Applications of TCSA to quenches
This is a modal window.
The media could not be loaded, either because the server or network failed or because the format is not supported.
Formal Metadata
| Title |
| |
| Title of Series | ||
| Number of Parts | 18 | |
| Author | ||
| Contributors | ||
| License | CC Attribution 3.0 Unported: You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor. | |
| Identifiers | 10.5446/46750 (DOI) | |
| Publisher | ||
| Release Date | ||
| Language |
Content Metadata
| Subject Area | ||
| Genre | ||
| Abstract |
|
1
13
14
17
18
00:00
Degrees of freedom (physics and chemistry)WärmestrahlungThermodynamic equilibriumPhysical systemQuantum field theoryReal numberMany-sorted logicCross-correlationCharacteristic polynomialLimit (category theory)TheoremINTEGRALThermodynamisches SystemMechanism designLocal ringUniverse (mathematics)Multiplication signDescriptive statisticsVariety (linguistics)InfinityScaling (geometry)MathematicsMereologyInvariant (mathematics)Game theoryHamiltonian (quantum mechanics)Boundary value problemParameter (computer programming)Operator (mathematics)Lattice (order)CalculationExplosionMusical ensembleClosed setFeldtheorieHypothesisRadical (chemistry)Eigenvalues and eigenvectorsLight coneStatistical mechanicsKörper <Algebra>Different (Kate Ryan album)Phase velocityFundamental theorem of algebraTime evolutionParticle statisticsTheory of relativityNumerical analysisDynamical systemDegree (graph theory)TangentLecture/Conference
08:24
Time evolutionMany-sorted logicKörper <Algebra>Physical systemTheoryPoint (geometry)Configuration spaceCondensationDegrees of freedom (physics and chemistry)Musical ensembleForm factor (electronics)Density matrixState of matterQuantum mechanicsHamiltonian (quantum mechanics)MathematicsMultilaterationIndependence (probability theory)Limit (category theory)FeldtheorieTerm (mathematics)Quantum chromodynamicsVariable (mathematics)FinitismusTrajectoryStatistical mechanicsThermal expansionLengthThermodynamisches SystemRenormalization groupMultiplication signMereologyDescriptive statisticsThermodynamic equilibriumScaling (geometry)Kritischer Punkt <Mathematik>Continuum hypothesisLokaler KörperAtomic numberStandard deviationGroup actionImage resolutionEigenvalues and eigenvectorsPlane (geometry)FrequencyOrbitUniverse (mathematics)VotingINTEGRALRange (statistics)Supersonic speedSampling (statistics)Sign (mathematics)Quantum field theoryFood energyPrice indexGrothendieck topologyLecture/Conference
16:48
Parameter (computer programming)Volume (thermodynamics)FeldtheorieComputabilityTheoremFactory (trading post)Eigenvalues and eigenvectorsDimensional analysisMultiplication signFinitismusMomentumElement (mathematics)MathematicsFormal power seriesPoint (geometry)TheoryS-MatrixState of matterTerm (mathematics)Game theory8 (number)Functional (mathematics)Quantum chromodynamicsAsymptotic analysisPopulation densityFood energyInvariant (mathematics)Basis <Mathematik>Group actionDifferent (Kate Ryan album)Numerical analysisCalculationSummierbarkeitCross-correlationWell-formed formulaModel theoryInfinityCurveExtension (kinesiology)Goodness of fitProjective planeCue sportsLocal ringOperator (mathematics)Hamiltonian (quantum mechanics)Power (physics)SpacetimeINTEGRALMultiplicationBoundary value problemMany-sorted logicAxiomatic systemBootstrap aggregatingThermal expansionForm factor (electronics)MereologyCartesian coordinate systemMassExistenceDivisorFiber bundleZustandsgrößeDescriptive statisticsEqualiser (mathematics)Körper <Algebra>Limit (category theory)WärmestrahlungModulformPropositional formulaFree groupConformal mapLecture/Conference
25:12
Thermal expansionRing (mathematics)Pairwise comparisonKörper <Algebra>ChainStandard deviationTransverse waveHamiltonian (quantum mechanics)Ising-ModellDimensional analysisTerm (mathematics)MassFree groupThermal expansionVolume (thermodynamics)Analytic setLimit (category theory)CalculationPopulation densityTheoryQuantum field theoryModel theoryHeat transferParameter (computer programming)Range (statistics)Lecture/Conference
26:14
Thermal expansionPairwise comparisonMoving averageAnalytic setTime seriesThermal expansionSquare numberMultiplication signMaß <Mathematik>Different (Kate Ryan album)CalculationFood energyExtrapolationEnergy levelSpacetimeMassConformal mapMereologyForm factor (electronics)Infinite setResultantTerm (mathematics)Line (geometry)Many-sorted logicTheoryCurveAbsolute valueMultilaterationVolume (thermodynamics)Scaling (geometry)Physical systemInfinityOperator (mathematics)ExponentiationTransformation (genetics)ModulformFrequencyState of matterValidity (statistics)Analytic setHamiltonian (quantum mechanics)Total S.A.ComputabilityPopulation densityHilbert spaceModel theoryDivisorAlgebraic structureINTEGRALFunctional (mathematics)MomentumVacuumCone penetration testPower (physics)ApproximationDot productCoefficientSign (mathematics)Curve fittingDiagram
30:18
Pairwise comparisonAnalytic setTime seriesExtrapolationThermal expansionCurveMatrix (mathematics)Point (geometry)Hamiltonian (quantum mechanics)MereologyAnalytic continuationTime evolutionDifferent (Kate Ryan album)Many-sorted logicVolume (thermodynamics)Conformal mapColor confinementOscillationSpacetimeExponentiationResultantMultiplication signLine (geometry)ExtrapolationKörper <Algebra>Kritischer Punkt <Mathematik>Light coneElement (mathematics)Basis <Mathematik>Boundary value problemIsing-ModellCalculationMaß <Mathematik>Physical systemGroup actionFrequencyFree groupVector spaceTransformation (genetics)Standard errorFood energyModel theoryClosed setNumerical analysisOrder (biology)EvoluteDot productChainDecimalBlock (periodic table)InfinityParameter (computer programming)Hill differential equationModulformTerm (mathematics)INTEGRALPhase transitionRight angleMassCoefficient of determinationState of matterPolar coordinate systemMathematicsMusical ensembleComputer animationDiagram
37:35
Form factor (electronics)ThetafunktionMatrix (mathematics)Quantum mechanicsFood energyNumerical analysisElement (mathematics)Volume (thermodynamics)Power (physics)Hamiltonian (quantum mechanics)ApproximationMultiplication signState of matterPotenz <Mathematik>Proof theoryTheoremDifferent (Kate Ryan album)Transformation (genetics)MomentumSpacetimeInfinityKörper <Algebra>Model theoryOperator (mathematics)Scaling (geometry)Order (biology)Point (geometry)Formal power seriesFinitismus2 (number)CoefficientMereologyCalculationComputabilityExponentiationCohen's kappaExpected valueSpectrum (functional analysis)ExtrapolationAnalytic setPredictabilityNormal subgroupResultantFrequencyLatent heatHilbert spaceGroup actionAlgebraic structureProduct (business)Conformal field theoryFinite setWell-formed formulaMaß <Mathematik>Lecture/Conference
44:43
