Many-body path integrals beyond tunnelling

Key claim

A many-body path integral sums over more than particle trajectories.

Its integration variables may be coherent-state amplitudes, Grassmann fields, order parameters, gauge fields, noise histories, replicas, or auxiliary fields that were absent from the original Hamiltonian.

Instantons form one saddle class within this functional integral.

Theme

Interacting path integrals: exact representations, collective fields, and nonperturbative saddles.

Guiding question

When you write an interacting quantum system as a path integral, which parts are:

1. an exact rewriting of the operator problem;
2. a controlled collective-field approximation;
3. a numerical sampling problem;
4. a real-time interference problem;
5. a search for nonperturbative saddles?

A Trotter identity, a saddle approximation, and a Monte Carlo estimator make different claims and fail for different reasons.

Setup: one model, several path integrals

Take the attractive two-site Bose–Hubbard model at fixed total particle number $N$:

Start from the equilibrium operator identity

Insert coherent-state resolutions of the identity between short imaginary-time steps. Continuum notation then gives

The definition still includes the time lattice, measure, boundary conditions, and operator ordering. The displayed continuum integral abbreviates their controlled limit.

At fixed $N$, introduce the population imbalance and relative phase,

On the spin-coherent-state sphere, the classical Hamiltonian is, up to a constant,

Define

For $\mathrm{\Lambda }>1$, mean-field theory has two degenerate minima,

The collective Euclidean action is

up to a Berry-phase gauge choice and boundary terms.

The Bose–Hubbard dimer supplies a concrete instanton example within a much wider path-integral framework.

The Path-Integral Ladder

Level 1: exact representation

Trotter decomposition and coherent-state insertion rewrite an operator trace or amplitude as a sum over histories, without assuming a saddle point.

Level 2: exact change or enlargement of variables

A Hubbard–Stratonovich transformation introduces an auxiliary field to linearize an interaction. The field need not correspond to a microscopic observable.

Level 3: controlled or uncontrolled truncation

A stationary auxiliary field gives mean-field theory. Its Hessian gives Gaussian fluctuations, collective modes, and often RPA-type response.

Control requires a small parameter, a large-$N$ limit, weak coupling, or a demonstrated separation of scales.

Level 4: nonperturbative sectors

An expansion about one vacuum misses instantons, vortices, skyrmions, bounces, and other complex saddles.

Boundary conditions and the integration contour decide which saddles contribute. The stationary equations only identify candidates.

Level 5: computational representation

Researchers may sample the discretized integral by Monte Carlo, reorganize it diagrammatically, deform its contour, or contract it as a tensor network. Formal exactness says nothing about computational cost.

Analysis

Derivation: interactions become fluctuating fields

Consider a schematic fermionic action

A continuous Hubbard–Stratonovich identity has the form

with contours and factors adjusted to the sign of $U$ and the chosen channel.

The fermions are now quadratic and can be integrated out:

where

The saddle equation

is a self-consistent mean-field or gap equation.

Expanding

gives

Zeros or poles of the inverse fluctuation kernel ${\mathrm{\Gamma }}^{(2)}$ determine collective modes and instabilities.

Depending on the chosen channel, the field $\varphi$ describes density, magnetization, pairing, or another collective degree of freedom.

Different exact HS decompositions reproduce the same untruncated integral. Once they are truncated at saddle level, however, they can disagree.

This channel dependence is the practical content of the Fierz ambiguity.

Scope and assumptions: why the measure may not be positive

After fermions are integrated out, auxiliary-field Monte Carlo often produces

Some symmetries force the determinant product to be nonnegative. The half-filled repulsive Hubbard model on a bipartite lattice is a standard example.

When those conditions fail, the weight may change sign or acquire a complex phase.

Sampling the absolute value gives

If $\mathrm{\Delta }f>0$, the signal-to-noise ratio decays exponentially with inverse temperature $\beta$ and volume $V$.

The generic sign problem is NP-hard. The result does not make every instance equally hard: symmetry removes the problem in several important model classes.

A Trotterized action retains discretization error until the time-step limit is taken or a continuous-time formulation is used.

Coherent-state actions also contain Berry phases and endpoint subtleties. Replacing operators naively by commuting numbers can miss ordering corrections.

Near a critical point, soft directions develop in the Hessian. A Gaussian expansion may then fail where collective fluctuations matter most.

Deriving $S_{\mathrm{e}\mathrm{f}\mathrm{f}}$ can be exact. Claiming that one smooth, real saddle dominates requires an additional argument.

Physical interpretation: choose the contour to match the question

Imaginary time computes equilibrium quantities:

From it one obtains thermodynamics, Matsubara correlators, and ground-state projection. Real-time response and nonequilibrium evolution instead use a forward-and-backward Schwinger–Keldysh contour:

The two fields track the ket and bra. Together they enforce normalization and causality.

