Random Hamiltonian in thermal equilibrium

0.36 MB·

by  Dorje C. Brody

Random Hamiltonian in thermal equilibrium
9 Dorje C Brody, David C P Ellis, and Darryl D Holm
0
Department of Mathematics, Imperial College London, London SW7 2AZ, UK
0
2
Abstract. A framework for the investigation of disordered quantum systems in thermal
n
equilibrium is proposed. The approach is based on a dynamical model—which consists of
a
J a combination of a double-bracket gradient flow and a uniform Brownian fluctuation—that
‘equilibrates’ the Hamiltonian into a canonical distribution. The resulting equilibrium state is
4
1 used to calculate quenchedand annealed averages of quantumobservables.

]
h
p
-
t 1. Introduction
n
In the conventional treatment of quantum statistical mechanics there is a natural division
a
u between (a) the system under study, which is treated quantum mechanically and whose states
q
are subject to thermal fluctuations, and (b) the Hamiltonian of the system, which is treated
[
essentially classically and is held fixed. For some quantum systems, however, the Hamiltonian

1 itself may fluctuate for one reason or another. Questions that interest us in this connection, in
v
particular, are: “How can a randomly fluctuating Hamiltonian approach its equilibrium state?”
5
2 and “What is the form of the equilibrium distribution, and how do we calculate observable
0 expectation values in equilibrium?” The former is a question of a dynamical nature, whereas
2
the latter is a question of a static nature. The purpose of the present paper is to propose an
.
1 approach to address these questions. Specifically, we shall derive a dynamical model having
0
the property that a given initial Hamiltonian evolves randomly—but isospectrally—in such a
9
0 way that the associated density function on the space of isospectral Hamiltonians approaches a
: steady state distribution given by the canonical ensemble. Furthermore, we apply the resulting
v
i equilibrium state to calculate thermal expectation values of other observables, leading to new
X
physical predictions in some limiting (quenched and annealed) cases.
r
a
Figure 1. Spin in fluctuating magnetic
B field. The space of pure states of a spin-
? 1 particle, in external magnetic field B,
2
is the surface of the Bloch sphere. In
statistical theory of quantum mechanics
the state is represented by statistical
distributions of the pure states, whereas
the magnetic field B that specifies the
Hamiltonian is held fixed. What happens
if the direction of the field B is itself
subject to a small fluctuation?
2. Approach to equilibrium
In classical statistical mechanics the notion of a gradient flow plays an important role in
describing the approach to equilibrium: A system immersed in a heat bath naturally tends
to release its energy into the environment and thus approach its minimum energy state, and
this tendency is characterised by a Hamiltonian gradient flow. An equilibrium state is attained
when this flow is on average counterbalanced by thermal noise due to a random interaction
with the bath. Here the magnitude of the noise is determined by the temperature of the bath.
Accordingly, the idea we are going to introduce here is a gradient flow equation on the space of
Hamiltonians with the property that the eigenstates of an arbitrary initial Hamiltonian H at
0
time t = 0 tend toward alignment with those of a reference Hamiltonian, denoted by G. Thus,
G plays the role of the ‘fixed’ Hamiltonian in conventional quantum statistical mechanics. The
eigenstates of H thus evolve toward those of G under the flow. By introducing a suitable noise
t
term, we are able to characterise the approach to an equilibrium distribution.
The dynamical model for characterising approach to equilibrium is given by
dH
t
= λ[H ,[H ,G]]+[H ,ω ], (1)
t t t t
dt −
where λ R . Here ω denotes a skew-symmetric matrix of independent white noise terms,
+ t
∈ { }
and [H ,ω ] is the Lie bracket of these with H (see also [1]). Hence [H ,ω ] is symmetric and
t t t t t
linear in both H and ω . The term [H ,[H ,G]] gives rise to the aforementioned gradient flow
t t t t
in the space of Hamiltonians. The Hermitian matrix G plays the role of the ‘Hamiltonian of
the Hamiltonians’ in the sense that G determines the motion in the space of Hamiltonians. In
particular, we can regard the linear function tr(HG) on the space of the totality of Hermitian
matrices H as representing the ‘energy’ function defined on that space.
