Invariance under Hamiltonian density redefinition

Chapter IX: Higher-dimensional generalizations of the Thouless charge pump 87

C.2 Invariance under Hamiltonian density redefinition

For a given Hamiltonian, there are many ways to define a Hamiltonian density. A typical example of this is the ambiguity in splitting an interaction term between two sites𝑝and𝑞into𝐻_𝑝and/or𝐻_𝑞. In this appendix, we will show that our microscopic formulas for physically observable transport coefficients are independent of the choice of the Hamiltonian density, even though individual terms in the microscopic formulas are not invariant. For some systems this can be used to simplify the microscopic formulas.

Invariance of the electric current

Consider the following change of the Hamiltonian density 𝐻𝑝 →𝐻𝑝+Õ

𝑟∈Λ

𝐴𝑟 𝑝, (C.5)

where 𝐴𝑟 𝑝 is skew-symmetric in𝑟, 𝑝. We want the final Hamiltonian to be𝑈(1)- invariant. Therefore, we have to impose

[𝑄,Õ

𝑟∈Λ

𝐴_{𝑟 𝑝}] =0. (C.6)

For a general choice of 𝐴𝑝𝑞 a stronger condition

[𝑄, 𝐴𝑝𝑞] =0, (C.7)

will not hold. However, one can always redefine 𝐴𝑝𝑞 (by subtracting the𝑈(1)-non- invariant part) in such a way that (C.7) holds without affecting𝐻_𝑝. In the following we will assume this was done and (C.7) is true.

Under the transformation (C.5) the electric current changes as 𝐽_𝑝𝑞^𝑁 → 𝐽_𝑝𝑞^𝑁 +𝑖Õ

𝑟∈Λ

[𝐴_𝑟𝑞, 𝑄_𝑝] − [𝐴_{𝑟 𝑝}, 𝑄_𝑞]

. (C.8)

Even though the current density changes, the net current through any section is invariant. Indeed,

𝐽^𝑁(𝛿 𝑓) →𝐽^𝑁(𝛿 𝑓) + 𝑖 2

Õ

𝑝,𝑞,𝑟∈Λ

[𝐴_𝑟𝑞, 𝑄_𝑝] − [𝐴_{𝑟 𝑝}, 𝑄_𝑞]

(𝑓(𝑞) − 𝑓(𝑝)), (C.9)

and the last term is zero since Õ

𝑝,𝑞,𝑟∈Λ

[𝐴_𝑟𝑞, 𝑄_𝑝] − [𝐴_{𝑟 𝑝}, 𝑄_𝑞]

(𝑓(𝑞) − 𝑓(𝑝))

= Õ

𝑝,𝑞,𝑟∈Λ

[𝐴_𝑟𝑞, 𝑄_𝑝] + [𝐴_𝑝𝑟, 𝑄_𝑞] + [𝐴_{𝑞 𝑝}, 𝑄_𝑟]

(𝑓(𝑞) − 𝑓(𝑝))

= 1 3

Õ

𝑝,𝑞,𝑟∈Λ

[𝐴_𝑟𝑞, 𝑄_𝑝] + [𝐴_𝑝𝑟, 𝑄_𝑞] + [𝐴_{𝑞 𝑝}, 𝑄_𝑟]

× (𝑓(𝑞) − 𝑓(𝑝) + 𝑓(𝑝) − 𝑓(𝑟) + 𝑓(𝑟) − 𝑓(𝑞))=0, (C.10) where we have used (C.7) and the skew-symmetry of [𝐴_𝑟𝑞, 𝑄_𝑝] + [𝐴_𝑝𝑟, 𝑄_𝑞] + [𝐴_{𝑞 𝑝}, 𝑄_𝑟].

Covariance of the energy current

Let us now consider the effect of the redefinition of the Hamiltonian density on the energy current. Imposing an energy analog of (C.6) or (C.7)

[𝐻,Õ

𝑟∈Λ

𝐴𝑟 𝑝] =^? 0, or [𝐻, 𝐴𝑝𝑞] =^? 0, (C.11) is far too restrictive, since it would only allow changes of the Hamiltoniain density by conserved quantities. For example, the difference between putting the interaction term between the two sites 𝑝 and 𝑞 either into 𝐻_𝑝 or into 𝐻_𝑞 corresponds to 𝐴_𝑝𝑞 equal to the interaction term. Obviously, interaction terms are not integrals of motion in general. Because of this we will not impose either of the equations in (C.11).

