arXiv:1506.04060v2 [cond-mat.stat-mech] 9 Oct 2015
Locality of temperature in spin chains
Senaida Hernández-Santana1 , Arnau Riera1 , Karen V. Hovhannisyan1 ,
Martı́ Perarnau-Llobet1 , Luca Tagliacozzo1 and Antonio Acı́n1,2
1
ICFO-Institut de Ciencies Fotoniques, Mediterranean Technology Park, 08860 Castelldefels
(Barcelona), Spain
2
ICREA-Institució Catalana de Recerca i Estudis Avançats, Lluis Companys 23, 08010
Barcelona, Spain
Abstract. In traditional thermodynamics, temperature is a local quantity: a subsystem of
a large thermal system is in a thermal state at the same temperature as the original system.
For strongly interacting systems, however, the locality of temperature breaks down. We study
the possibility of associating an effective thermal state to subsystems of infinite chains of
interacting spin particles of arbitrary finite dimension. We study the effect of correlations and
criticality in the definition of this effective thermal state and discuss the possible implications
for the classical simulation of thermal quantum systems.
2
Locality of temperature in spin chains
Contents
1
Introduction
2
2
Tensor network representation of the generalized covariance
2.1 Mapping the partition function of a D-dimensional quantum
contraction of tensor network of D + 1 dimensions . . . . . .
2.2 Generalized covariance as the contraction of tensor networks .
2.3 Transfer matrices . . . . . . . . . . . . . . . . . . . . . . . .
5
model to the
. . . . . . . .
. . . . . . . .
. . . . . . . .
5
7
8
3
Locality of temperature at non-zero temperature
4
Absolute zero temperature
4.1 Gapped systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Criticality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
10
11
5
A case study: The Ising chain
5.1 Generalized Covariance . . . . . . . . . . . . . . .
5.2 Locality of temperature in the quantum Ising chain
5.2.1 Non-zero temperatures . . . . . . . . . . .
5.2.2 Absolute zero temperature . . . . . . . . .
12
12
14
15
17
6
8
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Conclusions
18
Acknowledgments
19
Appendix A Proofs of the Lemmas
19
Appendix B Some bounds on the generalized covariance
22
Appendix C Solving quantum Ising model
23
References
25
1. Introduction
The question whether the standard notions of thermodynamics are still applicable in the
quantum regime has experienced a renewed interest in the recent years. This refreshed
motivation can be explained as the consequence of two successes. On the one hand, the
spectacular progress of the experiments encompassed in the so called quantum simulators
already allows for a direct observation of thermodynamic phenomena in many different
quantum systems, such as ultra cold atoms in optical lattices, ion traps, superconductor
qubits, etc [1, 2, 3, 4]. On the other hand, the inflow of ideas from quantum information
theory provided significant insight into the thermodynamics of quantum systems [5, 6, 7, 8].
Specifically, qualitative improvements have been made in understanding how the methods
Locality of temperature in spin chains
3
of statistical mechanics can be justified from quantum mechanics as its underlying theory
[2, 6, 9, 10, 11, 12].
One of the fundamental postulates of thermodynamics is the so called Zeroth Law: two bodies,
each in thermodynamic equilibrium with a third system, are in equilibrium with each other
[13, 14]. This is the law that stands behind the notion of temperature [13, 14]. In fact,
the above formulation of the Zeroth law consists of three parts: (i) there exists a thermal
equilibrium state which is characterized by a single parameter called temperature, and isolated
systems tend to this state [2, 6, 9, 10]; (ii) the temperature is local, namely, each part of the
whole is in a thermal state [13, 14]; and (iii) the temperature is an intensive quantity: if the
whole is in equilibrium, all the parts have the same temperature [13, 14, 15, 16, 17, 18].
The last two points are usually derived from statistical mechanics under the assumption of
weakly interacting systems. Nevertheless, when the interactions present in the system are
non-negligible, the points (ii) and (iii) need to be revised. Following the direction given
by Refs. [15, 16], in this work we concentrate on the clarification and generalization of the
aforementioned aspects of the Zeroth Law of the thermodynamics for spin chains with strong,
short range interactions.
The general setting of the problem is as follows. The system with Hamiltonian H is in thermal
equilibrium, described by a canonical state at inverse temperature β,
ω(H) =
e−βH
,
Tr (e−βH )
(1)
and we seek to understand the thermal properties of a finite part of the system.
Obviously, in the presence of strong interactions, the reduced density matrix of a subsystem
of the global system (especially in the quantum regime [18]) will not generally have the same
form as (1). In lattice systems, where the Hamiltonian is a sum of local terms interacting
according to some underlying graph, it is unclear how one can locally assign temperature to
a subsystem. More precisely, the reduced density matrix of the subsystem A (see Fig. 1) of a
global thermal state is described by
ρA = TrĀ (ω(H)) ,
(2)
which will not be thermal unless the particles in A do not interact with its environment Ā.
Hence, given only a subsystem state ρA and its Hamiltonian HA , it is not possible to assign a
temperature to it, since this would totally depend on the features of the environment and the
interactions that couple the subsystem to it.
In the context of quantum information, a first step to circumvent the problem of assigning
temperature to a subsystem was made in Ref. [15]. There, for harmonic lattices it was shown
that it is sufficient to extend the subsystem A by a boundary region B that, when traced out,
disregards the correlations and the boundary effects (see Fig. 1). If the size of such a boundary
region is independent of the total system size, temperature can still be said to be local.
More explicitly, given a lattice Hamiltonian H with a subsystem A, a shell region around it
B and its environment C = (A ∪ B)c , see Fig. 1, we aim to understand how the expectation
values of operators that act non-trivially only on A for the global thermal state ω(H) differ
4
Locality of temperature in spin chains
H
C
B
A
B
C
H0
AB := A ∪ B
Figure 1.
Scheme of the subsystem of interest A, the boundary region B, and their
environment C for a spin chain. Expectation values on A for the thermal state of the full
Hamiltonian H (above) are expected to be approximated by expectation values for thermal
state of the truncated Hamiltonian H0 (below) if the boundary region B is sufficiently large.
from those taken for the thermal state of the truncated Hamiltonian HAB (with AB = A ∪ B)
as the width ℓB of the boundary region B increases:
Tr[O ρ′A ] − Tr[O ρA ] ≤ ||O||∞ f (ℓB ) ,
(3)
where ρ′A = TrB ω(HAB ) is the state of A for the chain truncated to AB, and f (ℓB ) is expected
to be a monotonically decreasing function in ℓB . The width of the boundary region ℓB is
defined as the graph-distance between the sets of vertices (regions) A and C.
Surely, the differences (3) fully characterize the distance of ρ′A from ρA . Indeed, the trace
distance, a distinguishability measure for quantum states, D(ρ′A , ρA ) = 12 ||ρ′A − ρA ||1 , has the
following representation [19, 20]:
D(ρ′A , ρA ) = max Tr[O(ρ′A − ρA )] ≤ f (ℓB ),
0<O<I
(4)
where I is the identity operator in the Hilbert space of A.
In Ref. [17], it is proven that the correlations responsible for the distinguishability between
the truncated and non-truncated thermal states are quantified by a generalized covariance.