DivisorThermal expansionHill differential equationINTEGRALCurveModel theoryTranslation (relic)DivisorTheoryIndependence (probability theory)Körper <Algebra>S-MatrixNumerical analysisLine (geometry)Student's t-testVolume (thermodynamics)PredictabilityProduct (business)Many-sorted logicMaß <Mathematik>Dot productIsing-ModellSpacetimeMassPopulation densityGraph coloringLimit (category theory)Natural numberGame theoryState of matterParity (mathematics)Analytic setGroup actionForm factor (electronics)DampingTime evolutionParameter (computer programming)FinitismusLight conePerturbation theoryResultantAsymptotic analysisKritischer Punkt <Mathematik>Conformal field theoryFeldtheorieRenormalization groupSet theoryMultiplication signThermal expansionPotenz <Mathematik>Right angleComputational fluid dynamicsThetafunktionChinese remainder theoremPoint (geometry)Observational studyInvariant (mathematics)WeightTerm (mathematics)MomentumOscillationClassical physicsRange (statistics)Lecture/Conference
51:51
Thermal expansionThermal expansionMultiplication signResultantFigurate numberCombinatory logicCurveIsing-ModellModel theoryComplex numberINTEGRALTheoryProduct (business)Form factor (electronics)Perturbation theoryDescriptive statisticsDensity matrixInfinityParameter (computer programming)CalculationState of matterMereologyFrequencyFormal power seriesFood energyDifferent (Kate Ryan album)SpacetimeDivisorPhysical systemShift operatorPopulation densityLimit of a functionOrder (biology)Operator (mathematics)SummierbarkeitFree groupPlane (geometry)Limit (category theory)Term (mathematics)Functional (mathematics)Analytic setSeries (mathematics)Special unitary groupPole (complex analysis)Conformal mapDiagramLecture/Conference
59:12
Thermal expansionMathematicsFeldtheorieLine (geometry)Thermal expansionCalculationINTEGRALLimit (category theory)Körper <Algebra>TheoryPopulation densityCurveMultiplication signParameter (computer programming)Film editingPlane (geometry)Point (geometry)Lecture/Conference
01:02:14
SinePairwise comparisonVolume (thermodynamics)MereologyLine (geometry)MomentumGoodness of fitValidity (statistics)Population densityDot productOrder (biology)Algebraic structureFunctional (mathematics)State of matterCurveIsing-ModellVacuumTheoryStandard deviationComputabilityAbsolute valueCartesian coordinate systemPoint (geometry)Hamiltonian (quantum mechanics)Many-sorted logicLecture/Conference
Transcript: English(auto-generated)
00:16
Thank you for the invitation to this meeting at this very nice place.
00:21
And I want to present a status report of ongoing research, including also some extremely recent stuff.
00:40
Some of it is just a few days old, actually. But let's start with some introduction. My talk will be about quantum quenches in QFT, and more specifically about using Hamiltonian
01:04
truncation approach. Well, not just that, but it's a very big part of the thing that we're going to talk about to understand these quantum quenches. What is a quantum quench?
01:23
Probably many people are familiar with it, but anyway, some quick introduction doesn't do much harm. So we take a Hamiltonian of a system, which is given the system has this Hamiltonian until
01:40
some time, say time zero, and the time zero sort of like we turn a switch, and we change something abruptly. I mean, there's something could be a coupling in the Hamiltonian. It could be some local flip of some degree of freedom. It could be some change in boundary conditions, could be a lot of things.
02:01
I'm going to talk about here, mostly, well, not mostly, I think exclusively this time about global quantum quenches. Which means that I will suppose that H0 is translationally invariant, H is translationally invariant. In principle, it's an infinite system, which is not the case for the numerical calculations,
02:21
obviously. But in principle, it's an infinite system, and so this is called the global quantum quench, translationally invariant global quantum quench, and both of these are translationally invariant. What we are doing, actually, is we are adding some interaction, well, actually, integral
02:44
local operator to the Hamiltonian. What are we interested in? In general, this is a very big field with lots of different systems being investigated, but this touches some very fundamental cornerstones of quantum statistical mechanics.
03:02
Basically, what you are saying here is that I have a closed quantum system, and I bring it out of equilibrium. What I want to know is whether it equilibrates, whether it becomes thermal by itself. I won't go into specific details about why you think this is possible, but there
03:20
is something called the eigenstate thermalization hypothesis, which is basically states, a mechanism whereby non-integrable systems can become thermal even without an environment. If the system is large enough, then for its subsystems, it can act as its own environment, sort of.
03:40
If the system is integrable, then some other interesting things could happen, like non-thermalization or thermalization in a more general sense, equilibration, I should say, in a more general sense. The first question is whether there is equilibration, whether the equilibrium, if it exists, whether
04:05
it's thermal or not. If there is equilibration, and the next question is how does it reach equilibration, so it's the question of relaxation, so how does the time evolution actually happen towards
04:27
equilibrium, what are the relevant timescales, so actually I would say time evolution, timescales,
04:43
and related stuff, and while you might actually be, if there is no relaxation, that could even be more interesting, you could have some very interesting phenomena, some unexpected things. Normally, I should say, whenever I turn on a new system on the computer, just to look
05:03
at its behavior numerically right away, if I turn on a brand new system, I always found something which I didn't expect. So this was happening for quite some time, so I should say there are lots of interesting things. If one just looks into these behaviors, there's a variety of behaviors, but it's also interesting
05:28
to ask whether there are some universal features in these non-equilibrium behaviors. For example, such as the so-called light cone behavior, which is systems with nearest neighbor
05:52
or local, rather, I would say local interactions, statistical mechanics, there are nice theorems saying that, the Libra means on theorems basically, there are several versions of them,
06:03
saying that there is an upper limit to the propagation speed of excitations, and that means that correlations and tangent and whatever can only propagate inside the so-called light cone, where the light cone is some sort of characteristic speed, speed here, not necessarily light, but everybody calls it the light cone because it just looks like an characteristic light cone.
06:22
So these are the things. Now, next question. So this is a statistical mechanical question. So why do we think QFT? Well, maybe the simplest way to come to this idea is that QFT, as we know in equilibrium,
06:42
it's sort of like a universal description of statistical mechanical system. So you might want to look at universal description. However, I'm already giving here away part of the game, that is, this is dangerous.