Thereis aninvariant measure(stationary solution)associated withtheevolutionary equation
(1). This is given by the canonical density:
ρ(H) exp( λtr(HG)), (2)
∝ −
which can be used as a new basis for studying quantum statistical mechanics. In units ~ = 1
the coupling λ has dimension [Energy]−1, and has the interpretation of representing the inverse
temperature for the ‘Hamiltonian bath’ (not to be confused with the thermal bath in which
the system, and possibly also the apparatus determining the Hamiltonian, is immersed). The
canonical density (2) can also bederived by entropy maximisation subjectto the constraint that
the energy tr(HG) has a definite expectation value.
3. The gradient flow: double bracket equation
Let us begin by examining properties of the gradient term in (1). Specifically, consider the
following dynamical equation for Hermitian matrices:
dH
t
= λ[H ,[H ,G]]. (3)
t t
dt −
Note that in terms of the Hermitian matrix X = i[H,G] the double-bracket evolution (3) can
be rewritten as
dH
= iλ[H,X], (4)
dt
which formally is just the Heisenberg equation of motion. However, owing to the H-dependence
of X theevolution is nonunitary. TheHamiltonians H andG are bothassumed nondegenerate.
0
Theflow induced by (3)satisfies thefollowing properties: (i) the evolution is isospectral, i.e. the
eigenvalues of H are preserved; and (ii) the evolution gives the ‘alignment’ lim [H ,G] = 0.
0 t→∞ t
We remark that the double bracket flow was first introduced in the context of magnetism
(Landau-Lifshitz equation) [2]. In its modern form it was introduced by Brockett [3] and has
been successfully applied to many areas, such as optimal control, linear programming, sorting
algorithms, and dissipative systems (see references cited in [4]).
In the case of a 2 2 Hamiltonian the gradient flow equation (3) can be solved
×
straightforwardly. In this case we express the Hamiltonian in terms of the Pauli matrices:
H = 1u 1+ 1νσ n , (5)
t 2 t 2 · t
where n = (x ,y ,z ). Similarly for the reference Hamiltonian G we write
t t t t
G = 1v1+ 1 µσ g (6)
2 2 ·
for a unit vector g. A calculation shows that the solution to (3) reads
u νtanh(ωt c ) νsech(ωt c )e−iφ0
Ht = 12(cid:18) ν0s−ech(ωt c0)−eiφ00 u0+νtan−h(ω0t c0) (cid:19), (7)
− −
where ω = λνµ, and c = tanh−1(cosθ ) and φ are initial values [4]. Furthermore, the
0 0 0
eigenvalues of H are time-independent, and we have
t
u ν 0
tl→im∞Ht = 21 (cid:18) 00− u0+ν (cid:19). (8)
Thus, the Hamiltonian is asymptotically diagonalised in the G-basis. Observe that trH and
t
detH are conserved quantities. Therefore, the flow induced by (3) for fixed initial values u and
t 0
n is confined to a two-sphere , which is isomorphic to the state space of a two-level system.
0
| | L
It follows that in two dimensions we have the equivalence of the Schro¨dinger and Heisenberg
pictures, even though the dynamical equation is not unitary. Since u and n are constant, we
0 0
| |
fix these and focus our attention on parameterised by the dynamical coordinates (θ ,φ ).
t t
L
Figure 2. Gradient flow with unitary motion on the
sphere . The vector field generated by the unitarily-
L
modified gradient-flow (9) is plotted. The first term in (9)
generates a rotation around the G-axis (which is chosen
to be the z axis here), while the second term generates
geodesic flows toward the south pole. The axis n of the
0
initial Hamiltonian H spirals around the G-axis g and is
0
asymptotically aligned with the latter (south pole in this
example).
