2 e
−1 .
(47)
kAk∞ kBk∞
The fact that the condition β < β ∗ is equivalent to b(2 βJ) <
1 implies that b(2 βJ)L decays exponentially with L. In order
to obtain the desired exponential bound (11), we apply the
bound ∀x ∈ [0, x0 ] : exp(x) − 1 ≤ x (ex0 − 1)/x0 with the
b(2 βJ)L
choice x0 = ln(3). In order to have |∂A| 1−b(2
βJ) ≤ ln(3),
we impose
| ∂A|
/ ln(2 β J))
(48)
L ≥ ln
ln(3) (1 − b(2 βJ))
= ξ(β) ln ln(3) 1 − e−1/ξ(β) /|∂A|
(49)
=: L0 (β, |∂A|) .
(50)
This guarantees the exponential bound (11) and finishes the
proof.
C. Proofs of the fermionic versions of the main results
In order to also establish our main results for fermionic systems, we go through the proofs for spin systems and discuss
the necessary modifications.
Proof of Corollary 4. In Theorem 1, we choose H0 = H −
H∂B . As the local terms are all in G, we have that the thermal state of H0 factorizes, i.e., g0 = g[H↾B ] g[H↾B c ]. After
tracing over B c , the statement follows.
Proof of Theorem 4. We use the same tensor copy trick as in
the proof of Theorem 2. Eq. (40) still holds in the fermionic
setting. Note that the Hilbert space over which the trace is performed in Eq. (40) is not the Fock space of a system of 4 times
the number of modes but the tensor product of four identical
fermionic Fock spaces with the canonical inner product. This
Hilbert space can be interpreted as that of a system of four
types of fermionic particles that are each mutually indistinguishable and subject to (up to τ -dependent prefactors) identical Hamiltonians but do not interact with each other and can
be distinguished from each other. It is spanned by tensor products of Fock states. The state g[H̃ (+) ](β) is the thermal state
of this system. Eq. (42) with t(w) as defined as in Eq. (43)
still holds. Note that the swap operators swap tensor factors,
not fermionic modes. Thus, they still satisfy the symmetry relations that are used to prove that only terms corresponding to
words w ∈ C ≥L(∂A) can contribute to the covariance.
It remains to show that Lemma 1 still holds in the fermionic
setting. Lemmas 3 and 8 are purely combinatorial. Lemmas 4,
5, 6, 7, 9, and 10 only use the local boundedness of the Hamiltonian and that Hamiltonian terms with disjoint support commute. The same holds in the fermionic setting because the
Hamiltonian terms must be physical operators, i.e., even polynomials in the fermionic operators. Hence all lemmas used in
the proof of Lemma 1 carry over to the fermionic setting. It
is then straightforward to see that the proof itself also goes
through without any modifications.
Proof of Corollary 5. Tracing out B c in the second trace in
Eq. (32) and bounding the integral yields
Tr[A g(β)] − Tr[A g[H↾B ](β)]
≤ |β| sup
sup covτgs (β) (A, H∂B ) . (51)
s∈[0,1] τ ∈[0,1]
Taking the supremum over all A with kAk∞ = 1 and
supp(A) ⊆ S and using Theorem 4 finish the proof.
VI. CONCLUSIONS
In this work, we clarify the limitations of a universal concept of scale-independent temperature by showing that temperature is intensive on a given length scale if and only if
the correlations are negligible. The corresponding correlation measure turns out to also quantitatively capture the stability of thermal states against perturbations of the Hamiltonian. Moreover, we find a universal critical temperature above
which correlations always decay exponentially with the distance. We compare our results to known results on phase transitions, comment on recent advances concerning thermalization in closed quantum systems (e.g., concerning the eigenstate thermalization hypothesis), and discuss known matrix
product operator approximations of thermals states. More
concretely, our results imply that at high enough temperatures,
the error made when truncating a Hamiltonian at some distance away from the system of interest is exponentially suppressed with the distance. As a computational consequence,
expectation values of local observables can be approximated
efficiently.