Under the redefinition of the Hamiltonian density (C.5) the energy current changes as

𝐽_𝑝𝑞^𝐸 → 𝐽_𝑝𝑞^𝐸 +𝑖Õ

𝑟∈Λ

[𝐴_𝑟𝑞, 𝐻_𝑝] + [𝐻_𝑞, 𝐴_{𝑟 𝑝}]

, (C.12)

while the net current transforms as 𝐽^𝐸(𝛿 𝑓) →𝐽^𝐸(𝛿 𝑓) + 𝑖

2 Õ

𝑝,𝑞,𝑟∈Λ

[𝐴_𝑟𝑞, 𝐻_𝑝] + [𝐻_𝑞, 𝐴_{𝑟 𝑝}]

(𝑓(𝑞) − 𝑓(𝑝)). (C.13)

The last term can be rewritten as 𝑖

2 Õ

𝑝,𝑞,𝑟∈Λ

[𝐴_𝑟𝑞, 𝐻_𝑝] + [𝐻_𝑞, 𝐴_{𝑟 𝑝}]

(𝑓(𝑞) − 𝑓(𝑝))

= 𝑖 2

Õ

𝑝,𝑞,𝑟∈Λ

[𝐴_𝑟𝑞, 𝐻_𝑝] + [𝐴_𝑝𝑟, 𝐻_𝑞] + [𝐴_{𝑞 𝑝}, 𝐻_𝑟]

(𝑓(𝑞) − 𝑓(𝑝))

−𝑖 2

Õ

𝑝,𝑞,𝑟∈Λ

[𝐴_{𝑞 𝑝}, 𝐻_𝑟](𝑓(𝑞) −𝑓(𝑝)) = 𝑖 6

Õ

𝑝,𝑞,𝑟∈Λ

[𝐴_𝑟𝑞, 𝐻_𝑝] + [𝐴_𝑝𝑟, 𝐻_𝑞] + [𝐴_{𝑞 𝑝}, 𝐻_𝑟]

× (𝑓(𝑞) − 𝑓(𝑝) + 𝑓(𝑝) − 𝑓(𝑟) + 𝑓(𝑟) − 𝑓(𝑞)) − 𝑖 2

Õ

𝑝,𝑞∈Λ

[𝐻, 𝐴_𝑝𝑞](𝑓(𝑞) − 𝑓(𝑝))

=− ¤𝐴(𝛿 𝑓), where we have defined

𝐴(𝛿 𝑓)= 1 2

Õ

𝑝,𝑞∈Λ

𝐴_𝑝𝑞(𝑓(𝑞) − 𝑓(𝑝)). (C.14)

We find that the net energy current transforms as follows under a redefinition of the Hamiltonian density:

𝐽^𝐸(𝛿 𝑓) → 𝐽^𝐸(𝛿 𝑓) − ¤𝐴(𝛿 𝑓). (C.15) But this should be expected since a redefinition of the energy density changes how we define the energy of sub-regions and therefore should affect the net energy current.

Indeed, one can see that (C.13) is exactly the transformation needed in order to satisfy the energy conservation law

𝐻¤_𝑝=−Õ

𝑞∈Λ

𝐽^𝐸_𝑝𝑞 → 𝐻¤_𝑝+Õ

𝑞∈Λ

𝐴¤_{𝑞 𝑝} =−Õ

𝑞∈Λ

𝐽^𝐸_𝑝𝑞+Õ

𝑞∈Λ

𝐴¤_{𝑞 𝑝} (C.16) for the new energy density 𝐻𝑝+Í

𝑞∈Λ 𝐴𝑞 𝑝. By summing this transformation law over𝑝 weighted by a function 𝑓(𝑝)with a compact support we find that

¤

𝐻(𝑓) =−𝐽^𝐸(𝛿 𝑓) → 𝐻¤_𝑝+ ¤𝐴(𝛿 𝑓) =−𝐽^𝐸(𝛿 𝑓) + ¤𝐴(𝛿 𝑓), (C.17) which reproduces (C.15). Here we used an identity

Õ

𝑝,𝑞∈Λ

𝐴𝑝𝑞𝑓(𝑞) = 1 2

Õ

𝑝,𝑞∈Λ

𝐴𝑝𝑞(𝑓(𝑞) − 𝑓(𝑝)) = 𝐴(𝛿 𝑓) (C.18) which is true for any 𝑓 with a compact support.