We remark that the dynamical equation (3) can be modified to include a unitary term:
dH
t
= i[H ,G] λ[H ,[H ,G]] (9)
t t t
dt − −
without greatly affecting its physical characteristics. In the 2 2 example, the only change
×
occurs in the phase so that instead of φ = φ we have φ = φ +µt.
t 0 t 0
4. Elements of stochastic differential geometry
We now wish to introduce a Brownian term into the deterministic flow (3). Specifically, we
considerauniformBrownian fieldontheisospectralsubspaceof thespaceof Hermitian matrices
(the sphere in the 2 2 case). Before we proceed, however, it will be useful to recall how
L ×
stochastic motions can be defined on a manifold. The basic process we consider is the Wiener
process W defined on a filtered probability space (Ω, ,P). Here Ω is the sample space,
t
is a σ-fie{ld o}n Ω, and P is the probability measure. TheFfiltration of determines the causFal
structure of (Ω, ,P). This is given by a parameterised family F of nested σ-subfields
t 0≤t<∞
F {F }
satisfying for any s t < . We say that W is a Wiener process if: (a)
s t t
F ⊂ F ⊂ F ≤ ∞ { }
W = 0; and (b) W is Gaussian such that W W has mean zero and variance h (see
0 t t+h t
{ } − | |
[5]). A process σ is said to be adapted to the filtration generated by W if its random
t t t
{ } {F } { }
value at time t is determined by the history of W up to that time.
t
{ } t
If σ is -adapted, thenthestochastic integralM = σ dW exists, providedthat σ is
{ t} Ft t 0 s s { t}
almostsurelysquare-integrable. Ifthevarianceof Mt exisRts,then Mt satisfiesthemartingale
conditions E[M ] < and E[M ] = M , wh{ere }E[ ] denotes{expe}ctation with respect to
t t s s
the measure P|. T|he la∞tter conditio|nFimplies that given −the history of the Wiener process up to
time s the expectation of M for t s is given by its value at s.
t
≥
A general Ito process is defined by an integral of the form
t t
x = x + µ ds+ σ dW , (10)
t 0 s s s
Z Z
0 0
where µ and σ arecalledthedriftandthevolatility of x . Aconvenientwayofexpressing
t t t
{ } { } { }
(10) is to write dx = µ dt+σ dW , and to regard the initial condition x as implicit. In the
t t t t 0
special case µ = µ(x ) and σ = σ(x ), where µ(x) and σ(x) are prescribed functions, the
t t t t
process x is said to be a diffusion.
t
Thisanalysis can begeneralised tothecase ofadiffusion x taking valuesonamanifoldM,
t
{ }
driven by an m-dimensional Wiener process Wi . Let be a torsion-free connection
t i=1,...,m a
on M such that for any vector field ξa its cov{arian}t derivative in∇local coordinates is
a
∂ξ
δbδa( ξa) = +Γa ξc, (11)
b a ∇b ∂xb bc
a
where δ is the standard coordinate basis in a given coordinate patch. Suppose we have an Ito
a
process taking values in M. Let xa denote the coordinates of the process in a particular patch.
t
Then writing hab = σaσbi we define the drift process µa by
i
µadt= dxa+ 1Γa hbcdt σadWi. (12)
2 bc − i t
Alternatively, we can write the covariant Ito differential as dxa = δaa(dxa+ 12Γabchbcdt), where
δaa is the dual coordinate basis. Then (12) can be represented as
dxa = µadt+σadWi. (13)
i t
If µa(x) and σa(x) are m + 1 vector fields on M, then the general diffusion process on M is
i
governed by a stochastic differential equation dxa = µa(x)dt + σa(x)dWi, where dxa is the
i t
covariant Ito differential associated with the given connection (see Hughston [6]).
