On the path-independence of the relative thermal Hall conductance . 126

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

B.4 On the path-independence of the relative thermal Hall conductance . 126

Hall conductance is independent of the choice of the path connecting two points in the parameter space of 2d systems with finite correlation length. As explained in the body of the thesis, it is sufficient to show that the 1-form 𝜇^𝐸(𝛿 𝑓 ∪𝛿𝑔) is exact.

Here 𝑓(𝑝)and𝑔(𝑝)are smeared step-functions in the𝑥 and𝑦directions. Let 𝑓(𝑝) be constant except for𝑥(𝑝) '𝑎, and𝑔(𝑝) be constant except for𝑦(𝑝) '𝑏.

The first step is to make the 𝑦 direction periodic with period 𝐿, thereby replacing R²with a cylinderR𝑥 ×𝑆¹_𝑦. For 𝐿much larger than the correlation length this will change local quantities such as𝜇^𝐸_𝑝𝑞𝑟 by an amount of order𝐿^−∞. One complication is that the function𝑔(𝑝)is not periodic in the𝑦direction and thus does not descend to R𝑥 × 𝑆¹_𝑦. We deal with this by reinterpreting (𝛿𝑔)(𝑝, 𝑞) = 𝑔(𝑞) − 𝑔(𝑝) as a function onΛ×Λdefined only for |𝑝− 𝑞| < 𝐿/2. To make the evaluation of 𝜇^𝐸 on𝛿 𝑓 ∪𝛿𝑔well-defined, we truncate 𝜇^𝐸_𝑝𝑞𝑟 to zero whenever any two of the points

𝑝, 𝑞, 𝑟 are farther apart than𝐿/2. Let us denote the truncated energy magnetization by ˜𝜇^𝐸_𝑝𝑞𝑟.Because of truncation, we now have𝑑h𝐽_𝑝𝑞^𝐸 i =Í

𝑟𝜇˜^𝐸_𝑝𝑞𝑟+𝑂(𝐿^−∞). Or using the notation of Appendix A,

𝑑h𝐽^𝐸i =𝜕𝜇˜^𝐸 +𝑂(𝐿^−∞). (B.25) Naively, one can deduce the desired result using the Stokes’ theorem (A.7):

∫

Γ𝜇˜^𝐸(𝛿 𝑓 ∪𝛿𝑔) =

∫

Γ𝑑h𝐽^𝐸i(𝑓 ∪𝛿𝑔) +𝑂(𝐿^−∞) =𝑂(𝐿^−∞). (B.26) This argument is not correct because the 1-cochain 𝑓 ∪𝛿𝑔 is not cocontrolled (because (𝑓 ∪𝛿𝑔)(𝑝, 𝑞) does not vanish when 𝑥(𝑝) ' 𝑥(𝑞) and both 𝑥(𝑝) and 𝑥(𝑞)are large and negative), and the evaluation of𝑑h𝐽^𝐸ion such a 1-cochain is not well-defined. To fix this, we first modify the Hamiltonian for𝑥 < 𝑎−𝐿 by scaling it to zero. Since there are no phase transitions in 1d systems, the correlation length remains finite, and therefore the effect of such a modification on ˜𝜇^𝐸(𝛿 𝑓 ∪𝛿𝑔) will be of order 𝐿^−∞. Then the operator-valued chain𝐽^𝐸 also becomes zero for𝑥 𝑎, and the application of the Stokes’ theorem becomes legitimate. This concludes the argument.

Since by definition 𝜇^𝐸(𝛿 𝑓 ∪𝛿𝑔) is the differential of energy magnetization in the neighborhood of the point(𝑎, 𝑏), this result means that energy magnetization exists as a globally-defined function on the parameter space. This function is defined up to an additive constant.

B.5 The low-temperature behavior of the 1-formΨin a gapped system In this appendix we analyze the properties of the 1-form Ψ(𝑓 , 𝑔) whose integral defines the relative invariant of gapped 2d systems. We will have to use estimates on the behavior of certain correlation functions at low but non-zero temperature. More precisely, we will assume that if the𝑇 → 0 limit of a correlator is well-defined, then at sufficiently low temperature deviations from the𝑇 = 0 value are of order 𝑂(𝑒^{−𝑇∗/𝑇}) for some𝑇^∗ > 0. Physically, this is what one expects for a Hamiltonian with a gap for localized excitations.