For any two operators O and O′ , full-rank quantum state ρ, and parameter τ ∈ [0, 1], the
generalized covariance is defined as
(5)
covτρ (O, O′ ) := Tr ρτ O ρ1−τ O′ − Tr(ρ O) Tr(ρ O′ ) ,
and the average distinguishability of the two states measurements of some observable O can
provide reads as
Z 1 Z β/2
(6)
dt covt/β
ds
Tr[Oω(H0 )] − Tr[O ω(H)] = 2
ωs (HI , O),
0
0
where HI is the corresponding Hamiltonian term that couples B and C, H0 = H − HI is the
truncated Hamiltonian (see Fig. 1) and ωs = ω(H(s)) is the thermal state of the interpolating
Hamiltonian H(s) := H − (1 − s) HI . Hence, the generalized covariance is the quantity
that measures the response in a local operator of perturbing a thermal state and ultimately at
what length scales temperature can be defined.
Temperature is known to be a local quantity in a high temperature regime. More specifically,
in Ref. [17], it is shown that for any local Hamiltonian there is a threshold temperature
(that only depends on the connectivity of the underlying graph) above which the generalized
Locality of temperature in spin chains
5
covariance decays exponentially. Nevertheless, it is far from clear what occurs below the
threshold, and, especially, at low temperatures (β ≫ 1). Note that, in that case, the right
hand side of the truncation formula (6) could be significantly different from zero since the
integration runs up until β/2, while the covariance is expected to decay only algebraically for
critical systems.
In this work we show that, for one dimensional translation-invariant systems, temperature
is local for any β. Away from criticality, we rigorously bound the truncation formula (6)
by mapping the generalized covariance to the contraction of a tensor network and exploiting
some standard results in condensed matter. At criticality, we use some results from conformal
field theory [21, 22]. Finally, the results in [23], where the equivalence of microcanonical and
canonical ensembles is proven for translation-invariant lattices with short range interactions,
render our results valid also when, instead of being canonical, (1), the global state ω(H)
is, e.g., microcanonical. The latter is defined as an equiprobable mixture of all the energy
eigenstates in a narrow energy window (see [23] for details).
In condensed matter physics, this problem has been considered in the context of
approximating the expectation values of infinite systems by finite ones, receiving the name
of finite size scaling (see, e.g., [24] and references therein). Nevertheless, the finite-sizescaling methods are more focused to find the values for the critical exponents and the transition
temperature by observing how measured quantities vary for different lattice sizes.
The paper is organized as follows. In Sec. 2, the generalized covariance of 1D system is
mapped to the contraction of a 2D tensor network. In Sec. 3, we show that temperature
is local at non-zero temperature (β < ∞) by identifying in the tensor network a gapped
transfer matrix which leads to a clustering of correlations and ultimately to a clustering of
the generalized covariance. In Sec. 4, locality of temperature is proven at zero temperature
(β → ∞) using different methods for gapped and gapless systems. While transfer matrix
arguments work satisfactory for gapped systems, conformal field theory results have to be
used at criticality. In Sec. 5, all our results are illustrated in detail for the Ising model, for
which we study in addition the behaviour of the generalized covariance and compute explicitly
the physical distinguishability between the full and truncated Hamiltonians both at and off
criticality. Finally, we conclude.
2. Tensor network representation of the generalized covariance
2.1. Mapping the partition function of a D-dimensional quantum model to the contraction of
tensor network of D + 1 dimensions
Let us consider a system of spins described by a short range Hamiltonian. The structure of
the Hamiltonian is given by a graph G(V, E). The spins correspond to the set of vertices V
and the two-body interactions to the edges E. Such a Hamiltonian can be written as
X
H=
hu
(7)
u∈E
where hu are the Hamiltonian terms acting non-trivially on the adjacent vertices of u.
6
Locality of temperature in spin chains
a)
c)
b)
d)
Figure 2. Diagrammatic representation of (a) the operator e−βH with H the Hamiltonian of a
m
spin chain, (b) its decomposition (exp(−β/mH)) , (c) the tensor network that approximates
e−βH after performing the Trotter-Suzuki decomposition for a one dimensional short-ranged
Hamiltonian and (d) the same tensor network after a convenient arrangement of the tensors. We
use the Penrose notation: tensors are represented as geometric shapes, open legs represent their
indices and legs connecting different tensors encode their contraction over the corresponding
indices.
In Ref. [26, 27], it is shown that, for any error ε > 0, the matrix e−βH of a local Hamiltonian
can be approximated in one norm by its Trotter-Suzuki expansion,
!
! † m
Y
Y
β
β
e− 2m hu
ρ̃T N =
(8)
e− 2m hv ,
u∈E
v∈E
such that
ke−βH − ρ̃T N k1 ≤ εke−βH k1 ,
(9)
where m > 360β 2 |E|2 /ε and the products over u and v in Eq. (8) are realized in the same
order.
To illustrate the previous approximation, let us consider in detail the one dimensional case:
a spin chain with nearest neighbour interactions. By decomposing in the standard way the
Hamiltonian in its odd and even terms, the tensor network ρ̃T N becomes in this case
β
m
β
β
− 2m Hodd − m
Heven − 2m
Hodd
ρ̃T N = e
e
e
,
(10)
P
where Hodd/even = u∈odd/even hu and H = Hodd + Heven .
Let us think about each exp(β/mhu ) as a tensor. In this way, ρ̃T N can be seen as the
contraction of several of such tensors, that is, a tensor network see Fig. 2 (c). Starting from a
one dimensional quantum system, ρ̃T N can be interpreted as a tensor network spanning two
dimensions, with the extra dimension of length m. We will refer to this extra dimension as
the β direction, while the original dimension will be called spatial direction.
In Fig. 2 (a), the diagrammatic representation of ρ̃T N is presented. Its tensors can be
decomposed and arranged in order to form a square lattice of elementary tensors as shown
in Fig. 2 (b).
7
Locality of temperature in spin chains
a)
b)
Figure 3. Diagramatic representation of (a) the expectation value of a one site operator and (b)
the generalized covariance (a two-point correlation function) between two one site operators.
In both cases, the final result is computed as the ratio between the contraction of two tensor
networks.
2.2. Generalized covariance as the contraction of tensor networks
The expectation value of a local operator is given by
Tr(Oe−βH )
.
hOi = Tr(Oω(H)) =
Tr(e−βH )
(11)
By using Eq. (9), the fact that ke−βH k1 = Z and some elementary algebra, the expectation
value of a local operator A can be approximated by the ratio between the contraction of two
tensor networks
Tr(Oρ̃T N )
≤ 2kOk∞ ε .
(12)
hOi −
Tr(ρ̃T N )
This is represented diagramatically in Fig. 3(a).
The generalized covariance can be rewritten as
Tr Õe−τ βH Õ′ e−(1−τ )βH
covτωs (β) (O, O′ ) =
,
(13)
Tr (e−βH )
where Õ = O − Tr(Oω(H)) for any operator O. Hence, in a similar way as it has been made
for the expectation values, the generalized covariance can be also approximated as the ratio
between two tensor network contractions as shown in Fig. 3(b)
(1−τ )
covτωs (β) (O, O′ )
Tr(Õρ̃τT N Õ′ ρT N )
≤ 2kOk∞ kO′ k∞ ε .
−
Tr(ρ̃T N )
(14)
From this perspective, the generalized covariance can be seen as a two point correlation
function on a 2 dimensional lattice in which τ m is the separation in the β direction and the
distance between the non-trivial supports of O and O′ is the separation in the spatial direction
(see Fig. 3).
8
Locality of temperature in spin chains
Figure 4. Diagramatic representation of the transfer matrix in the spatial direction (left) and
the β direction (right).
This construction can be generalized to approximate expectation values of local operators
and n-point correlation functions of a D dimensional quantum model by the ratio of the
contraction of two D + 1 dimensional tensor networks.