VII. ACKNOWLEDGMENTS
We thank I. H. Kim, M. Holzäpfel, M. B. Hastings, B.
Nachtergaele, M. Friesdorf, and A. Werner for helpful feedback and discussions. This work was supported by the Studienstiftung des Deutschen Volkes, the Alexander von Humboldt Stiftung, the EU (Q-Essence, SIQS, RAQUEL), and the
ERC (TAQ).
VIII.
APPENDIX: CLUSTER EXPANSIONS AND PROOF
OF LEMMA 1
The following discussion of cluster expansions is expected
to be interesting in its own right, as it contains a rigorous formulation of the ideas outlined in Ref. [40]. We will provide a
proof of the original statement used to establish Theorem 3 as
well as of the tailored statement in Lemma 1, which is used to
prove Theorem 2 on the clustering of correlations.
12
Ak≥L (F ) the corresponding sets of k-fold animals, i.e.,
1. The original cluster expansion from Ref. [40]
]
k
Gj : Gj ∈ A≥L (F ) non-overlapping .
The original cluster expansion is similar to Lemma 1 with
just one copy of the system instead of two weighted ones:
Ak≥L (F ) :=
Lemma
2 (Truncated cluster expansion [40]). Let H =
P
λ∈E hλ be a local Hamiltonian with finite interaction
graph (V, E) having growth constant α and local interaction strength J = maxλ∈E khλ k∞ , and define b(x) :=
α e|x| e|x| − 1 . Moreover, let β be small enough such that
b(βJ) < 1. For some subset of edges F ⊂ E let C ≥L(F ) ⊂
E ∗ be the set of words containing at least one cluster c that
contains at least one letter of F and has size |c| ≥ L and
denote the corresponding truncated cluster expansion by
For a more compact notation, we write the terms in the exponential series as
Ω[H](β) :=
X
w∈C ≥L(F )
(−β)|w|
h(w) .
|w|!
(A.1)
j=1
f (w) :=
(−β)|w|
h(w) .
|w|!
We will frequently use the following fact: For any Hamiltonian with a finite interaction graph (V, E), the partial series
over any set of words W ⊆ E ∗ converges absolutely, i.e.,
X
f (w)
w∈W
∞
Then,
(A.2)
If one applies this lemma to the setting of Lemma 1, one obtains a bound similar as the one in Eq. (38) but with Z[H̃](β)
instead of Z(β), where the ratio Z[H̃](β)/Z[H](β) can be
exponentially large in the system size for τ ∈]0, 1[.
Lemma 2 was used in Ref. [40] to establish a mathematically (not algorithmically) constructive version of Theorem 3,
on MPO approximations, where the MPO in Eq. (17) is given
by
ρ(β, L) =
1
Z(β)
X
w∈E ∗ \C ≥L(E)
≤
≤
kΩ[H](β)k1
b(βJ)L
≤ exp |F |
− 1.
Z(β)
1 − b(βJ)
(−β)|w|
h(w) .
|w|!
(A.3)
2. Proofs of Lemmas 1 and 2
(A.4)
X (|β| J)|w|
|w|!
(A.5)
w∈W
X (|β| J)|w|
|w|!
∗
(A.6)
w∈E
= exp(|β| J |E|) .
(A.7)
In particular, this bound implies that the order of the terms in
the series over any subset of words W does not matter.
In the following proofs of Lemmas 1 and 2 we use several technical auxiliary lemmas, which we will only state and
prove subsequently.
Proof of Lemma 1. During this proof, we indicate quantities
corresponding to H̃ by a tilde accent, e.g., f˜(w) is defined as
in Eq. (A.4) but with respect to the local terms h̃λ of H̃ while
f (w) is defined with respect to the local terms hλ of H.