One could try to prove it by putting the system on a torus of finite size 𝐿. Then for a correlation function𝐶(𝑇) one can construct a finite-size analog𝐶(𝑇, 𝐿)such that𝐶(𝑇) =lim𝐿→∞𝐶(𝑇, 𝐿). The correlation function𝐶(𝑇, 𝐿) can be rewritten in terms of many-body Green’s function𝐺 = (𝑧−𝐻)⁻¹. For example, one can write

∫ 𝛽

0 h𝐴(−𝑖𝜏)𝐵i^𝐿𝑑𝜏=𝑍⁻¹

∮

𝑒^−𝛽𝑧 𝑑𝑧

2𝜋𝑖Tr(𝐺 𝐴𝐺 𝐵), (B.27)

where𝑍is the partition function, and the contour surrounds all the eigenvalues of𝐻.

Now if we deform the contour into a pair of contours, one surrounding𝑧 =𝐸₀and the other surrounding all other eigenvalues, we see that for low𝑇 the contribution of the first contour is exponentially close to its𝑇 → 0 limit, while the contribution of the second one is exponentially small at low𝑇. Thus𝐶(𝑇, 𝐿) −𝐶(0, 𝐿)is exponentially small at low𝑇. If we assume that the order of limits𝑇 → 0 and 𝐿 → ∞ can be interchanged, we can conclude that𝐶(𝑇) is exponentially small at low𝑇. These arguments are at best heuristic, since it is far from clear when interchanging the order of limits is legitimate.

For simplicity of presentation we will work onR²and simply assume that correlation functions in gapped phase at non-zero temperature are exponentially closed to their zero-temperature expectation value. Also, we will consider the system at a fixed non-zero temperature𝑇 and will vary only the Hamiltonian. As was explained in Section 5.3, rescaling the temperature is equivalent to rescaling the Hamiltonian.

Finally, let us fix some 𝐿 > 0 which is much larger than the correlation length and define the 𝐿-support of a 1-cochain 𝛼to be the set of points 𝑝 ∈ Λsuch that 𝛼(𝑝, 𝑞) ≠ 0 for at least for one𝑞such that|𝑝−𝑞| < 𝐿.

Consider the integral of Ψ(𝑓 , 𝑔) along a path connecting two zero-temperature phasesM andM⁰:

𝐼(M,M⁰) =

∫ _M⁰

M Ψ(𝑓 , 𝑔). (B.28)

We will argue that it converges, does not change under the shift of the end points M,M⁰ as long as they do not cross zero-temperature phase transitions, and does not change under suitable deformations of 𝑓 , 𝑔.

Let us start with the last property. We consider adding to 𝑓 a function of𝑥(𝑝)which has compact support (as a function of𝑥) . We need to show that

∫ _M⁰

M

Ψ(𝑓₀, 𝑔) =0, (B.29)

where 𝑓₀is as in Fig 5.1b. Since the path in the parameter space is away from phase transitions, the correlation length is finite everywhere along the path. Truncating 𝑓₀to zero a distance 𝐿away from the 𝐿-support of𝛿𝑔will introduce error of order 𝐿^−∞. Denote the truncated cochain e𝑓₀. It has compact support, and therefore we

can rewrite the magnetization term as 𝜇^𝐸

𝛿e𝑓₀∪𝛿𝑔

=𝜕𝜇^𝐸

e𝑓₀∪𝛿𝑔

=𝑑h𝐽^𝐸(e𝑓₀∪𝛿𝑔)i =−1

2𝑑h𝑖[𝐻(e𝑓₀), 𝐻(𝑔)]i, (B.30) where in the last step we have used the definition of𝐽^𝐸 and cup product. The Kubo term, on the other hand, can be rewritten as

𝜅^{K𝑢𝑏𝑜}_𝑥𝑦 (e𝑓₀, 𝑔) =−𝛽² lim

𝑠→0+

∫ _∞

0 𝑑𝑡 𝑒^−𝑠𝑡hh𝑑𝐻(e𝑓₀, 𝑡)

𝑑𝑡 ;𝐽^𝐸(𝛿𝑔)ii

= 𝛽²hh𝐻(e𝑓₀);𝐽^𝐸(𝛿𝑔)ii +𝛽² lim

𝑠→0+𝑠

∫ _∞

0 𝑑𝑡𝑒^−𝑠𝑡hh𝐻(e𝑓₀, 𝑡);𝐽^𝐸(𝛿𝑔)ii.