2.3. Transfer matrices
It is also very useful to define two extra objects: the transfer matrices along the spatial T and
β directions Tβ . The first is obtained by contracting a column of the elementary tensors of the
network, while the second is obtained by contracting several rows of elementary tensors, see
Fig. 4.
The number of rows that need to be contracted in order to obtain the transfer matrix in the
β direction, Tβ , is chosen such that its spectral gap between the largest and second-largest
eigenvalues is independent of both β and m. This can be achieved by contracting m/β rows,
leading to a transfer matrix with two largest eigenvalues µ1 and µ2
µ2
= e−∆
(15)
µ1
where ∆ is the gap of the Hamiltonian.
3. Locality of temperature at non-zero temperature
Let us consider now the case in which β is of order one. The physical distinguishability in A
between the full and the truncated Hamiltonians can be bounded by
Z1
Z1
τ
Tr[Oω(H0 )]−Tr[O ω(H)]≤β ds max covωs (HI , O)≤2β ds max max covτωs (hi , O), (16)
i
τ ∈[0,1]
0
τ ∈[0,1]
0
where HI = hL +hR , i ∈ {L, R}, and we have used the linearity of the generalized covariance
with respect of its operators.
Without loss of generality let us assume that the term of HI that maximizes the generalized
covariance is the one of the left, hL . Hence, the quantity to bound is covτωs (hL , O). In order
to do so, let us rewrite it as
covτωs (hL , O)=
h1L |Xs T ℓB Y T ℓB Ts |1R i h1L |Ts T ℓB Y T ℓB Ts |1R i h1L |Xs T 2ℓB +1 Ts |1R i
−
(17)
Zs
Zs
Zs
Locality of temperature in spin chains
9
Figure 5. Diagrammatic representation of a 3-point correlation function of the matrices Xs ,
Y and Ts for a system at zero temperature. The network is infinite in both directions.
where Zs := h1L |Ts T 2ℓB +1 Ts |1R i is the partition function, T is the transfer matrix in the
spatial direction, see Fig. 4 (left), and |1L/R i is the left/right dominant eigenvector of T
i. e. the eigenvector associated to its largest eigenvalue. The matrix Ts is the transfer matrix
corresponding to the boundaries BC where the elementary tensors of the network are different
from the rest for s < 1 and T1 = T . The matrix Y corresponds to the slice of the region
A where the operator O is supported, and the matrix Xs is the transfer matrix Ts with the
insertion of the operators hL located at a distance τ β from O in the transverse direction.
The diagrammatic representations of the matrices Xs , Y and Ts are shown in Fig. 5. To
simplify the calculations, the transfer matrix can always be normalized such that its dominant
eigenvalue is λ1 = 1.
To bound the generalized covariance (17) it is useful to rewrite it in terms of 2-point
correlation functions of the uniform system (s = 1)
covτωs (hL , O)
=
−
+
−
−
covT (ℓB ; Xs , Y T ℓB Ts )
Zs
hXs iT hTs iT covT (ℓB ; Ts , Y T ℓB Ts )
Zs
Zs
hXs iT hY T ℓB Ts iT covT (2ℓB + 1; Ts , Ts )
Zs
Zs
hTs iT hY T ℓB Ts iT covT (2ℓB + 1; Xs , Ts )
Zs
Zs
covT (ℓB ; Ts , Y T ℓB Ts ) covT (2ℓB + 1; Xs , Ts )
,
Zs
Zs
(18)
where
hXiT := h1L |X|1R i,
(19)
covT (ℓ; X, Y ) := h1L |XT ℓ Y |1R i − h1L |X|1R ih1L |Y |1R i ,
(20)
and we have used that h1L |T ℓB +1 Ts |1R i = hTs iT and covT (ℓB ; Xs , T ℓB +1 Ts ) = covT (2ℓB +
1; Xs , Ts ).
In short range one dimensional systems, the absence of phase transitions at non-zero
temperature [25] implies that the transfer matrix T is gapped, with a gap related to the spatial
correlation length as
ξ = − (ln |λ2 |)−1 > 0 ,
(21)
10
Locality of temperature in spin chains
where λ2 is the second largest eigenvalue of the transfer matrix T .
For gapped transfer matrices, the 2-point correlation function (20) can be proven to be upperbounded by
|covT (ℓ; X, Y )| ≤ kX|1L ikkY |1R ike−ℓ/ξ .
(22)
The complete proof of the previous statement can be found in Lemma 1 of the Appendix A.
Furthermore, Lemma 1 allows us to bound all the terms in Eq. (18), and, as it is shown in
Appendix B, the following inequality holds for the left hand side of Eq. (16):
Tr [O ω(HAB )] − Tr [O ω(H)] ≤ 2βckhL k∞ kOk∞ e−ℓB /ξ + O(e−2ℓB /ξ ), (23)
R1
where c = 1 + 0 dsσL (s), with σL (s) = (h1L |Ts Ts† |1L i − |hTs iT |2 )1/2 /|hTs iT | being the
relative standard deviation of Ts on the left dominant eigenstate |1L i. The quantity c is a
constant of order one that depends on the model considered. Hence, the temperature is proven
to be intensive for any one dimensional translationally invariant model at non-zero β.
4. Absolute zero temperature
4.1. Gapped systems
Given a Hamiltonian with gap ∆, here we study the regime in which β −1 ≪ ∆. This implies
that the lattice in its β direction is much larger than the correlation length β ≫ ξβ , with
−1
µ1
= ∆−1 .
(24)
ξβ := ln
µ2
In the limit of temperature tending to zero, the 2D network that represents the partition
function becomes infinite in the β direction (see Fig. 5).
In order to see that the temperature is also local in this case, let us decompose the integral
over t of the generalized covariance into two pieces
Z 1 Z β/2
dt covt/β
ds
Tr[O ω(HAB )] − Tr[O ω(H)] = 2
ωs (HI , O)
0
0
Z 1 Z β/2
Z 1 Z L
t/β
dt covt/β
ds
dt covωs (HI , O) + 2
ds
=2
ωs (HI , O) (25)
0
0
0
L
where L is a cut-off that will be chosen afterwards to minimize a bound on the right hand side,
and β will be made to tend to infinity.
Concerning the integral over 0 ≤ t ≤ L, we will exploit the fact that the system is gapped, and
hence its ground state is known to have a finite correlation length ξ in the spatial direction
and to be represented by a Matrix Product State of bond dimenson D, with D ∝ poly(ξ)
[40, 41, 42]. As argued in the previous section, a finite correlation length guarantees a gap in
the transfer matrix in the corresponding direction. By performing an analogous calculation to
the one described in the previous section, one obtains
Z 1 Z L
−ℓB /ξ
dt covt/β
.
(26)
ds
ωs (HI , O) ≤ 2ckhk∞ LkOk∞ e
0
0
Locality of temperature in spin chains
11
The second integral over t > L can be bounded by taking the transfer matrix in the β direction
which is also gapped for gapped Hamiltonians. More specifically, the generalized covariance
can be written as
hGS|OTβt HI |GSi hGS|OTβt |GSi hGS|Tβt HI |GSi
−
. (27)
covt/β
(H
,
O)
=
I
ωs
hGS|Tβt |GSi
hGS|Tβt |GSi hGS|Tβt |GSi
where we have identified Tβ as the transfer matrix for which the ground state of the
Hamiltonian |GSi is its dominant eigenvector. As previously, we make use of Lemma 1
in the Appendix A and obtain
−t/ξβ
|covt/β
.
