Quantum Condensed Matter Field Theory course of the University of Cambridge (Major topic (2015–16))

Let

[a, a^{†}] = 1

n = a^{†} a

, and take

S^{+} = (S^{-})^{†} = 2 S - n a

. The physical Holstein–Primakoff occupation constraint limits the Hilbert space, which contains only

n = 0, 1, \dots, 2 S

. Acting in this number basis gives

S^{-} ∣ n ⟩ = (n + 1) (2 S - n) ∣ n + 1 ⟩, S^{+} ∣ n ⟩ = n (2 S - n + 1) ∣ n - 1 ⟩ .

The upper and lower endpoints are annihilated as required. Consequently

[S^{+}, S^{-}] ∣ n ⟩ = {(n + 1) (2 S - n) - n (2 S - n + 1)} ∣ n ⟩ = 2 (S - n) ∣ n ⟩ = 2 S^{z} ∣ n ⟩ .

The change in

n

also gives

[S^{z}, S^{\pm}] = \pm S^{\pm}

. These matrix elements prove the spin algebra on the full physical spin-

S

space. Extending the oscillator space to arbitrary occupation would not give a physical finite spin: the occupation constraint and operator ordering in the square root matter. bosons on distinct sites have

[a_{m}, a_{n}^{†}] = δ_{mn}

and all other site commutators zero.

c

Solution

Rotate the spin on one sublattice through

π

about its

x

axis, using the unitary product of the corresponding spin-rotation operators. On a rotated site the old components obey

S_{old}^{x} = S^{x}

S_{old}^{y} = - S^{y}

S_{old}^{z} = - S^{z}

. Every bond joins opposite sublattices, so

S_{m, old}^{x} S_{m + 1, old}^{x} + S_{m, old}^{y} S_{m + 1, old}^{y} = S_{m}^{x} S_{m + 1}^{x} - S_{m}^{y} S_{m + 1}^{y} = \frac{1}{2} (S_{m}^{+} S_{m + 1}^{+} + S_{m}^{-} S_{m + 1}^{-}) .

The longitudinal product changes sign. Hence the rotated bond is $- J Δ S_{m}^{z} S_{m + 1}^{z} + \frac{J}{2} (S_{m}^{+} S_{m + 1}^{+} + S_{m}^{-} S_{m + 1}^{-})$ , giving the requested form. The canonical spin algebra is unchanged because the transformation is unitary. A staggered easy-axis reference configuration is now the uniform all-up state. At

α = 0

one first chooses a Néel direction, here

z

, before performing the same rotation.

d

Solution

For occupations small compared with

S

, use

S^{z} = S - n

S^{-} ≃ 2 S a^{†}

and

S^{+} ≃ 2 S a

. The longitudinal bond contributes

- J Δ S^{2} + J S Δ (n_{m} + n_{m + 1}) + O (S^{0})

; the transverse pair terms contribute

J S (a_{m} a_{m + 1} + a_{m}^{†} a_{m + 1}^{†}) + O (S^{0})

. Thus the direct real-space expansion is

H_{2} = - J Δ N S^{2} + J S \sum_{m} {2Δ n_{m} + a_{m} a_{m + 1} + a_{m}^{†} a_{m + 1}^{†}} .

Take

a_{m} = N^{- 1/2} \sum_{k} e^{ikm} a_{k}

, with

k = 2 πj / N

over one full Brillouin zone and the lattice spacing set to one in

k

. Symmetrizing

k

and

- k

changes the phase in the pair sum to

cos k

, giving

H_{2} = - J Δ N S^{2} + J S \sum_{k} {2Δ a_{k}^{†} a_{k} + cos k (a_{k} a_{- k} + a_{k}^{†} a_{- k}^{†})} .

normal ordering the second Nambu spinor component then yields

H_{2} = - J Δ NS (S + 1) + J S k \sum (a_{k}^{†} a_{- k}) (Δ cos k cos k Δ) (a _{- k}^{†} a _{k}) .

The printed matrix has a factor-of-two error on its diagonal. With a full-zone sum its printed

2Δ

would produce

4Δ a_{k}^{†} a_{k}

, and even its vacuum expectation would differ from the direct expansion by

J Δ NS

. A half-zone convention cannot repair both coefficients: one must then group each

k, - k

pair and double its off-diagonal pair coefficient as well. The formula above follows from the specified Hamiltonian and has the expected gapless isotropic limit; subsequent physical results use this consistent expansion. Terms quartic in bosons first enter at order

S^{0}

e

Solution

Write

u_{k} = cosh θ_{k}

v_{k} = sinh θ_{k}

, taking real even

θ_{k}

. The transformation gives

[b_{k}, b_{q}^{†}] = (u_{k} u_{q} - v_{k} v_{q}) δ_{k q} = δ_{k q}, [b_{k}, b_{q}] = (u_{k} v_{q} - v_{k} u_{q}) δ_{k, - q} = 0.

The second equality uses

u_{- k} = u_{k}

and

v_{- k} = v_{k}

, while the first uses

u_{k}^{2} - v_{k}^{2} = 1

. The creation-creation commutator vanishes by conjugation. Therefore the $b$ modes are canonical bosons. In the paired basis,

(b _{- k}^{†} b _{k}) = (u_{k} v_{k} v_{k} u_{k}) (a _{- k}^{†} a _{k}) .

Its determinant is one and its inverse reverses the signs of the off-diagonal entries. Hence

a_{k} = u_{k} b_{k} - v_{k} b_{- k}^{†}

. At self-inverse momenta

k = 0, π

this is the corresponding one-mode squeezing transformation, still canonical for finite

θ_{k}

f

Solution

For the easy-axis XXZ antiferromagnetic spin-wave spectrum, the consistent quadratic Hamiltonian in part (d) has

A = Δ

and

B_{k} = cos k

. The inverse transformation acts on the Nambu matrix by a congruence. Its diagonal and off-diagonal entries become

A cosh 2 θ_{k} - B_{k} sinh 2 θ_{k}

and

B_{k} cosh 2 θ_{k} - A sinh 2 θ_{k}

. The pair term vanishes when

tanh 2 θ_{k} = \frac{c o s k}{Δ}, Ω_{k} = Δ^{2} - cos^{2} k .

For

α > 0

the transformation is real and finite for every

k

. Summing the two Nambu number operators over the full zone gives

H = E_{0} + k \sum ϵ_{k} (b_{k}^{†} b_{k} + \frac{1}{2}) + O (S^{0}), E_{0} = - J Δ NS (S + 1), ϵ_{k} = 2 J S Δ^{2} - cos^{2} k .

Here

E_{0}

names the constant in the requested oscillator form; the actual quadratic ground-state energy is

E_{gs}^{(2)} = E_{0} + \frac{1}{2} \sum_{k} ϵ_{k} = - J Δ N S^{2} + J S \sum_{k} (Ω_{k} - Δ)

. This distinguishes the normal-ordering constant from the total zero-point contribution. bosonic Bogoliubov diagonalization is a canonical, not an ordinary unitary, diagonalization of the Nambu matrix.

α = 0

and

k \neq = 0, π

, the same formula holds with

Ω_{k} = ∣ sin k ∣

. The exact zero modes have

∣ tanh 2 θ_{k} ∣ = 1

, so no finite squeezing angle diagonalizes them as positive-frequency oscillators. Treat collective rotations separately or retain a small anisotropy regulator before taking the limit. Finally, if one instead diagonalizes the literal erroneous printed matrix, it gives

tanh 2 θ_{k} = cos k / (2Δ)

and

ϵ_{k}^{print} = 2 J S 4 Δ^{2} - cos^{2} k

. That result remains gapped at

α = 0

and is not the spin-wave spectrum of the Hamiltonian stated at the start.

g

Solution

For

α > 0

, the dispersion is periodic and has minima at

k = 0, π

in the rotated full-zone convention, with

ϵ_{m i n} = 2 J S (1 + α)^{2} - 1 = 2 J S α (2 + α) .

The maximum is

2 J S (1 + α)

k = \pm π /2

. The easy-axis state has no continuously broken symmetry, so a gap is allowed. Near a minimum the energy behaves as

2 J S α (2 + α) + (δ k)^{2}

. At

α = 0

it becomes $ϵ_{k} = 2 J S ∣ sin k ∣$ , linear near its zeros, with small-

k

slope

2 J S

in the chosen units. Restoring lattice spacing

a

and

ℏ

gives spin-wave speed

2 J S a /ℏ

. Zone folding reorganizes the same modes into the magnetic Brillouin zone.

Figure 1.
Consistent easy-axis, isotropic and easy-plane linear spin-wave spectra; illustrative anisotropies
. **Details**.

These are linear spin-wave predictions, not a statement that the exact one-dimensional quantum chain always has long-range Néel order. At the isotropic point the zero-point spin reduction contains

N^{- 1} \sum_{k} (∣ sin k ∣^{- 1} - 1) /2

, which diverges logarithmically as the infrared cutoff vanishes. This failure of the assumed ordered reference must not be mistaken for a calculation of the exact interacting spectrum.

h

Solution

In this part

0 < Δ = 1 + α < 1

, so the ordered reference lies in the

x y

plane. Choosing one direction breaks the continuous rotation symmetry within that plane. A gapless, linearly dispersing phase mode is therefore expected in the spin-wave approximation, while tilting the Néel axis out of the plane costs anisotropy energy. It is incorrect to continue the easy-axis expansion about an unstable

z

-ordered reference and interpret its imaginary frequencies as physical modes.

For the easy-plane XXZ antiferromagnetic spin-wave spectrum, take the Néel direction along old

x

, then rotate alternating local spins so that all longitudinal components point up. One such local frame gives the full-zone quadratic coefficients

A_{k} = 1 + \frac{Δ - 1}{2} cos k, B_{k} = \frac{Δ + 1}{2} cos k,

and therefore

ϵ_{k} = 2 J S (1 - cos k) (1 + Δ cos k) .

Near

k = 0

ϵ_{k} ≃ J S 2 (1 + Δ) ∣ k ∣

. The zone-edge mode has gap

ϵ_{π} = 2 J S 2 (1 - Δ)

. A different sublattice phase convention shifts the zero by

π

; folding into the magnetic zone displays an acoustic and a gapped polarization. The figure uses this explicitly chosen frame. One-dimensional long-wavelength fluctuations again limit the ordered-reference interpretation.

2

a

Solution

Take a uniform nearest-neighbour harmonic chain with displacement

u_{j}

, conjugate momentum

p_{j}

, mass

m_{0}

, spring stiffness

κ

, lattice spacing

a

and periodic length

ℓ = N a

. These specify a standard model for the requested notes, not additional numerical exam data. Its Lagrangian and Hamiltonian are

L = \sum_{j} [\frac{m _{0}}{2} \overset{u}{˙}_{j}^{2} - \frac{κ}{2} (u_{j + 1} - u_{j})^{2}], H = \sum_{j} [\frac{p _{j}^{2}}{2 m _{0}} + \frac{κ}{2} (u_{j + 1} - u_{j})^{2}] .