From the above discussion, one can see that energy current is not invariant but covariant under energy density redefinitions. If we choose 𝑓(𝑝) to be 1 when𝑝 is in some compact set𝐵 and zero otherwise, the physical meaning of (C.15) is very clear. It corresponds to ambiguities in the energy currents due to interaction terms along the boundary of 𝐵. Depending on how we distribute the interaction terms among𝐻_𝑝we can change the energy stored in the region𝐵as well as energy current through its boundary.

Invariance of the microscopic formulas for thermoelectic coefficients

In this section we will show that the coefficients 𝜈_𝑥𝑦 and 𝜂_𝑥𝑦 are invariant under a redefinition of the Hamiltonian density. We will start with skew-symmetric coefficients

𝑑𝜈^𝐴 = 1 2𝑑

𝜈^Kubo(𝛿 𝑓 , 𝛿𝑔) −𝜈^Kubo(𝛿𝑔, 𝛿 𝑓)

− 𝛽²𝜇^𝑁(𝛿 𝑓 ∪𝛿𝑔), (C.19) 𝑑𝜂^𝐴 = 1

2𝑑

𝜂^Kubo(𝛿 𝑓 , 𝛿𝑔) −𝜂^Kubo(𝛿𝑔, 𝛿 𝑓)

− 𝛽𝜇^𝑁(𝛿 𝑓 ∪𝛿𝑔). (C.20) Here we defined the Kubo parts as

𝜈^Kubo(𝛿 𝑓 , 𝛿𝑔)= 𝛽²lim

𝑠→0

∫ _∞

0 𝑑𝑡𝑒^−𝑠𝑡hh𝐽^𝑁(𝛿 𝑓 , 𝑡);𝐽^Q(𝛿𝑔)ii, (C.21) 𝜂^Kubo(𝛿 𝑓 , 𝛿𝑔)= 𝛽lim

𝑠→0

∫ _∞

0 𝑑𝑡𝑒^−𝑠𝑡hh𝐽^Q(𝛿 𝑓 , 𝑡);𝐽^𝑁(𝛿𝑔)ii. (C.22) Under Hamiltonian density redefinition the Kubo parts transform as

𝜈^Kubo(𝛿 𝑓 , 𝛿𝑔) → 𝜈^Kubo(𝛿 𝑓 , 𝛿𝑔) −𝛽²lim

𝑠→0

∫ _∞

0 𝑑𝑡𝑒^−𝑠𝑡hh𝐽^𝑁(𝛿 𝑓 , 𝑡);𝐴¤(𝛿𝑔)ii

=𝜈^Kubo(𝛿 𝑓 , 𝛿𝑔) −𝛽²hh𝐽^𝑁(𝛿 𝑓);𝐴(𝛿𝑔)ii,

(C.23) 𝜂^Kubo(𝛿 𝑓 , 𝛿𝑔) →𝜂^Kubo(𝛿 𝑓 , 𝛿𝑔) −𝛽lim

𝑠→0

∫ _∞

0 𝑑𝑡𝑒^−𝑠𝑡hh ¤𝐴(𝛿 𝑓 , 𝑡);𝐽^𝑁(𝛿𝑔);ii

=𝜈^Kubo(𝛿 𝑓 , 𝛿𝑔) +𝛽hh𝐴(𝛿 𝑓);𝐽^𝑁(𝛿𝑔)ii,

(C.24) where we used properties of the Kubo pairing.

Before finding the variation of the magnetization term it is useful to rewrite it

slightly:

𝜇^𝑁(𝛿 𝑓 ∪𝛿𝑔)

= 1 2

Õ

𝑝,𝑞∈Λ

"

1 3

Õ

𝑟∈Λ

𝜇𝑝𝑞𝑟(𝑔𝑝+𝑔𝑞+𝑔𝑟) − 1 2

Õ

𝑟∈Λ

𝜇𝑝𝑞𝑟(𝑔(𝑝) +𝑔(𝑞))

#

(𝑓(𝑞) − 𝑓(𝑝))

= 1 2

Õ

𝑝,𝑞∈Λ

"

1 3

Õ

𝑟∈Λ

𝜇𝑝𝑞𝑟(𝑔𝑝+𝑔𝑞+𝑔𝑟) − 1

2𝑑h𝐽^𝑁_𝑝𝑞i(𝑔(𝑝) +𝑔(𝑞))

#

(𝑓(𝑞) − 𝑓(𝑝)). (C.25) Note that one cannot expand the square brackets, since the two resulting sums over 𝑝, 𝑞will not converge separately.