ωs (HI , O)| ≤ kHI k∞ kOk∞ e
The integration is then bounded by
Z β/2 Z 1
−L/ξβ
ds covt/β
.
dt
lim
ωs (HI , O) ≤ ξβ kHI k∞ kOk∞ e
β→∞
L
(28)
(29)
0
Putting the previous bounds together, and after an optimization over L, we get
ℓB
Tr [O ω(HAB )] − Tr [O ω(H)] ≤ 4kOk∞ kHI k∞ c ξβ 1 +
− ln c e−ℓB /ξ , (30)
ξ
showing that temperature can be locally assigned to subsystems for arbitrarily large β and
gapped Hamiltonians.
4.2. Criticality
A system at zero temperature is said to be critical when the gap between the energy ground
state (space) and the first excited state closes to zero in the thermodynamic limit. The critical
exponents zν control how the spectral gap ∆ tends to zero
∆ ∝ N −zν ,
(31)
ξ ∝ Nν .
(32)
where N is the system’s size and ν is the critical exponent that controls the divergence of the
correlation length
The previous divergences are a signature of the scale invariance that the system experiences
at criticality. If the critical exponent z = 1, there is a further symmetry enhancement and the
system becomes conformal invariant. The group of conformal transformations includes, in
addition to scale transformations, translations and rotations.
In 1+1 dimensions, conformal symmetry completely dictates how correlation functions
behave and how local expectation values of local observables of infinite systems differ from
those taken for finite ones. Hence, conformal field theory establishes that
1
(33)
Tr[O ω(HAB )] − Tr[O ω(H)] ≃ y
ℓB
up to higher order terms, where y is the scaling dimension of the operator HI [38, 39]. If
HI is a standard Hamiltonian term, in the sense that the system is homogeneous, its leading
scaling dimension is y = 2.
Once more, we see that by increasing the buffer region temperature can be arbitrarily well
assigned.
Locality of temperature in spin chains
12
5. A case study: The Ising chain
Now we illustrate our results for the quantum Ising chain, which is described by the
Hamiltonian
N −1
N
hX i
1X i
i+1
σ ⊗ σx −
σ,
(34)
HN =
2 i=1 x
2 i=1 z
where σxi and σzi correspond to the Pauli matrices, h characterizes the strength of the magnetic
field and N is the number of spins. Notice that the interactions in the above Hamiltonian are of
finite range, a crucial assumption in our derivations, see (7). This model has a quantum phase
transition at h = 1, so it well exemplifies the different regimes discussed above: criticality
(only at zero temperature) and away from it (for zero and non-zero temperatures).
5.1. Generalized Covariance
First of all, as in the previous sections, we split the chain in three regions, which are shown
in Fig. 6. For such a splitting, and in the context of Eq. (6), we compute the generalized
N/2
t/β
covariance covωs (O, O′ ) taking for O a local operator in A, O = σz , and for O′ the
boundary Hamiltonian between B and C, O′ = HI , given by
1 N/2−3
HI =
(35)
σx
⊗ σxN/2−2 + σxN/2+2 ⊗ σxN/2+3 .
2
Figure 6. Scheme of the subsystem A, the boundary region B and their environment C. The
N/2
local operator σz acts on the subsystem A and the interaction term HI corresponds to the
red lines (connection between the subsystems AB and C).
t/β
N/2
In order to compute covωs (σz , HI ), we first diagonalize the Hamiltonian (34) using
standard techniques from statistical mechanics, such as the Jordan-Wigner and the
Bogoliubov transformation (see Appendix C). Once the Hamiltonian is diagonalized, we can
straightforwardly construct the corresponding thermal state for every large but finite N , and
t/β
N/2
compute covωs (σz , HI ) using expression (5).
N/2
t/β
Figure 7 shows covωs (σz , HI ) as a function of t/β for several temperatures (β =
5, 20, 1000), and for h = 0.9, 1 (i.e., near and at criticality). We take N = 40, which already
describes well the thermodynamic limit (recall that we are only interested on the local state,
and that the correlations decay exponentially). The area below the curves correspond to the
first integral in (6), which measures how well the local state in A can be approximated by a
thermal state in AB.
The results in Fig. 7 are in agreement with properties (i) and (ii) from Lemma 2 in Appendix
A. The first property implies that the covariance is symmetric with respect to t = β/2, and it
follows by taking l = t and n = β in (A.8). Second, property (ii) implies that it is bounded by
Locality of temperature in spin chains
13
a convex function of t with a maximum at t = 0 and t = β and with a minimum at t = β/2.
Therefore, the covariance satisfies the bound (A.9).
On the other hand, the covariance is not monotonic in s (see Fig. 8). This is somehow
counterintuitive, as it shows that the outcomes of two observables with no overlapping support
(located in A and in the intersection between B and C) do not always become more correlated
as s, which quantifies the strength of the interaction between B and C, increases.
Figure 7. Generalized covariance as a function of t for different values of s: s = 0, 1/3, 2/3, 1
for the dotted, dashed, black and thick lines. The figures correspond to inverse temperature
β = 5 (top), β = 20 (at the middle) and β = 100 (bottom) and field strength h = 0.9 (left)
and h = 1 (right). The grey area below the curves corresponds to the first integral of Eq. (6).
14
Locality of temperature in spin chains
Figure 8. Generalized covariance as a function of s for different values of t: t/β = 0, 1/3, 1/2
for the thick, black and dashed lines. The figures correspond to inverse temperature β = 5
(top), β = 20 (at the middle) and β = 100 (bottom) and field strength h = 0.9 (left) and h = 1
(right). The grey area below the curves corresponds to the second integral of Eq. (6). Notice
that, due to the symmetry in t, the values t/β = 2/3, 1 are also considered.
5.2. Locality of temperature in the quantum Ising chain
In our analytical findings, the generalized covariance naturally appeared as a tool to solve
the locality of temperature problem, see (6). This motivated the previous section, where we
studied its properties in the context of the quantum Ising chain. Nevertheless, in order to
t/β
obtain (6), one still needs to integrate covωs (O, HI ) over s and τ . While this approach is
useful when dealing with arbitrary generic systems, here we are dealing with a specific model
that is furthermore solvable, so we can take a more direct approach. Concretely, we first
compute
ρA = TrBC (ω(H)),
with H ≡ H∞
(36)
and
ρ′A = TrBC (ω(HAB )),
with HAB ≡ HN ,
(37)
15
Locality of temperature in spin chains
for different sizes N of the region AB. Secondly, we measure the distinguishability between
such states via the quantum fidelity, which is advantageous for computational reasons. For
two states, ρ′A and ρA , the fidelity is defined as [19]
q
√
′
′ √
(38)
ρA ρA ρA .
F [ρA , ρA ] = tr
It satisfies 0 ≤ F ≤ 1 and F [ρA , ρ′A ] = 1 if and only if ρA = ρ′A . In order to relate this
approach to our previous considerations, we note the following relation between the trace
distance, D[ρA , ρ′A ], and F (ρA , ρ′A ), given in [20],
√
1 − F ≤ D ≤ 1 − F 2.
(39)
Therefore, the fidelity provides us with upper and lower bounds to (4). In particular, when
D[ρA , ρ′A ] → 0 then F [ρA , ρ′A ] → 1, and in that case we say that the temperature is locally
well defined.
From now on, we take for A a two spin subsystem, an infinite chain as the total system, and
we compute F (ρA , ρ′A ) as a function of the size of AB, with N = 2 + 2lB , and the different
parameters of the Hamiltonian.