07:03
Because if you have a quantum quench, then an abrupt quantum quench, which you can actually do in a laboratory, so this is really an experimental relevant question that I'm talking about. So the abrupt quantum quench has a timescale tau, which is very short. You never do anything instantly. There's some timescale tau.
07:21
And your QFT, as applied to a statistical mechanical system, has a cut of lambda. So this is sort of like a timescale under which the switch happens from H0 to H. And
07:41
it's obviously a problem, or at least it seems so, when 1 over tau is larger than lambda. It doesn't necessarily need to be much larger, even if it's just comparable. That's a problem, because you are expecting to excite degrees of freedom, which are over the cutoff where your field theory description is not valid.
08:00
I'm giving away part of the game. I'm telling you already that this turned out to be not such a big problem, as it seems first. So the argument is very clever, this argument is fine. But for some reason that I'm going to tell you about, this is not such a big problem. However, there's another problem, which I think is still there, is that the QFT is not
08:25
something like a statistical mechanical system in the sense that statistical mechanical systems always, as we treat them, always have atoms. By atoms, I mean not just atoms in the sense of having some particles. But if you have a spin chain, your atom is basically a site.
08:41
And on the site, you already only have some discrete degrees of freedom. But the field theory, a local field theory, is a full continuum. A full continuum of degrees of freedom for which you can go any fine resolution down to any scale. It's a question whether relaxation can happen in such a case. And actually, the real problem is that, what is a field theory?
09:02
A field theory describes a critical point here, plus its environment. So from the statistical mechanical point of view, you are doing a very crazy thing. It's an infinitesimal environment. And actually what you are doing, you are sitting on R-G trajectories and scaling down to the critical point.
09:21
And the problem, you are changing parameters in such a way that you are keeping gaps finite and interaction quantities finite and all that. And the problem with that is, the quantum quench from statistical mechanical point of view happens, that you are looking at the QFT, happens in the vicinity of a critical point, actually an infinitely close vicinity, and it's an infinitely small quench from
09:41
statistical mechanical point of view. So it's sort of like a very, very non-trivial limit. And there are some indications, at least, that this changes things. I think Neil will talk about something like that as well, partially, Neil Robinson later.
10:00
So that our ideas, what we have from standard statistical mechanics, may not be directly or simply applicable to this situation. OK, so this is interesting. And last point, why is it interesting? Because this is experimentally realizable, at least so it's claimed. I actually did not understand the second point.
10:23
Yes? I prepare my system, I go to critical temperature, I go to some, you understand the first point about the cutoff, but what was the second point? The second point is basically, I can phrase it in relaxation times. If you think about relaxation times in the vicinity of a critical point, they tend to
10:44
blow up. So basically, what you do is relaxation time characterizes your sort of transitional period, while you are transiting between the behavior that you started to the behavior at the end point. And your relaxation time, as you are scaling your theory and continuum limit, they can blow up.
11:01
That's the way I like to think about it. So it could happen that the field theory is actually describing the transitional region. It's not describing the relaxation itself to the equilibrium. Because you just blew up this transitional region basically to occupy a utility long time. At least this could be the explanation for some behaviors that we see, that we directly
11:23
see. Obviously, if it's a non-integrable theory, I can't watch for this behavior to be exact in the sense it's numerically observed. It's calculated with certain expansions. But still, it looks very different from what you have in standard statistical systems.
11:41
And the basic problem is really that you can go up to arbitrarily small length scales and you scale your theory to that. I don't say I understand it. It's just my picture. So I just want to say there is another danger when you think about this as universal description. There's another danger that you are scaling in a very, very, very, very intricate limit
12:04
in which you can just lose part of the behavior that characterizes your statistical mechanical system simply because you are scaling it out of all ranges, more or less. And they are also claimed to be experimental realizable. So for example, Jorg Schmidt-Meyer in Vienna, which is very close to Budapest, where I
12:21
am, has this very nice system whereby there are two Seeger-like bosonic condensates, which are described. I mean, these are basically some bosonic atoms. And they are described by a Leibniger-like theory. But in certain limits, the relative phase between the condensates, which depends on
12:44
the coordinate, obeys a Sang-Gordan dynamics. At least so it's claimed, naively. So this phi variable obeys Sang-Gordan dynamics. And indeed, they have been able, for example, to see solitonic configurations in this condensate.
13:02
And some of the behavior which is related to Sang-Gordan seems to be there. But I spoke to Jorg and they also have some recent paper out which says that there are actual doubts whether this is actually described by Sang-Gordan. Some things that cannot be understood from Sang-Gordan.
13:20
It's not clear whether this is because it's not described by a field theory or because it's not exactly Sang-Gordan. But in any case, there are at least attempts to realize these quantum field theoretic quenches in a lab. If this is really quantum field theory, then maybe our methods could be applied to that.
13:46
All right. So what do we want to do? More detail. I have to find everything.
14:00
Next speaker would have it much easier. That's fine. So now with H0, I just said there's a starting Hamiltonian H0 while I can choose lots of
14:20
states to start with. I can start from the ground state of this Hamiltonian. I can start from thermal excited state. I can start from other excited states of this Hamiltonian. But I'm just doing it very simple. So I just say that I'm taking the simplest possible starting point when I'm just taking the ground state of the Hamiltonian, which I take to be zero energy.
14:44
So it's bounded from below by zero and that's it. And what we can do is that we can expand this state on the eigenstates of the post-quench Hamiltonian. And then we can obviously write down the time evolution of any observable.
15:03
It's very simple. There's an observable. Time evolution is simply just nT OM.
15:22
It's simple quantum mechanics. You just do the standard Hamiltonian and time evolution and you send it your state with the evolving state. So the first thing that it seems obvious that if there is any relaxation, if it goes to a time independent case, supposing that there are no degeneracies, then the T goes
15:44
to infinity, equilibrium should be something called the diagonal ensemble. So there is this density matrix and the thing is that the equilibrium is just this, basically
16:12
because the diagonal terms are time independent. The question is what is this diagonal ensemble, whether this can be replaced by something more statistical mechanical thing, because this is very microscopic and it contains all
16:22
the overlaps, microscopic overlaps, whether this is actually equivalent to a thermal ensemble in some sense, in a proper sense, which I'm not going into because that would take a lot of time, but you have to specify this very carefully or whether it is equivalent in a sense to some more so-called generalized ensemble or something.
16:46
But that actually lends itself to an approach by form factors when you consider field theory. So this approach is very interesting. I hope to convince you that, oh, there is a third one, thanks, but I wouldn't be able
17:06
to reach more than two of them anyway, so I just decided it doesn't make sense because I'm not, I'm just simply not tall enough to reach three blackboards at the same time.
17:23
So what you can do in a field theory, these states are just some multi-particle states. I'm talking about one-dimensional field theory here, so one plus one dimensions. Obviously, you can do all this in higher dimensions, but my computations that I'm going to present are in one plus one dimensions.