Variation gives

m_{0} \overset{u}{¨}_{j} = κ (u_{j + 1} - 2 u_{j} + u_{j - 1})

. For a mode

e^{i (kja - ω t)}

, periodicity requires

k = 2 πn / ℓ

and

ω_{k} = 2 κ / m_{0} ∣ sin (ka /2) ∣.

The lattice makes

k

periodic modulo

2 π / a

, so choose one Brillouin zone. At long wavelength

ω_{k} ≃ v ∣ k ∣

, where

v = a κ / m_{0}

is the sound speed. The continuum limit of a harmonic chain expands

u_{j + 1} - u_{j} ≃ a \partial_{x} u

and replaces

\sum_{j}

a^{- 1} \int d x

. Defining mass density

ρ = m_{0} / a

and one-dimensional elastic coefficient

K = κa

gives

L = \int_{0}^{ℓ} d x [\frac{ρ}{2} (\partial_{t} u)^{2} - \frac{K}{2} (\partial_{x} u)^{2}], ρ \partial_{t}^{2} u - K \partial_{x}^{2} u = 0, v^{2} = K / ρ .

The momentum density is

Π = ρ \partial_{t} u

, and the Hamiltonian density is

Π^{2} / (2 ρ) + K (\partial_{x} u)^{2} /2

. Translational invariance of

u

makes total momentum conserved. The continuum approximation describes acoustic wavelengths large compared with

a

; it is not a justification for extending the lattice dispersion and zero-point sums to arbitrarily large wavevector.^[ref]

canonical quantization promotes the coordinates and momenta to operators with

[u_{j}, p_{l}] = i ℏ δ_{j l}

and the other equal-time commutators zero. In the continuum,

[u (x), Π (x^{'})] = i ℏ δ_{per} (x - x^{'})

. The nonzero normal modes can be represented by independent phonon operators obeying

[b_{k}, b_{q}^{†}] = δ_{k q}

. A useful lattice expansion is

u_{j} = X + \frac{1}{N} \sum_{k \neq = 0} \frac{ℏ}{2 m _{0} ω _{k}} (b_{k} e^{ikja} + b_{k}^{†} e^{- ikja}),

where

X = N^{- 1} \sum_{j} u_{j}

is the centre-of-mass displacement. The corresponding momentum is

p_{j} = \frac{P}{N} - \frac{i}{N} \sum_{k \neq = 0} \frac{m _{0} ℏ ω _{k}}{2} (b_{k} e^{ikja} - b_{k}^{†} e^{- ikja}), [X, P] = i ℏ.

Including the centre-of-mass term restores the full site commutator: the nonzero-mode sum alone gives

i ℏ (δ_{j l} - 1/ N)

. Substitution diagonalizes the Hamiltonian,

H = \frac{P ^{2}}{2 N m _{0}} + k \neq = 0 \sum ℏ ω_{k} (b_{k}^{†} b_{k} + \frac{1}{2}) .

A phonon is one quantum of a collective normal mode, carrying energy

ℏ ω_{k}

and lattice momentum

ℏ k

modulo a reciprocal-lattice vector. Reality of

u_{j}

relates its Fourier coordinates by

u_{k}^{†} = u_{- k}

; it does not remove the independent oppositely travelling phonon modes. Using one complete Brillouin zone avoids double-counting real standing-wave coordinates.^[ref]

The free centre of mass has no restoring force and must not be treated as an oscillator of zero frequency. Fixing it, weakly pinning it, or specifying a finite centre-of-mass volume defines the corresponding zero-mode factor. With a fixed centre of mass, the internal thermal partition function and occupations are

Z_{int} = \prod_{k \neq = 0} \frac{1}{2 s i n h ( β ℏ ω _{k} /2 )}, ⟨ n_{k} ⟩ = \frac{1}{e ^{β ℏ ω_{k}} - 1} .

(7)

The internal energy is

\sum_{k \neq = 0} ℏ ω_{k} (⟨ n_{k} ⟩ + 1/2)

; its high-temperature heat capacity approaches

(N - 1) k_{B}

. For a large one-dimensional chain at temperatures below the lattice cutoff but above its finite-size mode spacing,

\frac{U _{T}}{ℓ} = \int \frac{d k}{2 π} \frac{ℏ v ∣ k ∣}{e ^{β ℏ v ∣ k ∣} - 1} = \frac{π ( k _{B} T ) ^{2}}{6ℏ v}, \frac{C}{ℓ} = \frac{π k _{B}^{2} T}{3ℏ v} .

(8)

This one-dimensional acoustic phonon heat capacity shows that quantization turns classical normal waves into bosonic excitations with a linear low-temperature heat capacity in one dimension. The Euclidean field action

S_{E} = \frac{1}{2} \int_{0}^{β ℏ} d τ d x [ρ (\partial_{τ} u)^{2} + K (\partial_{x} u)^{2}]

is Gaussian, with periodic imaginary-time boundary conditions; its Fourier modes reproduce these oscillators. Absolute displacement correlations require the zero-mode convention, and the lattice regulates short-distance fluctuations.

b

Solution

For one bosonic mode, define the unnormalized bosonic coherent state

∣ z ⟩ = e^{z a^{†}} ∣0 ⟩

. The commutator

[a, a^{†}] = 1

gives

a ∣ z ⟩ = z ∣ z ⟩

and

⟨ z^{'} ∣ z ⟩ = e^{\overset{z}{ˉ}^{'} z}

. The normalized state is

e^{- ∣ z ∣^{2} /2} ∣ z ⟩

; it is not orthogonal to a different coherent state. The coherent-state resolution of identity follows from Gaussian moments,

\int \frac{d ^{2} z}{π} e^{- ∣ z ∣^{2}} z^{n} \overset{z}{ˉ}^{m} = n! δ_{nm}, 1 = \int \frac{d ^{2} z}{π} e^{- ∣ z ∣^{2}} ∣ z ⟩ ⟨ z ∣.

For many modes take products of these states and measures. The complex amplitudes have the natural scale of square root occupation; coherent states turn normal-ordered matrix elements into classical symbols without replacing the quantum trace by a single classical configuration.

For fermions,

{c_{i}, c_{j}^{†}} = δ_{ij}

and a nilpotent annihilation operator cannot have an arbitrary nonzero ordinary complex eigenvalue. Introduce independent Grassmann variables

η_{i}, \overset{η}{ˉ}_{i}

, anticommuting among themselves and with the fermionic operators, and define

∣ η ⟩ = exp (- \sum_{i} η_{i} c_{i}^{†}) ∣0 ⟩

. Then

c_{i} ∣ η ⟩ = η_{i} ∣ η ⟩

and

⟨ \overset{η}{ˉ}^{'} ∣ η ⟩ = e^{\sum_{i} \overset{η}{ˉ}_{i}^{'} η_{i}}

. With an ordered Berezin integral measure satisfying

\int d \overset{η}{ˉ} d η \overset{η}{ˉ} η = - 1

for one pair,

1 = \int D (\overset{η}{ˉ}, η) e^{- \overset{η}{ˉ} η} ∣ η ⟩ ⟨ \overset{η}{ˉ} ∣, Tr O = \int D (\overset{η}{ˉ}, η) e^{- \overset{η}{ˉ} η} ⟨ - \overset{η}{ˉ} ∣ O ∣ η ⟩

for an even operator

O

; this is the fermionic coherent-state trace. In one mode the trace integrand is

e^{- \overset{η}{ˉ} η} (O_{00} - \overset{η}{ˉ} η O_{11})

, whose integral is

O_{00} + O_{11}

. Without the sign reversal it would be the supertrace, not the thermal trace. These formal Grassmann-valued coherent states do not represent ordinary complex-valued fermion amplitudes.

To construct the many-body thermal path integral, write

Z = Tr e^{- β K}

with

K = H - μ \hat{N}

and divide the imaginary-time interval

T = β ℏ

into slices of width

δ τ

. Insert the appropriate coherent-state resolution of identity between factors. For a normal-ordered Hamiltonian, the adjacent-slice kernel is

⟨ \overset{ˉ}{ψ}_{j + 1} ∣ e^{- δ τ K /ℏ} ∣ ψ_{j} ⟩ = exp [\overset{ˉ}{ψ}_{j + 1} ψ_{j} - \frac{δ τ}{ℏ} K_{N} (\overset{ˉ}{ψ}_{j + 1}, ψ_{j}) + O (δ τ^{2})] .

coherent-state time slicing makes the overlap and Gaussian measure combine into the first-order time derivative. The grand-canonical coherent-state action in the regulated continuum limit is

Z = \int D (\overset{ˉ}{ψ}, ψ) e^{- S_{E} /ℏ}, S_{E} = \int_{0}^{β ℏ} d τ [ℏ i \sum \overset{ˉ}{ψ}_{i} \partial_{τ} ψ_{i} + H_{N} (\overset{ˉ}{ψ}, ψ) - μ i \sum \overset{ˉ}{ψ}_{i} ψ_{i}] .

For bosons the integration variables are complex, with the conjugate labels on the original integration contour; for fermions the barred and unbarred Grassmann variables are independent. The continuum notation retains the time-sliced ordering prescription. For example,

n (n - 1) = a^{† 2} a^{2}

has normal symbol

\overset{z}{ˉ}^{2} z^{2}

, whereas

n^{2}

has that symbol plus

\overset{z}{ˉ} z

. Dropping that distinction changes a theory. The coherent-state thermal boundary conditions follow because the bosonic trace closes paths periodically and the fermionic sign in the trace closes them antiperiodically:

ψ (β ℏ) = ψ (0) (bosons), ψ (β ℏ) = - ψ (0) (fermions) .

The barred fields obey the same respective boundary conditions. Hence Matsubara frequencies are

2 πn / (β ℏ)

and

(2 n + 1) π / (β ℏ)

. The derivative term encodes the canonical structure; omitting it would remove the quantum dynamics.^[ref]

As a check, a free diagonal mode with

K = ξ a^{†} a

gives

Z_{B} = (1 - e^{- β ξ})^{- 1}

for

ξ > 0

, whereas

K = ξ c^{†} c

gives

Z_{F} = 1 + e^{- β ξ}

. Gaussian integration produces an inverse determinant for bosons and a determinant for independent complex fermions, with their respective temporal boundary conditions. Differentiating

lo g Z

recovers Bose-Einstein and Fermi-Dirac occupations. Interactions become higher powers in the normal-ordered symbols, and sources generate time-ordered correlations. The coherent-state construction preserves statistics, ordering and boundary conditions together; a formal continuum expression without its measure and temporal prescription is incomplete.

c

Solution

Use the contact-interaction Bose-gas field integral for a homogeneous three-dimensional weakly interacting Bose gas with repulsive short-range interaction as the standard model. Let particle mass be

m_{b}

, scattering length

a_{s} > 0

, physical coupling

g = 4 π ℏ^{2} a_{s} / m_{b}

and gas parameter

n a_{s}^{3} ≪ 1

. These are symbolic model definitions, not supplied numerical parameters. The grand-canonical coherent-state integral has periodic fields and action

S_{E} = \int_{0}^{β ℏ} d τ \int d^{3} x [\overset{ˉ}{ψ} (ℏ \partial_{τ} - \frac{ℏ ^{2} \nabla ^{2}}{2 m _{b}} - μ) ψ + \frac{g}{2} (\overset{ˉ}{ψ} ψ)^{2}] .