For the characterisation of the diffusion process it suffices to specify a connection on M, and
a metric is not required. The quadratic relation dxadxb = habdt, where hab = σaσbi, follows
i
fromtheItoidentities dt2 = 0, dtdWi = 0, anddWidWj = δijdt. Thenforany smoothfunction
t t t
φ(x) on M we define the associated process φ = φ(x ), and Ito’s formula takes the form
t t
dφ = ( φ)dxa+ 1( φ)dxadxb
t ∇a 2 ∇a∇b
= µa φ+ 1hab φ dt+σa φdWi. (14)
∇a 2 ∇a∇b i∇a t
(cid:16) (cid:17)
The probability law for x is characterised by a density function ρ(x,t) on M that satisfies the
t
Fokker-Planck equation
∂ρ
= (µaρ)+ 1 (habρ). (15)
∂t −∇a 2∇a∇b
Thediffusionis said tobenondegenerateif hab isof maximalrank. If g is aRiemannianmetric
ab
on M and is the associated Levi-Civita connection, then if hab = σ2gab, the process x is a
a t
Brownian m∇otion with drift on M, with volatility parameter σ.
5. Diffusion model for thermalisation
Consider a stochastic differential equation of the form
dxa = µadt+κσadWi (16)
i t
on a real manifold M. Here κ is a constant, the drift µa is a vector field on M, and the vectors
σa constitute an orthonormal basis in the tangent space of M. In this case the associated
{ i}
Fokker-Planck equation reads
∂
ρ (x) = (µaρ )+ 1κ2 2ρ . (17)
∂t t −∇a t 2 ∇ t
For our model we requirethat the driftvector µa represent the double-bracket gradient flow (3).
This is achieved by choosing
µa = 1κ2λ aG, (18)
−2 ∇
where G(x) is a function on M given by tr(HG). Then it follows that there exists a unique
stationary solution to (17), given by the canonical density
exp( λG(x))
ρ(x) = − . (19)
exp( λG(x))dV
M
−
R
If M is the space of pure states, then we have a model for thermalisation of quantum states
introduced by Brody & Hughston [7].
6. Two-dimensional case in more detail
To illustrate these results in more explicit terms we consider a system consisting of a single
spin-1 particle immersed in an external magnetic field. The Hamiltonian is then H = B S,
2 − ·
whereBdenotes thefieldandSthespinvector. Thedirection of thefieldB, however, issubject
to fluctuations around its stable direction, specified by G (directed along the z-axis). For the
dynamical equation we obtain [4]:
dθ = ω sinθ dt+√2ν(dW1 +dW2)
t t t t (20)
(cid:26) dφt = −sin1θt√2ν(dWt1−dWt2).
The associated Fokker-Planck equation reads
ρ˙ = ω(cosθ+sinθ∂ )ρ+2ν ∂2+ 1 ∂2 ρ, (21)
− θ θ sin2θ φ
(cid:16) (cid:17)
where ∂ = ∂/∂θ and ∂ = ∂/∂φ. The asymptotic solution is the canonical density function:
θ φ
λµ
ρ(θ,φ)= exp 1λµcosθ . (22)
2πsinh(1λµ) −2
2 (cid:0) (cid:1)
Direct substitution shows that (22) is the stationary solution to (21).
Itfollows from (22) and the useof the volume element dV = 1 sinθdθdφthat the equilibrium
4
mean Hamiltonian is
u +ν cosθ 0
H = 1 0 h iλ , (23)
h i 2 (cid:18) 0 u0 ν cosθ λ, (cid:19)
− h i
where
2 1
cosθ = . (24)
h iλ λµ − tanh(1λµ)
2
We may regard the parameter λ as representing the ‘inverse temperature’ for the Hamiltonian.
If the noise level is high (λ 1), then the direction of the external field B on the average lies
≪
close to the xy-plane so that cosθ 0. If the noise level is low λ 1, then the field B on
λ
h i ≃ ≫
the average is parallel to the z-axis and we have cosθ 1. We plot cosθ as a function
λ λ
h i ≃ − h i
of τ = 1/λ (see Fig. 3).
Figure 3. The plot of the expectation
cosθ as a function of the inverse
λ
h i
Hamiltonian temperature τ = 1/λ. For
τ 1 we have cosθ 1, whereas
λ
≪ h i ≃ −
for τ 1 we find cosθ 0.