17:41
And so these are so-called rapidity parameters which characterize the momentum, so the particles have momenta. For simplicity, I'm writing formulae with just one type of particles. I mean, in the models that I'm considering, sometimes there are even eight types of particles, but I'm just neglecting all this complication because it's unnecessary detail.
18:03
And basically, if you have H, the postbench Hamiltonian, then its eigenvalue is just the sum of the energies. And then what you can do, you can also expand your state on this basis, so then it becomes
18:31
something like, there are some amplitudes here, and so this case would replace the Cs,
18:44
these states would replace the Ns. That thing there is something called the foam factor of a local operator. In many integrable theories, we have explicit formulae for that. And here, I assume already that the theory is integrable. No, not yet.
19:02
This is very generic. The application needs, normally, if you want to apply this analytically, then you need integrability because you need an explicit idea about these matrix elements. But what is this state of N particles that you're assuming to have? It's asymptotic state, asymptotic states, but I need actually the gap.
19:23
If there's a mass gap in the theory, then I have the basis of asymptotic states, okay? Yeah, I can choose any. In-states and out-states are just particular ordering of rapidities in this formalism. And then I can extend the ordering analytically to any ordering of the rapidities, basically.
19:42
Actually, there should be a 1 over N factorial here if I do this extension, and I just integrate freely over all the rapidities, then there should be a 1 over N factorial, to properly normalize it. So I did not understand your answer. You're saying if the theory is gap, then you are guaranteed to have a good basis?
20:01
Yeah, asymptoticals, asymptotic scattering states. In infinite volume, you have the scattering states. So that's when it becomes a free theory? Well, every gapped theory has these asymptotic states. The theory is not free. I mean, you have a non-trivialized matrix, but asymptotically, I mean, you have in-states and out-states, you consider the dynamics, you consider infinitely past or infinite past
20:22
or infinite future, then you are basically guaranteed to have, there are even theorems long time ago, there is axiomatic field theory, there are theorems proven about existence of these sort of states. So if it's gapped, if it's not gapped, that's a tricky question. But I suppose that my field theory is gapped, which would be the case anyway for what I'm considering explicitly.
20:42
Yeah, but is this a useful basis, because now you have to expand this state psi? Very good question. Yes, I'm trying to convince you about that. This turns out to be a good basis for, at least for very interesting quenches that you would really like to consider. It's not a trivial question at all.
21:01
Yeah, it's not a trivial question, but there's such a simplistic, because if you think about a quench, and now comes the next issue. A quench is generating a finite energy density, simply because of translational invariance. Your state, psi zero, in terms of the post-quench Hamiltonian, will have finite energy density. So in principle, it's an infinite particle state,
21:21
contains infinitely many particles. Whether it is useful to expand it in this way, you will be amazed, I promise. This works extremely well, and I will try to explain why it works, but it's a very good question. It's not an obvious question whether this should work at all.
21:43
And for this question, is integrability important? No, for this question, integrability is not important. Integrability is important for being able to actually compute that sum there. And the tricky part of that sum, where my integrability comes in, is this.
22:00
You want to know these matrix elements. And in integrable field theories, in many of them, you know explicitly these matrix elements. They were computed long time, 20, 30 years ago, by all these people, form factor, bootstrap, all that, for many theories. Also, I should give another part of the game because of these questions,
22:20
is that c. Knowing c, I will come back to that, or knowing these k functions is a very non-trivial proposition. So I will come back to that. I must come back to that at some point. I'm confused about the order limit.
22:41
So there is an infinite volume, and then we have the synthetic states. And if you go to a finite volume, then we know that everything changes. Dramatically, the eigenstates are not, you cannot just write it like this simply. And also, we know that in finite volume, there's going to be finitely many particles on average,
23:02
and infinitely many particles. So why are you writing all these terms? Why is it not the term with n equal infinity, which is important? I will explain. Just try, I can't explain it before you see something more, OK? Yeah, very good. I mean, these are very good questions, exactly. I have to say something about that. I can't say it at this point, yes.
23:22
But just bear with me. This is the expansion. You put it in with all the obvious difficulties. I already listed a lot of them. You add this difficulty to those, OK? This is all a problem, OK? I agree. So what we actually do, OK, going to finite volume is not such a big problem,
23:40
because there's a theory of finite size effects, OK? So we know how the finite size effects go for energies. That's Lussher's theory, but if it's an integrable model, then we have even better description for this infinite volume, OK? For the finite volume dependence of these quantities,
24:02
we have formulae, which were worked out by Balazs, Pozhgai and me something like 10 years ago, that work exponentially well in the volume. So we have all captured all power-like corrections in the volume in field theory. So we know exactly. So it's basically up to exponential corrections. We know the finite volume dependence of the amplitudes.
24:23
And that's a formulae that already had lots of different applications to calculation of different things. So it's like thermal correlators and one point functions with boundaries and all sorts of things. So we understand that.
24:40
The real problem is for finite energy density, how we get around that problem, yes. So actually, first, just suspend belief and do it. And see what we get, OK? So that's where I'm going to need to project something, because plotting it all with the chalk would be, I mean, very awkward.
25:03
So the idea is that you do simulations with the truncated conformal space approach, or truncated space approach, which is very simple. I realize I have to describe this. What is up there is a very simple model, at least the upper side.
25:22
We just take the Ising quantum field theory, which is basically just a free Majorana fermion, one plus one dimension. And we do temperature quench or transverse field quench
25:41
if you think about it on the spin chain. Or in the Majorana term, in the Majorana field theory, this is just a mass quench, basically free quench in a mass quench, OK? And we just take this Majorana theory, which has this mass term here. So something like the Hamiltonian is one over, I think, it's 2 pi dx.
26:03
That's the standard. I think there's an i here. And basically, what we do is that we take some m0 before the quench. We quench to some m. You can calculate everything explicitly.
26:21
The quench is described by a Bogoliubov transformation. It's a little tricky to calculate magnetization in this model, because magnetization operator is non-local in terms of the fermion. But you can do that, OK? There are tricks. The analytic results here are actually by Estler and Schurich using some form factor, using the same sort of form factor expansion.
26:43
You need actually the form factor expansion to get the analytic results. You need to do a re-submation of certain terms. I will come back to that later. So you need some sort of re-submation of infinite sets of contributions, but that's done. So this is how it works. So what we do is that we just take the Hilbert space of this theory,
27:00
truncate it at some energy level, keep those states, and do the numerics. That's the truncated space approach in its simplicity. And these are lines with different truncations. m is the mass. Lambda is basically truncation level in energy. So m is unit of energy here. This is time in units of mass, or 1 over m.
27:21
And this is the so-called Loach-Miteko first. Loach-Miteko is simply that you take the initial state, overlap with the time-evolving state, and take its absolute value squared. That's Loach-Miteko for you. It's how well the initial state is preserved in time. What is the overlap with it as time goes?