The contact term is the normal symbol of

g \hat{ψ}^{† 2} \hat{ψ}^{2} /2

. In three dimensions a contact interaction needs a regulator and coupling matched to the measured scattering length; the displayed

g

is the low-energy physical coupling. This field integral is useful because an ordered saddle and its quantum/thermal fluctuations follow from the same action.

Stationarity with respect to

\overset{ˉ}{ψ}

gives the imaginary-time Gross–Pitaevskii equation. A uniform saddle exists at

μ > 0

with

ψ_{0} = n_{0} e^{i θ_{0}}

and condensate density

n_{0} = μ / g

. Its arbitrary phase reflects global

U (1)

symmetry; a selected condensate fixes a phase in the thermodynamic symmetry-breaking description. Write

ψ = n_{0} + η

after choosing that phase. Linear terms vanish, and the quadratic part is

S_{2} = \int d τ d^{3} x [\overset{η}{ˉ} (ℏ \partial_{τ} - \frac{ℏ ^{2} \nabla ^{2}}{2 m _{b}} + μ) η + \frac{μ}{2} (η^{2} + \overset{η}{ˉ}^{2})] .

With

ε_{k} = ℏ^{2} k^{2} / (2 m_{b})

and the Fourier convention

e^{i k \cdot x - i ω_{n} τ}

, the paired quadratic kernel is

M (k, ω_{n}) = (- i ℏ ω_{n} + ε_{k} + μ μ μ i ℏ ω_{n} + ε_{k} + μ), det M = (ℏ ω_{n})^{2} + ε_{k} (ε_{k} + 2 μ) .

The one-half in the Nambu spinor action removes pair double counting. Analytic continuation of the inverse kernel gives the Bogoliubov dispersion

E_{k} = ε_{k} (ε_{k} + 2 g n_{0}), E_{k} ≃ ℏ c_{s} k, c_{s} = g n_{0} / m_{b} .

At high momentum it approaches the free-particle kinetic energy plus its mean-field shift. The crossover scale is the healing length

ξ = ℏ/ 2 m_{b} g n_{0}

. Gaussian integration adds a fluctuation determinant to the saddle free energy. Expressed using the physical three-dimensional coupling, its zero-temperature integrand can be regularized as

\frac{1}{2} [E_{k} - ε_{k} - μ + μ^{2} / (2 ε_{k})]

; the subtraction follows from renormalization of a three-dimensional dilute-gas contact coupling, not discarding an infinite answer arbitrarily. The thermal contribution is

k_{B} T \int d^{3} k / (2 π)^{3} lo g (1 - e^{- β E_{k}})

. A calculation beyond mean field must keep that matching and its thermodynamic derivatives consistent.

The density-phase action of a Bose gas makes the physical content clear,

ψ = n_{0} + δ n e^{i θ}

. Up to a periodic density total derivative, the long-wavelength quadratic action contains

S_{2} = \int d τ d^{3} x [i ℏ δ n \partial_{τ} θ + \frac{g}{2} (δ n)^{2} + \frac{ℏ ^{2} n _{0}}{2 m _{b}} (\nabla θ)^{2} + \frac{ℏ ^{2}}{8 m _{b} n _{0}} (\nabla δ n)^{2}] .

number-phase conjugacy pairs the density fluctuation and phase in the quadratic density-phase action. Dropping the density-gradient term only in the long-wave limit and completing the density Gaussian square gives the Bose-gas phase-only action

S_{θ} = \frac{1}{2} \int d τ d^{3} x [\frac{ℏ ^{2}}{g} (\partial_{τ} θ)^{2} + \frac{ℏ ^{2} n _{0}}{m _{b}} (\nabla θ)^{2}] .

This has the same sound pole. A term

i ℏ n_{0} \partial_{τ} θ

is also present before restricting to the smooth zero-winding sector; although locally a total derivative, it cannot in general be ignored in topologically nontrivial sectors. The density and phase variables describe the same coupled Bogoliubov branch, not a second independent massive particle simply because the density has a static restoring cost.^[ref]

A phase gradient carries superflow

v_{s} = (ℏ/ m_{b}) \nabla θ

. Its energy cost defines the superfluid number density. At leading mean-field order the stiffness uses

n_{0}

, but condensate density and superfluid number density are different observables: for a Galilean-invariant continuum gas at zero temperature the latter is the full density even when interactions deplete the condensate. Bogoliubov coefficients give

v_{k}^{2} = [(ε_{k} + g n_{0}) / E_{k} - 1] /2

and a small zero-temperature depletion of relative order

n a_{s}^{3}

in three dimensions. Large depletion or strong long-wavelength phase fluctuations would invalidate the small-fluctuation saddle approximation.

The Landau criterion for creating excitations is

v_{L} = in f_{k > 0} E_{k} / (ℏ k) = c_{s}

for this homogeneous dispersion. Thus repulsion produces a linear sound mode and a positive critical velocity from the Landau criterion, whereas an ideal quadratic gas has zero value of this criterion. Vortices, obstacles and nucleation can reduce an actual flow threshold below

c_{s}

; the criterion alone is not a complete theory of metastable flow. At finite temperature thermal quasiparticles reduce the superfluid number density. Dimensionality and finite size also matter: the uniform three-dimensional weak-depletion condensate expansion must not be transferred unchanged to a one- or two-dimensional thermodynamic system. The field integral provides systematic corrections only within its dilute, stable and sufficiently weak-fluctuation regime.

3

a

Solution

Keep the electromagnetic units of the supplied Lagrangian, including its factor

c

, and regard

e

as the signed coupling used there. Set

a_{B} = e Φ_{B} / (2 π c)

. The canonical momentum is

p_{ϕ} = I \dot{ϕ} + a_{B}

, so the Legendre transform gives

H = \frac{( p _{ϕ} - a _{B} ) ^{2}}{2 I}, \hat{H} = \frac{1}{2 I} (- i ℏ \partial_{ϕ} - \frac{e Φ _{B}}{2 π c})^{2} .

Single-valued periodic wavefunctions have normalized eigenstates

e^{in ϕ} / 2 π

with integer

n

. Defining dimensionless flux

f = e Φ_{B} / (h c)

gives

E_{n} = ℏ^{2} (n - f)^{2} / (2 I)

, and the trace is therefore

Z = n \in Z \sum exp [- \frac{β ℏ ^{2}}{2 I} (n - f)^{2}] .

The flux-threaded quantum rotor spectrum is periodic under

f \mapsto f + 1

after relabelling

n

, corresponding to flux period

h c /∣ e ∣

in these units. A local gauge change can remove the magnetic vector potential from the differential operator but then imposes a twisted boundary condition with phase

e^{- 2 πi f}

; it cannot remove the flux dependence from both the Hamiltonian and boundary condition simultaneously. In SI conventions the flux coupling has no

c

, so one must not mix the two unit systems.

b

Solution

The trace integrates the diagonal thermal kernel over a single fundamental interval,

Z = \int_{0}^{2 π} d ϕ_{0} ⟨ ϕ_{0} ∣ e^{- β \hat{H}} ∣ ϕ_{0} ⟩

. On the circle a closed path can wind an integer number

m

of times. Lift it to the real covering coordinate so its endpoints are

ϕ (0) = ϕ_{0}

and

ϕ (T) = ϕ_{0} + 2 πm

, with

T = ℏ β

Under Wick rotation

t = - i τ

\dot{ϕ} = i \partial_{τ} ϕ

, so the real-time phase

i S /ℏ

becomes minus the Euclidean action divided by

ℏ

, with

S_{E} [ϕ] = \int_{0}^{T} d τ [\frac{I}{2} (\partial_{τ} ϕ)^{2} - i \frac{e Φ _{B}}{2 π c} \partial_{τ} ϕ] .

Thus sum the path integral

e^{- S_{E} /ℏ}

over these lifted endpoint conditions and integrate

ϕ_{0}

from zero to

2 π

. This gives the requested winding-sector representation of the trace. The flux term is a total derivative, so it leaves the local classical equation unchanged, but its weight is

e^{i 2 π f m}

and distinguishes different winding sectors. This is the Aharonov-Bohm phase in a thermal path integral, rather than a force from a magnetic field on the ring itself.

c

Solution

The Euclidean saddle in winding sector

m

is a constant-speed lifted path,

ϕ_{cl} (τ) = ϕ_{0} + 2 πm τ / T

. Write

ϕ = ϕ_{cl} + η

with Dirichlet fluctuation endpoints

η (0) = η (T) = 0

. The cross term in the kinetic action is proportional to

\int \partial_{τ} η d τ = 0

, and the flux term of

η

vanishes for the same reason. Therefore

\frac{S _{E}}{ℏ} = \frac{2 π ^{2} I}{ℏ ^{2} β} m^{2} - i 2 π f m + \frac{I}{2ℏ} \int_{0}^{T} (\partial_{τ} η)^{2} d τ .

Define the fluctuation prefactor by

Z_{0} = \int_{η (0) = η (T) = 0} D η exp [- \frac{I}{2ℏ} \int_{0}^{T} (\partial_{τ} η)^{2} d τ] .

It is the free-line diagonal Euclidean propagator for the angular coordinate with inertia

I

, or equivalently the Gaussian bridge amplitude about a given classical lifted path. It is independent of

m, ϕ_{0}

and flux. It is not the complete zero-flux ring partition function, which still includes every winding and the initial-angle integral. Translational independence supplies the factor

\int_{0}^{2 π} d ϕ_{0} = 2 π

, leaving

Z = 2 π Z_{0} m \in Z \sum exp [- \frac{2 π ^{2} I}{ℏ ^{2} β} m^{2} + i 2 π f m] .

The momentum-winding duality of a quantum rotor uses a Gaussian expansion that is exact here because the kinetic action is quadratic and the flux coupling is a total derivative. It is not merely a leading semiclassical approximation.

Figure 1.
Lifted winding sectors on a flux-threaded ring and the periodic flux dependence of quantum energy levels
. **Details**.

d

Solution

Continue the free-particle propagator to

t = - i T

and replace its coordinate mass by the angular inertia

I

. The Euclidean free-particle bridge amplitude at coincident endpoints is

Z_{0} = \frac{I}{2 π ℏ T} = \frac{I}{2 π ℏ ^{2} β} .

This normalization includes the time-sliced Gaussian measure, not just a formal determinant with an unspecified multiplicative constant. Set

A = 2 π^{2} I / (ℏ^{2} β) > 0

. The correct Poisson summation formula is

\sum_{m} F (m) = \sum_{n} \int_{- \infty}^{\infty} d x F (x) e^{- 2 πin x}

; the source PDF omits the integral sign before its

d θ

. For

F (x) = e^{- A x^{2} + i 2 π f x}

the Gaussian Fourier integral gives

\int_{- \infty}^{\infty} d x e^{- A x^{2} + i 2 π (f - n) x} = \frac{π}{A} e^{- π^{2} (n - f)^{2} / A} .