(B.31)

The last term is in general non-zero since hh𝐻(e𝑓₀, 𝑡);𝐽^𝐸(𝛿𝑔)ii does not have to converge to zero as 𝑡 → ∞. However, at zero temperature and for a gapped Hamiltonian one can explicitly check that this term is zero. Indeed, expanding the expression in the energy eigenbasis we get

𝑠lim→0+𝑠

∫ _∞

0 𝑑𝑡𝑒^−𝑠𝑡hh𝐻(e𝑓₀, 𝑡);𝐽^𝐸(𝛿𝑔)ii

=−𝑖 lim

𝑠→0+𝑠Õ

𝑛>0

h0|𝐻(e𝑓₀)|𝑛ih𝑛|𝐽^𝐸(𝛿𝑔)|0i − h0|𝐽^𝐸(𝛿𝑔)|𝑛ih𝑛|𝐻(e𝑓₀)|0i

(𝐸₀−𝐸_𝑛)² =0.

Therefore at small but non-zero temperature we expect the second term in (B.31) to be exponentially suppressed. The remaining term can be rewritten as

𝛽²𝑑hh𝐻(e𝑓₀);𝐽^𝐸(𝛿𝑔)ii =𝛽²𝑑hh𝐻(e𝑓₀);−𝑖[𝐻, 𝐻(𝑔)]ii =−𝛽𝑑h𝑖[𝐻(e𝑓₀), 𝐻(𝑔)]i. (B.32) This term cancels the energy magnetization contribution (B.30). ThereforeΨ(e𝑓₀, 𝑔) is a differential of a function which is exponentially small for𝑇 → 0. Hence the integral ofΨ(e𝑓₀, 𝑔) along a path connecting two gapped zero-temperature systems is zero. Therefore the integral of Ψ(𝑓₀, 𝑔) along the same path is of order 𝐿^−∞. Since𝐿 is arbitrary, we can take the limit𝐿 → ∞and conclude that the integral of Ψ(𝑓₀, 𝑔) along this path is zero. Similarly, one can prove that 𝐼(M,M⁰) does not change if we add to𝑔a compactly supported function of𝑦.

It is tempting to use the same argument with 𝑓₀ replaced with 𝑓 to show that 𝐼(M,M⁰) is zero. But the argument cannot be carried through because it is impossible to truncate 𝑓 and make its support compact in such a way that the

support of 𝛿 𝑓 ∪𝛿𝑔coincides with the support of 𝛿e𝑓 ∪𝛿𝑔. There will necessarily be additional intersections.

In order to show that the integral (B.28) defining𝐼(M,M⁰) converges and is inde- pendent of the precise choice of endpoints, consider a variation of the Hamiltonian supported in a quadrant of R². A general perturbation can be decomposed into a sum of four such perturbations. As discussed in Section 5.3, in order to show that 𝐼(M,M⁰)is independent of endpoints and converges it is sufficient to show that all components of the 1-form Ψ(𝑓 , 𝑔) are exponentially small as𝑇 → 0. Following the same logic as before, we can shift 𝑓 , 𝑔inΨ(𝑓 , 𝑔) away from the support of the variation introducing an error which is exponentially small in temperature. Recall that the 1-formΨis defined as

Ψ(𝑓 , 𝑔) = 𝛽²

𝑑

∫ _∞

0 𝛽𝑒^−𝑠𝑡hh𝐽^𝐸(𝛿 𝑓 , 𝑡);𝐽^𝐸(𝛿𝑔)ii𝑑𝑡−2𝜇^𝐸(𝛿𝛼∪𝛿𝛾)

. (B.33) Using the same arguments as in Section 5.2, one can show that expression in square brackets is zero at𝑇 =0. Therefore, it is exponentially small at zero temperature, and the same applies toΨ(𝑓 , 𝑔).