λ
≫ h i ≃
7. Quantum statistical mechanics of disordered systems
Now we consider how the statistical theory of Hamiltonians presented above can be applied to
quantum statistical mechanics, when the system and the apparatus specifying the Hamiltonian
are both immersed in a heat bath with inverse temperature β. In this context it is natural to
borrow ideas from the spin glass literature. We may take the averaged Hamiltonian H as the
λ
h i
starting point of the analysis—this gives the analogue of an annealed average:
tr Oe−βhHiλ
O = (25)
h iA tr(cid:0) e−βhHiλ (cid:1)
(cid:0) (cid:1)
of an observable O. Such an averaging, however, will change the eigenvalues of H.
Alternatively, we may use the ‘unaveraged’ Hamiltonian to compute the expectation of an
observable O, and then take its average:
tr(Oe−βH)
O = . (26)
h iQ (cid:28) tr(e−βH) (cid:29)
λ
This gives the analogue of a quenched average. The canonical quenched average of the
Hamiltonian G = σ is
z
G = 1µtanh 1βν 1 2 , (27)
h iQ 2 2 (cid:18)tanh(1λµ) − λµ(cid:19)
2
(cid:0) (cid:1)
whereas the canonical annealed average of G is
G = 1µtanh 1βν 1 2 . (28)
h iA 2 (cid:20)2 (cid:18)tanh(1λµ) − λµ(cid:19)(cid:21)
2
These results suggest a new line of studies on the extended quantum statistical mechanics of
disordered systems. The plot below shows the annealed (blue) and quenched (red) averages of
σ , as a function of the bath temperature T = 1/β, for fixed λ such that λ−1 = 0.1.
z
Figure 4. Quenched and annealed av-
erages of G. The functions G and
Q
h i
G are plotted against the tempera-
A
h i
ture T = β−1, where we set λ−1 = 0.1,
v = 0, ν = 1, and µ = 2 so that G =
σ . The ‘quenched magnetisation’ σ
z z Q
h i
does not attain the maximum value 1.0
at zero temperature unless λ−1 = 0.
8. Examination of higher-dimensional cases
The geometry of higher-dimensional Hermitian matrices is somewhat more intricate than the
two-dimensionalcaseexaminedabove. ThespaceofN N Hermitian matrices hasthestructure
of the product RN CPN CPN−1 CP1, wher×e CPk denotes the complex projective k-
space. This space is×consid×erably lar×ger··t·h×an the space CPN of pure states upon which N N
×
Hermitian matrices act, andas aconsequencetheequivalence of theSchro¨dinger andHeisenberg
pictures for a nonunitary motion would in general be lost.
A 3 3 Hermitian matrix H can be written as H = E E E +E E E +E E E ,
1 1 1 2 2 2 3 3 3
× | ih | | ih | | ih |
where E aretheeigenvalues and E aretheassociated eigenstates. Thedegreesoffreedom
i i
for the{ene}rgy eigenvalues correspo{n|d tio}the open space R3; the remaining degrees of freedom
corresponding to CP2 CP1 are encoded in the specification of the energy eigenstates. Writing
×
g for the eigenstates of G, we can express E in the form:
i 1
{| i} | i
E = sin 1ϑcos 1ϕg +sin 1ϑsin 1ϕeiξ g +cos 1ϑeiη g , (29)
| 1i 2 2 | 1i 2 2 | 2i 2 | 3i
which determines a point in CP2. There is a CP1 worth degrees of freedom left for E :
2
| i
E = cos 1αsin 1ϑcos 1ϕ sin 1αsin 1ϕeiβ g
| 2i 2 2 2 − 2 2 | 1i
(cid:16) (cid:17)
+ cos 1αcos 1ϑsin1ϕeiξ +sin 1αcos 1ϕei(β+ξ) g +cos 1αsin 1ϑeiη g . (30)
2 2 2 2 2 | 2i 2 2 | 3i
(cid:16) (cid:17)
The specifications of E and E leave no further freedom left for E and we have
1 2 3
| i | i | i
E = sin 1αcos 1ϑcos 1ϕ+cos 1αsin 1ϕeiβ g
| 3i 2 2 2 2 2 | 1i
(cid:16) (cid:17)
+ sin 1αcos 1ϑsin1ϕeiξ cos 1αcos 1ϕei(β+ξ) g +sin1αsin 1ϑeiη g . (31)
2 2 2 − 2 2 | 2i 2 2 | 3i
(cid:16) (cid:17)
In this manner we obtain the nine parameters required for the specification of an arbitrary 3 3
×
Hermitian matrix.