27:42
And you see here that it depends on the cutoff. But we understand a lot about RG, which was also worked by many different people, so people in this room like Slabarychko and his group.
28:01
We can basically apply this machinery of RG to the Loach-Miteko, and we can simply predict the exponent of lambda with which it scales. We can predict the exponent with the cutoff. That's simple. We could actually compute the coefficient as well, but it doesn't help too much. So we just simply predicted the exponent, which turns out to be minus 1.
28:20
It scales like lambda to the minus 1, and then we just extrapolate it. The curve here, the red curve, is the result of the extrapolation. But because this theory can be solved exactly, it's a free theory after all, we have these dots, which are the analytical results from Schurich and Dessler.
28:43
Apart from this little part here, which we know what it is, it's a truncated conformal space artifact. Because of the cutoff, there are some little wiggles in the line, and the extrapolation just doesn't work. It cannot catch these wiggles for short times. But the extrapolation is going a little wrong in here.
29:01
We see the extrapolation doesn't fit quite well, so we understand what is that. But the rest is fine. Is that because Schurich and ESSER truncate the expansion that's written there? So do you think that difference at short times is due to that truncation and that expansion?
29:23
How I know it's a truncation effect, it's basically, I can see these little wiggles, which you don't see because it's not magnified there. But I see these little wiggles, and then I look at their frequency. And their frequency is actually the cutoff frequency. And if I increase my cutoff, the frequency goes down. But the problem is that the wiggles don't fit together. If I increase the cutoff, then the wiggles will become the other frequency.
29:43
So they just interfere with the extrapolation, basically. But this only happens at short times where the wiggles are still there. At long times, you already have decoherence of modes. And the wiggles get washed out. And at long times, the extrapolation works much better. Basically, that's the thing that's happening here.
30:00
How big is this system? This system is 40 in size of 1 over m. And this is the infinite volume result that I'm plotting here. So also, I can do magnetization. Again, these curves are truncation curves.
30:21
So the dashed curves help. Go back. You'd eventually see the periodicity of the system. No, because there are some other effects obscuring the periodicity. So actually, we are just going to have the volume to be safe.
30:42
Basically, because of the light cone effect, if you have a finite volume, the light cone starts out. At L over 2, because my unit of speed of light is 1, at L over 2, it hits the boundary. Then my time evolution really deviates from infinite volume. So then I would have a problem.
31:04
But we are going to just half the volume. And here you see the relaxation of the magnetization. This is actually an exponential curve. It's just the first part of it. But it doesn't seem to curve too much.
31:21
But we could actually fit the relaxation exponent from this curve directly, and it works. So this is relaxation of magnetization. This is in a ferromagnetic regime, so where you start with the spontaneous magnetization. After a quench, it relaxes. And the way it relaxes is described by relaxation exponent. And you can fit it from this curve.
31:42
And this works also quite well. We get quite well the relaxation exponent that is predicted analytically. But here is a cheat line, which I don't say anything about this blue line, if you want. You can ask later. There's just no one to go into what we cheat here. We are cheating the TCSA a little.
32:01
But the vanilla TCSA is just a truncated Hamiltonian approach. Just these lines extrapolated here with the appropriate exponent that we know from RG, gives you the red curve, and then these dots are the analytic results. But this may not be so convincing.
32:20
Because after all, this is just a free model with a Bogoliubov transformation. You would say maybe Bogoliubov makes life so simple that after all, there is not much error in doing all this truncation. So the next is do non-integrable quenches. And the non-integrable quenches are basically the following.
32:41
We are quenching from a free Majorana fermion. But the end point to which we are quenching also includes in this Hamiltonian another term, which is an external magnetic field. Or if you like it in the Ising spin chain, quantum Ising spin chain, people would call this longitudinal magnetic field.
33:01
This is the transverse magnetic field on the quantum chain. And this is the longitudinal magnetic field in the quantum spin chain. OK, so you can do this either in ferromagnetic and paramagnetic regime. And then we saw these plots. So the lines here are the truncation results. So these are linotrivial, this is already non-integrable.
33:21
When we saw these plots, we really said there should be something wrong. Maybe we are not going to sufficiently long times here, because we saw no relaxation here. And actually, it turns out that here the relaxation is really problematic. I don't want to talk about the details of that again, because that would lead us too far away.
33:41
But in the ferromagnetic case, the relaxation, there is confinement. And confinement really interferes with relaxation. That would be a topic of a completely different talk. And it's in a completely different paper anyway. But the point is that you have this. And at first, you say, this just can't be right. This doesn't relax anywhere. If I continue this, you would see that this wiggling continues.
34:03
And then it comes back. And then it comes down. That's already quite long times. So what we did is that we asked a guy who already had the so-called ITBD, which is infinite volume time-evolved block decimation. I think some of the actual experts are sitting in this audience probably.
34:21
At least they are on the list. And so Mario Collura from CISA to produce spin-chain results directly on the chain using ITBD, which are the dots here. And that's very tricky, because in order to compare with the filter,
34:41
you have to scale yourself as close to the critical point as you can, scaling all the parameters, whatever. But ITBD goes all sorts of heavy. So actually, the filter calculation here takes something like a few minutes. And the ITBD calculation here takes days.
35:01
So filter here is much more effective, obviously. It's on turf, right? But you see that the blue line, the filter line, is very well on this line. And again, these little wiggles. Now you see the wiggles there. Those little wiggles are actually truncation artifacts coming magnified by the extrapolation. They would be much smaller if I showed you the non-extrapolated line.
35:22
But this is just the extrapolated result. And this is the paramagnetic phase, where you see some very nice oscillations with a frequency of almost, I mean, the period is almost 2 pi. The frequency is almost 1. And that's actually the particle mass, the frequency. So this is just the one particle excitation that does this. And there is a very, very slow decay here, which is not obvious.
35:43
But if I plotted more of this, I mean, the ITBD results are not available for longer times. The TCSA results are available for longer time. If I plotted it for longer time, you would have seen the decay. OK, so now one grows more confident. This is really going to work.
36:00
Come on. OK, so that's, I think... Can I ask a question? Yes. So in your results, when you truncate in the Phil Philis site, what it is that you are truncating? The number of states?
36:20
Yes, so basically, we use the same approach in the sense that we have the post-quench Hilbert space, the post-quench Hilbert space, which is the Hilbert space of this Hamiltonian without H. And we just truncate it in energy. And we represent the prequench state on this. Actually, we can do it numerically, or in some cases,
36:42
we know the exact representation so that we can put in the exact vector. But we did both, and it doesn't really matter. So the evolution is always by a free theory? No, that's the whole point. We are doing it on this Hilbert space, but to the Hamiltonian that we are actually doing, I mean, the basis of the Hilbert space, we are adding this.