The overall prefactor cancels exactly:

2 π Z_{0} π / A = 1

, and

π^{2} / A = β ℏ^{2} / (2 I)

. Hence

Z = n \sum e^{- β ℏ^{2} (n - f)^{2} / (2 I)},

in agreement with the angular-momentum trace. Pairing

m

and

- m

makes the winding expression real, and the momentum expression shows positivity. Both are periodic in flux and invariant under

f \mapsto - f

. At high temperature the zero-winding term gives

Z ≃ 2 π I / (β ℏ^{2})

, with exponentially small flux dependence; at low temperature the nearest integer angular-momentum state dominates, with two equal lowest terms at half-integer flux. These independent limits also check the normalization and the flux phase.

2015

Paper 1 (Quantum Condensed Matter Field Theory, 1/QCM)

Original solutions covering every option, including all three essay alternatives. ^[ref]

1

a

Solution

Use

ℏ = 1

and

S_{n}^{\pm} = S_{n}^{x} \pm i S_{n}^{y}

. Operators on different sites commute, and expanding the two products gives

S_{n}^{+} S_{n + 1}^{-} + S_{n}^{-} S_{n + 1}^{+} = 2 (S_{n}^{x} S_{n + 1}^{x} + S_{n}^{y} S_{n + 1}^{y})

. Therefore the quantum XX chain has

H = - \frac{J}{2} n = 1 \sum N (S_{n}^{+} S_{n + 1}^{-} + S_{n}^{-} S_{n + 1}^{+}) - h n = 1 \sum N S_{n}^{z} .

Each exchange moves a reversed spin between neighbours, conserving the total number of reversed spins. The factor of one half follows from defining

S^{\pm}

without a normalizing square root.

b

Solution

The spin commutation relations give

[S_{tot}^{z}, S_{n}^{\pm}] = \pm S_{n}^{\pm}

. Using the product rule for a commutator,

[S_{tot}^{z}, S_{n}^{+} S_{n + 1}^{-}] = S_{n}^{+} S_{n + 1}^{-} - S_{n}^{+} S_{n + 1}^{-} = 0,

and the conjugate term also vanishes. The longitudinal field commutes with every

S_{n}^{z}

, so $[H, S_{tot}^{z}] = 0$ . The Hamiltonian is invariant under the global spin rotation

U (θ) = e^{- i θ S_{tot}^{z}}

: its continuous symmetry group is

U (1)

. Simultaneous energy and total-magnetization eigenstates can be chosen, and evolution preserves each fixed-magnetization sector. Local magnetization can still flow between sites; generally

[H, S_{n}^{z}] \neq = 0

. In the fermion representation the conserved number will be

N_{f} = N /2 - S_{tot}^{z}

c

Solution

Use the canonical anticommutation relations

{c_{n}, c_{m}^{†}} = δ_{nm}

and

{c_{n}, c_{m}} = 0

. In particular,

n_{m} = c_{m}^{†} c_{m}

is a projector, so

(1 - 2 n_{m})^{2} = 1

. Define

K_{n} = \prod_{m < n} (1 - 2 n_{m})

. Then

K_{n}^{†} = K_{n}

K_{n}^{2} = 1

, and

K_{n}

commutes with

c_{n}

and

c_{n}^{†}

. Taking the adjoint of the given raising operator gives

S_{n}^{-} = c_{n}^{†} K_{n}

. Hence

[S_{n}^{+}, S_{n}^{-}] = K_{n} c_{n} c_{n}^{†} K_{n} - c_{n}^{†} K_{n}^{2} c_{n} = c_{n} c_{n}^{†} - c_{n}^{†} c_{n} = 1 - 2 n_{n} = 2 S_{n}^{z} .

Also

[n_{n}, c_{n}] = - c_{n}

and

[n_{n}, c_{n}^{†}] = c_{n}^{†}

imply

[S_{n}^{z}, S_{n}^{\pm}] = \pm S_{n}^{\pm}

. For different sites, exchanging two fermion operators contributes one minus sign and commuting an operator through the other site's parity string contributes another; their product is positive. Thus distinct spin algebras commute. The string supplies precisely the signs required for the Jordan–Wigner transformation. The fermion vacuum corresponds to spin up, while one occupied site corresponds to spin down; this convention fixes subsequent signs.

d

Solution

For a bulk bond,

K_{n + 1} = K_{n} (1 - 2 n_{n})

and

c_{n} (1 - 2 n_{n}) = - c_{n}

. Reordering the remaining different-site operators then gives

S_{n}^{+} S_{n + 1}^{-} = c_{n + 1}^{†} c_{n}, S_{n}^{-} S_{n + 1}^{+} = c_{n}^{†} c_{n + 1} (n < N) .

The periodic spin boundary condition requires separate attention. Set

P = \prod_{n = 1}^{N} (1 - 2 n_{n}) = (- 1)^{N_{f}}

. The end bond becomes

S_{N}^{+} S_{1}^{-} = - P c_{1}^{†} c_{N}

, with the conjugate relation for its adjoint. Consequently the exact finite-ring Hamiltonian is

H = - \frac{h N}{2} + h n = 1 \sum N n_{n} - \frac{J}{2} n = 1 \sum N - 1 (c_{n}^{†} c_{n + 1} + c_{n + 1}^{†} c_{n}) + \frac{J P}{2} (c_{N}^{†} c_{1} + c_{1}^{†} c_{N}) .

fermion parity is conserved, so diagonalize in a sector

P = \pm 1

. There the end bond can be written like every other bond provided

c_{N + 1} = - P c_{1}

. This parity-dependent boundary condition of the Jordan-Wigner chain is antiperiodic for even fermion number and periodic for odd fermion number. Simply assuming periodic fermions in both sectors changes the finite-size spectrum.

e

Solution

In a fixed fermion-parity sector use lattice spacing one and the unitary discrete Fourier transform

c_{n} = \frac{1}{N} \sum_{k} e^{ikn} c_{k}, c_{k} = \frac{1}{N} \sum_{n} e^{- ikn} c_{n} .

The allowed momenta satisfy

e^{ik N} = - P

: they are

k = 2 π (ℓ + 1/2) / N

for

P = + 1

and

k = 2 π ℓ / N

for

P = - 1

, with

N

distinct values modulo

2 π

. The hopping and number sums become

\sum_{k} 2 cos k c_{k}^{†} c_{k}

and

\sum_{k} c_{k}^{†} c_{k}

. Thus

H = E + k \sum ϵ_{k} c_{k}^{†} c_{k}, E = - h N /2, ϵ_{k} = h - J cos k .

Here

E

is the energy of the all-up spin configuration, not generally the ground-state energy. The dispersion has minimum

h - J

k = 0

, maximum

h + J

k = π

, and positive curvature at its minimum. Since the fermions do not interact, each mode can be occupied zero or one time. A negative-energy occupation lowers the many-body energy, so in the thermodynamic limit

∣ gs ⟩ = \prod_{∣ k ∣ < k_{F}} c_{k}^{†} ∣0 ⟩, k_{F} = arccos (h / J), 0 < h < J .

This is a filled Fermi sea containing many fermions; it is not obtained by putting a single particle into the band minimum. For

h > J

every mode has positive energy and the vacuum is the ground state. Its spin interpretation is complete longitudinal polarization. At

h = J

the zero-momentum one-particle excitation first becomes gapless.

For an exact finite ring use parity-consistent filling of a free-fermion band. In each momentum grid fill negative-energy levels, check that their particle number has the chosen parity, and if necessary toggle a mode with the smallest

∣ ϵ_{k} ∣

. Compare the two sector energies. A zero-energy mode allows degenerate occupations. This restores the physical spin Hilbert space without changing the stated band.

Figure 1.
Original XX-chain dispersion, occupied interval and thermodynamic magnetization; illustrative field ratios
. **Details**.

f

Solution

Because

S_{tot}^{z} = N /2 - N_{f}

, the exact finite-size answer is

m_{N} = 1/2 - N_{f}^{gs} / N

, with

N_{f}^{gs}

determined by the parity-consistent filling above. For the thermodynamic magnetization of the quantum XX chain, the occupied fraction is the length of the negative-energy interval divided by the Brillouin-zone length:

\frac{N _{f}}{N} ⟶ \frac{1}{2 π} \int_{- k_{F}}^{k_{F}} d k = \frac{k _{F}}{π} .

Therefore

m_{z} (h) = ⎩ ⎨ ⎧ \frac{1}{2} - \frac{1}{π} arccos (h / J) = \frac{1}{π} arcsin (h / J), \frac{1}{2}, 0 \leq h \leq J, h \geq J, h_{c} = J .

The zero-field limit has zero magnetization and the high-field limit has all spins up. The finite-ring curve consists of magnetization steps approaching this continuous limit. Strictly above

h_{c}

the fully polarized ground state is unique; exactly at

h_{c}

a zero-energy spin-flip state can be degenerate with it. This finite-size qualification does not alter the saturation field or limiting magnetization density.

2

a

Solution

The Holstein–Primakoff transformation converts a spin algebra into bosonic operators while retaining the finite physical spin Hilbert space. With

[a_{n}, a_{m}^{†}] = δ_{nm}

, write

S_{n}^{z} = S - a_{n}^{†} a_{n}, S_{n}^{+} = 2 S - a_{n}^{†} a_{n} a_{n}, S_{n}^{-} = a_{n}^{†} 2 S - a_{n}^{†} a_{n} .

The Holstein–Primakoff occupation constraint is

0 \leq n_{n} \leq 2 S

. Acting on

∣ n ⟩

, the raising matrix element is

n (2 S - n + 1)

and the lowering element is

(n + 1) (2 S - n)

. Their squared difference is

2 (S - n)

, reproducing

[S^{+}, S^{-}] = 2 S^{z}

. Operator ordering inside the square roots is essential. For low boson occupation relative to

S

, expand them to obtain

S^{+} ≃ 2 S a

and

S^{-} ≃ 2 S a^{†}

. Keeping quadratic terms gives the linear spin-wave approximation; higher terms describe magnon interactions. Large

S

helps control the expansion, but a low-dimensional infrared divergence can invalidate it even at large finite

S

For the ferromagnet choose

H_{F} = - J \sum_{n} S_{n} \cdot S_{n + 1}

J > 0

, with unit lattice spacing and

ℏ = 1

. Expand around the all-up state. A longitudinal bond contributes

- J S^{2} + J S (n_{n} + n_{n + 1})

, and the transverse exchange contributes

- J S (a_{n}^{†} a_{n + 1} + a_{n + 1}^{†} a_{n})

. After a Fourier transform,

H_{F}^{(2)} = - J N S^{2} + \sum_{k} ω_{F} (k) a_{k}^{†} a_{k}, ω_{F} (k) = 2 J S (1 - cos k) ≃ J S k^{2} .

