# Geometry, Causality & the Limits of Critical Cooling
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The geometry of a quench determines which excitations can reach a given region and when they arrive. This makes spacetime structure a control resource in critical systems. Agarwal and collaborators explore that resource through supersonically split condensates, hyperbolic quench trajectories, and moving fronts in long-range and two-dimensional models. The broader question is when a simple low-energy spacetime picture remains predictive for a finite, microscopic quantum simulator.
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### Figure to read
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![A moving front populates different propagation directions unequally.](https://arxiv.org/html/2404.02957v1/2Dconcept.svg)
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A moving front populates different propagation directions unequally. [Figure 1 in the source paper](https://arxiv.org/html/2404.02957v1#S1.F1). Bernier, Simon; Agarwal, Kartiek. *Spatiotemporal quenches for efficient critical ground state preparation in the two-dimensional transverse field Ising model*, Physical Review B 111, 054311 (2025). [Paper](https://doi.org/10.1103/PhysRevB.111.054311). [CC BY 4.0](https://creativecommons.org/licenses/by/4.0/). Original manuscript graphic; no alterations.
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Read the arrows as excitation trajectories, not as the velocity of the control pattern. Compare the cold modes behind the front with the hotter modes traveling alongside it; this is why local preparation quality and total sample energy are different tests.
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### Direction-dependent effective temperatures
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In a one-dimensional condensate split by a supersonically moving process, right- and left-moving excitations need not be populated equally. Agarwal, Dalla Torre, and collaborators analyze a prethermal state with direction-dependent Doppler-shifted temperatures. These temperatures characterize sectors of a nonequilibrium distribution; they do not imply that the entire system has reached a single Gibbs ensemble. The proposed interference measurements are designed to expose precisely that asymmetry. [1](https://doi.org/10.1103/PhysRevLett.113.190401)
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This is a useful conceptual precursor to critical cooling. The same effective Lorentz structure that organizes the split-condensate dynamics helps predict where a moving mass quench deposits energy. The connection is physical rather than merely terminological: both problems exploit how a spacetime-dependent perturbation populates counterpropagating modes differently. The original superluminal-front proposal then uses that imbalance to create a large low-energy region. [2](https://doi.org/10.1103/PhysRevLett.120.210604)
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### Why choose a hyperbola?
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Mitra, Ippoliti, Bhatt, Sondhi, and Agarwal consider trajectories of the form $t^2-x^2=T_0^2$ in units with the effective propagation speed set to one. In a relativistic one-dimensional theory, this geometry has useful symmetry properties. Starting from a related gapped system, the quench brings the evolution into a conformal regime whose stress-energy tensor is strongly constrained. The resulting cooling argument is broader than an exact solution of one free-particle model. [3](https://doi.org/10.1103/PhysRevB.99.104308)
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The qualification matters: “arbitrary” in the paper's title is bounded by its assumptions about the effective theory, dimension, initial state, and quench construction. It is not a method for preparing the ground state of every interacting lattice Hamiltonian. Singular lightlike trajectories and ultraviolet cutoffs also require attention. The continuum result is a guide to protocol design, while lattice simulations test whether the experimentally relevant scales preserve its benefits.
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### Long-range interactions test the causal picture
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Power-law interactions can change the low-energy dispersion and the relation between distance and propagation time. Bernier and Agarwal examine spatiotemporal quenches in long-range transverse-field Ising chains, asking when the cooling behavior associated with relativistic modes remains visible. The answer depends on the interaction regime, rather than simply on whether the interactions are labeled long-range. A single fitted front velocity may cease to characterize the relevant excitations. [4](https://doi.org/10.1103/PhysRevB.108.024310)
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This is an example of a productive failure test. If the effective theory predicts a velocity-dependent cooling minimum, deliberately varying dispersion provides a way to test that explanation. A missing minimum then carries information about the limits of the mechanism; it should not be concealed by presenting only the favorable parameter regime. For a practical simulator, it also indicates whether changing interaction range can help or hinder preparation.
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### The two-dimensional frontier
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The two-dimensional transverse-field Ising study extends the question to interacting systems where numerical simulation itself is demanding. Bernier and Agarwal compare excitation energy and correlation properties and identify an optimal front velocity near the emergent speed in their simulated geometries. Finite width, entanglement growth, and boundary effects limit the extrapolation. Low energy and long correlations are encouraging evidence, but they do not establish an unrestricted thermodynamic-limit guarantee. [5](https://doi.org/10.1103/PhysRevB.111.054311)
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The wider shortcuts-to-adiabaticity literature offers a useful resource comparison: geometrically structured control may avoid implementing complicated auxiliary operators, but requires reliable spatial addressing and timing. [6](https://doi.org/10.1103/RevModPhys.91.045001) An experimentally meaningful next step is to compare a uniform ramp, a linear front, and a curved trajectory under the same hardware constraints. Measurements should resolve where residual energy accumulates and whether the region of interest retains the intended correlations. This page connects [moving fronts](https://knowen.org/nodes/33542) to [thermalization](https://knowen.org/nodes/33538): a protocol can create locally cold or prethermal behavior without making the whole system globally equilibrated.
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### References
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1. Agarwal, Kartiek; Torre, Emanuele G. Dalla; Rauer, Bernhard; Langen, Tim; Schmiedmayer, Jörg; Demler, Eugene. [Chiral Prethermalization in supersonically split condensates](https://doi.org/10.1103/PhysRevLett.113.190401). Phys. Rev. Lett. 113, 190401 (2014). [Open manuscript](https://arxiv.org/abs/1402.6716).
2. Agarwal, Kartiek; Bhatt, R. N.; Sondhi, S. L.. [Fast Preparation of Critical Ground States Using Superluminal Fronts](https://doi.org/10.1103/PhysRevLett.120.210604). Physical Review Letters 120, 210604 (2018). [Open manuscript](https://arxiv.org/abs/1710.09840).
3. Mitra, Prahar; Ippoliti, Matteo; Bhatt, R. N.; Sondhi, S. L.; Agarwal, Kartiek. [Cooling arbitrary near-critical systems using hyperbolic quenches](https://doi.org/10.1103/PhysRevB.99.104308). Physical Review B 99, 104308 (2019). [Open manuscript](https://arxiv.org/abs/1809.01681).
4. Bernier, Simon; Agarwal, Kartiek. [Spatiotemporal Quenches in Long-Range Hamiltonians](https://doi.org/10.1103/PhysRevB.108.024310). Physical Review B 108, 024310 (2023). [Open manuscript](https://arxiv.org/abs/2212.07499).
5. Bernier, Simon; Agarwal, Kartiek. [Spatiotemporal quenches for efficient critical ground state preparation in the two-dimensional transverse field Ising model](https://doi.org/10.1103/PhysRevB.111.054311). Physical Review B 111, 054311 (2025). [Open manuscript](https://arxiv.org/abs/2404.02957).
6. Guéry-Odelin, D.; Ruschhaupt, A.; Kiely, A.; Torrontegui, E.; Martínez-Garaot, S.; Muga, J. G.. [Shortcuts to adiabaticity: concepts, methods, and applications](https://doi.org/10.1103/RevModPhys.91.045001). Rev. Mod. Phys. 91, 045001 (2019). [Open manuscript](https://arxiv.org/abs/1904.08448).
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*Independent research synthesis. Literature checked 5 September 2026; preprints are identified in the references.*
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# Parents
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* Non-equilibrium Control & Quantum State Preparation⏎