We start the proof by rearranging the terms in the series
over C ≥L(F ) in Eq. (37) according to the number of relevant
clusters they contain and use Lemma 3 with bk being the series
over C k≥L(F ) to obtain
Ω[H̃](β) =
∞
X
f˜(w)
X
(A.8)
k=1 w∈C k (F )
≥L
The purpose of this section is to prove Lemma 1. But,
along the way, we also prove Lemma 2. In order to do so,
we start with the introduction of some more notation, mainly
concerning clusters and lattice animals. For w ∈ E ∗ and any
sub-alphabet G ⊂ E, we write G ⊂ w if every letter in G
also occurs in w. By Gc := E \ G, we denote the complement of G ⊂ E. The extension of G is defined to be
G := {λ ∈ E | ∃λ′ ∈ G : λ′ ∩ λ 6= ∅} and, similarly as
for subsystems, its boundary is ∂G := G \ G. Throughout
the proof, we fix some subset of edges F ⊂ E. We denote by
C ≥L(F ) ⊂ E ∗ the set of words that contain at least one cluster c with c ∩ F 6= ∅ and |c| ≥ L, and we denote by C k≥L(F )
the set of words that contain exactly k such clusters. Note that
for an animal G ⊂ E, there exists a cluster c ∈ E ∗ such that
G = {λ ∈ c}, and if one imposes some order on G, one obtains a cluster. We denote by A=l (F ) and A≥L (F ) the sets
of animals that contain at least one edge of F and are of size
exactly l or at least L, respectively. Moreover, we denote by
=−
∞
X
(−1)
m
m=1
∞
X
k
m
k=m
X
f˜(w).
w∈C k
(F )
≥L
Lemmas 5, 6, and 9 are the core of the proof. They define
a series of operators (ρ̃m )∞
m=1 that have a particularly useful
form given in Lemma 10. This form exactly matches the series
over k in Eq. (A.8), which leads to the following identity:
Ω[H̃](β) = −
∞
X
(−1)m ρ̃m .
(A.9)
m=1
The operators ρ̃m are defined in Eq. (A.57) as series over
m-fold lattice animals G of operators ρ(G) [defined in
Eq. (A.31)]. This yields
X
Tr(S A B ρ̃m ) =
Tr(S A B ρ̃(G)) .
(A.10)
G∈Am
(F )
≥L
13
In the previous steps, the series over words has been rewritten
as a series over m-fold animals. Lemma 7 provides a bound
on ρ̃(G) that, together with Eqs. (A.9) and (A.10), yields
∞
X
X
Tr S A B Ω[H̃](β)
≤
y(βJ)|G| . (A.11)
kAk∞ kBk∞ Z(β)
m
m=1
G∈A≥L (F )
Now, a counting argument for lattice animals from Lemma 8
allows us to bound the series over m-fold animals G in terms
of a series of animals
m
∞
X
Tr S A B Ω[H̃](β)
1 X
≤
y(βJ)|G| .
kAk∞ kBk∞ Z(β)
m!
m=1
G∈A≥L (F )
Using that the number al [see Eq. (7)] of lattice animals G
with G ∩ F 6= ∅ and of size |G| = l is bounded by |F | al and
that al ≤ αl [see Eq. (8)], we obtain
!m
∞
∞
X
X
1
|F |
Tr S A B Ω[H̃](β) ≤ Z(β)
b(βJ)l
m!
m=1
l=L
with b(x) := α y(x). Performing the partial geometric series
over l with argument b(βJ) < 1 and the exponential series
over m yields Eq. (A.2) and completes the proof.
Similarly, we prove Lemma 2:
Proof of Lemma 2. By the same argument that led us to
Eq. (A.9) in the proof of Lemma 1, we obtain
Ω[H](β) = −
∞
X
(−1)m ρm .