The Keldysh rotation

separates classical histories from response or quantum components.

Integrating out an environment gives the reduced system an influence action:

The influence functional is generally nonlocal in time. Its retarded part encodes dissipation; its Keldysh component encodes fluctuations and noise.

A stochastic weak-noise process can also have

Its optimal rare path obeys saddle equations much like a Euclidean instanton.

The histories mean different things. A stochastic path carries probability; a quantum path contributes a complex amplitude that can interfere with other saddles.

The Instanton Sector: one nonperturbative use

Return to the attractive Bose–Hubbard dimer.

The two mean-field minima are approximately

A collective saddle connects $z=+z_0$ to $z=-z_0$ in Euclidean time.

The first-order Berry-phase action yields Hamilton equations in imaginary time rather than Newton’s equation for a particle in an inverted potential.

The phase $\varphi$ generally becomes complex along the connecting saddle. Eliminating it can produce a one-coordinate action for $z$, but with a coordinate-dependent kinetic term.

At large $N$,

Here

The factor $N$ reflects collective semiclassical scaling, not $N$ independent particles paying identical tunnelling actions.

The microscopic Fock-space picture

The hopping operator transfers only one boson:

The endpoint states first couple at order $N$ in $K$.

For the unperturbed interaction Hamiltonian,

The virtual states near $k=N/2$ are the most energetically expensive. The collective instanton packages this chain of matrix elements and denominators into a saddle action and prefactor.

When the one-instanton picture fails

Near $\mathrm{\Lambda }=1$, a collective mode softens. The fluctuation determinant becomes singular, so the standard dilute-instanton expansion is not uniform.

Coupling to a gapless environment may generate a memory kernel:

The bath can change the tunnelling exponent, destroy coherence, or produce a localization transition.

Multiple complex saddles can also interfere. The saddle equations find candidates; the original contour determines which ones contribute.

Frontier Map: beyond one saddle

1. Complex contours and Lefschetz thimbles

Holomorphic flow deforms the original integration manifold into complex field space without crossing singularities.

Phase fluctuations often shrink near a thimble. Hubbard-model studies report substantial improvement where the undeformed average sign is exponentially small.

Residual phases, Jacobians, multimodal sampling, singularities, and interference among several thimbles can still dominate the calculation.

2. Symmetry-designed sign-free formulations

Some auxiliary-field decompositions produce determinant pairs related by antiunitary symmetry:

Then

For these model classes, Monte Carlo weights are nonnegative exactly. Researchers are enlarging the classes through Majorana representations, symmetry criteria, basis changes, and redesigned interactions.

3. Generative and flow-based sampling

Normalizing flows learn an invertible map from a simple reference distribution to field configurations.

Reweighting or a Metropolis correction can keep the resulting sampler asymptotically exact.

Gauge-equivariant flows can build local symmetry into the architecture. Proof-of-principle work has reached four-dimensional $SU(3)$ lattice gauge theory.

Training cost, mode coverage, volume scaling, topology freezing, and transfer between couplings still limit these methods.

Learned proposals may explore the measure faster, but they leave the partition function unchanged and do not automatically cure a complex phase.

4. Temporal tensor networks

A discretized path integral forms a tensor network in spacetime. For an open system, TEMPO represents the influence functional as a matrix product operator along the time direction.

This compresses bath memory when temporal correlations have manageable bond dimension.

Recent extensions treat Grassmann influence functionals for fermionic impurity models. 2026 preprints extend the framework to more general bosonic couplings and superconducting fermionic baths.

Temporal entanglement or operator-space complexity sets the cost. Long memory and strong nonequilibrium correlations can force rapid bond-dimension growth.

The calculation compresses correlations among histories instead of sampling each history independently.

False claim to diagnose

Once an interacting system has been written as a path integral, the many-body problem has been solved; all that remains is approximation detail.

The rewriting can be exact while evaluation remains exponentially hard.

The action may be complex, nonlocal, multimodal, singular, or dominated by several interfering sectors.

Even the choice of variables can move difficulty between the action, measure, contour, and observable.

The narrower claim is:

A path integral reorganizes the many-body problem through fields, topology, saddles, stochastic samples, or tensor contractions. None is guaranteed to be efficient.

What follows — and what does not

Five layers are involved:

Layer	Object	Status
Operator theory	$\mathrm{Tr}{\mathrm{e}}^{-\beta H}$ or a real-time amplitude	Exact starting problem
Functional representation	Coherent, Grassmann, auxiliary, or contour fields	Exact when the measure and regulator are specified
Effective description	Saddle, gradient expansion, or low-energy field	Controlled only with a scale argument
Nonperturbative sector	Instantons, defects, complex saddles	Contribution depends on contour and boundary data
Numerical realization	Monte Carlo, flows, diagrams, tensor contraction	Accuracy and cost are method-dependent

The many-body instanton occupies the fourth row and cannot substitute for the other four.