In order to compute ρA and ρ′A , it is convenient to apply the Jordan-Wigner transformation
i
to (34), which maps spin operators σx,y,z
to fermionic operators ai , a†i (see Appendix C for
details). The Hamiltonian (34) takes then the form,
HN =
N
X
Aij ai a†j
i,j=1
N
1X
Bij (a†i a†j − ai aj ),
+
2 i,j=1
(40)
which is quadratic in terms of the fermionic operators. It follows that thermal states, as well
as their local states, are gaussian operators. Therefore it is possible to describe them by their
covariant matrix, whose size is only O(N 2 ). This allows us to compute ρ′A in (37) for finite
but large lB ; while in the limit N → ∞, i.e. to compute ρA in (36), we rely on the analytical
results from [36]. The explicit calculations are done in Appendix C.
5.2.1. Non-zero temperatures
Figure 9 shows F (ρA , ρ′A ) as a function of β and h, for N = 4 (left) and N = 20 (right).
Recall that N , with N = 2 + 2lB , defines the size of the boundary region which is used to
approximate ρA by ρ′A . Even if the boundary is small, N = 4, the fidelity is close to 1 for all
values of β and h, and thus the temperature is locally well-defined. As expected, F (ρA , ρ′A )
increases with N (see Fig. 10).
We also observe in Fig. 9 that the fidelity becomes minimal near h = 1, which is the phase
transition point. As N increases, this minimum is shifted to h = 1. At this point the spatial
correlations also increase, which suggests a relation between both quantities.
In order to further explore this connection, we compute the scaling of F (ρA , ρ′A ) with N , and
compare it to the decay of the correlations. The behaviour of F (ρA , ρ′A ) is plotted in Fig. 10,
which clearly shows that the fidelity follows an exponential law with N , given by
F (ρA , ρ′A ) ∼ 1 − e
N
− 2ξ
S
,
(41)
Locality of temperature in spin chains
16
where ξS is a parameter that characterizes the slope of the function. On the other hand, the
correlations between a local observable in A, σzi , and one in the intersection of B and C, σzi+d ,
corr(σzi , σzi+d ) = hσzi σzi+d i − hσzi ihσzi+d i,
(42)
can be obtained through the two-spin correlation function hσzi σzi+d i in [36]. Their asymptotic
behaviour is also exponential with d,
d
corr(σzi , σzi+d ) ∼ e− ξ ,
(43)
where ξ is the correlation length. Now, identifying d, the distance between particles, with
N/2, which is roughly the size of B, we obtain from the numerical results in Fig. 11 the
following simple relation,
ξ = 2ξS .
(44)
Roughly speaking, the quality of the approximation ρ′A is directly related to the strength of the
correlations in the system. This relation is in good agreement with previous considerations in
[43], where the correlation length is related to the error of the cluster approximation [43, 44].
In summary, temperature can be assigned to the local system for all h and non-zero β by
taking a small boundary region (with N ≥ 4, and thus lB ≥ 2). We have shown that this is
directly connected to the exponential decay of the correlations with the distance, which makes
the local state of a thermal state only be sensible to its closest boundary.
Figure 9. Fidelity F (ρA , ρ′A ) as a function of β and the strength of the magnetic field h for
N = 4 (left) and N = 20 (right). The temperature is locally well-defined provided that
F (ρA , ρ′A ) ≈ 1.
Locality of temperature in spin chains
17
Figure 10. Function of fidelity, − log(1 − F [ρA , ρ′A ]), as a function of N for h = 1 (left) and
h = 0.9 (right). The inverse temperature is β = 5, 20, 100, 200 for the grey, black, thick and
dashed lines.
Figure 11. Correlation length, ξ, as a function of h. The black spots correspond to the
numerical values for 2ξS . The inverse temperature is β = 5, 20, 75 for the black, dotted
and dashed lines.
5.2.2. Absolute zero temperature
The same conclusions apply at zero temperature, as the fidelity is also close to 1 for all h and
N ≥ 4. It also has a minimum near the critical point.
Nonetheless, the scaling of the fidelity (or more precisely 1 − F ) with N can differ from the
previous case. While the scaling is generally exponential at zero temperature, it becomes a
power law at the phase transition point (see Fig. 12),
F (ρA , ρ′A ) ∼ 1 − N −Cs .
(45)
This type of decay is also obtained for the correlations as a function of the distance, which
again shows a direct connection between the quality of the approximation (quantified by
F (ρA , ρ′A )) and the strength of the correlations.
Locality of temperature in spin chains
18
Figure 12. Function of fidelity, − log(1 − F (ρA , ρ′A )), as a function of N for β → ∞. The
field strength is h = 0.9, 1. for the dashed and black lines.
6. Conclusions
In this work we studied the locality aspect of the Zeroth law of thermodynamics for quantum
spin chains with strong but finite range interactions. Upon noting that in the presence of
strong interactions the marginal states of a global thermal state do not take the canonical form
themselves, we go on defining an effective thermal state for a subsystem. The latter being the
reduced density matrix of the subsystem considered as a part of a slightly bigger, enveloping
thermal system (see Fig. 1). Borrowing concepts from quantum information theory and
employing methods from quantum statistical mechanics, we relate the accuracy with which
the effective thermal state describes the actual state of the subsystem to the correlations
present in the whole system (see Eqs. (4, 5, 6) and the discussion around them). We further
utilize a Trotter approximation formula [26, 27] to build a tensor network representation of
the corresponding states of the subsystem to provide upper bounds on the aforementioned
accuracy, depending on the size of the enveloping thermal system, and such physical quantities
as the spectral gap of the global hamiltonian and the temperature of the parent chain. At the
quantum critical point, we use already existing asymptotical formulas from the conformal
field theory.
Lastly, we exemplify our analytical findings by analyzing the quantum Ising chain. The latter
is complex enough to have a quantum phase transition point, but simple enough to allow
for an exact diagonalization by standard tools of statistical mechanics, thereby serving as a
perfect testbed for our analytical upper bounds. In particular, we find that, e.g., away from
criticality, the envelope which is bigger than the system only by one layer of spins, is enough
to approximate the actual state with a rather high precision (see, e.g., Fig. 9).
Our results for one dimensional systems with finite range interactions suggest that
investigating the properties of the effective thermal states in higher dimensions and, possibly,
harbouring long range of interactions, is an interesting direction for further research, which
can have far-reaching implications in efficient simulation of the subsystems of large and
strongly interacting quantum systems. Another interesting open question beyond the scope
Locality of temperature in spin chains
19
of this work is whether these results can be generalized to other other types of equilibrium
states, e. g. the so called Generalized Gibbs Ensemble and steady states of local Liouvillians.
In a more practical vein, another field where our findings may find implications is quantum
thermometry with non-negligible interactions [45].
Acknowledgments
This work is supported by the IP project SIQS, the Spanish project FOQUS, and the
Generalitat de Catalunya (SGR 875). S.H.S. and M.P.L. acknowledge funding from the ”la
Caixa”-Severo Ochoa program, M.P.L is also supported by the Spanish grant FPU13/05988,
A.R. is supported by the Beatriu de Pinós fellowship (BP-DGR 2013), L.T. is supported by
the ERC AdG OSYRIS and A.A. is supported by the ERC CoG QITBOX. All authours thank
the EU COST Action MP1209 “Thermodynamics in the quantum regime”.