(A.12)
m=1
Applying the triangle inequality and using the bound on ρm
from Lemma 9 yields
!m
∞
∞
X
X
1
b(βJ)l
.
|F |
kΩ[H](β)k1 ≤ Z(β)
m!
m=1
l=L
(A.13)
Performing the partial geometric series over l with argument
b(βJ) < 1 and the exponential series over m yields Eq. (A.2)
and completes the proof.
We now prove various lemmas that are used in the previous
proofs of Lemmas 1 and 2.
Lemma 3. Let (bk )∞
k=1 be a sequence of complex matrices
AK :=
K
X
bk
(A.14)
Proof. Applying the binomial theorem to (1 − 1)k = 0 yields
k
X
(−1)l
l=0
k
=0,
l
(A.16)
which we will use. We prove the identity by induction. A1 =
B1 is easy to see. Under the assumption that AK = BK for
some K ∈ N, we obtain
K+1 K + 1
bK+1
(A.17)
BK+1 = BK − (−1)
K +1
K
X
m K +1
bK+1
(A.18)
(−1)
−
m
m=1
!
K
X
K
+
1
bK+1
(−1)m
= AK + −(−1)K+1 −
m
m=1
= AK+1 ,
(A.19)
where we have used Eq. (A.16) in the last step. This proves
the lemma.
The goal of the following lemmas is to show that ρm is
well-defined and to upper bound it in 1-norm. The order of
the lemmas is chosen in a way that makes clear that the two
quantities ρm and ρ(G), which will be defined shortly, are
actually well-defined.
We start with a 1-norm bound on the perturbed exponential
series.
Lemma 4 (Eq. (21) from Ref. [40]). Let H be a Hamiltonian with finite interaction graph (V, E). For any sequence
(Gj )kj=1 of sub-alphabets Gj ⊂ E,
e−β (H−
Pk
j=1
HGj )
1
≤ Z(β)
k
Y
e|β| HGj
∞
.
(A.20)
j=1
Proof. The lemma is essentially a consequence of the GoldenThompson inequality and the fact that the 1-norm of a positive operator coincides with its trace. Using first the GoldenThompson and then Hölder’s inequality, we obtain
−β (H−Pk−1 HG ) β HG
j=1
j e
k
≤
Tr
e
1
P
k−1
≤ Tr e−β (H− j=1 HGj ) e|β| HGk ∞ . (A.21)
e−β (H−
Pk
j=1
HGj )
Iteration completes the proof.
k=1
We will use the following lemma to bound the operator
norm of certain subseries of f (w).
and
BK := −
K
X
m=1
(−1)m
K
X
k
bk .
m
(A.15)
k=m
Then, AK = BK for all K ∈ N. In particular, if
both sequences converge, then their limits are the same, i.e.,
limK→∞ AK = limK→∞ BK .
Lemma 5. Let (V, E) be a finite graph and J ≥ 0. For any
G ⊂ E,
X
w∈G∗ :G⊂w
|G|
|βJ||w|
= e|βJ| − 1
.
|w|!
(A.22)
14
Proof. Ordering the words in the sum in Eq. (A.22) with respect to their length yields
X
w∈G∗ :G⊂w
∞
X
|βJ||w|
=
|w|!
X
l=|G| w∈Gl :G⊂w
=
|βJ||w|
|w|!
(A.23)
∞
X
|βJ|l
|{w ∈ Gl : G ⊂ w}| .
l!
l=|G|
(A.24)
From basic combinatorial considerations, we obtain
X
l
l
,
|{w ∈ G : G ⊂ w}| =
j
The η(Gj ) are supported on the single animals Gj composing
the m-fold animal G. As we will see, a norm bound for η(Gj )
follows immediately from the previous lemma, which, in turn,
also yields an upper bound on ρ(G). The form of ρ(G) given
in Eq. (A.33) together with this upper bound plays an important role in the main cluster expansion.