Exercise: recover the instanton exponent microscopically

Work in the strong-attraction regime

Starting from $|N,0⟩$, compute the leading off-diagonal matrix element connecting it to $|0,N⟩$ using $N$th-order degenerate perturbation theory.

Show that

At fixed

use Stirling’s approximation to extract the leading exponential behavior.

Then answer the transfer question:

Which part of this result would be missed by keeping only one Gaussian expansion around either minimum?

Hint 1

The $k$th hopping matrix element along the minimal Fock-space path is

$$-\frac{K}{2}\sqrt{k(N-k+1)}.$$Copy LaTeX

Hint 2

Use

$$\prod _{k=1}^{N-1}k(N-k)=[(N-1)!{]}^2.$$Copy LaTeX

Oral check 1. Why does the coupling first appear at order $N$, even though the Hamiltonian is only quadratic in creation and annihilation operators?

Oral check 2. In auxiliary-field Monte Carlo, are the sampled variables electron trajectories, microscopic observables, or decoupling fields?

Oral check 3. Why can a Keldysh path integral not be interpreted as an ordinary probability distribution over histories?

Solution

There is one minimal monotonic sequence:

$$|N,0⟩\to |N-1,1⟩\to \cdots \to |0,N⟩.$$Copy LaTeX

The product of hopping matrix elements is

$$\prod _{k=1}^N[-\frac{K}{2}\sqrt{k(N-k+1)}]={(-\frac{K}{2})}^{N}N!.$$Copy LaTeX

The $N-1$ intermediate denominators are

$$E_0-E_k=-gk(N-k).$$Copy LaTeX

Therefore,

$$|t_{\mathrm{e}\mathrm{f}\mathrm{f}}|=\frac{(K/2{)}^{N}N!}{g^{N-1}[(N-1)!{]}^2}.$$Copy LaTeX

The symmetric and antisymmetric combinations split by

$${\mathrm{\Delta }}_N=2|t_{\mathrm{e}\mathrm{f}\mathrm{f}}|.$$Copy LaTeX

Substitute

$$g=\frac{\mathrm{\Lambda }K}{N}.$$Copy LaTeX

Then

$${\mathrm{\Delta }}_N=K{2}^{1-N}\frac{N^N}{{\mathrm{\Lambda }}^{N-1}(N-1)!}.$$Copy LaTeX

Stirling’s formula gives

$$\boxed{{\displaystyle \log {\mathrm{\Delta }}_N=-N[\log (2\mathrm{\Lambda })-1]+O(\log N)}}.$$Copy LaTeX

Thus

$${\mathrm{\Delta }}_N\sim \mathrm{poly}(N)\exp \{-N[\log (2\mathrm{\Lambda })-1]\}.$$Copy LaTeX

The exponent is the strong-coupling limit of the collective instanton action.

The algebraic corrections belong to the prefactor and to finite-$N$ matching.

A Gaussian expansion around one minimum describes perturbative fluctuations confined to that basin. It cannot generate the exponentially small matrix element between distinct saddles.

The auxiliary-field variables in Monte Carlo are decoupling fields. Their relation to observables depends on the chosen channel and source derivatives.

A Keldysh weight is complex:

$${\mathrm{e}}^{\mathrm{i}S[{\varphi }_{+}]-\mathrm{i}S[{\varphi }_{-}]}.$$Copy LaTeX

It sums amplitudes with interference and therefore is not an ordinary positive history probability.

Check your understanding

Take one interacting problem you know and identify:

1. the exact path-integral variables;
2. any auxiliary or collective fields;
3. the small parameter behind a saddle expansion;
4. the main numerical obstruction;
5. whether the relevant contour is Euclidean, Keldysh, or complex-deformed.

Without item 3, the approximation has no stated control parameter.

You may also reply with “deeper,” “too easy,” “too hard,” or your perturbative derivation.

Further Reading

• Computational complexity and fundamental limitations of the fermion sign problem
• Taming the finite-density Hubbard sign problem with Lefschetz thimbles
• Normalizing flows for lattice gauge theory in arbitrary spacetime dimension
• Efficient non-Markovian dynamics using time-evolving matrix product operators
• Grassmann TEMPO for fermionic path-integral simulations
• Tensor-network influence functionals for general Gaussian bosonic baths
• Grassmann TEMPO for superconducting baths

Connections and next step

• Pressure point: a path integral is a representation, not an automatic solution.
• Exact layer: Trotterization, coherent states, and Hubbard–Stratonovich identities.
• Approximation layer: mean field, Gaussian fluctuations, and collective effective actions.
• Nonperturbative layer: instantons, defects, Berry phases, and complex saddles.
• Computational layer: sign-free symmetry, contour deformation, generative sampling, and temporal tensor networks.
• Revisit: derive the Keldysh influence action of an Ohmic bath in three runs.