37:00
We are using the full Hamilton. That's the truncated conformal space idea, normally that you are doing it in a sort of non-interacting conformal space, and then you are adding the interaction which you can't solve. But you know the matrix elements of this operator, because you can solve for the exact matrix elements. So you can represent the Hamiltonian as a matrix on this space. And then you just solve numerically for this time evolution.
37:23
There are some arcane tricks if you really want to be effective in this. But otherwise, it's a very simplistic idea. And then you have to take care about the cutoff. That's another thing. You have to add the RG. Without the RG, it wouldn't work. Gabor, but perhaps you should ask how many people in the room
37:41
don't know about the truncated conformal space. I mean, people from various approaches. And the first speaker, I thought. All right, OK. That little blackboard is more than sufficient. So how many people need a lightning introduction to truncate it? Hamiltonian approach.
38:01
That's quite a lot. So let's do it. It's extremely simple. You just take a Hamiltonian. This Hamiltonian, you know exactly. You put it in a finite volume. The spectrum of this Hamiltonian is then ground state, excited states, whatever. It's discrete because it's in a finite volume. It's discretized.
38:22
So basically, there is some 0, 1, 2, states of this Hamiltonian. OK, this is not the prequench Hamiltonian in this case. This is just some Hamiltonian of which you know the exact solution. So you know it in finite volume. You know the spectrum in finite volume. What you do is that you just take a cutoff, lambda, in some units, whichever units,
38:44
some upper energy cutoff. If you put this in because you have a discrete spectrum, you retain only finite number of states. Next assumption that you really must have is that you must know the exact matrix elements of this operator V in the space. In many cases, you know this from conformal field theory,
39:01
from form factor approach, from different things. People computed this. You don't even have to understand this. You just take the book from the shelf and take the formula or whatever. So here are these states n, and you have V and m. And what you do is just then this becomes a huge matrix.
39:20
5,000 times 5,000, 10,000 times 10,000, whatever. It just depends on where you put your cutoff. And then the nice thing is just to take this as a matrix, this approximation for the Hamiltonian in the finite volume. And then you just say, OK, well, we just do numerical quantum mechanics with that, matrix quantum mechanics.
39:40
It's very simple. Now, where all the tricky things start, for which this lecture wouldn't suffice, that would be a completely different lecture, but then you want to get real results for really complicated models, for which, for example, this Hilbert space grows too fast. So that you cannot put a very high cutoff,
40:01
or for which the cutoff dependence is very slow. So you would have to put a very high cutoff in order to get it well. And then you have to improve all this stuff by the normalization group methods. Basically, what you compute is the dependence of your results on the cutoff. If you can have analytical predictions, analytical predictions come in different flavors.
40:21
Sometimes you just know that your results, like an expectation value, would be like the expectation infinite cutoff plus something like c to lambda minus kappa where kappa is an exponent, and you know what is the exponent. Sometimes you can do such a detailed calculation that you even know c. Then you can even just really get rid of this.
40:42
You just subtract this dependence from your results and see that they scale on top of each other in the cutoff and all that. So this is really as simple as it gets, simple-minded thing. All the trick comes in really, first of all, implementing efficient approach to do this basic computation.
41:03
The second is implementing, doing DRG, which predicts you this exponent, and possibly, if this is possible, also the coefficients of the cutoff dependence. But this is all you have to think about. You don't need to do all the other intricate details.
41:23
So in your example, it's key that the initial state was well represented in that substrate? Yes, very good. Actually, that already comes down to an interesting question, is that these k amplitudes, they depend on the rapidities.
41:42
Basically, they depend on energy. So they should be decaying with energy in order for it to be very well represented. This is actually what happens. That's the whole point why it works. Because it's counterintuitive at first, because the energy scale could be at infinity in a sudden quench. But what happens normally in models is that this thing here,
42:03
normally, first of all, in many cases, which we call integrable quenches, this thing here is just factorized into two-particle overlaps. So this is just product of two-particle overlaps. That's a non-trivial thing, whether it happens or not.
42:21
It's not enough that the initial and the anti- Hamiltonian is integrable. The quench, this happens for specific quenches. For example, for the free-to-free quenches, this always happens. There's a non-collabor transformation why this happens. And then the question is that this two-particle overlap, let's call it k2, whether it decays with energy fast enough.
42:41
Now, the energy is basically e to the theta because it's cosh theta. I'm neglecting the e to the minus theta. But what happens is that this k2 normally is the power of this, an inverse power of this. So in many cases, this is just more or less 1 over e squared, energy squared, 1 over e to the 2 theta.
43:02
So it's a power-like decay. That's tricky. That's why you need extrapolation also because it's not fast enough. So you are really leaving out sizable chunks of your state from your Hilbert space. But fortunately, the chunks that you are leaving out have some very simple dependence in the energy. So you can extrapolate using the simple dependence.
43:23
So yes, you do leave some parts of your state out of your Hilbert space. But you can get around that. That's right. Yes? How does the map work from the multi-particle states that you're showing onto this? Yeah.
43:42
So normally, if this happens, if this happens at all, this happens only when you only have amplitudes like this. So this is the so-called pair state structure. When your quench state is just consisting of opposite-momentum particle pairs. This is what is drawn in, if anyone is familiar with quantum filter quench
44:04
formalism by Cardi and Calabrese, this is what drawn like this, that the initial state in a quantum quench, which is here, time is going this way, is basically emitting particle pairs of opposite momentum. If this is true, then you can prove, this is a theorem which we proved with
44:21
Spirosotriades and Giuseppe Musaldo, I think, yeah? Or maybe with David. I don't remember which paper the theorem actually was in. We have three papers about this problem, and I don't remember which paper. We actually have the proof of the theorem, and it's a theorem. The theorem is that if this is particle pairs,
44:41
then basically your state exponentiates. Then your initial state can be written as integral of k theta, creation amplitude of particle pairs, like a squeeze state. This is what you would call an integrable quench. An integrable quench is not just that the initial and the anti-Hamilton is integrable.
45:02
An integrable quench is, there is a recent paper by one of my former students and collaborators on this, Balazs Pozhgai, that this is what you would really call an integrable quench. This is like factorization. It means that the multi-particle overlaps are factorized in terms of two-particle overlaps. Basically, they are products of independent two-particle overlaps.
45:20
It's like S-matrix factorization in integrable models. It's like factorized scattering, more or less. If this is true, then there's lots of things you can do with this. For example, some TBA-like approach that Giuseppe Mussard and Davide Fioretti introduced some time ago, and things that you can do if this happens. Why do you need all this in a numerical calculation?
45:42
Why do you need to know all this? Not in the numerics directly. You need it because you don't trust your numerics to start with, because of all the issues that I told you in the beginning. So you want to get an alternative approach. I'm getting to that point. So what you really want, you want to have something to compare with. If you want to have something to compare with, it should rather be analytically calculable.
46:03
But can you come up with a set of theoretical arguments independent of integrability, which tell you when field theory works? Yes. OK, yes. So this is basically the result of the following.