These are ferromagnetic magnons. The polarized vacuum is an exact ground state of the full ferromagnetic Hamiltonian: every bond is in its maximum total-spin sector, minimizing that bond energy. It therefore has no quantum zero-point reduction of its magnetization. The one-magnon dispersion is also exact, although interacting many-magnon states need not be free bosons. Rotating the polarization generates the maximum-total-spin ground-state multiplet. Selecting a definite direction in a thermodynamic symmetry-breaking description breaks

S U (2)

U (1)

; the two transverse spin components form a canonical pair, producing a single quadratic type-B Goldstone boson. A small aligning field gaps it by the Zeeman energy.

At nonzero temperature the one-dimensional ferromagnet is different: its thermal magnon number contains

\int d k [e^{β J S k^{2}} - 1]^{- 1}

, with infrared behaviour

\int d k T / k^{2}

. The divergent magnetization reduction signals the absence of finite-temperature long-range order in an infinite short-range isotropic chain, consistent with the Mermin-Wagner theorem. It does not disallow the polarized zero-temperature ground states just constructed.^[ref]

For the antiferromagnet use

H_{A F} = + J \sum_{n} S_{n} \cdot S_{n + 1}

, still

J > 0

, on an even bipartite chain. The classical reference state alternates up and down. Rotate one sublattice through

π

so that both local quantization axes point along their reference spins. In unrotated physical components, to leading order

S_{A}^{z} = S - a^{†} a

S_{B}^{z} = - S + b^{†} b

S_{A}^{+} ≃ 2 S a

and

S_{B}^{+} ≃ 2 S b^{†}

. Choosing Fourier phases using the actual site positions gives

H_{A F}^{(2)} = - J N S^{2} + 2 J S \sum_{k \in RBZ} [a_{k}^{†} a_{k} + b_{k}^{†} b_{k} + cos k (a_{k} b_{- k} + a_{k}^{†} b_{- k}^{†})] .

The reduced Brillouin zone has

M = N /2

momenta and length

π

. The pair creation and annihilation terms express the fact that an antiparallel reference product state is not an eigenstate. A bosonic Bogoliubov transformation with

u_{k} = cosh θ_{k}

v_{k} = sinh θ_{k}

tanh 2 θ_{k} = cos k

diagonalizes each two-mode block. With the sign convention

a_{k} = u_{k} α_{k} - v_{k} β_{- k}^{†}

and

b_{- k} = u_{k} β_{- k} - v_{k} α_{k}^{†}

, the result is

H_{A F}^{(2)} = E_{A F}^{(2)} + \sum_{k \in RBZ} ω_{A F} (k) (α_{k}^{†} α_{k} + β_{k}^{†} β_{k}),

ω_{A F} (k) = 2 J S 1 - cos^{2} k = 2 J S ∣ sin k ∣ ≃ 2 J S ∣ k ∣, E_{A F}^{(2)} = - J N S^{2} + \sum_{k \in RBZ} [ω_{A F} (k) - 2 J S] .

Thus the harmonic antiferromagnetic modes have linear, rather than quadratic, long-wave dispersion. In an unfolded Brillouin-zone description the same result is represented by zeros at both zero and the antiferromagnetic wavevector; these are not extra independent degrees of freedom beyond the two-sublattice accounting.

The Bogoliubov vacuum has no

α

β

excitations but contains original spin bosons, with

v_{k}^{2} = \frac{1}{2} (∣ sin k ∣^{- 1} - 1)

. Its staggered magnetization is

m_{s} = S - M^{- 1} \sum_{k \in RBZ} v_{k}^{2}

. This quantum depletion of Néel order diverges logarithmically at small

k

in one dimension. Linear spin-wave theory therefore does not establish a stable ordered antiferromagnetic ground state for this chain. Its nominal gapless dispersion is a result of expanding around an invalid ordered reference, not a general proof of the exact excitation spectrum. For the nearest-neighbour isotropic chain, integer-spin systems have a Haldane gap, whereas half-integer-spin systems are gapless; the spin-one-half chain in particular has fractional spin excitations rather than a simple dilute gas of ordered-state magnons. This distinction requires physics beyond the harmonic expansion, including the topological term in the continuum theory.^[ref]

Figure 1.
Original harmonic chain dispersions; the antiferromagnetic curve is a spin-wave approximation, not the exact spectrum for every spin
. **Details**.

b

Solution

For one coordinate with Hamiltonian

H = \overset{p}{^}^{2} / (2 m) + V (\overset{x}{^})

, the propagator is

K (x_{f}, t_{f}; x_{i}, t_{i}) = ⟨ x_{f} ∣ e^{- i H T /ℏ} ∣ x_{i} ⟩

, where

T = t_{f} - t_{i} > 0

. Divide

T

into

M

intervals

δ t

, use the Trotter product formula to separate kinetic and potential evolution, and insert complete position and momentum bases at each slice. The momentum integral is Gaussian:

⟨ x_{j + 1} ∣ e^{- i δ t \overset{p}{^}^{2} / (2 m ℏ)} ∣ x_{j} ⟩ = \int \frac{d p _{j}}{2 π ℏ} exp [\frac{i}{ℏ} p_{j} (x_{j + 1} - x_{j}) - \frac{i δ t}{2 m ℏ} p_{j}^{2}] = (\frac{m}{2 πi ℏ δ t})^{1/2} e^{im (x_{j + 1} - x_{j})^{2} / (2ℏ δ t)} .

Multiplying the short-time kernels gives the time-sliced configuration-space path integral,

K = lim_{M \to \infty} (\frac{m}{2 πi ℏ δ t})^{M /2} \int \prod_{j = 1}^{M - 1} d x_{j} exp {\frac{i}{ℏ} \sum_{j = 0}^{M - 1} [\frac{m ( x _{j + 1} - x _{j} ) ^{2}}{2 δ t} - δ t V (x_{j})]},

with

x_{0} = x_{i}

x_{M} = x_{f}

. Continuum notation abbreviates this limit as

\int_{x_{i}}^{x_{f}} D x e^{i S [x] /ℏ}

, where

S = \int d t [m \overset{x}{˙}^{2} /2 - V (x)]

. The measure includes the displayed normalization; it is not an unweighted count of paths. A small damping or analytic-continuation prescription defines the oscillatory integrals. The paths contribute complex amplitudes, so alternative histories interfere before probabilities are calculated. This is the central difference from a classical statistical mixture of trajectories.^[ref]

When actions vary rapidly relative to

ℏ

, stationary phase selects classical paths. Put

x = x_{cl} + η

, with

η

zero at the endpoints. Vanishing first variation gives

m \overset{x}{¨}_{cl} + V^{'} (x_{cl}) = 0

. The quadratic term is

S [x] = S [x_{cl}] + \frac{1}{2} \int d t η (- m \frac{d ^{2}}{d t ^{2}} - V^{''} (x_{cl})) η + O (η^{3}) .

The resulting Gaussian fluctuation determinant is the semiclassical propagator prefactor. In one dimension, away from caustics,

K_{sc} = cl \sum (\frac{D _{cl}}{2 πi ℏ})^{1/2} exp [\frac{i S _{cl}}{ℏ} - \frac{iπ ν _{cl}}{2}], D_{cl} = - \frac{\partial ^{2} S _{cl}}{\partial x _{f} \partial x _{i}} .

The branch convention makes

ν = 0

for a direct free-particle path before any caustic. The Maslov index tracks the phase changes as fluctuations pass through a focal singularity. In several dimensions replace the endpoint derivative by the Van Vleck determinant. The approximation sums all relevant classical paths; real-time action need only be stationary, not a minimum. It fails in this simple form at a caustic or coalescing saddles, where a uniform approximation is needed. Quadratic actions have no higher fluctuation terms, making their Gaussian evaluation exact. For example

S_{cl} = m (x_{f} - x_{i})^{2} / (2 T)

gives the exact free-particle kernel

m / (2 πi ℏ T) e^{i S_{cl} /ℏ}

. Tunnelling involves classically forbidden regions and can be described by imaginary-time saddles rather than real classical trajectories.

Quantum statistical mechanics follows by Wick rotation

t = - i τ

. The imaginary-time evolution operator is

e^{- τ H /ℏ}

, with Euclidean action

S_{E} [x] = \int_{0}^{β ℏ} d τ [\frac{m}{2} (\partial_{τ} x)^{2} + V (x)] .

Taking a thermal trace identifies and integrates the endpoint positions:

Z = Tr e^{- β H} = \int d x \int_{x (0) = x}^{x (β ℏ) = x} D x e^{- S_{E} [x] /ℏ} .

This thermal trace as a closed Euclidean path integral explains the periodic imaginary-time direction. A density-matrix element instead retains open endpoints. For many identical particles the trace also sums over endpoint permutations, with a plus sign for bosons and permutation parity for fermions. In coherent-state language this becomes periodic bosonic fields and antiperiodic Grassmann fields, with corresponding Matsubara frequencies.

The Euclidean weight resembles a classical Boltzmann weight. A time-sliced local quantum field in

d

spatial dimensions becomes a classical-looking field model in

d + 1

dimensions, with anisotropic couplings and extra extent

β ℏ

. Thus zero-temperature quantum criticality can be treated using an infinitely extended imaginary-time direction, while nonzero temperature imposes a finite thickness. The dynamical scaling of time need not equal the spatial scaling: the dynamical critical exponent specifies their relationship. This quantum-to-classical imaginary-time mapping requires qualification when weights have signs or imaginary terms; such a functional integral is not automatically an ordinary positive classical probability distribution. Even with a positive weight, the mapping concerns equilibrium statistics, not literal dissipative real-time trajectories. For a free particle the Euclidean kernel is a diffusion kernel with diffusion coefficient

ℏ/ (2 m)

, illustrating the mathematical correspondence.

The classical high-temperature limit is a further approximation, rather than simply another name for imaginary time. If quantum level spacings and exchange effects are negligible, a one-particle partition function in

d

dimensions becomes

Z_{cl} ≃ \int \frac{d ^{d} x d ^{d} p}{( 2 π ℏ ) ^{d}} e^{- β [p^{2} / (2 m) + V (x)]} .

(7)

For indistinguishable particles there is a factor

1/ N!

in the dilute classical gas, and exchange is negligible when the number density times the thermal de Broglie wavelength to power

d

is small. For a harmonic oscillator, the exact quantum result

Z = [2 sinh (β ℏ ω /2)]^{- 1}

tends to

1/ (β ℏ ω)

β ℏ ω \to 0

, recovering the classical phase-space integral. stationary phase explains classical mechanics; the short imaginary-time, weak-exchange limit explains classical statistical mechanics. These are related but distinct limits of the same quantum formulation.

c

Solution

Consider a conventional spin-singlet, weak-coupling, three-dimensional superconductor. Set

ℏ = k_{B} = 1

in the microscopic action and restore electromagnetic units below. An attraction

λ > 0

in an energy shell about the Fermi surface gives a Grassmann action

S = \int d τ d^{3} x [\sum_{σ =↑, ↓} \overset{ˉ}{ψ}_{σ} (\partial_{τ} + ξ (- i \nabla)) ψ_{σ} - λ \overset{ˉ}{ψ}_{↑} \overset{ˉ}{ψ}_{↓} ψ_{↓} ψ_{↑}], ξ_{k} = ϵ_{k} - μ .

