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Physics 216: Special topics in many-body physics, Spring 2003:
http://socrates.berkeley.edu/˜ jemoore/phys216.html
Lecture XIX
This lecture introduces the imaginary-time path integral method as a way to calculate observable properties for systems in thermal equilibrium. We first present the general physical principle of how the measurable real-time response function R is obtained as the analytic continuation of a correlation function in imaginary time, which is applicable either to spin or Fermi systems, then later use the coherent-state representation to write an explicit form for the path integral for a single spin. Even though we are now talking about spins, what we are doing is logically a continuation of the same ideas used to develop Green’s functions at zero temperature for Fermi systems. The notation is that of Auerbach appendix B.
Let us consider a certain type of measurement on spin systems (similar “linear response” physics was covered in 212). We would like to know the response of the system to a weak perturbation jiα (t, q), which you can think of as a time- and space-dependent magnetic field. Here “response” means the change in the measured values of the observables of the system because of the pertur- bation. We also require that the perturbation was zero before some time t = 0. The Hamiltonian is then, in the Schrodinger representation,
H[j] = H 0 −
∑ q
jαi (t, q)Sαi (−q), (1)
and the evolution equation of some state ψ is
i
∂t |ψ(t)〉 = H[j]|ψ(t)〉. (2)
We can write the solution of this in terms of the unitary time translation operator,
|ψ(t)〉 = U [t, j]|ψ(0)〉, U [t, j] = Tt′^ exp
( −i
∫ (^) t
0
dt′^ H[j(t′)]
)
. (3)
Here the meaning of the time-ordered exponential is just as it was before: take the limit of a large number of steps in the interval [0, t], and within each step expand the exponential to linear order in order to see that U gives the correct time evolution.
The thermal expectation value of an operator (say the spin Siα in direction α on site i) follows from the assumption that the system is in thermal equilibrium at t = 0, with density matrix ρ = e−βH^0 : 〈Sαi (t)〉j = Z−^1 T r(ρU −^1 [t, j]Siα U [t, j]). (4)
Note that such a physical expectation value has to be invariant of representation (Schrodinger or Heisenberg). Now we make the linearity assumption and obtain
〈Siα (t)〉j = 〈Sαi (0)〉 + i
∫ (^) t
0
dt′^ 〈(−H[j(t′)]Siα (t) + Siα (t)H[j(t′)])〉 + O(j^2 )
= 〈Sαi (0)〉 + i
∫ (^) t
0
dt′^
∑
i′,α′
jα ′ i′^ (t
′)〈[Sα i (t), S
α′ t′^ (t
′)]〉 + O(j (^2) ). (5)
The response function Rαα ′ ii′^ (t^ −^ t ′) is the kernel of this integral:
Rαα ′ ii′^ (t^ −^ t
′) = −iθ(t − t′)〈[Sα i (t), S
α′ t′^ (t
′)]〉. (6)
Recall that the expectation values 〈〉 are taken with respect to H 0. So the response function is expressed as a correlation with respect to the unperturbed system. From now on we drop the directional indices α.
We can write R in terms of the exact many-body eigenstates of the unperturbed system. These form a complete set of states, and inserting two complete sets in the commutator gives
Rii′ (t) = −iθ(t)Z−^1
∑
αβ
e−Eα/T^
( ei(Eα−Eβ^ )t〈α|Si|β〉〈β|Si′ |α〉 − ei(Eβ^ −Eα)t〈α|Si′ |β〉〈β|Si|α〉
)
. (7)
Note that this is starting to look a little bit like the Green’s function we defined earlier, except that this is “causal” (only defined for one sign of time) and exists at nonzero temperature. Taking a Fourier space gives the more useful represenation
R(q, ω + i 0 +) = (N Z)−^1
∑
α,β
〈α|Sq|β〉〈β|S−q|α〉
e−Eα/T^ − e−Eβ^ /T Eα − Eβ + ω + i 0 +^
Our previous definition of the Green’s function (lecture 8), expressed in (ω, p) space using the basis of eigenstates, is
G(ω, p) ∼
( ∑ m
|〈 0 |cp|m〉|^2 ω − (Em − E 0 − μ) + iη
∑ m
|〈 0 |c† p|m〉|^2 ω + (Em − E 0 + μ) − iη
)
. (9)
Note the differences (aside from now using spin operators instead of electron annihilation or creation operators): the Green’s function contains terms corresponding to both signs of time, while the response function only contains one sign of time, and contains the thermal factors.
Experimentalists will probably be familiar with how to decompose the response function into real and imaginary parts, connected via Kramers-Kronig relations
< R(q, ω) = P
∫ (^) ∞
−∞
dω′ π
= R(q, ω′) ω′^ − ω
= R(q, ω) = −P
∫ (^) ∞
−∞
dω′ π
= R(q, ω′) ω′^ − ω
As a reminder, the imaginary part describes damping and dissipation by real excitations. An exercise on the final problem set relates the imaginary part of R to the spin structure factor 〈Si(t)Si′^ (t′)〉 in equilibrium (this is an example of a fluctuation-dissipation theorem):
S(q, ω) = −
1 − e−ω/T^
=R(q, ω) (11)
Now we introduce a theoretical method for calculating the response function. The idea is to define a generating functional in imaginary time whose correlation functions, continued to real time, will be related to the response function. Define the generating functional as (including a chemical potential, so this is the “grand canonical generating functional”)
Z[j, μ] = T r Tτ
( exp
[ −
∫ (^) β
0
dτ H[j(τ )]
])
. (12)
The path integral is mathematically imprecise and is consequently not as good for doing perturba- tion theory as the sort of operator methods we used earlier to justify Feynman diagrams. However, its advantage is that it makes clear the true quantum-mechanical geometry of problems, and often provides some insight into nonperturbative behavior that arises from this geometry. That will be our main goal in the following.
The basic ingredient is the representation of Z defined above, using the resolution of the identity in terms of coherent states:
Z[j] = lim Ne→∞
∫
∏
τ,i
d Ωˆiτ
∏^ β τ =
〈 Ω(ˆ τ )| Ω(ˆ τ − )〉[1 − H˜(τ )]. (19)
Here the so-called classical Hamiltonian H˜(τ ) is defined as the expectation value of H(τ ) in the coherent-state basis:
H˜(τ ) = 〈^ Ω(ˆ τ )|H(τ )| Ω(ˆτ − )〉 〈 Ω(ˆ τ )| Ω(ˆ τ − )〉
Also, we are now writing | Ωˆ〉 for a coherent state of many spins, which is made just by taking the product of many single-spin coherent states | Ωˆi〉 for the spin at site i.
Looking back at our representations for coherent states, we can write
〈 Ω(ˆ τ )| Ω(ˆτ − )〉 = exp
( −iS
∑
i
φ^ ˙i cos[θi(τ )] + ˙χi
)
. (21)
Here χ is an arbitrary phase convention which we set equal to a constant.
Finally, assuming that it is legitimate to pass to the continuum limit, we obtain the action
S[ Ω] =ˆ −iS
∑
i
ω[ Ωˆi] +
∫ (^) β
0
dτ H[ Ω(ˆ τ )] (22)
Here the geometric phase, which will be our main subject next time, for a single spin is
ω[ Ω] =ˆ −
∫ (^) β
0
dτ φ˙ cos θ. (23)