46:22
Your interacting field theory, let me suppose that it is a relevant perturbation of a free boson, or a free fermion. It works the same way. So it is basically at the ultraviolet it's free, and then there's some relevant operator perturbing that leads you out of the critical point. The relevant operator at high energies is small.
46:41
So your quench at very high energy is basically just quenching free to free. So you expect to reproduce at high energies, you expect to reproduce the asymptotic behavior of the free amplitude. That's it. The question is how high, obviously. So this can be far away, and then your truncation method is not very effective, because you can't extrapolate if it didn't get set in.
47:04
It's the question of the quench itself. I can also tell you that it turns out that the truncated method, the relevant small parameter that should be small, is actually the particle density post quench.
47:22
So you should have a small post quench density in the natural units. Natural units is, again, the mass related units of the smallest particle. That is basically telling you that this amplitude, pair amplitude squared, when you integrate it over all the rapidity range, this integral should be small enough.
47:42
So we work this out in many cases. And for example, in the Ising case, it turns out that in principle, our truncated conformal space could work out to 300 units in volume. In field theory, that's huge, because all finite size effects decay exponentially. So this is just a limitation of time evolution,
48:02
but the finite volume effects you can already forget. So the time evolution is limited because of the light cone effect. But otherwise, as regarding the volume, this is fine. So this turns out to be the real small parameter in the game. And this means that your quench is, in a sense, small.
48:23
This seems to be a real limitation. However, I would like to draw your attention that most theoretical approaches are limited to that. For example, there is the so-called semi-classical approach to quantum quenches in field theory, which is also limited to small densities. And the form factor expansion approaches are also limited to small densities.
48:40
Can I ask you a question on the k? Do you only have an even number of particles? No, that's also possible to have an odd number. Actually, I'm going to show you plots from a theory which has odd numbers. What could happen is that in this state, besides this integral, there is some term like this.
49:06
So because the translation invariance, if it's a single particle, it can only be zero momentum. So you can have overlaps with single particle states, and then they have their own amplitude, g. We call it g normally, this amplitude.
49:21
And that's also interesting. Actually, there are all sorts of complications related to that. We are working on some of these complications right now because they are plugging all sorts of analytical approaches to quenches. So we want to understand more about how the quenches analytical approaches work in the presence of this coupling. It's tricky. Yes.
49:41
So at the beginning, you said that you expected the method to work for any massive? No, I didn't expect it to work at all. But I expected it to work for massive, that's right. Massive with now certain qualifications, which I basically already spelled out. The basic qualification is that this sort of integral is small.
50:05
But you also now said that this would work if the UV, there was a CFT, asymptotically a CFT. I said free theory. If it's a CFT, it's an interacting CFT, then it's a tricky question. Why is it a tricky question?
50:26
It seems so. Is the interaction, is the relevance of the interaction in the UV, no? Because in an interacting CFT, you don't know what the overlaps would be. So you don't have the argument here. Depends very crucially that for a free theory, we have an explicit solution for the overlap.
50:44
So we know that it decays. For an interacting CFT, I can't vote for that. That's basically the problem. But you can still go ahead and apply the approach? Yeah, I will show.
51:00
So basically, I told you about the damping here. And we can also extract the damping. And the damping in these paramagnetic oscillations, the data points, numerical data points, are the color dots. And there is an explicit prediction for the case when this is integrable.
51:22
But the problem is that we are looking at the non-integrable. We are taking small values of this magnetic field. And we see as we scale this magnetic field to 0, then they really try to fall on that line. So this isn't bad. So even this damping, this very small damping that seems so small there, can be extracted
51:43
with very high precision from the curve. The curve is so precise that even a small damping can be extracted from it. And it agrees with theory. Right, yeah. So next, I have to tell you what to do if you don't have this simple-minded, free
52:03
Bogolyubov quench background in order to have a theory behind this. So then basically, what you do, it's on the blackboard. You just evaluate this sum if you want to do analytics.
52:24
You just evaluate this sum up to some terms. You have to be extremely careful there. You have to re-sum certain orders in order to actually get to the real results. But this can be done in some approaches. So there are actually two approaches on the market.
52:41
One was by Gesualdo Delfino in CISA. And his approach is basically that you quench from an integrable model, from integrable to non-integrable.
53:01
And on this side, because it's non-integrable, you don't actually know the small factors. You don't know the energies. You don't know nothing. But if it's a small quench from an integrable, you can think about this quench being perturbative. So you are doing perturbation theory around the prequench. You are doing a perturbation theory coupled with this expansion around the prequench.
53:24
So you can do expansion around prequench. And actually, that expansion predicts very well the amplitude on that plot.
53:42
The frequency is wrong, and it cannot get the decay rate well. But the amplitude is OK. There's a reason for that, actually. Maybe I won't have time to go into that. So the other possibility is by Esler and Schurich,
54:06
and in combination with some people like Bruno Bertini and maybe Axel Kubero this time. But basically, now it seems that Dirk is the stable person whose name you should watch if you want to get all these papers. And that is when you quench to an integrable theory.
54:27
And in that case, you can use the postquench exact form factors and everything. The only thing is that the integrable theory doesn't tell you the C's.
54:41
So there are differences between two approaches. Namely, first of all, what is the end state that they can treat? The other is that here, the K function, which is the C overlaps, K is input. But this approach is better in the sense that K is actually calculated perturbatively.
55:02
This is not what Aldo really does, but it is effectively the same as calculating K as well perturbatively. Because he's expressing everything in the prequench and does a perturbation theory, he actually captures all the things in this expansion in perturbation theory and the prequench.
55:22
The problem with his formalism that is already apparent is that it's going to be using the prequench energies, which means the prequench frequencies. And actually, the system oscillates in the postquench frequencies. That already gives a sizable shift to his results. And there are also some others. Such a form factor approach, because you are basically doing what?
55:43
You are summing one particle, two particle, three particle states. It's the low frequency part of that. So it should work, first of all, for T long enough. So T must be long enough. T must be larger than the relevant gap. So all these truncated expansions work for a long time.
56:01
For a short time, they wouldn't work. This is true for the second expansion. Let's call it the Schurichte. Actually, X slash Schurichte, whatever. But I just choose one of the names. Let's call it the Schurichte expansion. For the Delphino expansion, there's another limitation. It's that because he's doing it perturbatively in quench size,
56:21
let me call this quench size parameter lambda. There is some parameter lambda which specifies how big the quench is. Because it's perturbative, then he also has another limitation, that he must stay below one over lambda, because otherwise these things would come.
56:41
So we did what we call the calculation of so-called E8 quenches. And here is what we have. So E8 quenches are quenches in this theory, which are already much different from this. M is 0. We are taking H. That's a famous E8 integrable model of Zamolodchikov.