The attractive sign permits pairing even when the single-particle Fermi gas is otherwise stable. A complex Hubbard–Stratonovich transformation introduces a pair field

Δ

proportional at the saddle to

λ ⟨ ψ_{↓} ψ_{↑} ⟩

. The action is then quadratic in fermions. Integrating them out gives the BCS pairing-field effective action

S_{eff} [Δ] = \int d τ d^{3} x \frac{∣Δ ∣ ^{2}}{λ} - Tr ln G^{- 1} [Δ] + constant .

Here the trace uses one independent Nambu spinor pairing block; a fully doubled particle-hole representation instead carries its corresponding one-half factor. At a uniform saddle the Bogoliubov--de Gennes Hamiltonian can be written as

H_{BdG} (k) = (ξ_{k} Δ^{*} Δ - ξ_{k}), E_{k} = ξ_{k}^{2} + ∣Δ ∣^{2} .

The negative branch describes the paired vacuum; excitations are Bogoliubov quasiparticles mixing particles and holes. Extremizing the effective action gives

\frac{1}{λ} = \int_{shell} \frac{d ^{3} k}{( 2 π ) ^{3}} \frac{tanh ( E _{k} /2 T )}{2 E _{k}} .

The physical cutoff or a renormalized interaction is part of the microscopic model; an unregulated contact integral must not be used indiscriminately. With constant density of states

N (0)

per spin and shell half-width

Ω_{c}

, the zero-temperature weak-coupling solution is

Δ_{0} ≃ 2 Ω_{c} e^{- 1/ [N (0) λ]}

. The exponentially small scale follows from the logarithmic pairing susceptibility of states near the Fermi surface. For this ideal weak-coupling model,

2 Δ_{0} / (k_{B} T_{c}) ≃ 3.53

. The microscopic theory relates the gap, pairing and thermodynamic transition.^[ref]

Close to a continuous transition, for fields varying slowly compared with microscopic scales, expand the effective action in powers and gradients of

Δ

. Its static part is the Ginzburg-Landau theory of superconductivity:

F = \int d^{3} x [a (T) ∣Δ ∣^{2} + \frac{b}{2} ∣Δ ∣^{4} + K (- i \nabla - \frac{q}{ℏ} A) Δ^{2} + \frac{∣ B ∣ ^{2}}{2 μ _{0}}], q = - 2 e, b, K > 0.

, condensation free-energy density

The normalization of

Δ

fixes the coefficients; here

a (T)

changes sign at

T_{c}

. The uniform minimum has

∣ Δ_{0} ∣^{2} = - a / b

for

a < 0

- a^{2} / (2 b)

, and characteristic coherence-length scale

ξ = K /∣ a ∣

. Above the transition the minimum is at zero. The quartic stabilization and gradient cost are precisely the ingredients of Landau theory for a complex order parameter. This controlled near-transition expansion supplies a bridge between microscopic BCS theory and macroscopic magnetic response; its coefficients and validity depend on the material and fluctuation regime.^[ref]

Write

Δ = (∣ Δ_{0} ∣ + η) e^{i ϕ}

. The radial fluctuation has positive local restoring curvature, whereas the phase has no local potential in a neutral theory with global

U (1)

symmetry. Freezing the amplitude gives

F_{ϕ} = \frac{ρ _{s}}{2} \int d^{3} x (\nabla ϕ - \frac{q}{ℏ} A)^{2}, ρ_{s} = 2 K ∣ Δ_{0} ∣^{2} .

The neutral case supports a gapless collective phase mode. In the charged case the electromagnetic field couples to this phase rigidity. Under

A \mapsto A + \nabla χ

and

ϕ \mapsto ϕ + q χ /ℏ

, the displayed combination is invariant. Locally choosing a gauge with constant phase leaves a quadratic term in

A

. Varying the free energy gives the transverse London current and magnetic screening scale,

j = - \frac{q ^{2} ρ _{s}}{ℏ ^{2}} A, λ_{L}^{- 2} = \frac{μ _{0} q ^{2} ρ _{s}}{ℏ ^{2}} .

(7)

Combining this relation with static Maxwell equations gives exponential field penetration, the Meissner effect. Single-valuedness of the pair field around a loop with negligible current gives flux in units

Φ_{0} = 2 π ℏ/∣ q ∣ = π ℏ/ e

. These are gauge-invariant physical consequences of the condensate stiffness.

For the dynamics, the neutral phase and density mode is reorganized by the Coulomb interaction and the dynamical electromagnetic field. In a bulk three-dimensional system the long-wave longitudinal mode has a plasma gap, rather than remaining the neutral sound mode. This is the Anderson-Higgs mechanism in a superconductor: a phase coordinate becomes part of the electromagnetic collective response instead of surviving as an independent massless physical particle. Local gauge transformations are redundancies of the description, so a literal breaking of a local redundancy is not an observable. Phase coherence, current response and magnetic screening provide the physical statements.^[ref]

A static Landau functional alone cannot determine that full dynamical spectrum. A two-dimensional film, for example, need not have the same long-wave plasma dispersion as a bulk charged system. Nor does a positive radial curvature automatically guarantee a sharp Higgs amplitude resonance: coupling to quasiparticles and other modes can damp it. The chosen mean-field phase is also a thermodynamic symmetry-breaking description; an isolated finite system can be formulated with definite particle number. The microscopic pairing saddle, the near-transition Landau expansion and the electromagnetic Anderson-Higgs response describe successive scales of the same superconducting state, with distinct assumptions at each step.

3

a

Solution

For a normal-ordered bosonic Hamiltonian, insert coherent-state resolutions between short imaginary-time evolution factors. The overlaps generate

\overset{ˉ}{ψ} \partial_{τ} ψ

and the Hamiltonian symbol is obtained by replacing creation and annihilation operators by

\overset{ˉ}{ψ}

and

ψ

. Including the chemical potential gives, in the stated

ℏ = 1

convention,

S = \int_{0}^{β} d τ \int_{V} d^{d} r [\overset{ˉ}{ψ} (\partial_{τ} - \frac{\nabla ^{2}}{2 m} - μ) ψ + \frac{g}{2} (\overset{ˉ}{ψ} ψ)^{2}] .

The trace imposes periodic imaginary-time boundary conditions,

ψ (r, β) = ψ (r, 0), \overset{ˉ}{ψ} (r, β) = \overset{ˉ}{ψ} (r, 0),

so bosonic Matsubara frequencies are

ω_{n} = 2 πn / β

. The two complex coherent-state integration variables are treated independently before imposing the physical saddle contour. There is no extra zero-point ordering term: the supplied interaction is already normal ordered.

The spatial confinement is not specified beyond the volume. The uniform bulk results below use a homogeneous box with periodic spatial boundaries, or the interior of a large vessel where boundary effects can be neglected. A hard wall instead requires a vanishing field at that wall and an inhomogeneous saddle near it. Temporal periodicity follows from the thermal trace and does not by itself impose periodic spatial boundaries.

b

Solution

For a uniform time-independent saddle, extremizing the action gives

(- μ + g ∣ ψ_{0} ∣^{2}) ψ_{0} = 0

. Its grand-potential density is

f (ρ) = - μ ρ + g ρ^{2} /2

, with

ρ = ∣ ψ_{0} ∣^{2} \geq 0

. Since

g > 0

, the minimum at positive density for

μ > 0

ρ_{0} = μ / g, ψ_{0} = ρ_{0} e^{i ϕ_{0}}, f_{m i n} = - μ^{2} / (2 g) .

For

μ \leq 0

the uniform mean-field minimum is the vacuum, rather than a positive-density condensate. At fixed nonzero density in the weakly interacting condensed regime, the saddle relation sets

μ ≃ g ρ_{0}

. The mean-field one-body density matrix factorizes as

⟨ a^{†} (r) a (r^{'})⟩ ≃ ρ_{0}

, giving macroscopic occupation of the uniform mode. Depletion corrections are omitted at this stage.

The action has the global particle-number symmetry

ψ \mapsto e^{i α} ψ

\overset{ˉ}{ψ} \mapsto e^{- i α} \overset{ˉ}{ψ}

. Choosing one saddle phase breaks this

U (1)

symmetry; the degenerate mean-field minima form the circle $U (1) ≃ S^{1}$ . There is no energetic preference for a uniform value of

ϕ_{0}

, but spatially varying phase costs gradient energy. Consequently a neutral condensate has a gapless Goldstone mode and a nonzero phase stiffness. The mode becomes sound once density dynamics is included. With

ℏ = 1

the superfluid velocity is

\nabla ϕ / m

, so a single-valued field allows circulation in units

2 π / m

around a closed loop that avoids zeros of the amplitude. This phase coherence and rigidity are the physical content of the broken-symmetry description.

The claim is a mean-field saddle statement, not an assertion that every

d

and temperature in the question supports true long-range order. In a finite number-conserving system

⟨ a ⟩ = 0

; the symmetry-breaking order parameter is defined with the thermodynamic limit taken before an infinitesimal source is removed. In a homogeneous infinite short-range gas at nonzero temperature, infrared phase fluctuations prohibit true condensation in

d \leq 2

. In one dimension they also destroy true condensate long-range order at zero temperature. A two-dimensional gas can nevertheless have superfluidity and algebraically decaying correlations below a Berezinskii–Kosterlitz–Thouless transition. A finite trap can have substantial coherence without contradicting these thermodynamic statements.^[ref]

c

Solution

On the physical density-phase contour put

\overset{ˉ}{ψ} = ρ e^{- i ϕ}

. Direct differentiation gives

\overset{ˉ}{ψ} \partial_{τ} ψ = \frac{1}{2} \partial_{τ} ρ + i ρ \partial_{τ} ϕ, ∣\nabla ψ ∣^{2} = \frac{∣\nabla ρ ∣ ^{2}}{4 ρ} + ρ ∣\nabla ϕ ∣^{2} .

Integrating the Laplacian by parts with the stated bulk boundary convention, and dropping the periodic total derivative of

ρ

, produces the density-phase action of a Bose gas,

S = \int d τ d^{d} r [i ρ \partial_{τ} ϕ + \frac{∣\nabla ρ ∣ ^{2}}{8 m ρ} + \frac{ρ}{2 m} ∣\nabla ϕ ∣^{2} + \frac{g}{2} (ρ - ρ_{0})^{2} - \frac{μ ^{2}}{2 g}] .

Write

ρ = ρ_{0} + δ ρ

and

ϕ = ϕ_{0} + φ

, where

ϕ_{0}

is constant. To quadratic order,

S_{2} = \int d τ d^{d} r [\frac{g}{2} (δ ρ)^{2} + i δ ρ \partial_{τ} φ + \frac{∣\nabla δ ρ ∣ ^{2}}{8 m ρ _{0}} + \frac{ρ _{0}}{2 m} ∣\nabla φ ∣^{2}] .