Let us find the variation of 1

2h𝐽^𝑁_𝑝𝑞i(𝑔(𝑝) + 𝑔(𝑞)) under a Hamiltonian density redefinition. It reads

1

2h𝐽_𝑝𝑞^𝑁 i(𝑔(𝑝) +𝑔(𝑞)) → 1

2h𝐽_𝑝𝑞^𝑁i(𝑔(𝑝) +𝑔(𝑞)) + 𝑖

2 Õ

𝑟∈Λ

h[𝐴_𝑟𝑞, 𝑄_𝑝] − [𝐴_{𝑟 𝑝}, 𝑄_𝑞]i(𝑔(𝑝) +𝑔(𝑞)). (C.26) The last term can be rewritten as follows:

𝑖 2

Õ

𝑟∈Λ

h[𝐴𝑟𝑞, 𝑄𝑝] − [𝐴𝑟 𝑝, 𝑄𝑞]i(𝑔(𝑝) +𝑔(𝑞))

= 𝛽 2

Õ

𝑟∈Λ

hh𝑔(𝑝) ¤𝑄_𝑝;𝐴_𝑟𝑞ii + 𝛽 2

Õ

𝑟∈Λ

hh ¤𝑄_𝑝;𝑔(𝑞)𝐴_𝑟𝑞ii − (𝑝 ↔ 𝑞), (C.27) where we used the properties of the Kubo pairing. The first term in this expression can be rewritten as

Õ

𝑟∈Λ

hh𝑔(𝑝) ¤𝑄_𝑝;𝐴_𝑟𝑞ii − (𝑝↔ 𝑞)=−1 2

Õ

𝑠,𝑟∈Λ

hh𝐽_𝑠𝑝(𝑔(𝑠) +𝑔(𝑝));𝐴_𝑟𝑞ii − (𝑝 ↔𝑞)

=−1 2

Õ

𝑠,𝑟∈Λ

hh𝐽_𝑠𝑝^𝑁(𝑔(𝑠)+𝑔(𝑝))+𝐽_𝑝𝑠^𝑁(𝑔(𝑠)−𝑔(𝑝));𝐴_𝑟𝑞ii−(𝑝↔ 𝑞)= hh𝐽^𝑁(𝛿𝑔);𝐴_{𝑞 𝑝}ii

− 1 2

Õ

𝑟,𝑠∈Λ

"

hh𝐽_{𝑟 𝑝}^𝑁(𝑔(𝑟) +𝑔(𝑝));𝐴_𝑠𝑞ii + hh𝐽_𝑝𝑠^𝑁(𝑔(𝑠) −𝑔(𝑝));𝐴_𝑟𝑞ii +2 perms

# , (C.28)

where "2 perms" means the two cyclic permutations in 𝑝, 𝑞, 𝑟. Note that the term in square brackets is skew-symmetric in 𝑝, 𝑞, 𝑟. The second term can be rewritten as

Õ

𝑟∈Λ

hh ¤𝑄𝑝;𝑔(𝑞)𝐴𝑟𝑞ii − (𝑝↔ 𝑞)=− Õ

𝑠,𝑟∈Λ

hh𝐽_𝑠𝑝^𝑁;𝑔(𝑞)𝐴𝑟𝑞ii − (𝑝 ↔𝑞)

=−1 2

Õ

𝑠,𝑟∈Λ

hh𝐽_𝑠𝑝^𝑁;𝐴𝑟𝑞(𝑔(𝑞)+𝑔(𝑟))+𝐴𝑞𝑟(𝑔(𝑟)−𝑔(𝑞))ii−(𝑝 ↔ 𝑞) = hh𝐽^𝑁_𝑝𝑞;𝐴(𝛿𝑔)ii

− 1 2

Õ

𝑠,𝑟∈Λ

hhh𝐽_𝑠𝑝^𝑁;𝐴𝑟𝑞(𝑔(𝑞) +𝑔(𝑟))ii + hh𝐽_{𝑟 𝑝}^𝑁;𝐴𝑞𝑟(𝑔(𝑟) −𝑔(𝑞))ii +2 permsi . (C.29) Note that term in square brackets is skew-symmetric in𝑝, 𝑞, 𝑟