Appendix A. Proofs of the Lemmas
In this section, we present the proofs of the lemma’s used in Secs. 3 and 4 to get statements
on the locality of temperature for gapped systems. They consist of how different covariances
decay for one dimensional systems with a gapped transfer matrix T .
Lemma 1. [Infinite chain] Given a gapped transfer matrix T with eigenvalues λk labelled
in decreasing order, i. e. |λk | ≥ |λk′ | for all k < k ′ , a right (left) dominant eigenvector |1R i
(h1L |), the largest eigenvalue λ1 = 1, and a covariance between any two operators O and O′
separated by a distance ℓ defined as
cov(ℓ; O, O′ , T ) = h1L |O† T ℓ O′ |1R i − h1L |O† T ℓ |1R ih1L |T ℓ O′ |1R i .
(A.1)
Then, the covariance can be proven to decay exponentially in ℓ
|cov(ℓ; O, O′ , T )| ≤ kO|1L ikkO′ |1R ike−ℓ/ξ
(A.2)
where ξ = − (ln |λ2 |)−1 > 0 is the correlation length and the k|ϕik = hϕ|ϕi1/2 is the norm
of the vector |ϕi.
Proof. Let us first introduce the new transfer matrix T̃ = T − |1R ih1L | to rewrite the
covariance as
cov(ℓ; O, O′ , T ) = h1L |O† T̃ ℓ O′ |1R i ,
(A.3)
where we have used that T̃ ℓ = T ℓ − |1R ih1L |. By using the Cauchy-Schwarz inequality one
gets
1/2
h1L |O† T̃ ℓ O′ |1R i ≤ kO′ |1R ik h1L |O† T̃ ℓ (T̃ † )ℓ O|1L i
.
(A.4)
20
Locality of temperature in spin chains
Let us now consider second factor separately. By inserting a resolution of the identity, a
straight forward calculation leads to
X
h1L |O† T̃ ℓ (T̃ † )ℓ O|1L i =
|λk |2ℓ h1L |O† |kR ihkR |O|1L i
k≥2
≤ |λ2 |2ℓ
X
k≥2
h1L |O† |kR ihkR |O|1L i ≤ |λ2 |2ℓ kO|1L ik2
(A.5)
where we have used that |λ2 | is an upper-bound for all the |λk | with k ≥ 2 and the Parseval
inequality.
Finally, we put everything together and get
|cov(ℓ; O, O′ , T )| ≤ kO|1L ikkO′ |1R ike−ℓ/ξ
(A.6)
where the 2nd largest eigenvalue |λ2 | has been written in terms of the correlation length ξ.
Lemma 2. [Periodic boundary conditions] Given a system with periodic boundary
conditions, an Hermitian transfer matrix T with a gap ∆ and a covariance between any two
operators O and O′ separated by a distance ℓ defined as
cov(ℓ; n, O, O′ , T ) =
Tr(OT ℓ O′ T n−ℓ ) Tr(OT n ) Tr(O′ T n )
−
.
Tr(T n )
Tr(T n ) Tr(T n )
(A.7)
where 0 ≤ ℓ ≤ n and n is the system size. Then, the covariance cov(ℓ) = cov(ℓ; n, O, O′ , T )
as a function of ℓ fulfills following properties:
(i) Its real part is symmetric respect to the n/2 and the interchange of A and B, i. e.
cov(n−ℓ; n, O, O′ , T ) = cov(ℓ; n, O, O′ , T )∗ = cov(ℓ; n, O, O′ , T ) .(A.8)
′
(ii) Given two operators O and O′ , there always exist two other operators OM and OM
such
that
′
|cov(ℓ; n, O, O′ , T )| ≤ cov(ℓ; n, OM , OM
,T)
(A.9)
′
and where cov(ℓ; n, OM , OM
, T ) is a convex function in ℓ that is maximum at ℓ = 0 and
1, and reaches its minimum at ℓ = n/2.
Proof. Statement (i) is a simple consequence of the following elementary equalities
∗
Tr(OT ℓ O′ T n−ℓ ) = Tr (OT ℓ O′ T n−ℓ )†
= Tr(T n−ℓ O′ T ℓ O) = Tr(O′ T ℓ OT n−ℓ ) .
(A.10)
In order to prove (ii), let us focus on the first term in Eq. (A.7), since note that the second one
does not depend on ℓ. With this aim, we define
Tr(OT ℓ O′ T n−ℓ )
′
f (ℓ; O, O ) = ℜ
.
(A.11)
Tr(T n )
Locality of temperature in spin chains
21
By introducing the transfer matrix in its spectral representation, f (ℓ) can be written as
"
ℓ n−ℓ !
n
X
λk
s
λk
1
h1|O|1ih1|O′ |1i +
ck
+
f (ℓ; O, O′ ) =
n
Tr(T )
λ1
λ1
k≥2
#
ℓ n−ℓ
X
λk
λk ′
+
dkk′
,
(A.12)
λ1
λ1
k,k′ ≥2
where ck = ℜ (h1|O|kihk|O′ |1i) and dkk′ = ℜ (hk|O|k ′ ihk ′ |O′ |ki). Note now that
ℓ n−ℓ
λk
ℓ − n/2
λk
− 2ξn
,
+
= 2e k cosh
λ1
λ1
ξk
where the correlation length ξk is defined as
λ1
−1
ξk := ln
.
λk
(A.13)
(A.14)
Note that as the eigenvalues of the transfer matrix are ordered, a larger k implies a shorter
correlation length ξk .
In a similar way, we can also simplify the terms in the last sum in Eq. (A.12). Note that
ℓ n−ℓ ℓ n−ℓ
λk ′
λk
λk ′
λk
− n − ℓ
−n+ ℓ
+
= e ξk′ ξkk′ + e ξk ξkk′
λ1
λ1
λ1
λ1
ℓ
−
n/2
− 2ξn
.(A.15)
= 2e kk′ cosh
ξkk′
−1
−1
−1
where the length ξkk′ has been defined as ξkk
′ = ξk − ξk ′ . Puting the previous steps together,
we get
"
X −n
λn1
ℓ
−
n/2
f (ℓ; A, B) =
h1|A|1ih1|B|1i + 2
e 2ξk ck cosh
Tr(T n )
ξk
k≥2
#
X
X
ℓ
−
n/2
− 2ξn
− ξn
dkk′ e kk′ cosh
. (A.16)
+ 2
dkk e k + 2
ξkk′
k≥2
2≤k<k′
Note that in general the covariance could oscillate in ℓ, since ck and dkk′ could take negative
values for some k and k ′ . Nevertheless, given two operators A and B for which some
′
ck and dkk′ are negative, there always exist two operators OM and OM
such that their
′
˜
respective c̃k = |ck | and dkk′ = |dkk′ |. For instance, hk|OM |k i = −hk|O|k ′ i for the k ′
′
and k-s with negative coefficents and hk|OM |k ′ i = hk|OM |k ′ i otherwise, and OM
= O′ .
′
This covariance f (ℓ; OM , OM
) is an upper bound to the absolute value of the previous one
′
covariance f (ℓ; O, O ),
"
n
X −n
λ
ℓ
−
n/2
1
′
′
f (ℓ; OM , OM ) =
|h1|O|1ih1|O |1i| + 2
e 2ξk |ck | cosh
Tr(T n )
ξk
k≥2
#
X
X
ℓ
−
n/2
− 2ξn
− ξn
|dkk′ |e kk′ cosh
+ 2
|dkk |e k + 2
ξkk′
k≥2
2≤k<k′
≥ f (ℓ; O, O′ ) .
