For calculations at a finite temperatureT = βk1
B, which are presented in Sec. 11.3, two ingredients are necessary. First, a state for which we can define a temperature, with respect to a given Hamiltonian, and second, a way to change the corresponding temperature. In this section, we summarize a way to represent a mixed state with an MPS that is called purification and present two different ways to obtain a state at infinite temperature, i.e., β= 0. Afterwards, the imaginary-time evolution, which is analog to the real-time evolution (see chapter 9), can be used as a tool to reduce the temperature of such a state.
Purification
5.4.1
A quantum-mechanical state at a finite temperature is represented by an ensemble, which can be represented by a mixed state. MPSs are wave functions and hence can only represent pure states. It is therefore necessary to use a subsystem of an MPS, because if two subsystems are entangled, their reduced density operators represent mixed states. To be more precise, in the purification approach it is necessary to double the system and define one half of the sites as the physical subsystem P and the other half as the auxiliary subsystemQ. The desired properties then follow directly from the Schmidt decomposition,
|ψβ=0i=X
s
Σs|siP |siQ ⇒ρˆP =X
s
Σ2s|siPhs|P . (5.41) For the following consideration, we need to assume that we can obtain a state at infinite tem-perature|ψβ=0iwith the corresponding density operatorρˆ0, which we show below. The thermal density operator ρˆβ can then be derived via
ˆ
ρβ = e−βHˆ Z(β)
Z(0)ˆρ0= ˆId
= Z(0) Z(β)TrQ
e−β/2Hˆ|ψβ=0i hψβ=0|e−β/2Hˆ
, (5.42)
with the partition function Z(β) = TrPe−βHˆ. The partial trace can be expanded around the exponentials, because the Hamiltonian does not act on the auxiliary subsystem Q. Because β ∈R, we can define|ψβi=e−β2Hˆ |ψβ=0i, which shows that by using an imaginary-time evolution up to timeτ =−iβ2, we obtain a state at temperature T = βk1
B.
3If a truncation sweep to the right is performed and the spectrum at bondjshall be obtained, the singular values of the SVD of the active sitej+ 1need to be considered.
Section 5.4. Finite-Temperature Representation 43
In order to evaluate expectation values, we need to employ the fact that the density opera-tor contains all physically measurable information. Hence, we can obtain thermally averaged expectation values via
hOˆiβ = Z(0) Z(β)TrP
Oˆρˆβ
= Z(0)
Z(β)hψβ|Oˆ|ψβi . (5.43) Note thatZ(0) =dL, andZ(β) is given by the normalization factors of the purified state in the progress of the imaginary-time evolution. Hence, we can use the method introduced in Sec. 6.2 in order to calculate expectation values of the thermal ensemble.
Infinite-Temperature States
5.4.2
In order to fulfill the assumption that an infinite-temperature state can be obtained, here we present three different ways to accomplish this task. Note that we identify an infinite-temperature state with a maximally entangled mixed state.
Consider the factorization
ˆ ρ0 = Idˆ
dL = Idˆ d
!⊗L
, (5.44)
which yields an enlarged local sitej that consists of one physical sitep(j) and one auxiliary site q(j),
Idˆj
d =TrQ|ψijhψ|j |ψij = 1
√d X
σ
|σip(j)⊗ |σiq(j) . (5.45) At this point, the methodically simplest approach for the infinite-temperature state is to directly apply Eq. (5.45). This means, for a spin-1/2 system
|ψij = 1
√2
|↑ip(j)⊗ |↑iq(j)+|↓ip(j)⊗ |↓iq(j)
. (5.46)
This strategy is surely expandable to arbitrary systems, but the resulting states also lack prop-erties necessary to employ multiple conserved quantum numbers. Hence, for systems with larger local basis or multiple conserved quantum numbers, more sophisticated methods are needed to obtain a canonical infinite-temperature state. Two of those methods are presented in the following.
Entangler Hamiltonian
In Ref. [NA16], a way to obtain a state that conserves multiple U(1) symmetries of a model, e.g., total spin and particle-number conservation, independently within the physical and the auxiliary system is presented. The idea is to formulate a so-called entangler-Hamiltonian or, short, entangler, whose ground state is the desired state atβ = 0, and to perform a ground-state search (see chapter 8) with that operator. Note, that the entangler is constructed only by fixing the particle statistics, e.g., spin-1/2 fermions. Therefore, the same entangler can be used for the construction of infinite-temperature states as long as the local basis matches.
44 Chapter 5. Matrix-Product States
Here we follow [NA16] and construct the entangler-Hamiltonian for a system consisting of spin-1/2
fermions by choosing
HˆC2Spin-12-fermions=− X
j6=j0, σ=↑,↓
Λˆ†σ,jΛˆσ,j0+h.c.
, (5.47)
with
Λˆσ,j = ˆcσ,p(j)ˆc¯σ,q(j)Pˆjσ (5.48) and
Pˆjσ =|1−nˆ¯σ,p(j)−nˆσ,q(j)|. (5.49) Still,p(j)labels physical sites andq(j)the corresponding auxiliary sites. σ¯ denotes the opposite spin direction of σ.
In Ref. [NA16], further entangler-Hamiltonians are presented, but we refrain from reviewing them here, because they are not relevant for this thesis. Moreover, the following approach turns out to be conceptually easier and computationally faster, because no ground-state search is necessary.
Iterative Filling
Another method to create a canonical infinite-temperature state is introduced in Ref. [Bar16].
The main idea is to start from a vacuum state and to homogenously fill the system iteratively with maximally entangled states on the corresponding rungs until the desired quantum numbers are realized. This filling is accomplished by applying a global operator, which, in the case of a spin-1/2-fermion system, is given by
Cˆtot† = XL j=1
hcˆ†↑,p(j)⊗cˆ†↓,q(j)+ ˆc†↓,p(j)⊗ˆc†↑,q(j)i
. (5.50)
This operator already ensures that the total spin is zero in the full system as well as in the individual subsystems. In order to assure the constant spin, it is necessary to add two particles in either subsystem with every application of the operator Eq. (5.50) distributed over all sites.
If a different total spin is desired, another operator is needed. In Sec. 10.4.4, we review this approach in the context of our QCS and present an intuitive way how to perform the necessary operations.