Lemma 6. Let H be a Hamiltonian
Umwith finite interaction
graph (V, E). For G ⊂ E, let G = j=1 Gj be the decomposition of G into non-overlapping animals Gj ⊂ E and define
ρ(G) := e−β H(G)c
(A.25)
j1 ,j2 ,...,jn ≥1,
j1 +j2 +···+jn =l
γ(n, l)
(A.26)
l=n
|βJ|
l!
l
X
j1 ,j2 ,...,jn ≥1,
j1 +j2 +···+jn =l
l
j
j1 ,j2 ,...,jn ≥1,
j1 +j2 +···+jn =l
(A.27)
j2
j1
X
=
∞
X
X
l=n j1 ,j2 ,...,jn ≥1,
j1 +j2 +···+jn =l
=
∞ l−(n−1)
X |βJ|j1
X
l=n
|βJ| |βJ|
j1 !
j2 !
|βJ|
...
jn !
.
j1 =1
j1 !
|βJ|jn
|βJ|j1 |βJ|j2
...
j1 !
j2 !
jn !
X
|βJ|jn
|βJ|j2
...
j2 !
jn !
j2 ,...,jn ≥1,
j2 +···+jn =l−j1
∞ X
l
X
|βJ|j1
=
γ(n − 1, l + n − 1 − j1 )
j1 !
j =1
l=1
(A.28)
1
and, after realizing that the last series is a Cauchy product,
∞
∞
X
|βJ|j1 X
γ(n) =
γ(n − 1, l)
j1 !
j =1
1
X
f (w) .
(A.32)
w∈G∗ :G⊂w
Then,
ρ(G) =
X
f (w) .
(A.33)
(A.29)
l=n−1
= (e|βJ| − 1) γ(n − 1) .
Proof. To simplify the notation, we denote the relevant set of
words that contain no letters in ∂G and each letter in G at least
once by
W ⊃G := {w ∈ [(∂ G)c ]∗ : G ⊂ w} .
jn
Then,
γ(n) =
η(G) :=
w∈[(∂ G)c ]∗ : G⊂w
with
γ(n, l) :=
(A.31)
with
where j is a multinomial coefficient. Therefore, the righthand side of Eq. (A.24) only depends on n := |G| and we
denote it by
γ(n) :=
η(Gj )
j=1
l
∞
X
m
Y
(A.30)
We note that γ(1) = e|βJ| −1, and iteration finishes the proof.
The following lemma provides a factorization of the series ρ(G) in Eq. (A.33) over words that have no letters on
the boundary of an m-fold animal G ∈ Am
=l (F ) and contain
all letters in G, into exp(−β H(G)c ), whose norm we have
bounded in Lemma 4, times a product of operators η(Gj ).
(A.34)
The idea is to group these words into subsets [w] ⊂ W ⊃G
that coincide on the connected components of G and on (G)c
and correspondingly split up the series (A.33). We formalize
this idea by introducing an equivalence relation on W ⊃G . For
v, w ∈ W, we define
(
v ↾ Gc = w ↾ Gc
v ∼ w :⇔
,
v ↾ Gj = w ↾ Gj ∀j = 1, 2, . . . , k
where, for any sub-alphabet G′ ⊂ E, the restriction w ↾ G′
of a word w ∈ E ∗ is obtained from w by omitting all letters
that are not in G′ . Then, the size of each equivalence class
[w] ∈ W ⊃G / ∼ is given by the multinomial coefficient
|w|
.
(A.35)
|[w]| =
(|w ↾ Gc |, |w ↾ G1 |, . . . , |w ↾ Gk |)
c Qk
Note also that h([w]) := h(w ↾ G ) j=1 h(w ↾ Gj ) = h(w)
is well-defined as a function on the classes. Let us denote the
set of words over the alphabet Gj that contain all letters at
least once by
W =Gj := {w ∈ (Gj )∗ : Gj ⊂ w} .