The uniform static density fluctuation has positive curvature

g

. Its inverse static susceptibility at wavevector

q

A_{q} = g + q^{2} / (4 m ρ_{0})

, which remains nonzero at

q = 0

. This is the sense in which the amplitude or density fluctuation is massive. The phase has no term proportional to

φ^{2}

without derivatives; a uniform phase rotation costs no action. It is the massless direction required by global

U (1)

symmetry.

The term

i δ ρ \partial_{τ} φ

shows density-phase conjugacy. It is important to distinguish static density stiffness versus an excitation gap: the density and phase do not form two independent normal-mode branches with one gapped and one gapless particle. In this nonrelativistic first-order theory they supply the conjugate variables of one positive Bogoliubov branch, derived below. The omitted cubic terms include

δ ρ ∣\nabla φ ∣^{2} / (2 m)

and corrections from the density-gradient denominator.

There is also a background term

i ρ_{0} \int d τ d^{d} r \partial_{τ} φ

. It vanishes for smooth fluctuations with zero temporal winding. Periodicity of

ψ

, however, permits

φ (β) - φ (0) = 2 π w

. The compact phase winding term must be retained when such sectors are included; it is not generally the same constant in all sectors. The local Gaussian phonon calculation concerns the smooth zero-winding sector and does not describe vortices or points where the density vanishes.

d

Solution

At wavelengths with

q^{2} / (4 m ρ_{0}) ≪ g

, neglect the density-gradient term in the quadratic density-phase action. Complete the square in its remaining density terms:

\frac{g}{2} (δ ρ)^{2} + i δ ρ \partial_{τ} φ = \frac{g}{2} (δ ρ + \frac{i}{g} \partial_{τ} φ)^{2} + \frac{1}{2 g} (\partial_{τ} φ)^{2} .

The convergent Gaussian integral can be performed by a contour shift or by its Fourier-transform identity. Its determinant is independent of

φ

at this order. Combining that normalization with the saddle action into a constant

S_{0}

gives

Z = e^{- S_{0}} \int D φ e^{- S_{eff} [φ]}, S_{eff} = \frac{1}{2} \int_{0}^{β} d τ \int_{V} d^{d} r [\frac{( \partial _{τ} φ ) ^{2}}{g} + \frac{ρ _{0}}{m} ∣\nabla φ ∣^{2}] .

The saddle contribution to

S_{0}

- β V μ^{2} / (2 g)

; the additional measure-dependent constant need not be assigned an invented numerical value. The density Gaussian extends the fluctuation integration beyond its exact positivity domain, which is justified only while fluctuations around

ρ_{0} > 0

are small. The smooth zero-winding assumption is also required as discussed above. A constant global phase zero mode gives a finite compact-group volume, handled separately from nonzero phonon modes.

Fourier transforming the Bose-gas phase-only action gives a kernel

ω_{n}^{2} / g + (ρ_{0} / m) q^{2}

. Analytically continuing

ω_{n} \mapsto - i (ω + i 0^{+})

gives its real-frequency poles,

ω (q) = c ∣ q ∣, c = g ρ_{0} / m .

In the present units the excitation energy equals

ω

. Each real Fourier coordinate is a harmonic oscillator with inertia coefficient

1/ g

and spring coefficient

ρ_{0} q^{2} / m

, hence frequency equal to the square root of their ratio. This is the direct relation to a quantum harmonic chain. The mode is a gapless sound phonon; its finite speed reflects both the compressibility

\partial ρ / \partial μ = 1/ g

and the phase stiffness

ρ_{0} / m

Retaining the density-gradient term provides a useful check. Its integration instead gives a temporal coefficient

1/ A_{q}

, so

ω^{2} (q) = \frac{ρ _{0} q ^{2}}{m} (g + \frac{q ^{2}}{4 m ρ _{0}}) = c^{2} q^{2} + \frac{q ^{4}}{4 m ^{2}}, ω (q) = ϵ_{q} (ϵ_{q} + 2 g ρ_{0}), ϵ_{q} = \frac{q ^{2}}{2 m} .

This is the Bogoliubov spectrum. Its small-

q

limit is the derived sound branch; at large

q

within the contact model it crosses towards particle-like behaviour. The crossover in the squared dispersion is

q \sim 2 m g ρ_{0}

, precisely the scale beyond which the discarded density gradient is not small.

The same action checks the dimensional restriction in the saddle analysis. At nonzero temperature its zero Matsubara mode gives phase fluctuations proportional to

\int d^{d} q / q^{2}

; at zero temperature integrating the frequency gives

\int d^{d} q /∣ q ∣

. Their infrared divergences invalidate uniform true long-range order for

d \leq 2

at nonzero temperature and for

d = 1

at zero temperature. The long-wave collective description can remain useful for correlations even where a global condensate saddle is not a literal symmetry-broken state. The contact coupling here is an effective low-energy parameter; this calculation does not determine ultraviolet renormalization or trap-specific surface modes.

Figure 1.
Original dimensionless Bogoliubov dispersion and its sound and particle limits
. **Details**.

2014

Paper 1 (Quantum Condensed Matter Field Theory, Paper 160)

Original examination PDF, physical pages 651–653

1

Solution

Use the Holstein–Primakoff transformation only on the physical occupation sector

0 \leq n \leq 2 S

, with

n = a^{†} a

and

S^{-} = a^{†} 2 S - n

. Acting on number states gives

S^{+} ∣ n ⟩ = n (2 S - n + 1) ∣ n - 1 ⟩, S^{-} ∣ n ⟩ = (n + 1) (2 S - n) ∣ n + 1 ⟩ .

Consequently

S^{+} S^{-} = (n + 1) (2 S - n)

and

S^{-} S^{+} = n (2 S - n + 1)

, so

[S^{+}, S^{-}] = 2 (S - n) = 2 S^{z}, [S^{z}, S^{\pm}] = \pm S^{\pm} .

The endpoint factors keep the spin operators inside the physical sector. The square-root ordering matters; replacing it by an unrestricted linear oscillator would not give the exact finite-spin algebra.

a

Solution

For the even periodic chain, every classical Néel state

S_{m} = (- 1)^{m} S n

has energy

- J N S^{2}

, with arbitrary unit vector

n

. Thus there is a continuous sphere of classical orientations, not merely two Ising-like states. Opposite staggered directions are connected by a global spin rotation.

A quantum Néel product is not an eigenstate: transverse exchange flips opposite neighbouring spins. The ground state therefore contains quantum superpositions and zero-point spin waves, reducing the classical staggered moment. A finite isotropic balanced chain has a symmetry-invariant ground state; a symmetry-broken reference is an approximation for analysing fluctuations, not its exact finite-system eigenstate.

b

Solution

Rotate alternate sites by

π

about a transverse axis so all local reference spins point up. Choosing the transverse axes to give positive anomalous coupling, the linear spin-wave approximation yields

H = - J N S^{2} + J S \sum_{m} [2 a_{m}^{†} a_{m} + a_{m} a_{m + 1} + a_{m}^{†} a_{m + 1}^{†}] + O (S^{0}) .

Use lattice spacing one and

a_{m} = N^{- 1/2} \sum_{k} e^{ikm} a_{k}

, with all $N$ momenta

k = 2 πj / N

j = - N /2, \dots, N /2 - 1

, in

[- π, π)

. Then

γ_{k} = cos k

and

H = - J N S^{2} + J S \sum_{k} [2 a_{k}^{†} a_{k} + γ_{k} (a_{k} a_{- k} + a_{k}^{†} a_{- k}^{†})] .

Reordering

a_{- k} a_{- k}^{†} = a_{- k}^{†} a_{- k} + 1

gives the requested Nambu form with constant

- J NS (S + 1)

. Its extra

J SN

restores

- J N S^{2}

before diagonalization. This full-zone convention has no additional sublattice or half-zone factor.

c

Solution

For real even

θ_{k}

, the Bogoliubov transformation gives

[α_{k}, α_{q}^{†}] = (cosh^{2} θ_{k} - sinh^{2} θ_{k}) δ_{k q} = δ_{k q}, [α_{k}, α_{q}] = 0.

Equivalently, its Nambu matrix obeys $T_{k} σ_{z} T_{k}^{†} = σ_{z}$ . The bosonic canonical metric is preserved;

T

need not be unitary in the ordinary positive-definite metric.

d

Solution

The inverse transformation has negative off-diagonal entries. Choosing

tanh 2 θ_{k} = γ_{k}

removes anomalous terms and gives

ω_{k} = 2 J S 1 - γ_{k}^{2} = 2 J S ∣ sin k ∣, H = - J NS (S + 1) + k \sum ω_{k} (α_{k}^{†} α_{k} + 1/2) + O (S^{0}) .

There are gapless points at

k = 0

and the zone boundary; near zero,

ω_{k} ≃ 2 J S ∣ k ∣

, a linear antiferromagnetic spin wave with velocity

2 J S

in the stated units. The zero modes are collective rotations: their divergent squeezing cannot be treated as ordinary normalized independent oscillator vacua. A finite-size cutoff or infinitesimal symmetry-breaking field regulates them.

As a constant-term check, the continuum spin-wave vacuum energy per site is

- J S (S + 1) + (2/ π) J S

. This is a spin-wave prediction, not an exact spectrum for every finite-

S

one-dimensional chain.

Figure 1.
Linear antiferromagnetic spin-wave dispersion and logarithmically growing one-dimensional vacuum depletion
. **Source**. **Details**.

e

Solution

Choose the reference staggered direction so its classical moment is positive. The Bogoliubov vacuum has

⟨ a_{k}^{†} a_{k} ⟩ = sinh^{2} θ_{k}

, hence

\frac{M _{s}}{N} = S - δ S, δ S = \frac{1}{2 N} k \sum [\frac{1}{1 - γ _{k}^{2}} - 1] .

Near each soft point the integrand is proportional to

1/∣ k ∣

, so the one-dimensional thermodynamic integral diverges logarithmically. The ordered-reference expansion fails; it does not predict a physical negative magnetization. Long-range Néel order is destroyed in this isotropic one-dimensional setting. The exact finite symmetric ground state also has zero staggered expectation without selecting a direction.

For linearly dispersing modes in

d

dimensions, zero-temperature depletion behaves as

\int_{0} d k k^{d - 2}

and is infrared finite for

d > 1

, allowing order in suitable two- and three-dimensional antiferromagnets. Finite depletion is not a guarantee for every frustrated quantum magnet. At nonzero temperature, the extra Bose occupation changes the singularity to

\int d k k^{d - 3}

: short-range isotropic systems cannot sustain continuous-symmetry long-range order for

d \leq 2

. This thermal conclusion is distinct from the one-dimensional zero-point argument.

2

Solution

A Wick rotation

t = - i τ_{E}

changes real-time evolution into

e^{- H τ_{E} /ℏ}

. At

τ_{E} = β ℏ

, taking the trace integrates equal endpoints:

Z = \int d q K_{E} (q, q; β ℏ) = \int_{q (0) = q (β ℏ)} D q e^{- S_{E} [q] /ℏ}, S_{E} = \int_{0}^{β ℏ} [\frac{m}{2} \overset{q}{˙}^{2} + V (q)] d τ_{E} .