57:02
And we are quenching in H. So pre-quench and post-quench are both integrable. Both methods are applicable to this. And I think the results speak for themselves. So the TCSA curve in the upper figures is the red curve. This is the S dash Schurichte expansion. And this is Aldo's expansion, Delphino's expansion.
57:23
Actually, this is a very small quench. If I put in a much larger quench, the result would be even more pronounced. But that figure is not in the production stage yet. So I don't have production quality figure about that yet. This is magnetization operator.
57:46
And this is the density of psi bar psi. This is so-called epsilon. Energy density operator in the Ising model. So you see that this is another quench.
58:03
So you see that, first of all, what you see is that the TCSA is very good in the sense, in the following sense. It follows very soon. First of all, Schurichte expansion is also very good in the sense that even at very short times, the Schurichte expansion is very precise. We are only keeping one particle pieces.
58:20
We are not even going to two particles in these expansion. We are just keeping the first one particle pieces here. So if you take together the truncated conformal space and the Schurichte expansion, we basically describe this quench O to infinite times. Because the Schurichte expansion gets even better if you go to higher times. That's not a problem for it.
58:41
And the only difference here is this little tiny part, which you can barely see that the TCSA result starts from here, but actually Schurichte expansion starts as well. And Schurichte expansion goes here at zero time.
59:01
If you piece it together, then you have an infinite time description of this quench. And for other quenches as well. If we quench away, yeah.
59:24
Just let me tell you about the upper line because that's easy. What you can also do is that you keep H the same and you switch on M. That breaks integrability. That means we call it quenching away from the E8 line. Then we don't have the Schurichte expansion because it uses post-quench integrability.
59:41
But then we have Aldo's expansion, which again doesn't perform quite well. And the upshot is what I would say is not that we should throw it out. You shouldn't throw out the baby with the bathwater, right?
01:00:00
For the time being, this is the only expansion that would work for a post-quench non-integrable case. So I think it has to be improved. And I think it can be improved. We have some ideas about how to go around that. So actually, I'm hoping that instead of saying that this just disagrees, I'm hoping that we can get this curve
01:00:21
to line up with that with some improvement of the actual expansion. It just needs to be done in a much more comprehensive way, including lots of other stuff in it. OK, yeah, so that's the end.
01:00:41
So let me just conclude. So I think that for field theory quenches, first of all, they are interesting because we know much less about them than statistical mechanical quenches. I think this is true. And the relaxation doesn't seem to happen so easily in field theories.
01:01:00
And there are less approaches to really calculate field theoretic quenches. What I first want to say is that truncated Hamiltonian methods seem to be effective, at least for low density quenches. This is the real small parameter. If it's a low density quench, then the truncation method is applicable, even after all its limitations.
01:01:22
That it has to work in finite volume, that it has a cut of whatever. Even after all these limitations, it is applicable. And I think that on the theory side, having analytical expansions, there are some very nice works. But a lot more needs to be done because of their limitations. So basically, at this point, we
01:01:42
don't have a good analytical calculation when the postquench theory is non-integrable. And this is one of the most outstanding questions. For when the postquench theory is integrable, I think the Schurich expansion is, apart from little details, maybe, but the idea is fine. But for the postquench theory, non-integrable,
01:02:01
which is actually a very interesting case, obviously, a non-integrable case, we need to do much more in the theory. So thank you for your attention. Questions? I didn't understand very well this part about the structure
01:02:22
of the initial state. So for example, in this case, where you quench to a final amount that is not integrable, do you still claim that it has this perfect? No. No, I don't claim. However, there is an interesting observation, is that if the density is small, you can imagine that these particles are created
01:02:41
far away from each other. So you can imagine that the leading part is still a more or less factorized particle per creation. It's just that it's not exactly factorized. So for small densities, you can imagine that this is still a good approximation, even if the quench is not integrable. So hold it. And you can measure these k factors experimentally,
01:03:01
by the way. Experimental, I would say, TCSA. Experimental in truncated Hamiltonian, you can not measure, numerically compute it. So the k factors, you can numerically compute in truncated Hamiltonian approach. You can compute just at the expense of, here is such a computation with the theory on it.
01:03:21
This is an integrable quench, but you can do it in non-integrable case. So you can measure these overlaps. So you can check whether your actual assumptions about this integral being small is valid or not. You can check it right inside the approach itself. So it's an internal check. You don't need to know it from something else. What do you show in these plots?
01:03:42
It's just an overlap calculation. A k function, absolute value of a k function as momentum. You see the power line decay. It's an overlap between what and what? Actually, it's an overlap in sine-gordon theory after a quench between the initial state and the lowest
01:04:02
two-particle state as the function of the momentum of the two-particle state. And the blue line is a theoretical curve that we got together with Spiros Sotriades and Giuseppe Musarto. The red line would be free theory, and the dots are the measured values of the overlaps. I just want to say that this can be measured inside
01:04:21
the approximation itself. So you can validate this approximation from inside. You don't need to know anything from outside about the overlaps. So this is to check two parts, but here, your statement was that it exponentiates, right? So you usually check. Actually, you have to remember that smallness can also be checked from the following.
01:04:41
The total overlap should be 1, because it's a normalized state. So if you check that your vacuum overlap is still sizable, that means that all the rest should be small. So that validates your numerics. It's maybe an even easier check. I showed you something which is more, so to say,
01:05:01
theoretically based, but actually, if you just check that in the given volume, your vacuum overlap is still sizable, then you are on a safe ground. Actually, a quick question. So OK, quenches are, of course, fascinating.
01:05:22
But when you were working on quenches, from the methodological point of view, did you learn something about the method which could be used to improve its functioning even in other situations, not just as far as quenches are
01:05:43
concerned, but in the classic applications of the method, which are spectroscopy and then extending it to other theories, not just the good old Ising CFT? I learned more about the details of its validity. It turns out that the validity for these questions is the same as this one.
01:06:01
So the validity criterion of TCSA is the smallness of this sort of integrals, because even then, there you are not doing quenches, but you have your H0 theory from which you start from, and that has a ground state. And then you are looking at the overlaps of this ground state with the perturbed excited states,
01:06:24
and they should decay fast enough, basically. And you can even specify this in more details. You can even give the highest volume to which you can go with the given cutoff. I mean, for example, in the Ising theory, in the standard Ising truncation,
01:06:42
I think what we can do is 300 in the volume. That's what I quoted, 300 in the volume. That is not a quench related thing. That is just truncated Hamiltonian as its functioning, its precision. And basically, we also learned, but you already, I remember we discussed this, so you already guessed this yourself or observed this yourself. We learned about how the multi-particle overlaps
01:07:02
are scaling with parameters, and while they are scaling, it's not too, okay, there's not much time, but I can tell you in private, but it's not too surprising, but we confirmed all that they are scaling in the proper way that you would expect in order for all this overlap should sum up to one. They should scale with the volume,
01:07:21
some non-trivial way in order for this to be possible.