By combining equations (C.25-C.29) we find that the magnetization contribution changes under a redefinition of the Hamiltonian density as follows:

𝜇^𝑁(𝛿 𝑓 ∪𝛿𝑔) → 𝜇^𝑁(𝛿 𝑓 ∪𝛿𝑔) − 𝛽

2𝑑hh𝐽^𝑁(𝛿 𝑓);𝐴(𝛿𝑔)ii + 𝛽

2𝑑hh𝐽^𝑁(𝛿𝑔);𝐴(𝛿 𝑓)ii + 1

2 Õ

𝑝,𝑞,𝑟∈Λ

𝐶𝑝𝑞𝑟(𝑓(𝑞) − 𝑓(𝑝)),

(C.30) where𝐶_𝑝𝑞𝑟 is a skew-symmetric function of 𝑝, 𝑞, 𝑟 which is combination of skew- symmetric parts (and their derivatives) in the right-hand sides of Eqs. (C.25-C.29).

Due to its skew-symmetry we find that 1

2 Õ

𝑝,𝑞,𝑟∈Λ

𝐶𝑝𝑞𝑟(𝑓(𝑞) − 𝑓(𝑝))

= 1 6

Õ

𝑝,𝑞,𝑟∈Λ

𝐶_𝑝𝑞𝑟(𝑓(𝑞) − 𝑓(𝑝) + 𝑓(𝑝) − 𝑓(𝑠) + 𝑓(𝑠) − 𝑓(𝑞)) =0.

(C.31)

We see that the variation of the magnetization exactly compensates the variation of the Kubo parts. Thus the skew-symmetric parts of the thermoelectric tensors are invariant under a redefinition of the Hamiltonian density.

Now let us consider the symmetric parts 𝜈^𝑆_𝑥𝑦 = 1

2

𝜈^Kubo(𝛿 𝑓 , 𝛿𝑔) +𝜈^Kubo(𝛿𝑔, 𝛿 𝑓)

+𝛽𝑈(𝛿 𝑓 , 𝛿𝑔), (C.32) 𝜂^𝑆_𝑥𝑦 = 1

2

𝜂^Kubo(𝛿 𝑓 , 𝛿𝑔) +𝜂^Kubo(𝛿𝑔, 𝛿 𝑓)

−𝑈(𝛿 𝑓 , 𝛿𝑔). (C.33)

The variation of Kubo parts were already determined before, so we focus on the transformation of𝑈. Under (C.5) it transforms as follows:

𝑈(𝛿 𝑓 , 𝛿𝑔) →𝑈(𝛿 𝑓 , 𝛿𝑔) + 𝑖

4 Õ

𝑝,𝑞∈Λ

h[𝜕 𝐴_𝑞, 𝑄_𝑝] + [𝜕 𝐴_𝑝, 𝑄_𝑞]i(𝑓(𝑞) − 𝑓(𝑝))(𝑔(𝑞) −𝑔(𝑝)). (C.34) We can rewrite this equation by noticing that

𝑖

2h[𝜕 𝐴𝑞, 𝑄𝑝] + [𝜕 𝐴𝑝, 𝑄𝑞]i(𝑔(𝑞) −𝑔(𝑝))=−𝛽

2hh𝑔(𝑝) ¤𝑄𝑝;𝜕 𝐴𝑞ii + 𝛽

2hh ¤𝑄𝑝;𝜕𝑔(𝑞)𝐴𝑞ii − (𝑝 ↔ 𝑞). (C.35)

Then using eqs. (C.28, C.29) we find 𝑈(𝛿 𝑓 , 𝛿𝑔) →𝑈(𝛿 𝑓 , 𝛿𝑔) + 𝛽

2hh𝐽^𝑁(𝛿 𝑓);𝐴(𝛿𝑔)ii + 𝛽

2hh𝐽^𝑁(𝛿𝑔);𝐴(𝛿 𝑓)ii (C.36) We see that the variation of this term cancels the varitions of the Kubo parts.

One can do the same checks for the thermal Hall conductivity and verify that the microscopic formula for it is in invariant under a redefinition of the Hamiltonian density. To linear order in𝐴𝑝𝑞 all the manipulations are almost the same except for the replacement𝑄𝑝 → 𝐻𝑝and𝐽^𝑁 → 𝐽^𝐸.