It should be remarked that the foregoing procedure is merely an example of how one might
parameteriseagenericN N Hermitianmatrixinasystematic manner;theschemeissomewhat
×
unconventional in that a generic 2 2 Hermitian matrix in this parameterisation is written
×
1 0 cosθ sinθe−iφ
H = ω +ω , (32)
+(cid:18) 0 1 (cid:19) −(cid:18) sinθeiφ cosθ (cid:19)
−
where ω = 1(E E ). This set of coordinates is nevertheless convenient because it isolates
± 2 1 ± 2
invariant quantities ω from the coordinates (θ,φ) of the isospectral submanifolds.
±
For our application in quantum statistical mechanics of disordered systems we are required
to determine the partition function
Z(λ) = e−λtr(HG)dV. (33)
Z
In the case of a 3 3 Hamiltonian, since the dynamical equation (1) preserves the three
×
eigenvalues of H, the dynamical motion stays on the isospectral submanifold spanned by the six
angular variables ϑ,ϕ,α [0,π] and ξ,η,β [0,2π], with volume element
∈ ∈
dV = 1 sinαsinϑ(1 cosϑ)sinϕdαdβdϑdϕdξdη. (34)
128 −
SinceweareworkingintheG-basis,thecalculationoftr(HG)isstraightforward,andtheintegral
(33) reduces to an expression analogous to the integral representation for a Bessel function.
The choice of parameterisation adapted here for a generic N N Hermitian matrix need not
×
be the most adequate for our purpose. An alternative approach is to write H = U†EU, where
E is a diagonal matrix with eigenvalues E , and U is a unitary matrix. Then the calculation
i
{ }
of the partition function becomes semi-Gaussian, and we are left with an integration over the
invariant Haar measure [8]. Ideas from matrix theory or random matrices might prove useful in
the statistical analysis of disordered quantum systems introduced here.
Acknowledgments
TheauthorsthankR.Brockett andE.J.Brodyforcommentsandstimulatingdiscussions. DDH
thanks the Royal Society of London for partial support by its Wolfson Merit Award.
[1] Brockett, R. W. “Notes on stochastic processes on manifolds” In Systems and Control in the Twenty-First
Century (C. Byrnes, et al., eds.) pp.75-101, (Boston: Birkh¨auser, 1997).
[2] Landau,L.D.andLifshitz,E.M.“Onthetheoryofthedispersionofmagneticpermeabilityinferromagnetic
bodies” Phys. Z. Sowietunion 8 153-169 (1935).
[3] Brockett, R. W. “Dynamical systems that sort lists, diagonalise matrices, and solve linear programming
problems” Lin. Alg. Appl. 146 79-91 (1991).
[4] Brody,D.C.,Ellis,D.C.P.,andHolm,D.D.“Hamiltonianstatisticalmechanics”J.Phys.A41502002(2008).
[5] Hida,T. Brownian Motion. (Berlin, Germany: Springer, 1980).
[6] Hughston, L. P. “Geometry of stochastic state vector reduction” Proc. Roy. Soc. London A452 953-979
(1996).
[7] Brody,D. C. and Hughston,L. P. “Thermalisation of quantum states” J. Math. Phys. 40 12-18 (1999).
[8] Sugiura, M. Unitary Representations and Hamonic Analysis (Amsterdam: North-Holland,1990).