(A.36)
Then, the quotient set can be identified with a Cartesian product of these sets
W ⊃G /∼ ∼
= [(G)c ]∗ ×
k
×W
j=1
=Gj
.
(A.37)
15
For each equivalence class K ∈ W ⊃G /∼ we pick an arbitrary representative wK ∈ W ⊃G , use the definition of f in Eq. (A.4),
and determine that k is the number of connected components of G. This yields
X
f (w) =
w∈[(∂ G)c ]∗ : G⊂w
X
K∈W ⊃G /∼
|K|
(−β)|wK |
h(wK )
|wK |!
(A.38)
Pk
Pk
k
Y
|v| + j=1 |wj |
(−β)|v|+ j=1 |wj |
···
h(wj )
h(v)
=
P
(|v|, |w1 |, . . . , |wk |) (|v| + kj=1 |wj |)!
=Gk
=G1 w ∈W =G2
c
∗
j=1
w
∈W
w
∈W
v∈[(G) ]
2
1
k
X
X
X
X
=
(A.39)
f (wk ) .
f (w2 ) . . .
f (w1 )
f (v)
Using the definition of η from Eq. (A.32) on the last factors
yields
w∈[(∂
X
f (w) = e
−βH(G)c
k
Y
Eq. (A.31) but with respect to H̃. Then,
Tr SA(1) B (2) e−β H̃Gc ρ̃(G)
≤ y(βJ)|G| ,
kAk∞ kBk∞ Z(β)
where y(x) := e|x| e|x| − 1 .
η(Gj ) = ρ(G).
j=1
G)c ]∗ : G⊂w
wk ∈W =Gk
w2 ∈W =G2
w1 ∈W =G1
v∈[(G)c ]∗
X
X
X
X
(A.40)
(A.41)
(1)
(1)
(2)
(2)
Proof. Let us denote kλ := τ hλ and kλ := (1 − τ )hλ .
For w ∈ E ∗ and v ∈ {1, 2}|w| , we define h̃(w, v) :=
v
v1 v2
kw
. Then, by expanding the product h̃(w), it
k . . . kw|w|
|w|
1 w2
can be written as
X
h̃(w, v).
(A.42)
h̃(w) =
The following lemma is a tighter variant of some of the
original arguments leading to Lemma 2 for Hamiltonians consisting of two weighted copies of a local Hamiltonian. Its
purpose is to provide a specialized tighter bound on ρ(G),
which turns out to be sufficient for our purposes. The central idea of the lemma is to expand ρ(G) in the left-hand side
of Eq. (A.41) in order to be able to bound the trace using the
generalized Hölder’s inequality.
v∈{1,2}|w|
Importantly, we can reorder the terms in h̃(w, v) so that
h̃(w, v) = h̃(1) (w, v)h̃(2) (w, v) ,
(A.43)
where h̃(1) (w, v) = h(i) (w, v) ⊗ 1 and h̃(2) (w, v) = 1 ⊗
h(ii) (w, v). Factorizing the operators and using the swap-trick
(39), we obtain
Lemma 7. Let τ , H, H̃, A, and B be as in Lemma S
1 and let
m
G ∈ A=m
(F
)
be
an
m-fold
lattice
animal
with
G
=
≥L
j=1 Gj
and Gj ∈ A≥L (F ). Moreover, let ρ̃(G) be defined as ρ(G) in
Tr SA(1) B (2) e−β H̃Gc h̃(w, v) = Tr S A e−βτ HGc h(i) (w, v) ⊗ B e−β(1−τ )HGc h(ii) (w, v)
= Tr A e−βτ HGc h(i) (w, v) B e−β(1−τ )HGc h(ii) (w, v) .