Thus temperature sets the imaginary-time circumference, and bosonic coordinate paths are periodic. The measure is fixed by the short-time propagator. The trace sums paths, not just a single stationary trajectory; a quadratic action is special because the saddle plus Gaussian fluctuations is exact.

a

Solution

Put

T = β ℏ

and

x = ω T

. With equal endpoints

q

, the Euclidean classical path and its action are

q_{cl} (τ_{E}) = q \frac{c o s h [ ω ( τ _{E} - T /2 )]}{c o s h ( x /2 )}, S_{cl} = mω q^{2} tanh (x /2) .

Write

q (τ_{E}) = q_{cl} + η

, where

η (0) = η (T) = 0

. Sine modes have fluctuation eigenvalues

(nπ / T)^{2} + ω^{2}

. Relative to the free-particle determinant, their Gaussian integral contributes

\prod_{n = 1}^{\infty} [1 + (\frac{x}{nπ})^{2}]^{- 1/2} = \frac{x}{s i n h x} .

Multiplying the free prefactor

m / (2 π ℏ T)

gives the Euclidean oscillator diagonal kernel

K_{E} (q, q; β ℏ) = \frac{mω}{2 π ℏ sinh x} exp [- \frac{mω q ^{2}}{ℏ} tanh (x /2)] .

Integrating this over the common endpoint is precisely the trace. All fluctuations here are Dirichlet fluctuations about a fixed-endpoint path; periodicity is imposed on the full trace after endpoint integration.

b

Solution

The endpoint integral is a Gaussian with coefficient

A = (mω /ℏ) tanh (x /2)

. Therefore

Z = \frac{mω}{2 π ℏ s i n h x} \frac{π}{A} = \frac{1}{2 s i n h x t a n h ( x /2 )} = \frac{1}{2 sinh ( x /2 )} .

It agrees with

\sum_{n \geq 0} e^{- x (n + 1/2)}

, tends to

1/ x

at high temperature, and to

e^{- x /2}

at low temperature. The last limit retains the oscillator zero-point energy.

c

Solution

For unnormalized coherent states,

∣ z ⟩ = e^{z a^{†}} ∣0 ⟩

⟨ w ∣ z ⟩ = e^{\overset{w}{ˉ} z}

and

1 = \int d^{2} z e^{- ∣ z ∣^{2}} ∣ z ⟩ ⟨ z ∣/ π

. Insert this identity between short imaginary-time evolutions. The overlaps generate

\overset{ˉ}{ϕ} \partial_{τ} ϕ

, while the normal-ordered symbol of

H = ℏ ω (a^{†} a + 1/2)

gives

Z = e^{- β ℏ ω /2} \int_{ϕ (β) = ϕ (0)} D ϕ exp [- \int_{0}^{β} \overset{ˉ}{ϕ} (\partial_{τ} + ℏ ω) ϕ d τ]_{normal} .

Here

τ = τ_{E} /ℏ

runs from zero to

β

; the exponent is dimensionless. The bosonic trace makes the complex fields periodic, not antiperiodic.

The printed expression omits the explicit vacuum factor. It represents the full oscillator only if a symmetric/Weyl determinant prescription implicitly includes that factor. With the ordinary normal-ordered time-sliced coherent-state measure, it instead represents

H = ℏ ω a^{†} a

. The continuum formula alone does not specify the energy-zero convention.

d

Solution

Let

ε = ℏ ω

, use

M

time slices and

q = e^{- βε / M}

. The exact normal-ordered coherent Gaussian has cyclic matrix

A = 1 - qP

, where

P

shifts one slice. Its eigenvalues are

1 - q e^{2 πin / M}

and

det A = 1 - q^{M} = 1 - e^{- βε}, Z_{normal} = \frac{1}{1 - e ^{- βε}} .

Restoring the constant energy gives

Z = e^{- βε /2} Z_{normal} = \frac{1}{2 sinh ( βε /2 )},

consistent with the coordinate path integral. Alternatively, pairing positive and negative Matsubara frequencies symmetrically yields a determinant proportional to

x \prod_{n \geq 1} [1 + (x /2 πn)^{2}] = 2 sinh (x /2)

. This symmetric prescription already incorporates the zero-point convention and must not receive the vacuum factor twice. A naive unregulated frequency product can hide this difference.

3

a

Solution

The real-time path integral sums

e^{i S /ℏ}

over paths. In a classically forbidden region, the dominant semiclassical contribution is clearer after Wick rotation: the weight becomes

e^{- S_{E} /ℏ}

and stationary paths obey

m \overset{q}{¨} = V^{'} (q)

, equivalent to motion in the inverted potential. A finite-action path joining degenerate minima is an instanton.

For

V (q) = λ (q^{2} - a^{2})^{2}

λ > 0

, zero Euclidean mechanical energy gives

m \overset{q}{˙}^{2} /2 = V

. Thus

q_{I} (τ_{E}) = a tanh [\frac{ω _{0} ( τ _{E} - τ _{0} )}{2}], ω_{0}^{2} = 8 λ a^{2} / m, S_{0} = \int_{- a}^{a} 2 mV (q) d q = \frac{4}{3} a^{3} 2 mλ .

The reversed trajectory is an anti-instanton.^[ref] Expanding

q = q_{I} + η

produces the fluctuation operator

- \partial_{τ_{E}}^{2} + V^{''} (q_{I}) / m

. Its translation zero mode is handled by integrating the collective centre

τ_{0}

, not by dividing by a zero determinant. The remaining determinant and Jacobian form an attempt-frequency prefactor

A

When

S_{0} /ℏ ≫ 1

, well-separated events form a dilute gas with rate scale

κ = A e^{- S_{0} /ℏ}

. Paths returning to the same well have an even number of events, while paths ending in the other well have an odd number. The sums yield

cosh (κ T)

and

sinh (κ T)

, giving

E_{odd} - E_{even} ≃ 2ℏ κ .

The ground state is the lower symmetric combination of localized well states: tunnelling lifts the apparent classical degeneracy. This splitting is nonperturbative in

ℏ

, beyond a power series about either minimum. In a metastable well a bounce, with a negative fluctuation mode, describes decay rather than the stable doublet splitting. Finite temperature, overlapping events or a low barrier require improvements beyond the dilute approximation.

Figure 1.
An original quartic double well and its Euclidean instanton trajectory
. **Source**. **Details**.

b

Solution

A coherent state is an eigenstate of an annihilation operator. Bosonic unnormalized states have exponential overlap and the Gaussian resolution of identity used above; their overcompleteness supplies a phase-space integration measure, not an orthonormal state count. For a normal-ordered many-body Hamiltonian, coherent-state insertion gives the bosonic coherent-state path integral

Z = \int_{periodic} D (\overset{ˉ}{ϕ}, ϕ) exp [- \int_{0}^{β} d τ (\sum_{i} \overset{ˉ}{ϕ}_{i} \partial_{τ} ϕ_{i} + H_{N} (\overset{ˉ}{ϕ}, ϕ) - μ \sum_{i} \overset{ˉ}{ϕ}_{i} ϕ_{i})] .

Constants in

H_{N}

must be retained. The exact time-sliced prescription places creation and annihilation symbols on adjacent slices; discarding that prescription can change ordering corrections, as the oscillator example demonstrates. The derivative term is first order because the fields describe phase space.

Fermionic coherent states instead use Grassmann variables. Their Gaussian identity is

\int d \overset{ˉ}{ψ} d ψ e^{- \overset{ˉ}{ψ} A ψ} = det A

, whereas complex bosons give

(det A)^{- 1}

. Anticommutation in the trace produces antiperiodic fields. Accordingly bosons have

ν_{n} = 2 πn / β

, fermions

ν_{n} = (2 n + 1) π / β

, in these inverse-energy time units.

For one mode of grand-canonical energy

ξ

, the products give

Z_{B} = (1 - e^{- β ξ})^{- 1}

and

Z_{F} = 1 + e^{- β ξ}

; convergence requires

ξ > 0

for the bosonic Gaussian. Correlation functions follow by source differentiation. Quartic interactions can be treated perturbatively or decoupled with a Hubbard–Stratonovich transformation; a saddle then describes an order parameter and quadratic fluctuations its collective modes. Such saddles require stability and an appropriate control parameter; the functional notation alone is not an exact solution of an interacting problem.

c

Solution

For a dilute three-dimensional weakly interacting Bose gas with repulsive short-range scattering, use a contact coupling

g = 4 π ℏ^{2} a_{s} / m > 0

and

n a_{s}^{3} ≪ 1

. In imaginary time

τ_{E}

, the grand-canonical action is

S_{E} = \int_{0}^{β ℏ} d τ_{E} \int d^{3} x [\overset{ˉ}{ψ} ℏ \partial_{τ_{E}} ψ + \frac{ℏ ^{2}}{2 m} ∣\nabla ψ ∣^{2} - μ ∣ ψ ∣^{2} + \frac{g}{2} ∣ ψ ∣^{4}], Z = \int D ψ e^{- S_{E} /ℏ} .

The uniform saddle has

n_{0} = μ / g

. Its real-time equation is the Gross–Pitaevskii equation; quantum depletion means condensate and total density need not coincide exactly.

Writing

ψ = n_{0} + δ ψ

and retaining quadratic fluctuations gives a Nambu kernel with diagonal

ϵ_{k} + g n_{0}

, anomalous coupling

g n_{0}

and

ϵ_{k} = ℏ^{2} k^{2} / (2 m)

. The Bogoliubov quasiparticle dispersion is

E_{k} = ϵ_{k} (ϵ_{k} + 2 g n_{0}), E_{k} ≃ ℏ c_{s} ∣ k ∣, c_{s} = g n_{0} / m .

Phase and density fluctuations form the gapless sound mode associated with broken global

U (1)

symmetry. At large momentum the spectrum crosses towards particle behaviour; the healing length is

ξ = ℏ/ 2 m g n_{0}

. The depletion integral

\int d^{3} k v_{k}^{2} / (2 π)^{3}

is finite in three dimensions and gives

(n - n_{0}) / n ≃ 8 n a_{s}^{3} / (3 π)

to leading order. The contact coupling denotes a scattering-length-renormalized interaction; naive ultraviolet zero-point integrals require renormalization.

The condensate phase gives superfluid velocity

v_{s} = (ℏ/ m) \nabla θ

and quantized circulation

\oint v_{s} \cdot d l = 2 π ℏ ℓ / m

. The Landau criterion,

v_{c} = min_{k} E_{k} / (ℏ k)

, gives

c_{s}

for this homogeneous Bogoliubov spectrum. Boundaries and vortex nucleation can cause dissipation below that value, so it is not a universal measured critical speed. Condensation and superfluid stiffness are related but distinct; dimensional fluctuations can remove true condensate order while retaining superfluid responses.

Figure 1.
The dilute-gas Bogoliubov spectrum crosses from sound to particle-like excitations
. **Source**. **Details**.

2013

Paper 1 (Quantum Condensed Matter Field Theory, Paper 160)