(A.17)
22
Locality of temperature in spin chains
As the sum in Eq. (A.17) is a linear combination of convex functions with positive coefficents,
′
f (ℓ; OM , OM
) is also convex. It is also obvious from the properties of the cosh() function,
′
that f (ℓ; OM , OM
) reaches its maximum at ℓ = 0 and n, and its minimum at ℓ = n/2.
Appendix B. Some bounds on the generalized covariance
In this appendix we bound the different terms in Eq. (18) to provide an upper-bound for the
generalized covariance used in the Section 3.
Let us start with the first term in Eq. (18). By using Lemma 1, we get
|covT (ℓB ; Xs , Y T ℓB Ts )| ≤ kXs |1L ikkY TBℓ Ts |1R ike−ℓB /ξ .
(B.1)
The coefficients can be bounded by
kXs |1L ik2 = h1L |Xs Xs† |1L i ≤ khL k2∞ h1L |Ts Ts† |1L i
kY T ℓB Ts |1R ik2 ≤ kOk2∞ kT ℓB +1 Ts |1R ik2
≤ kOk2∞ |hTs iT |2 + h1R |Ts† Ts |1R ie−2(ℓ+1)/ξ ,
(B.2)
(B.3)
where we have used the structure of Xs and Y and the fact that Tr(Oρ) ≤ kOk∞ Tr(ρ) for
any ρ ≥ 0. It is also necessary to upper bound the inverse of the partition function. With that
aim, let us write the partition function as
Zs = hTs i2T + h1L |Ts (T − |1R ih1L |)2ℓB +1 Ts |1R i
(B.4)
|Zs | ≥ |hTs iT |2 − kTs† |1L ikkTs |1R ike−(2ℓB +1)/ξ ,
(B.5)
where we have used that (T − |1R ih1L |)ℓ = T ℓ − |1R ih1L |. By proceeding similarly as in the
proof of Lemma 1, the partition function Zs can be lower bounded by
and its inverse is upper bounded by
kT † |1L ikkTs |1R ik −(2ℓB +1)/ξ
−1
−2
1+ s
|Zs | ≤ |hTs iT |
e
|hTs iT |4
kTs† |1L ik2 kTs |1R ik2 −(4ℓB +2)/ξ
+
.
e
|hTs iT |8
By putting the previous bounds together we get
(B.6)
covT (ℓB ; Xs , Y T ℓB Ts )
≤ kOk∞ khL k∞ (1+σL (s))e−ℓB /ξ +O(e−2ℓB /ξ ), (B.7)
Zs
where σL (s) = (h1L |Ts Ts† |1L i − |hTs iT |2 )1/2 /|hTs iT | is the relative standard deviation of Ts
on the left dominant eigenstate |1L i and we have omitted for simplicity the second order terms
in e−ℓB /ξ .
In a similar way, the second term in Eq. (18) can be bounded by
hXs iT hTs iT covT (ℓB ; Ts , Y T ℓB Ts )
≤ khL k∞ kOk∞ (1 + σL (s))e−ℓB /ξ
Zs
Zs
+ O(e−2ℓB /ξ ),
where we have used that |hXs iT /hTs iT | ≤ khL k∞ .
(B.8)
23
Locality of temperature in spin chains
The rest of terms in Eq. (18) can be analogously bounded. Note that they will only contribute
to the second order. Putting everything together in Eq. (16), the physical distinguishability on
the region A between the truncated and untruncated thermal states is upper-bounded by
Tr [O ω(HAB )] − Tr [O ω(H)] ≤ 2βckhL k∞ kOk∞ e−ℓB /ξ + O(e−2ℓB /ξ ),
R1
where c = 1 + 0 dsσL (s) is a constant of order one, depending on the model.
(B.9)
Appendix C. Solving quantum Ising model
In this appendix we find the states (36) and (37) using formalism of covariance matrices.
Jordan-Wigner transformation
Let us first apply the Jordan-Wigner transformation, σxi ⊗ σxi+1 = (a†i − ai )(ai+1 + a†i+1 ) and
σzi = ai a†i − a†i ai , to the Hamiltonian (34). We obtain,
Hn =
N
X
i,j=1
Aij ai a†j +
N
1X
Bij (a†i a†j − ai aj ),
2 i,j=1
(C.1)
with Aij = hδi,j + 12 (δi+1,j + δi,j+1 ) and Bij = 12 (δi+1,j − δi,j+1 ) and where ai and a†i denote
annihilation and creation operators, respectively. From this form of the Hamiltonian, we
notice it is quadratic, and thus the thermal state (and their marginal states) are gaussian states.
Therefore we can deal with them using the covariance matrix formalism.
The correlation matrix
In this formalism, we define the global correlation matrix, Γ, as
Γ(X) = hXX † i =
k hai a†j i kN ×N
k hai aj i kN ×N
k ha†i a†j i kN ×N
k ha†i aj i kN ×N
a1
..
.
aN
with X =
a† , (C.2)
1
..
.
a†N
where k ... kN ×N refers to a N × N matrix. Given Γ, we can obtain the correlation matrix
corresponding to a reduced state by just selecting the corresponding matrix elements of Γ. For
example, the correlation matrix of the fermions k, k + 1 is given by,
hak ak i
hak ak+1 i
hak a†k+1 i
hak a†k i
ha a† i ha a† i ha a i ha a i
k+1 k
k+1 k+1
k+1 k+1
k
(C.3)
Γk,k+1 = k+1
.
ha†k ak+1 i
ha†k ak i
ha†k a†k+1 i
ha†k a†k i
ha†k+1 a†k i ha†k+1 a†k+1 i ha†k+1 ak i ha†k+1 ak+1 i
Since the Jordan Wigner transformation is local, in the sense that it maps the kth fermion to
the kth spin in the chain, this correlation matrix also corresponds to the two-spin subsystem
24
Locality of temperature in spin chains
at sites k and k + 1. This subsystem is precisely the region of interest A in section 5.2, and
thus (C.3) corresponds to the correlation matrix of ρA in (36).
Given the reduced correlation matrix, the explicit form of ρA can be easily obtained. As the
reduced state of a thermal state is gaussian, there is a one to one connection between (C.3)
and ρA . Indeed, for any gaussian state, with
†
e−X M X
ρ=
with M a coefficient matrix,
T r[e−X † M X ]
it is straightforward to prove that, provided that M is diagonalizable,
1
Γ(X) =
(1 + e−2M )
(C.4)
(C.5)
or, equivalently, that
1
M = − log(Γ(X)−1 − 1).
2
(C.6)
Explicit computation
Now we explicitly compute (C.3) for a finite and an infinite chain, in order to obtain ρ′A and
ρA , respectively, using relation (C.6).
• Finite chain
For the case of a finite chain, we need to obtain the correlation matrix (C.10) corresponding to the global state. It is then useful to first diagonalize the Hamiltonian (C.1)
by applying the Bogoliubov transformation
N
X
1
1
(φjk + ψjk )ai − (φjk − ψjk )a†i ,
(C.7)
2
2
k=1
P
PN
2
2
where φ and ψ are real matrices and verify N
φ
=
jk
k=1
k=1 ψjk = 1. The Hamiltonian
can then take the form,
bj =
H=
N
X
k=1
ξk (b†k bk − 1/2),
(C.8)
where ξk are the fermionic excitation energies and bk and b†k denote annihilation and
creation operators, respectively. The excitation energies, ξk , and the matrices φ and ψ
are obtained by solving the equation
(A − B)φ = ψD,
(C.9)
where D is a diagonal matrix whose entries correspond to the excitation energies, ξk .