(A.44)
(A.45)
Bounding the trace by the 1-norm and applying Hölder’s inequality generalized to several operators yields
Tr SA(1) B (2) e−β H̃Gc h̃(w, v) ≤ kAk∞ kBk∞ e−βτ HGc
≤ kAk∞ kBk∞ e
−βHGc
1/τ
1
1/p
where in the second step, we have used that kXkp = k|X|p k1
(1)
J
e−β(1−τ )HGc
|w| n(1) (v)
τ
h(i) (w, v)
1/(1−τ )
(1 − τ )
n(2) (v)
∞
h(ii) (w, v)
,
∞
(A.46)
and that with n(j) (v) := |{vk : vk = j}| for j ∈ {1, 2} the
bounds kh(i) (w, v)k∞ ≤ (τ J)n (v) and kh(ii) (w, v)k∞ ≤ ((1 − τ ) J)n
ke|β|HGj k∞ ≤ e|β|J|Gj | to arrive at
(2)
(v)
hold. Now, we apply Lemma 4 and use that
(1)
(2)
Tr SA(1) B (2) e−β H̃Gc h̃(w, v) ≤ kAk∞ kBk∞ Z(β) e|β| J |G| J |w| τ n (v) (1 − τ )n (v) .
(A.47)
16
From the definition of η in Eq. (A.32), it follows that
m
Y
X
η̃(Gj ) =
j=1
(
w(j) ∈G∗
j:
Gj ⊂w(j)
=
(
w
(j)
kAk∞ kBk∞ Z(β)
∈G∗
j:
(j)
Gj ⊂w
where w := w(1) w(2) . . . w(m) and hence h̃(w) =
Qm
Tr SA(1) B (2) e−β H̃Gc j=1 η̃(Gj )
X
i=1
(
w
w(i) !
i=1
)m
(−β)|w|
Qm
(i) !
i=1 w
(j)
X
ρ̃(G)
Tr SA B e
kAk∞ kBk∞ Z(β)
(i)
m
Y
X
(|β| J)|w |
|β| J |G|
=e
w(i) !
)m
(
(j)
∗
i=1
w
= e|β| J |G|
m
Y
i=1
≤ e|β| J |G|
∈G∗
j:
(j)
Gj ⊂w
(A.51)
m
Y
i=1
X
w(i) ∈G∗
i:
Gi ⊂w(i)
(|β|J) w
w(i) !
!
)m
j=1
(A.53)
Lemma 8. Let (V, E) be a finite (hyper)graph and y ∈ [0, 1[.
Then, for any F ⊂ E,
G∈Am
(F )
≥L
y |G|
X
G∈A≥L (F )
τn
(1)
(v)
v∈{1,2}|w|
(1 − τ )n
(2)
(v)
. (A.50)
X
G∈Am
(F )
≥L
y |G| ≤
X
m−1
G1 ∈A≥L
(F )
=
X
y |G1 |+|G2 |
G2 ∈A≥L (F )
X
m−1
G∈A≥L
(F )
X
y |G|
G∈A≥L (F )
y |G| .
X
y |G|
1
≤
m!
X
G∈A≥L (F )
m
y |G| .
(A.56)
(A.54)
We will need the following combinatorial lemma:
1
≤
m!
X
m−1
A≥L
(F ) and G2 ∈ A≥L (F ) such that G = G1 ⊎ G2 ; hence,
G∈Am
(F )
≥L
|Gi |
e|β| J − 1
,
(A.49)
By iterating this inequality, we obtain
j=1
(i)
h̃(w, v) ,
v∈{1,2}|w|
(A.52)
where in the second to last step, we have factorized the series
and in the last step, we have used Lemma 5.
X
X
∈Gj :
Gj ⊂w(j)
(A.48)
|β||w|
Qm
J |w|
(i) !
i=1 w
m
(2) −β H̃Gc
h w(i) . Together with the bound (A.47), this yields
Using the definition (A.31) of ρ̃(G) and the multinomial formula yields
(1)
h̃ w(i)
j=1
j=1
Qm
≤ e|β| J |G|
)m
(i)
m
Y
(−β)|w |
m
y |G| .
In the following lemma, we define a family of operators ρm
and bound their 1-norms. The bounds, in particular, guarantee
that the ρm are well-defined. In addition, they are useful for
the proof of Lemma 2, albeit they are not explicitly needed for
the proof of Lemma 1.
Lemma 9. Let ρ(G) be defined as in Lemma 6 with respect
to a Hamiltonian H having a finite interaction (hyper)graph
(V, E) with growth constant α and let
ρm :=
X
ρ(G)
(A.57)
G∈Am
(F )
≥L
(A.55)
Proof. Remember that Am
≥L (F ) is the set of m-fold (edge)
animals of size at least L that contain a letter from F . For
every G ∈ Am
≥L (F ), one finds m pairs (G1 , G2 ) with G1 ∈
for some F ⊂ E. Then,
Z(β)
kρm k1 ≤
m!
|F |
∞
X
l=L
where b(x) := α e|x| (e|x| − 1).
b(βJ)
l
!m
,
(A.58)
17
Proof. Consider a k-fold animal G ∈ Ak≥L (F ) and decompose it into its k non-overlapping animals Gj ∈ A≥L (F ) as
Uk
G = j=1 Gj ⊂ E. Then, Eq. (A.31) and Hölder’s inequality imply
kρ(G)k1 ≤ e−β H(G)c
1
k
Y
j=1
kη(Gj )k∞ ,
(A.59)
and it follows from Lemma 4 and Lemma 5 in conjunction
with the definition of η in Eq. (A.32) that
kρ(G)k1 ≤ Z(β) y(βJ)|G| .
Lemma 10. Let ρm be defined as in Eq. (A.57). Then
ρm
k=m
≤
Z(β)
m!
X
G∈A≥L (F )
m
y(βJ)|G| .
(A.62)
By decomposing the set of animals of size at least L into
a union of sets of animals of fixed size l, i.e., A≥L (F ) =
U∞
l=L A=l (F ), we can write
!m
∞
Z(β) X
l
kρm k1 ≤
.
(A.63)
|A=l (F )| y(βJ)
m!
l=L
The bound (8) on the number of lattice animals, the fact that
the number |F | of edges in F upper bounds the number of
possibilities of translating an animal G such that G ⊂ F , and
b = α y finish the proof.
While the last lemma provides a bound on ρm and, in particular, implies that ρm is well-defined, the next lemma provides a useful form of ρm .
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f (w) .
(A.64)
w∈C k
(F )
≥L
W(G) := {w ∈ [(∂ G)c ]∗ : G ⊂ w} .
(A.65)
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X
X
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(A.66)
G∈Am
(F ) w∈[(∂ G)c ]∗ :G⊂w
≥L
As
[
G∈Am
(F )
≥L
X
Proof. For G ∈ Am
≥L (F ), let
(A.60)
Hence, by the definition from Eq. (A.57) and Lemma 8, we
obtain
X
kρm k1 ≤ Z(β)
y(βJ)|G|
(A.61)
∞
X
k
=
m
G∈Am
(F )
≥L
W(G) =
]∞
k=m
C k≥L(F ) ,
(A.67)
the sums in Eqs. (A.57) and (A.64) contain the same terms.
It remains to show that the multiplicities are correct, i.e., are
given by the binomial factor. Every word in W(G) contains
at least m maximal clusters of size at least L, each of which
contains a letter in F . The key is to decompose this set as
W(G) =
with
]∞
k=m
W k (G)
(A.68)
W k (G) := {w ∈ W(G) : ∃ exactly k maximal clusters
c ⊂ w : c ∈ C ≥L(F )} ,
i.e., into sets of words having exactly k ≥ m such clusters.
Then, the observation
that for every w ∈ W k (G) there are
k
exactly m many m-fold animals G′ ∈ Am
≥L (F ) with w ⊂
G′ completes the proof.
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