Locality of temperature in spin chains
25
Once the Hamiltonian is diagonalized, it is easy to compute the correlation matrix of a
thermal state at inverse temperature β in the diagonalized basis, obtaining
b1
..
.
1
0N ×N
1+e−βD
bN
Γ(Y ) =
(C.10)
with Y =
b† ,
1
1
0N ×N
..
1+eβD
.
b†N
where the non-zero matrices are diagonal.
From that expression we can obtain the correlation matrix in the original basis, Γ(X),
via
Γ(X) = T † Γ(Y )T ,
(C.11)
where T is the transformation matrix defined by the Bogoliubov transformation (C.7).
That is, Y = T X, with
"
#
γ µ
T =
.
(C.12)
µ∗ γ ∗
and
1
1
γ = (φ + ψ) and µ = − (φ − ψ).
2
2
(C.13)
• Infinite chain (N → ∞)
In the case of an infinite chain, (C.3) can be obtained relying on the analytical results
from [36]. The partial state of a two-spin subsystem is
"
#
3
X
1
ρn→∞
=
1 + hσzk i(σzk + σzk+1 ) +
hσlk σlk+1 iσlk ⊗ σlk+1 , (C.14)
2
4
l=x,y,z
where the average hσzk i and the two-spin correlation functions {hσlk σlk+1 i}l={x,y,z} are
given by [36]. In order to express the state in the fermionic basis, we can compute the
reduced correlation matrix (C.3) from this state,
2(1 + α) −(β + γ)
0
−(β − γ)
β−γ
0
−(β + γ) 2(1 + α)
(C.15)
Γk,k+1 =
,
0
β−γ
2(1 − α)
β+γ
−(β − γ)
0
β+γ
2(1 − α)
with α = hσzk i, β = hσxk σxk+1 i and γ = hσyk σyk+1 i.
Locality of temperature in spin chains
26
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
[21]
[22]
[23]
[24]
[25]
[26]
[27]
[28]
[29]
[30]
[31]
[32]
[33]
[34]
[35]
[36]
[37]
[38]
[39]
[40]
[41]
[42]
[43]
I. Bloch, J. Dalibard, and S. Nascimbène, Nat. Phys. 8, 267 (2012).
A. Polkovnikov, K. Sengupta, A. Silva, and M. Vengalattore, Rev. Mod. Phys. 83, 863 (2011).
J. Kai, A. Retzker, F. Jelezko, and M.B. Plenio, Nat. Phys. 9, 168 (2013).
J. Simon, et al, Nature 472, 307 (2011).
K. Maruyama, F. Nori, and V. Vedral, Rev. Mod. Phys. 81, 1 (2009).
S. Popescu, A.J. Short, and A. Winter, Nat. Phys. 2, 754 (2006).
M. Horodecki, and J. Oppenheim, Nat. Commun. 4, 2059 (2013).
L. del Rio, et al, Nature 474, 61 (2011).
S. Goldstein, J.L. Lebowitz, R. Tumulka, and N. Zanghi, Phys. Rev. Lett. 96, 050403 (2006).
J. Gemmer, M. Michel, and G. Mahler, Quantum Thermodynamics, (Berlin: Springer, 2004).
A.J. Short, and T.C. Farrelly, New J. Phys. 14, 013063 (2012).
F.G.S.L. Brandão, and M. Cramer, Equivalence of Statistical Mechanical Ensembles for Non-Critical
Quantum Systems, arXiv:1502.03263 [quant-ph].
L.E. Reichl, A Modern Course in Statistical Physics (New York: Wiley, 1998).
R. Balian, From Microphysics to Macrophysics, (Heidelberg: Springer, 2007).
A. Ferraro, A. Garcı́a-Saez, and A. Acı́n, Europhys. Lett. 98, 10009 (2012).
A. Garcı́a-Saez, A. Ferraro, and A. Acı́n, Phys. Rev. A 79, 052340 (2009).
M. Kliesch, C. Gogolin, M.J. Kastoryano, A. Riera, and J. Eisert, Phys. Rev. X 4, 031019 (2014).
L.A. Pachon, J.F. Triana, D. Zueco, and P. Brumer, Uncertainty Principle Consequences at Thermal
Equilibrium, arXiv:1401.1418 [quant-ph].
M.A. Nielsen, and I.L. Chuang, Quantum Computation and Quantum Information (Cambridge: Cambridge
University Press, 2000).
C.A. Fuchs, and J. van de Graaf, Trans IEEE. Inf. Theory 45, 121627 (1999).
P. Di Francesco, P. Mathieu, and D. Senechal, Conformal Field Theory (New York: Springer, 1997).
J. Cardy, Boundary conformal field theory, Encyclopedia of mathematical Physics ed J-P Franoise,GL
Naber and T S Tsun (Amsterdam: Elsevier).
M.P. Mueller, E. Adlam, L. Masanes, and N. Wiebe, Thermalization and canonical typicality in translationinvariant quantum lattice systems, arXiv:1312.7420 [quant-ph].
J. Cardy, Finite-size scaling (Amsterdam: Elsevier, 2012).
A. Gelfert, and W. Nolting, J. Phys.: Condens. Matter 13 505-24 (2001).
M.B. Hastings, Phys. Rev. B 73, 085115 (2006).
A. Molnár, N. Schuch, F. Verstraete, and J.I. Cirac, Phys. Rev. B 91, 045138 (2014).
T. Kinoshita, T. Wenger, and D.S. Weiss, Nature 440, 900 (2006).
S. Hofferberth et al., Nature 449, 324 (2007).
I. Bloch, J. Dalibard, and W. Zwerger, Rev. Mod. Phys. 80, 885 (2008).
M. Rigol, V. Dunjko, and M. Olshanii, Nature 452, 854 (2008).
M.C. Bañuls, J.I. Cirac, and M.B. Hastings, Phys. Rev. Lett. 106, 050405 (2011).
N. Linden, S. Popescu, and P. Skrzypczyk, Phys. Rev. Lett. 105, 130401 (2010).
C. Gogolin, M.P. Mueller, and J. Eisert, Phys. Rev. Lett. 10, 040401 (2011).
T.J. Osborne, and M.A. Nielsen, Phys. Rev. A 66, 032110 (2002).
E. Barouch, and B.M. McCoy, Phys. Rev. A 3, 786 (1970).
M.A. Nielsen, The Fermionic canonical commutation relations and the Jordan-Wigner transform, School
of Physical Sciences (The University of Queensland, 2005).
J. Cardy. Nucl. Phys. B 240, 514-32 (1984).
J. Cardy. Nucl. Phys. B 275, 200-18 (1986).
D. Perez-Garcia, F. Verstraete, M. M. Wolf, and J. I. Cirac. Quantum Info. Comput. 7 , 401-30 (2007).
B. Pirvu, G. Vidal, F. Verstraete, and L. Tagliacozzo. Phys. Rev. B, 86 075117 (2012).
F. Verstraete, and J. I. Cirac. Phys. Rev. B, 73 094423 (2006).
M. Lubasch, J.I. Cirac, M.C. Bañuls. Phys. Rev. B 90, 064425 (2014).
Locality of temperature in spin chains
27
[44] M. Lubasch, J.I. Cirac, M.C. Bañuls. New J. Phys. 16, 033014 (2014)
[45] A. De Pasquale, D. Rossini, R. Fazio, V. Giovannetti. Local quantum thermometry arXiv:1504.07787
[quant-ph].